id	sid	tid	token	lemma	pos
asir-3636	1	1	applied	apply	VERB
asir-3636	1	2	science	science	NOUN
asir-3636	1	3	and	and	CCONJ
asir-3636	1	4	innovative	innovative	ADJ
asir-3636	1	5	research	research	NOUN
asir-3636	1	6	issn	issn	VERB
asir-3636	1	7	2474	2474	NUM
asir-3636	1	8	-	-	SYM
asir-3636	1	9	4972	4972	NUM
asir-3636	1	10	(	(	PUNCT
asir-3636	1	11	print	print	NOUN
asir-3636	1	12	)	)	PUNCT
asir-3636	1	13	issn	issn	VERB
asir-3636	1	14	2474	2474	NUM
asir-3636	1	15	-	-	SYM
asir-3636	1	16	4980	4980	NUM
asir-3636	1	17	(	(	PUNCT
asir-3636	1	18	online	online	ADJ
asir-3636	1	19	)	)	PUNCT
asir-3636	1	20	vol	vol	NOUN
asir-3636	1	21	.	.	PROPN
asir-3636	2	1	5	5	NUM
asir-3636	2	2	,	,	PUNCT
asir-3636	2	3	no	no	INTJ
asir-3636	2	4	.	.	NOUN
asir-3636	2	5	1	1	NUM
asir-3636	2	6	,	,	PUNCT
asir-3636	2	7	2021	2021	NUM
asir-3636	2	8	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	VERB
asir-3636	2	9	20	20	NUM
asir-3636	2	10	original	original	ADJ
asir-3636	2	11	paper	paper	NOUN
asir-3636	2	12	validity	validity	NOUN
asir-3636	2	13	of	of	ADP
asir-3636	2	14	closed	closed	ADJ
asir-3636	2	15	ideals	ideal	NOUN
asir-3636	2	16	in	in	ADP
asir-3636	2	17	algebras	algebra	NOUN
asir-3636	2	18	of	of	ADP
asir-3636	2	19	series	series	NOUN
asir-3636	2	20	of	of	ADP
asir-3636	2	21	square	square	ADJ
asir-3636	2	22	analytic	analytic	ADJ
asir-3636	2	23	functions	function	NOUN
asir-3636	2	24	musa	musa	PROPN
asir-3636	2	25	siddig1	siddig1	PROPN
asir-3636	2	26	*	*	PROPN
asir-3636	2	27	,	,	PUNCT
asir-3636	2	28	shawgy	shawgy	ADJ
asir-3636	2	29	hussein2	hussein2	PROPN
asir-3636	2	30	&	&	CCONJ
asir-3636	2	31	amani	amani	PROPN
asir-3636	2	32	elseid3	elseid3	PROPN
asir-3636	2	33	1	1	NUM
asir-3636	2	34	department	department	NOUN
asir-3636	2	35	of	of	ADP
asir-3636	2	36	mathematics	mathematic	NOUN
asir-3636	2	37	,	,	PUNCT
asir-3636	2	38	faculty	faculty	NOUN
asir-3636	2	39	of	of	ADP
asir-3636	2	40	science	science	NOUN
asir-3636	2	41	,	,	PUNCT
asir-3636	2	42	university	university	PROPN
asir-3636	2	43	of	of	ADP
asir-3636	2	44	kordofan	kordofan	PROPN
asir-3636	2	45	,	,	PUNCT
asir-3636	2	46	sudan	sudan	PROPN
asir-3636	2	47	2	2	NUM
asir-3636	2	48	department	department	NOUN
asir-3636	2	49	of	of	ADP
asir-3636	2	50	mathematics	mathematic	NOUN
asir-3636	2	51	,	,	PUNCT
asir-3636	2	52	college	college	NOUN
asir-3636	2	53	of	of	ADP
asir-3636	2	54	science	science	PROPN
asir-3636	2	55	,	,	PUNCT
asir-3636	2	56	sudan	sudan	PROPN
asir-3636	2	57	university	university	PROPN
asir-3636	2	58	of	of	ADP
asir-3636	2	59	science	science	NOUN
asir-3636	2	60	and	and	CCONJ
asir-3636	2	61	technology	technology	NOUN
asir-3636	2	62	,	,	PUNCT
asir-3636	2	63	sudan	sudan	PROPN
asir-3636	2	64	3	3	NUM
asir-3636	2	65	aldayer	aldayer	PROPN
asir-3636	2	66	university	university	PROPN
asir-3636	2	67	college	college	PROPN
asir-3636	2	68	,	,	PUNCT
asir-3636	2	69	jazan	jazan	PROPN
asir-3636	2	70	university	university	PROPN
asir-3636	2	71	,	,	PUNCT
asir-3636	2	72	saudi	saudi	PROPN
asir-3636	2	73	arabia	arabia	PROPN
asir-3636	2	74	*	*	PUNCT
asir-3636	2	75	musa	musa	PROPN
asir-3636	2	76	siddig	siddig	PROPN
asir-3636	2	77	,	,	PUNCT
asir-3636	2	78	department	department	NOUN
asir-3636	2	79	of	of	ADP
asir-3636	2	80	mathematics	mathematic	NOUN
asir-3636	2	81	,	,	PUNCT
asir-3636	2	82	faculty	faculty	NOUN
asir-3636	2	83	of	of	ADP
asir-3636	2	84	science	science	NOUN
asir-3636	2	85	,	,	PUNCT
asir-3636	2	86	university	university	PROPN
asir-3636	2	87	of	of	ADP
asir-3636	2	88	kordofan	kordofan	PROPN
asir-3636	2	89	,	,	PUNCT
asir-3636	2	90	sudan	sudan	PROPN
asir-3636	2	91	received	receive	VERB
asir-3636	2	92	:	:	PUNCT
asir-3636	2	93	december	december	PROPN
asir-3636	2	94	31	31	NUM
asir-3636	2	95	,	,	PUNCT
asir-3636	2	96	2020	2020	NUM
asir-3636	2	97	accepted	accept	VERB
asir-3636	2	98	:	:	PUNCT
asir-3636	2	99	january	january	PROPN
asir-3636	2	100	16	16	NUM
asir-3636	2	101	,	,	PUNCT
asir-3636	2	102	2021	2021	NUM
asir-3636	2	103	online	online	ADV
asir-3636	2	104	published	publish	VERB
asir-3636	2	105	:	:	PUNCT
asir-3636	2	106	january	january	PROPN
asir-3636	2	107	22	22	NUM
asir-3636	2	108	,	,	PUNCT
asir-3636	2	109	2021	2021	NUM
asir-3636	2	110	doi:10.22158	doi:10.22158	NOUN
asir-3636	2	111	/	/	SYM
asir-3636	2	112	asir.v5n1p20	asir.v5n1p20	NOUN
asir-3636	2	113	url	url	PROPN
asir-3636	2	114	:	:	PUNCT
asir-3636	3	1	http://doi.org/10.22158/asir.v5n1p20	http://doi.org/10.22158/asir.v5n1p20	NOUN
asir-3636	3	2	abstract	abstract	ADJ
asir-3636	3	3	we	we	PRON
asir-3636	3	4	show	show	VERB
asir-3636	3	5	the	the	DET
asir-3636	3	6	validity	validity	NOUN
asir-3636	3	7	of	of	ADP
asir-3636	3	8	a	a	DET
asir-3636	3	9	complete	complete	ADJ
asir-3636	3	10	description	description	NOUN
asir-3636	3	11	of	of	ADP
asir-3636	3	12	closed	closed	ADJ
asir-3636	3	13	ideals	ideal	NOUN
asir-3636	3	14	of	of	ADP
asir-3636	3	15	the	the	DET
asir-3636	3	16	algebra	algebra	NOUN
asir-3636	3	17	which	which	PRON
asir-3636	3	18	is	be	AUX
asir-3636	3	19	a	a	DET
asir-3636	3	20	commutative	commutative	ADJ
asir-3636	3	21	banach	banach	NOUN
asir-3636	3	22	algebra	algebra	NOUN
asir-3636	3	23	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	3	24	2	2	NUM
asir-3636	3	25	,	,	PUNCT
asir-3636	3	26	that	that	PRON
asir-3636	3	27	endowed	endow	VERB
asir-3636	3	28	with	with	ADP
asir-3636	3	29	a	a	DET
asir-3636	3	30	pointwise	pointwise	PROPN
asir-3636	3	31	operations	operation	NOUN
asir-3636	3	32	act	act	NOUN
asir-3636	3	33	on	on	ADP
asir-3636	3	34	dirichlet	dirichlet	PROPN
asir-3636	3	35	space	space	NOUN
asir-3636	3	36	of	of	ADP
asir-3636	3	37	algebra	algebra	NOUN
asir-3636	3	38	of	of	ADP
asir-3636	3	39	series	series	NOUN
asir-3636	3	40	of	of	ADP
asir-3636	3	41	analytic	analytic	ADJ
asir-3636	3	42	functions	function	NOUN
asir-3636	3	43	on	on	ADP
asir-3636	3	44	the	the	DET
asir-3636	3	45	unit	unit	NOUN
asir-3636	3	46	disk	disk	NOUN
asir-3636	3	47	𝔻	𝔻	PROPN
asir-3636	3	48	satisfying	satisfy	VERB
asir-3636	3	49	the	the	DET
asir-3636	3	50	lipscitz	lipscitz	NOUN
asir-3636	3	51	condition	condition	NOUN
asir-3636	3	52	of	of	ADP
asir-3636	3	53	order	order	NOUN
asir-3636	3	54	of	of	ADP
asir-3636	3	55	square	square	ADJ
asir-3636	3	56	sequence	sequence	NOUN
asir-3636	3	57	𝛼𝑗	𝛼𝑗	ADP
asir-3636	3	58	2	2	NUM
asir-3636	3	59	obtained	obtain	VERB
asir-3636	3	60	by	by	ADP
asir-3636	3	61	(	(	PUNCT
asir-3636	3	62	brahim	brahim	PROPN
asir-3636	3	63	bouya	bouya	PROPN
asir-3636	3	64	,	,	PUNCT
asir-3636	3	65	2008	2008	NUM
asir-3636	3	66	)	)	PUNCT
asir-3636	3	67	,	,	PUNCT
asir-3636	3	68	we	we	PRON
asir-3636	3	69	introduce	introduce	VERB
asir-3636	3	70	and	and	CCONJ
asir-3636	3	71	deal	deal	VERB
asir-3636	3	72	with	with	ADP
asir-3636	3	73	approximation	approximation	NOUN
asir-3636	3	74	square	square	ADJ
asir-3636	3	75	functions	function	NOUN
asir-3636	3	76	which	which	PRON
asir-3636	3	77	is	be	AUX
asir-3636	3	78	an	an	DET
asir-3636	3	79	outer	outer	ADJ
asir-3636	3	80	functions	function	NOUN
asir-3636	3	81	to	to	PART
asir-3636	3	82	produce	produce	VERB
asir-3636	3	83	and	and	CCONJ
asir-3636	3	84	show	show	VERB
asir-3636	3	85	results	result	NOUN
asir-3636	3	86	in	in	ADP
asir-3636	3	87	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	3	88	2	2	NUM
asir-3636	3	89	.	.	PUNCT
asir-3636	3	90	keywords	keyword	VERB
asir-3636	3	91	dirichlet	dirichlet	PROPN
asir-3636	3	92	space	space	NOUN
asir-3636	3	93	,	,	PUNCT
asir-3636	3	94	lipschitz	lipschitz	VERB
asir-3636	3	95	condition	condition	NOUN
asir-3636	3	96	,	,	PUNCT
asir-3636	3	97	banach	banach	NOUN
asir-3636	3	98	algebra	algebra	NOUN
asir-3636	3	99	,	,	PUNCT
asir-3636	3	100	besov	besov	NOUN
asir-3636	3	101	algebras	algebra	NOUN
asir-3636	3	102	,	,	PUNCT
asir-3636	3	103	beurling	beurle	VERB
asir-3636	3	104	-	-	PUNCT
asir-3636	3	105	rudin	rudin	VERB
asir-3636	3	106	characterization	characterization	NOUN
asir-3636	3	107	,	,	PUNCT
asir-3636	3	108	beurling	beurle	VERB
asir-3636	3	109	-	-	PUNCT
asir-3636	3	110	carleman	carleman	ADJ
asir-3636	3	111	-	-	PUNCT
asir-3636	3	112	domar	domar	ADJ
asir-3636	3	113	resolvent	resolvent	ADJ
asir-3636	3	114	method	method	NOUN
asir-3636	3	115	,	,	PUNCT
asir-3636	3	116	f	f	X
asir-3636	3	117	-	-	PUNCT
asir-3636	3	118	property	property	NOUN
asir-3636	3	119	1	1	NUM
asir-3636	3	120	.	.	PUNCT
asir-3636	4	1	introduction	introduction	NOUN
asir-3636	4	2	the	the	DET
asir-3636	4	3	dirichlet	dirichlet	PROPN
asir-3636	4	4	space	space	NOUN
asir-3636	4	5	𝒟	𝒟	PROPN
asir-3636	4	6	consists	consist	VERB
asir-3636	4	7	of	of	ADP
asir-3636	4	8	the	the	DET
asir-3636	4	9	sequence	sequence	NOUN
asir-3636	4	10	of	of	ADP
asir-3636	4	11	square	square	ADJ
asir-3636	4	12	complex	complex	ADV
asir-3636	4	13	-	-	PUNCT
asir-3636	4	14	valued	value	VERB
asir-3636	4	15	analytic	analytic	ADJ
asir-3636	4	16	functions	function	NOUN
asir-3636	4	17	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	4	18	2	2	NUM
asir-3636	4	19	on	on	ADP
asir-3636	4	20	the	the	DET
asir-3636	4	21	unit	unit	NOUN
asir-3636	4	22	disk	disk	NOUN
asir-3636	4	23	𝔻	𝔻	PROPN
asir-3636	4	24	with	with	ADP
asir-3636	4	25	finite	finite	ADJ
asir-3636	4	26	dirichlet	dirichlet	PROPN
asir-3636	4	27	integral	integral	ADJ
asir-3636	4	28	∑𝐷(𝑓𝑗	∑𝐷(𝑓𝑗	PROPN
asir-3636	4	29	2	2	NUM
asir-3636	4	30	)	)	PUNCT
asir-3636	4	31	𝑗	𝑗	NOUN
asir-3636	4	32	:	:	PUNCT
asir-3636	5	1	=	=	SYM
asir-3636	5	2	∫	∫	PROPN
asir-3636	5	3	∑	∑	PROPN
asir-3636	5	4	𝑗	𝑗	PROPN
asir-3636	5	5	|(𝑓𝑗	|(𝑓𝑗	NUM
asir-3636	5	6	2	2	NUM
asir-3636	5	7	)	)	PUNCT
asir-3636	5	8	′	′	NOUN
asir-3636	5	9	(	(	PUNCT
asir-3636	5	10	𝑧)|	𝑧)|	ADV
asir-3636	5	11	2	2	NUM
asir-3636	5	12	𝔻	𝔻	PROPN
asir-3636	5	13	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	5	14	)	)	PUNCT
asir-3636	5	15	<	<	X
asir-3636	6	1	+	+	PROPN
asir-3636	6	2	∞	∞	PROPN
asir-3636	6	3	,	,	PUNCT
asir-3636	6	4	where	where	SCONJ
asir-3636	6	5	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
asir-3636	6	6	)	)	PUNCT
asir-3636	6	7	=	=	SYM
asir-3636	6	8	1	1	NUM
asir-3636	6	9	𝜋	𝜋	NOUN
asir-3636	6	10	(	(	PUNCT
asir-3636	6	11	1	1	NUM
asir-3636	6	12	−	−	NOUN
asir-3636	6	13	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	6	14	−	−	PROPN
asir-3636	6	15	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	6	16	denotes	denote	VERB
asir-3636	6	17	the	the	DET
asir-3636	6	18	normalized	normalized	ADJ
asir-3636	6	19	area	area	NOUN
asir-3636	6	20	measure	measure	NOUN
asir-3636	6	21	on	on	ADP
asir-3636	6	22	𝔻.	𝔻.	NOUN
asir-3636	6	23	equipped	equip	VERB
asir-3636	6	24	with	with	ADP
asir-3636	6	25	the	the	DET
asir-3636	6	26	pointwise	pointwise	ADJ
asir-3636	6	27	algebraic	algebraic	ADJ
asir-3636	6	28	operations	operation	NOUN
asir-3636	6	29	and	and	CCONJ
asir-3636	6	30	the	the	DET
asir-3636	6	31	series	series	NOUN
asir-3636	6	32	of	of	ADP
asir-3636	6	33	norms	norm	NOUN
asir-3636	6	34	∑‖𝑓𝑗	∑‖𝑓𝑗	PROPN
asir-3636	6	35	2‖	2‖	PROPN
asir-3636	6	36	𝒟	𝒟	NOUN
asir-3636	6	37	2	2	NUM
asir-3636	6	38	𝑗	𝑗	NOUN
asir-3636	6	39	≔	≔	NOUN
asir-3636	6	40	1	1	NUM
asir-3636	6	41	2𝜋	2𝜋	NUM
asir-3636	6	42	∫	∫	NOUN
asir-3636	7	1	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	7	2	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	7	3	2	2	X
asir-3636	7	4	)	)	PUNCT
asir-3636	7	5	|	|	ADV
asir-3636	7	6	2	2	NUM
asir-3636	7	7	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	7	8	+	+	CCONJ
asir-3636	7	9	𝐷(𝑓𝑗	𝐷(𝑓𝑗	NOUN
asir-3636	7	10	2	2	NUM
asir-3636	7	11	)	)	PUNCT
asir-3636	7	12	𝑗	𝑗	NOUN
asir-3636	8	1	=	=	PUNCT
asir-3636	8	2	∑∑(1	∑∑(1	PROPN
asir-3636	8	3	+	+	CCONJ
asir-3636	8	4	𝑛)|𝑓𝑗	𝑛)|𝑓𝑗	NOUN
asir-3636	8	5	2̂(𝑛)|	2̂(𝑛)|	NUM
asir-3636	8	6	2	2	NUM
asir-3636	8	7	𝑗	𝑗	NOUN
asir-3636	8	8	∞	∞	NUM
asir-3636	8	9	𝑛=0	𝑛=0	PROPN
asir-3636	8	10	2𝜋	2𝜋	NOUN
asir-3636	8	11	0	0	NUM
asir-3636	8	12	,	,	PUNCT
asir-3636	8	13	𝒟	𝒟	PROPN
asir-3636	8	14	becomes	become	VERB
asir-3636	8	15	a	a	DET
asir-3636	8	16	hilbert	hilbert	NOUN
asir-3636	8	17	space	space	NOUN
asir-3636	8	18	.	.	PUNCT
asir-3636	9	1	for	for	ADP
asir-3636	9	2	0	0	NUM
asir-3636	9	3	<	<	X
asir-3636	9	4	𝛼𝑗	𝛼𝑗	PROPN
asir-3636	9	5	2	2	NUM
asir-3636	9	6	≤	≤	NUM
asir-3636	9	7	1	1	NUM
asir-3636	9	8	,	,	PUNCT
asir-3636	9	9	let	let	VERB
asir-3636	9	10	lip𝛼𝑗	lip𝛼𝑗	NOUN
asir-3636	9	11	2	2	NUM
asir-3636	9	12	be	be	AUX
asir-3636	9	13	the	the	DET
asir-3636	9	14	algebra	algebra	NOUN
asir-3636	9	15	of	of	ADP
asir-3636	9	16	sequence	sequence	NOUN
asir-3636	9	17	of	of	ADP
asir-3636	9	18	square	square	ADJ
asir-3636	9	19	analytic	analytic	ADJ
asir-3636	9	20	functions	function	NOUN
asir-3636	9	21	𝑓𝑗	𝑓𝑗	VERB
asir-3636	9	22	2	2	NUM
asir-3636	9	23	on	on	ADP
asir-3636	9	24	𝔻	𝔻	PROPN
asir-3636	9	25	that	that	PRON
asir-3636	9	26	are	be	AUX
asir-3636	9	27	continuous	continuous	ADJ
asir-3636	9	28	on	on	ADP
asir-3636	9	29	�	�	PROPN
asir-3636	9	30	̅	̅	NOUN
asir-3636	9	31	�	�	NOUN
asir-3636	9	32	satisfing	satisfe	VERB
asir-3636	9	33	the	the	DET
asir-3636	9	34	lipschitz	lipschitz	ADJ
asir-3636	9	35	condition	condition	NOUN
asir-3636	9	36	of	of	ADP
asir-3636	9	37	order	order	NOUN
asir-3636	9	38	𝛼𝑗	𝛼𝑗	NOUN
asir-3636	9	39	2	2	NUM
asir-3636	9	40	on	on	ADP
asir-3636	9	41	�	�	NOUN
asir-3636	9	42	̅	̅	NOUN
asir-3636	9	43	�	�	NOUN
asir-3636	9	44	:	:	PUNCT
asir-3636	9	45	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	9	46	applied	apply	VERB
asir-3636	9	47	science	science	NOUN
asir-3636	9	48	and	and	CCONJ
asir-3636	9	49	innovative	innovative	ADJ
asir-3636	9	50	research	research	NOUN
asir-3636	9	51	vol	vol	NOUN
asir-3636	9	52	.	.	PROPN
asir-3636	10	1	5	5	NUM
asir-3636	10	2	,	,	PUNCT
asir-3636	10	3	no	no	INTJ
asir-3636	10	4	.	.	NOUN
asir-3636	10	5	1	1	NUM
asir-3636	10	6	,	,	PUNCT
asir-3636	10	7	2021	2021	NUM
asir-3636	10	8	21	21	NUM
asir-3636	10	9	published	publish	VERB
asir-3636	10	10	by	by	ADP
asir-3636	10	11	scholink	scholink	PROPN
asir-3636	10	12	inc	inc	PROPN
asir-3636	10	13	.	.	PUNCT
asir-3636	11	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	11	2	2(𝑧	2(𝑧	NUM
asir-3636	11	3	)	)	PUNCT
asir-3636	11	4	−	−	ADP
asir-3636	11	5	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	11	6	2(𝑧	2(𝑧	NUM
asir-3636	11	7	−	−	NOUN
asir-3636	11	8	𝜖)|	𝜖)|	NOUN
asir-3636	11	9	𝑗	𝑗	PROPN
asir-3636	11	10	=	=	X
asir-3636	11	11	∑𝑜	∑𝑜	ADJ
asir-3636	11	12	(	(	PUNCT
asir-3636	11	13	|𝜖|𝛼𝑗	|𝜖|𝛼𝑗	NOUN
asir-3636	11	14	2	2	X
asir-3636	11	15	)	)	PUNCT
asir-3636	11	16	𝑗	𝑗	PROPN
asir-3636	11	17	(	(	PUNCT
asir-3636	11	18	|𝜖|	|𝜖|	PROPN
asir-3636	11	19	→	→	SYM
asir-3636	11	20	0	0	NUM
asir-3636	11	21	)	)	PUNCT
asir-3636	11	22	.	.	PUNCT
asir-3636	12	1	note	note	VERB
asir-3636	12	2	that	that	SCONJ
asir-3636	12	3	this	this	DET
asir-3636	12	4	condition	condition	NOUN
asir-3636	12	5	is	be	AUX
asir-3636	12	6	equivalent	equivalent	ADJ
asir-3636	12	7	to	to	ADP
asir-3636	12	8	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	13	1	2)′(𝑧)|	2)′(𝑧)|	NUM
asir-3636	13	2	𝑗	𝑗	NOUN
asir-3636	13	3	=	=	ADJ
asir-3636	13	4	∑𝑜((1	∑𝑜((1	NOUN
asir-3636	13	5	−	−	PROPN
asir-3636	13	6	|𝑧|)𝛼𝑗	|𝑧|)𝛼𝑗	PROPN
asir-3636	13	7	2−1	2−1	NOUN
asir-3636	13	8	)	)	PUNCT
asir-3636	13	9	𝑗	𝑗	PROPN
asir-3636	13	10	(	(	PUNCT
asir-3636	13	11	|𝑧|	|𝑧|	PROPN
asir-3636	13	12	→	→	SYM
asir-3636	13	13	1−	1−	NUM
asir-3636	13	14	)	)	PUNCT
asir-3636	13	15	.	.	PUNCT
asir-3636	14	1	then	then	ADV
asir-3636	14	2	,	,	PUNCT
asir-3636	14	3	𝑙𝑖𝑝𝛼𝑗	𝑙𝑖𝑝𝛼𝑗	NOUN
asir-3636	14	4	2	2	NUM
asir-3636	14	5	is	be	AUX
asir-3636	14	6	a	a	DET
asir-3636	14	7	banach	banach	NOUN
asir-3636	14	8	algebra	algebra	NOUN
asir-3636	14	9	when	when	SCONJ
asir-3636	14	10	equipped	equip	VERB
asir-3636	14	11	with	with	ADP
asir-3636	14	12	series	series	NOUN
asir-3636	14	13	of	of	ADP
asir-3636	14	14	norms	norm	NOUN
asir-3636	14	15	∑‖𝑓𝑗	∑‖𝑓𝑗	PROPN
asir-3636	14	16	2‖	2‖	PROPN
asir-3636	14	17	𝛼𝑗	𝛼𝑗	ADJ
asir-3636	14	18	2	2	NUM
asir-3636	14	19	𝑗	𝑗	NOUN
asir-3636	14	20	∶=∑‖𝑓𝑗	∶=∑‖𝑓𝑗	NOUN
asir-3636	15	1	2‖	2‖	NUM
asir-3636	15	2	∞	∞	NUM
asir-3636	15	3	𝑗	𝑗	PROPN
asir-3636	16	1	+	+	CCONJ
asir-3636	16	2	sup∑{(1	sup∑{(1	PROPN
asir-3636	16	3	−	−	NOUN
asir-3636	16	4	|𝑧|)1−𝛼𝑗	|𝑧|)1−𝛼𝑗	NOUN
asir-3636	16	5	2	2	NUM
asir-3636	16	6	|(𝑓𝑗	|(𝑓𝑗	NUM
asir-3636	16	7	2)′(𝑧)|	2)′(𝑧)|	NUM
asir-3636	16	8	j	j	NOUN
asir-3636	17	1	∶	∶	NOUN
asir-3636	17	2	𝑧	𝑧	PRON
asir-3636	17	3	∈	∈	PROPN
asir-3636	17	4	𝔻	𝔻	ADJ
asir-3636	17	5	}	}	PUNCT
asir-3636	17	6	.	.	PUNCT
asir-3636	18	1	here	here	ADV
asir-3636	18	2	∑	∑	PUNCT
asir-3636	18	3	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	18	4	2‖	2‖	PROPN
asir-3636	18	5	∞𝑗	∞𝑗	NUM
asir-3636	18	6	∶=	∶=	NUM
asir-3636	18	7	sup𝑧∈𝔻∑	sup𝑧∈𝔻∑	NOUN
asir-3636	18	8	|𝑓𝑗	|𝑓𝑗	ADP
asir-3636	18	9	2(𝑧)|𝑗	2(𝑧)|𝑗	NUM
asir-3636	18	10	.	.	PUNCT
asir-3636	19	1	unlike	unlike	ADP
asir-3636	19	2	as	as	ADP
asir-3636	19	3	for	for	ADP
asir-3636	19	4	the	the	DET
asir-3636	19	5	case	case	NOUN
asir-3636	19	6	when	when	SCONJ
asir-3636	19	7	0	0	X
asir-3636	19	8	<	<	X
asir-3636	19	9	𝛼𝑗	𝛼𝑗	PROPN
asir-3636	19	10	2	2	NUM
asir-3636	19	11	≤	≤	NUM
asir-3636	19	12	1	1	NUM
asir-3636	19	13	4	4	NUM
asir-3636	19	14	,	,	PUNCT
asir-3636	19	15	the	the	DET
asir-3636	19	16	inclusion	inclusion	NOUN
asir-3636	19	17	lip𝛼𝑗	lip𝛼𝑗	NOUN
asir-3636	19	18	2	2	NUM
asir-3636	19	19	⊂	⊂	PROPN
asir-3636	19	20	𝒟	𝒟	PROPN
asir-3636	19	21	always	always	ADV
asir-3636	19	22	holds	hold	VERB
asir-3636	19	23	provided	provide	VERB
asir-3636	19	24	that	that	SCONJ
asir-3636	19	25	1	1	NUM
asir-3636	19	26	4	4	NUM
asir-3636	19	27	<	<	X
asir-3636	19	28	𝛼𝑗	𝛼𝑗	PROPN
asir-3636	19	29	2	2	NUM
asir-3636	19	30	≤	≤	NUM
asir-3636	19	31	1	1	NUM
asir-3636	19	32	.	.	PUNCT
asir-3636	20	1	in	in	ADP
asir-3636	20	2	what	what	PRON
asir-3636	20	3	follows	follow	VERB
asir-3636	20	4	,	,	PUNCT
asir-3636	20	5	let	let	VERB
asir-3636	20	6	0	0	PUNCT
asir-3636	20	7	<	<	X
asir-3636	20	8	𝛼𝑗	𝛼𝑗	PROPN
asir-3636	20	9	2	2	NUM
asir-3636	20	10	≤	≤	NUM
asir-3636	20	11	1	1	NUM
asir-3636	20	12	4	4	NUM
asir-3636	20	13	and	and	CCONJ
asir-3636	20	14	define	define	VERB
asir-3636	20	15	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	20	16	2	2	NUM
asir-3636	20	17	∶=	∶=	NOUN
asir-3636	20	18	𝒟	𝒟	NOUN
asir-3636	20	19	∩	∩	NOUN
asir-3636	20	20	lip𝛼𝑗	lip𝛼𝑗	NOUN
asir-3636	20	21	2	2	NUM
asir-3636	20	22	.	.	PUNCT
asir-3636	21	1	it	it	PRON
asir-3636	21	2	is	be	AUX
asir-3636	21	3	easy	easy	ADJ
asir-3636	21	4	to	to	PART
asir-3636	21	5	check	check	VERB
asir-3636	21	6	that	that	SCONJ
asir-3636	21	7	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	21	8	2	2	NUM
asir-3636	21	9	is	be	AUX
asir-3636	21	10	a	a	DET
asir-3636	21	11	commutative	commutative	ADJ
asir-3636	21	12	banach	banach	NOUN
asir-3636	21	13	algebra	algebra	NOUN
asir-3636	21	14	when	when	SCONJ
asir-3636	21	15	it	it	PRON
asir-3636	21	16	is	be	AUX
asir-3636	21	17	endowed	endow	VERB
asir-3636	21	18	with	with	ADP
asir-3636	21	19	the	the	DET
asir-3636	21	20	pointwise	pointwise	ADJ
asir-3636	21	21	algebraic	algebraic	ADJ
asir-3636	21	22	operations	operation	NOUN
asir-3636	21	23	and	and	CCONJ
asir-3636	21	24	series	serie	NOUN
asir-3636	21	25	of	of	ADP
asir-3636	21	26	norms	norm	NOUN
asir-3636	21	27	∑	∑	PUNCT
asir-3636	21	28	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	21	29	2‖	2‖	PROPN
asir-3636	21	30	𝒜	𝒜	NOUN
asir-3636	21	31	αj	αj	NOUN
asir-3636	21	32	2	2	NUM
asir-3636	21	33	𝑗	𝑗	NOUN
asir-3636	21	34	∶=	∶=	NUM
asir-3636	21	35	∑	∑	PROPN
asir-3636	21	36	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	21	37	2‖	2‖	PROPN
asir-3636	21	38	αj	αj	ADP
asir-3636	21	39	2𝑗	2𝑗	NOUN
asir-3636	21	40	+	+	CCONJ
asir-3636	21	41	∑	∑	PROPN
asir-3636	21	42	𝐷	𝐷	PROPN
asir-3636	21	43	1	1	NUM
asir-3636	21	44	2(𝑓𝑗	2(𝑓𝑗	NUM
asir-3636	21	45	2)𝑗	2)𝑗	NOUN
asir-3636	21	46	,	,	PUNCT
asir-3636	21	47	(	(	PUNCT
asir-3636	21	48	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	21	49	2	2	NUM
asir-3636	21	50	∈	∈	NOUN
asir-3636	21	51	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	21	52	2	2	NUM
asir-3636	21	53	)	)	PUNCT
asir-3636	21	54	.	.	PUNCT
asir-3636	22	1	in	in	ADP
asir-3636	22	2	order	order	NOUN
asir-3636	22	3	to	to	PART
asir-3636	22	4	describe	describe	VERB
asir-3636	22	5	the	the	DET
asir-3636	22	6	closed	closed	ADJ
asir-3636	22	7	ideals	ideal	NOUN
asir-3636	22	8	in	in	ADP
asir-3636	22	9	subalgebras	subalgebra	NOUN
asir-3636	22	10	of	of	ADP
asir-3636	22	11	the	the	DET
asir-3636	22	12	disc	disc	NOUN
asir-3636	22	13	algebra	algebra	PROPN
asir-3636	22	14	𝐴(𝔻	𝐴(𝔻	PROPN
asir-3636	22	15	)	)	PUNCT
asir-3636	22	16	,	,	PUNCT
asir-3636	22	17	it	it	PRON
asir-3636	22	18	is	be	AUX
asir-3636	22	19	natural	natural	ADJ
asir-3636	22	20	to	to	PART
asir-3636	22	21	make	make	VERB
asir-3636	22	22	use	use	NOUN
asir-3636	22	23	of	of	ADP
asir-3636	22	24	nevanlinna	nevanlinna	NOUN
asir-3636	22	25	’s	’s	PART
asir-3636	22	26	factorization	factorization	NOUN
asir-3636	22	27	theory	theory	NOUN
asir-3636	22	28	.	.	PUNCT
asir-3636	23	1	for	for	ADP
asir-3636	23	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	23	3	2	2	NUM
asir-3636	23	4	∈	∈	PROPN
asir-3636	23	5	𝐴(𝔻	𝐴(𝔻	NOUN
asir-3636	23	6	)	)	PUNCT
asir-3636	23	7	there	there	PRON
asir-3636	23	8	is	be	VERB
asir-3636	23	9	a	a	DET
asir-3636	23	10	canonical	canonical	ADJ
asir-3636	23	11	factorization	factorization	NOUN
asir-3636	23	12	=	=	PUNCT
asir-3636	23	13	𝐶𝑓𝑗	𝐶𝑓𝑗	PROPN
asir-3636	23	14	2𝑈𝑓𝑗	2𝑈𝑓𝑗	NUM
asir-3636	23	15	2𝑂𝑓𝑗	2𝑂𝑓𝑗	NUM
asir-3636	23	16	2	2	NUM
asir-3636	23	17	,	,	PUNCT
asir-3636	23	18	where	where	SCONJ
asir-3636	23	19	𝐶𝑓𝑗	𝐶𝑓𝑗	PROPN
asir-3636	23	20	2	2	NUM
asir-3636	23	21	is	be	AUX
asir-3636	23	22	a	a	DET
asir-3636	23	23	constant	constant	ADJ
asir-3636	23	24	,	,	PUNCT
asir-3636	23	25	𝑈𝑓𝑗	𝑈𝑓𝑗	PROPN
asir-3636	23	26	2	2	NUM
asir-3636	23	27	a	a	DET
asir-3636	23	28	sequence	sequence	NOUN
asir-3636	23	29	of	of	ADP
asir-3636	23	30	square	square	ADJ
asir-3636	23	31	inner	inner	ADJ
asir-3636	23	32	functions	function	NOUN
asir-3636	23	33	that	that	PRON
asir-3636	23	34	is	be	AUX
asir-3636	23	35	∑	∑	PROPN
asir-3636	23	36	|𝑈𝑓𝑗	|𝑈𝑓𝑗	PROPN
asir-3636	23	37	2|𝑗	2|𝑗	NUM
asir-3636	23	38	=	=	SYM
asir-3636	23	39	1	1	NUM
asir-3636	23	40	a.e	a.e	NOUN
asir-3636	23	41	on	on	ADP
asir-3636	23	42	𝕋	𝕋	NOUN
asir-3636	23	43	and	and	CCONJ
asir-3636	23	44	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	23	45	2	2	NUM
asir-3636	23	46	the	the	DET
asir-3636	23	47	sequence	sequence	NOUN
asir-3636	23	48	of	of	ADP
asir-3636	23	49	square	square	ADJ
asir-3636	23	50	outer	outer	ADJ
asir-3636	23	51	functions	function	NOUN
asir-3636	23	52	given	give	VERB
asir-3636	23	53	by	by	ADP
asir-3636	23	54	∑𝑂𝑓𝑗	∑𝑂𝑓𝑗	ADV
asir-3636	23	55	2(𝑧	2(𝑧	NUM
asir-3636	23	56	)	)	PUNCT
asir-3636	23	57	𝑗	𝑗	PROPN
asir-3636	23	58	=	=	SYM
asir-3636	23	59	exp	exp	NOUN
asir-3636	23	60	{	{	PUNCT
asir-3636	23	61	1	1	NUM
asir-3636	23	62	2𝜋	2𝜋	NUM
asir-3636	23	63	∫	∫	NOUN
asir-3636	23	64	2𝜋	2𝜋	NOUN
asir-3636	23	65	0	0	PUNCT
asir-3636	23	66	∑	∑	PROPN
asir-3636	23	67	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	23	68	2	2	NUM
asir-3636	23	69	+	+	CCONJ
asir-3636	23	70	𝑧	𝑧	PROPN
asir-3636	23	71	𝑒𝑖𝜃	𝑒𝑖𝜃	NUM
asir-3636	23	72	2	2	NUM
asir-3636	23	73	−	−	PROPN
asir-3636	23	74	𝑧	𝑧	PRON
asir-3636	23	75	𝑗	𝑗	INTJ
asir-3636	23	76	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	23	77	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	23	78	2	2	NUM
asir-3636	23	79	)	)	PUNCT
asir-3636	23	80	|𝑑𝜃2	|𝑑𝜃2	NUM
asir-3636	23	81	}	}	PUNCT
asir-3636	23	82	.	.	PUNCT
asir-3636	24	1	denote	denote	VERB
asir-3636	24	2	by	by	ADP
asir-3636	24	3	ℋ∞(𝔻	ℋ∞(𝔻	PROPN
asir-3636	24	4	)	)	PUNCT
asir-3636	24	5	the	the	DET
asir-3636	24	6	algebra	algebra	NOUN
asir-3636	24	7	of	of	ADP
asir-3636	24	8	bounded	bounded	ADJ
asir-3636	24	9	analytic	analytic	ADJ
asir-3636	24	10	functions	function	NOUN
asir-3636	24	11	.	.	PUNCT
asir-3636	25	1	note	note	VERB
asir-3636	25	2	that	that	SCONJ
asir-3636	25	3	𝒜𝛼𝑗	𝒜𝛼𝑗	PROPN
asir-3636	25	4	2	2	NUM
asir-3636	25	5	has	have	VERB
asir-3636	25	6	the	the	DET
asir-3636	25	7	so	so	ADV
asir-3636	25	8	-	-	PUNCT
asir-3636	25	9	called	call	VERB
asir-3636	25	10	f	f	NOUN
asir-3636	25	11	-	-	PUNCT
asir-3636	25	12	property	property	NOUN
asir-3636	25	13	(	(	PUNCT
asir-3636	25	14	shirokov	shirokov	NOUN
asir-3636	25	15	,	,	PUNCT
asir-3636	25	16	1988	1988	NUM
asir-3636	25	17	;	;	PUNCT
asir-3636	26	1	carleson	carleson	PROPN
asir-3636	26	2	,	,	PUNCT
asir-3636	26	3	1960	1960	NUM
asir-3636	26	4	):	):	PUNCT
asir-3636	26	5	if	if	SCONJ
asir-3636	26	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	26	7	2	2	NUM
asir-3636	26	8	∈	∈	NOUN
asir-3636	26	9	𝒜𝛼𝑗	𝒜𝛼𝑗	ADP
asir-3636	26	10	2	2	NUM
asir-3636	26	11	and	and	CCONJ
asir-3636	26	12	𝑈	𝑈	PROPN
asir-3636	26	13	is	be	AUX
asir-3636	26	14	an	an	DET
asir-3636	26	15	inner	inner	ADJ
asir-3636	26	16	function	function	NOUN
asir-3636	26	17	such	such	ADJ
asir-3636	26	18	that	that	DET
asir-3636	26	19	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	26	20	2/𝑈	2/𝑈	NUM
asir-3636	26	21	∈	∈	PROPN
asir-3636	26	22	ℋ∞(𝔻	ℋ∞(𝔻	NOUN
asir-3636	26	23	)	)	PUNCT
asir-3636	26	24	then	then	ADV
asir-3636	26	25	𝑓𝑗	𝑓𝑗	VERB
asir-3636	26	26	2/𝑈	2/𝑈	NUM
asir-3636	26	27	∈	∈	PROPN
asir-3636	26	28	𝒜αj	𝒜αj	PROPN
asir-3636	26	29	2	2	NUM
asir-3636	26	30	and	and	CCONJ
asir-3636	26	31	∑	∑	NOUN
asir-3636	26	32	‖𝑓𝑗	‖𝑓𝑗	NUM
asir-3636	26	33	2/𝑈‖	2/𝑈‖	NUM
asir-3636	26	34	𝒜	𝒜	NOUN
asir-3636	26	35	αj	αj	NOUN
asir-3636	26	36	2	2	NUM
asir-3636	26	37	𝑗	𝑗	NOUN
asir-3636	26	38	≤	≤	NOUN
asir-3636	26	39	∑	∑	PUNCT
asir-3636	26	40	𝐶𝛼𝑗	𝐶𝛼𝑗	ADJ
asir-3636	26	41	2‖𝑓𝑗	2‖𝑓𝑗	NOUN
asir-3636	26	42	2‖	2‖	X
asir-3636	26	43	𝒜	𝒜	NOUN
asir-3636	26	44	αj	αj	NOUN
asir-3636	26	45	2	2	NUM
asir-3636	26	46	𝑗	𝑗	NOUN
asir-3636	26	47	,	,	PUNCT
asir-3636	26	48	where	where	SCONJ
asir-3636	26	49	𝐶𝛼𝑗	𝐶𝛼𝑗	ADJ
asir-3636	26	50	2	2	NUM
asir-3636	26	51	is	be	AUX
asir-3636	26	52	independent	independent	ADJ
asir-3636	26	53	of	of	ADP
asir-3636	26	54	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	26	55	2	2	NUM
asir-3636	26	56	.	.	PUNCT
asir-3636	26	57	korenblum	korenblum	PROPN
asir-3636	26	58	(	(	PUNCT
asir-3636	26	59	1972	1972	NUM
asir-3636	26	60	)	)	PUNCT
asir-3636	26	61	has	have	AUX
asir-3636	26	62	described	describe	VERB
asir-3636	26	63	the	the	DET
asir-3636	26	64	closed	close	VERB
asir-3636	26	65	ideals	ideal	NOUN
asir-3636	26	66	of	of	ADP
asir-3636	26	67	the	the	DET
asir-3636	26	68	algebra	algebra	NOUN
asir-3636	26	69	𝐻1	𝐻1	PROPN
asir-3636	26	70	2	2	NUM
asir-3636	26	71	of	of	ADP
asir-3636	26	72	sequence	sequence	NOUN
asir-3636	26	73	of	of	ADP
asir-3636	26	74	square	square	ADJ
asir-3636	26	75	analytic	analytic	ADJ
asir-3636	26	76	functions	function	NOUN
asir-3636	26	77	𝑓𝑗	𝑓𝑗	VERB
asir-3636	26	78	2	2	NUM
asir-3636	26	79	such	such	ADJ
asir-3636	26	80	that	that	PRON
asir-3636	26	81	(	(	PUNCT
asir-3636	26	82	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	26	83	2)′	2)′	NUM
asir-3636	26	84	∈	∈	PROPN
asir-3636	26	85	𝐻2	𝐻2	NOUN
asir-3636	26	86	,	,	PUNCT
asir-3636	26	87	where	where	SCONJ
asir-3636	26	88	𝐻2	𝐻2	PROPN
asir-3636	26	89	is	be	AUX
asir-3636	26	90	the	the	DET
asir-3636	26	91	hardy	hardy	ADJ
asir-3636	26	92	space	space	NOUN
asir-3636	26	93	.	.	PUNCT
asir-3636	27	1	this	this	DET
asir-3636	27	2	result	result	NOUN
asir-3636	27	3	has	have	AUX
asir-3636	27	4	been	be	AUX
asir-3636	27	5	extended	extend	VERB
asir-3636	27	6	to	to	ADP
asir-3636	27	7	some	some	DET
asir-3636	27	8	other	other	ADJ
asir-3636	27	9	banach	banach	NOUN
asir-3636	27	10	algebras	algebra	NOUN
asir-3636	27	11	of	of	ADP
asir-3636	27	12	sequence	sequence	NOUN
asir-3636	27	13	of	of	ADP
asir-3636	27	14	square	square	ADJ
asir-3636	27	15	analytic	analytic	ADJ
asir-3636	27	16	functions	function	NOUN
asir-3636	27	17	,	,	PUNCT
asir-3636	27	18	by	by	ADP
asir-3636	27	19	matheson	matheson	PROPN
asir-3636	27	20	(	(	PUNCT
asir-3636	27	21	1978	1978	NUM
asir-3636	27	22	)	)	PUNCT
asir-3636	27	23	for	for	ADP
asir-3636	27	24	lip𝛼𝑗	lip𝛼𝑗	NOUN
asir-3636	27	25	2	2	NUM
asir-3636	27	26	and	and	CCONJ
asir-3636	27	27	by	by	ADP
asir-3636	27	28	shamoyan	shamoyan	PROPN
asir-3636	27	29	(	(	PUNCT
asir-3636	27	30	1994	1994	NUM
asir-3636	27	31	)	)	PUNCT
asir-3636	27	32	for	for	ADP
asir-3636	27	33	the	the	DET
asir-3636	27	34	algebra	algebra	NOUN
asir-3636	27	35	𝜆𝑧−𝜖	𝜆𝑧−𝜖	PROPN
asir-3636	27	36	(	(	PUNCT
asir-3636	27	37	𝑛	𝑛	NOUN
asir-3636	27	38	)	)	PUNCT
asir-3636	27	39	of	of	ADP
asir-3636	27	40	sequence	sequence	NOUN
asir-3636	27	41	of	of	ADP
asir-3636	27	42	square	square	ADJ
asir-3636	27	43	analytic	analytic	ADJ
asir-3636	27	44	functions	function	NOUN
asir-3636	27	45	𝑓𝑗	𝑓𝑗	VERB
asir-3636	27	46	2	2	NUM
asir-3636	27	47	on	on	ADP
asir-3636	27	48	𝔻	𝔻	PROPN
asir-3636	27	49	such	such	ADJ
asir-3636	27	50	that	that	SCONJ
asir-3636	27	51	∑	∑	ADP
asir-3636	27	52	|𝑓𝑗	|𝑓𝑗	PROPN
asir-3636	27	53	2)(𝑛)((𝑧	2)(𝑛)((𝑧	NUM
asir-3636	27	54	−	−	PROPN
asir-3636	27	55	2𝜖)1	2𝜖)1	NUM
asir-3636	27	56	)	)	PUNCT
asir-3636	28	1	−	−	PROPN
asir-3636	28	2	(	(	PUNCT
asir-3636	28	3	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	28	4	2)(𝑛)((𝑧	2)(𝑛)((𝑧	NUM
asir-3636	28	5	−	−	NUM
asir-3636	28	6	2𝜖)1	2𝜖)1	NUM
asir-3636	28	7	−	−	PROPN
asir-3636	28	8	𝜖)|𝑗	𝜖)|𝑗	NUM
asir-3636	28	9	=	=	SYM
asir-3636	28	10	𝑜(𝜔(|𝜖|	𝑜(𝜔(|𝜖|	NOUN
asir-3636	28	11	)	)	PUNCT
asir-3636	28	12	)	)	PUNCT
asir-3636	29	1	as	as	ADP
asir-3636	29	2	|𝜖|	|𝜖|	PROPN
asir-3636	29	3	→	→	SYM
asir-3636	29	4	0	0	NUM
asir-3636	29	5	,	,	PUNCT
asir-3636	29	6	where	where	SCONJ
asir-3636	29	7	𝑛	𝑛	PROPN
asir-3636	29	8	is	be	AUX
asir-3636	29	9	a	a	DET
asir-3636	29	10	non	non	ADJ
asir-3636	29	11	negative	negative	ADJ
asir-3636	29	12	integer	integer	NOUN
asir-3636	29	13	and	and	CCONJ
asir-3636	29	14	𝜔	𝜔	AUX
asir-3636	29	15	an	an	DET
asir-3636	29	16	arbitrary	arbitrary	ADJ
asir-3636	29	17	nonnegative	nonnegative	ADJ
asir-3636	29	18	non	non	ADJ
asir-3636	29	19	decreasing	decrease	VERB
asir-3636	29	20	subadditive	subadditive	ADJ
asir-3636	29	21	function	function	NOUN
asir-3636	29	22	on	on	ADP
asir-3636	29	23	(	(	PUNCT
asir-3636	29	24	0	0	NUM
asir-3636	29	25	,	,	PUNCT
asir-3636	29	26	+	+	NOUN
asir-3636	29	27	∞	∞	NOUN
asir-3636	29	28	)	)	PUNCT
asir-3636	29	29	.	.	PUNCT
asir-3636	29	30	shirokov	shirokov	PROPN
asir-3636	29	31	(	(	PUNCT
asir-3636	29	32	1982	1982	NUM
asir-3636	29	33	,	,	PUNCT
asir-3636	29	34	1988	1988	NUM
asir-3636	29	35	)	)	PUNCT
asir-3636	29	36	had	have	AUX
asir-3636	29	37	given	give	VERB
asir-3636	29	38	a	a	DET
asir-3636	29	39	complete	complete	ADJ
asir-3636	29	40	description	description	NOUN
asir-3636	29	41	of	of	ADP
asir-3636	29	42	closed	closed	ADJ
asir-3636	29	43	ideals	ideal	NOUN
asir-3636	29	44	for	for	ADP
asir-3636	29	45	besov	besov	NOUN
asir-3636	29	46	algebras	algebras	PROPN
asir-3636	29	47	𝐴𝐵1+𝜖,1+𝜖	𝐴𝐵1+𝜖,1+𝜖	PROPN
asir-3636	29	48	(	(	PUNCT
asir-3636	29	49	1	1	NUM
asir-3636	29	50	2	2	NUM
asir-3636	29	51	+	+	NOUN
asir-3636	29	52	𝜖	𝜖	X
asir-3636	29	53	)	)	PUNCT
asir-3636	29	54	of	of	ADP
asir-3636	29	55	sequence	sequence	NOUN
asir-3636	29	56	of	of	ADP
asir-3636	29	57	square	square	ADJ
asir-3636	29	58	analytic	analytic	ADJ
asir-3636	29	59	functions	function	NOUN
asir-3636	29	60	and	and	CCONJ
asir-3636	29	61	particularly	particularly	ADV
asir-3636	29	62	for	for	ADP
asir-3636	29	63	the	the	DET
asir-3636	29	64	case	case	NOUN
asir-3636	29	65	𝜖	𝜖	X
asir-3636	29	66	>	>	X
asir-3636	29	67	0	0	NUM
asir-3636	29	68	.	.	NOUN
asir-3636	29	69	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-3636	29	70	applied	apply	VERB
asir-3636	29	71	science	science	NOUN
asir-3636	29	72	and	and	CCONJ
asir-3636	29	73	innovative	innovative	ADJ
asir-3636	29	74	research	research	NOUN
asir-3636	29	75	vol	vol	NOUN
asir-3636	29	76	.	.	PROPN
asir-3636	30	1	5	5	NUM
asir-3636	30	2	,	,	PUNCT
asir-3636	30	3	no	no	INTJ
asir-3636	30	4	.	.	NOUN
asir-3636	30	5	1	1	NUM
asir-3636	30	6	,	,	PUNCT
asir-3636	30	7	2021	2021	NUM
asir-3636	30	8	22	22	NUM
asir-3636	30	9	published	publish	VERB
asir-3636	30	10	by	by	ADP
asir-3636	30	11	scholink	scholink	PROPN
asir-3636	30	12	inc	inc	PROPN
asir-3636	30	13	.	.	PROPN
asir-3636	30	14	𝐴𝐵2,2	𝐴𝐵2,2	PROPN
asir-3636	30	15	(	(	PUNCT
asir-3636	30	16	1	1	NUM
asir-3636	30	17	2	2	NUM
asir-3636	30	18	+	+	NOUN
asir-3636	30	19	𝜖	𝜖	X
asir-3636	30	20	)	)	PUNCT
asir-3636	30	21	=	=	SYM
asir-3636	30	22	{	{	PUNCT
asir-3636	30	23	(	(	PUNCT
asir-3636	30	24	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	30	25	2	2	NUM
asir-3636	30	26	∈	∈	NOUN
asir-3636	30	27	𝐴(𝔻):∑∑|𝑓𝑗	𝐴(𝔻):∑∑|𝑓𝑗	NOUN
asir-3636	30	28	2̂(𝑛)|	2̂(𝑛)|	NUM
asir-3636	30	29	2	2	NUM
asir-3636	30	30	𝑗	𝑗	NOUN
asir-3636	30	31	(	(	PUNCT
asir-3636	30	32	1	1	NUM
asir-3636	30	33	+	+	NUM
asir-3636	30	34	𝑛)(1	𝑛)(1	NOUN
asir-3636	30	35	+	+	SYM
asir-3636	30	36	2𝜖	2𝜖	NUM
asir-3636	30	37	)	)	PUNCT
asir-3636	30	38	<	<	X
asir-3636	30	39	∞	∞	PROPN
asir-3636	30	40	𝑛≥0	𝑛≥0	PROPN
asir-3636	30	41	}	}	PUNCT
asir-3636	30	42	.	.	PUNCT
asir-3636	31	1	note	note	VERB
asir-3636	31	2	that	that	SCONJ
asir-3636	31	3	the	the	DET
asir-3636	31	4	case	case	NOUN
asir-3636	31	5	of	of	ADP
asir-3636	31	6	𝐴𝐵2,2	𝐴𝐵2,2	PROPN
asir-3636	31	7	1	1	NUM
asir-3636	31	8	2	2	NUM
asir-3636	31	9	=	=	SYM
asir-3636	31	10	𝐴(𝔻	𝐴(𝔻	NOUN
asir-3636	31	11	)	)	PUNCT
asir-3636	31	12	∩	∩	NOUN
asir-3636	31	13	𝒟	𝒟	NOUN
asir-3636	31	14	the	the	DET
asir-3636	31	15	problem	problem	NOUN
asir-3636	31	16	of	of	ADP
asir-3636	31	17	description	description	NOUN
asir-3636	31	18	of	of	ADP
asir-3636	31	19	closed	closed	ADJ
asir-3636	31	20	ideals	ideal	NOUN
asir-3636	31	21	appears	appear	VERB
asir-3636	31	22	to	to	PART
asir-3636	31	23	be	be	AUX
asir-3636	31	24	much	much	ADV
asir-3636	31	25	more	more	ADV
asir-3636	31	26	difficult	difficult	ADJ
asir-3636	31	27	(	(	PUNCT
asir-3636	31	28	see	see	VERB
asir-3636	31	29	hedenmalm	hedenmalm	PROPN
asir-3636	31	30	&	&	CCONJ
asir-3636	31	31	shields	shield	NOUN
asir-3636	31	32	,	,	PUNCT
asir-3636	31	33	1990	1990	NUM
asir-3636	31	34	;	;	PUNCT
asir-3636	31	35	el	el	PROPN
asir-3636	31	36	-	-	PUNCT
asir-3636	31	37	fallah	fallah	ADJ
asir-3636	31	38	,	,	PUNCT
asir-3636	31	39	kellay	kellay	NOUN
asir-3636	31	40	,	,	PUNCT
asir-3636	31	41	&	&	CCONJ
asir-3636	31	42	ransford	ransford	PROPN
asir-3636	31	43	,	,	PUNCT
asir-3636	31	44	2006	2006	NUM
asir-3636	31	45	)	)	PUNCT
asir-3636	31	46	.	.	PUNCT
asir-3636	32	1	brahim	brahim	PROPN
asir-3636	32	2	bouya	bouya	PROPN
asir-3636	32	3	(	(	PUNCT
asir-3636	32	4	2008	2008	NUM
asir-3636	32	5	)	)	PUNCT
asir-3636	32	6	described	describe	VERB
asir-3636	32	7	the	the	DET
asir-3636	32	8	structure	structure	NOUN
asir-3636	32	9	of	of	ADP
asir-3636	32	10	the	the	DET
asir-3636	32	11	closed	closed	ADJ
asir-3636	32	12	ideals	ideal	NOUN
asir-3636	32	13	of	of	ADP
asir-3636	32	14	the	the	DET
asir-3636	32	15	banach	banach	NOUN
asir-3636	32	16	algebras	algebra	VERB
asir-3636	32	17	𝒜αj	𝒜αj	PROPN
asir-3636	32	18	2	2	X
asir-3636	32	19	.	.	PUNCT
asir-3636	32	20	more	more	ADV
asir-3636	32	21	precisely	precisely	ADV
asir-3636	32	22	he	he	PRON
asir-3636	32	23	proved	prove	VERB
asir-3636	32	24	that	that	SCONJ
asir-3636	32	25	these	these	DET
asir-3636	32	26	ideals	ideal	NOUN
asir-3636	32	27	are	be	AUX
asir-3636	32	28	standard	standard	ADJ
asir-3636	32	29	in	in	ADP
asir-3636	32	30	the	the	DET
asir-3636	32	31	sense	sense	NOUN
asir-3636	32	32	of	of	ADP
asir-3636	32	33	the	the	DET
asir-3636	32	34	beurling	beurle	VERB
asir-3636	32	35	-	-	PUNCT
asir-3636	32	36	rudin	rudin	ADJ
asir-3636	32	37	characterization	characterization	NOUN
asir-3636	32	38	of	of	ADP
asir-3636	32	39	the	the	DET
asir-3636	32	40	closed	close	VERB
asir-3636	32	41	ideals	ideal	NOUN
asir-3636	32	42	in	in	ADP
asir-3636	32	43	the	the	DET
asir-3636	32	44	disc	disc	NOUN
asir-3636	32	45	algebra	algebra	NOUN
asir-3636	32	46	(	(	PUNCT
asir-3636	32	47	hoffman	hoffman	NOUN
asir-3636	32	48	,	,	PUNCT
asir-3636	32	49	1988	1988	NUM
asir-3636	32	50	)	)	PUNCT
asir-3636	32	51	,	,	PUNCT
asir-3636	32	52	we	we	PRON
asir-3636	32	53	show	show	VERB
asir-3636	32	54	the	the	DET
asir-3636	32	55	general	general	ADJ
asir-3636	32	56	validation	validation	NOUN
asir-3636	32	57	following	follow	VERB
asir-3636	32	58	(	(	PUNCT
asir-3636	32	59	brahim	brahim	PROPN
asir-3636	32	60	bouya	bouya	PROPN
asir-3636	32	61	,	,	PUNCT
asir-3636	32	62	2008	2008	NUM
asir-3636	32	63	):	):	PUNCT
asir-3636	32	64	theorem	theorem	NOUN
asir-3636	32	65	(	(	PUNCT
asir-3636	32	66	1.1	1.1	NUM
asir-3636	32	67	):	):	PUNCT
asir-3636	32	68	if	if	SCONJ
asir-3636	32	69	i	i	PRON
asir-3636	32	70	is	be	AUX
asir-3636	32	71	closed	closed	ADJ
asir-3636	32	72	ideal	ideal	NOUN
asir-3636	32	73	of	of	ADP
asir-3636	32	74	𝒜αj	𝒜αj	PROPN
asir-3636	32	75	2	2	NUM
asir-3636	32	76	,	,	PUNCT
asir-3636	32	77	then	then	ADV
asir-3636	32	78	𝔗	𝔗	PROPN
asir-3636	32	79	=	=	PUNCT
asir-3636	32	80	{	{	PUNCT
asir-3636	32	81	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	32	82	2	2	NUM
asir-3636	32	83	∈	∈	NOUN
asir-3636	32	84	𝒜αj	𝒜αj	PROPN
asir-3636	32	85	2	2	NUM
asir-3636	32	86	:	:	PUNCT
asir-3636	32	87	(	(	PUNCT
asir-3636	32	88	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	32	89	2)∖𝐸𝔗	2)∖𝐸𝔗	NUM
asir-3636	32	90	=	=	SYM
asir-3636	32	91	0	0	NUM
asir-3636	32	92	and	and	CCONJ
asir-3636	32	93	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	32	94	2/𝑈𝔗	2/𝑈𝔗	NUM
asir-3636	32	95	∈	∈	PROPN
asir-3636	32	96	ℋ	ℋ	PROPN
asir-3636	32	97	∞(𝔻	∞(𝔻	NOUN
asir-3636	32	98	)	)	PUNCT
asir-3636	32	99	}	}	PUNCT
asir-3636	32	100	,	,	PUNCT
asir-3636	32	101	where	where	SCONJ
asir-3636	32	102	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	32	103	≔	≔	NOUN
asir-3636	32	104	{	{	PUNCT
asir-3636	32	105	𝑧	𝑧	PRON
asir-3636	32	106	∈	∈	PROPN
asir-3636	32	107	𝕋	𝕋	NOUN
asir-3636	32	108	∶	∶	NOUN
asir-3636	32	109	∑	∑	ADP
asir-3636	32	110	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	32	111	2(𝑧)𝑗	2(𝑧)𝑗	NOUN
asir-3636	32	112	=	=	SYM
asir-3636	32	113	0	0	NUM
asir-3636	32	114	,	,	PUNCT
asir-3636	32	115	∀𝑓𝑗	∀𝑓𝑗	NOUN
asir-3636	32	116	2	2	NUM
asir-3636	32	117	∈	∈	PROPN
asir-3636	32	118	𝔗	𝔗	PROPN
asir-3636	32	119	}	}	PUNCT
asir-3636	32	120	and	and	CCONJ
asir-3636	32	121	𝑈𝔗	𝑈𝔗	PROPN
asir-3636	32	122	is	be	AUX
asir-3636	32	123	the	the	DET
asir-3636	32	124	greatest	great	ADJ
asir-3636	32	125	common	common	ADJ
asir-3636	32	126	divisor	divisor	NOUN
asir-3636	32	127	of	of	ADP
asir-3636	32	128	the	the	DET
asir-3636	32	129	inner	inner	ADJ
asir-3636	32	130	parts	part	NOUN
asir-3636	32	131	of	of	ADP
asir-3636	32	132	the	the	DET
asir-3636	32	133	non	non	ADJ
asir-3636	32	134	-	-	ADJ
asir-3636	32	135	zero	zero	NUM
asir-3636	32	136	functions	function	NOUN
asir-3636	32	137	in	in	ADP
asir-3636	32	138	𝔗.	𝔗.	PROPN
asir-3636	32	139	such	such	ADJ
asir-3636	32	140	characterization	characterization	NOUN
asir-3636	32	141	of	of	ADP
asir-3636	32	142	closed	closed	ADJ
asir-3636	32	143	ideals	ideal	NOUN
asir-3636	32	144	can	can	AUX
asir-3636	32	145	be	be	AUX
asir-3636	32	146	reduced	reduce	VERB
asir-3636	32	147	further	far	ADV
asir-3636	32	148	to	to	ADP
asir-3636	32	149	a	a	DET
asir-3636	32	150	problem	problem	NOUN
asir-3636	32	151	of	of	ADP
asir-3636	32	152	approximation	approximation	NOUN
asir-3636	32	153	of	of	ADP
asir-3636	32	154	outer	outer	ADJ
asir-3636	32	155	functions	function	NOUN
asir-3636	32	156	using	use	VERB
asir-3636	32	157	the	the	DET
asir-3636	32	158	beurling	beurling	NOUN
asir-3636	32	159	–	–	PUNCT
asir-3636	32	160	carleman	carleman	ADJ
asir-3636	32	161	–	–	PUNCT
asir-3636	32	162	domar	domar	ADJ
asir-3636	32	163	resolvent	resolvent	ADJ
asir-3636	32	164	method	method	NOUN
asir-3636	32	165	.	.	PUNCT
asir-3636	33	1	define	define	VERB
asir-3636	33	2	𝑑(𝜉	𝑑(𝜉	PROPN
asir-3636	33	3	,	,	PUNCT
asir-3636	33	4	𝐸	𝐸	PROPN
asir-3636	33	5	)	)	PUNCT
asir-3636	33	6	to	to	PART
asir-3636	33	7	be	be	AUX
asir-3636	33	8	the	the	DET
asir-3636	33	9	distance	distance	NOUN
asir-3636	33	10	from	from	ADP
asir-3636	33	11	𝜉	𝜉	PROPN
asir-3636	33	12	∈	∈	PROPN
asir-3636	33	13	𝑇	𝑇	PROPN
asir-3636	33	14	to	to	ADP
asir-3636	33	15	the	the	DET
asir-3636	33	16	set	set	NOUN
asir-3636	33	17	𝐸	𝐸	PROPN
asir-3636	33	18	⊂	⊂	ADJ
asir-3636	33	19	𝕋.	𝕋.	NOUN
asir-3636	33	20	suppose	suppose	VERB
asir-3636	33	21	that	that	SCONJ
asir-3636	33	22	𝔗	𝔗	PROPN
asir-3636	33	23	is	be	AUX
asir-3636	33	24	a	a	DET
asir-3636	33	25	closed	closed	ADJ
asir-3636	33	26	ideal	ideal	NOUN
asir-3636	33	27	in	in	ADP
asir-3636	33	28	𝒜αj	𝒜αj	PROPN
asir-3636	33	29	2	2	NUM
asir-3636	33	30	such	such	ADJ
asir-3636	33	31	that	that	SCONJ
asir-3636	33	32	𝑈𝔗	𝑈𝔗	PROPN
asir-3636	33	33	=	=	SYM
asir-3636	34	1	1	1	X
asir-3636	34	2	.	.	X
asir-3636	35	1	we	we	PRON
asir-3636	35	2	have	have	VERB
asir-3636	35	3	𝑍𝔗	𝑍𝔗	PROPN
asir-3636	35	4	=	=	SYM
asir-3636	35	5	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	35	6	,	,	PUNCT
asir-3636	35	7	where	where	SCONJ
asir-3636	35	8	𝑍𝔗	𝑍𝔗	PROPN
asir-3636	35	9	≔	≔	VERB
asir-3636	35	10	{	{	PUNCT
asir-3636	35	11	𝑧	𝑧	PROPN
asir-3636	35	12	∈	∈	PROPN
asir-3636	35	13	�	�	PROPN
asir-3636	35	14	̅	̅	NOUN
asir-3636	35	15	�	�	NOUN
asir-3636	35	16	:∑𝑓𝑗	:∑𝑓𝑗	NOUN
asir-3636	35	17	2(𝑧	2(𝑧	NUM
asir-3636	35	18	)	)	PUNCT
asir-3636	35	19	𝑗	𝑗	PROPN
asir-3636	35	20	=	=	SYM
asir-3636	35	21	0	0	NUM
asir-3636	35	22	,	,	PUNCT
asir-3636	35	23	∀𝑓𝑗	∀𝑓𝑗	NOUN
asir-3636	35	24	2	2	NUM
asir-3636	35	25	∈	∈	PROPN
asir-3636	35	26	𝔗	𝔗	PROPN
asir-3636	35	27	}	}	PUNCT
asir-3636	35	28	.	.	PUNCT
asir-3636	36	1	next	next	ADV
asir-3636	36	2	,	,	PUNCT
asir-3636	36	3	for	for	ADP
asir-3636	36	4	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	36	5	2	2	NUM
asir-3636	36	6	∈	∈	NOUN
asir-3636	36	7	𝒜αj	𝒜αj	PROPN
asir-3636	36	8	2	2	NUM
asir-3636	36	9	such	such	ADJ
asir-3636	36	10	that	that	SCONJ
asir-3636	36	11	∑	∑	ADP
asir-3636	36	12	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	36	13	2(𝜉)|𝑗	2(𝜉)|𝑗	NUM
asir-3636	36	14	≤	≤	NUM
asir-3636	36	15	∑	∑	PUNCT
asir-3636	36	16	𝐶𝑑(𝜉	𝐶𝑑(𝜉	NOUN
asir-3636	36	17	,	,	PUNCT
asir-3636	36	18	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	36	19	)	)	PUNCT
asir-3636	36	20	𝑀	𝑀	PROPN
asir-3636	36	21	αj	αj	AUX
asir-3636	36	22	2	2	NUM
asir-3636	36	23	𝑗	𝑗	NOUN
asir-3636	36	24	(	(	PUNCT
asir-3636	36	25	𝜉	𝜉	NOUN
asir-3636	36	26	∈	∈	PROPN
asir-3636	36	27	𝕋	𝕋	PROPN
asir-3636	36	28	)	)	PUNCT
asir-3636	36	29	,	,	PUNCT
asir-3636	36	30	where	where	SCONJ
asir-3636	36	31	𝑀αj	𝑀αj	PROPN
asir-3636	36	32	2	2	NUM
asir-3636	36	33	is	be	AUX
asir-3636	36	34	a	a	DET
asir-3636	36	35	positive	positive	ADJ
asir-3636	36	36	constant	constant	ADJ
asir-3636	36	37	depending	depend	VERB
asir-3636	36	38	only	only	ADV
asir-3636	36	39	on	on	ADP
asir-3636	36	40	𝒜αj	𝒜αj	PROPN
asir-3636	36	41	2	2	NUM
asir-3636	36	42	,	,	PUNCT
asir-3636	36	43	we	we	PRON
asir-3636	36	44	have	have	VERB
asir-3636	36	45	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	36	46	2	2	NUM
asir-3636	36	47	∈	∈	NOUN
asir-3636	36	48	𝔗	𝔗	PROPN
asir-3636	36	49	(	(	PUNCT
asir-3636	36	50	see	see	PROPN
asir-3636	36	51	section	section	NOUN
asir-3636	36	52	3	3	NUM
asir-3636	36	53	for	for	ADP
asir-3636	36	54	more	more	ADJ
asir-3636	36	55	precisions	precision	NOUN
asir-3636	36	56	)	)	PUNCT
asir-3636	36	57	.	.	PUNCT
asir-3636	37	1	now	now	ADV
asir-3636	37	2	,	,	PUNCT
asir-3636	37	3	to	to	PART
asir-3636	37	4	show	show	VERB
asir-3636	37	5	theorem	theorem	ADJ
asir-3636	37	6	(	(	PUNCT
asir-3636	37	7	1.1	1.1	NUM
asir-3636	37	8	)	)	PUNCT
asir-3636	37	9	we	we	PRON
asir-3636	37	10	need	need	VERB
asir-3636	37	11	theorem	theorem	ADJ
asir-3636	37	12	(	(	PUNCT
asir-3636	37	13	1.2	1.2	NUM
asir-3636	37	14	)	)	PUNCT
asir-3636	37	15	below	below	ADV
asir-3636	37	16	,	,	PUNCT
asir-3636	37	17	which	which	PRON
asir-3636	37	18	states	state	VERB
asir-3636	37	19	that	that	SCONJ
asir-3636	37	20	every	every	DET
asir-3636	37	21	function	function	NOUN
asir-3636	37	22	in	in	ADP
asir-3636	37	23	𝒜αj	𝒜αj	PROPN
asir-3636	37	24	2\	2\	PROPN
asir-3636	37	25	{	{	PUNCT
asir-3636	37	26	0	0	NUM
asir-3636	37	27	}	}	PUNCT
asir-3636	37	28	can	can	AUX
asir-3636	37	29	be	be	AUX
asir-3636	37	30	approximated	approximate	VERB
asir-3636	37	31	in	in	ADP
asir-3636	37	32	𝒜αj	𝒜αj	PROPN
asir-3636	37	33	2	2	NUM
asir-3636	37	34	by	by	ADP
asir-3636	37	35	functions	function	NOUN
asir-3636	37	36	with	with	ADP
asir-3636	37	37	boundary	boundary	ADJ
asir-3636	37	38	zeros	zero	NOUN
asir-3636	37	39	of	of	ADP
asir-3636	37	40	arbitrary	arbitrary	ADJ
asir-3636	37	41	high	high	ADJ
asir-3636	37	42	order	order	NOUN
asir-3636	37	43	.	.	PUNCT
asir-3636	38	1	theorem	theorem	NOUN
asir-3636	38	2	(	(	PUNCT
asir-3636	38	3	1.2	1.2	NUM
asir-3636	38	4	):	):	PUNCT
asir-3636	38	5	let	let	VERB
asir-3636	38	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	38	7	2	2	NUM
asir-3636	38	8	be	be	AUX
asir-3636	38	9	a	a	DET
asir-3636	38	10	function	function	NOUN
asir-3636	38	11	in	in	ADP
asir-3636	38	12	𝒜αj	𝒜αj	PROPN
asir-3636	38	13	2\	2\	PROPN
asir-3636	38	14	{	{	PUNCT
asir-3636	38	15	0	0	NUM
asir-3636	38	16	}	}	PUNCT
asir-3636	38	17	and	and	CCONJ
asir-3636	38	18	let	let	VERB
asir-3636	38	19	𝜖	𝜖	PROPN
asir-3636	38	20	≥	≥	PRON
asir-3636	38	21	0	0	NUM
asir-3636	38	22	.	.	PUNCT
asir-3636	39	1	there	there	PRON
asir-3636	39	2	exists	exist	VERB
asir-3636	39	3	a	a	DET
asir-3636	39	4	sequence	sequence	NOUN
asir-3636	39	5	of	of	ADP
asir-3636	39	6	functions	function	NOUN
asir-3636	39	7	{	{	PUNCT
asir-3636	39	8	(	(	PUNCT
asir-3636	39	9	𝑔𝑗)𝑛}𝑛=1	𝑔𝑗)𝑛}𝑛=1	NOUN
asir-3636	39	10	∞	∞	NUM
asir-3636	39	11	⊂	⊂	PROPN
asir-3636	39	12	𝐴(𝔻	𝐴(𝔻	PROPN
asir-3636	39	13	)	)	PUNCT
asir-3636	39	14	such	such	ADJ
asir-3636	39	15	that	that	SCONJ
asir-3636	39	16	(	(	PUNCT
asir-3636	39	17	i	i	NOUN
asir-3636	39	18	)	)	PUNCT
asir-3636	39	19	for	for	ADP
asir-3636	39	20	all	all	DET
asir-3636	39	21	𝑛	𝑛	DET
asir-3636	39	22	∈	∈	PROPN
asir-3636	39	23	ℕ	ℕ	PROPN
asir-3636	39	24	,	,	PUNCT
asir-3636	39	25	we	we	PRON
asir-3636	39	26	have	have	VERB
asir-3636	39	27	∑	∑	ADV
asir-3636	39	28	(	(	PUNCT
asir-3636	39	29	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	39	30	2)𝑛𝑗	2)𝑛𝑗	PROPN
asir-3636	39	31	=	=	SYM
asir-3636	39	32	∑	∑	PUNCT
asir-3636	39	33	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	39	34	2(𝑔𝑗	2(𝑔𝑗	NUM
asir-3636	39	35	2)𝑛𝑗	2)𝑛𝑗	PROPN
asir-3636	39	36	∈	∈	PROPN
asir-3636	39	37	𝒜αj	𝒜αj	PROPN
asir-3636	39	38	2	2	NUM
asir-3636	39	39	and	and	CCONJ
asir-3636	39	40	𝐿𝑖𝑚𝑛→∞∑	𝐿𝑖𝑚𝑛→∞∑	NOUN
asir-3636	39	41	‖(𝑓𝑗	‖(𝑓𝑗	PROPN
asir-3636	39	42	2)𝑛	2)𝑛	NUM
asir-3636	39	43	−	−	ADP
asir-3636	39	44	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	39	45	2‖	2‖	PROPN
asir-3636	39	46	𝒜	𝒜	NOUN
asir-3636	39	47	αj	αj	NOUN
asir-3636	39	48	2	2	NUM
asir-3636	39	49	𝑗	𝑗	NOUN
asir-3636	39	50	=	=	NOUN
asir-3636	39	51	0	0	PROPN
asir-3636	39	52	.	.	PUNCT
asir-3636	39	53	(	(	PUNCT
asir-3636	39	54	ii	ii	NOUN
asir-3636	39	55	)	)	PUNCT
asir-3636	39	56	∑	∑	ADV
asir-3636	39	57	|(𝑔𝑗	|(𝑔𝑗	NUM
asir-3636	39	58	2)(𝜉)|𝑗	2)(𝜉)|𝑗	NUM
asir-3636	39	59	≤	≤	NOUN
asir-3636	39	60	∑	∑	PUNCT
asir-3636	39	61	𝐶𝑛𝑑	𝐶𝑛𝑑	PROPN
asir-3636	39	62	1+𝜖	1+𝜖	NUM
asir-3636	39	63	(	(	PUNCT
asir-3636	39	64	𝜉	𝜉	X
asir-3636	39	65	,	,	PUNCT
asir-3636	39	66	𝐸𝑓𝑗	𝐸𝑓𝑗	PROPN
asir-3636	39	67	2)𝑗	2)𝑗	NOUN
asir-3636	39	68	(	(	PUNCT
asir-3636	39	69	𝜉	𝜉	NOUN
asir-3636	39	70	∈	∈	NOUN
asir-3636	39	71	𝑇),where	𝑇),where	X
asir-3636	40	1	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	40	2	2	2	NUM
asir-3636	40	3	∶=	∶=	NUM
asir-3636	40	4	{	{	PUNCT
asir-3636	40	5	𝜉	𝜉	PROPN
asir-3636	40	6	∈	∈	PROPN
asir-3636	40	7	𝑇	𝑇	PROPN
asir-3636	40	8	∶	∶	PROPN
asir-3636	40	9	∑	∑	PUNCT
asir-3636	40	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	40	11	2(𝜉)𝑗	2(𝜉)𝑗	NUM
asir-3636	40	12	=	=	NOUN
asir-3636	40	13	0	0	NUM
asir-3636	40	14	}	}	PUNCT
asir-3636	40	15	.	.	PUNCT
asir-3636	41	1	to	to	PART
asir-3636	41	2	show	show	VERB
asir-3636	41	3	this	this	DET
asir-3636	41	4	theorem	theorem	NOUN
asir-3636	41	5	,	,	PUNCT
asir-3636	41	6	we	we	PRON
asir-3636	41	7	give	give	VERB
asir-3636	41	8	a	a	DET
asir-3636	41	9	refinement	refinement	NOUN
asir-3636	41	10	of	of	ADP
asir-3636	41	11	the	the	DET
asir-3636	41	12	classical	classical	ADJ
asir-3636	41	13	korenblum	korenblum	NOUN
asir-3636	41	14	approximation	approximation	NOUN
asir-3636	41	15	theory	theory	NOUN
asir-3636	41	16	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	41	17	applied	apply	VERB
asir-3636	41	18	science	science	NOUN
asir-3636	41	19	and	and	CCONJ
asir-3636	41	20	innovative	innovative	ADJ
asir-3636	41	21	research	research	NOUN
asir-3636	41	22	vol	vol	NOUN
asir-3636	41	23	.	.	PROPN
asir-3636	42	1	5	5	NUM
asir-3636	42	2	,	,	PUNCT
asir-3636	42	3	no	no	INTJ
asir-3636	42	4	.	.	NOUN
asir-3636	42	5	1	1	NUM
asir-3636	42	6	,	,	PUNCT
asir-3636	42	7	2021	2021	NUM
asir-3636	42	8	23	23	NUM
asir-3636	42	9	published	publish	VERB
asir-3636	42	10	by	by	ADP
asir-3636	42	11	scholink	scholink	PROPN
asir-3636	42	12	inc	inc	PROPN
asir-3636	42	13	.	.	PROPN
asir-3636	42	14	(	(	PUNCT
asir-3636	42	15	korenblum	korenblum	PROPN
asir-3636	42	16	,	,	PUNCT
asir-3636	42	17	1972	1972	NUM
asir-3636	42	18	;	;	PUNCT
asir-3636	42	19	matheson	matheson	PROPN
asir-3636	42	20	,	,	PUNCT
asir-3636	42	21	1978	1978	NUM
asir-3636	42	22	;	;	PUNCT
asir-3636	42	23	shamoyan	shamoyan	VERB
asir-3636	42	24	,	,	PUNCT
asir-3636	42	25	1994	1994	NUM
asir-3636	42	26	;	;	PUNCT
asir-3636	42	27	shirokov	shirokov	NOUN
asir-3636	42	28	,	,	PUNCT
asir-3636	42	29	1982	1982	NUM
asir-3636	42	30	;	;	PUNCT
asir-3636	42	31	shirokov	shirokov	NOUN
asir-3636	42	32	,	,	PUNCT
asir-3636	42	33	1988	1988	NUM
asir-3636	42	34	)	)	PUNCT
asir-3636	42	35	.	.	PUNCT
asir-3636	43	1	2	2	X
asir-3636	43	2	.	.	X
asir-3636	43	3	main	main	ADJ
asir-3636	43	4	result	result	NOUN
asir-3636	43	5	on	on	ADP
asir-3636	43	6	approximation	approximation	NOUN
asir-3636	43	7	of	of	ADP
asir-3636	43	8	functions	function	NOUN
asir-3636	43	9	in	in	ADP
asir-3636	43	10	𝓐𝛂𝐣	𝓐𝛂𝐣	PROPN
asir-3636	43	11	𝟐	𝟐	NUM
asir-3636	43	12	let	let	VERB
asir-3636	43	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	43	14	2	2	NUM
asir-3636	43	15	∈	∈	NOUN
asir-3636	43	16	𝒜αj	𝒜αj	PROPN
asir-3636	43	17	2	2	NUM
asir-3636	43	18	and	and	CCONJ
asir-3636	43	19	let	let	VERB
asir-3636	43	20	{	{	PUNCT
asir-3636	43	21	𝛾𝑛	𝛾𝑛	NOUN
asir-3636	43	22	∶=	∶=	NUM
asir-3636	43	23	(	(	PUNCT
asir-3636	43	24	𝑎𝑛	𝑎𝑛	PROPN
asir-3636	43	25	,	,	PUNCT
asir-3636	43	26	(	(	PUNCT
asir-3636	43	27	𝑎	𝑎	X
asir-3636	43	28	+	+	NUM
asir-3636	43	29	𝜖)𝑛)}𝑛≥0	𝜖)𝑛)}𝑛≥0	NOUN
asir-3636	43	30	be	be	VERB
asir-3636	43	31	the	the	DET
asir-3636	43	32	countable	countable	ADJ
asir-3636	43	33	collection	collection	NOUN
asir-3636	43	34	of	of	ADP
asir-3636	43	35	the	the	DET
asir-3636	43	36	(	(	PUNCT
asir-3636	43	37	disjoint	disjoint	NOUN
asir-3636	43	38	open	open	ADJ
asir-3636	43	39	)	)	PUNCT
asir-3636	43	40	arcs	arc	NOUN
asir-3636	43	41	of	of	ADP
asir-3636	43	42	𝕋	𝕋	PROPN
asir-3636	43	43	\𝐸𝑓𝑗	\𝐸𝑓𝑗	PROPN
asir-3636	43	44	2	2	NUM
asir-3636	43	45	.	.	PUNCT
asir-3636	44	1	we	we	PRON
asir-3636	44	2	can	can	AUX
asir-3636	44	3	suppose	suppose	VERB
asir-3636	44	4	that	that	SCONJ
asir-3636	44	5	the	the	DET
asir-3636	44	6	arc	arc	NOUN
asir-3636	44	7	lengths	length	NOUN
asir-3636	44	8	of	of	ADP
asir-3636	44	9	𝛾𝑛	𝛾𝑛	NOUN
asir-3636	44	10	are	be	AUX
asir-3636	44	11	less	less	ADJ
asir-3636	44	12	than	than	ADP
asir-3636	44	13	1	1	NUM
asir-3636	44	14	2	2	NUM
asir-3636	44	15	.	.	PUNCT
asir-3636	45	1	in	in	ADP
asir-3636	45	2	what	what	PRON
asir-3636	45	3	follows	follow	VERB
asir-3636	45	4	,	,	PUNCT
asir-3636	45	5	we	we	PRON
asir-3636	45	6	denote	denote	VERB
asir-3636	45	7	by	by	ADP
asir-3636	45	8	γ	γ	PROPN
asir-3636	45	9	the	the	DET
asir-3636	45	10	union	union	NOUN
asir-3636	45	11	of	of	ADP
asir-3636	45	12	a	a	DET
asir-3636	45	13	family	family	NOUN
asir-3636	45	14	of	of	ADP
asir-3636	45	15	arcs	arc	NOUN
asir-3636	45	16	𝛾𝑛.	𝛾𝑛.	AUX
asir-3636	45	17	define	define	VERB
asir-3636	45	18	∑(𝑓𝑗	∑(𝑓𝑗	PROPN
asir-3636	45	19	2	2	NUM
asir-3636	45	20	)	)	PUNCT
asir-3636	45	21	γ	γ	X
asir-3636	45	22	(	(	PUNCT
asir-3636	45	23	𝑧	𝑧	NOUN
asir-3636	45	24	)	)	PUNCT
asir-3636	45	25	𝑗	𝑗	PROPN
asir-3636	45	26	≔	≔	NOUN
asir-3636	45	27	exp	exp	NOUN
asir-3636	45	28	{	{	PUNCT
asir-3636	45	29	1	1	NUM
asir-3636	45	30	2𝜋	2𝜋	NUM
asir-3636	45	31	∫	∫	PROPN
asir-3636	45	32	∑	∑	PROPN
asir-3636	45	33	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	45	34	2	2	NUM
asir-3636	46	1	+	+	CCONJ
asir-3636	46	2	𝑧	𝑧	PROPN
asir-3636	46	3	𝑒𝑖𝜃	𝑒𝑖𝜃	NUM
asir-3636	46	4	2	2	NUM
asir-3636	46	5	−	−	NOUN
asir-3636	46	6	𝑧	𝑧	VERB
asir-3636	46	7	𝑗	𝑗	INTJ
asir-3636	46	8	γ	γ	X
asir-3636	46	9	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	46	10	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	46	11	2	2	NUM
asir-3636	46	12	)	)	PUNCT
asir-3636	46	13	|𝑑𝜃2	|𝑑𝜃2	NUM
asir-3636	46	14	}	}	PUNCT
asir-3636	46	15	.	.	PUNCT
asir-3636	47	1	the	the	DET
asir-3636	47	2	difficult	difficult	ADJ
asir-3636	47	3	part	part	NOUN
asir-3636	47	4	in	in	ADP
asir-3636	47	5	the	the	DET
asir-3636	47	6	proof	proof	NOUN
asir-3636	47	7	of	of	ADP
asir-3636	47	8	theorem	theorem	NOUN
asir-3636	47	9	(	(	PUNCT
asir-3636	47	10	1.2	1.2	NUM
asir-3636	47	11	)	)	PUNCT
asir-3636	47	12	is	be	AUX
asir-3636	47	13	to	to	PART
asir-3636	47	14	establish	establish	VERB
asir-3636	47	15	the	the	DET
asir-3636	47	16	following	following	ADJ
asir-3636	47	17	theorem	theorem	NOUN
asir-3636	47	18	(	(	PUNCT
asir-3636	47	19	2.1	2.1	NUM
asir-3636	47	20	):	):	PUNCT
asir-3636	47	21	let	let	VERB
asir-3636	47	22	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	47	23	2	2	NUM
asir-3636	47	24	∈	∈	NOUN
asir-3636	47	25	𝒜αj	𝒜αj	PROPN
asir-3636	47	26	2\{0	2\{0	NUM
asir-3636	47	27	}	}	PUNCT
asir-3636	47	28	be	be	AUX
asir-3636	47	29	an	an	DET
asir-3636	47	30	outer	outer	ADJ
asir-3636	47	31	function	function	NOUN
asir-3636	47	32	such	such	ADJ
asir-3636	47	33	that	that	SCONJ
asir-3636	47	34	∑	∑	PROPN
asir-3636	47	35	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	47	36	2‖	2‖	PROPN
asir-3636	47	37	𝒜	𝒜	NOUN
asir-3636	47	38	αj	αj	NOUN
asir-3636	47	39	2	2	NUM
asir-3636	47	40	𝑗	𝑗	NOUN
asir-3636	47	41	≤	≤	NUM
asir-3636	47	42	1	1	NUM
asir-3636	47	43	and	and	CCONJ
asir-3636	47	44	let	let	VERB
asir-3636	47	45	𝜖	𝜖	PROPN
asir-3636	47	46	≥	≥	X
asir-3636	47	47	1	1	NUM
asir-3636	47	48	and	and	CCONJ
asir-3636	47	49	𝜖	𝜖	X
asir-3636	47	50	>	>	X
asir-3636	47	51	0	0	NUM
asir-3636	47	52	.	.	PUNCT
asir-3636	48	1	then	then	ADV
asir-3636	48	2	we	we	PRON
asir-3636	48	3	have	have	VERB
asir-3636	48	4	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	48	5	2(1+𝜖	2(1+𝜖	NUM
asir-3636	48	6	)	)	PUNCT
asir-3636	48	7	(	(	PUNCT
asir-3636	48	8	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	48	9	2(1+𝜖	2(1+𝜖	NUM
asir-3636	48	10	)	)	PUNCT
asir-3636	48	11	∈	∈	PROPN
asir-3636	49	1	𝒜αj	𝒜αj	PROPN
asir-3636	49	2	2	2	NUM
asir-3636	49	3	and	and	CCONJ
asir-3636	49	4	supγ∑	supγ∑	NOUN
asir-3636	49	5	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	49	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	49	7	)	)	PUNCT
asir-3636	49	8	(	(	PUNCT
asir-3636	49	9	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	49	10	2(1+𝜖	2(1+𝜖	NUM
asir-3636	49	11	)	)	PUNCT
asir-3636	49	12	‖	‖	PROPN
asir-3636	49	13	𝒜	𝒜	NOUN
asir-3636	49	14	αj	αj	NOUN
asir-3636	49	15	2	2	NUM
asir-3636	49	16	𝑗	𝑗	NOUN
asir-3636	49	17	≤	≤	NOUN
asir-3636	49	18	𝐶1+𝜖,1+𝜖	𝐶1+𝜖,1+𝜖	VERB
asir-3636	49	19	,	,	PUNCT
asir-3636	49	20	(	(	PUNCT
asir-3636	49	21	1	1	X
asir-3636	49	22	)	)	PUNCT
asir-3636	49	23	where	where	SCONJ
asir-3636	49	24	𝐶1+𝜖,1+𝜖	𝐶1+𝜖,1+𝜖	PROPN
asir-3636	49	25	is	be	AUX
asir-3636	49	26	a	a	DET
asir-3636	49	27	positive	positive	ADJ
asir-3636	49	28	constant	constant	ADJ
asir-3636	49	29	independent	independent	NOUN
asir-3636	49	30	of	of	ADP
asir-3636	49	31	γ	γ	PROPN
asir-3636	49	32	.	.	PROPN
asir-3636	49	33	remark	remark	PROPN
asir-3636	49	34	(	(	PUNCT
asir-3636	49	35	2.2	2.2	NUM
asir-3636	49	36	):	):	PUNCT
asir-3636	49	37	for	for	ADP
asir-3636	49	38	a	a	DET
asir-3636	49	39	set	set	NOUN
asir-3636	49	40	𝑆	𝑆	PROPN
asir-3636	49	41	⊂	⊂	PROPN
asir-3636	49	42	𝐴(𝔻	𝐴(𝔻	PROPN
asir-3636	49	43	)	)	PUNCT
asir-3636	49	44	,	,	PUNCT
asir-3636	49	45	we	we	PRON
asir-3636	49	46	denote	denote	VERB
asir-3636	49	47	by	by	ADP
asir-3636	49	48	𝑐𝑜(𝑆	𝑐𝑜(𝑆	NOUN
asir-3636	49	49	)	)	PUNCT
asir-3636	49	50	the	the	DET
asir-3636	49	51	convex	convex	PROPN
asir-3636	49	52	hull	hull	NOUN
asir-3636	49	53	of	of	ADP
asir-3636	49	54	𝑆	𝑆	PROPN
asir-3636	49	55	consisting	consist	VERB
asir-3636	49	56	of	of	ADP
asir-3636	49	57	the	the	DET
asir-3636	49	58	intersection	intersection	NOUN
asir-3636	49	59	of	of	ADP
asir-3636	49	60	all	all	DET
asir-3636	49	61	convex	convex	NOUN
asir-3636	49	62	sets	set	NOUN
asir-3636	49	63	that	that	PRON
asir-3636	49	64	contain	contain	VERB
asir-3636	49	65	𝑆.	𝑆.	NOUN
asir-3636	49	66	set	set	NOUN
asir-3636	49	67	𝛤𝑛	𝛤𝑛	PROPN
asir-3636	49	68	=	=	SYM
asir-3636	49	69	∪𝜖≥0	∪𝜖≥0	NOUN
asir-3636	49	70	𝛾𝑛+𝜖	𝛾𝑛+𝜖	NOUN
asir-3636	49	71	and	and	CCONJ
asir-3636	49	72	let	let	VERB
asir-3636	49	73	𝑓𝑗	𝑓𝑗	NOUN
asir-3636	49	74	2	2	NUM
asir-3636	49	75	be	be	AUX
asir-3636	49	76	as	as	ADP
asir-3636	49	77	in	in	ADP
asir-3636	49	78	the	the	DET
asir-3636	49	79	theorem	theorem	NOUN
asir-3636	49	80	(	(	PUNCT
asir-3636	49	81	2.1	2.1	NUM
asir-3636	49	82	)	)	PUNCT
asir-3636	49	83	it	it	PRON
asir-3636	49	84	is	be	AUX
asir-3636	49	85	clear	clear	ADJ
asir-3636	49	86	that	that	SCONJ
asir-3636	49	87	the	the	DET
asir-3636	49	88	sequence	sequence	NOUN
asir-3636	49	89	(	(	PUNCT
asir-3636	49	90	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	49	91	2(1+𝜖)(𝑓𝑗)γn	2(1+𝜖)(𝑓𝑗)γn	NUM
asir-3636	49	92	2(1+𝜖	2(1+𝜖	NUM
asir-3636	49	93	)	)	PUNCT
asir-3636	49	94	)	)	PUNCT
asir-3636	49	95	converges	converge	VERB
asir-3636	49	96	uniformly	uniformly	ADV
asir-3636	49	97	on	on	ADP
asir-3636	49	98	compact	compact	ADJ
asir-3636	49	99	subsets	subset	NOUN
asir-3636	49	100	of	of	ADP
asir-3636	49	101	𝔻	𝔻	PRON
asir-3636	49	102	to	to	PART
asir-3636	49	103	𝑓𝑗	𝑓𝑗	VERB
asir-3636	49	104	2(1+𝜖	2(1+𝜖	NUM
asir-3636	49	105	)	)	PUNCT
asir-3636	49	106	.	.	PUNCT
asir-3636	50	1	we	we	PRON
asir-3636	50	2	use	use	VERB
asir-3636	50	3	(	(	PUNCT
asir-3636	50	4	2.1	2.1	NUM
asir-3636	50	5	)	)	PUNCT
asir-3636	50	6	to	to	PART
asir-3636	50	7	deduce	deduce	VERB
asir-3636	50	8	,	,	PUNCT
asir-3636	50	9	by	by	ADP
asir-3636	50	10	the	the	DET
asir-3636	50	11	hilbertian	hilbertian	ADJ
asir-3636	50	12	structure	structure	NOUN
asir-3636	50	13	of	of	ADP
asir-3636	50	14	𝒟	𝒟	PROPN
asir-3636	50	15	,	,	PUNCT
asir-3636	50	16	that	that	SCONJ
asir-3636	50	17	there	there	PRON
asir-3636	50	18	is	be	VERB
asir-3636	50	19	a	a	DET
asir-3636	50	20	sequence	sequence	NOUN
asir-3636	50	21	(	(	PUNCT
asir-3636	50	22	ℎ𝑗	ℎ𝑗	PROPN
asir-3636	50	23	2)𝑛	2)𝑛	NUM
asir-3636	50	24	∈	∈	PROPN
asir-3636	50	25	𝑐𝑜({𝑓𝑗	𝑐𝑜({𝑓𝑗	VERB
asir-3636	50	26	2(1+𝜖)(𝑓𝑗)γ1+𝜖	2(1+𝜖)(𝑓𝑗)γ1+𝜖	NUM
asir-3636	50	27	2(1+𝜖	2(1+𝜖	NUM
asir-3636	50	28	)	)	PUNCT
asir-3636	50	29	}	}	PUNCT
asir-3636	50	30	𝜖=0	𝜖=0	NUM
asir-3636	50	31	∞	∞	NUM
asir-3636	50	32	)	)	PUNCT
asir-3636	50	33	converging	converge	VERB
asir-3636	50	34	to	to	ADP
asir-3636	50	35	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	50	36	2(1+𝜖	2(1+𝜖	NUM
asir-3636	50	37	)	)	PUNCT
asir-3636	50	38	in	in	ADP
asir-3636	50	39	𝒟.	𝒟.	PROPN
asir-3636	50	40	also	also	ADV
asir-3636	50	41	,	,	PUNCT
asir-3636	50	42	by	by	ADP
asir-3636	50	43	(	(	PUNCT
asir-3636	50	44	matheson	matheson	PROPN
asir-3636	50	45	,	,	PUNCT
asir-3636	50	46	1978	1978	NUM
asir-3636	50	47	,	,	PUNCT
asir-3636	50	48	section	section	NOUN
asir-3636	50	49	4	4	NUM
asir-3636	50	50	)	)	PUNCT
asir-3636	50	51	,	,	PUNCT
asir-3636	50	52	we	we	PRON
asir-3636	50	53	obtain	obtain	VERB
asir-3636	50	54	that	that	PRON
asir-3636	50	55	(	(	PUNCT
asir-3636	50	56	ℎ𝑗	ℎ𝑗	PRON
asir-3636	50	57	2)𝑛	2)𝑛	NUM
asir-3636	50	58	converges	converge	VERB
asir-3636	50	59	to	to	ADP
asir-3636	50	60	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	50	61	2(1+𝜖	2(1+𝜖	NUM
asir-3636	50	62	)	)	PUNCT
asir-3636	50	63	in	in	ADP
asir-3636	50	64	lipαj	lipαj	NOUN
asir-3636	50	65	2	2	NUM
asir-3636	50	66	,	,	PUNCT
asir-3636	50	67	for	for	ADP
asir-3636	50	68	sufficiently	sufficiently	ADV
asir-3636	50	69	large	large	ADJ
asir-3636	50	70	(	(	PUNCT
asir-3636	50	71	1	1	NUM
asir-3636	50	72	+	+	CCONJ
asir-3636	50	73	𝜖	𝜖	NOUN
asir-3636	50	74	)	)	PUNCT
asir-3636	50	75	(	(	PUNCT
asir-3636	50	76	in	in	ADP
asir-3636	50	77	fact	fact	NOUN
asir-3636	50	78	,	,	PUNCT
asir-3636	50	79	we	we	PRON
asir-3636	50	80	can	can	AUX
asir-3636	50	81	show	show	VERB
asir-3636	50	82	that	that	SCONJ
asir-3636	50	83	this	this	DET
asir-3636	50	84	result	result	NOUN
asir-3636	50	85	remains	remain	VERB
asir-3636	50	86	true	true	ADJ
asir-3636	50	87	for	for	ADP
asir-3636	50	88	every	every	DET
asir-3636	50	89	𝜖	𝜖	X
asir-3636	50	90	≥	≥	NOUN
asir-3636	50	91	0	0	NUM
asir-3636	50	92	)	)	PUNCT
asir-3636	50	93	.	.	PUNCT
asir-3636	51	1	therefore	therefore	ADV
asir-3636	51	2	∑	∑	PROPN
asir-3636	51	3	‖(ℎ𝑗	‖(ℎ𝑗	PROPN
asir-3636	51	4	2)𝑛	2)𝑛	NUM
asir-3636	51	5	−	−	ADP
asir-3636	51	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	51	7	2(1+𝜖	2(1+𝜖	NUM
asir-3636	51	8	)	)	PUNCT
asir-3636	51	9	‖	‖	PROPN
asir-3636	51	10	𝒜	𝒜	NOUN
asir-3636	51	11	αj	αj	NOUN
asir-3636	51	12	2	2	NUM
asir-3636	51	13	→	→	SYM
asir-3636	51	14	0𝑗	0𝑗	NOUN
asir-3636	51	15	,	,	PUNCT
asir-3636	51	16	as	as	SCONJ
asir-3636	51	17	𝑛	𝑛	PROPN
asir-3636	51	18	→	→	SYM
asir-3636	51	19	∞.	∞.	PROPN
asir-3636	51	20	define	define	VERB
asir-3636	51	21	𝒥(𝐹	𝒥(𝐹	NOUN
asir-3636	51	22	)	)	PUNCT
asir-3636	51	23	to	to	PART
asir-3636	51	24	be	be	AUX
asir-3636	51	25	the	the	DET
asir-3636	51	26	closed	closed	ADJ
asir-3636	51	27	ideal	ideal	NOUN
asir-3636	51	28	of	of	ADP
asir-3636	51	29	all	all	DET
asir-3636	51	30	functions	function	NOUN
asir-3636	51	31	in	in	ADP
asir-3636	51	32	𝒜αj	𝒜αj	PROPN
asir-3636	51	33	2	2	NUM
asir-3636	51	34	that	that	PRON
asir-3636	51	35	vanish	vanish	VERB
asir-3636	51	36	on	on	ADP
asir-3636	51	37	𝐹	𝐹	PROPN
asir-3636	51	38	⊂	⊂	PROPN
asir-3636	51	39	�	�	PROPN
asir-3636	51	40	̅	̅	NOUN
asir-3636	51	41	�	�	PROPN
asir-3636	51	42	.	.	PUNCT
asir-3636	52	1	in	in	ADP
asir-3636	52	2	the	the	DET
asir-3636	52	3	proof	proof	NOUN
asir-3636	52	4	of	of	ADP
asir-3636	52	5	theorem	theorem	NOUN
asir-3636	52	6	(	(	PUNCT
asir-3636	52	7	1.2	1.2	NUM
asir-3636	52	8	)	)	PUNCT
asir-3636	52	9	,	,	PUNCT
asir-3636	52	10	we	we	PRON
asir-3636	52	11	need	need	VERB
asir-3636	52	12	the	the	DET
asir-3636	52	13	following	follow	VERB
asir-3636	52	14	classical	classical	ADJ
asir-3636	52	15	lemma	lemma	PROPN
asir-3636	52	16	(	(	PUNCT
asir-3636	52	17	see	see	VERB
asir-3636	52	18	brahim	brahim	PROPN
asir-3636	52	19	bouya	bouya	PROPN
asir-3636	52	20	,	,	PUNCT
asir-3636	52	21	2008	2008	NUM
asir-3636	52	22	)	)	PUNCT
asir-3636	52	23	,	,	PUNCT
asir-3636	52	24	see	see	VERB
asir-3636	52	25	for	for	ADP
asir-3636	52	26	instance	instance	NOUN
asir-3636	52	27	(	(	PUNCT
asir-3636	52	28	matheson	matheson	PROPN
asir-3636	52	29	,	,	PUNCT
asir-3636	52	30	1978	1978	NUM
asir-3636	52	31	,	,	PUNCT
asir-3636	52	32	lemma	lemma	PROPN
asir-3636	52	33	4	4	NUM
asir-3636	52	34	)	)	PUNCT
asir-3636	52	35	and	and	CCONJ
asir-3636	52	36	(	(	PUNCT
asir-3636	52	37	korenblum	korenblum	PROPN
asir-3636	52	38	,	,	PUNCT
asir-3636	52	39	1972	1972	NUM
asir-3636	52	40	,	,	PUNCT
asir-3636	52	41	lemma	lemma	PROPN
asir-3636	52	42	24	24	NUM
asir-3636	52	43	)	)	PUNCT
asir-3636	52	44	.	.	PUNCT
asir-3636	53	1	lemma	lemma	PROPN
asir-3636	53	2	(	(	PUNCT
asir-3636	53	3	2.3	2.3	NUM
asir-3636	53	4	):	):	PUNCT
asir-3636	53	5	let	let	VERB
asir-3636	53	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	53	7	2	2	NUM
asir-3636	53	8	∈	∈	NOUN
asir-3636	53	9	𝒜αj	𝒜αj	PROPN
asir-3636	53	10	2	2	NUM
asir-3636	53	11	and	and	CCONJ
asir-3636	53	12	𝐸′	𝐸′	PROPN
asir-3636	53	13	be	be	AUX
asir-3636	53	14	a	a	DET
asir-3636	53	15	finite	finite	NOUN
asir-3636	53	16	subset	subset	NOUN
asir-3636	53	17	of	of	ADP
asir-3636	53	18	𝕋	𝕋	PRON
asir-3636	53	19	such	such	ADJ
asir-3636	53	20	that	that	SCONJ
asir-3636	53	21	∑	∑	ADP
asir-3636	53	22	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	53	23	2|𝐸′𝑗	2|𝐸′𝑗	NUM
asir-3636	53	24	=	=	SYM
asir-3636	53	25	0	0	PROPN
asir-3636	53	26	.	.	PUNCT
asir-3636	54	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
asir-3636	54	2	𝜖	𝜖	PROPN
asir-3636	54	3	≥	≥	X
asir-3636	54	4	0	0	NUM
asir-3636	54	5	be	be	AUX
asir-3636	54	6	given	give	VERB
asir-3636	54	7	.	.	PUNCT
asir-3636	55	1	for	for	ADP
asir-3636	55	2	every	every	DET
asir-3636	55	3	𝜀	𝜀	NOUN
asir-3636	55	4	>	>	X
asir-3636	55	5	0	0	PUNCT
asir-3636	55	6	there	there	PRON
asir-3636	55	7	is	be	VERB
asir-3636	55	8	an	an	DET
asir-3636	55	9	outer	outer	ADJ
asir-3636	55	10	function	function	NOUN
asir-3636	55	11	𝐹	𝐹	PROPN
asir-3636	55	12	in	in	ADP
asir-3636	55	13	𝒥(𝐸′	𝒥(𝐸′	NOUN
asir-3636	55	14	)	)	PUNCT
asir-3636	55	15	such	such	ADJ
asir-3636	55	16	that	that	SCONJ
asir-3636	55	17	(	(	PUNCT
asir-3636	55	18	i	i	NOUN
asir-3636	55	19	)	)	PUNCT
asir-3636	55	20	∑	∑	PUNCT
asir-3636	56	1	‖𝐹𝑓𝑗	‖𝐹𝑓𝑗	NOUN
asir-3636	56	2	2	2	NUM
asir-3636	56	3	−	−	NOUN
asir-3636	56	4	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	56	5	2‖	2‖	PROPN
asir-3636	56	6	𝒜	𝒜	NOUN
asir-3636	56	7	αj	αj	NOUN
asir-3636	56	8	2	2	NUM
asir-3636	56	9	𝑗	𝑗	PROPN
asir-3636	56	10	≤	≤	NUM
asir-3636	56	11	𝜀	𝜀	PROPN
asir-3636	56	12	,	,	PUNCT
asir-3636	56	13	(	(	PUNCT
asir-3636	56	14	ii	ii	NOUN
asir-3636	56	15	)	)	PUNCT
asir-3636	56	16	|𝐹(𝜉)|	|𝐹(𝜉)|	PROPN
asir-3636	56	17	≤	≤	PROPN
asir-3636	56	18	𝐶𝑑1+𝜖(𝜉	𝐶𝑑1+𝜖(𝜉	PROPN
asir-3636	56	19	,	,	PUNCT
asir-3636	56	20	𝐸′	𝐸′	NOUN
asir-3636	56	21	)	)	PUNCT
asir-3636	56	22	(	(	PUNCT
asir-3636	56	23	𝜉	𝜉	PROPN
asir-3636	56	24	∈	∈	PROPN
asir-3636	56	25	𝕋	𝕋	PROPN
asir-3636	56	26	)	)	PUNCT
asir-3636	56	27	.	.	PUNCT
asir-3636	57	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-3636	57	2	applied	apply	VERB
asir-3636	57	3	science	science	NOUN
asir-3636	57	4	and	and	CCONJ
asir-3636	57	5	innovative	innovative	ADJ
asir-3636	57	6	research	research	NOUN
asir-3636	57	7	vol	vol	NOUN
asir-3636	57	8	.	.	PROPN
asir-3636	58	1	5	5	NUM
asir-3636	58	2	,	,	PUNCT
asir-3636	58	3	no	no	INTJ
asir-3636	58	4	.	.	NOUN
asir-3636	58	5	1	1	NUM
asir-3636	58	6	,	,	PUNCT
asir-3636	58	7	2021	2021	NUM
asir-3636	58	8	24	24	NUM
asir-3636	58	9	published	publish	VERB
asir-3636	58	10	by	by	ADP
asir-3636	58	11	scholink	scholink	PROPN
asir-3636	58	12	inc	inc	PROPN
asir-3636	58	13	.	.	PROPN
asir-3636	58	14	proof	proof	NOUN
asir-3636	58	15	of	of	ADP
asir-3636	58	16	theorem	theorem	NOUN
asir-3636	58	17	(	(	PUNCT
asir-3636	58	18	1.2	1.2	NUM
asir-3636	58	19	):	):	PUNCT
asir-3636	58	20	now	now	ADV
asir-3636	58	21	,	,	PUNCT
asir-3636	58	22	we	we	PRON
asir-3636	58	23	can	can	AUX
asir-3636	58	24	deduce	deduce	VERB
asir-3636	58	25	the	the	DET
asir-3636	58	26	proof	proof	NOUN
asir-3636	58	27	of	of	ADP
asir-3636	58	28	theorem	theorem	NOUN
asir-3636	58	29	(	(	PUNCT
asir-3636	58	30	1.2	1.2	NUM
asir-3636	58	31	)	)	PUNCT
asir-3636	58	32	by	by	ADP
asir-3636	58	33	using	use	VERB
asir-3636	58	34	theorem	theorem	NOUN
asir-3636	58	35	(	(	PUNCT
asir-3636	58	36	2.1	2.1	NUM
asir-3636	58	37	)	)	PUNCT
asir-3636	58	38	and	and	CCONJ
asir-3636	58	39	lemma	lemma	PROPN
asir-3636	58	40	(	(	PUNCT
asir-3636	58	41	2.3	2.3	NUM
asir-3636	58	42	)	)	PUNCT
asir-3636	58	43	indeed	indeed	ADV
asir-3636	58	44	,	,	PUNCT
asir-3636	58	45	let	let	VERB
asir-3636	58	46	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	58	47	2	2	NUM
asir-3636	58	48	be	be	AUX
asir-3636	58	49	a	a	DET
asir-3636	58	50	sequence	sequence	NOUN
asir-3636	58	51	of	of	ADP
asir-3636	58	52	functions	function	NOUN
asir-3636	58	53	in	in	ADP
asir-3636	58	54	𝒜αj	𝒜αj	PROPN
asir-3636	58	55	2\{0	2\{0	NUM
asir-3636	58	56	}	}	PUNCT
asir-3636	58	57	such	such	ADJ
asir-3636	58	58	that	that	SCONJ
asir-3636	58	59	∑	∑	PROPN
asir-3636	58	60	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	58	61	2‖	2‖	PROPN
asir-3636	58	62	𝒜	𝒜	NOUN
asir-3636	58	63	αj	αj	NOUN
asir-3636	58	64	2	2	NUM
asir-3636	58	65	𝑗	𝑗	NOUN
asir-3636	58	66	≤	≤	NUM
asir-3636	58	67	1	1	NUM
asir-3636	58	68	and	and	CCONJ
asir-3636	58	69	let	let	VERB
asir-3636	58	70	𝜖	𝜖	PROPN
asir-3636	58	71	>	>	X
asir-3636	58	72	0	0	NUM
asir-3636	58	73	.	.	PUNCT
asir-3636	59	1	for	for	ADP
asir-3636	59	2	𝜖	𝜖	PROPN
asir-3636	59	3	≥	≥	NUM
asir-3636	59	4	0	0	NUM
asir-3636	59	5	we	we	PRON
asir-3636	59	6	have	have	VERB
asir-3636	59	7	∑(𝑓𝑗	∑(𝑓𝑗	PROPN
asir-3636	59	8	2𝑂	2𝑂	NOUN
asir-3636	59	9	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	59	10	2	2	NUM
asir-3636	59	11	1	1	NUM
asir-3636	59	12	1+𝜖	1+𝜖	NUM
asir-3636	59	13	−	−	NOUN
asir-3636	59	14	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	59	15	2	2	NUM
asir-3636	59	16	)	)	PUNCT
asir-3636	59	17	′	′	NUM
asir-3636	60	1	𝑗	𝑗	NOUN
asir-3636	61	1	=	=	PUNCT
asir-3636	61	2	∑(𝑂	∑(𝑂	ADJ
asir-3636	61	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	61	4	2	2	NUM
asir-3636	61	5	1	1	NUM
asir-3636	61	6	1+𝜖	1+𝜖	NUM
asir-3636	61	7	−	−	NOUN
asir-3636	61	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	61	9	2	2	NUM
asir-3636	61	10	)	)	PUNCT
asir-3636	61	11	(	(	PUNCT
asir-3636	61	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	61	13	2)′	2)′	NUM
asir-3636	61	14	𝑗	𝑗	NOUN
asir-3636	61	15	+	+	NOUN
asir-3636	61	16	∑	∑	PROPN
asir-3636	61	17	1	1	NUM
asir-3636	61	18	1+𝜖	1+𝜖	NUM
asir-3636	61	19	𝑈𝑓𝑗	𝑈𝑓𝑗	NOUN
asir-3636	61	20	2𝑂	2𝑂	NOUN
asir-3636	61	21	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	61	22	2	2	NUM
asir-3636	61	23	1	1	NUM
asir-3636	61	24	1+𝜖𝑂	1+𝜖𝑂	NUM
asir-3636	61	25	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	61	26	2	2	NUM
asir-3636	61	27	′	′	NUM
asir-3636	61	28	𝑗	𝑗	INTJ
asir-3636	61	29	.	.	PUNCT
asir-3636	62	1	the	the	DET
asir-3636	62	2	f	f	NOUN
asir-3636	62	3	-	-	PUNCT
asir-3636	62	4	property	property	NOUN
asir-3636	62	5	of	of	ADP
asir-3636	62	6	𝒜αj	𝒜αj	PROPN
asir-3636	62	7	2	2	NUM
asir-3636	62	8	implies	imply	VERB
asir-3636	62	9	that	that	SCONJ
asir-3636	62	10	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	62	11	2	2	NUM
asir-3636	62	12	∈	∈	NOUN
asir-3636	62	13	𝒜αj	𝒜αj	PROPN
asir-3636	62	14	2	2	X
asir-3636	62	15	.	.	PUNCT
asir-3636	63	1	then	then	ADV
asir-3636	63	2	,	,	PUNCT
asir-3636	63	3	there	there	PRON
asir-3636	63	4	exists	exist	VERB
asir-3636	63	5	𝜂0	𝜂0	PROPN
asir-3636	63	6	∈	∈	PROPN
asir-3636	63	7	ℕ	ℕ	PROPN
asir-3636	63	8	such	such	ADJ
asir-3636	63	9	that	that	DET
asir-3636	63	10	∑‖𝑓𝑗	∑‖𝑓𝑗	NOUN
asir-3636	63	11	2𝑂	2𝑂	NOUN
asir-3636	63	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	63	13	2	2	NUM
asir-3636	63	14	1	1	NUM
asir-3636	63	15	1+𝜖	1+𝜖	NUM
asir-3636	63	16	−	−	NOUN
asir-3636	63	17	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	63	18	2‖	2‖	PROPN
asir-3636	63	19	𝒜	𝒜	NOUN
asir-3636	63	20	αj	αj	NOUN
asir-3636	63	21	2	2	NUM
asir-3636	63	22	𝑗	𝑗	NOUN
asir-3636	63	23	<	<	X
asir-3636	63	24	𝜖	𝜖	X
asir-3636	63	25	3	3	NUM
asir-3636	63	26	(	(	PUNCT
asir-3636	63	27	𝜖	𝜖	X
asir-3636	63	28	≥	≥	NOUN
asir-3636	63	29	0	0	NUM
asir-3636	63	30	)	)	PUNCT
asir-3636	63	31	.	.	PUNCT
asir-3636	64	1	set	set	VERB
asir-3636	64	2	𝛤𝑛	𝛤𝑛	PROPN
asir-3636	64	3	=	=	PUNCT
asir-3636	64	4	∪1+𝜖≥𝑛	∪1+𝜖≥𝑛	PROPN
asir-3636	64	5	𝛾1+𝜖	𝛾1+𝜖	NOUN
asir-3636	64	6	and	and	CCONJ
asir-3636	64	7	αj	αj	ADP
asir-3636	64	8	2	2	NUM
asir-3636	64	9	≤	≤	NUM
asir-3636	64	10	1	1	NUM
asir-3636	64	11	for	for	ADP
asir-3636	64	12	a	a	DET
asir-3636	64	13	given	give	VERB
asir-3636	64	14	𝜖	𝜖	PROPN
asir-3636	64	15	≥	≥	NUM
asir-3636	64	16	0	0	NUM
asir-3636	64	17	.	.	PUNCT
asir-3636	65	1	by	by	ADP
asir-3636	65	2	remark	remark	NOUN
asir-3636	65	3	(	(	PUNCT
asir-3636	65	4	2.2	2.2	NUM
asir-3636	65	5	)	)	PUNCT
asir-3636	65	6	applied	apply	VERB
asir-3636	65	7	to	to	ADP
asir-3636	65	8	𝑂𝑓𝑗	𝑂𝑓𝑗	PROPN
asir-3636	65	9	2	2	NUM
asir-3636	65	10	(	(	PUNCT
asir-3636	65	11	with	with	ADP
asir-3636	65	12	𝜖	𝜖	PROPN
asir-3636	65	13	=	=	NOUN
asir-3636	65	14	>	>	X
asir-3636	65	15	0	0	NUM
asir-3636	65	16	)	)	PUNCT
asir-3636	65	17	,	,	PUNCT
asir-3636	65	18	there	there	PRON
asir-3636	65	19	is	be	VERB
asir-3636	65	20	a	a	DET
asir-3636	65	21	sequence	sequence	NOUN
asir-3636	65	22	𝑘𝑛,1+𝜖	𝑘𝑛,1+𝜖	PROPN
asir-3636	65	23	∈	∈	PROPN
asir-3636	66	1	𝑐𝑜	𝑐𝑜	INTJ
asir-3636	66	2	(	(	PUNCT
asir-3636	66	3	{	{	PUNCT
asir-3636	66	4	(	(	PUNCT
asir-3636	66	5	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	66	6	)	)	PUNCT
asir-3636	66	7	𝛤1+𝜖	𝛤1+𝜖	NOUN
asir-3636	66	8	1+𝜖	1+𝜖	NUM
asir-3636	66	9	}	}	PUNCT
asir-3636	66	10	𝜖=0	𝜖=0	NUM
asir-3636	66	11	∞	∞	NUM
asir-3636	66	12	)	)	PUNCT
asir-3636	66	13	such	such	ADJ
asir-3636	66	14	that	that	SCONJ
asir-3636	66	15	∑‖𝑂	∑‖𝑂	PROPN
asir-3636	66	16	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	66	17	2	2	NUM
asir-3636	66	18	2+𝜖	2+𝜖	NUM
asir-3636	66	19	1+𝜖	1+𝜖	NUM
asir-3636	66	20	𝑘𝑛,1+𝜖	𝑘𝑛,1+𝜖	NOUN
asir-3636	66	21	−	−	NOUN
asir-3636	66	22	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	66	23	2	2	NUM
asir-3636	66	24	2+𝜖	2+𝜖	NUM
asir-3636	66	25	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	66	26	𝒜	𝒜	PROPN
asir-3636	66	27	α2𝑗	α2𝑗	ADP
asir-3636	66	28	<	<	X
asir-3636	66	29	1	1	NUM
asir-3636	66	30	1	1	NUM
asir-3636	66	31	+	+	CCONJ
asir-3636	66	32	𝜖	𝜖	X
asir-3636	66	33	(	(	PUNCT
asir-3636	66	34	𝑛	𝑛	PROPN
asir-3636	66	35	∈	∈	PROPN
asir-3636	66	36	ℕ	ℕ	PROPN
asir-3636	66	37	,	,	PUNCT
asir-3636	66	38	𝜖	𝜖	X
asir-3636	66	39	≥	≥	NOUN
asir-3636	66	40	0	0	NUM
asir-3636	66	41	)	)	PUNCT
asir-3636	66	42	.	.	PUNCT
asir-3636	67	1	it	it	PRON
asir-3636	67	2	is	be	AUX
asir-3636	67	3	clear	clear	ADJ
asir-3636	67	4	that	that	SCONJ
asir-3636	67	5	∑‖𝑂	∑‖𝑂	PROPN
asir-3636	67	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	67	7	2	2	NUM
asir-3636	67	8	1	1	NUM
asir-3636	67	9	1+𝜖(𝑓𝑗	1+𝜖(𝑓𝑗	NUM
asir-3636	67	10	)	)	PUNCT
asir-3636	68	1	𝛤𝑛	𝛤𝑛	NOUN
asir-3636	68	2	2(1+𝜖	2(1+𝜖	NUM
asir-3636	68	3	)	)	PUNCT
asir-3636	68	4	−	−	NOUN
asir-3636	69	1	𝑂	𝑂	NOUN
asir-3636	69	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	69	3	2	2	NUM
asir-3636	69	4	1	1	NUM
asir-3636	69	5	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	69	6	∞𝑗	∞𝑗	NUM
asir-3636	69	7	⟶	⟶	NOUN
asir-3636	69	8	0	0	NUM
asir-3636	69	9	(	(	PUNCT
asir-3636	69	10	𝑛	𝑛	PROPN
asir-3636	69	11	⟶	⟶	NOUN
asir-3636	69	12	+	+	PROPN
asir-3636	69	13	∞	∞	NUM
asir-3636	69	14	)	)	PUNCT
asir-3636	69	15	.	.	PUNCT
asir-3636	70	1	then	then	ADV
asir-3636	70	2	for	for	ADP
asir-3636	70	3	every	every	DET
asir-3636	70	4	𝜖	𝜖	X
asir-3636	70	5	≥	≥	NUM
asir-3636	70	6	0	0	NUM
asir-3636	70	7	we	we	PRON
asir-3636	70	8	get	get	VERB
asir-3636	70	9	∑‖𝑂	∑‖𝑂	ADJ
asir-3636	70	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	70	11	2	2	NUM
asir-3636	70	12	1	1	NUM
asir-3636	70	13	1+𝜖	1+𝜖	NUM
asir-3636	70	14	𝑘𝑛,1+𝜖	𝑘𝑛,1+𝜖	NOUN
asir-3636	70	15	−	−	NOUN
asir-3636	70	16	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	70	17	2	2	NUM
asir-3636	70	18	1	1	NUM
asir-3636	70	19	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	70	20	∞𝑗	∞𝑗	NUM
asir-3636	70	21	⟶	⟶	NOUN
asir-3636	70	22	0	0	NUM
asir-3636	70	23	(	(	PUNCT
asir-3636	70	24	𝑛	𝑛	PROPN
asir-3636	70	25	⟶	⟶	NOUN
asir-3636	70	26	+	+	PROPN
asir-3636	70	27	∞	∞	NOUN
asir-3636	70	28	)	)	PUNCT
asir-3636	70	29	.	.	PUNCT
asir-3636	71	1	so	so	ADV
asir-3636	71	2	,	,	PUNCT
asir-3636	71	3	there	there	PRON
asir-3636	71	4	is	be	VERB
asir-3636	71	5	a	a	DET
asir-3636	71	6	sequence	sequence	NOUN
asir-3636	71	7	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	71	8	∈	∈	PROPN
asir-3636	71	9	𝑐𝑜	𝑐𝑜	INTJ
asir-3636	71	10	(	(	PUNCT
asir-3636	71	11	{	{	PUNCT
asir-3636	71	12	(	(	PUNCT
asir-3636	71	13	𝑓𝑗)𝛤1+𝜖	𝑓𝑗)𝛤1+𝜖	X
asir-3636	71	14	2(1+𝜖	2(1+𝜖	NUM
asir-3636	71	15	)	)	PUNCT
asir-3636	71	16	}	}	PUNCT
asir-3636	71	17	0	0	NUM
asir-3636	71	18	∞	∞	NUM
asir-3636	71	19	)	)	PUNCT
asir-3636	71	20	such	such	ADJ
asir-3636	71	21	that	that	SCONJ
asir-3636	71	22	{	{	PUNCT
asir-3636	71	23	∑‖𝑂	∑‖𝑂	PROPN
asir-3636	71	24	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	71	25	2	2	NUM
asir-3636	71	26	2+𝜖	2+𝜖	NUM
asir-3636	71	27	1+𝜖	1+𝜖	NUM
asir-3636	71	28	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	71	29	−	−	PROPN
asir-3636	71	30	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	71	31	2	2	NUM
asir-3636	71	32	2+𝜖	2+𝜖	NUM
asir-3636	71	33	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	71	34	𝒜	𝒜	NOUN
asir-3636	71	35	αj	αj	NOUN
asir-3636	71	36	2	2	NUM
asir-3636	71	37	𝑗	𝑗	NOUN
asir-3636	71	38	≤	≤	NUM
asir-3636	71	39	1	1	NUM
asir-3636	71	40	1	1	NUM
asir-3636	71	41	+	+	CCONJ
asir-3636	71	42	𝜖	𝜖	X
asir-3636	71	43	(	(	PUNCT
asir-3636	71	44	𝜖	𝜖	X
asir-3636	71	45	≥	≥	NUM
asir-3636	71	46	0	0	NUM
asir-3636	71	47	)	)	PUNCT
asir-3636	71	48	,	,	PUNCT
asir-3636	71	49	∑‖𝑂	∑‖𝑂	PROPN
asir-3636	71	50	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	71	51	2	2	NUM
asir-3636	71	52	1	1	NUM
asir-3636	71	53	1+𝜖	1+𝜖	NUM
asir-3636	71	54	𝑘1+𝜖	𝑘1+𝜖	ADP
asir-3636	71	55	−	−	PROPN
asir-3636	71	56	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	71	57	2	2	NUM
asir-3636	71	58	1	1	NUM
asir-3636	71	59	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	71	60	∞𝑗	∞𝑗	NUM
asir-3636	72	1	≤	≤	NUM
asir-3636	72	2	1	1	NUM
asir-3636	72	3	1	1	NUM
asir-3636	72	4	+	+	CCONJ
asir-3636	72	5	𝜖	𝜖	X
asir-3636	72	6	(	(	PUNCT
asir-3636	72	7	𝜖	𝜖	X
asir-3636	72	8	≥	≥	NOUN
asir-3636	72	9	0	0	NUM
asir-3636	72	10	)	)	PUNCT
asir-3636	72	11	.	.	PUNCT
asir-3636	73	1	we	we	PRON
asir-3636	73	2	have	have	VERB
asir-3636	73	3	∑	∑	ADV
asir-3636	73	4	(	(	PUNCT
asir-3636	73	5	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	73	6	2𝑂	2𝑂	NOUN
asir-3636	73	7	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	73	8	2	2	NUM
asir-3636	73	9	1	1	NUM
asir-3636	73	10	1+𝜖	1+𝜖	NUM
asir-3636	73	11	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	73	12	−	−	NOUN
asir-3636	73	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	73	14	2𝑂	2𝑂	NOUN
asir-3636	73	15	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	73	16	2	2	NUM
asir-3636	73	17	1	1	NUM
asir-3636	73	18	1+𝜖)𝑗	1+𝜖)𝑗	NUM
asir-3636	73	19	′	′	NUM
asir-3636	74	1	=	=	PUNCT
asir-3636	74	2	∑	∑	PUNCT
asir-3636	74	3	(	(	PUNCT
asir-3636	74	4	(	(	PUNCT
asir-3636	74	5	𝑓𝑗	𝑓𝑗	NOUN
asir-3636	75	1	2)′	2)′	NUM
asir-3636	75	2	−	−	NOUN
asir-3636	76	1	𝑈𝑓𝑗	𝑈𝑓𝑗	NOUN
asir-3636	76	2	2𝑂	2𝑂	NOUN
asir-3636	76	3	𝑓𝑗	𝑓𝑗	VERB
asir-3636	76	4	2	2	NUM
asir-3636	76	5	′	′	NUM
asir-3636	76	6	)	)	PUNCT
asir-3636	77	1	(	(	PUNCT
asir-3636	77	2	𝑂	𝑂	PROPN
asir-3636	77	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	77	4	2	2	NUM
asir-3636	77	5	1	1	NUM
asir-3636	77	6	1+𝜖	1+𝜖	NUM
asir-3636	77	7	𝑘1+𝜖	𝑘1+𝜖	ADP
asir-3636	77	8	−	−	PROPN
asir-3636	77	9	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	77	10	2	2	NUM
asir-3636	77	11	1	1	NUM
asir-3636	77	12	1+𝜖)𝑗	1+𝜖)𝑗	NUM
asir-3636	78	1	+	+	CCONJ
asir-3636	78	2	∑	∑	PUNCT
asir-3636	78	3	(	(	PUNCT
asir-3636	78	4	𝑈𝑓𝑗	𝑈𝑓𝑗	NOUN
asir-3636	78	5	2𝑂	2𝑂	NOUN
asir-3636	78	6	𝑓𝑗	𝑓𝑗	VERB
asir-3636	78	7	2	2	NUM
asir-3636	78	8	2+𝜖	2+𝜖	NUM
asir-3636	78	9	1+𝜖	1+𝜖	NUM
asir-3636	78	10	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	78	11	−𝑗	−𝑗	PRON
asir-3636	78	12	𝑂	𝑂	PROPN
asir-3636	78	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	78	14	2	2	NUM
asir-3636	78	15	2+𝜖	2+𝜖	NUM
asir-3636	78	16	1+𝜖	1+𝜖	NUM
asir-3636	78	17	)	)	PUNCT
asir-3636	78	18	′	′	NOUN
asir-3636	79	1	since	since	SCONJ
asir-3636	79	2	∑	∑	ADP
asir-3636	79	3	‖𝑂𝑓𝑗	‖𝑂𝑓𝑗	PROPN
asir-3636	79	4	2‖	2‖	PROPN
asir-3636	79	5	𝒜	𝒜	NOUN
asir-3636	79	6	αj	αj	NOUN
asir-3636	79	7	2	2	NUM
asir-3636	79	8	𝑗	𝑗	NOUN
asir-3636	79	9	≤	≤	NOUN
asir-3636	79	10	∑	∑	ADP
asir-3636	79	11	𝐶αj	𝐶αj	ADJ
asir-3636	79	12	2‖𝑓𝑗	2‖𝑓𝑗	NOUN
asir-3636	79	13	2‖	2‖	PROPN
asir-3636	79	14	αj	αj	PART
asir-3636	79	15	2𝑗	2𝑗	NOUN
asir-3636	79	16	≤	≤	NOUN
asir-3636	79	17	∑	∑	ADP
asir-3636	79	18	𝐶αj	𝐶αj	ADJ
asir-3636	79	19	2𝑗	2𝑗	NOUN
asir-3636	79	20	,	,	PUNCT
asir-3636	79	21	we	we	PRON
asir-3636	79	22	obtain	obtain	VERB
asir-3636	79	23	∑	∑	PUNCT
asir-3636	80	1	‖𝑓𝑗	‖𝑓𝑗	NUM
asir-3636	80	2	2𝑂	2𝑂	NOUN
asir-3636	80	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	80	4	2	2	NUM
asir-3636	80	5	1	1	NUM
asir-3636	80	6	1+𝜖	1+𝜖	NUM
asir-3636	80	7	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	80	8	−	−	NOUN
asir-3636	80	9	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	80	10	2𝑂	2𝑂	NOUN
asir-3636	80	11	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	80	12	2	2	NUM
asir-3636	80	13	1	1	NUM
asir-3636	80	14	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	80	15	𝒜	𝒜	NOUN
asir-3636	80	16	αj	αj	NOUN
asir-3636	80	17	2	2	NUM
asir-3636	80	18	∑	∑	PUNCT
asir-3636	80	19	‖𝑓𝑗	‖𝑓𝑗	NUM
asir-3636	81	1	2𝑂	2𝑂	NOUN
asir-3636	81	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	81	3	2	2	NUM
asir-3636	81	4	1	1	NUM
asir-3636	81	5	1+𝜖	1+𝜖	NUM
asir-3636	81	6	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	81	7	−	−	NOUN
asir-3636	81	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	81	9	2𝑂	2𝑂	NOUN
asir-3636	81	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	81	11	2	2	NUM
asir-3636	81	12	1	1	NUM
asir-3636	81	13	1+𝜖‖	1+𝜖‖	NUM
asir-3636	81	14	∞	∞	NUM
asir-3636	81	15	𝑗𝑗	𝑗𝑗	PROPN
asir-3636	82	1	+	+	CCONJ
asir-3636	82	2	𝑠𝑢𝑝𝑧∈𝔻	𝑠𝑢𝑝𝑧∈𝔻	X
asir-3636	82	3	{	{	PUNCT
asir-3636	82	4	∑	∑	PROPN
asir-3636	82	5	(	(	PUNCT
asir-3636	82	6	1	1	NUM
asir-3636	82	7	−	−	PRON
asir-3636	82	8	|𝑧|)1−αj	|𝑧|)1−αj	NOUN
asir-3636	82	9	2	2	NUM
asir-3636	82	10	|(𝑓𝑗	|(𝑓𝑗	NUM
asir-3636	82	11	2𝑂	2𝑂	NOUN
asir-3636	82	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	13	2	2	NUM
asir-3636	82	14	1	1	NUM
asir-3636	82	15	1+𝜖	1+𝜖	NUM
asir-3636	82	16	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	82	17	−	−	NOUN
asir-3636	82	18	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	19	2𝑂	2𝑂	NOUN
asir-3636	82	20	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	21	2	2	NUM
asir-3636	82	22	1	1	NUM
asir-3636	82	23	1+𝜖	1+𝜖	NUM
asir-3636	82	24	)	)	PUNCT
asir-3636	82	25	′	′	NUM
asir-3636	82	26	(	(	PUNCT
asir-3636	82	27	𝑧)|𝑗	𝑧)|𝑗	X
asir-3636	82	28	}	}	PUNCT
asir-3636	82	29	+	+	CCONJ
asir-3636	82	30	∑	∑	PROPN
asir-3636	82	31	𝐷	𝐷	PROPN
asir-3636	82	32	1	1	NUM
asir-3636	82	33	2	2	NUM
asir-3636	82	34	(	(	PUNCT
asir-3636	82	35	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	36	2𝑂	2𝑂	NOUN
asir-3636	82	37	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	38	2	2	NUM
asir-3636	82	39	1	1	NUM
asir-3636	82	40	1+𝜖	1+𝜖	NUM
asir-3636	82	41	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	82	42	−	−	NOUN
asir-3636	82	43	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	44	2𝑂	2𝑂	NOUN
asir-3636	82	45	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	82	46	2	2	NUM
asir-3636	82	47	1	1	NUM
asir-3636	82	48	1+𝜖)𝑗	1+𝜖)𝑗	NUM
asir-3636	82	49	≤	≤	NOUN
asir-3636	82	50	∑	∑	PUNCT
asir-3636	83	1	‖𝑓𝑗	‖𝑓𝑗	NUM
asir-3636	83	2	2𝑂	2𝑂	NOUN
asir-3636	83	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	83	4	2	2	NUM
asir-3636	83	5	1	1	NUM
asir-3636	83	6	1+𝜖	1+𝜖	NUM
asir-3636	83	7	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	83	8	−	−	NOUN
asir-3636	83	9	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	83	10	2𝑂	2𝑂	NOUN
asir-3636	83	11	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	83	12	2	2	NUM
asir-3636	83	13	1	1	NUM
asir-3636	83	14	1+𝜖‖𝑗	1+𝜖‖𝑗	NUM
asir-3636	83	15	∞	∞	NUM
asir-3636	83	16	+	+	CCONJ
asir-3636	83	17	∑	∑	PROPN
asir-3636	83	18	𝐶αj	𝐶αj	ADJ
asir-3636	83	19	2‖𝑓𝑗	2‖𝑓𝑗	NOUN
asir-3636	83	20	2‖	2‖	PROPN
asir-3636	83	21	αj	αj	ADP
asir-3636	83	22	2	2	NUM
asir-3636	83	23	‖𝑂𝑓𝑗	‖𝑂𝑓𝑗	ADP
asir-3636	83	24	2	2	NUM
asir-3636	83	25	1	1	NUM
asir-3636	83	26	1+𝜖	1+𝜖	NUM
asir-3636	83	27	𝑘1+𝜖	𝑘1+𝜖	ADP
asir-3636	83	28	−	−	PROPN
asir-3636	83	29	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	83	30	2	2	NUM
asir-3636	83	31	1	1	NUM
asir-3636	83	32	1+𝜖‖	1+𝜖‖	NUM
asir-3636	83	33	∞	∞	NUM
asir-3636	83	34	𝑗	𝑗	PRON
asir-3636	83	35	+	+	NUM
asir-3636	83	36	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	83	37	applied	apply	VERB
asir-3636	83	38	science	science	NOUN
asir-3636	83	39	and	and	CCONJ
asir-3636	83	40	innovative	innovative	ADJ
asir-3636	83	41	research	research	NOUN
asir-3636	83	42	vol	vol	NOUN
asir-3636	83	43	.	.	PROPN
asir-3636	84	1	5	5	NUM
asir-3636	84	2	,	,	PUNCT
asir-3636	84	3	no	no	INTJ
asir-3636	84	4	.	.	NOUN
asir-3636	84	5	1	1	NUM
asir-3636	84	6	,	,	PUNCT
asir-3636	84	7	2021	2021	NUM
asir-3636	84	8	25	25	NUM
asir-3636	84	9	published	publish	VERB
asir-3636	84	10	by	by	ADP
asir-3636	84	11	scholink	scholink	PROPN
asir-3636	84	12	inc	inc	PROPN
asir-3636	84	13	.	.	PROPN
asir-3636	84	14	𝑠𝑢𝑝𝑧∈𝔻	𝑠𝑢𝑝𝑧∈𝔻	PROPN
asir-3636	84	15	{	{	PUNCT
asir-3636	84	16	∑	∑	PROPN
asir-3636	84	17	(	(	PUNCT
asir-3636	84	18	1	1	NUM
asir-3636	84	19	−	−	PRON
asir-3636	84	20	|𝑧|)1−αj	|𝑧|)1−αj	NOUN
asir-3636	84	21	2	2	NUM
asir-3636	85	1	|(𝑂	|(𝑂	ADJ
asir-3636	85	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	85	3	2	2	NUM
asir-3636	85	4	2+𝜖	2+𝜖	NUM
asir-3636	85	5	1+𝜖	1+𝜖	NUM
asir-3636	85	6	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	85	7	−	−	PROPN
asir-3636	85	8	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	85	9	2	2	NUM
asir-3636	85	10	2+𝜖	2+𝜖	NUM
asir-3636	85	11	1+𝜖	1+𝜖	NUM
asir-3636	85	12	)	)	PUNCT
asir-3636	85	13	′	′	NUM
asir-3636	85	14	(	(	PUNCT
asir-3636	85	15	𝑧)|𝑗	𝑧)|𝑗	X
asir-3636	85	16	}	}	PUNCT
asir-3636	86	1	+	+	CCONJ
asir-3636	86	2	𝐶	𝐶	PROPN
asir-3636	86	3	∑	∑	PUNCT
asir-3636	86	4	‖𝑂	‖𝑂	PROPN
asir-3636	86	5	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	86	6	2	2	NUM
asir-3636	86	7	1	1	NUM
asir-3636	86	8	1+𝜖	1+𝜖	NUM
asir-3636	86	9	𝑘1+𝜖	𝑘1+𝜖	ADP
asir-3636	86	10	−	−	PROPN
asir-3636	86	11	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	86	12	2	2	NUM
asir-3636	86	13	1	1	NUM
asir-3636	86	14	1+𝜖‖	1+𝜖‖	NUM
asir-3636	86	15	∞	∞	NUM
asir-3636	86	16	𝑗	𝑗	PROPN
asir-3636	86	17	+	+	CCONJ
asir-3636	86	18	∑	∑	PROPN
asir-3636	86	19	𝐷	𝐷	PROPN
asir-3636	86	20	1	1	NUM
asir-3636	86	21	2(𝑓𝑗	2(𝑓𝑗	NUM
asir-3636	86	22	2)𝑗	2)𝑗	NOUN
asir-3636	86	23	+	+	CCONJ
asir-3636	86	24	𝐶𝐷	𝐶𝐷	PROPN
asir-3636	86	25	1	1	NUM
asir-3636	86	26	2∑	2∑	NOUN
asir-3636	86	27	(	(	PUNCT
asir-3636	86	28	𝑂	𝑂	NOUN
asir-3636	86	29	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	86	30	2	2	NUM
asir-3636	86	31	2+𝜖	2+𝜖	NUM
asir-3636	86	32	1+𝜖	1+𝜖	NUM
asir-3636	86	33	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	86	34	−	−	PROPN
asir-3636	86	35	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	86	36	2	2	NUM
asir-3636	86	37	2+𝜖	2+𝜖	NUM
asir-3636	86	38	1+𝜖)𝑗	1+𝜖)𝑗	NUM
asir-3636	86	39	≤	≤	NOUN
asir-3636	86	40	∑	∑	PUNCT
asir-3636	86	41	𝐶αj	𝐶αj	PROPN
asir-3636	86	42	2	2	NUM
asir-3636	86	43	‖𝑂	‖𝑂	NOUN
asir-3636	86	44	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	86	45	2	2	NUM
asir-3636	86	46	1	1	NUM
asir-3636	86	47	1+𝜖	1+𝜖	NUM
asir-3636	86	48	𝑘1+𝜖	𝑘1+𝜖	ADP
asir-3636	86	49	−	−	PROPN
asir-3636	86	50	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	86	51	2	2	NUM
asir-3636	86	52	1	1	NUM
asir-3636	86	53	1+𝜖‖	1+𝜖‖	NUM
asir-3636	86	54	∞	∞	NUM
asir-3636	86	55	𝑗	𝑗	PROPN
asir-3636	86	56	+	+	CCONJ
asir-3636	86	57	𝐶	𝐶	PROPN
asir-3636	86	58	∑	∑	PUNCT
asir-3636	86	59	‖𝑂	‖𝑂	PROPN
asir-3636	86	60	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	86	61	2	2	NUM
asir-3636	86	62	2+𝜖	2+𝜖	NUM
asir-3636	86	63	1+𝜖	1+𝜖	NUM
asir-3636	86	64	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	86	65	−	−	PROPN
asir-3636	86	66	𝑂𝑓𝑗	𝑂𝑓𝑗	ADV
asir-3636	86	67	2	2	NUM
asir-3636	86	68	2+𝜖	2+𝜖	NUM
asir-3636	86	69	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	86	70	𝒜	𝒜	NOUN
asir-3636	86	71	αj	αj	NOUN
asir-3636	86	72	2	2	NUM
asir-3636	86	73	𝑗	𝑗	NOUN
asir-3636	86	74	≤	≤	X
asir-3636	86	75	∑	∑	PUNCT
asir-3636	86	76	𝐶	𝐶	PROPN
asir-3636	86	77	αj	αj	ADP
asir-3636	86	78	2	2	NUM
asir-3636	86	79	1+𝜖𝑗	1+𝜖𝑗	NUM
asir-3636	86	80	then	then	ADV
asir-3636	86	81	,	,	PUNCT
asir-3636	86	82	fix	fix	VERB
asir-3636	86	83	𝜖	𝜖	X
asir-3636	86	84	≥	≥	NUM
asir-3636	86	85	0	0	NUM
asir-3636	86	86	such	such	ADJ
asir-3636	86	87	that	that	SCONJ
asir-3636	86	88	∑‖𝑓𝑗	∑‖𝑓𝑗	NOUN
asir-3636	87	1	2𝑂	2𝑂	NOUN
asir-3636	87	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	87	3	2	2	NUM
asir-3636	87	4	1	1	NUM
asir-3636	87	5	1+𝜖	1+𝜖	NUM
asir-3636	87	6	𝑘1+𝜖	𝑘1+𝜖	INTJ
asir-3636	87	7	−	−	NOUN
asir-3636	87	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	87	9	2𝑂	2𝑂	NOUN
asir-3636	87	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	87	11	2	2	NUM
asir-3636	87	12	1	1	NUM
asir-3636	87	13	1+𝜖‖	1+𝜖‖	PROPN
asir-3636	87	14	𝒜	𝒜	NOUN
asir-3636	87	15	αj	αj	NOUN
asir-3636	87	16	2𝑗	2𝑗	NOUN
asir-3636	87	17	<	<	X
asir-3636	87	18	𝜖	𝜖	PROPN
asir-3636	87	19	3⁄	3⁄	NUM
asir-3636	87	20	(	(	PUNCT
asir-3636	87	21	𝜖	𝜖	X
asir-3636	87	22	≥	≥	NOUN
asir-3636	87	23	0	0	NUM
asir-3636	87	24	)	)	PUNCT
asir-3636	87	25	.	.	PUNCT
asir-3636	88	1	we	we	PRON
asir-3636	88	2	have	have	VERB
asir-3636	88	3	𝑘1+𝜖	𝑘1+𝜖	NOUN
asir-3636	88	4	=	=	SYM
asir-3636	88	5	∑	∑	PUNCT
asir-3636	88	6	∑	∑	PUNCT
asir-3636	88	7	𝑐𝑖𝑓γ𝑖	𝑐𝑖𝑓γ𝑖	NOUN
asir-3636	88	8	2(1+𝜖	2(1+𝜖	NUM
asir-3636	88	9	)	)	PUNCT
asir-3636	88	10	𝑗𝑖≤𝑗1+𝜖	𝑗𝑖≤𝑗1+𝜖	NOUN
asir-3636	88	11	,	,	PUNCT
asir-3636	88	12	where	where	SCONJ
asir-3636	88	13	∑	∑	PUNCT
asir-3636	88	14	𝑐𝑖	𝑐𝑖	NOUN
asir-3636	88	15	=	=	SYM
asir-3636	88	16	1.𝑖≤𝑗1+𝜖	1.𝑖≤𝑗1+𝜖	NUM
asir-3636	88	17	set	set	VERB
asir-3636	88	18	𝐸1+𝜖	𝐸1+𝜖	ADJ
asir-3636	89	1	′	′	NOUN
asir-3636	89	2	=	=	PUNCT
asir-3636	89	3	∪𝑖≤𝑗1+𝜖	∪𝑖≤𝑗1+𝜖	NOUN
asir-3636	89	4	𝜕𝛾𝑖	𝜕𝛾𝑖	PUNCT
asir-3636	89	5	.	.	PUNCT
asir-3636	90	1	using	use	VERB
asir-3636	90	2	lemma	lemma	PROPN
asir-3636	90	3	(	(	PUNCT
asir-3636	90	4	2.3	2.3	NUM
asir-3636	90	5	)	)	PUNCT
asir-3636	90	6	,	,	PUNCT
asir-3636	90	7	we	we	PRON
asir-3636	90	8	obtain	obtain	VERB
asir-3636	90	9	an	an	DET
asir-3636	90	10	outer	outer	ADJ
asir-3636	90	11	function	function	NOUN
asir-3636	90	12	𝐹1+𝜖	𝐹1+𝜖	X
asir-3636	90	13	∈	∈	PROPN
asir-3636	91	1	𝒥(𝐸1+𝜖	𝒥(𝐸1+𝜖	X
asir-3636	91	2	′	′	NUM
asir-3636	91	3	)	)	PUNCT
asir-3636	92	1	such	such	ADJ
asir-3636	92	2	that	that	SCONJ
asir-3636	92	3	|𝐹1+𝜖(𝜁)|	|𝐹1+𝜖(𝜁)|	DET
asir-3636	92	4	≤	≤	NOUN
asir-3636	92	5	𝐶1+𝜖𝑑	𝐶1+𝜖𝑑	ADJ
asir-3636	92	6	1+𝜖(𝜁	1+𝜖(𝜁	ADJ
asir-3636	92	7	,	,	PUNCT
asir-3636	92	8	𝐸1+𝜖	𝐸1+𝜖	ADJ
asir-3636	92	9	′	′	NUM
asir-3636	92	10	)	)	PUNCT
asir-3636	92	11	for	for	ADP
asir-3636	92	12	𝜁	𝜁	PROPN
asir-3636	92	13	∈	∈	PROPN
asir-3636	92	14	𝑇	𝑇	PROPN
asir-3636	92	15	and	and	CCONJ
asir-3636	92	16	∑‖𝑓𝑗	∑‖𝑓𝑗	VERB
asir-3636	92	17	2𝑂	2𝑂	NOUN
asir-3636	92	18	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	92	19	2	2	NUM
asir-3636	92	20	1	1	NUM
asir-3636	92	21	1+𝜖	1+𝜖	NUM
asir-3636	92	22	𝑘1+𝜖𝐹1+𝜖	𝑘1+𝜖𝐹1+𝜖	NOUN
asir-3636	92	23	−	−	NOUN
asir-3636	92	24	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	92	25	2𝑂	2𝑂	NOUN
asir-3636	92	26	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	92	27	2	2	NUM
asir-3636	92	28	1	1	NUM
asir-3636	92	29	1+𝜖	1+𝜖	NUM
asir-3636	92	30	𝑘1+𝜖‖	𝑘1+𝜖‖	PROPN
asir-3636	92	31	𝒜	𝒜	NOUN
asir-3636	92	32	αj	αj	NOUN
asir-3636	92	33	2𝑗	2𝑗	NOUN
asir-3636	92	34	<	<	X
asir-3636	92	35	1	1	NUM
asir-3636	92	36	1	1	NUM
asir-3636	92	37	+	+	CCONJ
asir-3636	92	38	𝜖	𝜖	PROPN
asir-3636	92	39	,	,	PUNCT
asir-3636	92	40	(	(	PUNCT
asir-3636	92	41	𝜖	𝜖	X
asir-3636	92	42	≥	≥	NUM
asir-3636	92	43	1	1	NUM
asir-3636	92	44	)	)	PUNCT
asir-3636	92	45	.	.	PUNCT
asir-3636	93	1	then	then	ADV
asir-3636	93	2	fix	fix	VERB
asir-3636	93	3	𝜖	𝜖	X
asir-3636	93	4	≥	≥	NOUN
asir-3636	93	5	0	0	NUM
asir-3636	93	6	such	such	ADJ
asir-3636	93	7	that	that	SCONJ
asir-3636	93	8	∑‖𝑓𝑗	∑‖𝑓𝑗	NOUN
asir-3636	93	9	2𝑂	2𝑂	NOUN
asir-3636	93	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	93	11	2	2	NUM
asir-3636	93	12	1	1	NUM
asir-3636	93	13	1+𝜖	1+𝜖	NUM
asir-3636	93	14	𝑘1+𝜖𝐹1+𝜖	𝑘1+𝜖𝐹1+𝜖	NOUN
asir-3636	93	15	−	−	NOUN
asir-3636	93	16	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	93	17	2𝑂	2𝑂	NOUN
asir-3636	93	18	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	93	19	2	2	NUM
asir-3636	93	20	1	1	NUM
asir-3636	93	21	1+𝜖	1+𝜖	NUM
asir-3636	93	22	𝑘1+𝜖‖	𝑘1+𝜖‖	PROPN
asir-3636	93	23	𝒜	𝒜	NOUN
asir-3636	93	24	αj	αj	NOUN
asir-3636	93	25	2𝑗	2𝑗	NOUN
asir-3636	93	26	<	<	X
asir-3636	93	27	𝜖	𝜖	PROPN
asir-3636	93	28	3⁄	3⁄	NUM
asir-3636	93	29	(	(	PUNCT
asir-3636	93	30	𝜖	𝜖	X
asir-3636	93	31	≥	≥	NOUN
asir-3636	93	32	0	0	NUM
asir-3636	93	33	)	)	PUNCT
asir-3636	93	34	.	.	PUNCT
asir-3636	94	1	consequently	consequently	ADV
asir-3636	94	2	we	we	PRON
asir-3636	94	3	obtain	obtain	VERB
asir-3636	94	4	∑‖𝑓𝑗	∑‖𝑓𝑗	NOUN
asir-3636	95	1	2𝑂	2𝑂	NOUN
asir-3636	95	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	95	3	2	2	NUM
asir-3636	95	4	1	1	NUM
asir-3636	95	5	1+𝜖	1+𝜖	NUM
asir-3636	95	6	𝑘1+𝜖𝐹1+𝜖	𝑘1+𝜖𝐹1+𝜖	NOUN
asir-3636	95	7	−	−	NOUN
asir-3636	96	1	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	97	1	2‖	2‖	PROPN
asir-3636	97	2	𝒜	𝒜	NOUN
asir-3636	97	3	αj	αj	NOUN
asir-3636	97	4	2𝑗	2𝑗	NOUN
asir-3636	97	5	<	<	X
asir-3636	97	6	𝜖	𝜖	X
asir-3636	97	7	(	(	PUNCT
asir-3636	97	8	𝜖	𝜖	X
asir-3636	97	9	≥	≥	NOUN
asir-3636	97	10	0	0	NUM
asir-3636	97	11	)	)	PUNCT
asir-3636	97	12	.	.	PUNCT
asir-3636	98	1	it	it	PRON
asir-3636	98	2	is	be	AUX
asir-3636	98	3	not	not	PART
asir-3636	98	4	hard	hard	ADJ
asir-3636	98	5	to	to	PART
asir-3636	98	6	see	see	VERB
asir-3636	98	7	that	that	DET
asir-3636	98	8	∑|𝑂	∑|𝑂	NOUN
asir-3636	98	9	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	98	10	2	2	NUM
asir-3636	98	11	1	1	NUM
asir-3636	98	12	1+𝜖	1+𝜖	NUM
asir-3636	98	13	𝑘1+𝜖𝐹1+𝜖(𝜉)|	𝑘1+𝜖𝐹1+𝜖(𝜉)|	NOUN
asir-3636	98	14	𝑗	𝑗	INTJ
asir-3636	98	15	≤∑𝐶1+𝜖𝑑	≤∑𝐶1+𝜖𝑑	PROPN
asir-3636	98	16	1+𝜖	1+𝜖	NUM
asir-3636	98	17	(	(	PUNCT
asir-3636	98	18	𝜉	𝜉	X
asir-3636	98	19	,	,	PUNCT
asir-3636	98	20	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	98	21	2	2	NUM
asir-3636	98	22	)	)	PUNCT
asir-3636	98	23	𝑗	𝑗	NOUN
asir-3636	98	24	(	(	PUNCT
asir-3636	98	25	𝜉	𝜉	NOUN
asir-3636	98	26	∈	∈	PROPN
asir-3636	98	27	𝕋	𝕋	PROPN
asir-3636	98	28	)	)	PUNCT
asir-3636	98	29	.	.	PUNCT
asir-3636	99	1	therefore	therefore	ADV
asir-3636	99	2	∑	∑	INTJ
asir-3636	99	3	(	(	PUNCT
asir-3636	99	4	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	99	5	2)1+𝜖	2)1+𝜖	NUM
asir-3636	99	6	𝑗	𝑗	NOUN
asir-3636	99	7	=	=	X
asir-3636	99	8	∑	∑	PUNCT
asir-3636	99	9	𝑂	𝑂	PROPN
asir-3636	99	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	99	11	2	2	NUM
asir-3636	99	12	1	1	NUM
asir-3636	99	13	1+𝜖	1+𝜖	NUM
asir-3636	99	14	𝑘1+𝜖𝐹1+𝜖𝑗	𝑘1+𝜖𝐹1+𝜖𝑗	NOUN
asir-3636	99	15	is	be	AUX
asir-3636	99	16	the	the	DET
asir-3636	99	17	desired	desire	VERB
asir-3636	99	18	series	series	NOUN
asir-3636	99	19	of	of	ADP
asir-3636	99	20	sequence	sequence	NOUN
asir-3636	99	21	,	,	PUNCT
asir-3636	99	22	which	which	PRON
asir-3636	99	23	completes	complete	VERB
asir-3636	99	24	the	the	DET
asir-3636	99	25	proof	proof	NOUN
asir-3636	99	26	of	of	ADP
asir-3636	99	27	theorem	theorem	NOUN
asir-3636	99	28	(	(	PUNCT
asir-3636	99	29	1.2	1.2	NUM
asir-3636	99	30	)	)	PUNCT
asir-3636	99	31	.	.	PUNCT
asir-3636	100	1	3	3	X
asir-3636	100	2	.	.	X
asir-3636	100	3	beurling	beurle	VERB
asir-3636	100	4	–	–	PUNCT
asir-3636	100	5	carleman	carleman	NOUN
asir-3636	100	6	–	–	PUNCT
asir-3636	100	7	domar	domar	ADJ
asir-3636	100	8	resolvent	resolvent	NOUN
asir-3636	100	9	methed	methe	VERB
asir-3636	100	10	since	since	SCONJ
asir-3636	100	11	𝒜αj	𝒜αj	PROPN
asir-3636	100	12	2	2	NUM
asir-3636	100	13	⊂	⊂	PROPN
asir-3636	100	14	lipαj	lipαj	NOUN
asir-3636	100	15	2	2	NUM
asir-3636	100	16	,	,	PUNCT
asir-3636	100	17	then	then	ADV
asir-3636	100	18	for	for	ADP
asir-3636	100	19	all	all	PRON
asir-3636	100	20	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	100	21	2	2	NUM
asir-3636	100	22	∈	∈	NOUN
asir-3636	100	23	𝒜αj	𝒜αj	PROPN
asir-3636	100	24	2	2	NUM
asir-3636	100	25	,	,	PUNCT
asir-3636	100	26	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	100	27	2	2	NUM
asir-3636	100	28	satisfies	satisfy	VERB
asir-3636	100	29	the	the	DET
asir-3636	100	30	carleson	carleson	NOUN
asir-3636	100	31	condition	condition	NOUN
asir-3636	100	32	∫∑log	∫∑log	PROPN
asir-3636	100	33	1	1	NUM
asir-3636	100	34	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	100	35	2	2	NUM
asir-3636	100	36	,	,	PUNCT
asir-3636	100	37	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	100	38	2	2	NUM
asir-3636	100	39	)	)	PUNCT
asir-3636	100	40	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	100	41	j	j	NOUN
asir-3636	100	42	<	<	X
asir-3636	100	43	+	+	X
asir-3636	100	44	∞.	∞.	PROPN
asir-3636	100	45	𝕋	𝕋	NOUN
asir-3636	100	46	for	for	ADP
asir-3636	100	47	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	100	48	2	2	NUM
asir-3636	100	49	∈	∈	NOUN
asir-3636	100	50	𝒜αj	𝒜αj	PROPN
asir-3636	100	51	2	2	NUM
asir-3636	100	52	,	,	PUNCT
asir-3636	100	53	we	we	PRON
asir-3636	100	54	denote	denote	VERB
asir-3636	100	55	by	by	ADP
asir-3636	100	56	𝐵𝑓𝑗	𝐵𝑓𝑗	PROPN
asir-3636	100	57	2	2	NUM
asir-3636	100	58	the	the	DET
asir-3636	100	59	blashke	blashke	ADJ
asir-3636	100	60	product	product	NOUN
asir-3636	100	61	with	with	ADP
asir-3636	100	62	zeros	zero	NOUN
asir-3636	101	1	𝑍𝑓𝑗	𝑍𝑓𝑗	PROPN
asir-3636	101	2	2\𝐸𝑓𝑗	2\𝐸𝑓𝑗	NUM
asir-3636	101	3	2	2	NUM
asir-3636	101	4	,	,	PUNCT
asir-3636	101	5	where	where	SCONJ
asir-3636	101	6	𝑍𝑓𝑗	𝑍𝑓𝑗	PROPN
asir-3636	101	7	2	2	NUM
asir-3636	101	8	∶=	∶=	NUM
asir-3636	101	9	{	{	PUNCT
asir-3636	101	10	𝑧	𝑧	PROPN
asir-3636	101	11	∈	∈	PROPN
asir-3636	101	12	�	�	NOUN
asir-3636	101	13	̅	̅	NOUN
asir-3636	101	14	�	�	PROPN
asir-3636	101	15	∶	∶	NOUN
asir-3636	101	16	∑	∑	ADP
asir-3636	101	17	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	101	18	2(𝑧)𝑗	2(𝑧)𝑗	NOUN
asir-3636	101	19	=	=	SYM
asir-3636	101	20	0	0	NUM
asir-3636	101	21	}	}	PUNCT
asir-3636	101	22	.	.	PUNCT
asir-3636	102	1	we	we	PRON
asir-3636	102	2	begin	begin	VERB
asir-3636	102	3	with	with	ADP
asir-3636	102	4	following	follow	VERB
asir-3636	102	5	lemma	lemma	PROPN
asir-3636	102	6	(	(	PUNCT
asir-3636	102	7	see	see	VERB
asir-3636	102	8	brahim	brahim	PROPN
asir-3636	102	9	bouya	bouya	PROPN
asir-3636	102	10	,	,	PUNCT
asir-3636	102	11	2008	2008	NUM
asir-3636	102	12	)	)	PUNCT
asir-3636	102	13	.	.	PUNCT
asir-3636	103	1	lemma	lemma	PROPN
asir-3636	103	2	(	(	PUNCT
asir-3636	103	3	3.1	3.1	NUM
asir-3636	103	4	):	):	PUNCT
asir-3636	103	5	let	let	VERB
asir-3636	103	6	𝔗	𝔗	PRON
asir-3636	103	7	be	be	AUX
asir-3636	103	8	a	a	DET
asir-3636	103	9	closed	closed	ADJ
asir-3636	103	10	ideal	ideal	NOUN
asir-3636	103	11	of	of	ADP
asir-3636	103	12	𝒜αj	𝒜αj	PROPN
asir-3636	103	13	2	2	X
asir-3636	103	14	.	.	PUNCT
asir-3636	103	15	define	define	VERB
asir-3636	103	16	𝐵𝔗	𝐵𝔗	PROPN
asir-3636	103	17	to	to	PART
asir-3636	103	18	be	be	AUX
asir-3636	103	19	the	the	DET
asir-3636	103	20	blashke	blashke	ADJ
asir-3636	103	21	product	product	NOUN
asir-3636	103	22	with	with	ADP
asir-3636	103	23	zeros	zero	NOUN
asir-3636	103	24	𝑍𝔗\𝐸𝔗.	𝑍𝔗\𝐸𝔗.	ADV
asir-3636	103	25	there	there	PRON
asir-3636	103	26	is	be	VERB
asir-3636	103	27	a	a	DET
asir-3636	103	28	sequence	sequence	NOUN
asir-3636	103	29	of	of	ADP
asir-3636	103	30	functions	function	NOUN
asir-3636	103	31	𝑓𝑗	𝑓𝑗	VERB
asir-3636	103	32	2	2	NUM
asir-3636	103	33	∈	∈	NOUN
asir-3636	103	34	𝔗	𝔗	NOUN
asir-3636	103	35	such	such	ADJ
asir-3636	103	36	that	that	SCONJ
asir-3636	103	37	𝐵𝑓𝑗	𝐵𝑓𝑗	PROPN
asir-3636	103	38	2	2	NUM
asir-3636	103	39	=	=	SYM
asir-3636	103	40	𝐵𝔗.	𝐵𝔗.	X
asir-3636	103	41	proof	proof	NOUN
asir-3636	103	42	.	.	PUNCT
asir-3636	104	1	let	let	VERB
asir-3636	104	2	𝑔𝑗	𝑔𝑗	ADV
asir-3636	104	3	2	2	NUM
asir-3636	104	4	∈	∈	NOUN
asir-3636	104	5	𝔗	𝔗	NOUN
asir-3636	104	6	and	and	CCONJ
asir-3636	104	7	let	let	VERB
asir-3636	104	8	𝐵𝑛	𝐵𝑛	PROPN
asir-3636	104	9	be	be	AUX
asir-3636	104	10	the	the	DET
asir-3636	104	11	blashke	blashke	ADJ
asir-3636	104	12	product	product	NOUN
asir-3636	104	13	with	with	ADP
asir-3636	104	14	zeros	zero	NOUN
asir-3636	104	15	𝑍𝑔𝑗	𝑍𝑔𝑗	PROPN
asir-3636	104	16	2	2	NUM
asir-3636	104	17	∩	∩	NOUN
asir-3636	104	18	𝔻𝑛	𝔻𝑛	PROPN
asir-3636	104	19	,	,	PUNCT
asir-3636	104	20	where	where	SCONJ
asir-3636	104	21	𝔻𝑛	𝔻𝑛	ADV
asir-3636	104	22	≔	≔	VERB
asir-3636	104	23	{	{	PUNCT
asir-3636	104	24	𝑧	𝑧	PRON
asir-3636	104	25	∈	∈	PROPN
asir-3636	104	26	𝔻	𝔻	ADJ
asir-3636	104	27	∶	∶	NOUN
asir-3636	104	28	|𝑧|	|𝑧|	NOUN
asir-3636	104	29	<	<	X
asir-3636	104	30	𝑛−1	𝑛−1	NUM
asir-3636	104	31	𝑛	𝑛	PROPN
asir-3636	104	32	,	,	PUNCT
asir-3636	104	33	𝑛	𝑛	PROPN
asir-3636	104	34	∈	∈	PROPN
asir-3636	104	35	ℕ	ℕ	PROPN
asir-3636	104	36	}	}	PUNCT
asir-3636	104	37	.	.	PUNCT
asir-3636	105	1	set	set	VERB
asir-3636	105	2	∑	∑	PROPN
asir-3636	105	3	(	(	PUNCT
asir-3636	105	4	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	105	5	2)𝑗	2)𝑗	PROPN
asir-3636	105	6	𝑛	𝑛	NOUN
asir-3636	105	7	=	=	PUNCT
asir-3636	105	8	∑	∑	PUNCT
asir-3636	105	9	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	105	10	2/𝐾𝑛𝑗	2/𝐾𝑛𝑗	NUM
asir-3636	105	11	,	,	PUNCT
asir-3636	105	12	where	where	SCONJ
asir-3636	105	13	𝐾𝑛	𝐾𝑛	PROPN
asir-3636	105	14	=	=	PUNCT
asir-3636	105	15	𝐵𝑛/𝐼𝑛	𝐵𝑛/𝐼𝑛	NOUN
asir-3636	105	16	and	and	CCONJ
asir-3636	105	17	𝐼𝑛	𝐼𝑛	PROPN
asir-3636	105	18	is	be	AUX
asir-3636	105	19	the	the	DET
asir-3636	105	20	blashke	blashke	ADJ
asir-3636	105	21	product	product	NOUN
asir-3636	105	22	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	105	23	applied	apply	VERB
asir-3636	105	24	science	science	NOUN
asir-3636	105	25	and	and	CCONJ
asir-3636	105	26	innovative	innovative	ADJ
asir-3636	105	27	research	research	NOUN
asir-3636	105	28	vol	vol	NOUN
asir-3636	105	29	.	.	PROPN
asir-3636	106	1	5	5	NUM
asir-3636	106	2	,	,	PUNCT
asir-3636	106	3	no	no	INTJ
asir-3636	106	4	.	.	NOUN
asir-3636	106	5	1	1	NUM
asir-3636	106	6	,	,	PUNCT
asir-3636	106	7	2021	2021	NUM
asir-3636	106	8	26	26	NUM
asir-3636	106	9	published	publish	VERB
asir-3636	106	10	by	by	ADP
asir-3636	106	11	scholink	scholink	PROPN
asir-3636	106	12	inc	inc	PROPN
asir-3636	106	13	.	.	PROPN
asir-3636	106	14	with	with	ADP
asir-3636	106	15	zeros	zero	NOUN
asir-3636	106	16	𝑍𝑔𝑗	𝑍𝑔𝑗	PROPN
asir-3636	106	17	2	2	NUM
asir-3636	106	18	∩	∩	NOUN
asir-3636	107	1	𝔻𝑛	𝔻𝑛	PROPN
asir-3636	107	2	.we	.we	PUNCT
asir-3636	107	3	have	have	VERB
asir-3636	107	4	(	(	PUNCT
asir-3636	107	5	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	107	6	2)𝑛	2)𝑛	NUM
asir-3636	107	7	∈	∈	NOUN
asir-3636	107	8	𝐼	𝐼	NOUN
asir-3636	107	9	for	for	ADP
asir-3636	107	10	every	every	DET
asir-3636	107	11	𝑛.	𝑛.	NOUN
asir-3636	107	12	indeed	indeed	ADV
asir-3636	107	13	,	,	PUNCT
asir-3636	107	14	fix	fix	VERB
asir-3636	107	15	𝑛	𝑛	DET
asir-3636	107	16	∈	∈	PROPN
asir-3636	107	17	ℕ.	ℕ.	PROPN
asir-3636	107	18	it	it	PRON
asir-3636	107	19	is	be	AUX
asir-3636	107	20	permissible	permissible	ADJ
asir-3636	107	21	to	to	PART
asir-3636	107	22	assume	assume	VERB
asir-3636	107	23	that	that	SCONJ
asir-3636	107	24	𝑍𝐾𝑛	𝑍𝐾𝑛	NOUN
asir-3636	107	25	consists	consist	VERB
asir-3636	107	26	of	of	ADP
asir-3636	107	27	a	a	DET
asir-3636	107	28	single	single	ADJ
asir-3636	107	29	point	point	NOUN
asir-3636	107	30	,	,	PUNCT
asir-3636	107	31	say	say	VERB
asir-3636	107	32	𝑍𝐾𝑛	𝑍𝐾𝑛	NOUN
asir-3636	107	33	=	=	SYM
asir-3636	107	34	{	{	PUNCT
asir-3636	107	35	𝑧	𝑧	DET
asir-3636	107	36	−	−	PROPN
asir-3636	107	37	𝜖	𝜖	NOUN
asir-3636	107	38	}	}	PUNCT
asir-3636	107	39	.	.	PUNCT
asir-3636	108	1	let	let	VERB
asir-3636	108	2	𝜋	𝜋	PRON
asir-3636	108	3	∶	∶	VERB
asir-3636	108	4	𝒜αj	𝒜αj	PROPN
asir-3636	108	5	2	2	NUM
asir-3636	108	6	→	→	SYM
asir-3636	108	7	𝒜αj	𝒜αj	PROPN
asir-3636	108	8	2/𝔗	2/𝔗	NUM
asir-3636	108	9	be	be	AUX
asir-3636	108	10	the	the	DET
asir-3636	108	11	canonical	canonical	ADJ
asir-3636	108	12	quotient	quotient	NOUN
asir-3636	108	13	map	map	NOUN
asir-3636	108	14	.	.	PUNCT
asir-3636	109	1	first	first	ADV
asir-3636	109	2	suppose	suppose	VERB
asir-3636	109	3	(	(	PUNCT
asir-3636	109	4	𝑧	𝑧	PROPN
asir-3636	109	5	−	−	PROPN
asir-3636	109	6	𝜖	𝜖	PROPN
asir-3636	109	7	)	)	PUNCT
asir-3636	109	8	∉	∉	PROPN
asir-3636	109	9	𝑍𝔗	𝑍𝔗	PROPN
asir-3636	109	10	,	,	PUNCT
asir-3636	109	11	then	then	ADV
asir-3636	109	12	𝜋(𝐾𝑛	𝜋(𝐾𝑛	PROPN
asir-3636	109	13	)	)	PUNCT
asir-3636	109	14	is	be	AUX
asir-3636	109	15	invertible	invertible	ADJ
asir-3636	109	16	in	in	ADP
asir-3636	109	17	𝒜αj	𝒜αj	PROPN
asir-3636	109	18	2/𝔗.	2/𝔗.	NUM
asir-3636	109	19	it	it	PRON
asir-3636	109	20	follows	follow	VERB
asir-3636	109	21	that	that	SCONJ
asir-3636	109	22	∑	∑	ADP
asir-3636	110	1	𝜋(𝑔𝑗	𝜋(𝑔𝑗	NOUN
asir-3636	110	2	2)𝑛	2)𝑛	NUM
asir-3636	110	3	𝑗	𝑗	NOUN
asir-3636	110	4	=	=	SYM
asir-3636	110	5	∑	∑	PUNCT
asir-3636	110	6	𝜋(𝑔𝑗	𝜋(𝑔𝑗	PROPN
asir-3636	110	7	2)𝜋−1(𝐾𝑛)𝑗	2)𝜋−1(𝐾𝑛)𝑗	NUM
asir-3636	110	8	=	=	SYM
asir-3636	110	9	0	0	NUM
asir-3636	110	10	,	,	PUNCT
asir-3636	110	11	hence	hence	ADV
asir-3636	110	12	(	(	PUNCT
asir-3636	110	13	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	110	14	2)𝑛	2)𝑛	NUM
asir-3636	110	15	∈	∈	PROPN
asir-3636	110	16	𝔗.	𝔗.	PROPN
asir-3636	110	17	if	if	SCONJ
asir-3636	110	18	(	(	PUNCT
asir-3636	110	19	𝑧	𝑧	PROPN
asir-3636	110	20	−	−	PROPN
asir-3636	110	21	𝜖	𝜖	PROPN
asir-3636	110	22	)	)	PUNCT
asir-3636	110	23	∈	∈	PROPN
asir-3636	110	24	𝑍𝔗	𝑍𝔗	PROPN
asir-3636	110	25	,	,	PUNCT
asir-3636	110	26	we	we	PRON
asir-3636	110	27	consider	consider	VERB
asir-3636	110	28	the	the	DET
asir-3636	110	29	following	follow	VERB
asir-3636	110	30	ideal	ideal	ADJ
asir-3636	110	31	𝒥𝑧−𝜖	𝒥𝑧−𝜖	NOUN
asir-3636	110	32	∶=	∶=	NUM
asir-3636	110	33	{	{	PUNCT
asir-3636	110	34	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	110	35	2	2	NUM
asir-3636	110	36	∈	∈	NOUN
asir-3636	110	37	𝒜αj	𝒜αj	PROPN
asir-3636	110	38	2	2	NUM
asir-3636	110	39	∶	∶	NOUN
asir-3636	110	40	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	110	41	2𝐼𝑛	2𝐼𝑛	NUM
asir-3636	110	42	∈	∈	PROPN
asir-3636	110	43	𝔗	𝔗	PROPN
asir-3636	110	44	}	}	PUNCT
asir-3636	110	45	.	.	PUNCT
asir-3636	111	1	it	it	PRON
asir-3636	111	2	is	be	AUX
asir-3636	111	3	clear	clear	ADJ
asir-3636	111	4	that	that	SCONJ
asir-3636	111	5	𝒥𝑧−𝜖	𝒥𝑧−𝜖	VERB
asir-3636	111	6	is	be	AUX
asir-3636	111	7	closed	closed	ADJ
asir-3636	111	8	.	.	PUNCT
asir-3636	112	1	since	since	SCONJ
asir-3636	112	2	(	(	PUNCT
asir-3636	112	3	𝑧	𝑧	PROPN
asir-3636	112	4	−	−	PROPN
asir-3636	112	5	𝜖	𝜖	NOUN
asir-3636	112	6	)	)	PUNCT
asir-3636	112	7	∉	∉	PROPN
asir-3636	112	8	𝑍𝒥𝑧−𝜖	𝑍𝒥𝑧−𝜖	NOUN
asir-3636	112	9	,	,	PUNCT
asir-3636	112	10	it	it	PRON
asir-3636	112	11	follows	follow	VERB
asir-3636	112	12	that	that	SCONJ
asir-3636	112	13	𝐾𝑛	𝐾𝑛	PROPN
asir-3636	112	14	is	be	AUX
asir-3636	112	15	invertible	invertible	ADJ
asir-3636	112	16	in	in	ADP
asir-3636	112	17	the	the	DET
asir-3636	112	18	quotient	quotient	NOUN
asir-3636	112	19	algebra	algebra	VERB
asir-3636	112	20	𝒜αj	𝒜αj	PROPN
asir-3636	112	21	2/𝒥𝑧−𝜖	2/𝒥𝑧−𝜖	NUM
asir-3636	112	22	and	and	CCONJ
asir-3636	112	23	so	so	ADV
asir-3636	112	24	𝑔𝑗	𝑔𝑗	ADJ
asir-3636	112	25	2/(𝐼𝑛𝐾𝑛	2/(𝐼𝑛𝐾𝑛	NOUN
asir-3636	112	26	)	)	PUNCT
asir-3636	112	27	∈	∈	PROPN
asir-3636	112	28	𝒥𝑧−𝜖.	𝒥𝑧−𝜖.	PROPN
asir-3636	112	29	hence	hence	ADV
asir-3636	112	30	(	(	PUNCT
asir-3636	112	31	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	112	32	2)𝑛	2)𝑛	NUM
asir-3636	112	33	∈	∈	PROPN
asir-3636	112	34	𝔗.	𝔗.	PROPN
asir-3636	113	1	it	it	PRON
asir-3636	113	2	is	be	AUX
asir-3636	113	3	clear	clear	ADJ
asir-3636	113	4	that	that	SCONJ
asir-3636	113	5	(	(	PUNCT
asir-3636	113	6	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	113	7	2)𝑛	2)𝑛	NUM
asir-3636	113	8	converges	converge	VERB
asir-3636	113	9	uniformly	uniformly	ADV
asir-3636	113	10	on	on	ADP
asir-3636	113	11	compact	compact	ADJ
asir-3636	113	12	subsets	subset	NOUN
asir-3636	113	13	of	of	ADP
asir-3636	113	14	𝔻	𝔻	PROPN
asir-3636	113	15	to	to	PART
asir-3636	113	16	∑	∑	PART
asir-3636	113	17	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	113	18	2	2	NUM
asir-3636	113	19	𝑗	𝑗	NOUN
asir-3636	113	20	=	=	X
asir-3636	113	21	∑	∑	PUNCT
asir-3636	113	22	(	(	PUNCT
asir-3636	113	23	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	113	24	2/𝐵𝑔𝑗	2/𝐵𝑔𝑗	NUM
asir-3636	113	25	2)𝐵𝔗𝐽	2)𝐵𝔗𝐽	NOUN
asir-3636	114	1	and	and	CCONJ
asir-3636	114	2	we	we	PRON
asir-3636	114	3	have	have	VERB
asir-3636	114	4	∑	∑	PROPN
asir-3636	114	5	𝐵𝑓𝑗	𝐵𝑓𝑗	PROPN
asir-3636	114	6	2𝐽	2𝐽	NOUN
asir-3636	115	1	=	=	PUNCT
asir-3636	115	2	𝐵𝔗.	𝐵𝔗.	X
asir-3636	115	3	in	in	ADP
asir-3636	115	4	the	the	DET
asir-3636	115	5	sequel	sequel	NOUN
asir-3636	115	6	we	we	PRON
asir-3636	115	7	prove	prove	VERB
asir-3636	115	8	that	that	SCONJ
asir-3636	115	9	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	115	10	2	2	NUM
asir-3636	115	11	∈	∈	NOUN
asir-3636	115	12	𝔗.	𝔗.	NOUN
asir-3636	115	13	if	if	SCONJ
asir-3636	115	14	we	we	PRON
asir-3636	115	15	obtain	obtain	VERB
asir-3636	115	16	∑|((𝑔𝑗	∑|((𝑔𝑗	ADJ
asir-3636	115	17	2	2	NUM
asir-3636	115	18	)	)	PUNCT
asir-3636	115	19	𝑛	𝑛	PROPN
asir-3636	115	20	)	)	PUNCT
asir-3636	115	21	′	′	NUM
asir-3636	116	1	(	(	PUNCT
asir-3636	116	2	𝑧)|	𝑧)|	PROPN
asir-3636	116	3	𝑗	𝑗	PROPN
asir-3636	116	4	≤∑𝜊	≤∑𝜊	PROPN
asir-3636	116	5	(	(	PUNCT
asir-3636	116	6	1	1	NUM
asir-3636	116	7	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	116	8	2	2	NUM
asir-3636	116	9	)	)	PUNCT
asir-3636	116	10	𝑗	𝑗	NOUN
asir-3636	116	11	(	(	PUNCT
asir-3636	116	12	𝑧	𝑧	PROPN
asir-3636	116	13	∈	∈	PROPN
asir-3636	116	14	𝔻	𝔻	NOUN
asir-3636	116	15	)	)	PUNCT
asir-3636	116	16	,	,	PUNCT
asir-3636	116	17	uniformly	uniformly	ADV
asir-3636	116	18	with	with	ADP
asir-3636	116	19	respect	respect	NOUN
asir-3636	116	20	to	to	ADP
asir-3636	116	21	n	n	CCONJ
asir-3636	116	22	,	,	PUNCT
asir-3636	116	23	we	we	PRON
asir-3636	116	24	can	can	AUX
asir-3636	116	25	deduce	deduce	VERB
asir-3636	116	26	by	by	ADP
asir-3636	116	27	using	use	VERB
asir-3636	116	28	(	(	PUNCT
asir-3636	116	29	matheson	matheson	PROPN
asir-3636	116	30	,	,	PUNCT
asir-3636	116	31	1978	1978	NUM
asir-3636	116	32	,	,	PUNCT
asir-3636	116	33	lemma	lemma	PROPN
asir-3636	116	34	1	1	NUM
asir-3636	116	35	)	)	PUNCT
asir-3636	116	36	that	that	SCONJ
asir-3636	116	37	lim𝑛→+∞∑	lim𝑛→+∞∑	PROPN
asir-3636	116	38	‖(𝑔𝑗	‖(𝑔𝑗	CCONJ
asir-3636	116	39	2	2	NUM
asir-3636	116	40	)	)	PUNCT
asir-3636	117	1	𝑛	𝑛	PRON
asir-3636	117	2	−	−	NOUN
asir-3636	117	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	117	4	2‖𝑗	2‖𝑗	NUM
asir-3636	117	5	αj	αj	NOUN
asir-3636	117	6	2	2	NUM
asir-3636	117	7	=	=	SYM
asir-3636	117	8	0	0	NUM
asir-3636	117	9	.	.	PUNCT
asir-3636	118	1	indeed	indeed	ADV
asir-3636	118	2	,	,	PUNCT
asir-3636	118	3	by	by	ADP
asir-3636	118	4	the	the	DET
asir-3636	118	5	cauchy	cauchy	ADJ
asir-3636	118	6	integral	integral	ADJ
asir-3636	118	7	formula	formula	NOUN
asir-3636	118	8	∑((𝑔𝑗	∑((𝑔𝑗	NOUN
asir-3636	118	9	2	2	NUM
asir-3636	118	10	)	)	PUNCT
asir-3636	118	11	𝑛	𝑛	PROPN
asir-3636	118	12	)	)	PUNCT
asir-3636	118	13	′	′	NUM
asir-3636	118	14	(	(	PUNCT
asir-3636	118	15	𝑧	𝑧	X
asir-3636	118	16	)	)	PUNCT
asir-3636	118	17	𝑗	𝑗	NOUN
asir-3636	118	18	=	=	SYM
asir-3636	118	19	1	1	NUM
asir-3636	118	20	2𝜋𝑖	2𝜋𝑖	ADJ
asir-3636	118	21	∫	∫	PROPN
asir-3636	118	22	∑	∑	PROPN
asir-3636	118	23	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	118	24	2(𝑧	2(𝑧	NUM
asir-3636	118	25	−	−	PROPN
asir-3636	118	26	2𝜖)𝐾𝑛(𝑧	2𝜖)𝐾𝑛(𝑧	NUM
asir-3636	118	27	−	−	PROPN
asir-3636	118	28	2𝜖)̅̅	2𝜖)̅̅	NUM
asir-3636	118	29	̅̅	̅̅	PROPN
asir-3636	118	30	̅̅	̅̅	PROPN
asir-3636	118	31	̅̅	̅̅	PROPN
asir-3636	118	32	̅̅	̅̅	PROPN
asir-3636	118	33	̅̅	̅̅	PROPN
asir-3636	118	34	̅̅	̅̅	PROPN
asir-3636	118	35	4𝜖2	4𝜖2	NUM
asir-3636	119	1	𝑗	𝑗	PROPN
asir-3636	119	2	𝕋	𝕋	NOUN
asir-3636	119	3	𝑑(𝑧	𝑑(𝑧	NOUN
asir-3636	119	4	−	−	NOUN
asir-3636	119	5	2	2	NUM
asir-3636	119	6	=	=	SYM
asir-3636	119	7	1	1	NUM
asir-3636	119	8	2𝜋𝑖	2𝜋𝑖	ADJ
asir-3636	119	9	∫	∫	PROPN
asir-3636	119	10	∑	∑	PROPN
asir-3636	119	11	(	(	PUNCT
asir-3636	119	12	𝑔𝑗	𝑔𝑗	ADV
asir-3636	119	13	2(𝑧	2(𝑧	NUM
asir-3636	119	14	−	−	NOUN
asir-3636	119	15	2𝜖	2𝜖	NUM
asir-3636	119	16	)	)	PUNCT
asir-3636	120	1	−	−	ADP
asir-3636	120	2	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	120	3	2(𝑧	2(𝑧	NUM
asir-3636	120	4	∕	∕	NOUN
asir-3636	120	5	|𝑧|))𝐾𝑛(𝑧	|𝑧|))𝐾𝑛(𝑧	ADV
asir-3636	120	6	−	−	PROPN
asir-3636	120	7	2𝜖)̅̅	2𝜖)̅̅	NUM
asir-3636	120	8	̅̅	̅̅	PROPN
asir-3636	120	9	̅̅	̅̅	PROPN
asir-3636	120	10	̅̅	̅̅	PROPN
asir-3636	120	11	̅̅	̅̅	PROPN
asir-3636	120	12	̅̅	̅̅	PROPN
asir-3636	120	13	̅̅	̅̅	PROPN
asir-3636	120	14	4𝜖2	4𝜖2	NUM
asir-3636	120	15	𝑗	𝑗	PROPN
asir-3636	120	16	𝕋	𝕋	PROPN
asir-3636	120	17	𝑑(𝑧	𝑑(𝑧	PROPN
asir-3636	120	18	−	−	NOUN
asir-3636	120	19	2𝜖	2𝜖	NUM
asir-3636	120	20	)	)	PUNCT
asir-3636	120	21	(	(	PUNCT
asir-3636	120	22	𝑧	𝑧	PRON
asir-3636	120	23	∈	∈	PROPN
asir-3636	120	24	𝔻	𝔻	PROPN
asir-3636	120	25	)	)	PUNCT
asir-3636	120	26	.	.	PUNCT
asir-3636	121	1	then	then	ADV
asir-3636	121	2	,	,	PUNCT
asir-3636	121	3	for	for	ADP
asir-3636	121	4	𝑧	𝑧	X
asir-3636	121	5	=	=	X
asir-3636	121	6	(	(	PUNCT
asir-3636	121	7	1	1	NUM
asir-3636	121	8	−	−	PROPN
asir-3636	121	9	𝜖)𝑒𝑖𝜃	𝜖)𝑒𝑖𝜃	PROPN
asir-3636	121	10	2	2	NUM
asir-3636	121	11	∈	∈	NOUN
asir-3636	121	12	𝔻	𝔻	ADJ
asir-3636	121	13	∑((𝑔𝑗	∑((𝑔𝑗	NOUN
asir-3636	121	14	2	2	NUM
asir-3636	121	15	)	)	PUNCT
asir-3636	121	16	𝑛	𝑛	PROPN
asir-3636	121	17	)	)	PUNCT
asir-3636	121	18	′	′	NUM
asir-3636	122	1	(	(	PUNCT
asir-3636	122	2	𝑧	𝑧	X
asir-3636	122	3	)	)	PUNCT
asir-3636	122	4	𝑗	𝑗	NOUN
asir-3636	122	5	≤	≤	NUM
asir-3636	122	6	‖𝐾𝑛‖∞	‖𝐾𝑛‖∞	PUNCT
asir-3636	122	7	2𝜋	2𝜋	NOUN
asir-3636	122	8	∫	∫	PROPN
asir-3636	122	9	∑	∑	PROPN
asir-3636	122	10	|𝑔𝑗	|𝑔𝑗	NUM
asir-3636	122	11	2(𝑧	2(𝑧	NUM
asir-3636	122	12	−	−	NOUN
asir-3636	122	13	2𝜖	2𝜖	NUM
asir-3636	122	14	)	)	PUNCT
asir-3636	123	1	−	−	ADP
asir-3636	123	2	𝑔𝑗	𝑔𝑗	ADV
asir-3636	123	3	2(𝑧	2(𝑧	NUM
asir-3636	123	4	∕	∕	NOUN
asir-3636	123	5	|𝑧|)|	|𝑧|)|	PUNCT
asir-3636	123	6	4|𝜖|2	4|𝜖|2	NUM
asir-3636	123	7	𝑗	𝑗	NOUN
asir-3636	123	8	𝕋	𝕋	NOUN
asir-3636	123	9	|𝑑(𝑧	|𝑑(𝑧	PROPN
asir-3636	123	10	−	−	NOUN
asir-3636	123	11	2𝜖)|	2𝜖)|	NUM
asir-3636	123	12	=	=	SYM
asir-3636	123	13	1	1	NUM
asir-3636	123	14	2𝜋	2𝜋	NUM
asir-3636	123	15	∫	∫	PROPN
asir-3636	123	16	∑	∑	PROPN
asir-3636	123	17	|𝑔𝑗	|𝑔𝑗	NUM
asir-3636	123	18	2(𝑒𝑖(𝑡	2(𝑒𝑖(𝑡	NUM
asir-3636	123	19	2+𝜃2	2+𝜃2	NUM
asir-3636	123	20	)	)	PUNCT
asir-3636	123	21	)	)	PUNCT
asir-3636	123	22	−	−	ADP
asir-3636	124	1	𝑔𝑗	𝑔𝑗	ADV
asir-3636	124	2	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	124	3	2	2	NUM
asir-3636	124	4	)	)	PUNCT
asir-3636	124	5	|	|	CCONJ
asir-3636	124	6	(	(	PUNCT
asir-3636	124	7	2𝜖	2𝜖	NUM
asir-3636	124	8	−	−	PROPN
asir-3636	124	9	1	1	X
asir-3636	124	10	)	)	PUNCT
asir-3636	124	11	cos	cos	ADP
asir-3636	124	12	𝑡2	𝑡2	PROPN
asir-3636	124	13	+	+	CCONJ
asir-3636	124	14	(	(	PUNCT
asir-3636	124	15	1	1	NUM
asir-3636	124	16	−	−	PROPN
asir-3636	124	17	𝜖)2	𝜖)2	PROPN
asir-3636	124	18	𝑗	𝑗	INTJ
asir-3636	124	19	𝜋	𝜋	NOUN
asir-3636	124	20	−𝜋	−𝜋	ADJ
asir-3636	124	21	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	124	22	.	.	PUNCT
asir-3636	125	1	for	for	ADP
asir-3636	125	2	all	all	DET
asir-3636	125	3	𝜀	𝜀	NOUN
asir-3636	125	4	>	>	X
asir-3636	125	5	0	0	NUM
asir-3636	125	6	,	,	PUNCT
asir-3636	125	7	there	there	PRON
asir-3636	125	8	is	be	VERB
asir-3636	125	9	𝜂	𝜂	NOUN
asir-3636	125	10	>	>	X
asir-3636	125	11	0	0	NUM
asir-3636	125	12	such	such	ADJ
asir-3636	125	13	that	that	SCONJ
asir-3636	125	14	if	if	SCONJ
asir-3636	125	15	|𝑡2|	|𝑡2|	VERB
asir-3636	125	16	≤	≤	ADJ
asir-3636	125	17	𝜂	𝜂	NOUN
asir-3636	125	18	,	,	PUNCT
asir-3636	125	19	we	we	PRON
asir-3636	125	20	have	have	VERB
asir-3636	125	21	∑	∑	PROPN
asir-3636	126	1	|𝑔𝑗	|𝑔𝑗	NUM
asir-3636	126	2	2(𝑒𝑖(𝑡	2(𝑒𝑖(𝑡	NUM
asir-3636	126	3	2+𝜃2	2+𝜃2	NUM
asir-3636	126	4	)	)	PUNCT
asir-3636	126	5	)	)	PUNCT
asir-3636	127	1	−	−	ADP
asir-3636	127	2	𝑔𝑗	𝑔𝑗	DET
asir-3636	127	3	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	127	4	2	2	NUM
asir-3636	127	5	)	)	PUNCT
asir-3636	128	1	|𝑗	|𝑗	VERB
asir-3636	128	2	≤	≤	NUM
asir-3636	128	3	∑	∑	PROPN
asir-3636	128	4	𝜀|𝑡2|αj	𝜀|𝑡2|αj	ADP
asir-3636	128	5	2	2	NUM
asir-3636	128	6	𝑗	𝑗	NOUN
asir-3636	128	7	(	(	PUNCT
asir-3636	128	8	𝜃2	𝜃2	PROPN
asir-3636	128	9	∈	∈	PROPN
asir-3636	129	1	[	[	X
asir-3636	129	2	−𝜋,+𝜋	−𝜋,+𝜋	NUM
asir-3636	129	3	]	]	PUNCT
asir-3636	129	4	)	)	PUNCT
asir-3636	129	5	.	.	PUNCT
asir-3636	130	1	then	then	ADV
asir-3636	130	2	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	VERB
asir-3636	130	3	applied	apply	VERB
asir-3636	130	4	science	science	NOUN
asir-3636	130	5	and	and	CCONJ
asir-3636	130	6	innovative	innovative	ADJ
asir-3636	130	7	research	research	NOUN
asir-3636	130	8	vol	vol	NOUN
asir-3636	130	9	.	.	PROPN
asir-3636	131	1	5	5	NUM
asir-3636	131	2	,	,	PUNCT
asir-3636	131	3	no	no	INTJ
asir-3636	131	4	.	.	NOUN
asir-3636	131	5	1	1	NUM
asir-3636	131	6	,	,	PUNCT
asir-3636	131	7	2021	2021	NUM
asir-3636	131	8	27	27	NUM
asir-3636	131	9	published	publish	VERB
asir-3636	131	10	by	by	ADP
asir-3636	131	11	scholink	scholink	PROPN
asir-3636	131	12	inc	inc	PROPN
asir-3636	131	13	.	.	PROPN
asir-3636	131	14	∫	∫	PROPN
asir-3636	131	15	∑	∑	PROPN
asir-3636	132	1	|𝑔𝑗	|𝑔𝑗	NUM
asir-3636	132	2	2(𝑒𝑖(𝑡	2(𝑒𝑖(𝑡	NUM
asir-3636	132	3	2+𝜃2	2+𝜃2	NUM
asir-3636	132	4	)	)	PUNCT
asir-3636	132	5	)	)	PUNCT
asir-3636	133	1	−	−	ADP
asir-3636	134	1	𝑔𝑗	𝑔𝑗	ADV
asir-3636	134	2	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	134	3	2	2	NUM
asir-3636	134	4	)	)	PUNCT
asir-3636	134	5	|	|	CCONJ
asir-3636	134	6	(	(	PUNCT
asir-3636	134	7	2𝜖	2𝜖	NUM
asir-3636	134	8	−	−	PROPN
asir-3636	134	9	1	1	X
asir-3636	134	10	)	)	PUNCT
asir-3636	134	11	cos	cos	ADP
asir-3636	134	12	𝑡2	𝑡2	PROPN
asir-3636	134	13	+	+	CCONJ
asir-3636	134	14	(	(	PUNCT
asir-3636	134	15	1	1	NUM
asir-3636	134	16	−	−	PROPN
asir-3636	134	17	𝜖)2	𝜖)2	PROPN
asir-3636	134	18	𝑗	𝑗	INTJ
asir-3636	134	19	𝜋	𝜋	NOUN
asir-3636	134	20	−𝜋	−𝜋	ADJ
asir-3636	134	21	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	134	22	≤	≤	NOUN
asir-3636	134	23	𝜀∫	𝜀∫	NOUN
asir-3636	134	24	∑	∑	ADP
asir-3636	134	25	|𝑡2|αj	|𝑡2|αj	PROPN
asir-3636	134	26	2	2	NUM
asir-3636	134	27	𝜖2	𝜖2	NOUN
asir-3636	134	28	+	+	NUM
asir-3636	134	29	4(1	4(1	NUM
asir-3636	135	1	−	−	ADP
asir-3636	135	2	𝜖)𝑡2	𝜖)𝑡2	PROPN
asir-3636	135	3	∕	∕	PROPN
asir-3636	135	4	𝜋2	𝜋2	PROPN
asir-3636	135	5	𝑗	𝑗	PROPN
asir-3636	135	6	|𝑡2|≤𝜂	|𝑡2|≤𝜂	NOUN
asir-3636	135	7	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	136	1	+	+	PROPN
asir-3636	136	2	∑‖𝑔𝑗	∑‖𝑔𝑗	PROPN
asir-3636	136	3	2‖	2‖	PROPN
asir-3636	136	4	αj	αj	PART
asir-3636	136	5	2	2	NUM
asir-3636	136	6	𝑗	𝑗	NOUN
asir-3636	136	7	∫	∫	NOUN
asir-3636	136	8	∑	∑	PROPN
asir-3636	136	9	|𝑡2|αj	|𝑡2|αj	PROPN
asir-3636	136	10	2	2	NUM
asir-3636	136	11	𝜖2	𝜖2	NOUN
asir-3636	136	12	+	+	NUM
asir-3636	136	13	4(1	4(1	NUM
asir-3636	137	1	−	−	ADP
asir-3636	137	2	𝜖)𝑡2	𝜖)𝑡2	PROPN
asir-3636	137	3	∕	∕	PROPN
asir-3636	137	4	𝜋2	𝜋2	PROPN
asir-3636	137	5	𝑗	𝑗	PROPN
asir-3636	137	6	|𝑡2|≤𝜂	|𝑡2|≤𝜂	NOUN
asir-3636	137	7	𝑑𝑡2	𝑑𝑡2	PROPN
asir-3636	137	8	≤∑	≤∑	PROPN
asir-3636	137	9	𝜀	𝜀	PROPN
asir-3636	137	10	(	(	PUNCT
asir-3636	137	11	1	1	NUM
asir-3636	137	12	−	−	PROPN
asir-3636	137	13	𝜖	𝜖	PROPN
asir-3636	137	14	)	)	PUNCT
asir-3636	137	15	1+αj	1+αj	NUM
asir-3636	137	16	2	2	NUM
asir-3636	137	17	2	2	NUM
asir-3636	137	18	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	137	19	2	2	NUM
asir-3636	137	20	𝑗	𝑗	NOUN
asir-3636	137	21	∫	∫	NOUN
asir-3636	137	22	∑	∑	PART
asir-3636	137	23	𝑢αj	𝑢αj	NOUN
asir-3636	137	24	2	2	NUM
asir-3636	137	25	1	1	NUM
asir-3636	137	26	+	+	CCONJ
asir-3636	137	27	(	(	PUNCT
asir-3636	137	28	2𝑢	2𝑢	NOUN
asir-3636	137	29	∕	∕	NOUN
asir-3636	137	30	𝜋)2	𝜋)2	ADV
asir-3636	137	31	𝑗	𝑗	NOUN
asir-3636	138	1	+	+	NOUN
asir-3636	138	2	∞	∞	NUM
asir-3636	138	3	0	0	NUM
asir-3636	138	4	𝑑𝑢	𝑑𝑢	PUNCT
asir-3636	139	1	+	+	ADJ
asir-3636	139	2	∑	∑	PROPN
asir-3636	139	3	‖𝑔𝑗	‖𝑔𝑗	NUM
asir-3636	139	4	2‖	2‖	PROPN
asir-3636	139	5	αj	αj	ADP
asir-3636	139	6	2	2	NUM
asir-3636	139	7	(	(	PUNCT
asir-3636	139	8	1	1	NUM
asir-3636	139	9	−	−	PROPN
asir-3636	139	10	𝜖	𝜖	PROPN
asir-3636	139	11	)	)	PUNCT
asir-3636	139	12	1+αj	1+αj	NUM
asir-3636	139	13	2	2	NUM
asir-3636	139	14	2	2	NUM
asir-3636	139	15	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	139	16	2	2	NUM
asir-3636	139	17	𝑗	𝑗	NOUN
asir-3636	139	18	∫	∫	NOUN
asir-3636	139	19	∑	∑	PART
asir-3636	139	20	𝑢αj	𝑢αj	NOUN
asir-3636	139	21	2	2	NUM
asir-3636	139	22	1	1	NUM
asir-3636	139	23	+	+	CCONJ
asir-3636	139	24	(	(	PUNCT
asir-3636	139	25	2𝑢	2𝑢	NOUN
asir-3636	139	26	∕	∕	NOUN
asir-3636	139	27	𝜋)2	𝜋)2	ADV
asir-3636	139	28	𝑗	𝑗	VERB
asir-3636	139	29	|𝑢|≥	|𝑢|≥	NOUN
asir-3636	139	30	𝜂√1−𝜖	𝜂√1−𝜖	NOUN
asir-3636	139	31	𝜖	𝜖	PROPN
asir-3636	139	32	𝑑𝑢	𝑑𝑢	X
asir-3636	139	33	≤∑𝜀𝑂	≤∑𝜀𝑂	NOUN
asir-3636	139	34	(	(	PUNCT
asir-3636	139	35	1	1	NUM
asir-3636	139	36	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	139	37	2	2	NUM
asir-3636	139	38	)	)	PUNCT
asir-3636	139	39	𝑗	𝑗	NOUN
asir-3636	140	1	+	+	ADJ
asir-3636	140	2	∑‖𝑔𝑗	∑‖𝑔𝑗	PROPN
asir-3636	140	3	2‖	2‖	PROPN
asir-3636	140	4	αj	αj	ADP
asir-3636	140	5	2𝑂	2𝑂	NOUN
asir-3636	140	6	(	(	PUNCT
asir-3636	140	7	1	1	NUM
asir-3636	140	8	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	140	9	2	2	NUM
asir-3636	140	10	)	)	PUNCT
asir-3636	140	11	𝑗	𝑗	NOUN
asir-3636	140	12	.	.	PUNCT
asir-3636	141	1	we	we	PRON
asir-3636	141	2	obtain	obtain	VERB
asir-3636	141	3	∫	∫	PROPN
asir-3636	141	4	∑	∑	PROPN
asir-3636	141	5	|𝑔𝑗	|𝑔𝑗	PROPN
asir-3636	141	6	2(𝑒𝑖(𝑡	2(𝑒𝑖(𝑡	NUM
asir-3636	141	7	2+𝜃2))−𝑔𝑗	2+𝜃2))−𝑔𝑗	NUM
asir-3636	141	8	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	141	9	2	2	NUM
asir-3636	141	10	)	)	PUNCT
asir-3636	141	11	|	|	CCONJ
asir-3636	141	12	(	(	PUNCT
asir-3636	141	13	2𝜖−1	2𝜖−1	NUM
asir-3636	141	14	)	)	PUNCT
asir-3636	141	15	cos	cos	PROPN
asir-3636	141	16	𝑡2+(1−𝜖)2𝑗	𝑡2+(1−𝜖)2𝑗	PROPN
asir-3636	141	17	𝜋	𝜋	PRON
asir-3636	141	18	−𝜋	−𝜋	ADJ
asir-3636	141	19	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	141	20	≤	≤	NOUN
asir-3636	141	21	∑	∑	PUNCT
asir-3636	141	22	‖𝑔𝑗	‖𝑔𝑗	NUM
asir-3636	141	23	2‖	2‖	PROPN
asir-3636	142	1	αj	αj	NOUN
asir-3636	142	2	2𝑂	2𝑂	NOUN
asir-3636	142	3	(	(	PUNCT
asir-3636	142	4	1	1	NUM
asir-3636	142	5	𝜖	𝜖	PROPN
asir-3636	142	6	1−αj	1−αj	NUM
asir-3636	142	7	2)𝑗	2)𝑗	NOUN
asir-3636	142	8	.	.	PUNCT
asir-3636	143	1	(	(	PUNCT
asir-3636	143	2	2	2	X
asir-3636	143	3	)	)	PUNCT
asir-3636	143	4	consequently	consequently	ADV
asir-3636	143	5	∑|((𝑔𝑗	∑|((𝑔𝑗	ADJ
asir-3636	143	6	2	2	NUM
asir-3636	143	7	)	)	PUNCT
asir-3636	143	8	𝑛	𝑛	PROPN
asir-3636	143	9	)	)	PUNCT
asir-3636	143	10	′	′	NUM
asir-3636	144	1	(	(	PUNCT
asir-3636	144	2	𝑧)|	𝑧)|	NOUN
asir-3636	144	3	𝑗	𝑗	PRON
asir-3636	144	4	≤	≤	PROPN
asir-3636	144	5	∑‖𝑔𝑗	∑‖𝑔𝑗	NUM
asir-3636	144	6	2‖	2‖	PROPN
asir-3636	144	7	αj	αj	ADP
asir-3636	145	1	2𝑂	2𝑂	NOUN
asir-3636	145	2	(	(	PUNCT
asir-3636	145	3	1	1	NUM
asir-3636	145	4	𝜖1−αj	𝜖1−αj	NOUN
asir-3636	145	5	2	2	NUM
asir-3636	145	6	)	)	PUNCT
asir-3636	145	7	𝑗	𝑗	NOUN
asir-3636	145	8	(	(	PUNCT
asir-3636	145	9	𝑧	𝑧	PROPN
asir-3636	145	10	∈	∈	PROPN
asir-3636	145	11	𝔻	𝔻	PROPN
asir-3636	145	12	)	)	PUNCT
asir-3636	145	13	.	.	PUNCT
asir-3636	146	1	by	by	ADP
asir-3636	146	2	the	the	DET
asir-3636	146	3	f	f	NOUN
asir-3636	146	4	-	-	PUNCT
asir-3636	146	5	property	property	NOUN
asir-3636	146	6	of	of	ADP
asir-3636	146	7	𝒜αj	𝒜αj	PROPN
asir-3636	146	8	2	2	NUM
asir-3636	146	9	,	,	PUNCT
asir-3636	146	10	we	we	PRON
asir-3636	146	11	have	have	VERB
asir-3636	146	12	∑	∑	ADV
asir-3636	146	13	‖(𝑔𝑗	‖(𝑔𝑗	NUM
asir-3636	146	14	2	2	NUM
asir-3636	146	15	)	)	PUNCT
asir-3636	146	16	𝑛	𝑛	DET
asir-3636	146	17	‖𝑗	‖𝑗	NOUN
asir-3636	146	18	≤	≤	NOUN
asir-3636	146	19	∑	∑	PUNCT
asir-3636	146	20	𝐶αj	𝐶αj	PROPN
asir-3636	146	21	2	2	NUM
asir-3636	146	22	‖(𝑔𝑗	‖(𝑔𝑗	NUM
asir-3636	146	23	2	2	NUM
asir-3636	146	24	)	)	PUNCT
asir-3636	146	25	𝑛	𝑛	DET
asir-3636	146	26	‖	‖	PROPN
asir-3636	146	27	𝒜	𝒜	NOUN
asir-3636	146	28	αj	αj	NOUN
asir-3636	146	29	2	2	NUM
asir-3636	146	30	𝑗	𝑗	NOUN
asir-3636	146	31	.	.	PUNCT
asir-3636	147	1	using	use	VERB
asir-3636	147	2	the	the	DET
asir-3636	147	3	hilbertian	hilbertian	ADJ
asir-3636	147	4	structure	structure	NOUN
asir-3636	147	5	of	of	ADP
asir-3636	147	6	𝒟	𝒟	PROPN
asir-3636	147	7	,	,	PUNCT
asir-3636	147	8	we	we	PRON
asir-3636	147	9	deduce	deduce	VERB
asir-3636	147	10	that	that	SCONJ
asir-3636	147	11	there	there	PRON
asir-3636	147	12	is	be	VERB
asir-3636	147	13	a	a	DET
asir-3636	147	14	sequence	sequence	NOUN
asir-3636	147	15	(	(	PUNCT
asir-3636	147	16	ℎ𝑗	ℎ𝑗	PROPN
asir-3636	147	17	2)𝑛	2)𝑛	NUM
asir-3636	147	18	∈	∈	PROPN
asir-3636	147	19	𝑐𝑜({(𝑔𝑗	𝑐𝑜({(𝑔𝑗	NOUN
asir-3636	147	20	2	2	NUM
asir-3636	147	21	)	)	PUNCT
asir-3636	147	22	𝑘	𝑘	PRON
asir-3636	147	23	}	}	PUNCT
asir-3636	147	24	𝑘=𝑛	𝑘=𝑛	PROPN
asir-3636	147	25	∞	∞	NUM
asir-3636	147	26	)	)	PUNCT
asir-3636	147	27	converging	converge	VERB
asir-3636	147	28	to	to	ADP
asir-3636	147	29	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	147	30	2	2	NUM
asir-3636	147	31	in	in	ADP
asir-3636	147	32	𝒟.	𝒟.	PROPN
asir-3636	147	33	it	it	PRON
asir-3636	147	34	is	be	AUX
asir-3636	147	35	clear	clear	ADJ
asir-3636	147	36	that	that	SCONJ
asir-3636	147	37	(	(	PUNCT
asir-3636	147	38	ℎ𝑗	ℎ𝑗	NOUN
asir-3636	147	39	2	2	NUM
asir-3636	147	40	)	)	PUNCT
asir-3636	147	41	𝑛	𝑛	DET
asir-3636	147	42	∈	∈	PROPN
asir-3636	147	43	𝔗	𝔗	PROPN
asir-3636	147	44	and	and	CCONJ
asir-3636	147	45	lim𝑛→+∞∑	lim𝑛→+∞∑	PROPN
asir-3636	147	46	‖(ℎ𝑗	‖(ℎ𝑗	ADP
asir-3636	147	47	2	2	NUM
asir-3636	147	48	)	)	PUNCT
asir-3636	147	49	𝑛	𝑛	DET
asir-3636	147	50	−	−	NOUN
asir-3636	147	51	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	147	52	2‖	2‖	NUM
asir-3636	147	53	αj	αj	NOUN
asir-3636	147	54	2𝑗	2𝑗	NOUN
asir-3636	147	55	=	=	NOUN
asir-3636	147	56	0	0	NUM
asir-3636	147	57	.	.	PUNCT
asir-3636	148	1	then	then	ADV
asir-3636	148	2	lim𝑛→+∞∑	lim𝑛→+∞∑	PROPN
asir-3636	148	3	‖(ℎ𝑗	‖(ℎ𝑗	ADV
asir-3636	148	4	2	2	NUM
asir-3636	148	5	)	)	PUNCT
asir-3636	148	6	𝑛	𝑛	DET
asir-3636	148	7	−𝑗	−𝑗	PRON
asir-3636	148	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	148	9	2‖	2‖	PROPN
asir-3636	148	10	𝒜	𝒜	NOUN
asir-3636	148	11	αj	αj	NOUN
asir-3636	148	12	2	2	NUM
asir-3636	148	13	=	=	SYM
asir-3636	148	14	0	0	NUM
asir-3636	148	15	.	.	PUNCT
asir-3636	149	1	thus	thus	ADV
asir-3636	149	2	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	149	3	2	2	NUM
asir-3636	149	4	∈	∈	NOUN
asir-3636	149	5	𝔗.	𝔗.	NOUN
asir-3636	149	6	this	this	PRON
asir-3636	149	7	completes	complete	VERB
asir-3636	149	8	the	the	DET
asir-3636	149	9	proof	proof	NOUN
asir-3636	149	10	of	of	ADP
asir-3636	149	11	the	the	DET
asir-3636	149	12	lemma	lemma	PROPN
asir-3636	149	13	.	.	PUNCT
asir-3636	150	1	we	we	PRON
asir-3636	150	2	can	can	AUX
asir-3636	150	3	see	see	VERB
asir-3636	150	4	that	that	SCONJ
asir-3636	150	5	∑	∑	ADP
asir-3636	150	6	‖(𝑔𝑗	‖(𝑔𝑗	PROPN
asir-3636	150	7	2	2	NUM
asir-3636	150	8	)	)	PUNCT
asir-3636	151	1	𝑛	𝑛	PRON
asir-3636	151	2	‖	‖	PROPN
asir-3636	151	3	αj	αj	NOUN
asir-3636	151	4	2	2	NUM
asir-3636	151	5	𝑂	𝑂	NOUN
asir-3636	151	6	(	(	PUNCT
asir-3636	151	7	1	1	NUM
asir-3636	151	8	𝜖	𝜖	PROPN
asir-3636	151	9	1−αj	1−αj	NUM
asir-3636	151	10	2)𝑗	2)𝑗	NOUN
asir-3636	151	11	=	=	SYM
asir-3636	151	12	∑	∑	PUNCT
asir-3636	151	13	𝑂	𝑂	PROPN
asir-3636	151	14	(	(	PUNCT
asir-3636	151	15	1	1	NUM
asir-3636	151	16	𝜖	𝜖	PROPN
asir-3636	151	17	1−αj	1−αj	NUM
asir-3636	151	18	2)𝑗	2)𝑗	NOUN
asir-3636	151	19	.	.	PUNCT
asir-3636	152	1	as	as	ADP
asir-3636	152	2	a	a	DET
asir-3636	152	3	consequence	consequence	NOUN
asir-3636	152	4	of	of	ADP
asir-3636	152	5	theorem	theorem	NOUN
asir-3636	152	6	(	(	PUNCT
asir-3636	152	7	1.2	1.2	NUM
asir-3636	152	8	)	)	PUNCT
asir-3636	152	9	,	,	PUNCT
asir-3636	152	10	we	we	PRON
asir-3636	152	11	can	can	AUX
asir-3636	152	12	show	show	VERB
asir-3636	152	13	theorem	theorem	ADJ
asir-3636	152	14	(	(	PUNCT
asir-3636	152	15	1.1	1.1	NUM
asir-3636	152	16	)	)	PUNCT
asir-3636	152	17	and	and	CCONJ
asir-3636	152	18	deduce	deduce	VERB
asir-3636	152	19	that	that	SCONJ
asir-3636	152	20	each	each	DET
asir-3636	152	21	closed	close	VERB
asir-3636	152	22	ideal	ideal	NOUN
asir-3636	152	23	of	of	ADP
asir-3636	152	24	𝒜αj	𝒜αj	PROPN
asir-3636	152	25	2	2	NUM
asir-3636	152	26	is	be	AUX
asir-3636	152	27	standard	standard	ADJ
asir-3636	152	28	.	.	PUNCT
asir-3636	153	1	for	for	ADP
asir-3636	153	2	the	the	DET
asir-3636	153	3	sake	sake	NOUN
asir-3636	153	4	of	of	ADP
asir-3636	153	5	completeness	completeness	NOUN
asir-3636	153	6	,	,	PUNCT
asir-3636	153	7	we	we	PRON
asir-3636	153	8	sketch	sketch	VERB
asir-3636	153	9	here	here	ADV
asir-3636	153	10	the	the	DET
asir-3636	153	11	proof	proof	NOUN
asir-3636	153	12	,	,	PUNCT
asir-3636	153	13	(	(	PUNCT
asir-3636	153	14	see	see	VERB
asir-3636	153	15	brahim	brahim	PROPN
asir-3636	153	16	bouya	bouya	PROPN
asir-3636	153	17	,	,	PUNCT
asir-3636	153	18	2008	2008	NUM
asir-3636	153	19	)	)	PUNCT
asir-3636	153	20	.	.	PUNCT
asir-3636	154	1	proof	proof	NOUN
asir-3636	154	2	of	of	ADP
asir-3636	154	3	theorem	theorem	NOUN
asir-3636	154	4	(	(	PUNCT
asir-3636	154	5	1.1	1.1	NUM
asir-3636	154	6	):	):	PUNCT
asir-3636	154	7	define	define	VERB
asir-3636	154	8	𝛾	𝛾	PROPN
asir-3636	154	9	on	on	ADP
asir-3636	154	10	𝔻	𝔻	NOUN
asir-3636	154	11	by	by	ADP
asir-3636	154	12	𝛾(𝑧	𝛾(𝑧	NOUN
asir-3636	154	13	)	)	PUNCT
asir-3636	154	14	=	=	SYM
asir-3636	154	15	𝑧	𝑧	PROPN
asir-3636	154	16	and	and	CCONJ
asir-3636	154	17	let	let	VERB
asir-3636	154	18	𝜋	𝜋	PRON
asir-3636	154	19	∶	∶	VERB
asir-3636	154	20	𝒜αj	𝒜αj	PROPN
asir-3636	154	21	2	2	NUM
asir-3636	154	22	→	→	SYM
asir-3636	154	23	𝒜αj	𝒜αj	PROPN
asir-3636	154	24	2/𝔗	2/𝔗	NUM
asir-3636	154	25	be	be	AUX
asir-3636	154	26	the	the	DET
asir-3636	154	27	canonical	canonical	ADJ
asir-3636	154	28	quotient	quotient	NOUN
asir-3636	154	29	map	map	NOUN
asir-3636	154	30	.	.	PUNCT
asir-3636	155	1	also	also	ADV
asir-3636	155	2	,	,	PUNCT
asir-3636	155	3	let	let	VERB
asir-3636	155	4	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	155	5	2	2	NUM
asir-3636	155	6	∈	∈	NOUN
asir-3636	155	7	𝒥(𝐸𝔗	𝒥(𝐸𝔗	PROPN
asir-3636	155	8	)	)	PUNCT
asir-3636	155	9	be	be	VERB
asir-3636	155	10	such	such	ADJ
asir-3636	155	11	that	that	SCONJ
asir-3636	155	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	155	13	2/𝑈𝔗	2/𝑈𝔗	NUM
asir-3636	155	14	∈	∈	PROPN
asir-3636	155	15	ℋ	ℋ	PROPN
asir-3636	155	16	∞(𝔻	∞(𝔻	PROPN
asir-3636	155	17	)	)	PUNCT
asir-3636	155	18	and	and	CCONJ
asir-3636	155	19	(	(	PUNCT
asir-3636	155	20	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	155	21	2)𝑛	2)𝑛	NUM
asir-3636	155	22	be	be	AUX
asir-3636	155	23	the	the	DET
asir-3636	155	24	sequence	sequence	NOUN
asir-3636	155	25	in	in	ADP
asir-3636	155	26	theorem	theorem	NOUN
asir-3636	155	27	(	(	PUNCT
asir-3636	155	28	1.2	1.2	NUM
asir-3636	155	29	)	)	PUNCT
asir-3636	155	30	associated	associate	VERB
asir-3636	155	31	to	to	ADP
asir-3636	155	32	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	155	33	2	2	NUM
asir-3636	155	34	with	with	ADP
asir-3636	155	35	𝜖	𝜖	PROPN
asir-3636	155	36	≥	≥	NUM
asir-3636	155	37	2	2	NUM
asir-3636	155	38	.	.	PUNCT
asir-3636	156	1	more	more	ADV
asir-3636	156	2	exactly	exactly	ADV
asir-3636	156	3	,	,	PUNCT
asir-3636	156	4	we	we	PRON
asir-3636	156	5	have	have	VERB
asir-3636	156	6	∑	∑	ADV
asir-3636	156	7	(	(	PUNCT
asir-3636	156	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	156	9	2)𝑛	2)𝑛	NUM
asir-3636	156	10	𝑗	𝑗	NOUN
asir-3636	156	11	=	=	PUNCT
asir-3636	156	12	∑	∑	PUNCT
asir-3636	156	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	156	14	2(𝑔𝑗	2(𝑔𝑗	NUM
asir-3636	156	15	2)𝑛	2)𝑛	NUM
asir-3636	156	16	𝑗	𝑗	INTJ
asir-3636	156	17	,	,	PUNCT
asir-3636	156	18	where	where	SCONJ
asir-3636	156	19	∑	∑	PUNCT
asir-3636	156	20	|(𝑔𝑗	|(𝑔𝑗	NUM
asir-3636	156	21	2	2	NUM
asir-3636	156	22	)	)	PUNCT
asir-3636	156	23	𝑛	𝑛	PROPN
asir-3636	156	24	(	(	PUNCT
asir-3636	156	25	𝜉)|𝑗	𝜉)|𝑗	PROPN
asir-3636	156	26	≤	≤	NOUN
asir-3636	156	27	∑	∑	PROPN
asir-3636	156	28	𝑑3(𝜉	𝑑3(𝜉	ADJ
asir-3636	156	29	,	,	PUNCT
asir-3636	156	30	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	156	31	2)𝑗	2)𝑗	NOUN
asir-3636	157	1	≤	≤	ADV
asir-3636	158	1	𝑑3(𝜉	𝑑3(𝜉	PROPN
asir-3636	158	2	,	,	PUNCT
asir-3636	158	3	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	158	4	)	)	PUNCT
asir-3636	158	5	.	.	PUNCT
asir-3636	159	1	define	define	VERB
asir-3636	159	2	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	159	3	applied	apply	VERB
asir-3636	159	4	science	science	NOUN
asir-3636	159	5	and	and	CCONJ
asir-3636	159	6	innovative	innovative	ADJ
asir-3636	159	7	research	research	NOUN
asir-3636	159	8	vol	vol	NOUN
asir-3636	159	9	.	.	PROPN
asir-3636	160	1	5	5	NUM
asir-3636	160	2	,	,	PUNCT
asir-3636	160	3	no	no	INTJ
asir-3636	160	4	.	.	NOUN
asir-3636	160	5	1	1	NUM
asir-3636	160	6	,	,	PUNCT
asir-3636	160	7	2021	2021	NUM
asir-3636	160	8	28	28	NUM
asir-3636	160	9	published	publish	VERB
asir-3636	160	10	by	by	ADP
asir-3636	160	11	scholink	scholink	PROPN
asir-3636	160	12	inc	inc	PROPN
asir-3636	160	13	.	.	PROPN
asir-3636	160	14	∑𝐿𝜆(𝑓𝑗	∑𝐿𝜆(𝑓𝑗	PROPN
asir-3636	160	15	2)(𝑧	2)(𝑧	X
asir-3636	160	16	)	)	PUNCT
asir-3636	160	17	𝑗	𝑗	PRON
asir-3636	160	18	≔	≔	NOUN
asir-3636	160	19	{	{	PUNCT
asir-3636	160	20	∑	∑	ADV
asir-3636	160	21	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	160	22	2(𝑧	2(𝑧	NUM
asir-3636	160	23	)	)	PUNCT
asir-3636	160	24	−	−	ADP
asir-3636	160	25	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	160	26	2(𝜆	2(𝜆	NUM
asir-3636	160	27	)	)	PUNCT
asir-3636	160	28	𝑧	𝑧	DET
asir-3636	160	29	−	−	NOUN
asir-3636	161	1	𝜆	𝜆	ADP
asir-3636	161	2	𝑗	𝑗	INTJ
asir-3636	161	3	if	if	SCONJ
asir-3636	161	4	𝑧	𝑧	PRON
asir-3636	161	5	≠	≠	PROPN
asir-3636	161	6	𝜆	𝜆	PROPN
asir-3636	161	7	,	,	PUNCT
asir-3636	161	8	∑(𝑓𝑗	∑(𝑓𝑗	PROPN
asir-3636	161	9	2)′(𝜆	2)′(𝜆	NUM
asir-3636	161	10	)	)	PUNCT
asir-3636	161	11	𝑗	𝑗	INTJ
asir-3636	161	12	if	if	SCONJ
asir-3636	161	13	𝑧	𝑧	ADP
asir-3636	161	14	=	=	PUNCT
asir-3636	161	15	𝜆.	𝜆.	NOUN
asir-3636	161	16	then	then	ADV
asir-3636	161	17	∑	∑	PUNCT
asir-3636	161	18	𝜋(𝑓𝑗	𝜋(𝑓𝑗	PROPN
asir-3636	161	19	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	161	20	)	)	PUNCT
asir-3636	162	1	−	−	PROPN
asir-3636	162	2	𝜆)−1𝑗	𝜆)−1𝑗	PUNCT
asir-3636	163	1	=	=	PUNCT
asir-3636	163	2	∑	∑	PUNCT
asir-3636	163	3	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	163	4	2(𝜆)(𝜋(𝛾	2(𝜆)(𝜋(𝛾	NUM
asir-3636	163	5	)	)	PUNCT
asir-3636	163	6	−	−	PROPN
asir-3636	163	7	𝜆)−1𝑗	𝜆)−1𝑗	PUNCT
asir-3636	164	1	+	+	CCONJ
asir-3636	164	2	∑	∑	PROPN
asir-3636	164	3	𝜋	𝜋	PROPN
asir-3636	164	4	(	(	PUNCT
asir-3636	164	5	𝐿𝜆(𝑓𝑗	𝐿𝜆(𝑓𝑗	PROPN
asir-3636	164	6	2))𝑗	2))𝑗	NUM
asir-3636	164	7	.	.	PUNCT
asir-3636	165	1	(	(	PUNCT
asir-3636	165	2	3	3	X
asir-3636	165	3	)	)	PUNCT
asir-3636	165	4	it	it	PRON
asir-3636	165	5	is	be	AUX
asir-3636	165	6	clear	clear	ADJ
asir-3636	165	7	that	that	SCONJ
asir-3636	165	8	(	(	PUNCT
asir-3636	165	9	𝜋(𝛾	𝜋(𝛾	NOUN
asir-3636	165	10	)	)	PUNCT
asir-3636	166	1	−	−	PROPN
asir-3636	166	2	𝜆)−1	𝜆)−1	NOUN
asir-3636	166	3	is	be	AUX
asir-3636	166	4	an	an	DET
asir-3636	166	5	analytic	analytic	ADJ
asir-3636	166	6	function	function	NOUN
asir-3636	166	7	on	on	ADP
asir-3636	166	8	ℂ\𝑍𝔗.	ℂ\𝑍𝔗.	NOUN
asir-3636	166	9	note	note	VERB
asir-3636	166	10	that	that	SCONJ
asir-3636	166	11	the	the	DET
asir-3636	166	12	multiplicity	multiplicity	NOUN
asir-3636	166	13	of	of	ADP
asir-3636	166	14	the	the	DET
asir-3636	166	15	pole	pole	NOUN
asir-3636	166	16	𝑧0	𝑧0	PROPN
asir-3636	166	17	∈	∈	PROPN
asir-3636	166	18	𝑍𝔗	𝑍𝔗	PROPN
asir-3636	166	19	∩	∩	NOUN
asir-3636	166	20	𝔻	𝔻	PROPN
asir-3636	166	21	of	of	ADP
asir-3636	166	22	(	(	PUNCT
asir-3636	166	23	𝜋(𝛾	𝜋(𝛾	NOUN
asir-3636	166	24	)	)	PUNCT
asir-3636	166	25	−	−	PROPN
asir-3636	166	26	𝜆	𝜆	NOUN
asir-3636	166	27	)	)	PUNCT
asir-3636	166	28	−1	−1	NOUN
asir-3636	166	29	is	be	AUX
asir-3636	166	30	equal	equal	ADJ
asir-3636	166	31	to	to	ADP
asir-3636	166	32	the	the	DET
asir-3636	166	33	multiplicity	multiplicity	NOUN
asir-3636	166	34	of	of	ADP
asir-3636	166	35	the	the	DET
asir-3636	166	36	zero	zero	NUM
asir-3636	166	37	𝑧0	𝑧0	NOUN
asir-3636	166	38	of	of	ADP
asir-3636	166	39	𝑈𝔗.	𝑈𝔗.	NOUN
asir-3636	166	40	since	since	SCONJ
asir-3636	166	41	𝑈𝔗	𝑈𝔗	PROPN
asir-3636	166	42	divides	divide	VERB
asir-3636	166	43	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	166	44	2	2	NUM
asir-3636	166	45	,	,	PUNCT
asir-3636	166	46	then	then	ADV
asir-3636	166	47	according	accord	VERB
asir-3636	166	48	to	to	ADP
asir-3636	166	49	(	(	PUNCT
asir-3636	166	50	3	3	X
asir-3636	166	51	)	)	PUNCT
asir-3636	166	52	we	we	PRON
asir-3636	166	53	can	can	AUX
asir-3636	166	54	deduce	deduce	VERB
asir-3636	166	55	that	that	SCONJ
asir-3636	166	56	∑	∑	ADP
asir-3636	166	57	𝜋(𝑓𝑗	𝜋(𝑓𝑗	PROPN
asir-3636	166	58	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	166	59	)	)	PUNCT
asir-3636	167	1	−	−	PROPN
asir-3636	167	2	𝜆)−1𝑗	𝜆)−1𝑗	PROPN
asir-3636	167	3	is	be	AUX
asir-3636	167	4	a	a	DET
asir-3636	167	5	series	series	NOUN
asir-3636	167	6	of	of	ADP
asir-3636	167	7	square	square	ADJ
asir-3636	167	8	analytic	analytic	ADJ
asir-3636	167	9	functions	function	NOUN
asir-3636	167	10	on	on	ADP
asir-3636	167	11	ℂ\𝐸𝔗.	ℂ\𝐸𝔗.	ADV
asir-3636	167	12	let	let	VERB
asir-3636	167	13	|𝜆|	|𝜆|	VERB
asir-3636	167	14	>	>	X
asir-3636	167	15	1	1	NUM
asir-3636	167	16	,	,	PUNCT
asir-3636	167	17	we	we	PRON
asir-3636	167	18	have	have	VERB
asir-3636	167	19	∑	∑	PROPN
asir-3636	167	20	‖𝜋(𝑓𝑗	‖𝜋(𝑓𝑗	PROPN
asir-3636	167	21	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	167	22	)	)	PUNCT
asir-3636	168	1	−	−	PROPN
asir-3636	168	2	𝜆)−1‖	𝜆)−1‖	NOUN
asir-3636	168	3	𝒜	𝒜	NOUN
asir-3636	168	4	αj	αj	NOUN
asir-3636	168	5	2	2	NUM
asir-3636	168	6	𝑗	𝑗	NOUN
asir-3636	168	7	≤	≤	NOUN
asir-3636	168	8	∑	∑	PUNCT
asir-3636	168	9	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	168	10	2‖	2‖	PROPN
asir-3636	168	11	𝒜	𝒜	NOUN
asir-3636	168	12	αj	αj	NOUN
asir-3636	168	13	2	2	NUM
asir-3636	168	14	𝑗	𝑗	NOUN
asir-3636	168	15	∑	∑	PROPN
asir-3636	168	16	∑	∑	PROPN
asir-3636	168	17	‖𝛾𝑛‖𝒜	‖𝛾𝑛‖𝒜	PROPN
asir-3636	168	18	αj	αj	ADP
asir-3636	168	19	2	2	NUM
asir-3636	168	20	|𝜆|−𝑛−1𝑗	|𝜆|−𝑛−1𝑗	PROPN
asir-3636	168	21	≤	≤	NOUN
asir-3636	168	22	∑	∑	PUNCT
asir-3636	168	23	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	168	24	2‖	2‖	PROPN
asir-3636	168	25	𝒜	𝒜	NOUN
asir-3636	168	26	αj	αj	NOUN
asir-3636	168	27	2	2	NUM
asir-3636	168	28	𝑗	𝑗	PROPN
asir-3636	168	29	𝐶	𝐶	PROPN
asir-3636	168	30	(	(	PUNCT
asir-3636	168	31	|𝜆|−1	|𝜆|−1	PROPN
asir-3636	168	32	)	)	PUNCT
asir-3636	168	33	3	3	NUM
asir-3636	168	34	2	2	NUM
asir-3636	168	35	∞	∞	NUM
asir-3636	168	36	𝑛=0	𝑛=0	PROPN
asir-3636	168	37	.	.	PUNCT
asir-3636	169	1	(	(	PUNCT
asir-3636	169	2	4	4	X
asir-3636	169	3	)	)	PUNCT
asir-3636	169	4	by	by	ADP
asir-3636	169	5	lemma	lemma	PROPN
asir-3636	169	6	(	(	PUNCT
asir-3636	169	7	3.1	3.1	NUM
asir-3636	169	8	)	)	PUNCT
asir-3636	169	9	,	,	PUNCT
asir-3636	169	10	there	there	PRON
asir-3636	169	11	is	be	VERB
asir-3636	169	12	𝑔𝑗	𝑔𝑗	NUM
asir-3636	169	13	2	2	NUM
asir-3636	169	14	∈	∈	NOUN
asir-3636	169	15	𝔗	𝔗	PRON
asir-3636	169	16	such	such	ADJ
asir-3636	169	17	that	that	DET
asir-3636	169	18	𝐵𝑔𝑗	𝐵𝑔𝑗	PROPN
asir-3636	169	19	2	2	NUM
asir-3636	169	20	=	=	SYM
asir-3636	169	21	𝐵𝔗	𝐵𝔗	PROPN
asir-3636	169	22	.	.	PUNCT
asir-3636	170	1	let	let	VERB
asir-3636	170	2	𝑘	𝑘	X
asir-3636	170	3	=	=	PUNCT
asir-3636	170	4	∑	∑	PUNCT
asir-3636	170	5	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	170	6	2(𝑔𝑗	2(𝑔𝑗	NUM
asir-3636	170	7	2/𝐵𝑔𝑗	2/𝐵𝑔𝑗	NUM
asir-3636	170	8	2)𝑗	2)𝑗	NOUN
asir-3636	170	9	.	.	PUNCT
asir-3636	171	1	then	then	ADV
asir-3636	171	2	,	,	PUNCT
asir-3636	171	3	𝑘	𝑘	PROPN
asir-3636	171	4	=	=	X
asir-3636	171	5	∑	∑	PUNCT
asir-3636	171	6	(	(	PUNCT
asir-3636	171	7	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	171	8	2/𝐵𝔗)𝑔𝑗	2/𝐵𝔗)𝑔𝑗	NUM
asir-3636	171	9	2	2	NUM
asir-3636	171	10	𝑗	𝑗	NOUN
asir-3636	171	11	∈	∈	PROPN
asir-3636	171	12	𝔗	𝔗	PROPN
asir-3636	171	13	and	and	CCONJ
asir-3636	171	14	for	for	ADP
asir-3636	171	15	|𝜆|	|𝜆|	ADP
asir-3636	171	16	<	<	X
asir-3636	171	17	1	1	NUM
asir-3636	171	18	,	,	PUNCT
asir-3636	171	19	we	we	PRON
asir-3636	171	20	have	have	VERB
asir-3636	171	21	𝑘(𝜆)(𝜋(𝛾	𝑘(𝜆)(𝜋(𝛾	PROPN
asir-3636	171	22	)	)	PUNCT
asir-3636	172	1	−	−	PROPN
asir-3636	172	2	𝜆)−1	𝜆)−1	NOUN
asir-3636	172	3	=	=	SYM
asir-3636	172	4	−𝜋(𝐿𝜆(𝑘	−𝜋(𝐿𝜆(𝑘	NOUN
asir-3636	172	5	)	)	PUNCT
asir-3636	172	6	)	)	PUNCT
asir-3636	172	7	.	.	PUNCT
asir-3636	173	1	therefore	therefore	ADV
asir-3636	173	2	∑	∑	PUNCT
asir-3636	173	3	‖𝜋(𝑓𝑗	‖𝜋(𝑓𝑗	PROPN
asir-3636	173	4	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	173	5	)	)	PUNCT
asir-3636	174	1	−	−	PROPN
asir-3636	174	2	𝜆)−1‖	𝜆)−1‖	NOUN
asir-3636	174	3	𝒜	𝒜	NOUN
asir-3636	174	4	αj	αj	NOUN
asir-3636	174	5	2	2	NUM
asir-3636	174	6	𝑗	𝑗	NOUN
asir-3636	174	7	≤	≤	X
asir-3636	174	8	∑	∑	PUNCT
asir-3636	174	9	|𝑓𝑗	|𝑓𝑗	PROPN
asir-3636	174	10	2(𝜆)|‖(𝜋(𝛾	2(𝜆)|‖(𝜋(𝛾	NOUN
asir-3636	174	11	)	)	PUNCT
asir-3636	175	1	−	−	PROPN
asir-3636	175	2	𝜆)−1‖𝒜	𝜆)−1‖𝒜	NOUN
asir-3636	176	1	αj	αj	NOUN
asir-3636	176	2	2𝑗	2𝑗	NOUN
asir-3636	176	3	+	+	CCONJ
asir-3636	176	4	∑	∑	PROPN
asir-3636	176	5	‖𝐿𝜆(𝑓𝑗	‖𝐿𝜆(𝑓𝑗	PROPN
asir-3636	176	6	2)‖	2)‖	PROPN
asir-3636	176	7	𝒜	𝒜	NOUN
asir-3636	176	8	αj	αj	NOUN
asir-3636	176	9	2	2	NUM
asir-3636	176	10	𝑗	𝑗	NOUN
asir-3636	176	11	≤	≤	NOUN
asir-3636	176	12	∑	∑	PUNCT
asir-3636	176	13	‖𝐿𝜆(𝑘)‖𝒜	‖𝐿𝜆(𝑘)‖𝒜	NOUN
asir-3636	176	14	αj	αj	NOUN
asir-3636	176	15	2	2	NUM
asir-3636	176	16	|𝑔𝑗	|𝑔𝑗	NUM
asir-3636	176	17	2/𝐵	2/𝐵	NUM
asir-3636	176	18	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	176	19	2|(𝜆	2|(𝜆	NUM
asir-3636	176	20	)	)	PUNCT
asir-3636	176	21	𝑗	𝑗	PROPN
asir-3636	177	1	+	+	ADJ
asir-3636	177	2	∑	∑	PROPN
asir-3636	177	3	‖𝐿𝜆(𝑓𝑗	‖𝐿𝜆(𝑓𝑗	PROPN
asir-3636	177	4	2)‖	2)‖	PROPN
asir-3636	177	5	𝒜	𝒜	NOUN
asir-3636	177	6	αj	αj	NOUN
asir-3636	177	7	2	2	NUM
asir-3636	177	8	𝑗	𝑗	NOUN
asir-3636	177	9	≤	≤	ADV
asir-3636	177	10	∑	∑	PROPN
asir-3636	177	11	𝐶(𝑓𝑗	𝐶(𝑓𝑗	NOUN
asir-3636	177	12	2,𝑘	2,𝑘	NUM
asir-3636	177	13	)	)	PUNCT
asir-3636	177	14	(	(	PUNCT
asir-3636	177	15	1−|𝜆|)|𝑔𝑗	1−|𝜆|)|𝑔𝑗	NUM
asir-3636	177	16	2/𝐵	2/𝐵	NUM
asir-3636	177	17	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	177	18	2|(𝜆	2|(𝜆	NUM
asir-3636	177	19	)	)	PUNCT
asir-3636	177	20	𝑗	𝑗	PRON
asir-3636	177	21	≤	≤	NUM
asir-3636	177	22	∑	∑	PROPN
asir-3636	177	23	𝐶(𝑓𝑗	𝐶(𝑓𝑗	NOUN
asir-3636	177	24	2	2	NUM
asir-3636	177	25	,	,	PUNCT
asir-3636	177	26	𝑘)𝑒	𝑘)𝑒	X
asir-3636	177	27	𝐶	𝐶	PROPN
asir-3636	177	28	1−|𝜆|	1−|𝜆|	PROPN
asir-3636	177	29	𝑗	𝑗	INTJ
asir-3636	177	30	(	(	PUNCT
asir-3636	177	31	|𝜆|	|𝜆|	ADP
asir-3636	177	32	<	<	X
asir-3636	177	33	1	1	NUM
asir-3636	177	34	)	)	PUNCT
asir-3636	177	35	.	.	PUNCT
asir-3636	178	1	(	(	PUNCT
asir-3636	178	2	5	5	X
asir-3636	178	3	)	)	PUNCT
asir-3636	178	4	we	we	PRON
asir-3636	178	5	use	use	VERB
asir-3636	178	6	(	(	PUNCT
asir-3636	178	7	taylor	taylor	PROPN
asir-3636	178	8	&	&	CCONJ
asir-3636	178	9	williams,1970	williams,1970	PROPN
asir-3636	178	10	,	,	PUNCT
asir-3636	178	11	lemmas	lemma	VERB
asir-3636	178	12	5.8	5.8	NUM
asir-3636	178	13	and	and	CCONJ
asir-3636	178	14	5.9	5.9	NUM
asir-3636	178	15	)	)	PUNCT
asir-3636	178	16	to	to	PART
asir-3636	178	17	deduce	deduce	VERB
asir-3636	178	18	∑‖𝜋(𝑓𝑗	∑‖𝜋(𝑓𝑗	NOUN
asir-3636	178	19	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	178	20	)	)	PUNCT
asir-3636	179	1	−	−	NOUN
asir-3636	179	2	𝜉)−1‖	𝜉)−1‖	VERB
asir-3636	179	3	𝑗	𝑗	PRON
asir-3636	179	4	≤∑	≤∑	PROPN
asir-3636	179	5	𝐶(𝑓𝑗	𝐶(𝑓𝑗	NOUN
asir-3636	179	6	2	2	NUM
asir-3636	179	7	,	,	PUNCT
asir-3636	179	8	𝑘	𝑘	NOUN
asir-3636	179	9	)	)	PUNCT
asir-3636	179	10	𝑑(𝜉	𝑑(𝜉	PROPN
asir-3636	179	11	,	,	PUNCT
asir-3636	179	12	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	179	13	)	)	PUNCT
asir-3636	179	14	3	3	NUM
asir-3636	179	15	𝑗	𝑗	NOUN
asir-3636	179	16	(	(	PUNCT
asir-3636	179	17	1	1	NUM
asir-3636	179	18	≤	≤	NUM
asir-3636	179	19	|𝜉|	|𝜉|	NOUN
asir-3636	179	20	≤	≤	NOUN
asir-3636	179	21	2	2	NUM
asir-3636	179	22	,	,	PUNCT
asir-3636	179	23	𝜉	𝜉	X
asir-3636	179	24	∉	∉	PROPN
asir-3636	179	25	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	179	26	)	)	PUNCT
asir-3636	179	27	.	.	PUNCT
asir-3636	180	1	then	then	ADV
asir-3636	180	2	,	,	PUNCT
asir-3636	180	3	we	we	PRON
asir-3636	180	4	obtain	obtain	VERB
asir-3636	180	5	𝜉	𝜉	ADP
asir-3636	180	6	⟼	⟼	PROPN
asir-3636	180	7	∑	∑	PROPN
asir-3636	180	8	|((𝑔𝑗	|((𝑔𝑗	PROPN
asir-3636	180	9	2)𝑛	2)𝑛	NUM
asir-3636	180	10	)	)	PUNCT
asir-3636	180	11	(	(	PUNCT
asir-3636	180	12	𝜉)|‖𝜋(𝑓𝑗	𝜉)|‖𝜋(𝑓𝑗	NOUN
asir-3636	180	13	2)(𝜋(𝛾	2)(𝜋(𝛾	NUM
asir-3636	180	14	)	)	PUNCT
asir-3636	180	15	−	−	PROPN
asir-3636	181	1	𝜉)−1‖𝑗	𝜉)−1‖𝑗	PROPN
asir-3636	181	2	∈	∈	PROPN
asir-3636	181	3	𝐿∞(𝕋	𝐿∞(𝕋	PROPN
asir-3636	181	4	)	)	PUNCT
asir-3636	181	5	.	.	PUNCT
asir-3636	182	1	with	with	ADP
asir-3636	182	2	a	a	DET
asir-3636	182	3	simple	simple	ADJ
asir-3636	182	4	calculation	calculation	NOUN
asir-3636	182	5	as	as	ADP
asir-3636	182	6	in	in	ADP
asir-3636	182	7	(	(	PUNCT
asir-3636	182	8	esterle	esterle	PROPN
asir-3636	182	9	,	,	PUNCT
asir-3636	182	10	strouse	strouse	NOUN
asir-3636	182	11	,	,	PUNCT
asir-3636	182	12	&	&	CCONJ
asir-3636	182	13	zouakia	zouakia	PROPN
asir-3636	182	14	,	,	PUNCT
asir-3636	182	15	1994	1994	NUM
asir-3636	182	16	,	,	PUNCT
asir-3636	182	17	lemma	lemma	PROPN
asir-3636	182	18	2.4	2.4	NUM
asir-3636	182	19	)	)	PUNCT
asir-3636	182	20	,	,	PUNCT
asir-3636	182	21	we	we	PRON
asir-3636	182	22	can	can	AUX
asir-3636	182	23	deduce	deduce	VERB
asir-3636	182	24	that	that	PRON
asir-3636	182	25	∑𝜋((𝑓𝑗	∑𝜋((𝑓𝑗	PROPN
asir-3636	182	26	2)𝑛	2)𝑛	NUM
asir-3636	182	27	)	)	PUNCT
asir-3636	183	1	𝑗	𝑗	NOUN
asir-3636	183	2	=	=	SYM
asir-3636	183	3	1	1	NUM
asir-3636	183	4	2𝜋𝑖	2𝜋𝑖	ADJ
asir-3636	183	5	∫	∫	PROPN
asir-3636	183	6	∑((𝑔𝑗	∑((𝑔𝑗	NOUN
asir-3636	183	7	2)𝑛	2)𝑛	NUM
asir-3636	183	8	)	)	PUNCT
asir-3636	183	9	(	(	PUNCT
asir-3636	183	10	𝜉)(𝜋(𝛾	𝜉)(𝜋(𝛾	NOUN
asir-3636	183	11	)	)	PUNCT
asir-3636	183	12	−	−	NOUN
asir-3636	183	13	𝜉)−1𝑑𝜉	𝜉)−1𝑑𝜉	NOUN
asir-3636	183	14	𝑗	𝑗	INTJ
asir-3636	183	15	.	.	PUNCT
asir-3636	184	1	𝕋	𝕋	PRON
asir-3636	184	2	denote	denote	VERB
asir-3636	184	3	𝔗𝑈𝔗	𝔗𝑈𝔗	PROPN
asir-3636	184	4	∞	∞	NUM
asir-3636	184	5	(	(	PUNCT
asir-3636	184	6	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	184	7	)	)	PUNCT
asir-3636	184	8	≔	≔	NOUN
asir-3636	184	9	{	{	PUNCT
asir-3636	184	10	ℎ𝑗	ℎ𝑗	PROPN
asir-3636	184	11	2	2	NUM
asir-3636	184	12	∈	∈	PROPN
asir-3636	184	13	𝐴(𝔻	𝐴(𝔻	NOUN
asir-3636	184	14	):	):	PUNCT
asir-3636	184	15	(	(	PUNCT
asir-3636	184	16	ℎ𝑗	ℎ𝑗	DET
asir-3636	184	17	2)∖𝐸𝔗	2)∖𝐸𝔗	NUM
asir-3636	184	18	=	=	SYM
asir-3636	184	19	0	0	NUM
asir-3636	184	20	and	and	CCONJ
asir-3636	184	21	ℎ𝑗	ℎ𝑗	DET
asir-3636	184	22	2	2	NUM
asir-3636	184	23	∕	∕	NOUN
asir-3636	184	24	𝑈𝔗	𝑈𝔗	PROPN
asir-3636	184	25	∈	∈	PROPN
asir-3636	184	26	𝐴(𝔻	𝐴(𝔻	PROPN
asir-3636	184	27	)	)	PUNCT
asir-3636	184	28	}	}	PUNCT
asir-3636	184	29	.	.	PUNCT
asir-3636	185	1	from	from	ADP
asir-3636	185	2	(	(	PUNCT
asir-3636	185	3	hoffman	hoffman	NOUN
asir-3636	185	4	,	,	PUNCT
asir-3636	185	5	1988	1988	NUM
asir-3636	185	6	,	,	PUNCT
asir-3636	185	7	p.	p.	NOUN
asir-3636	185	8	81	81	NUM
asir-3636	185	9	)	)	PUNCT
asir-3636	185	10	,	,	PUNCT
asir-3636	185	11	we	we	PRON
asir-3636	185	12	know	know	VERB
asir-3636	185	13	that	that	SCONJ
asir-3636	185	14	𝔗𝑈𝔗	𝔗𝑈𝔗	PROPN
asir-3636	185	15	∞	∞	NUM
asir-3636	185	16	(	(	PUNCT
asir-3636	185	17	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	185	18	)	)	PUNCT
asir-3636	185	19	has	have	VERB
asir-3636	185	20	an	an	DET
asir-3636	185	21	approximate	approximate	ADJ
asir-3636	185	22	identity	identity	NOUN
asir-3636	185	23	(	(	PUNCT
asir-3636	185	24	𝑒1+𝜖)𝜖≥0	𝑒1+𝜖)𝜖≥0	PROPN
asir-3636	185	25	∈	∈	PROPN
asir-3636	185	26	𝔗	𝔗	PRON
asir-3636	185	27	such	such	ADJ
asir-3636	185	28	that	that	PRON
asir-3636	185	29	‖𝑒1+𝜖‖∞	‖𝑒1+𝜖‖∞	PUNCT
asir-3636	185	30	≤	≤	NUM
asir-3636	185	31	1	1	NUM
asir-3636	185	32	.	.	PUNCT
asir-3636	186	1	𝔗	𝔗	PROPN
asir-3636	186	2	is	be	AUX
asir-3636	186	3	dense	dense	ADJ
asir-3636	186	4	in	in	ADP
asir-3636	186	5	𝔗𝑈𝔗	𝔗𝑈𝔗	PROPN
asir-3636	186	6	∞	∞	NUM
asir-3636	186	7	(	(	PUNCT
asir-3636	186	8	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	186	9	)	)	PUNCT
asir-3636	186	10	with	with	ADP
asir-3636	186	11	respect	respect	NOUN
asir-3636	186	12	to	to	ADP
asir-3636	186	13	the	the	DET
asir-3636	186	14	sup	sup	NOUN
asir-3636	186	15	norm	norm	NOUN
asir-3636	186	16	‖∙‖∞	‖∙‖∞	NOUN
asir-3636	186	17	,	,	PUNCT
asir-3636	186	18	so	so	SCONJ
asir-3636	186	19	there	there	PRON
asir-3636	186	20	exists	exist	VERB
asir-3636	186	21	(	(	PUNCT
asir-3636	186	22	𝑢1+𝜖)𝜖≥0	𝑢1+𝜖)𝜖≥0	NOUN
asir-3636	186	23	∈	∈	PROPN
asir-3636	186	24	𝔗	𝔗	NOUN
asir-3636	186	25	with	with	ADP
asir-3636	186	26	‖𝑢1+𝜖‖∞	‖𝑢1+𝜖‖∞	NOUN
asir-3636	186	27	≤	≤	ADJ
asir-3636	186	28	1	1	NUM
asir-3636	186	29	and	and	CCONJ
asir-3636	186	30	lim1+𝜖→∞𝑢1+𝜖(𝜉	lim1+𝜖→∞𝑢1+𝜖(𝜉	PROPN
asir-3636	186	31	)	)	PUNCT
asir-3636	186	32	=	=	SYM
asir-3636	186	33	1	1	NUM
asir-3636	186	34	for	for	ADP
asir-3636	186	35	𝜉	𝜉	X
asir-3636	186	36	∈	∈	NOUN
asir-3636	186	37	𝕋\𝐸𝔗.	𝕋\𝐸𝔗.	PROPN
asir-3636	186	38	therefore	therefore	ADV
asir-3636	186	39	∑	∑	PUNCT
asir-3636	186	40	𝜋((𝑓𝑗	𝜋((𝑓𝑗	PROPN
asir-3636	186	41	2)𝑛	2)𝑛	NUM
asir-3636	186	42	)	)	PUNCT
asir-3636	186	43	𝑗	𝑗	PROPN
asir-3636	187	1	=	=	PUNCT
asir-3636	187	2	∑	∑	PUNCT
asir-3636	187	3	𝜋	𝜋	X
asir-3636	187	4	(	(	PUNCT
asir-3636	187	5	(	(	PUNCT
asir-3636	187	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	187	7	2)𝑛	2)𝑛	NUM
asir-3636	187	8	−	−	PROPN
asir-3636	187	9	(	(	PUNCT
asir-3636	187	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	187	11	2)𝑛	2)𝑛	NUM
asir-3636	187	12	𝑢1+𝜖)𝑗	𝑢1+𝜖)𝑗	NOUN
asir-3636	187	13	→	→	SYM
asir-3636	187	14	0	0	PUNCT
asir-3636	187	15	as	as	SCONJ
asir-3636	187	16	𝜖	𝜖	PROPN
asir-3636	187	17	→	→	SYM
asir-3636	187	18	∞.	∞.	PROPN
asir-3636	187	19	then	then	ADV
asir-3636	187	20	(	(	PUNCT
asir-3636	187	21	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	187	22	2)𝑛	2)𝑛	NUM
asir-3636	187	23	∈	∈	PROPN
asir-3636	187	24	𝔗	𝔗	PROPN
asir-3636	187	25	and	and	CCONJ
asir-3636	187	26	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	187	27	2	2	NUM
asir-3636	187	28	∈	∈	PROPN
asir-3636	187	29	𝔗.	𝔗.	NOUN
asir-3636	187	30	note	note	NOUN
asir-3636	187	31	that	that	SCONJ
asir-3636	187	32	:	:	PUNCT
asir-3636	187	33	if	if	SCONJ
asir-3636	187	34	lim𝑛→∞∑	lim𝑛→∞∑	VERB
asir-3636	187	35	|(𝑔𝑗	|(𝑔𝑗	NUM
asir-3636	187	36	2)𝑛	2)𝑛	NUM
asir-3636	187	37	(	(	PUNCT
asir-3636	187	38	𝜉)|𝑗	𝜉)|𝑗	VERB
asir-3636	187	39	=	=	SYM
asir-3636	187	40	∑	∑	PROPN
asir-3636	187	41	|(𝑔𝑗	|(𝑔𝑗	NUM
asir-3636	187	42	2)|	2)|	NUM
asir-3636	187	43	|𝜉|𝑗	|𝜉|𝑗	CCONJ
asir-3636	187	44	then	then	ADV
asir-3636	187	45	,	,	PUNCT
asir-3636	187	46	∑	∑	ADV
asir-3636	187	47	𝑐𝑑1+𝜖(𝜉	𝑐𝑑1+𝜖(𝜉	NOUN
asir-3636	187	48	,	,	PUNCT
asir-3636	187	49	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	187	50	2	2	NUM
asir-3636	187	51	)	)	PUNCT
asir-3636	187	52	𝑗	𝑗	NOUN
asir-3636	187	53	=	=	PUNCT
asir-3636	187	54	∑	∑	PUNCT
asir-3636	187	55	𝑑3(𝜉	𝑑3(𝜉	ADJ
asir-3636	187	56	,	,	PUNCT
asir-3636	187	57	𝐸𝑓𝑗	𝐸𝑓𝑗	NOUN
asir-3636	187	58	2	2	NUM
asir-3636	187	59	)	)	PUNCT
asir-3636	187	60	𝑗	𝑗	NOUN
asir-3636	187	61	.	.	PUNCT
asir-3636	188	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-3636	188	2	applied	apply	VERB
asir-3636	188	3	science	science	NOUN
asir-3636	188	4	and	and	CCONJ
asir-3636	188	5	innovative	innovative	ADJ
asir-3636	188	6	research	research	NOUN
asir-3636	188	7	vol	vol	NOUN
asir-3636	188	8	.	.	PROPN
asir-3636	189	1	5	5	NUM
asir-3636	189	2	,	,	PUNCT
asir-3636	189	3	no	no	INTJ
asir-3636	189	4	.	.	NOUN
asir-3636	189	5	1	1	NUM
asir-3636	189	6	,	,	PUNCT
asir-3636	189	7	2021	2021	NUM
asir-3636	189	8	29	29	NUM
asir-3636	189	9	published	publish	VERB
asir-3636	189	10	by	by	ADP
asir-3636	189	11	scholink	scholink	PROPN
asir-3636	189	12	inc	inc	PROPN
asir-3636	189	13	.	.	PROPN
asir-3636	189	14	4	4	NUM
asir-3636	189	15	.	.	X
asir-3636	190	1	proof	proof	NOUN
asir-3636	190	2	of	of	ADP
asir-3636	190	3	theorem	theorem	NOUN
asir-3636	190	4	(	(	PUNCT
asir-3636	190	5	2.1	2.1	NUM
asir-3636	190	6	)	)	PUNCT
asir-3636	190	7	the	the	DET
asir-3636	190	8	proof	proof	NOUN
asir-3636	190	9	of	of	ADP
asir-3636	190	10	theorem	theorem	NOUN
asir-3636	190	11	(	(	PUNCT
asir-3636	190	12	2.1	2.1	NUM
asir-3636	190	13	)	)	PUNCT
asir-3636	190	14	is	be	AUX
asir-3636	190	15	based	base	VERB
asir-3636	190	16	on	on	ADP
asir-3636	190	17	a	a	DET
asir-3636	190	18	series	series	NOUN
asir-3636	190	19	of	of	ADP
asir-3636	190	20	lemmas	lemmas	PROPN
asir-3636	190	21	.	.	PUNCT
asir-3636	191	1	in	in	ADP
asir-3636	191	2	what	what	PRON
asir-3636	191	3	follows	follow	VERB
asir-3636	191	4	,	,	PUNCT
asir-3636	191	5	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	191	6	will	will	AUX
asir-3636	191	7	denote	denote	VERB
asir-3636	191	8	a	a	DET
asir-3636	191	9	positive	positive	ADJ
asir-3636	191	10	number	number	NOUN
asir-3636	191	11	that	that	PRON
asir-3636	191	12	depends	depend	VERB
asir-3636	191	13	only	only	ADV
asir-3636	191	14	on	on	ADP
asir-3636	191	15	1	1	NUM
asir-3636	191	16	+	+	CCONJ
asir-3636	191	17	𝜖	𝜖	PROPN
asir-3636	191	18	,	,	PUNCT
asir-3636	191	19	not	not	PART
asir-3636	191	20	necessarily	necessarily	ADV
asir-3636	191	21	the	the	DET
asir-3636	191	22	same	same	ADJ
asir-3636	191	23	at	at	ADP
asir-3636	191	24	each	each	DET
asir-3636	191	25	occurrence	occurrence	NOUN
asir-3636	191	26	.	.	PUNCT
asir-3636	192	1	for	for	ADP
asir-3636	192	2	an	an	DET
asir-3636	192	3	open	open	ADJ
asir-3636	192	4	subset	subset	ADJ
asir-3636	192	5	δ	δ	PROPN
asir-3636	192	6	of	of	ADP
asir-3636	192	7	𝔻	𝔻	PROPN
asir-3636	192	8	,	,	PUNCT
asir-3636	192	9	we	we	PRON
asir-3636	192	10	put	put	VERB
asir-3636	192	11	∑‖((ℎ𝑗	∑‖((ℎ𝑗	VERB
asir-3636	192	12	2)′‖	2)′‖	NUM
asir-3636	192	13	𝐿2(δ	𝐿2(δ	CCONJ
asir-3636	192	14	)	)	PUNCT
asir-3636	192	15	2	2	NUM
asir-3636	192	16	𝑗	𝑗	NOUN
asir-3636	192	17	≔	≔	NOUN
asir-3636	192	18	∫∑|(𝑓𝑗	∫∑|(𝑓𝑗	PUNCT
asir-3636	192	19	2)′(𝑧)|	2)′(𝑧)|	NUM
asir-3636	192	20	2	2	NUM
asir-3636	192	21	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
asir-3636	192	22	)	)	PUNCT
asir-3636	192	23	𝑗	𝑗	INTJ
asir-3636	192	24	.	.	PUNCT
asir-3636	193	1	δ	δ	X
asir-3636	193	2	we	we	PRON
asir-3636	193	3	begin	begin	VERB
asir-3636	193	4	with	with	ADP
asir-3636	193	5	the	the	DET
asir-3636	193	6	following	follow	VERB
asir-3636	193	7	key	key	ADJ
asir-3636	193	8	lemma	lemma	PROPN
asir-3636	193	9	(	(	PUNCT
asir-3636	193	10	see	see	VERB
asir-3636	193	11	brahim	brahim	PROPN
asir-3636	193	12	bouya	bouya	PROPN
asir-3636	193	13	,	,	PUNCT
asir-3636	193	14	2008	2008	NUM
asir-3636	193	15	)	)	PUNCT
asir-3636	193	16	.	.	PUNCT
asir-3636	194	1	lemma	lemma	PROPN
asir-3636	194	2	(	(	PUNCT
asir-3636	194	3	4.1	4.1	NUM
asir-3636	194	4	):	):	PUNCT
asir-3636	194	5	let	let	VERB
asir-3636	194	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	194	7	2	2	NUM
asir-3636	194	8	∈	∈	NOUN
asir-3636	194	9	𝒜𝑓𝑗	𝒜𝑓𝑗	PROPN
asir-3636	194	10	2	2	NUM
asir-3636	194	11	be	be	AUX
asir-3636	194	12	such	such	ADJ
asir-3636	194	13	that	that	SCONJ
asir-3636	194	14	∑	∑	PROPN
asir-3636	194	15	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	194	16	2‖	2‖	PROPN
asir-3636	194	17	𝒜	𝒜	NOUN
asir-3636	194	18	αj	αj	NOUN
asir-3636	194	19	2	2	NUM
asir-3636	194	20	𝑗	𝑗	NOUN
asir-3636	194	21	≤	≤	NUM
asir-3636	194	22	1	1	NUM
asir-3636	194	23	and	and	CCONJ
asir-3636	194	24	let	let	VERB
asir-3636	194	25	𝜖	𝜖	PROPN
asir-3636	194	26	>	>	X
asir-3636	194	27	0	0	PUNCT
asir-3636	194	28	be	be	AUX
asir-3636	194	29	given	give	VERB
asir-3636	194	30	.	.	PUNCT
asir-3636	195	1	then	then	ADV
asir-3636	195	2	∫∑	∫∑	VERB
asir-3636	195	3	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	195	4	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	195	5	2	2	NUM
asir-3636	195	6	)	)	PUNCT
asir-3636	195	7	|	|	ADV
asir-3636	195	8	2(1+𝜖	2(1+𝜖	NUM
asir-3636	195	9	)	)	PUNCT
asir-3636	195	10	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	195	11	2	2	NUM
asir-3636	195	12	)	)	PUNCT
asir-3636	195	13	𝑗	𝑗	PROPN
asir-3636	195	14	𝛾	𝛾	ADP
asir-3636	195	15	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	195	16	≤∑𝐶1+𝜖‖(𝑓𝑗	≤∑𝐶1+𝜖‖(𝑓𝑗	PUNCT
asir-3636	196	1	2)′‖	2)′‖	NUM
asir-3636	196	2	𝐿2(γ	𝐿2(γ	NUM
asir-3636	196	3	)	)	PUNCT
asir-3636	196	4	2	2	NUM
asir-3636	196	5	𝑗	𝑗	NOUN
asir-3636	196	6	,	,	PUNCT
asir-3636	196	7	where	where	SCONJ
asir-3636	196	8	𝑎	𝑎	X
asir-3636	196	9	,	,	PUNCT
asir-3636	196	10	𝑎	𝑎	X
asir-3636	196	11	+	+	NUM
asir-3636	196	12	𝜖	𝜖	PROPN
asir-3636	196	13	∈	∈	PROPN
asir-3636	196	14	𝐸𝔗	𝐸𝔗	PROPN
asir-3636	196	15	,	,	PUNCT
asir-3636	196	16	𝛾	𝛾	NOUN
asir-3636	196	17	=	=	SYM
asir-3636	196	18	(	(	PUNCT
asir-3636	196	19	𝑎	𝑎	X
asir-3636	196	20	,	,	PUNCT
asir-3636	196	21	𝑎	𝑎	PRON
asir-3636	196	22	+	+	X
asir-3636	196	23	𝜖	𝜖	X
asir-3636	196	24	)	)	PUNCT
asir-3636	196	25	⊂	⊂	PROPN
asir-3636	196	26	𝕋\𝐸𝑓𝑗	𝕋\𝐸𝑓𝑗	PROPN
asir-3636	196	27	2	2	NUM
asir-3636	196	28	,	,	PUNCT
asir-3636	196	29	𝑑(𝑧	𝑑(𝑧	ADJ
asir-3636	196	30	)	)	PUNCT
asir-3636	196	31	∶=	∶=	NUM
asir-3636	196	32	min{|𝑧	min{|𝑧	NOUN
asir-3636	196	33	−	−	PROPN
asir-3636	196	34	𝑎|	𝑎|	PROPN
asir-3636	196	35	,	,	PUNCT
asir-3636	196	36	|𝑧	|𝑧	NOUN
asir-3636	196	37	−	−	PROPN
asir-3636	196	38	(	(	PUNCT
asir-3636	196	39	𝑎	𝑎	PROPN
asir-3636	196	40	+	+	X
asir-3636	196	41	𝜖)|	𝜖)|	NOUN
asir-3636	196	42	}	}	PUNCT
asir-3636	196	43	and	and	CCONJ
asir-3636	196	44	∆𝛾≔	∆𝛾≔	NOUN
asir-3636	196	45	{	{	PUNCT
asir-3636	196	46	𝑧	𝑧	PROPN
asir-3636	196	47	∈	∈	PROPN
asir-3636	196	48	𝐷	𝐷	PROPN
asir-3636	196	49	:	:	PUNCT
asir-3636	196	50	𝑧/|𝑧|	𝑧/|𝑧|	ADP
asir-3636	196	51	∈	∈	PROPN
asir-3636	196	52	𝛾	𝛾	NOUN
asir-3636	196	53	}	}	PUNCT
asir-3636	196	54	.	.	PUNCT
asir-3636	197	1	proof	proof	NOUN
asir-3636	197	2	:	:	PUNCT
asir-3636	197	3	let	let	VERB
asir-3636	197	4	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	197	5	2	2	NUM
asir-3636	197	6	∈	∈	NOUN
asir-3636	197	7	𝛾	𝛾	NOUN
asir-3636	197	8	and	and	CCONJ
asir-3636	197	9	define	define	VERB
asir-3636	197	10	𝑧𝑡2	𝑧𝑡2	NOUN
asir-3636	197	11	∶=	∶=	NUM
asir-3636	197	12	(	(	PUNCT
asir-3636	197	13	1	1	NUM
asir-3636	197	14	−	−	PROPN
asir-3636	197	15	𝑑(𝑒	𝑑(𝑒	NOUN
asir-3636	197	16	𝑖𝑡2))𝑒𝑖𝑡	𝑖𝑡2))𝑒𝑖𝑡	X
asir-3636	197	17	2	2	NUM
asir-3636	197	18	.	.	PUNCT
asir-3636	198	1	since	since	SCONJ
asir-3636	198	2	|𝛾|	|𝛾|	NOUN
asir-3636	198	3	<	<	X
asir-3636	198	4	1/2	1/2	NUM
asir-3636	198	5	,	,	PUNCT
asir-3636	198	6	we	we	PRON
asir-3636	198	7	obtain	obtain	VERB
asir-3636	198	8	|𝑧𝑡2|	|𝑧𝑡2|	PUNCT
asir-3636	198	9	>	>	SYM
asir-3636	198	10	1	1	NUM
asir-3636	198	11	2	2	NUM
asir-3636	198	12	.	.	PUNCT
asir-3636	199	1	we	we	PRON
asir-3636	199	2	have	have	VERB
asir-3636	199	3	∑	∑	DET
asir-3636	199	4	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	199	5	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	199	6	2	2	NUM
asir-3636	199	7	)	)	PUNCT
asir-3636	199	8	|2(1+𝜖)𝑗	|2(1+𝜖)𝑗	ADP
asir-3636	199	9	≤	≤	NOUN
asir-3636	199	10	∑	∑	PUNCT
asir-3636	199	11	22𝜖+1(|𝑓𝑗	22𝜖+1(|𝑓𝑗	PROPN
asir-3636	199	12	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	199	13	2	2	NUM
asir-3636	199	14	)	)	PUNCT
asir-3636	199	15	−	−	ADP
asir-3636	199	16	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	199	17	2(𝑧𝑡2)|	2(𝑧𝑡2)|	NOUN
asir-3636	199	18	2(1+𝜖	2(1+𝜖	NUM
asir-3636	199	19	)	)	PUNCT
asir-3636	200	1	+	+	CCONJ
asir-3636	200	2	|𝑓𝑗	|𝑓𝑗	PROPN
asir-3636	200	3	2(𝑧𝑡2)|	2(𝑧𝑡2)|	NUM
asir-3636	200	4	2(1+𝜖))𝑗	2(1+𝜖))𝑗	NUM
asir-3636	200	5	.	.	PUNCT
asir-3636	201	1	(	(	PUNCT
asir-3636	201	2	6	6	NUM
asir-3636	201	3	)	)	PUNCT
asir-3636	201	4	by	by	ADP
asir-3636	201	5	holder	holder	NOUN
asir-3636	201	6	’s	’s	PART
asir-3636	201	7	inequality	inequality	NOUN
asir-3636	201	8	combined	combine	VERB
asir-3636	201	9	with	with	ADP
asir-3636	201	10	the	the	DET
asir-3636	201	11	fact	fact	NOUN
asir-3636	201	12	that	that	SCONJ
asir-3636	201	13	∑	∑	PROPN
asir-3636	201	14	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	201	15	2‖	2‖	PROPN
asir-3636	201	16	∞𝑗	∞𝑗	NUM
asir-3636	201	17	≤	≤	NUM
asir-3636	201	18	∑	∑	PUNCT
asir-3636	201	19	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	201	20	2‖	2‖	PROPN
asir-3636	202	1	𝒜	𝒜	NOUN
asir-3636	202	2	αj	αj	NOUN
asir-3636	202	3	2	2	NUM
asir-3636	202	4	𝑗	𝑗	PROPN
asir-3636	202	5	≤	≤	NUM
asir-3636	202	6	1	1	NUM
asir-3636	202	7	,	,	PUNCT
asir-3636	202	8	we	we	PRON
asir-3636	202	9	get	get	VERB
asir-3636	202	10	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	202	11	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	ADJ
asir-3636	202	12	2	2	NUM
asir-3636	202	13	)	)	PUNCT
asir-3636	202	14	−	−	ADP
asir-3636	202	15	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	202	16	2(𝑧𝑡2)|	2(𝑧𝑡2)|	NOUN
asir-3636	202	17	2(1+𝜖	2(1+𝜖	NUM
asir-3636	202	18	)	)	PUNCT
asir-3636	202	19	𝑗	𝑗	NOUN
asir-3636	203	1	=	=	VERB
asir-3636	203	2	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	203	3	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	203	4	2	2	X
asir-3636	203	5	)	)	PUNCT
asir-3636	203	6	−	−	ADP
asir-3636	203	7	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	203	8	2(𝑧𝑡2)|	2(𝑧𝑡2)|	NUM
asir-3636	203	9	2𝜖|𝑓𝑗	2𝜖|𝑓𝑗	NUM
asir-3636	203	10	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	203	11	2	2	NUM
asir-3636	203	12	)	)	PUNCT
asir-3636	203	13	−	−	ADP
asir-3636	203	14	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	203	15	2(𝑧𝑡2)|	2(𝑧𝑡2)|	NUM
asir-3636	203	16	2	2	NUM
asir-3636	203	17	𝑗	𝑗	PROPN
asir-3636	203	18	≤	≤	PUNCT
asir-3636	203	19	22𝜖(1	22𝜖(1	NUM
asir-3636	203	20	−	−	PROPN
asir-3636	203	21	|𝑧𝑡2|)∫	|𝑧𝑡2|)∫	NOUN
asir-3636	203	22	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	204	1	2)′((1	2)′((1	NUM
asir-3636	204	2	−	−	PROPN
asir-3636	204	3	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	204	4	2	2	NUM
asir-3636	204	5	)	)	PUNCT
asir-3636	204	6	|	|	ADV
asir-3636	204	7	2	2	NUM
asir-3636	204	8	𝑗	𝑗	NOUN
asir-3636	204	9	1	1	NUM
asir-3636	204	10	|𝑧𝑡2|	|𝑧𝑡2|	PUNCT
asir-3636	204	11	(	(	PUNCT
asir-3636	204	12	1	1	NUM
asir-3636	204	13	−	−	NOUN
asir-3636	204	14	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	204	15	−	−	NUM
asir-3636	204	16	𝜖	𝜖	NOUN
asir-3636	204	17	)	)	PUNCT
asir-3636	204	18	≤	≤	NOUN
asir-3636	204	19	22𝜖+1𝑑(𝑒𝑖𝑡	22𝜖+1𝑑(𝑒𝑖𝑡	NUM
asir-3636	204	20	2	2	NUM
asir-3636	204	21	)	)	PUNCT
asir-3636	204	22	∫	∫	PROPN
asir-3636	204	23	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	205	1	2)′((1	2)′((1	NUM
asir-3636	205	2	−	−	PROPN
asir-3636	205	3	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	205	4	2	2	NUM
asir-3636	205	5	)	)	PUNCT
asir-3636	206	1	|	|	ADV
asir-3636	206	2	𝑗	𝑗	INTJ
asir-3636	206	3	21	21	NUM
asir-3636	206	4	0	0	NUM
asir-3636	206	5	(	(	PUNCT
asir-3636	206	6	1	1	NUM
asir-3636	206	7	−	−	NOUN
asir-3636	206	8	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	206	9	−	−	PROPN
asir-3636	206	10	𝜖	𝜖	NOUN
asir-3636	206	11	)	)	PUNCT
asir-3636	206	12	.	.	PUNCT
asir-3636	207	1	hence	hence	ADV
asir-3636	207	2	∫	∫	PROPN
asir-3636	207	3	∑	∑	PROPN
asir-3636	207	4	|𝑓𝑗	|𝑓𝑗	PROPN
asir-3636	207	5	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	NUM
asir-3636	207	6	2	2	NUM
asir-3636	207	7	)	)	PUNCT
asir-3636	207	8	−𝑓𝑗	−𝑓𝑗	VERB
asir-3636	207	9	2(𝑧	2(𝑧	NUM
asir-3636	207	10	𝑡2	𝑡2	NOUN
asir-3636	207	11	)	)	PUNCT
asir-3636	207	12	|	|	ADV
asir-3636	207	13	2(1+𝜖	2(1+𝜖	NUM
asir-3636	207	14	)	)	PUNCT
asir-3636	207	15	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	207	16	2	2	NUM
asir-3636	207	17	)	)	PUNCT
asir-3636	207	18	𝑗	𝑗	PRON
asir-3636	207	19	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	207	20	≤	≤	NOUN
asir-3636	207	21	𝛾	𝛾	ADP
asir-3636	207	22	2(2𝜖+1	2(2𝜖+1	NUM
asir-3636	207	23	)	)	PUNCT
asir-3636	208	1	∫	∫	PROPN
asir-3636	208	2	∫	∫	PROPN
asir-3636	208	3	∑	∑	PROPN
asir-3636	208	4	|(𝑓𝑗	|(𝑓𝑗	PROPN
asir-3636	208	5	2)′(𝑟𝑒𝑖𝑡	2)′(𝑟𝑒𝑖𝑡	NUM
asir-3636	208	6	2	2	NUM
asir-3636	208	7	)	)	PUNCT
asir-3636	208	8	|𝑗	|𝑗	NOUN
asir-3636	208	9	2	2	NUM
asir-3636	208	10	(	(	PUNCT
asir-3636	208	11	1	1	NUM
asir-3636	208	12	−	−	NOUN
asir-3636	208	13	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	208	14	−	−	PROPN
asir-3636	208	15	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	208	16	≤	≤	ADV
asir-3636	208	17	1	1	NUM
asir-3636	208	18	0	0	NUM
asir-3636	208	19	𝛾	𝛾	ADP
asir-3636	208	20	∑	∑	PROPN
asir-3636	208	21	2(2𝜖+1)𝜋‖(𝑓𝑗	2(2𝜖+1)𝜋‖(𝑓𝑗	NUM
asir-3636	208	22	2)′‖	2)′‖	NUM
asir-3636	208	23	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	208	24	)	)	PUNCT
asir-3636	208	25	.	.	PUNCT
asir-3636	209	1	2	2	NUM
asir-3636	209	2	𝑗	𝑗	NOUN
asir-3636	209	3	(	(	PUNCT
asir-3636	209	4	7	7	NUM
asir-3636	209	5	)	)	PUNCT
asir-3636	209	6	since	since	SCONJ
asir-3636	209	7	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	209	8	2	2	NUM
asir-3636	209	9	)	)	PUNCT
asir-3636	209	10	≤	≤	NUM
asir-3636	209	11	1/2	1/2	NUM
asir-3636	209	12	,	,	PUNCT
asir-3636	209	13	we	we	PRON
asir-3636	209	14	obtain	obtain	VERB
asir-3636	209	15	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	209	16	2	2	NUM
asir-3636	209	17	)	)	PUNCT
asir-3636	209	18	√2	√2	NOUN
asir-3636	209	19	≤	≤	NUM
asir-3636	209	20	𝑑(𝑧𝑡2	𝑑(𝑧𝑡2	NOUN
asir-3636	209	21	)	)	PUNCT
asir-3636	209	22	≤	≤	NOUN
asir-3636	209	23	√2𝑑(𝑒	√2𝑑(𝑒	NUM
asir-3636	209	24	𝑖𝑡2	𝑖𝑡2	NOUN
asir-3636	209	25	)	)	PUNCT
asir-3636	209	26	.	.	PUNCT
asir-3636	210	1	put	put	VERB
asir-3636	210	2	𝑑(𝑧𝑡2	𝑑(𝑧𝑡2	NOUN
asir-3636	210	3	)	)	PUNCT
asir-3636	211	1	=	=	PUNCT
asir-3636	211	2	|𝑧𝑡2	|𝑧𝑡2	PART
asir-3636	212	1	−	−	PROPN
asir-3636	212	2	𝜉|	𝜉|	PROPN
asir-3636	212	3	and	and	CCONJ
asir-3636	212	4	note	note	VERB
asir-3636	212	5	that	that	SCONJ
asir-3636	212	6	either	either	CCONJ
asir-3636	212	7	𝜉	𝜉	X
asir-3636	212	8	=	=	SYM
asir-3636	212	9	𝑎	𝑎	PROPN
asir-3636	212	10	or	or	CCONJ
asir-3636	212	11	𝜉	𝜉	X
asir-3636	212	12	=	=	SYM
asir-3636	212	13	𝑎	𝑎	PROPN
asir-3636	212	14	+	+	X
asir-3636	212	15	𝜖.	𝜖.	NOUN
asir-3636	212	16	let	let	VERB
asir-3636	212	17	𝑧𝑡2(𝑢	𝑧𝑡2(𝑢	PROPN
asir-3636	212	18	)	)	PUNCT
asir-3636	213	1	=	=	PUNCT
asir-3636	214	1	(	(	PUNCT
asir-3636	214	2	1	1	NUM
asir-3636	214	3	−	−	NOUN
asir-3636	214	4	𝑢)𝑧𝑡2	𝑢)𝑧𝑡2	NOUN
asir-3636	214	5	+	+	CCONJ
asir-3636	214	6	𝑢𝜉	𝑢𝜉	NOUN
asir-3636	214	7	(	(	PUNCT
asir-3636	214	8	0	0	NUM
asir-3636	214	9	≤	≤	NUM
asir-3636	214	10	𝑢	𝑢	PRON
asir-3636	214	11	≤	≤	NUM
asir-3636	214	12	1	1	NUM
asir-3636	214	13	)	)	PUNCT
asir-3636	214	14	.	.	PUNCT
asir-3636	215	1	with	with	ADP
asir-3636	215	2	a	a	DET
asir-3636	215	3	simple	simple	ADJ
asir-3636	215	4	calculation	calculation	NOUN
asir-3636	215	5	,	,	PUNCT
asir-3636	215	6	we	we	PRON
asir-3636	215	7	can	can	AUX
asir-3636	215	8	prove	prove	VERB
asir-3636	215	9	that	that	SCONJ
asir-3636	215	10	for	for	ADP
asir-3636	215	11	all	all	DET
asir-3636	215	12	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	215	13	2	2	NUM
asir-3636	215	14	∈	∈	NOUN
asir-3636	215	15	𝛾	𝛾	NOUN
asir-3636	215	16	and	and	CCONJ
asir-3636	215	17	for	for	ADP
asir-3636	215	18	all	all	DET
asir-3636	215	19	𝑢	𝑢	NOUN
asir-3636	215	20	,	,	PUNCT
asir-3636	215	21	0	0	NUM
asir-3636	215	22	≤	≤	NUM
asir-3636	215	23	𝑢	𝑢	PRON
asir-3636	215	24	≤	≤	NUM
asir-3636	215	25	1	1	NUM
asir-3636	215	26	,	,	PUNCT
asir-3636	215	27	we	we	PRON
asir-3636	215	28	have	have	VERB
asir-3636	215	29	|𝑧𝑡2(𝑢	|𝑧𝑡2(𝑢	PROPN
asir-3636	215	30	)	)	PUNCT
asir-3636	215	31	−	−	PROPN
asir-3636	216	1	𝑤|	𝑤|	PROPN
asir-3636	216	2	>	>	X
asir-3636	216	3	1	1	NUM
asir-3636	216	4	2	2	NUM
asir-3636	216	5	(	(	PUNCT
asir-3636	216	6	1	1	NUM
asir-3636	216	7	−	−	NOUN
asir-3636	216	8	𝑢)𝑑(𝑒𝑖𝑡	𝑢)𝑑(𝑒𝑖𝑡	NUM
asir-3636	216	9	2	2	NUM
asir-3636	216	10	)	)	PUNCT
asir-3636	216	11	(	(	PUNCT
asir-3636	216	12	𝑤	𝑤	ADP
asir-3636	216	13	∈	∈	PROPN
asir-3636	216	14	𝜕∆𝛾	𝜕∆𝛾	PROPN
asir-3636	216	15	)	)	PUNCT
asir-3636	216	16	,	,	PUNCT
asir-3636	216	17	where	where	SCONJ
asir-3636	216	18	𝜕∆𝛾	𝜕∆𝛾	PRON
asir-3636	216	19	is	be	AUX
asir-3636	216	20	the	the	DET
asir-3636	216	21	boundary	boundary	NOUN
asir-3636	216	22	of	of	ADP
asir-3636	216	23	∆𝛾.	∆𝛾.	PROPN
asir-3636	216	24	then	then	ADV
asir-3636	216	25	𝔻𝑡2,𝑢	𝔻𝑡2,𝑢	NUM
asir-3636	216	26	∶=	∶=	NUM
asir-3636	216	27	{	{	PUNCT
asir-3636	216	28	𝑧	𝑧	PRON
asir-3636	216	29	∈	∈	PROPN
asir-3636	216	30	𝔻	𝔻	PROPN
asir-3636	216	31	:	:	PUNCT
asir-3636	216	32	|𝑧	|𝑧	NOUN
asir-3636	216	33	−	−	PROPN
asir-3636	216	34	𝑧𝑡2𝑡	𝑧𝑡2𝑡	PROPN
asir-3636	216	35	2(𝑢)|	2(𝑢)|	NUM
asir-3636	216	36	≤	≤	NUM
asir-3636	216	37	1	1	NUM
asir-3636	216	38	2	2	NUM
asir-3636	216	39	(	(	PUNCT
asir-3636	216	40	1	1	NUM
asir-3636	216	41	−	−	NOUN
asir-3636	216	42	𝑢)𝑑(𝑒𝑖𝑡	𝑢)𝑑(𝑒𝑖𝑡	NUM
asir-3636	216	43	2	2	NUM
asir-3636	216	44	)	)	PUNCT
asir-3636	216	45	}	}	PUNCT
asir-3636	216	46	⊂	⊂	PROPN
asir-3636	216	47	∆𝛾	∆𝛾	PROPN
asir-3636	216	48	,	,	PUNCT
asir-3636	216	49	for	for	ADP
asir-3636	216	50	all	all	DET
asir-3636	216	51	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	216	52	2	2	NUM
asir-3636	216	53	∈	∈	NOUN
asir-3636	216	54	𝛾	𝛾	NOUN
asir-3636	216	55	and	and	CCONJ
asir-3636	216	56	for	for	ADP
asir-3636	216	57	all	all	DET
asir-3636	216	58	𝑢	𝑢	NOUN
asir-3636	216	59	,	,	PUNCT
asir-3636	216	60	0	0	NUM
asir-3636	216	61	≤	≤	NUM
asir-3636	216	62	𝑢	𝑢	PRON
asir-3636	216	63	≤	≤	ADJ
asir-3636	216	64	1	1	NUM
asir-3636	216	65	.	.	PUNCT
asir-3636	217	1	since	since	SCONJ
asir-3636	217	2	∑	∑	PUNCT
asir-3636	217	3	|(𝑓𝑗	|(𝑓𝑗	PROPN
asir-3636	217	4	2)′(𝑧)|𝑗	2)′(𝑧)|𝑗	NUM
asir-3636	217	5	is	be	AUX
asir-3636	217	6	a	a	DET
asir-3636	217	7	series	series	NOUN
asir-3636	217	8	of	of	ADP
asir-3636	217	9	subharmonic	subharmonic	NOUN
asir-3636	217	10	on	on	ADP
asir-3636	217	11	𝔻	𝔻	PROPN
asir-3636	217	12	,	,	PUNCT
asir-3636	217	13	it	it	PRON
asir-3636	217	14	follows	follow	VERB
asir-3636	217	15	that	that	SCONJ
asir-3636	217	16	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	217	17	applied	apply	VERB
asir-3636	217	18	science	science	NOUN
asir-3636	217	19	and	and	CCONJ
asir-3636	217	20	innovative	innovative	ADJ
asir-3636	217	21	research	research	NOUN
asir-3636	217	22	vol	vol	NOUN
asir-3636	217	23	.	.	PROPN
asir-3636	218	1	5	5	NUM
asir-3636	218	2	,	,	PUNCT
asir-3636	218	3	no	no	INTJ
asir-3636	218	4	.	.	NOUN
asir-3636	218	5	1	1	NUM
asir-3636	218	6	,	,	PUNCT
asir-3636	218	7	2021	2021	NUM
asir-3636	218	8	30	30	NUM
asir-3636	218	9	published	publish	VERB
asir-3636	218	10	by	by	ADP
asir-3636	218	11	scholink	scholink	PROPN
asir-3636	218	12	inc	inc	PROPN
asir-3636	218	13	.	.	PROPN
asir-3636	218	14	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	219	1	2)′(𝑧𝑡2(𝑢))|	2)′(𝑧𝑡2(𝑢))|	NUM
asir-3636	219	2	𝑗	𝑗	X
asir-3636	219	3	≤	≤	NUM
asir-3636	219	4	4	4	NUM
asir-3636	219	5	𝜋(1	𝜋(1	NOUN
asir-3636	219	6	−	−	PROPN
asir-3636	219	7	𝑢)2𝑑2(𝑒𝑖𝑡	𝑢)2𝑑2(𝑒𝑖𝑡	PROPN
asir-3636	219	8	2	2	NUM
asir-3636	219	9	)	)	PUNCT
asir-3636	219	10	∫	∫	PROPN
asir-3636	219	11	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	219	12	2)′(𝑧)|𝑑𝐴(𝑧	2)′(𝑧)|𝑑𝐴(𝑧	NUM
asir-3636	219	13	)	)	PUNCT
asir-3636	220	1	𝑗	𝑗	PROPN
asir-3636	220	2	𝔻𝑡,𝑢	𝔻𝑡,𝑢	PROPN
asir-3636	220	3	≤	≤	NOUN
asir-3636	221	1	2	2	NUM
asir-3636	221	2	𝜋	𝜋	NOUN
asir-3636	221	3	1	1	NUM
asir-3636	221	4	2(1	2(1	NUM
asir-3636	221	5	−	−	NOUN
asir-3636	221	6	𝑢	𝑢	NOUN
asir-3636	221	7	)	)	PUNCT
asir-3636	221	8	𝑑	𝑑	PROPN
asir-3636	221	9	(	(	PUNCT
asir-3636	221	10	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	221	11	2	2	NUM
asir-3636	221	12	)	)	PUNCT
asir-3636	221	13	∑‖(𝑓𝑗	∑‖(𝑓𝑗	VERB
asir-3636	221	14	2)′‖	2)′‖	NUM
asir-3636	221	15	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	221	16	)	)	PUNCT
asir-3636	221	17	𝑗	𝑗	INTJ
asir-3636	221	18	.	.	PUNCT
asir-3636	222	1	set	set	PROPN
asir-3636	222	2	𝜀(1+𝜖	𝜀(1+𝜖	NOUN
asir-3636	222	3	)	)	PUNCT
asir-3636	223	1	=	=	PUNCT
asir-3636	223	2	2αj	2αj	NOUN
asir-3636	223	3	2𝜖.	2𝜖.	NUM
asir-3636	223	4	we	we	PRON
asir-3636	223	5	have	have	VERB
asir-3636	223	6	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	223	7	2(1+𝜖	2(1+𝜖	NUM
asir-3636	223	8	)	)	PUNCT
asir-3636	223	9	(	(	PUNCT
asir-3636	223	10	𝑧𝑡2)|	𝑧𝑡2)|	PROPN
asir-3636	223	11	2	2	NUM
asir-3636	223	12	𝑗	𝑗	NOUN
asir-3636	223	13	=	=	ADJ
asir-3636	223	14	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	223	15	2(1+𝜖)(𝑧𝑡2	2(1+𝜖)(𝑧𝑡2	NUM
asir-3636	223	16	)	)	PUNCT
asir-3636	223	17	−	−	ADP
asir-3636	223	18	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	223	19	2(1+𝜖	2(1+𝜖	NUM
asir-3636	223	20	)	)	PUNCT
asir-3636	223	21	(	(	PUNCT
asir-3636	223	22	𝜉)|	𝜉)|	NOUN
asir-3636	223	23	2	2	NUM
asir-3636	223	24	𝑗	𝑗	NOUN
asir-3636	223	25	=	=	X
asir-3636	223	26	(	(	PUNCT
asir-3636	223	27	1	1	NUM
asir-3636	223	28	+	+	NUM
asir-3636	223	29	𝜖)2|𝑧𝑡2	𝜖)2|𝑧𝑡2	NOUN
asir-3636	223	30	−	−	PROPN
asir-3636	223	31	𝜉|	𝜉|	PROPN
asir-3636	223	32	2	2	NUM
asir-3636	223	33	|∫	|∫	NOUN
asir-3636	223	34	∑𝑓𝑗	∑𝑓𝑗	NUM
asir-3636	224	1	2𝜖(𝑧𝑡2(𝑢))(𝑓𝑗	2𝜖(𝑧𝑡2(𝑢))(𝑓𝑗	NUM
asir-3636	225	1	2)′(𝑧𝑡2(𝑢))𝑑𝑢	2)′(𝑧𝑡2(𝑢))𝑑𝑢	NUM
asir-3636	225	2	𝑗	𝑗	PRON
asir-3636	225	3	1	1	NUM
asir-3636	225	4	0	0	NUM
asir-3636	225	5	|	|	CCONJ
asir-3636	225	6	2	2	NUM
asir-3636	225	7	≤	≤	NOUN
asir-3636	225	8	𝐶1+𝜖𝑑	𝐶1+𝜖𝑑	VERB
asir-3636	225	9	2(𝑒𝑖𝑡	2(𝑒𝑖𝑡	ADJ
asir-3636	225	10	2	2	NUM
asir-3636	225	11	)	)	PUNCT
asir-3636	225	12	(	(	PUNCT
asir-3636	225	13	∫	∫	PROPN
asir-3636	225	14	∑|𝑧𝑡2(𝑢	∑|𝑧𝑡2(𝑢	PROPN
asir-3636	225	15	)	)	PUNCT
asir-3636	226	1	−	−	PROPN
asir-3636	226	2	𝜉|	𝜉|	PROPN
asir-3636	226	3	𝜀1+𝜖	𝜀1+𝜖	VERB
asir-3636	226	4	2	2	NUM
asir-3636	226	5	|(𝑓𝑗	|(𝑓𝑗	NUM
asir-3636	226	6	2)′(𝑧𝑡2(𝑢))|𝑑𝑢	2)′(𝑧𝑡2(𝑢))|𝑑𝑢	NUM
asir-3636	226	7	𝑗	𝑗	NOUN
asir-3636	226	8	1	1	NUM
asir-3636	226	9	0	0	NUM
asir-3636	226	10	)	)	PUNCT
asir-3636	226	11	2	2	NUM
asir-3636	226	12	≤	≤	NOUN
asir-3636	226	13	𝐶1+𝜖𝑑	𝐶1+𝜖𝑑	NOUN
asir-3636	226	14	𝜀1+𝜖(𝑒𝑖𝑡	𝜀1+𝜖(𝑒𝑖𝑡	VERB
asir-3636	226	15	2	2	NUM
asir-3636	226	16	)	)	PUNCT
asir-3636	226	17	(	(	PUNCT
asir-3636	226	18	∫	∫	PROPN
asir-3636	226	19	1	1	NUM
asir-3636	226	20	(	(	PUNCT
asir-3636	226	21	1	1	NUM
asir-3636	226	22	−	−	PRON
asir-3636	226	23	𝑢)1−	𝑢)1−	NOUN
asir-3636	226	24	𝜀1+𝜖	𝜀1+𝜖	NOUN
asir-3636	226	25	2	2	NUM
asir-3636	226	26	𝑑𝑢	𝑑𝑢	NOUN
asir-3636	226	27	1	1	NUM
asir-3636	226	28	0	0	NUM
asir-3636	226	29	)	)	PUNCT
asir-3636	226	30	2	2	NUM
asir-3636	226	31	∑‖(𝑓𝑗	∑‖(𝑓𝑗	VERB
asir-3636	226	32	2)′‖	2)′‖	NUM
asir-3636	226	33	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	226	34	)	)	PUNCT
asir-3636	226	35	2	2	NUM
asir-3636	226	36	𝑗	𝑗	PROPN
asir-3636	226	37	≤	≤	NOUN
asir-3636	226	38	𝐶1+𝜖𝑑	𝐶1+𝜖𝑑	NOUN
asir-3636	226	39	𝜀1+𝜖(𝑒𝑖𝑡	𝜀1+𝜖(𝑒𝑖𝑡	VERB
asir-3636	226	40	2	2	NUM
asir-3636	226	41	)	)	PUNCT
asir-3636	226	42	∑‖(𝑓𝑗	∑‖(𝑓𝑗	VERB
asir-3636	226	43	2)′‖	2)′‖	NUM
asir-3636	226	44	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	226	45	)	)	PUNCT
asir-3636	226	46	2	2	NUM
asir-3636	226	47	𝑗	𝑗	NOUN
asir-3636	226	48	.	.	PUNCT
asir-3636	227	1	hence	hence	ADV
asir-3636	227	2	∫	∫	PROPN
asir-3636	227	3	∑	∑	PROPN
asir-3636	227	4	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	227	5	2(𝑧	2(𝑧	NUM
asir-3636	227	6	𝑡2	𝑡2	PROPN
asir-3636	227	7	)	)	PUNCT
asir-3636	227	8	|	|	ADV
asir-3636	227	9	2(1+𝜖	2(1+𝜖	NUM
asir-3636	227	10	)	)	PUNCT
asir-3636	227	11	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	227	12	2	2	NUM
asir-3636	227	13	)	)	PUNCT
asir-3636	228	1	𝑗	𝑗	PROPN
asir-3636	228	2	𝛾	𝛾	NOUN
asir-3636	228	3	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	228	4	≤	≤	NOUN
asir-3636	228	5	∑	∑	PUNCT
asir-3636	228	6	𝐶𝜌‖(𝑓𝑗	𝐶𝜌‖(𝑓𝑗	PROPN
asir-3636	228	7	2)′‖	2)′‖	NUM
asir-3636	228	8	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	228	9	)	)	PUNCT
asir-3636	228	10	2	2	NUM
asir-3636	228	11	𝑗	𝑗	NOUN
asir-3636	228	12	.	.	PUNCT
asir-3636	229	1	(	(	PUNCT
asir-3636	229	2	8)	8)	NUM
asir-3636	229	3	therefore	therefore	ADV
asir-3636	229	4	the	the	DET
asir-3636	229	5	result	result	NOUN
asir-3636	229	6	follows	follow	VERB
asir-3636	229	7	from	from	ADP
asir-3636	229	8	(	(	PUNCT
asir-3636	229	9	6	6	NUM
asir-3636	229	10	)	)	PUNCT
asir-3636	229	11	,	,	PUNCT
asir-3636	229	12	(	(	PUNCT
asir-3636	229	13	7	7	X
asir-3636	229	14	)	)	PUNCT
asir-3636	229	15	and	and	CCONJ
asir-3636	229	16	(	(	PUNCT
asir-3636	229	17	8)	8)	NUM
asir-3636	229	18	.	.	PUNCT
asir-3636	230	1	in	in	ADP
asir-3636	230	2	the	the	DET
asir-3636	230	3	sequel	sequel	NOUN
asir-3636	230	4	,	,	PUNCT
asir-3636	230	5	we	we	PRON
asir-3636	230	6	denote	denote	VERB
asir-3636	230	7	by	by	ADP
asir-3636	230	8	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	230	9	2	2	NUM
asir-3636	230	10	a	a	DET
asir-3636	230	11	series	series	NOUN
asir-3636	230	12	of	of	ADP
asir-3636	230	13	square	square	ADJ
asir-3636	230	14	outer	outer	ADJ
asir-3636	230	15	functions	function	NOUN
asir-3636	230	16	in	in	ADP
asir-3636	230	17	𝒜αj	𝒜αj	PROPN
asir-3636	230	18	2	2	NUM
asir-3636	230	19	such	such	ADJ
asir-3636	230	20	that	that	SCONJ
asir-3636	230	21	∑	∑	PROPN
asir-3636	230	22	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	230	23	2‖	2‖	PROPN
asir-3636	230	24	𝒜	𝒜	NOUN
asir-3636	230	25	αj	αj	NOUN
asir-3636	230	26	2	2	NUM
asir-3636	230	27	𝑗	𝑗	NOUN
asir-3636	230	28	≤	≤	NUM
asir-3636	230	29	1	1	NUM
asir-3636	231	1	and	and	CCONJ
asir-3636	231	2	we	we	PRON
asir-3636	231	3	fix	fix	VERB
asir-3636	231	4	a	a	DET
asir-3636	231	5	constant	constant	ADJ
asir-3636	231	6	1	1	NUM
asir-3636	231	7	+	+	CCONJ
asir-3636	231	8	𝜖	𝜖	PROPN
asir-3636	231	9	,	,	PUNCT
asir-3636	231	10	0	0	PUNCT
asir-3636	231	11	<	<	X
asir-3636	231	12	𝜖	𝜖	X
asir-3636	231	13	≤	≤	NUM
asir-3636	231	14	1	1	NUM
asir-3636	231	15	.	.	PUNCT
asir-3636	232	1	by	by	ADP
asir-3636	232	2	(	(	PUNCT
asir-3636	232	3	matheson	matheson	PROPN
asir-3636	232	4	,	,	PUNCT
asir-3636	232	5	1978	1978	NUM
asir-3636	232	6	theorem	theorem	NOUN
asir-3636	232	7	b	b	NOUN
asir-3636	232	8	)	)	PUNCT
asir-3636	232	9	,	,	PUNCT
asir-3636	232	10	we	we	PRON
asir-3636	232	11	have	have	VERB
asir-3636	232	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	232	13	2(1+𝜖	2(1+𝜖	NUM
asir-3636	232	14	)	)	PUNCT
asir-3636	232	15	(	(	PUNCT
asir-3636	232	16	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	232	17	2(1+𝜖	2(1+𝜖	NUM
asir-3636	232	18	)	)	PUNCT
asir-3636	232	19	∈	∈	PROPN
asir-3636	232	20	lipαj	lipαj	NOUN
asir-3636	232	21	2	2	NUM
asir-3636	232	22	and	and	CCONJ
asir-3636	232	23	∑	∑	ADP
asir-3636	232	24	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	232	25	2(1+𝜖	2(1+𝜖	NUM
asir-3636	232	26	)	)	PUNCT
asir-3636	232	27	(	(	PUNCT
asir-3636	232	28	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	232	29	2(1+𝜖	2(1+𝜖	NUM
asir-3636	232	30	)	)	PUNCT
asir-3636	233	1	‖	‖	PROPN
asir-3636	233	2	lip	lip	NOUN
asir-3636	233	3	αj	αj	ADP
asir-3636	233	4	2	2	NUM
asir-3636	233	5	𝑗	𝑗	PROPN
asir-3636	233	6	≤	≤	NUM
asir-3636	233	7	𝐶1+𝜖,1+𝜖.	𝐶1+𝜖,1+𝜖.	NUM
asir-3636	233	8	to	to	PART
asir-3636	233	9	prove	prove	VERB
asir-3636	233	10	theorem	theorem	ADJ
asir-3636	233	11	(	(	PUNCT
asir-3636	233	12	2.1	2.1	NUM
asir-3636	233	13	)	)	PUNCT
asir-3636	233	14	we	we	PRON
asir-3636	233	15	need	need	VERB
asir-3636	233	16	to	to	PART
asir-3636	233	17	estimate	estimate	VERB
asir-3636	233	18	the	the	DET
asir-3636	233	19	integral	integral	ADJ
asir-3636	233	20	∫	∫	NOUN
asir-3636	233	21	∑	∑	PROPN
asir-3636	233	22	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	233	23	2(1+𝜖	2(1+𝜖	NUM
asir-3636	233	24	)	)	PUNCT
asir-3636	233	25	(	(	PUNCT
asir-3636	233	26	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	233	27	2(1+𝜖	2(1+𝜖	NUM
asir-3636	233	28	)	)	PUNCT
asir-3636	233	29	)	)	PUNCT
asir-3636	234	1	′|𝑗	′|𝑗	CCONJ
asir-3636	234	2	2	2	NUM
asir-3636	234	3	𝔻	𝔻	PROPN
asir-3636	234	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	234	5	)	)	PUNCT
asir-3636	234	6	.	.	PUNCT
asir-3636	235	1	define	define	VERB
asir-3636	235	2	∑	∑	ADV
asir-3636	235	3	(	(	PUNCT
asir-3636	235	4	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	235	5	2	2	NUM
asir-3636	235	6	)	)	PUNCT
asir-3636	235	7	γ	γ	NOUN
asir-3636	235	8	(	(	PUNCT
asir-3636	235	9	𝑧)𝑗	𝑧)𝑗	NOUN
asir-3636	235	10	≔	≔	NOUN
asir-3636	235	11	1	1	NUM
asir-3636	235	12	𝜋	𝜋	NOUN
asir-3636	235	13	∫	∫	PROPN
asir-3636	235	14	∑	∑	PROPN
asir-3636	235	15	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	235	16	2	2	NUM
asir-3636	235	17	(	(	PUNCT
asir-3636	235	18	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	235	19	2	2	NUM
asir-3636	235	20	−𝑧)2	−𝑧)2	NOUN
asir-3636	235	21	𝑙𝑜𝑔|𝑓𝑗	𝑙𝑜𝑔|𝑓𝑗	ADJ
asir-3636	235	22	2(𝑒𝑖𝜃	2(𝑒𝑖𝜃	NUM
asir-3636	235	23	2	2	NUM
asir-3636	235	24	)	)	PUNCT
asir-3636	235	25	|𝑗	|𝑗	VERB
asir-3636	235	26	γ	γ	X
asir-3636	235	27	𝑑𝜃2	𝑑𝜃2	NOUN
asir-3636	235	28	.	.	PUNCT
asir-3636	236	1	(	(	PUNCT
asir-3636	236	2	9	9	X
asir-3636	236	3	)	)	PUNCT
asir-3636	236	4	clearly	clearly	ADV
asir-3636	236	5	we	we	PRON
asir-3636	236	6	have	have	VERB
asir-3636	236	7	∑	∑	ADV
asir-3636	236	8	(	(	PUNCT
asir-3636	236	9	𝑓𝑗	𝑓𝑗	PROPN
asir-3636	236	10	2)′𝑗	2)′𝑗	NUM
asir-3636	236	11	=	=	SYM
asir-3636	236	12	∑	∑	PUNCT
asir-3636	236	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	236	14	2((𝑔𝑗	2((𝑔𝑗	NUM
asir-3636	236	15	2)γ	2)γ	NUM
asir-3636	236	16	+	+	CCONJ
asir-3636	236	17	(	(	PUNCT
asir-3636	236	18	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	236	19	2)𝕋\γ	2)𝕋\γ	NUM
asir-3636	236	20	)	)	PUNCT
asir-3636	236	21	𝑗	𝑗	PROPN
asir-3636	236	22	and	and	CCONJ
asir-3636	236	23	∑	∑	ADP
asir-3636	236	24	(	(	PUNCT
asir-3636	236	25	(	(	PUNCT
asir-3636	236	26	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	236	27	2(1+𝜖	2(1+𝜖	NUM
asir-3636	236	28	)	)	PUNCT
asir-3636	236	29	)	)	PUNCT
asir-3636	237	1	𝑗	𝑗	NOUN
asir-3636	238	1	′	′	NOUN
asir-3636	238	2	=	=	PUNCT
asir-3636	238	3	∑	∑	PUNCT
asir-3636	238	4	(	(	PUNCT
asir-3636	238	5	1	1	NUM
asir-3636	238	6	+	+	CCONJ
asir-3636	238	7	𝜖)(𝑓𝑗)γ	𝜖)(𝑓𝑗)γ	PROPN
asir-3636	238	8	2(1+𝜖	2(1+𝜖	NUM
asir-3636	238	9	)	)	PUNCT
asir-3636	238	10	(	(	PUNCT
asir-3636	238	11	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	238	12	2)γ	2)γ	NUM
asir-3636	238	13	𝑗	𝑗	INTJ
asir-3636	238	14	,	,	PUNCT
asir-3636	238	15	∑	∑	ADV
asir-3636	238	16	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	238	17	2(1+𝜖	2(1+𝜖	NUM
asir-3636	238	18	)	)	PUNCT
asir-3636	238	19	(	(	PUNCT
asir-3636	238	20	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	238	21	2(1+𝜖	2(1+𝜖	NUM
asir-3636	238	22	)	)	PUNCT
asir-3636	238	23	)	)	PUNCT
asir-3636	239	1	′𝑗	′𝑗	PROPN
asir-3636	239	2	=	=	PUNCT
asir-3636	239	3	∑	∑	PUNCT
asir-3636	239	4	(	(	PUNCT
asir-3636	239	5	1	1	NUM
asir-3636	239	6	+	+	NUM
asir-3636	239	7	𝜖)𝑓𝑗	𝜖)𝑓𝑗	NOUN
asir-3636	239	8	2(1+𝜖	2(1+𝜖	NUM
asir-3636	239	9	)	)	PUNCT
asir-3636	239	10	(	(	PUNCT
asir-3636	239	11	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	239	12	2(1+𝜖	2(1+𝜖	NUM
asir-3636	239	13	)	)	PUNCT
asir-3636	239	14	(	(	PUNCT
asir-3636	239	15	𝑔𝑗	𝑔𝑗	NOUN
asir-3636	239	16	2)γ	2)γ	NUM
asir-3636	239	17	𝑗	𝑗	X
asir-3636	239	18	(	(	PUNCT
asir-3636	239	19	10	10	NUM
asir-3636	239	20	)	)	PUNCT
asir-3636	239	21	=	=	PUNCT
asir-3636	239	22	∑	∑	PUNCT
asir-3636	239	23	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	239	24	2𝜖(1	2𝜖(1	NUM
asir-3636	239	25	+	+	CCONJ
asir-3636	239	26	𝜖)(𝑓𝑗	𝜖)(𝑓𝑗	NOUN
asir-3636	240	1	2)′(𝑓𝑗)γ	2)′(𝑓𝑗)γ	NUM
asir-3636	240	2	(	(	PUNCT
asir-3636	240	3	1+𝜖	1+𝜖	NUM
asir-3636	240	4	)	)	PUNCT
asir-3636	240	5	𝑗	𝑗	NOUN
asir-3636	240	6	−	−	NOUN
asir-3636	240	7	∑	∑	INTJ
asir-3636	240	8	(	(	PUNCT
asir-3636	240	9	1	1	NUM
asir-3636	240	10	+	+	NUM
asir-3636	240	11	𝜖)𝑓𝑗	𝜖)𝑓𝑗	NOUN
asir-3636	240	12	2(1+𝜖	2(1+𝜖	NUM
asir-3636	240	13	)	)	PUNCT
asir-3636	240	14	(	(	PUNCT
asir-3636	240	15	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	240	16	2(1+𝜖	2(1+𝜖	NUM
asir-3636	240	17	)	)	PUNCT
asir-3636	240	18	(	(	PUNCT
asir-3636	240	19	𝑔𝑗	𝑔𝑗	PROPN
asir-3636	240	20	2)𝕋\γ	2)𝕋\γ	NUM
asir-3636	240	21	𝑗	𝑗	NOUN
asir-3636	240	22	.	.	PUNCT
asir-3636	241	1	(	(	PUNCT
asir-3636	241	2	11	11	NUM
asir-3636	241	3	)	)	PUNCT
asir-3636	241	4	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	241	5	applied	apply	VERB
asir-3636	241	6	science	science	NOUN
asir-3636	241	7	and	and	CCONJ
asir-3636	241	8	innovative	innovative	ADJ
asir-3636	241	9	research	research	NOUN
asir-3636	241	10	vol	vol	NOUN
asir-3636	241	11	.	.	PROPN
asir-3636	242	1	5	5	NUM
asir-3636	242	2	,	,	PUNCT
asir-3636	242	3	no	no	INTJ
asir-3636	242	4	.	.	NOUN
asir-3636	242	5	1	1	NUM
asir-3636	242	6	,	,	PUNCT
asir-3636	242	7	2021	2021	NUM
asir-3636	242	8	31	31	NUM
asir-3636	242	9	published	publish	VERB
asir-3636	242	10	by	by	ADP
asir-3636	242	11	scholink	scholink	PROPN
asir-3636	242	12	inc	inc	PROPN
asir-3636	242	13	.	.	PROPN
asir-3636	243	1	since	since	SCONJ
asir-3636	243	2	∑	∑	PROPN
asir-3636	243	3	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	243	4	2‖	2‖	PROPN
asir-3636	243	5	∞𝑗	∞𝑗	NUM
asir-3636	243	6	≤	≤	NUM
asir-3636	243	7	1	1	NUM
asir-3636	243	8	,	,	PUNCT
asir-3636	243	9	it	it	PRON
asir-3636	243	10	is	be	AUX
asir-3636	243	11	obvious	obvious	ADJ
asir-3636	243	12	that	that	SCONJ
asir-3636	243	13	∑	∑	PROPN
asir-3636	243	14	‖(𝑓𝑗)γ	‖(𝑓𝑗)γ	X
asir-3636	243	15	2(1+𝜖	2(1+𝜖	NUM
asir-3636	243	16	)	)	PUNCT
asir-3636	243	17	‖	‖	PROPN
asir-3636	243	18	∞	∞	PROPN
asir-3636	243	19	𝑗	𝑗	PROPN
asir-3636	243	20	≤	≤	ADJ
asir-3636	243	21	1	1	NUM
asir-3636	243	22	and	and	CCONJ
asir-3636	243	23	∑	∑	PUNCT
asir-3636	243	24	‖𝑓𝑗	‖𝑓𝑗	PROPN
asir-3636	243	25	2𝜖‖	2𝜖‖	PROPN
asir-3636	243	26	∞𝑗	∞𝑗	NUM
asir-3636	243	27	≤	≤	NUM
asir-3636	243	28	1	1	NUM
asir-3636	243	29	.	.	PUNCT
asir-3636	244	1	hence	hence	ADV
asir-3636	244	2	,	,	PUNCT
asir-3636	244	3	by	by	ADP
asir-3636	244	4	(	(	PUNCT
asir-3636	244	5	11	11	NUM
asir-3636	244	6	)	)	PUNCT
asir-3636	244	7	we	we	PRON
asir-3636	244	8	get	get	VERB
asir-3636	244	9	∫	∫	PROPN
asir-3636	244	10	∑	∑	PROPN
asir-3636	244	11	|(𝑓𝑗	|(𝑓𝑗	PROPN
asir-3636	244	12	2(1+𝜖	2(1+𝜖	NUM
asir-3636	244	13	)	)	PUNCT
asir-3636	244	14	(	(	PUNCT
asir-3636	244	15	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	244	16	2(1+𝜖	2(1+𝜖	NUM
asir-3636	244	17	)	)	PUNCT
asir-3636	244	18	)	)	PUNCT
asir-3636	245	1	′	′	NUM
asir-3636	245	2	|𝑗	|𝑗	NOUN
asir-3636	245	3	2	2	NUM
asir-3636	245	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
asir-3636	245	5	)	)	PUNCT
asir-3636	245	6	≤	≤	NOUN
asir-3636	245	7	2(1	2(1	NUM
asir-3636	246	1	+	+	CCONJ
asir-3636	246	2	𝜖)2	𝜖)2	PROPN
asir-3636	246	3	𝔻	𝔻	ADJ
asir-3636	246	4	∫	∫	PROPN
asir-3636	246	5	∑	∑	PROPN
asir-3636	246	6	|(𝑓𝑗	|(𝑓𝑗	PROPN
asir-3636	246	7	2(1+𝜖	2(1+𝜖	NUM
asir-3636	246	8	)	)	PUNCT
asir-3636	246	9	(	(	PUNCT
asir-3636	246	10	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	246	11	2(1+𝜖	2(1+𝜖	NUM
asir-3636	246	12	)	)	PUNCT
asir-3636	246	13	)	)	PUNCT
asir-3636	247	1	′	′	NUM
asir-3636	247	2	|𝑗	|𝑗	NOUN
asir-3636	247	3	2	2	NUM
asir-3636	247	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	247	5	)	)	PUNCT
asir-3636	247	6	.	.	PUNCT
asir-3636	248	1	𝔻	𝔻	PROPN
asir-3636	248	2	(	(	PUNCT
asir-3636	248	3	12	12	NUM
asir-3636	248	4	)	)	PUNCT
asir-3636	248	5	we	we	PRON
asir-3636	248	6	fix	fix	VERB
asir-3636	248	7	𝛾	𝛾	ADP
asir-3636	248	8	=	=	SYM
asir-3636	248	9	(	(	PUNCT
asir-3636	248	10	𝑎	𝑎	X
asir-3636	248	11	,	,	PUNCT
asir-3636	248	12	𝑎	𝑎	PRON
asir-3636	248	13	+	+	X
asir-3636	248	14	𝜖	𝜖	X
asir-3636	248	15	)	)	PUNCT
asir-3636	248	16	⊂	⊂	PROPN
asir-3636	249	1	𝑇\𝐸𝑓𝑗	𝑇\𝐸𝑓𝑗	ADP
asir-3636	249	2	2	2	NUM
asir-3636	249	3	such	such	ADJ
asir-3636	249	4	that	that	SCONJ
asir-3636	249	5	∑	∑	ADP
asir-3636	249	6	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	249	7	2(𝑎)𝑗	2(𝑎)𝑗	NUM
asir-3636	249	8	=	=	SYM
asir-3636	249	9	∑	∑	PUNCT
asir-3636	249	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	249	11	2(𝑎	2(𝑎	NUM
asir-3636	249	12	+	+	CCONJ
asir-3636	249	13	𝜖)𝑗	𝜖)𝑗	NOUN
asir-3636	249	14	=	=	SYM
asir-3636	249	15	0	0	X
asir-3636	249	16	.	.	PUNCT
asir-3636	250	1	our	our	PRON
asir-3636	250	2	purpose	purpose	NOUN
asir-3636	250	3	in	in	ADP
asir-3636	250	4	what	what	PRON
asir-3636	250	5	follows	follow	VERB
asir-3636	250	6	is	be	AUX
asir-3636	250	7	to	to	PART
asir-3636	250	8	estimate	estimate	VERB
asir-3636	250	9	the	the	DET
asir-3636	250	10	integral	integral	ADJ
asir-3636	250	11	∫	∫	PROPN
asir-3636	250	12	∑	∑	PROPN
asir-3636	250	13	|(𝑓𝑗	|(𝑓𝑗	PROPN
asir-3636	250	14	2(1+𝜖	2(1+𝜖	NUM
asir-3636	250	15	)	)	PUNCT
asir-3636	250	16	(	(	PUNCT
asir-3636	250	17	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	250	18	2(1+𝜖	2(1+𝜖	NUM
asir-3636	250	19	)	)	PUNCT
asir-3636	250	20	)	)	PUNCT
asir-3636	251	1	′	′	NUM
asir-3636	251	2	|𝑗	|𝑗	NOUN
asir-3636	251	3	2	2	NUM
asir-3636	251	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	251	5	)	)	PUNCT
asir-3636	251	6	∆𝛾	∆𝛾	PROPN
asir-3636	251	7	(	(	PUNCT
asir-3636	251	8	13	13	NUM
asir-3636	251	9	)	)	PUNCT
asir-3636	251	10	which	which	PRON
asir-3636	251	11	we	we	PRON
asir-3636	251	12	can	can	AUX
asir-3636	251	13	rewrite	rewrite	VERB
asir-3636	251	14	as	as	ADP
asir-3636	251	15	∫	∫	PROPN
asir-3636	251	16	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	251	17	2(1+𝜖	2(1+𝜖	NUM
asir-3636	251	18	)	)	PUNCT
asir-3636	251	19	(	(	PUNCT
asir-3636	251	20	𝑓𝑗)γ	𝑓𝑗)γ	PROPN
asir-3636	251	21	2(1+𝜖	2(1+𝜖	NUM
asir-3636	251	22	)	)	PUNCT
asir-3636	251	23	)	)	PUNCT
asir-3636	252	1	′	′	NUM
asir-3636	253	1	|	|	ADV
asir-3636	253	2	𝑗	𝑗	INTJ
asir-3636	253	3	2	2	NUM
asir-3636	253	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
asir-3636	253	5	)	)	PUNCT
asir-3636	253	6	∆𝛾	∆𝛾	PROPN
asir-3636	253	7	=	=	SYM
asir-3636	253	8	∫	∫	PROPN
asir-3636	254	1	+	+	NUM
asir-3636	254	2	∫	∫	PROPN
asir-3636	254	3	,	,	PUNCT
asir-3636	254	4	∆𝛾	∆𝛾	PROPN
asir-3636	254	5	2	2	NUM
asir-3636	254	6	∆𝛾	∆𝛾	PROPN
asir-3636	254	7	1	1	NUM
asir-3636	254	8	where	where	SCONJ
asir-3636	254	9	∆𝛾	∆𝛾	PROPN
asir-3636	254	10	1≔	1≔	NUM
asir-3636	254	11	{	{	PUNCT
asir-3636	254	12	𝑧	𝑧	PROPN
asir-3636	254	13	∈	∈	PROPN
asir-3636	254	14	∆𝛾	∆𝛾	PROPN
asir-3636	254	15	:	:	PUNCT
asir-3636	254	16	𝑑(𝑧	𝑑(𝑧	ADJ
asir-3636	254	17	)	)	PUNCT
asir-3636	254	18	<	<	X
asir-3636	254	19	2(1	2(1	NUM
asir-3636	254	20	−	−	NOUN
asir-3636	254	21	|𝑧|	|𝑧|	NOUN
asir-3636	254	22	)	)	PUNCT
asir-3636	254	23	}	}	PUNCT
asir-3636	254	24	∆𝛾	∆𝛾	PROPN
asir-3636	254	25	2≔	2≔	NUM
asir-3636	254	26	{	{	PUNCT
asir-3636	254	27	𝑧	𝑧	PROPN
asir-3636	254	28	∈	∈	PROPN
asir-3636	254	29	∆𝛾	∆𝛾	PROPN
asir-3636	254	30	:	:	PUNCT
asir-3636	254	31	𝑑(𝑧	𝑑(𝑧	ADJ
asir-3636	254	32	)	)	PUNCT
asir-3636	254	33	≥	≥	NOUN
asir-3636	254	34	2(1	2(1	NUM
asir-3636	254	35	−	−	NOUN
asir-3636	254	36	|𝑧|	|𝑧|	NOUN
asir-3636	254	37	)	)	PUNCT
asir-3636	254	38	}	}	PUNCT
asir-3636	254	39	.	.	PUNCT
asir-3636	255	1	the	the	DET
asir-3636	255	2	integral	integral	ADJ
asir-3636	255	3	on	on	ADP
asir-3636	255	4	the	the	DET
asir-3636	255	5	region	region	NOUN
asir-3636	255	6	∆𝛾	∆𝛾	PROPN
asir-3636	255	7	1	1	NUM
asir-3636	255	8	.	.	PUNCT
asir-3636	256	1	we	we	PRON
asir-3636	256	2	begin	begin	VERB
asir-3636	256	3	with	with	ADP
asir-3636	256	4	the	the	DET
asir-3636	256	5	following	follow	VERB
asir-3636	256	6	lemma	lemma	PROPN
asir-3636	256	7	(	(	PUNCT
asir-3636	256	8	see	see	VERB
asir-3636	256	9	brahim	brahim	PROPN
asir-3636	256	10	bouya	bouya	PROPN
asir-3636	256	11	,	,	PUNCT
asir-3636	256	12	2008	2008	NUM
asir-3636	256	13	)	)	PUNCT
asir-3636	256	14	.	.	PUNCT
asir-3636	257	1	lemma	lemma	PROPN
asir-3636	257	2	(	(	PUNCT
asir-3636	257	3	4.2	4.2	NUM
asir-3636	257	4	):	):	PUNCT
asir-3636	257	5	∫	∫	PROPN
asir-3636	257	6	∑	∑	PROPN
asir-3636	257	7	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	257	8	2	2	NUM
asir-3636	257	9	(	(	PUNCT
asir-3636	257	10	𝑧	𝑧	NOUN
asir-3636	257	11	)	)	PUNCT
asir-3636	257	12	−	−	PROPN
asir-3636	257	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	257	14	2	2	NUM
asir-3636	257	15	(	(	PUNCT
asir-3636	257	16	𝑧	𝑧	PROPN
asir-3636	257	17	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	257	18	)	)	PUNCT
asir-3636	257	19	|	|	ADV
asir-3636	257	20	2(1+𝜖	2(1+𝜖	NUM
asir-3636	257	21	)	)	PUNCT
asir-3636	257	22	(	(	PUNCT
asir-3636	257	23	1	1	NUM
asir-3636	257	24	−	−	PROPN
asir-3636	257	25	|𝑧|)2	|𝑧|)2	NUM
asir-3636	257	26	𝒋	𝒋	SYM
asir-3636	257	27	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	257	28	)	)	PUNCT
asir-3636	257	29	≤∑	≤∑	NOUN
asir-3636	257	30	1	1	NUM
asir-3636	257	31	2αj	2αj	ADJ
asir-3636	257	32	2	2	NUM
asir-3636	257	33	𝜖	𝜖	X
asir-3636	257	34	‖(𝑓𝑗	‖(𝑓𝑗	X
asir-3636	257	35	2	2	NUM
asir-3636	257	36	)	)	PUNCT
asir-3636	257	37	′‖	′‖	X
asir-3636	257	38	𝐿2(∆𝛾	𝐿2(∆𝛾	PROPN
asir-3636	257	39	)	)	PUNCT
asir-3636	257	40	𝒋	𝒋	X
asir-3636	257	41	∆𝛾	∆𝛾	PROPN
asir-3636	257	42	.	.	PUNCT
asir-3636	258	1	proof	proof	NOUN
asir-3636	258	2	:	:	PUNCT
asir-3636	258	3	let	let	VERB
asir-3636	258	4	𝑧	𝑧	VERB
asir-3636	258	5	=	=	PUNCT
asir-3636	258	6	(	(	PUNCT
asir-3636	258	7	1	1	NUM
asir-3636	258	8	−	−	NOUN
asir-3636	258	9	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	NOUN
asir-3636	258	10	2	2	NUM
asir-3636	258	11	∈	∈	PROPN
asir-3636	258	12	∆𝛾	∆𝛾	PROPN
asir-3636	258	13	and	and	CCONJ
asir-3636	258	14	put	put	VERB
asir-3636	258	15	𝜀1+𝜖	𝜀1+𝜖	NOUN
asir-3636	258	16	=	=	SYM
asir-3636	258	17	2αj	2αj	NOUN
asir-3636	258	18	2𝜖.	2𝜖.	NUM
asir-3636	258	19	we	we	PRON
asir-3636	258	20	have	have	VERB
asir-3636	258	21	∑(1	∑(1	NUM
asir-3636	258	22	−	−	PROPN
asir-3636	258	23	𝜖	𝜖	PROPN
asir-3636	258	24	)	)	PUNCT
asir-3636	258	25	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	258	26	2	2	NUM
asir-3636	258	27	(	(	PUNCT
asir-3636	258	28	(	(	PUNCT
asir-3636	258	29	1	1	NUM
asir-3636	258	30	−	−	PROPN
asir-3636	258	31	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	258	32	2	2	NUM
asir-3636	258	33	)	)	PUNCT
asir-3636	258	34	–	–	PUNCT
asir-3636	258	35	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	258	36	2	2	NUM
asir-3636	258	37	(	(	PUNCT
asir-3636	258	38	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	258	39	2	2	NUM
asir-3636	258	40	)	)	PUNCT
asir-3636	258	41	|	|	ADV
asir-3636	258	42	2(1+𝜖	2(1+𝜖	NUM
asir-3636	258	43	)	)	PUNCT
asir-3636	258	44	𝑗	𝑗	NOUN
asir-3636	259	1	=	=	PUNCT
asir-3636	259	2	∑(1	∑(1	PROPN
asir-3636	259	3	−	−	NOUN
asir-3636	259	4	𝜖)|𝑓𝑗	𝜖)|𝑓𝑗	PROPN
asir-3636	259	5	2	2	NUM
asir-3636	259	6	(	(	PUNCT
asir-3636	259	7	(	(	PUNCT
asir-3636	259	8	1	1	NUM
asir-3636	259	9	−	−	PROPN
asir-3636	259	10	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	259	11	2	2	NUM
asir-3636	259	12	)	)	PUNCT
asir-3636	259	13	–	–	PUNCT
asir-3636	259	14	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	259	15	2	2	NUM
asir-3636	259	16	(	(	PUNCT
asir-3636	259	17	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	259	18	2	2	NUM
asir-3636	259	19	)	)	PUNCT
asir-3636	259	20	|	|	CCONJ
asir-3636	259	21	2𝜖	2𝜖	VERB
asir-3636	259	22	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	259	23	2	2	NUM
asir-3636	259	24	(	(	PUNCT
asir-3636	259	25	(	(	PUNCT
asir-3636	259	26	1	1	NUM
asir-3636	259	27	−	−	PROPN
asir-3636	259	28	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	259	29	2	2	NUM
asir-3636	259	30	)	)	PUNCT
asir-3636	259	31	–	–	PUNCT
asir-3636	259	32	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	259	33	2	2	NUM
asir-3636	259	34	(	(	PUNCT
asir-3636	259	35	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	259	36	2	2	NUM
asir-3636	259	37	)	)	PUNCT
asir-3636	259	38	|	|	ADV
asir-3636	259	39	2	2	NUM
asir-3636	259	40	𝑗	𝑗	NOUN
asir-3636	259	41	≤	≤	NUM
asir-3636	259	42	(	(	PUNCT
asir-3636	259	43	1	1	NUM
asir-3636	259	44	−	−	PROPN
asir-3636	259	45	𝜖)𝜖1+𝜀(1+𝜖)∫	𝜖)𝜖1+𝜀(1+𝜖)∫	ADJ
asir-3636	259	46	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	259	47	2	2	X
asir-3636	259	48	)	)	PUNCT
asir-3636	259	49	′	′	NOUN
asir-3636	259	50	(	(	PUNCT
asir-3636	259	51	(	(	PUNCT
asir-3636	259	52	1	1	NUM
asir-3636	259	53	2	2	NUM
asir-3636	259	54	+	+	NOUN
asir-3636	259	55	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	259	56	2	2	NUM
asir-3636	259	57	)	)	PUNCT
asir-3636	259	58	|	|	ADV
asir-3636	259	59	2	2	NUM
asir-3636	259	60	𝑑	𝑑	NOUN
asir-3636	259	61	(	(	PUNCT
asir-3636	259	62	1	1	NUM
asir-3636	259	63	2	2	NUM
asir-3636	259	64	+	+	NOUN
asir-3636	259	65	𝜖	𝜖	X
asir-3636	259	66	)	)	PUNCT
asir-3636	259	67	𝑗	𝑗	PROPN
asir-3636	259	68	≤	≤	ADJ
asir-3636	259	69	1	1	NUM
asir-3636	259	70	(	(	PUNCT
asir-3636	259	71	1−𝜖	1−𝜖	NUM
asir-3636	259	72	)	)	PUNCT
asir-3636	259	73	(	(	PUNCT
asir-3636	259	74	1	1	NUM
asir-3636	259	75	−	−	PROPN
asir-3636	259	76	𝜖)𝜖1+𝜀(1+𝜖)∫	𝜖)𝜖1+𝜀(1+𝜖)∫	ADJ
asir-3636	259	77	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	259	78	2	2	X
asir-3636	259	79	)	)	PUNCT
asir-3636	259	80	′	′	NOUN
asir-3636	259	81	(	(	PUNCT
asir-3636	259	82	(	(	PUNCT
asir-3636	259	83	1	1	NUM
asir-3636	259	84	2	2	NUM
asir-3636	259	85	+	+	NOUN
asir-3636	259	86	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	259	87	2	2	NUM
asir-3636	259	88	)	)	PUNCT
asir-3636	259	89	|	|	ADV
asir-3636	259	90	𝑗	𝑗	INTJ
asir-3636	259	91	2	2	NUM
asir-3636	259	92	(	(	PUNCT
asir-3636	259	93	1	1	NUM
asir-3636	259	94	2	2	NUM
asir-3636	259	95	+	+	NOUN
asir-3636	259	96	𝜖	𝜖	X
asir-3636	259	97	)	)	PUNCT
asir-3636	259	98	𝑑	𝑑	NOUN
asir-3636	259	99	(	(	PUNCT
asir-3636	259	100	1	1	NUM
asir-3636	259	101	2	2	NUM
asir-3636	259	102	+	+	NOUN
asir-3636	259	103	𝜖	𝜖	NOUN
asir-3636	259	104	)	)	PUNCT
asir-3636	259	105	1	1	NUM
asir-3636	259	106	(	(	PUNCT
asir-3636	259	107	1−𝜖	1−𝜖	NUM
asir-3636	259	108	)	)	PUNCT
asir-3636	259	109	.	.	PUNCT
asir-3636	260	1	therefore	therefore	ADV
asir-3636	260	2	∫	∫	PROPN
asir-3636	260	3	∑	∑	PROPN
asir-3636	260	4	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	260	5	2	2	NUM
asir-3636	260	6	(	(	PUNCT
asir-3636	260	7	𝑧	𝑧	NOUN
asir-3636	260	8	)	)	PUNCT
asir-3636	260	9	−	−	PROPN
asir-3636	260	10	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	260	11	2	2	NUM
asir-3636	260	12	(	(	PUNCT
asir-3636	260	13	𝑧	𝑧	PROPN
asir-3636	260	14	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	260	15	)	)	PUNCT
asir-3636	260	16	|	|	ADV
asir-3636	260	17	2(1+𝜖	2(1+𝜖	NUM
asir-3636	260	18	)	)	PUNCT
asir-3636	260	19	(	(	PUNCT
asir-3636	260	20	1	1	NUM
asir-3636	260	21	−	−	NOUN
asir-3636	260	22	|𝑧|)2	|𝑧|)2	NOUN
asir-3636	260	23	𝑗	𝑗	X
asir-3636	260	24	∆𝛾	∆𝛾	PROPN
asir-3636	260	25	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	260	26	)	)	PUNCT
asir-3636	260	27	=	=	SYM
asir-3636	261	1	∫	∫	PROPN
asir-3636	261	2	(	(	PUNCT
asir-3636	261	3	∫	∫	PROPN
asir-3636	261	4	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	261	5	2	2	NUM
asir-3636	261	6	(	(	PUNCT
asir-3636	261	7	(	(	PUNCT
asir-3636	261	8	1	1	NUM
asir-3636	261	9	−	−	PROPN
asir-3636	261	10	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	261	11	2	2	NUM
asir-3636	261	12	)	)	PUNCT
asir-3636	261	13	–	–	PUNCT
asir-3636	261	14	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	261	15	2	2	NUM
asir-3636	261	16	(	(	PUNCT
asir-3636	261	17	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	261	18	2	2	NUM
asir-3636	261	19	)	)	PUNCT
asir-3636	262	1	|	|	ADV
asir-3636	262	2	𝑗	𝑗	NOUN
asir-3636	262	3	2(1+𝜖	2(1+𝜖	NUM
asir-3636	262	4	)	)	PUNCT
asir-3636	262	5	(	(	PUNCT
asir-3636	262	6	1	1	NUM
asir-3636	262	7	−	−	PROPN
asir-3636	262	8	𝜖)𝑑𝑡	𝜖)𝑑𝑡	PROPN
asir-3636	262	9	𝜋	𝜋	NOUN
asir-3636	262	10	𝛾	𝛾	NOUN
asir-3636	262	11	)	)	PUNCT
asir-3636	262	12	1	1	NUM
asir-3636	262	13	0	0	X
asir-3636	262	14	𝑑(1	𝑑(1	NOUN
asir-3636	262	15	−	−	NUM
asir-3636	262	16	𝜖	𝜖	NOUN
asir-3636	262	17	)	)	PUNCT
asir-3636	262	18	𝜖2	𝜖2	PROPN
asir-3636	262	19	≤∑‖(𝑓𝑗	≤∑‖(𝑓𝑗	PROPN
asir-3636	262	20	2	2	NUM
asir-3636	262	21	)	)	PUNCT
asir-3636	262	22	′‖	′‖	X
asir-3636	262	23	𝐿2(∆𝛾	𝐿2(∆𝛾	PROPN
asir-3636	262	24	)	)	PUNCT
asir-3636	262	25	∫	∫	PROPN
asir-3636	262	26	1	1	NUM
asir-3636	262	27	ϵ1−ε(1+𝜖	ϵ1−ε(1+𝜖	NUM
asir-3636	262	28	)	)	PUNCT
asir-3636	262	29	𝟏	𝟏	PUNCT
asir-3636	263	1	𝟎	𝟎	X
asir-3636	263	2	𝑑(1	𝑑(1	PROPN
asir-3636	263	3	−	−	NUM
asir-3636	263	4	𝜖	𝜖	X
asir-3636	263	5	)	)	PUNCT
asir-3636	263	6	𝑗	𝑗	INTJ
asir-3636	263	7	.	.	PUNCT
asir-3636	264	1	this	this	PRON
asir-3636	264	2	completes	complete	VERB
asir-3636	264	3	the	the	DET
asir-3636	264	4	proof	proof	NOUN
asir-3636	264	5	.	.	PUNCT
asir-3636	265	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-3636	265	2	applied	apply	VERB
asir-3636	265	3	science	science	NOUN
asir-3636	265	4	and	and	CCONJ
asir-3636	265	5	innovative	innovative	ADJ
asir-3636	265	6	research	research	NOUN
asir-3636	265	7	vol	vol	NOUN
asir-3636	265	8	.	.	PROPN
asir-3636	266	1	5	5	NUM
asir-3636	266	2	,	,	PUNCT
asir-3636	266	3	no	no	INTJ
asir-3636	266	4	.	.	NOUN
asir-3636	266	5	1	1	NUM
asir-3636	266	6	,	,	PUNCT
asir-3636	266	7	2021	2021	NUM
asir-3636	266	8	32	32	NUM
asir-3636	266	9	published	publish	VERB
asir-3636	266	10	by	by	ADP
asir-3636	266	11	scholink	scholink	PROPN
asir-3636	266	12	inc	inc	PROPN
asir-3636	266	13	.	.	PUNCT
asir-3636	267	1	now	now	ADV
asir-3636	267	2	,	,	PUNCT
asir-3636	267	3	we	we	PRON
asir-3636	267	4	can	can	AUX
asir-3636	267	5	state	state	VERB
asir-3636	267	6	the	the	DET
asir-3636	267	7	following	following	ADJ
asir-3636	267	8	result	result	NOUN
asir-3636	267	9	(	(	PUNCT
asir-3636	267	10	see	see	VERB
asir-3636	267	11	brahim	brahim	PROPN
asir-3636	267	12	bouya	bouya	PROPN
asir-3636	267	13	,	,	PUNCT
asir-3636	267	14	2008	2008	NUM
asir-3636	267	15	)	)	PUNCT
asir-3636	267	16	.	.	PUNCT
asir-3636	268	1	lemma	lemma	PROPN
asir-3636	268	2	(	(	PUNCT
asir-3636	268	3	4.3	4.3	NUM
asir-3636	268	4	):	):	PUNCT
asir-3636	268	5	∫	∫	PROPN
asir-3636	268	6	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	268	7	2	2	NUM
asir-3636	268	8	(	(	PUNCT
asir-3636	268	9	𝑧)|	𝑧)|	NOUN
asir-3636	268	10	2(1+𝜖	2(1+𝜖	NUM
asir-3636	268	11	)	)	PUNCT
asir-3636	268	12	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	268	13	2	2	X
asir-3636	268	14	)	)	PUNCT
asir-3636	268	15	γ	γ	NOUN
asir-3636	268	16	)	)	PUNCT
asir-3636	268	17	′	′	NUM
asir-3636	269	1	(	(	PUNCT
asir-3636	269	2	𝑧)|	𝑧)|	ADJ
asir-3636	269	3	2	2	NUM
asir-3636	269	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
asir-3636	269	5	)	)	PUNCT
asir-3636	269	6	𝑗	𝑗	PROPN
asir-3636	269	7	≤	≤	PROPN
asir-3636	269	8	∆𝛾	∆𝛾	PROPN
asir-3636	269	9	1	1	NUM
asir-3636	269	10	∑𝐶(1+𝜖	∑𝐶(1+𝜖	PROPN
asir-3636	269	11	)	)	PUNCT
asir-3636	269	12	𝒋	𝒋	X
asir-3636	269	13	‖(𝑓𝑗	‖(𝑓𝑗	PROPN
asir-3636	269	14	2	2	NUM
asir-3636	269	15	)	)	PUNCT
asir-3636	269	16	′‖𝐿2(∆𝛾	′‖𝐿2(∆𝛾	PROPN
asir-3636	269	17	)	)	PUNCT
asir-3636	269	18	2	2	NUM
asir-3636	269	19	.	.	PUNCT
asir-3636	270	1	proof	proof	NOUN
asir-3636	270	2	:	:	PUNCT
asir-3636	270	3	.	.	PUNCT
asir-3636	271	1	by	by	ADP
asir-3636	271	2	cauchy	cauchy	PROPN
asir-3636	271	3	’s	’s	PART
asir-3636	271	4	estimate	estimate	NOUN
asir-3636	271	5	,	,	PUNCT
asir-3636	271	6	it	it	PRON
asir-3636	271	7	follows	follow	VERB
asir-3636	271	8	that	that	SCONJ
asir-3636	271	9	∑	∑	PROPN
asir-3636	271	10	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	271	11	2	2	NUM
asir-3636	271	12	)	)	PUNCT
asir-3636	271	13	γ	γ	NOUN
asir-3636	271	14	)	)	PUNCT
asir-3636	271	15	′((1	′((1	PROPN
asir-3636	271	16	−	−	PROPN
asir-3636	271	17	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	271	18	2	2	NUM
asir-3636	271	19	)	)	PUNCT
asir-3636	271	20	|𝑗	|𝑗	VERB
asir-3636	271	21	≤	≤	NUM
asir-3636	271	22	1	1	NUM
asir-3636	271	23	𝜖	𝜖	X
asir-3636	271	24	.	.	PUNCT
asir-3636	272	1	using	use	VERB
asir-3636	272	2	lemma	lemma	PROPN
asir-3636	272	3	(	(	PUNCT
asir-3636	272	4	4.2	4.2	NUM
asir-3636	272	5	)	)	PUNCT
asir-3636	272	6	,	,	PUNCT
asir-3636	272	7	we	we	PRON
asir-3636	272	8	get	get	VERB
asir-3636	272	9	∫	∫	PROPN
asir-3636	272	10	∑	∑	PROPN
asir-3636	272	11	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	272	12	2	2	NUM
asir-3636	272	13	(	(	PUNCT
asir-3636	272	14	𝑧)|	𝑧)|	ADJ
asir-3636	272	15	2(1+𝜖	2(1+𝜖	NUM
asir-3636	272	16	)	)	PUNCT
asir-3636	272	17	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	272	18	2	2	NUM
asir-3636	272	19	)	)	PUNCT
asir-3636	272	20	γ	γ	NOUN
asir-3636	272	21	)	)	PUNCT
asir-3636	272	22	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	272	23	2	2	NUM
asir-3636	272	24	𝑑𝐴(𝑧)𝒋	𝑑𝐴(𝑧)𝒋	PROPN
asir-3636	272	25	∆𝛾	∆𝛾	PROPN
asir-3636	272	26	1	1	NUM
asir-3636	272	27	≤	≤	NUM
asir-3636	272	28	∫	∫	PROPN
asir-3636	272	29	∑	∑	PROPN
asir-3636	272	30	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	272	31	2	2	NUM
asir-3636	272	32	(	(	PUNCT
asir-3636	272	33	𝑧)|	𝑧)|	NOUN
asir-3636	272	34	2(1+𝜖	2(1+𝜖	NUM
asir-3636	272	35	)	)	PUNCT
asir-3636	272	36	(	(	PUNCT
asir-3636	272	37	1−|𝑧|)2𝒋	1−|𝑧|)2𝒋	NUM
asir-3636	272	38	∆𝛾	∆𝛾	PROPN
asir-3636	272	39	1	1	NUM
asir-3636	272	40	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	272	41	)	)	PUNCT
asir-3636	272	42	≤	≤	NOUN
asir-3636	272	43	∑	∑	PUNCT
asir-3636	272	44	𝐶(1+𝜖)‖(𝑓𝑗	𝐶(1+𝜖)‖(𝑓𝑗	PROPN
asir-3636	272	45	2	2	NUM
asir-3636	272	46	)	)	PUNCT
asir-3636	272	47	′‖	′‖	X
asir-3636	272	48	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	272	49	)	)	PUNCT
asir-3636	272	50	2	2	NUM
asir-3636	272	51	𝑗	𝑗	NOUN
asir-3636	272	52	+	+	NOUN
asir-3636	272	53	2(2𝜖+1	2(2𝜖+1	NUM
asir-3636	272	54	)	)	PUNCT
asir-3636	272	55	∫	∫	PROPN
asir-3636	272	56	∑	∑	PROPN
asir-3636	272	57	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	272	58	2	2	NUM
asir-3636	272	59	(	(	PUNCT
asir-3636	272	60	𝑧	𝑧	PROPN
asir-3636	272	61	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	272	62	)	)	PUNCT
asir-3636	272	63	|	|	ADV
asir-3636	272	64	2(1+𝜖	2(1+𝜖	NUM
asir-3636	272	65	)	)	PUNCT
asir-3636	272	66	(	(	PUNCT
asir-3636	272	67	1−|𝑧|)2𝒋	1−|𝑧|)2𝒋	NUM
asir-3636	272	68	∆𝛾	∆𝛾	PROPN
asir-3636	272	69	1	1	NUM
asir-3636	272	70	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	272	71	)	)	PUNCT
asir-3636	272	72	.	.	PUNCT
asir-3636	273	1	(	(	PUNCT
asir-3636	273	2	14	14	NUM
asir-3636	273	3	)	)	PUNCT
asir-3636	273	4	using	use	VERB
asir-3636	273	5	lemma	lemma	PROPN
asir-3636	273	6	(	(	PUNCT
asir-3636	273	7	4.1	4.1	NUM
asir-3636	273	8	)	)	PUNCT
asir-3636	273	9	,	,	PUNCT
asir-3636	273	10	we	we	PRON
asir-3636	273	11	obtain	obtain	VERB
asir-3636	273	12	∫	∫	PROPN
asir-3636	273	13	∑	∑	PROPN
asir-3636	273	14	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	273	15	2	2	NUM
asir-3636	273	16	(	(	PUNCT
asir-3636	273	17	𝑧	𝑧	PROPN
asir-3636	273	18	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	273	19	)	)	PUNCT
asir-3636	273	20	|	|	ADV
asir-3636	273	21	2(1+𝜖	2(1+𝜖	NUM
asir-3636	273	22	)	)	PUNCT
asir-3636	274	1	(	(	PUNCT
asir-3636	274	2	1−|𝑧|)2𝒋	1−|𝑧|)2𝒋	NUM
asir-3636	274	3	∆𝛾	∆𝛾	PROPN
asir-3636	274	4	1	1	NUM
asir-3636	274	5	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	274	6	)	)	PUNCT
asir-3636	274	7	=	=	SYM
asir-3636	274	8	1	1	NUM
asir-3636	274	9	𝜇	𝜇	ADP
asir-3636	274	10	∫	∫	PROPN
asir-3636	274	11	∑	∑	PROPN
asir-3636	274	12	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	274	13	2	2	NUM
asir-3636	274	14	(	(	PUNCT
asir-3636	274	15	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	274	16	2	2	NUM
asir-3636	274	17	)	)	PUNCT
asir-3636	274	18	|	|	ADV
asir-3636	274	19	2(1+𝜖	2(1+𝜖	NUM
asir-3636	274	20	)	)	PUNCT
asir-3636	274	21	𝝐2𝒋	𝝐2𝒋	PUNCT
asir-3636	274	22	(	(	PUNCT
asir-3636	274	23	1	1	NUM
asir-3636	274	24	−	−	NOUN
asir-3636	274	25	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	274	26	−	−	PROPN
asir-3636	274	27	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	274	28	∆𝛾	∆𝛾	PROPN
asir-3636	274	29	1	1	NUM
asir-3636	274	30	≤	≤	NOUN
asir-3636	274	31	𝐶	𝐶	PROPN
asir-3636	274	32	𝜋	𝜋	X
asir-3636	274	33	∫	∫	PROPN
asir-3636	274	34	∑	∑	PROPN
asir-3636	274	35	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	274	36	2	2	NUM
asir-3636	274	37	(	(	PUNCT
asir-3636	274	38	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	274	39	2	2	NUM
asir-3636	274	40	)	)	PUNCT
asir-3636	274	41	|	|	ADV
asir-3636	274	42	2(1+𝜖	2(1+𝜖	NUM
asir-3636	274	43	)	)	PUNCT
asir-3636	274	44	𝝐2𝒋	𝝐2𝒋	NOUN
asir-3636	274	45	𝑑𝑡2	𝑑𝑡2	VERB
asir-3636	274	46	≤	≤	NOUN
asir-3636	274	47	∑	∑	PUNCT
asir-3636	274	48	𝐶(1+𝜖)‖(𝑓𝑗	𝐶(1+𝜖)‖(𝑓𝑗	PROPN
asir-3636	274	49	2	2	NUM
asir-3636	274	50	)	)	PUNCT
asir-3636	274	51	′‖	′‖	X
asir-3636	274	52	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	274	53	)	)	PUNCT
asir-3636	274	54	2	2	NUM
asir-3636	274	55	𝑗	𝑗	NOUN
asir-3636	274	56	.	.	PUNCT
asir-3636	275	1	𝛾	𝛾	X
asir-3636	275	2	(	(	PUNCT
asir-3636	275	3	15	15	NUM
asir-3636	275	4	)	)	PUNCT
asir-3636	275	5	the	the	DET
asir-3636	275	6	result	result	NOUN
asir-3636	275	7	of	of	ADP
asir-3636	275	8	our	our	PRON
asir-3636	275	9	lemma	lemma	PROPN
asir-3636	275	10	follows	follow	VERB
asir-3636	275	11	by	by	ADP
asir-3636	275	12	combining	combine	VERB
asir-3636	275	13	the	the	DET
asir-3636	275	14	estimates	estimate	NOUN
asir-3636	275	15	(	(	PUNCT
asir-3636	275	16	14	14	NUM
asir-3636	275	17	)	)	PUNCT
asir-3636	275	18	and	and	CCONJ
asir-3636	275	19	(	(	PUNCT
asir-3636	275	20	15	15	NUM
asir-3636	275	21	)	)	PUNCT
asir-3636	275	22	.	.	PUNCT
asir-3636	276	1	the	the	DET
asir-3636	276	2	integral	integral	ADJ
asir-3636	276	3	on	on	ADP
asir-3636	276	4	the	the	DET
asir-3636	276	5	region	region	NOUN
asir-3636	276	6	∆𝛾	∆𝛾	PROPN
asir-3636	276	7	2	2	NUM
asir-3636	276	8	.	.	PUNCT
asir-3636	277	1	in	in	ADP
asir-3636	277	2	this	this	DET
asir-3636	277	3	subsection	subsection	NOUN
asir-3636	277	4	,	,	PUNCT
asir-3636	277	5	we	we	PRON
asir-3636	277	6	estimate	estimate	VERB
asir-3636	277	7	the	the	DET
asir-3636	277	8	integral	integral	ADJ
asir-3636	277	9	∫	∫	NOUN
asir-3636	277	10	∑	∑	PROPN
asir-3636	277	11	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	277	12	2	2	NUM
asir-3636	277	13	(	(	PUNCT
asir-3636	277	14	𝑧)|	𝑧)|	ADJ
asir-3636	277	15	2(1+𝜖	2(1+𝜖	NUM
asir-3636	277	16	)	)	PUNCT
asir-3636	277	17	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	277	18	2	2	NUM
asir-3636	277	19	)	)	PUNCT
asir-3636	277	20	γ	γ	NOUN
asir-3636	277	21	)	)	PUNCT
asir-3636	277	22	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	277	23	2	2	NUM
asir-3636	277	24	𝑑𝐴(𝑧)𝒋	𝑑𝐴(𝑧)𝒋	PROPN
asir-3636	277	25	∆𝛾	∆𝛾	PROPN
asir-3636	277	26	2	2	NUM
asir-3636	277	27	.	.	PUNCT
asir-3636	278	1	before	before	ADP
asir-3636	278	2	this	this	PRON
asir-3636	278	3	,	,	PUNCT
asir-3636	278	4	we	we	PRON
asir-3636	278	5	make	make	VERB
asir-3636	278	6	some	some	DET
asir-3636	278	7	remarks	remark	NOUN
asir-3636	278	8	.	.	PUNCT
asir-3636	279	1	for	for	ADP
asir-3636	279	2	𝑧	𝑧	PRON
asir-3636	279	3	∈	∈	PROPN
asir-3636	279	4	𝔻	𝔻	PROPN
asir-3636	279	5	define	define	VERB
asir-3636	279	6	𝑎𝛾(𝑧	𝑎𝛾(𝑧	NOUN
asir-3636	279	7	)	)	PUNCT
asir-3636	279	8	≔	≔	NOUN
asir-3636	279	9	{	{	PUNCT
asir-3636	279	10	1	1	NUM
asir-3636	279	11	2𝜋	2𝜋	NUM
asir-3636	279	12	∫	∫	NOUN
asir-3636	279	13	∑	∑	PROPN
asir-3636	279	14	−log|𝑓𝑗	−log|𝑓𝑗	PROPN
asir-3636	279	15	2	2	NUM
asir-3636	279	16	(	(	PUNCT
asir-3636	279	17	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	279	18	2	2	NUM
asir-3636	279	19	)	)	PUNCT
asir-3636	279	20	|	|	ADV
asir-3636	279	21	|𝑒𝑖𝜃	|𝑒𝑖𝜃	NOUN
asir-3636	279	22	2	2	NUM
asir-3636	279	23	−	−	NOUN
asir-3636	279	24	𝑧|	𝑧|	ADV
asir-3636	279	25	2	2	NUM
asir-3636	279	26	𝑑𝜃2	𝑑𝜃2	NOUN
asir-3636	279	27	𝑗	𝑗	INTJ
asir-3636	279	28	𝑖𝑓	𝑖𝑓	NOUN
asir-3636	279	29	𝛾	𝛾	ADP
asir-3636	279	30	⊈	⊈	PROPN
asir-3636	279	31	γ	γ	X
asir-3636	279	32	γ	γ	X
asir-3636	279	33	1	1	NUM
asir-3636	279	34	2𝜋	2𝜋	NUM
asir-3636	279	35	∫	∫	NOUN
asir-3636	279	36	∑	∑	PROPN
asir-3636	279	37	−log|𝑓𝑗	−log|𝑓𝑗	PROPN
asir-3636	279	38	2	2	NUM
asir-3636	279	39	(	(	PUNCT
asir-3636	279	40	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	279	41	2	2	NUM
asir-3636	279	42	)	)	PUNCT
asir-3636	279	43	|	|	ADV
asir-3636	279	44	|𝑒𝑖𝜃	|𝑒𝑖𝜃	NOUN
asir-3636	279	45	2	2	NUM
asir-3636	279	46	−	−	NOUN
asir-3636	279	47	𝑧|	𝑧|	ADV
asir-3636	279	48	2	2	NUM
asir-3636	279	49	𝑑𝜃2	𝑑𝜃2	NOUN
asir-3636	279	50	𝑗	𝑗	INTJ
asir-3636	279	51	𝑖𝑓	𝑖𝑓	NOUN
asir-3636	279	52	𝛾	𝛾	ADP
asir-3636	279	53	⊈	⊈	PROPN
asir-3636	279	54	γ	γ	X
asir-3636	279	55	.	.	PUNCT
asir-3636	279	56	𝕋∖γ	𝕋∖γ	NOUN
asir-3636	279	57	using	use	VERB
asir-3636	279	58	the	the	DET
asir-3636	279	59	equation	equation	NOUN
asir-3636	279	60	(	(	PUNCT
asir-3636	279	61	10	10	NUM
asir-3636	279	62	)	)	PUNCT
asir-3636	279	63	,	,	PUNCT
asir-3636	279	64	it	it	PRON
asir-3636	279	65	is	be	AUX
asir-3636	279	66	easy	easy	ADJ
asir-3636	279	67	to	to	PART
asir-3636	279	68	see	see	VERB
asir-3636	279	69	that	that	SCONJ
asir-3636	279	70	∑	∑	ADP
asir-3636	279	71	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	279	72	2	2	NUM
asir-3636	279	73	(	(	PUNCT
asir-3636	279	74	𝑧)1+𝜖((𝑓𝑗	𝑧)1+𝜖((𝑓𝑗	PROPN
asir-3636	279	75	2	2	NUM
asir-3636	279	76	)	)	PUNCT
asir-3636	279	77	γ	γ	NOUN
asir-3636	279	78	)	)	PUNCT
asir-3636	279	79	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	279	80	2	2	NUM
asir-3636	279	81	𝑗	𝑗	X
asir-3636	279	82	≤	≤	NUM
asir-3636	279	83	4∑	4∑	NOUN
asir-3636	279	84	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	279	85	2	2	NUM
asir-3636	279	86	(	(	PUNCT
asir-3636	279	87	𝑧)1+𝜖	𝑧)1+𝜖	ADJ
asir-3636	279	88	1	1	NUM
asir-3636	279	89	2𝜋	2𝜋	NUM
asir-3636	279	90	∫	∫	NOUN
asir-3636	279	91	−log|𝑓𝑗	−log|𝑓𝑗	PROPN
asir-3636	279	92	2	2	NUM
asir-3636	279	93	(	(	PUNCT
asir-3636	279	94	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	279	95	2	2	NUM
asir-3636	279	96	)	)	PUNCT
asir-3636	279	97	|	|	ADV
asir-3636	279	98	|𝑒𝑖𝜃	|𝑒𝑖𝜃	PROPN
asir-3636	279	99	2	2	NUM
asir-3636	279	100	−𝑧|	−𝑧|	NUM
asir-3636	279	101	2	2	NUM
asir-3636	279	102	𝑑𝜃2	𝑑𝜃2	NOUN
asir-3636	279	103	γ	γ	X
asir-3636	279	104	|	|	ADV
asir-3636	279	105	2	2	NUM
asir-3636	279	106	𝑗	𝑗	NOUN
asir-3636	279	107	.	.	PUNCT
asir-3636	280	1	(	(	PUNCT
asir-3636	280	2	16	16	NUM
asir-3636	280	3	)	)	PUNCT
asir-3636	280	4	using	use	VERB
asir-3636	280	5	the	the	DET
asir-3636	280	6	equation	equation	NOUN
asir-3636	280	7	(	(	PUNCT
asir-3636	280	8	11	11	NUM
asir-3636	280	9	)	)	PUNCT
asir-3636	280	10	,	,	PUNCT
asir-3636	280	11	it	it	PRON
asir-3636	280	12	is	be	AUX
asir-3636	280	13	clear	clear	ADJ
asir-3636	280	14	that	that	SCONJ
asir-3636	280	15	∑	∑	ADP
asir-3636	280	16	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	280	17	2	2	NUM
asir-3636	280	18	(	(	PUNCT
asir-3636	280	19	𝑧)1+𝜖((𝑓𝑗	𝑧)1+𝜖((𝑓𝑗	PROPN
asir-3636	280	20	2	2	NUM
asir-3636	280	21	)	)	PUNCT
asir-3636	280	22	γ	γ	PROPN
asir-3636	280	23	)	)	PUNCT
asir-3636	280	24	′(𝑧)|𝑗	′(𝑧)|𝑗	VERB
asir-3636	280	25	2	2	NUM
asir-3636	280	26	≤	≤	NUM
asir-3636	280	27	2∑	2∑	NUM
asir-3636	280	28	|(𝑓𝑗	|(𝑓𝑗	SYM
asir-3636	280	29	2	2	NUM
asir-3636	280	30	)	)	PUNCT
asir-3636	280	31	′(𝑧)|𝑗	′(𝑧)|𝑗	NOUN
asir-3636	280	32	2	2	NUM
asir-3636	280	33	+	+	CCONJ
asir-3636	280	34	8∑	8∑	NUM
asir-3636	280	35	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	280	36	2	2	NUM
asir-3636	280	37	(	(	PUNCT
asir-3636	280	38	𝑧)1+𝜖	𝑧)1+𝜖	ADJ
asir-3636	280	39	1	1	NUM
asir-3636	280	40	2𝜋	2𝜋	NUM
asir-3636	280	41	∫	∫	NOUN
asir-3636	280	42	−log|𝑓𝑗	−log|𝑓𝑗	PROPN
asir-3636	280	43	2	2	NUM
asir-3636	280	44	(	(	PUNCT
asir-3636	280	45	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	280	46	2	2	NUM
asir-3636	280	47	)	)	PUNCT
asir-3636	280	48	|	|	ADV
asir-3636	280	49	|𝑒𝑖𝜃	|𝑒𝑖𝜃	PROPN
asir-3636	280	50	2	2	NUM
asir-3636	280	51	−𝑧|	−𝑧|	NUM
asir-3636	280	52	2	2	NUM
asir-3636	280	53	𝑑𝜃2	𝑑𝜃2	NOUN
asir-3636	280	54	𝕋∖γ	𝕋∖γ	NOUN
asir-3636	280	55	|	|	ADV
asir-3636	280	56	2	2	NUM
asir-3636	280	57	𝑗	𝑗	NOUN
asir-3636	280	58	.	.	PUNCT
asir-3636	281	1	(	(	PUNCT
asir-3636	281	2	17	17	NUM
asir-3636	281	3	)	)	PUNCT
asir-3636	281	4	then	then	ADV
asir-3636	281	5	∫	∫	PROPN
asir-3636	281	6	∑	∑	PROPN
asir-3636	281	7	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	281	8	2	2	NUM
asir-3636	281	9	(	(	PUNCT
asir-3636	281	10	𝑧)|	𝑧)|	ADJ
asir-3636	281	11	2(1+𝜖	2(1+𝜖	NUM
asir-3636	281	12	)	)	PUNCT
asir-3636	281	13	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	281	14	2	2	NUM
asir-3636	281	15	)	)	PUNCT
asir-3636	281	16	γ	γ	NOUN
asir-3636	281	17	)	)	PUNCT
asir-3636	281	18	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	281	19	2	2	NUM
asir-3636	281	20	𝑑𝐴(𝑧)𝒋	𝑑𝐴(𝑧)𝒋	PROPN
asir-3636	281	21	∆𝛾	∆𝛾	PROPN
asir-3636	281	22	2	2	NUM
asir-3636	281	23	≤	≤	NOUN
asir-3636	281	24	2∑	2∑	NUM
asir-3636	281	25	‖(𝑓𝑗	‖(𝑓𝑗	ADP
asir-3636	281	26	2	2	NUM
asir-3636	281	27	)	)	PUNCT
asir-3636	281	28	′‖	′‖	X
asir-3636	281	29	𝐿2(∆𝛾	𝐿2(∆𝛾	PROPN
asir-3636	281	30	)	)	PUNCT
asir-3636	281	31	2	2	NUM
asir-3636	281	32	j	j	NOUN
asir-3636	281	33	+	+	CCONJ
asir-3636	281	34	8∫	8∫	NUM
asir-3636	281	35	∑	∑	SYM
asir-3636	281	36	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	281	37	2	2	NUM
asir-3636	281	38	(	(	PUNCT
asir-3636	281	39	𝑧)2(1+𝜖)𝑎𝛾	𝑧)2(1+𝜖)𝑎𝛾	VERB
asir-3636	281	40	2(𝑧)𝑑𝐴(𝑧)𝒋	2(𝑧)𝑑𝐴(𝑧)𝒋	PROPN
asir-3636	281	41	∆𝛾	∆𝛾	PROPN
asir-3636	281	42	2	2	NUM
asir-3636	281	43	.	.	PUNCT
asir-3636	282	1	(	(	PUNCT
asir-3636	282	2	18	18	NUM
asir-3636	282	3	)	)	PUNCT
asir-3636	282	4	since	since	SCONJ
asir-3636	282	5	log	log	VERB
asir-3636	282	6	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	282	7	2	2	NUM
asir-3636	282	8	|	|	NOUN
asir-3636	282	9	∈	∈	PROPN
asir-3636	282	10	𝐿1(𝕋	𝐿1(𝕋	NOUN
asir-3636	282	11	)	)	PUNCT
asir-3636	282	12	,	,	PUNCT
asir-3636	282	13	we	we	PRON
asir-3636	282	14	have	have	VERB
asir-3636	282	15	𝑎𝛾(𝑧	𝑎𝛾(𝑧	NOUN
asir-3636	282	16	)	)	PUNCT
asir-3636	283	1	≤	≤	NUM
asir-3636	284	1	𝐶	𝐶	PROPN
asir-3636	284	2	𝑑2(𝑧	𝑑2(𝑧	PROPN
asir-3636	284	3	)	)	PUNCT
asir-3636	284	4	(	(	PUNCT
asir-3636	284	5	𝑧	𝑧	PROPN
asir-3636	284	6	∈	∈	PROPN
asir-3636	284	7	∆𝛾	∆𝛾	PROPN
asir-3636	284	8	)	)	PUNCT
asir-3636	284	9	(	(	PUNCT
asir-3636	284	10	19	19	NUM
asir-3636	284	11	)	)	PUNCT
asir-3636	284	12	given	give	VERB
asir-3636	284	13	such	such	ADJ
asir-3636	284	14	inequality	inequality	NOUN
asir-3636	285	1	,	,	PUNCT
asir-3636	285	2	it	it	PRON
asir-3636	285	3	is	be	AUX
asir-3636	285	4	not	not	PART
asir-3636	285	5	easy	easy	ADJ
asir-3636	285	6	to	to	PART
asir-3636	285	7	estimate	estimate	VERB
asir-3636	285	8	immediately	immediately	ADV
asir-3636	285	9	the	the	DET
asir-3636	285	10	integral	integral	NOUN
asir-3636	285	11	of	of	ADP
asir-3636	285	12	the	the	DET
asir-3636	285	13	series	series	NOUN
asir-3636	285	14	of	of	ADP
asir-3636	285	15	functions	function	NOUN
asir-3636	285	16	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	VERB
asir-3636	285	17	applied	apply	VERB
asir-3636	285	18	science	science	NOUN
asir-3636	285	19	and	and	CCONJ
asir-3636	285	20	innovative	innovative	ADJ
asir-3636	285	21	research	research	NOUN
asir-3636	285	22	vol	vol	NOUN
asir-3636	285	23	.	.	PROPN
asir-3636	286	1	5	5	NUM
asir-3636	286	2	,	,	PUNCT
asir-3636	286	3	no	no	INTJ
asir-3636	286	4	.	.	NOUN
asir-3636	286	5	1	1	NUM
asir-3636	286	6	,	,	PUNCT
asir-3636	286	7	2021	2021	NUM
asir-3636	286	8	33	33	NUM
asir-3636	286	9	published	publish	VERB
asir-3636	286	10	by	by	ADP
asir-3636	286	11	scholink	scholink	PROPN
asir-3636	286	12	inc	inc	PROPN
asir-3636	286	13	.	.	PROPN
asir-3636	286	14	∑	∑	ADP
asir-3636	286	15	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	286	16	2	2	NUM
asir-3636	286	17	(	(	PUNCT
asir-3636	286	18	𝑧)|2(1+𝜖)𝑎𝛾	𝑧)|2(1+𝜖)𝑎𝛾	X
asir-3636	286	19	2(𝑧)𝑗	2(𝑧)𝑗	NOUN
asir-3636	286	20	on	on	ADP
asir-3636	286	21	the	the	DET
asir-3636	286	22	whole	whole	ADJ
asir-3636	286	23	∆𝛾	∆𝛾	PROPN
asir-3636	286	24	2	2	NUM
asir-3636	286	25	.	.	PUNCT
asir-3636	287	1	in	in	ADP
asir-3636	287	2	what	what	PRON
asir-3636	287	3	follows	follow	VERB
asir-3636	287	4	,	,	PUNCT
asir-3636	287	5	we	we	PRON
asir-3636	287	6	give	give	VERB
asir-3636	287	7	a	a	DET
asir-3636	287	8	partition	partition	NOUN
asir-3636	287	9	of	of	ADP
asir-3636	287	10	∆𝛾	∆𝛾	PROPN
asir-3636	287	11	2	2	NUM
asir-3636	287	12	into	into	ADP
asir-3636	287	13	three	three	NUM
asir-3636	287	14	parts	part	NOUN
asir-3636	287	15	so	so	SCONJ
asir-3636	287	16	that	that	SCONJ
asir-3636	287	17	one	one	PRON
asir-3636	287	18	can	can	AUX
asir-3636	287	19	estimate	estimate	VERB
asir-3636	287	20	the	the	DET
asir-3636	287	21	integral	integral	ADJ
asir-3636	287	22	∫	∫	NOUN
asir-3636	287	23	∑	∑	PROPN
asir-3636	287	24	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	287	25	2	2	NUM
asir-3636	287	26	(	(	PUNCT
asir-3636	287	27	𝑧)|	𝑧)|	NOUN
asir-3636	287	28	2(1+𝜖	2(1+𝜖	NUM
asir-3636	287	29	)	)	PUNCT
asir-3636	287	30	𝑎𝛾	𝑎𝛾	ADP
asir-3636	287	31	2(𝑧)𝑑𝐴(𝑧)𝑗	2(𝑧)𝑑𝐴(𝑧)𝑗	NUM
asir-3636	287	32	on	on	ADP
asir-3636	287	33	each	each	DET
asir-3636	287	34	part	part	NOUN
asir-3636	287	35	.	.	PUNCT
asir-3636	288	1	let	let	VERB
asir-3636	288	2	𝑧	𝑧	PRON
asir-3636	288	3	∈	∈	PROPN
asir-3636	288	4	∆𝛾	∆𝛾	PROPN
asir-3636	288	5	2	2	NUM
asir-3636	288	6	,	,	PUNCT
asir-3636	288	7	three	three	NUM
asir-3636	288	8	situations	situation	NOUN
asir-3636	288	9	are	be	AUX
asir-3636	288	10	possible	possible	ADJ
asir-3636	288	11	:	:	PUNCT
asir-3636	288	12	𝑎𝛾(𝑧	𝑎𝛾(𝑧	X
asir-3636	288	13	)	)	PUNCT
asir-3636	288	14	≤	≤	NUM
asir-3636	288	15	8	8	NUM
asir-3636	288	16	|log	|log	NOUN
asir-3636	288	17	(	(	PUNCT
asir-3636	288	18	𝑑(𝑧))|	𝑑(𝑧))|	PROPN
asir-3636	288	19	𝑑(𝑧	𝑑(𝑧	PROPN
asir-3636	288	20	)	)	PUNCT
asir-3636	288	21	,	,	PUNCT
asir-3636	288	22	(	(	PUNCT
asir-3636	288	23	20	20	NUM
asir-3636	288	24	)	)	PUNCT
asir-3636	288	25	8	8	NUM
asir-3636	288	26	|log	|log	X
asir-3636	288	27	(	(	PUNCT
asir-3636	288	28	𝑑(𝑧))|	𝑑(𝑧))|	PROPN
asir-3636	288	29	𝑑(𝑧	𝑑(𝑧	PROPN
asir-3636	288	30	)	)	PUNCT
asir-3636	288	31	<	<	X
asir-3636	288	32	𝑎𝛾(𝑧	𝑎𝛾(𝑧	NOUN
asir-3636	288	33	)	)	PUNCT
asir-3636	288	34	<	<	X
asir-3636	288	35	8	8	NUM
asir-3636	288	36	|log	|log	X
asir-3636	288	37	(	(	PUNCT
asir-3636	288	38	𝑑(𝑧))|	𝑑(𝑧))|	PRON
asir-3636	288	39	𝜖	𝜖	PROPN
asir-3636	288	40	(	(	PUNCT
asir-3636	288	41	21	21	NUM
asir-3636	288	42	)	)	PUNCT
asir-3636	288	43	8	8	NUM
asir-3636	288	44	|log	|log	NOUN
asir-3636	288	45	(	(	PUNCT
asir-3636	288	46	𝑑(𝑧))|	𝑑(𝑧))|	X
asir-3636	288	47	𝜖	𝜖	X
asir-3636	288	48	≤	≤	NOUN
asir-3636	288	49	𝑎𝛾(𝑧	𝑎𝛾(𝑧	PUNCT
asir-3636	288	50	)	)	PUNCT
asir-3636	288	51	(	(	PUNCT
asir-3636	288	52	22	22	X
asir-3636	288	53	)	)	PUNCT
asir-3636	288	54	we	we	PRON
asir-3636	288	55	can	can	AUX
asir-3636	288	56	now	now	ADV
asir-3636	288	57	divide	divide	VERB
asir-3636	288	58	∆𝛾	∆𝛾	PROPN
asir-3636	288	59	2	2	NUM
asir-3636	288	60	into	into	ADP
asir-3636	288	61	the	the	DET
asir-3636	288	62	following	follow	VERB
asir-3636	288	63	three	three	NUM
asir-3636	288	64	parts	part	NOUN
asir-3636	288	65	∆𝛾	∆𝛾	PROPN
asir-3636	288	66	21≔	21≔	NUM
asir-3636	288	67	{	{	PUNCT
asir-3636	288	68	𝑧	𝑧	PROPN
asir-3636	288	69	∈	∈	PROPN
asir-3636	288	70	∆𝛾	∆𝛾	PROPN
asir-3636	288	71	2	2	NUM
asir-3636	288	72	:	:	PUNCT
asir-3636	288	73	𝑧	𝑧	X
asir-3636	288	74	satisfying	satisfying	ADJ
asir-3636	288	75	(	(	PUNCT
asir-3636	288	76	20	20	NUM
asir-3636	288	77	)	)	PUNCT
asir-3636	288	78	}	}	PUNCT
asir-3636	288	79	,	,	PUNCT
asir-3636	288	80	∆𝛾	∆𝛾	PROPN
asir-3636	288	81	22≔	22≔	NUM
asir-3636	288	82	{	{	PUNCT
asir-3636	288	83	𝑧	𝑧	PROPN
asir-3636	288	84	∈	∈	PROPN
asir-3636	288	85	∆𝛾	∆𝛾	PROPN
asir-3636	288	86	2	2	NUM
asir-3636	288	87	:	:	PUNCT
asir-3636	288	88	𝑧	𝑧	X
asir-3636	288	89	satisfying	satisfying	ADJ
asir-3636	288	90	(	(	PUNCT
asir-3636	288	91	21	21	NUM
asir-3636	288	92	)	)	PUNCT
asir-3636	288	93	}	}	PUNCT
asir-3636	288	94	,	,	PUNCT
asir-3636	288	95	∆𝛾	∆𝛾	PROPN
asir-3636	288	96	23≔	23≔	NUM
asir-3636	288	97	{	{	PUNCT
asir-3636	288	98	𝑧	𝑧	PROPN
asir-3636	288	99	∈	∈	PROPN
asir-3636	288	100	∆𝛾	∆𝛾	PROPN
asir-3636	288	101	2	2	NUM
asir-3636	288	102	:	:	PUNCT
asir-3636	288	103	𝑧	𝑧	X
asir-3636	288	104	satisfying	satisfying	ADJ
asir-3636	288	105	(	(	PUNCT
asir-3636	288	106	22	22	NUM
asir-3636	288	107	)	)	PUNCT
asir-3636	288	108	}	}	PUNCT
asir-3636	288	109	,	,	PUNCT
asir-3636	288	110	the	the	DET
asir-3636	288	111	integral	integral	ADJ
asir-3636	288	112	on	on	ADP
asir-3636	288	113	the	the	DET
asir-3636	288	114	regions	region	NOUN
asir-3636	288	115	∆𝛾	∆𝛾	PROPN
asir-3636	288	116	21	21	NUM
asir-3636	288	117	and	and	CCONJ
asir-3636	288	118	∆𝛾	∆𝛾	PROPN
asir-3636	288	119	23	23	NUM
asir-3636	288	120	.	.	PUNCT
asir-3636	289	1	in	in	ADP
asir-3636	289	2	this	this	DET
asir-3636	289	3	case	case	NOUN
asir-3636	289	4	we	we	PRON
asir-3636	289	5	begin	begin	VERB
asir-3636	289	6	by	by	ADP
asir-3636	289	7	the	the	DET
asir-3636	289	8	following	follow	VERB
asir-3636	289	9	(	(	PUNCT
asir-3636	289	10	see	see	VERB
asir-3636	289	11	brahim	brahim	PROPN
asir-3636	289	12	bouya	bouya	PROPN
asir-3636	289	13	,	,	PUNCT
asir-3636	289	14	2008	2008	NUM
asir-3636	289	15	)	)	PUNCT
asir-3636	289	16	.	.	PUNCT
asir-3636	290	1	lemma	lemma	PROPN
asir-3636	290	2	(	(	PUNCT
asir-3636	290	3	4.4	4.4	NUM
asir-3636	290	4	):	):	PUNCT
asir-3636	290	5	∫	∫	PROPN
asir-3636	291	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	291	2	2	2	NUM
asir-3636	291	3	(	(	PUNCT
asir-3636	291	4	𝑧)|	𝑧)|	NOUN
asir-3636	291	5	2(1+𝜖	2(1+𝜖	NUM
asir-3636	291	6	)	)	PUNCT
asir-3636	291	7	𝑎𝛾	𝑎𝛾	ADP
asir-3636	291	8	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	291	9	)	)	PUNCT
asir-3636	292	1	𝑗	𝑗	VERB
asir-3636	292	2	≤	≤	NUM
asir-3636	292	3	∆𝜸	∆𝜸	PROPN
asir-3636	292	4	𝟐𝟏	𝟐𝟏	NUM
asir-3636	292	5	∑𝐶(1+𝜖)‖(𝑓𝑗	∑𝐶(1+𝜖)‖(𝑓𝑗	PROPN
asir-3636	292	6	2	2	NUM
asir-3636	292	7	)	)	PUNCT
asir-3636	292	8	′‖	′‖	X
asir-3636	292	9	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	292	10	)	)	PUNCT
asir-3636	292	11	2	2	NUM
asir-3636	292	12	𝒋	𝒋	NOUN
asir-3636	292	13	.	.	PUNCT
asir-3636	293	1	proof	proof	NOUN
asir-3636	293	2	:	:	PUNCT
asir-3636	293	3	using	use	VERB
asir-3636	293	4	lemma	lemma	PROPN
asir-3636	293	5	(	(	PUNCT
asir-3636	293	6	4.2	4.2	NUM
asir-3636	293	7	)	)	PUNCT
asir-3636	293	8	,	,	PUNCT
asir-3636	293	9	we	we	PRON
asir-3636	293	10	get	get	VERB
asir-3636	293	11	∫	∫	NOUN
asir-3636	294	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	294	2	2	2	NUM
asir-3636	294	3	(	(	PUNCT
asir-3636	294	4	𝑧)|	𝑧)|	NOUN
asir-3636	294	5	2(1+𝜖	2(1+𝜖	NUM
asir-3636	294	6	)	)	PUNCT
asir-3636	294	7	𝑎𝛾	𝑎𝛾	ADP
asir-3636	294	8	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	294	9	)	)	PUNCT
asir-3636	294	10	𝒋	𝒋	PRON
asir-3636	294	11	∆𝜸	∆𝜸	PROPN
asir-3636	294	12	𝟐𝟏	𝟐𝟏	NUM
asir-3636	294	13	≤	≤	NUM
asir-3636	294	14	2(1+𝜖)∫	2(1+𝜖)∫	NUM
asir-3636	295	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	295	2	2	2	NUM
asir-3636	295	3	(	(	PUNCT
asir-3636	295	4	𝑧)|	𝑧)|	PROPN
asir-3636	295	5	𝜖	𝜖	PROPN
asir-3636	295	6	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	295	7	2	2	NUM
asir-3636	295	8	(	(	PUNCT
asir-3636	295	9	𝑧	𝑧	NOUN
asir-3636	295	10	)	)	PUNCT
asir-3636	295	11	−	−	PROPN
asir-3636	295	12	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	295	13	2	2	NUM
asir-3636	295	14	(	(	PUNCT
asir-3636	295	15	𝑧	𝑧	PROPN
asir-3636	295	16	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	295	17	)	)	PUNCT
asir-3636	295	18	|	|	CCONJ
asir-3636	295	19	(	(	PUNCT
asir-3636	295	20	𝜖+2	𝜖+2	NOUN
asir-3636	295	21	)	)	PUNCT
asir-3636	295	22	𝑎𝛾	𝑎𝛾	ADP
asir-3636	295	23	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	295	24	)	)	PUNCT
asir-3636	296	1	𝑗	𝑗	PROPN
asir-3636	296	2	∆𝜸	∆𝜸	PROPN
asir-3636	296	3	𝟐𝟏	𝟐𝟏	NUM
asir-3636	296	4	+	+	NUM
asir-3636	296	5	2(1+𝜖)∫	2(1+𝜖)∫	ADJ
asir-3636	296	6	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	296	7	2	2	NUM
asir-3636	296	8	(	(	PUNCT
asir-3636	296	9	𝑧)|	𝑧)|	ADV
asir-3636	296	10	𝑗	𝑗	PROPN
asir-3636	296	11	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	296	12	2	2	NUM
asir-3636	296	13	(	(	PUNCT
asir-3636	296	14	𝑧	𝑧	PROPN
asir-3636	296	15	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	296	16	)	)	PUNCT
asir-3636	297	1	|	|	ADV
asir-3636	297	2	𝜖+2	𝜖+2	ADV
asir-3636	297	3	𝑎𝛾	𝑎𝛾	PROPN
asir-3636	297	4	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	297	5	)	)	PUNCT
asir-3636	297	6	𝒋	𝒋	PRON
asir-3636	297	7	∆𝜸	∆𝜸	PROPN
asir-3636	297	8	𝟐𝟏	𝟐𝟏	NUM
asir-3636	297	9	≤	≤	PROPN
asir-3636	297	10	𝐶1+𝜖∫	𝐶1+𝜖∫	ADP
asir-3636	297	11	∑	∑	PUNCT
asir-3636	297	12	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	297	13	2	2	NUM
asir-3636	297	14	(	(	PUNCT
asir-3636	297	15	𝑧	𝑧	NOUN
asir-3636	297	16	)	)	PUNCT
asir-3636	297	17	−	−	PROPN
asir-3636	297	18	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	297	19	2	2	NUM
asir-3636	297	20	(	(	PUNCT
asir-3636	297	21	𝑧	𝑧	PROPN
asir-3636	297	22	|𝑧|⁄	|𝑧|⁄	NUM
asir-3636	297	23	)	)	PUNCT
asir-3636	298	1	|	|	ADV
asir-3636	298	2	𝜖+2	𝜖+2	NUM
asir-3636	298	3	(	(	PUNCT
asir-3636	298	4	1	1	NUM
asir-3636	298	5	−	−	NOUN
asir-3636	298	6	|𝑧|)2	|𝑧|)2	NOUN
asir-3636	298	7	𝑗	𝑗	X
asir-3636	298	8	∆𝛾	∆𝛾	PROPN
asir-3636	298	9	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
asir-3636	298	10	)	)	PUNCT
asir-3636	298	11	+	+	CCONJ
asir-3636	298	12	𝐶1+𝜖∫	𝐶1+𝜖∫	ADP
asir-3636	298	13	∑	∑	ADP
asir-3636	298	14	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	298	15	2	2	NUM
asir-3636	298	16	(	(	PUNCT
asir-3636	298	17	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	298	18	2	2	NUM
asir-3636	298	19	)	)	PUNCT
asir-3636	298	20	|	|	ADV
asir-3636	298	21	𝜖+2	𝜖+2	ADV
asir-3636	298	22	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	298	23	2	2	NUM
asir-3636	298	24	)	)	PUNCT
asir-3636	298	25	(	(	PUNCT
asir-3636	298	26	1	1	NUM
asir-3636	298	27	−	−	NOUN
asir-3636	298	28	𝜖)𝑑(1	𝜖)𝑑(1	ADJ
asir-3636	298	29	−	−	PROPN
asir-3636	298	30	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	NOUN
asir-3636	298	31	𝑗	𝑗	PRON
asir-3636	298	32	∆𝜸	∆𝜸	PROPN
asir-3636	298	33	𝟐𝟏	𝟐𝟏	NUM
asir-3636	298	34	≤∑𝐶1+𝜖‖(𝑓𝑗	≤∑𝐶1+𝜖‖(𝑓𝑗	SYM
asir-3636	298	35	2	2	NUM
asir-3636	298	36	)	)	PUNCT
asir-3636	298	37	′‖	′‖	X
asir-3636	298	38	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	298	39	)	)	PUNCT
asir-3636	298	40	2	2	NUM
asir-3636	298	41	𝑗	𝑗	NOUN
asir-3636	298	42	+	+	X
asir-3636	298	43	𝐶1+𝜖∫	𝐶1+𝜖∫	ADP
asir-3636	298	44	∑	∑	ADP
asir-3636	298	45	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	298	46	2	2	NUM
asir-3636	298	47	(	(	PUNCT
asir-3636	298	48	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	298	49	2	2	NUM
asir-3636	298	50	)	)	PUNCT
asir-3636	298	51	|	|	ADV
asir-3636	298	52	𝜖+2	𝜖+2	ADV
asir-3636	298	53	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	298	54	2	2	NUM
asir-3636	298	55	)	)	PUNCT
asir-3636	298	56	𝑑(1	𝑑(1	NOUN
asir-3636	298	57	−	−	NOUN
asir-3636	298	58	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	NOUN
asir-3636	298	59	𝑗	𝑗	PRON
asir-3636	298	60	∆𝜸	∆𝜸	PROPN
asir-3636	298	61	𝟐𝟏	𝟐𝟏	NUM
asir-3636	298	62	=	=	SYM
asir-3636	298	63	𝐼2,1	𝐼2,1	NOUN
asir-3636	298	64	.	.	PUNCT
asir-3636	299	1	let	let	VERB
asir-3636	299	2	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	299	3	2	2	NUM
asir-3636	299	4	∈	∈	NOUN
asir-3636	299	5	𝛾	𝛾	NOUN
asir-3636	299	6	and	and	CCONJ
asir-3636	299	7	denote	denote	VERB
asir-3636	299	8	by	by	ADP
asir-3636	299	9	(	(	PUNCT
asir-3636	299	10	𝑧	𝑧	DET
asir-3636	299	11	−	−	PROPN
asir-3636	299	12	2𝜖)𝑡2	2𝜖)𝑡2	NOUN
asir-3636	299	13	the	the	DET
asir-3636	299	14	point	point	NOUN
asir-3636	299	15	of	of	ADP
asir-3636	299	16	𝜕∆𝛾	𝜕∆𝛾	NOUN
asir-3636	299	17	2	2	NUM
asir-3636	299	18	∩𝔻	∩𝔻	NOUN
asir-3636	299	19	such	such	ADJ
asir-3636	299	20	that	that	PRON
asir-3636	299	21	(	(	PUNCT
asir-3636	299	22	𝑧	𝑧	PROPN
asir-3636	299	23	−	−	PROPN
asir-3636	299	24	2𝜖)𝑡2/|(𝑧	2𝜖)𝑡2/|(𝑧	NUM
asir-3636	299	25	−	−	NOUN
asir-3636	299	26	2𝜖)𝑡2|	2𝜖)𝑡2|	NUM
asir-3636	299	27	=	=	SYM
asir-3636	299	28	𝑒𝑖𝑡	𝑒𝑖𝑡	PROPN
asir-3636	299	29	2	2	NUM
asir-3636	299	30	.	.	PUNCT
asir-3636	300	1	we	we	PRON
asir-3636	300	2	have	have	VERB
asir-3636	300	3	|𝑒𝑖𝑡	|𝑒𝑖𝑡	PROPN
asir-3636	300	4	2	2	NUM
asir-3636	300	5	−	−	NOUN
asir-3636	301	1	(	(	PUNCT
asir-3636	301	2	𝑧	𝑧	PROPN
asir-3636	301	3	−	−	PROPN
asir-3636	301	4	2𝜖)𝑡2|	2𝜖)𝑡2|	NUM
asir-3636	301	5	=	=	SYM
asir-3636	301	6	1	1	NUM
asir-3636	301	7	−	−	NOUN
asir-3636	301	8	|(𝑧	|(𝑧	NOUN
asir-3636	301	9	−	−	PROPN
asir-3636	302	1	2𝜖)𝑡2|	2𝜖)𝑡2|	NUM
asir-3636	302	2	=	=	SYM
asir-3636	302	3	𝑑((𝑧	𝑑((𝑧	NOUN
asir-3636	302	4	−	−	NOUN
asir-3636	302	5	2𝜖)𝑡2	2𝜖)𝑡2	NUM
asir-3636	302	6	)	)	PUNCT
asir-3636	302	7	2	2	NUM
asir-3636	302	8	≤	≤	NOUN
asir-3636	302	9	𝑑(𝑒𝑖𝑡	𝑑(𝑒𝑖𝑡	PROPN
asir-3636	302	10	2	2	NUM
asir-3636	302	11	)	)	PUNCT
asir-3636	302	12	.	.	PUNCT
asir-3636	303	1	then	then	ADV
asir-3636	303	2	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	VERB
asir-3636	303	3	applied	apply	VERB
asir-3636	303	4	science	science	NOUN
asir-3636	303	5	and	and	CCONJ
asir-3636	303	6	innovative	innovative	ADJ
asir-3636	303	7	research	research	NOUN
asir-3636	303	8	vol	vol	NOUN
asir-3636	303	9	.	.	PROPN
asir-3636	304	1	5	5	NUM
asir-3636	304	2	,	,	PUNCT
asir-3636	304	3	no	no	INTJ
asir-3636	304	4	.	.	NOUN
asir-3636	304	5	1	1	NUM
asir-3636	304	6	,	,	PUNCT
asir-3636	304	7	2021	2021	NUM
asir-3636	304	8	34	34	NUM
asir-3636	304	9	published	publish	VERB
asir-3636	304	10	by	by	ADP
asir-3636	304	11	scholink	scholink	PROPN
asir-3636	304	12	inc	inc	PROPN
asir-3636	304	13	.	.	PROPN
asir-3636	304	14	∫	∫	PROPN
asir-3636	305	1	∑	∑	PROPN
asir-3636	305	2	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	305	3	2	2	NUM
asir-3636	305	4	(	(	PUNCT
asir-3636	305	5	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	305	6	2	2	NUM
asir-3636	305	7	)	)	PUNCT
asir-3636	305	8	|	|	ADV
asir-3636	305	9	𝜖+2	𝜖+2	ADV
asir-3636	305	10	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	305	11	2	2	NUM
asir-3636	305	12	)	)	PUNCT
asir-3636	305	13	𝑑(1	𝑑(1	NOUN
asir-3636	305	14	−	−	NOUN
asir-3636	305	15	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	NOUN
asir-3636	305	16	𝑗	𝑗	PRON
asir-3636	305	17	∆𝜸	∆𝜸	PROPN
asir-3636	305	18	𝟐𝟏	𝟐𝟏	NUM
asir-3636	305	19	≤	≤	NUM
asir-3636	305	20	∫	∫	PROPN
asir-3636	305	21	∑	∑	PROPN
asir-3636	305	22	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	305	23	2	2	NUM
asir-3636	305	24	(	(	PUNCT
asir-3636	305	25	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	305	26	2	2	NUM
asir-3636	305	27	)	)	PUNCT
asir-3636	305	28	|	|	ADV
asir-3636	305	29	𝜖+2	𝜖+2	ADV
asir-3636	305	30	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	305	31	2	2	NUM
asir-3636	305	32	)	)	PUNCT
asir-3636	305	33	𝑑(1	𝑑(1	NOUN
asir-3636	305	34	−	−	NOUN
asir-3636	305	35	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	NOUN
asir-3636	305	36	𝑗	𝑗	PRON
asir-3636	305	37	∆𝜸	∆𝜸	PROPN
asir-3636	305	38	𝟐	𝟐	NUM
asir-3636	305	39	=	=	SYM
asir-3636	305	40	∫	∫	PROPN
asir-3636	305	41	∑	∑	PROPN
asir-3636	305	42	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	305	43	2	2	NUM
asir-3636	305	44	(	(	PUNCT
asir-3636	305	45	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	305	46	2	2	NUM
asir-3636	305	47	)	)	PUNCT
asir-3636	305	48	|	|	ADV
asir-3636	305	49	𝜖+2	𝜖+2	ADV
asir-3636	305	50	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	305	51	2	2	NUM
asir-3636	305	52	)	)	PUNCT
asir-3636	305	53	𝑗	𝑗	PRON
asir-3636	305	54	∫	∫	NOUN
asir-3636	305	55	𝑑(1	𝑑(1	NOUN
asir-3636	305	56	−	−	PROPN
asir-3636	306	1	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	306	2	1	1	NUM
asir-3636	306	3	|(𝑧−2𝜖)𝑡2|	|(𝑧−2𝜖)𝑡2|	ADP
asir-3636	306	4	𝛾	𝛾	PROPN
asir-3636	306	5	≤	≤	NUM
asir-3636	306	6	∫	∫	PROPN
asir-3636	306	7	∑	∑	PROPN
asir-3636	306	8	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	306	9	2	2	NUM
asir-3636	306	10	(	(	PUNCT
asir-3636	306	11	𝑒𝑖𝑡	𝑒𝑖𝑡	NOUN
asir-3636	306	12	2	2	NUM
asir-3636	306	13	)	)	PUNCT
asir-3636	306	14	|	|	ADV
asir-3636	306	15	𝜖+2	𝜖+2	ADV
asir-3636	306	16	𝑑2(𝑒𝑖𝑡	𝑑2(𝑒𝑖𝑡	VERB
asir-3636	306	17	2	2	NUM
asir-3636	306	18	)	)	PUNCT
asir-3636	306	19	𝑗	𝑗	PROPN
asir-3636	306	20	𝛾	𝛾	NOUN
asir-3636	306	21	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	306	22	.	.	PUNCT
asir-3636	307	1	using	use	VERB
asir-3636	307	2	lemma	lemma	PROPN
asir-3636	307	3	(	(	PUNCT
asir-3636	307	4	4.1	4.1	NUM
asir-3636	307	5	)	)	PUNCT
asir-3636	307	6	,	,	PUNCT
asir-3636	307	7	we	we	PRON
asir-3636	307	8	get	get	VERB
asir-3636	307	9	𝐼2,1	𝐼2,1	NOUN
asir-3636	307	10	≤	≤	NOUN
asir-3636	307	11	∑	∑	PUNCT
asir-3636	307	12	𝐶1+𝜖‖(𝑓𝑗	𝐶1+𝜖‖(𝑓𝑗	PROPN
asir-3636	307	13	2	2	NUM
asir-3636	307	14	)	)	PUNCT
asir-3636	307	15	′‖	′‖	X
asir-3636	307	16	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	307	17	)	)	PUNCT
asir-3636	307	18	2	2	NUM
asir-3636	307	19	𝑗	𝑗	NOUN
asir-3636	307	20	.	.	PUNCT
asir-3636	308	1	this	this	PRON
asir-3636	308	2	proves	prove	VERB
asir-3636	308	3	the	the	DET
asir-3636	308	4	result	result	NOUN
asir-3636	308	5	.	.	PUNCT
asir-3636	309	1	lemma	lemma	PROPN
asir-3636	309	2	(	(	PUNCT
asir-3636	309	3	4.5	4.5	NUM
asir-3636	309	4	):	):	PUNCT
asir-3636	309	5	∫	∫	PROPN
asir-3636	310	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	310	2	2	2	NUM
asir-3636	310	3	(	(	PUNCT
asir-3636	310	4	𝑧)|	𝑧)|	ADV
asir-3636	310	5	𝑗	𝑗	ADJ
asir-3636	310	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	310	7	)	)	PUNCT
asir-3636	310	8	𝑎𝛾	𝑎𝛾	ADP
asir-3636	310	9	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	310	10	)	)	PUNCT
asir-3636	311	1	≤	≤	PUNCT
asir-3636	311	2	∆𝜸	∆𝜸	PROPN
asir-3636	311	3	𝟐𝟑	𝟐𝟑	NUM
asir-3636	311	4	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	311	5	)	)	PUNCT
asir-3636	311	6	,	,	PUNCT
asir-3636	311	7	where	where	SCONJ
asir-3636	311	8	𝐴(∆𝛾)is	𝐴(∆𝛾)is	ADJ
asir-3636	311	9	the	the	DET
asir-3636	311	10	area	area	NOUN
asir-3636	311	11	measure	measure	NOUN
asir-3636	311	12	of	of	ADP
asir-3636	311	13	∆𝛾.	∆𝛾.	NOUN
asir-3636	311	14	proof	proof	NOUN
asir-3636	311	15	:	:	PUNCT
asir-3636	311	16	set	set	VERB
asir-3636	311	17	λγ	λγ	NOUN
asir-3636	311	18	≔	≔	NOUN
asir-3636	311	19	{	{	PUNCT
asir-3636	311	20	γ	γ	X
asir-3636	311	21	for	for	ADP
asir-3636	311	22	γ	γ	X
asir-3636	311	23	⊈	⊈	PROPN
asir-3636	311	24	γ	γ	X
asir-3636	311	25	,	,	PUNCT
asir-3636	311	26	𝕋	𝕋	PROPN
asir-3636	311	27	∖	∖	X
asir-3636	311	28	γ	γ	NOUN
asir-3636	311	29	for	for	ADP
asir-3636	311	30	γ	γ	PROPN
asir-3636	311	31	⊆	⊆	NUM
asir-3636	311	32	γ	γ	X
asir-3636	311	33	.	.	PUNCT
asir-3636	312	1	let	let	VERB
asir-3636	312	2	𝑧	𝑧	PRON
asir-3636	312	3	∈	∈	PROPN
asir-3636	312	4	∆𝛾	∆𝛾	PROPN
asir-3636	312	5	23	23	NUM
asir-3636	312	6	.	.	PUNCT
asir-3636	313	1	we	we	PRON
asir-3636	313	2	have	have	VERB
asir-3636	313	3	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	313	4	2	2	NUM
asir-3636	313	5	(	(	PUNCT
asir-3636	313	6	𝑧)|	𝑧)|	PRON
asir-3636	313	7	𝑗	𝑗	X
asir-3636	313	8	=	=	SYM
asir-3636	313	9	exp	exp	NOUN
asir-3636	313	10	{	{	PUNCT
asir-3636	313	11	1	1	NUM
asir-3636	313	12	2𝜋	2𝜋	NUM
asir-3636	313	13	∫	∫	NOUN
asir-3636	313	14	∑	∑	PUNCT
asir-3636	313	15	2𝜖	2𝜖	VERB
asir-3636	313	16	−	−	PROPN
asir-3636	313	17	𝜖2	𝜖2	NOUN
asir-3636	313	18	|𝑒𝑖𝜃	|𝑒𝑖𝜃	X
asir-3636	313	19	2	2	NUM
asir-3636	313	20	−	−	ADP
asir-3636	313	21	𝑧|	𝑧|	ADJ
asir-3636	313	22	2	2	NUM
asir-3636	313	23	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	313	24	2	2	NUM
asir-3636	313	25	(	(	PUNCT
asir-3636	313	26	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	313	27	2	2	NUM
asir-3636	313	28	)	)	PUNCT
asir-3636	313	29	|𝑑𝜃2	|𝑑𝜃2	NOUN
asir-3636	313	30	𝑗	𝑗	PROPN
asir-3636	313	31	2𝜋	2𝜋	NOUN
asir-3636	313	32	0	0	NUM
asir-3636	313	33	}	}	PUNCT
asir-3636	313	34	≤	≤	NUM
asir-3636	313	35	exp	exp	NOUN
asir-3636	313	36	{	{	PUNCT
asir-3636	313	37	1	1	NUM
asir-3636	313	38	2𝜋	2𝜋	NUM
asir-3636	313	39	∫	∫	NOUN
asir-3636	313	40	∑	∑	PUNCT
asir-3636	313	41	2𝜖	2𝜖	VERB
asir-3636	313	42	−	−	PROPN
asir-3636	313	43	𝜖2	𝜖2	NOUN
asir-3636	313	44	|𝑒𝑖𝜃	|𝑒𝑖𝜃	X
asir-3636	313	45	2	2	NUM
asir-3636	313	46	−	−	ADP
asir-3636	313	47	𝑧|	𝑧|	ADJ
asir-3636	313	48	2	2	NUM
asir-3636	313	49	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	313	50	2	2	NUM
asir-3636	313	51	(	(	PUNCT
asir-3636	313	52	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	313	53	2	2	NUM
asir-3636	313	54	)	)	PUNCT
asir-3636	313	55	|𝑑𝜃2	|𝑑𝜃2	NOUN
asir-3636	313	56	𝑗	𝑗	PROPN
asir-3636	313	57	λγ	λγ	NOUN
asir-3636	313	58	}	}	PUNCT
asir-3636	313	59	=	=	SYM
asir-3636	313	60	exp{−𝜖𝑎𝛾(𝑧	exp{−𝜖𝑎𝛾(𝑧	NOUN
asir-3636	313	61	)	)	PUNCT
asir-3636	313	62	}	}	PUNCT
asir-3636	313	63	≤	≤	PROPN
asir-3636	313	64	𝑑	𝑑	PROPN
asir-3636	313	65	8(𝑧	8(𝑧	NUM
asir-3636	313	66	)	)	PUNCT
asir-3636	313	67	.	.	PUNCT
asir-3636	314	1	using	use	VERB
asir-3636	314	2	(	(	PUNCT
asir-3636	314	3	19	19	NUM
asir-3636	314	4	)	)	PUNCT
asir-3636	314	5	,	,	PUNCT
asir-3636	314	6	we	we	PRON
asir-3636	314	7	obtain	obtain	VERB
asir-3636	314	8	the	the	DET
asir-3636	314	9	result	result	NOUN
asir-3636	314	10	.	.	PUNCT
asir-3636	315	1	the	the	DET
asir-3636	315	2	integral	integral	ADJ
asir-3636	315	3	on	on	ADP
asir-3636	315	4	the	the	DET
asir-3636	315	5	region	region	NOUN
asir-3636	315	6	∆𝛾	∆𝛾	PROPN
asir-3636	315	7	23	23	NUM
asir-3636	315	8	.	.	PUNCT
asir-3636	316	1	here	here	ADV
asir-3636	316	2	,	,	PUNCT
asir-3636	316	3	we	we	PRON
asir-3636	316	4	will	will	AUX
asir-3636	316	5	give	give	VERB
asir-3636	316	6	an	an	DET
asir-3636	316	7	estimate	estimate	NOUN
asir-3636	316	8	of	of	ADP
asir-3636	316	9	the	the	DET
asir-3636	316	10	following	follow	VERB
asir-3636	316	11	integral	integral	ADJ
asir-3636	316	12	∫	∫	NOUN
asir-3636	316	13	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	316	14	2	2	NUM
asir-3636	316	15	(	(	PUNCT
asir-3636	316	16	𝑧)|	𝑧)|	ADV
asir-3636	316	17	𝑗	𝑗	ADJ
asir-3636	316	18	2(1+𝜖	2(1+𝜖	NUM
asir-3636	316	19	)	)	PUNCT
asir-3636	316	20	𝑎𝛾	𝑎𝛾	ADP
asir-3636	316	21	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	316	22	)	)	PUNCT
asir-3636	316	23	∆𝜸	∆𝜸	PROPN
asir-3636	316	24	𝟐𝟐	𝟐𝟐	X
asir-3636	316	25	.	.	PUNCT
asir-3636	317	1	before	before	ADP
asir-3636	317	2	doing	do	VERB
asir-3636	317	3	this	this	PRON
asir-3636	317	4	,	,	PUNCT
asir-3636	317	5	we	we	PRON
asir-3636	317	6	begin	begin	VERB
asir-3636	317	7	with	with	ADP
asir-3636	317	8	some	some	DET
asir-3636	317	9	lemmas	lemma	NOUN
asir-3636	317	10	(	(	PUNCT
asir-3636	317	11	see	see	VERB
asir-3636	317	12	brahim	brahim	PROPN
asir-3636	317	13	bouya	bouya	PROPN
asir-3636	317	14	,	,	PUNCT
asir-3636	317	15	2008	2008	NUM
asir-3636	317	16	)	)	PUNCT
asir-3636	317	17	.	.	PUNCT
asir-3636	318	1	the	the	DET
asir-3636	318	2	next	next	ADJ
asir-3636	318	3	one	one	NOUN
asir-3636	318	4	is	be	AUX
asir-3636	318	5	essential	essential	ADJ
asir-3636	318	6	for	for	ADP
asir-3636	318	7	what	what	PRON
asir-3636	318	8	follows	follow	VERB
asir-3636	318	9	.	.	PUNCT
asir-3636	319	1	note	note	VERB
asir-3636	319	2	that	that	SCONJ
asir-3636	319	3	a	a	DET
asir-3636	319	4	similar	similar	ADJ
asir-3636	319	5	result	result	NOUN
asir-3636	319	6	is	be	AUX
asir-3636	319	7	used	use	VERB
asir-3636	319	8	by	by	ADP
asir-3636	319	9	different	different	ADJ
asir-3636	319	10	authors	author	NOUN
asir-3636	319	11	:	:	PUNCT
asir-3636	319	12	korenblum	korenblum	PROPN
asir-3636	319	13	(	(	PUNCT
asir-3636	319	14	1972	1972	NUM
asir-3636	319	15	)	)	PUNCT
asir-3636	319	16	,	,	PUNCT
asir-3636	319	17	matheson	matheson	PROPN
asir-3636	319	18	(	(	PUNCT
asir-3636	319	19	1978	1978	NUM
asir-3636	319	20	)	)	PUNCT
asir-3636	319	21	,	,	PUNCT
asir-3636	319	22	shamoyan	shamoyan	ADJ
asir-3636	319	23	(	(	PUNCT
asir-3636	319	24	1994	1994	NUM
asir-3636	319	25	)	)	PUNCT
asir-3636	319	26	and	and	CCONJ
asir-3636	319	27	shirokov	shirokov	NOUN
asir-3636	319	28	(	(	PUNCT
asir-3636	319	29	1982	1982	NUM
asir-3636	319	30	,	,	PUNCT
asir-3636	319	31	1988	1988	NUM
asir-3636	319	32	)	)	PUNCT
asir-3636	319	33	.	.	PUNCT
asir-3636	320	1	lemma	lemma	PROPN
asir-3636	320	2	(	(	PUNCT
asir-3636	320	3	4.6	4.6	NUM
asir-3636	320	4	):	):	PUNCT
asir-3636	320	5	let	let	VERB
asir-3636	320	6	𝑧	𝑧	PRON
asir-3636	320	7	∈	∈	PROPN
asir-3636	320	8	∆𝜸	∆𝜸	PROPN
asir-3636	320	9	𝟐𝟐	𝟐𝟐	NOUN
asir-3636	320	10	and	and	CCONJ
asir-3636	320	11	let	let	VERB
asir-3636	320	12	𝜇𝑧	𝜇𝑧	NOUN
asir-3636	320	13	=	=	SYM
asir-3636	320	14	1	1	NUM
asir-3636	320	15	−	−	PROPN
asir-3636	320	16	8|log	8|log	NUM
asir-3636	320	17	(	(	PUNCT
asir-3636	320	18	𝑑(𝑧))|	𝑑(𝑧))|	PRON
asir-3636	320	19	𝑎𝛾(𝑧	𝑎𝛾(𝑧	ADJ
asir-3636	320	20	)	)	PUNCT
asir-3636	320	21	.	.	PUNCT
asir-3636	321	1	then	then	ADV
asir-3636	321	2	∑	∑	PUNCT
asir-3636	321	3	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	321	4	2	2	NUM
asir-3636	321	5	(	(	PUNCT
asir-3636	321	6	𝜇𝑧𝑧)|	𝜇𝑧𝑧)|	PROPN
asir-3636	321	7	≤	≤	PROPN
asir-3636	321	8	𝑑	𝑑	NOUN
asir-3636	321	9	2(𝑧)𝑗	2(𝑧)𝑗	NOUN
asir-3636	321	10	.	.	PUNCT
asir-3636	322	1	(	(	PUNCT
asir-3636	322	2	23	23	X
asir-3636	322	3	)	)	PUNCT
asir-3636	322	4	proof	proof	NOUN
asir-3636	322	5	:	:	PUNCT
asir-3636	322	6	let	let	VERB
asir-3636	322	7	𝑧	𝑧	DET
asir-3636	322	8	∈	∈	PROPN
asir-3636	322	9	∆𝜸	∆𝜸	PROPN
asir-3636	322	10	and	and	CCONJ
asir-3636	322	11	let	let	VERB
asir-3636	322	12	𝜇	𝜇	ADP
asir-3636	322	13	<	<	X
asir-3636	322	14	1	1	NUM
asir-3636	322	15	.	.	PUNCT
asir-3636	323	1	we	we	PRON
asir-3636	323	2	have	have	VERB
asir-3636	323	3	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	323	4	applied	apply	VERB
asir-3636	323	5	science	science	NOUN
asir-3636	323	6	and	and	CCONJ
asir-3636	323	7	innovative	innovative	ADJ
asir-3636	323	8	research	research	NOUN
asir-3636	323	9	vol	vol	NOUN
asir-3636	323	10	.	.	PROPN
asir-3636	324	1	5	5	NUM
asir-3636	324	2	,	,	PUNCT
asir-3636	324	3	no	no	INTJ
asir-3636	324	4	.	.	NOUN
asir-3636	324	5	1	1	NUM
asir-3636	324	6	,	,	PUNCT
asir-3636	324	7	2021	2021	NUM
asir-3636	324	8	35	35	NUM
asir-3636	324	9	published	publish	VERB
asir-3636	324	10	by	by	ADP
asir-3636	324	11	scholink	scholink	PROPN
asir-3636	324	12	inc	inc	PROPN
asir-3636	324	13	.	.	PUNCT
asir-3636	325	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	325	2	2	2	NUM
asir-3636	325	3	(	(	PUNCT
asir-3636	325	4	𝜇𝑧)|	𝜇𝑧)|	NOUN
asir-3636	325	5	𝑗	𝑗	NOUN
asir-3636	325	6	=	=	SYM
asir-3636	325	7	exp	exp	NOUN
asir-3636	325	8	{	{	PUNCT
asir-3636	325	9	1	1	NUM
asir-3636	325	10	2𝜋	2𝜋	NUM
asir-3636	325	11	∫	∫	NOUN
asir-3636	325	12	∑	∑	PROPN
asir-3636	325	13	1−	1−	NUM
asir-3636	325	14	(	(	PUNCT
asir-3636	325	15	𝜇(1	𝜇(1	PROPN
asir-3636	325	16	−	−	PROPN
asir-3636	325	17	𝜖))2	𝜖))2	PROPN
asir-3636	325	18	|𝑒𝑖𝜃	|𝑒𝑖𝜃	PUNCT
asir-3636	325	19	2	2	NUM
asir-3636	325	20	−	−	NOUN
asir-3636	325	21	𝜇𝑧|	𝜇𝑧|	NOUN
asir-3636	325	22	2	2	NUM
asir-3636	325	23	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	325	24	2	2	NUM
asir-3636	325	25	(	(	PUNCT
asir-3636	325	26	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	325	27	2	2	NUM
asir-3636	325	28	)	)	PUNCT
asir-3636	325	29	|𝑑𝜃2	|𝑑𝜃2	NOUN
asir-3636	325	30	𝑗	𝑗	PROPN
asir-3636	325	31	2𝜋	2𝜋	NOUN
asir-3636	325	32	0	0	NUM
asir-3636	325	33	}	}	PUNCT
asir-3636	325	34	≤	≤	NUM
asir-3636	325	35	exp	exp	NOUN
asir-3636	325	36	{	{	PUNCT
asir-3636	325	37	1	1	NUM
asir-3636	325	38	2𝜋	2𝜋	NUM
asir-3636	325	39	∫	∫	NOUN
asir-3636	325	40	∑	∑	PROPN
asir-3636	325	41	1−	1−	NUM
asir-3636	325	42	(	(	PUNCT
asir-3636	325	43	𝜇(1	𝜇(1	PROPN
asir-3636	325	44	−	−	PROPN
asir-3636	325	45	𝜖))2	𝜖))2	PROPN
asir-3636	325	46	|𝑒𝑖𝜃	|𝑒𝑖𝜃	PUNCT
asir-3636	325	47	2	2	NUM
asir-3636	325	48	−	−	NOUN
asir-3636	325	49	𝜇𝑧|	𝜇𝑧|	NOUN
asir-3636	325	50	2	2	NUM
asir-3636	325	51	log|𝑓𝑗	log|𝑓𝑗	ADJ
asir-3636	325	52	2	2	NUM
asir-3636	325	53	(	(	PUNCT
asir-3636	325	54	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	325	55	2	2	NUM
asir-3636	325	56	)	)	PUNCT
asir-3636	325	57	|𝑑𝜃2	|𝑑𝜃2	NOUN
asir-3636	325	58	𝑗	𝑗	PROPN
asir-3636	325	59	λγ	λγ	NOUN
asir-3636	325	60	}	}	PUNCT
asir-3636	325	61	=	=	SYM
asir-3636	325	62	exp	exp	NOUN
asir-3636	325	63	{	{	PUNCT
asir-3636	325	64	−(1	−(1	NOUN
asir-3636	325	65	−	−	PROPN
asir-3636	326	1	𝜇(1	𝜇(1	PROPN
asir-3636	326	2	−	−	PROPN
asir-3636	326	3	𝜖	𝜖	PROPN
asir-3636	326	4	)	)	PUNCT
asir-3636	326	5	)	)	PUNCT
asir-3636	327	1	inf𝜃2∈λγ	inf𝜃2∈λγ	PROPN
asir-3636	328	1	|	|	INTJ
asir-3636	328	2	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	328	3	2	2	NUM
asir-3636	328	4	−	−	PROPN
asir-3636	328	5	𝑧	𝑧	PROPN
asir-3636	328	6	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	328	7	2	2	NUM
asir-3636	328	8	−	−	NOUN
asir-3636	329	1	𝜇𝑧	𝜇𝑧	NOUN
asir-3636	329	2	|	|	ADV
asir-3636	329	3	2	2	NUM
asir-3636	329	4	𝑎𝛾(𝑧	𝑎𝛾(𝑧	NOUN
asir-3636	329	5	)	)	PUNCT
asir-3636	329	6	}	}	PUNCT
asir-3636	329	7	.	.	PUNCT
asir-3636	330	1	for	for	ADP
asir-3636	330	2	𝑧	𝑧	PRON
asir-3636	330	3	∈	∈	PROPN
asir-3636	330	4	∆𝜸	∆𝜸	NOUN
asir-3636	330	5	𝟐𝟐	𝟐𝟐	NUM
asir-3636	331	1	it	it	PRON
asir-3636	331	2	is	be	AUX
asir-3636	331	3	clear	clear	ADJ
asir-3636	331	4	that	that	SCONJ
asir-3636	331	5	1	1	NUM
asir-3636	331	6	−	−	NOUN
asir-3636	331	7	𝜇𝑧	𝜇𝑧	PROPN
asir-3636	331	8	≤	≤	PUNCT
asir-3636	331	9	𝑑(𝑧	𝑑(𝑧	ADJ
asir-3636	331	10	)	)	PUNCT
asir-3636	331	11	≤	≤	NOUN
asir-3636	331	12	|𝑒𝑖𝜃	|𝑒𝑖𝜃	PUNCT
asir-3636	331	13	2	2	NUM
asir-3636	331	14	−	−	NOUN
asir-3636	331	15	𝑧|	𝑧|	ADV
asir-3636	331	16	for	for	ADP
asir-3636	331	17	all	all	DET
asir-3636	331	18	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	331	19	2	2	NUM
asir-3636	331	20	∈	∈	PROPN
asir-3636	331	21	λγ	λγ	NOUN
asir-3636	331	22	.	.	PUNCT
asir-3636	332	1	then	then	ADV
asir-3636	332	2	inf𝜃2∈λγ	inf𝜃2∈λγ	VERB
asir-3636	332	3	|	|	ADV
asir-3636	332	4	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	332	5	2	2	NUM
asir-3636	332	6	−	−	PROPN
asir-3636	332	7	𝑧	𝑧	PROPN
asir-3636	332	8	𝑒𝑖𝜃	𝑒𝑖𝜃	PROPN
asir-3636	332	9	2	2	NUM
asir-3636	332	10	−	−	NOUN
asir-3636	332	11	𝜇𝑧	𝜇𝑧	NOUN
asir-3636	332	12	|	|	ADV
asir-3636	332	13	2	2	NUM
asir-3636	332	14	≥	≥	NOUN
asir-3636	332	15	1	1	NUM
asir-3636	332	16	2	2	NUM
asir-3636	332	17	(	(	PUNCT
asir-3636	332	18	𝑧	𝑧	PRON
asir-3636	332	19	∈	∈	PROPN
asir-3636	332	20	∆𝜸	∆𝜸	PROPN
asir-3636	332	21	𝟐𝟐	𝟐𝟐	NUM
asir-3636	332	22	)	)	PUNCT
asir-3636	332	23	.	.	PUNCT
asir-3636	333	1	thus	thus	ADV
asir-3636	333	2	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	333	3	2	2	NUM
asir-3636	333	4	(	(	PUNCT
asir-3636	333	5	𝜇𝑧𝑧)|	𝜇𝑧𝑧)|	VERB
asir-3636	333	6	𝑗	𝑗	PRON
asir-3636	333	7	≤	≤	NUM
asir-3636	333	8	exp	exp	NOUN
asir-3636	333	9	{	{	PUNCT
asir-3636	333	10	−	−	PROPN
asir-3636	333	11	1	1	NUM
asir-3636	333	12	−	−	NOUN
asir-3636	333	13	𝜇𝑧	𝜇𝑧	NOUN
asir-3636	333	14	4	4	NUM
asir-3636	333	15	𝑎𝛾(𝑧	𝑎𝛾(𝑧	NUM
asir-3636	333	16	)	)	PUNCT
asir-3636	333	17	}	}	PUNCT
asir-3636	333	18	(	(	PUNCT
asir-3636	333	19	𝑧	𝑧	PRON
asir-3636	333	20	∈	∈	PROPN
asir-3636	333	21	∆𝜸	∆𝜸	PROPN
asir-3636	333	22	𝟐𝟐	𝟐𝟐	NUM
asir-3636	333	23	)	)	PUNCT
asir-3636	333	24	.	.	PUNCT
asir-3636	334	1	then	then	ADV
asir-3636	334	2	,	,	PUNCT
asir-3636	334	3	we	we	PRON
asir-3636	334	4	have	have	VERB
asir-3636	334	5	∑	∑	ADV
asir-3636	334	6	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	334	7	2	2	NUM
asir-3636	334	8	(	(	PUNCT
asir-3636	334	9	𝜇𝑧𝑧)|𝑗	𝜇𝑧𝑧)|𝑗	NOUN
asir-3636	334	10	≤	≤	NOUN
asir-3636	334	11	exp	exp	NOUN
asir-3636	334	12	{	{	PUNCT
asir-3636	334	13	−	−	PROPN
asir-3636	334	14	1	1	NUM
asir-3636	334	15	4	4	NUM
asir-3636	334	16	(	(	PUNCT
asir-3636	334	17	1	1	NUM
asir-3636	334	18	−	−	NOUN
asir-3636	334	19	𝜇𝑧)𝑎𝛾(𝑧	𝜇𝑧)𝑎𝛾(𝑧	NOUN
asir-3636	334	20	)	)	PUNCT
asir-3636	334	21	}	}	PUNCT
asir-3636	334	22	=	=	SYM
asir-3636	334	23	𝑑	𝑑	NOUN
asir-3636	334	24	2(𝑧	2(𝑧	NUM
asir-3636	334	25	)	)	PUNCT
asir-3636	334	26	(	(	PUNCT
asir-3636	334	27	𝑧	𝑧	PRON
asir-3636	334	28	∈	∈	PROPN
asir-3636	334	29	∆𝜸	∆𝜸	PROPN
asir-3636	334	30	𝟐𝟐	𝟐𝟐	NUM
asir-3636	334	31	)	)	PUNCT
asir-3636	334	32	,	,	PUNCT
asir-3636	334	33	which	which	PRON
asir-3636	334	34	yields	yield	VERB
asir-3636	334	35	(	(	PUNCT
asir-3636	334	36	23	23	NUM
asir-3636	334	37	)	)	PUNCT
asir-3636	334	38	.	.	PUNCT
asir-3636	335	1	for	for	ADP
asir-3636	335	2	𝜖	𝜖	PROPN
asir-3636	335	3	>	>	X
asir-3636	335	4	0	0	PUNCT
asir-3636	335	5	define	define	VERB
asir-3636	335	6	𝛾(1−𝜖	𝛾(1−𝜖	NOUN
asir-3636	335	7	)	)	PUNCT
asir-3636	335	8	≔	≔	NOUN
asir-3636	335	9	{	{	PUNCT
asir-3636	335	10	𝑧	𝑧	PRON
asir-3636	335	11	∈	∈	PROPN
asir-3636	335	12	𝔻	𝔻	ADJ
asir-3636	335	13	:	:	PUNCT
asir-3636	335	14	|𝑧|	|𝑧|	NOUN
asir-3636	335	15	=	=	SYM
asir-3636	336	1	1	1	NUM
asir-3636	336	2	−	−	NOUN
asir-3636	336	3	𝜖	𝜖	X
asir-3636	336	4	and	and	CCONJ
asir-3636	336	5	𝑧/|𝑧|	𝑧/|𝑧|	ADV
asir-3636	336	6	∈	∈	PROPN
asir-3636	336	7	𝛾	𝛾	NOUN
asir-3636	336	8	}	}	PUNCT
asir-3636	336	9	.	.	PUNCT
asir-3636	337	1	without	without	ADP
asir-3636	337	2	loss	loss	NOUN
asir-3636	337	3	of	of	ADP
asir-3636	337	4	generality	generality	NOUN
asir-3636	337	5	,	,	PUNCT
asir-3636	337	6	we	we	PRON
asir-3636	337	7	can	can	AUX
asir-3636	337	8	suppose	suppose	VERB
asir-3636	337	9	that	that	SCONJ
asir-3636	337	10	𝑑(𝑧	𝑑(𝑧	NOUN
asir-3636	337	11	)	)	PUNCT
asir-3636	337	12	≤	≤	NUM
asir-3636	337	13	1	1	NUM
asir-3636	337	14	2	2	NUM
asir-3636	337	15	,	,	PUNCT
asir-3636	337	16	𝑧	𝑧	DET
asir-3636	337	17	∈	∈	NOUN
asir-3636	337	18	∆𝜸	∆𝜸	NOUN
asir-3636	337	19	𝟐.	𝟐.	X
asir-3636	337	20	we	we	PRON
asir-3636	337	21	need	need	VERB
asir-3636	337	22	the	the	DET
asir-3636	337	23	following	follow	VERB
asir-3636	337	24	(	(	PUNCT
asir-3636	337	25	see	see	VERB
asir-3636	337	26	brahim	brahim	PROPN
asir-3636	337	27	bouya	bouya	PROPN
asir-3636	337	28	,	,	PUNCT
asir-3636	337	29	2008	2008	NUM
asir-3636	337	30	)	)	PUNCT
asir-3636	337	31	.	.	PUNCT
asir-3636	338	1	note	note	VERB
asir-3636	338	2	that	that	SCONJ
asir-3636	338	3	:	:	PUNCT
asir-3636	338	4	we	we	PRON
asir-3636	338	5	deduce	deduce	VERB
asir-3636	338	6	that	that	SCONJ
asir-3636	338	7	∑	∑	ADP
asir-3636	338	8	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	338	9	2	2	NUM
asir-3636	338	10	(	(	PUNCT
asir-3636	338	11	𝜇𝑧𝑧)|𝑗	𝜇𝑧𝑧)|𝑗	NOUN
asir-3636	338	12	≤	≤	X
asir-3636	338	13	𝑐′	𝑐′	NUM
asir-3636	338	14	‖log	‖log	PROPN
asir-3636	338	15	(	(	PUNCT
asir-3636	338	16	1	1	NUM
asir-3636	338	17	2	2	NUM
asir-3636	338	18	)	)	PUNCT
asir-3636	338	19	‖	‖	PROPN
asir-3636	338	20	where	where	SCONJ
asir-3636	338	21	𝑐′	𝑐′	ADP
asir-3636	338	22	=	=	SYM
asir-3636	338	23	𝑐	𝑐	PROPN
asir-3636	338	24	16	16	NUM
asir-3636	338	25	.	.	PUNCT
asir-3636	339	1	lemma	lemma	PROPN
asir-3636	339	2	(	(	PUNCT
asir-3636	339	3	4.7	4.7	NUM
asir-3636	339	4	):	):	PUNCT
asir-3636	339	5	let	let	VERB
asir-3636	339	6	𝜖	𝜖	PROPN
asir-3636	339	7	>	>	X
asir-3636	339	8	0	0	NUM
asir-3636	339	9	.	.	PUNCT
asir-3636	340	1	then	then	ADV
asir-3636	340	2	∫	∫	PROPN
asir-3636	341	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	341	2	2	2	NUM
asir-3636	341	3	(	(	PUNCT
asir-3636	341	4	(	(	PUNCT
asir-3636	341	5	1	1	NUM
asir-3636	341	6	−	−	PROPN
asir-3636	341	7	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	341	8	2	2	NUM
asir-3636	341	9	)	)	PUNCT
asir-3636	341	10	−	−	ADP
asir-3636	341	11	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	341	12	2	2	NUM
asir-3636	341	13	(	(	PUNCT
asir-3636	341	14	𝜇	𝜇	X
asir-3636	341	15	(	(	PUNCT
asir-3636	341	16	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	341	17	2(1	2(1	NUM
asir-3636	341	18	−	−	NOUN
asir-3636	341	19	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	341	20	2	2	NUM
asir-3636	341	21	)	)	PUNCT
asir-3636	341	22	|	|	ADV
asir-3636	341	23	𝑗	𝑗	NOUN
asir-3636	341	24	2(1+𝜖	2(1+𝜖	NUM
asir-3636	341	25	)	)	PUNCT
asir-3636	341	26	𝑎𝛾	𝑎𝛾	ADP
asir-3636	341	27	2((1	2((1	NUM
asir-3636	341	28	−	−	PROPN
asir-3636	341	29	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	341	30	2	2	NUM
asir-3636	341	31	)	)	PUNCT
asir-3636	341	32	(	(	PUNCT
asir-3636	341	33	1	1	NUM
asir-3636	341	34	−	−	PROPN
asir-3636	341	35	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	341	36	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	PROPN
asir-3636	341	37	𝟐𝟐	𝟐𝟐	NUM
asir-3636	341	38	≤∑	≤∑	PROPN
asir-3636	341	39	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	341	40	𝜖1−𝜀(1+𝜖	𝜖1−𝜀(1+𝜖	ADV
asir-3636	341	41	)	)	PUNCT
asir-3636	341	42	‖(𝑓𝑗	‖(𝑓𝑗	PROPN
asir-3636	341	43	2	2	NUM
asir-3636	341	44	)	)	PUNCT
asir-3636	341	45	′‖	′‖	X
asir-3636	341	46	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	341	47	)	)	PUNCT
asir-3636	341	48	2	2	NUM
asir-3636	341	49	𝑗	𝑗	NOUN
asir-3636	341	50	,	,	PUNCT
asir-3636	341	51	where	where	SCONJ
asir-3636	341	52	𝜀(1+𝜖	𝜀(1+𝜖	NOUN
asir-3636	341	53	)	)	PUNCT
asir-3636	341	54	=	=	PUNCT
asir-3636	342	1	α	α	NOUN
asir-3636	342	2	2𝜖.	2𝜖.	NUM
asir-3636	342	3	proof	proof	NOUN
asir-3636	342	4	:	:	PUNCT
asir-3636	342	5	let	let	VERB
asir-3636	342	6	(	(	PUNCT
asir-3636	342	7	1	1	NUM
asir-3636	342	8	−	−	NOUN
asir-3636	342	9	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	NOUN
asir-3636	342	10	2	2	NUM
asir-3636	342	11	∈	∈	PROPN
asir-3636	342	12	∆𝜸	∆𝜸	NOUN
asir-3636	343	1	𝟐𝟐.	𝟐𝟐.	ADV
asir-3636	343	2	then	then	ADV
asir-3636	343	3	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	343	4	2	2	NUM
asir-3636	343	5	(	(	PUNCT
asir-3636	343	6	(	(	PUNCT
asir-3636	343	7	1	1	NUM
asir-3636	343	8	−	−	PROPN
asir-3636	343	9	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	343	10	2	2	NUM
asir-3636	343	11	)	)	PUNCT
asir-3636	343	12	−	−	ADP
asir-3636	343	13	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	343	14	2	2	NUM
asir-3636	343	15	(	(	PUNCT
asir-3636	343	16	𝜇	𝜇	X
asir-3636	343	17	(	(	PUNCT
asir-3636	343	18	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	343	19	2(1	2(1	NUM
asir-3636	343	20	−	−	NOUN
asir-3636	343	21	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	343	22	2	2	NUM
asir-3636	343	23	)	)	PUNCT
asir-3636	343	24	|	|	ADV
asir-3636	343	25	𝜖	𝜖	X
asir-3636	344	1	[	[	X
asir-3636	344	2	(	(	PUNCT
asir-3636	344	3	1	1	NUM
asir-3636	344	4	−	−	NOUN
asir-3636	344	5	𝜇	𝜇	X
asir-3636	344	6	(	(	PUNCT
asir-3636	344	7	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	344	8	2	2	NUM
asir-3636	344	9	)	)	PUNCT
asir-3636	344	10	𝑎𝛾((1	𝑎𝛾((1	NUM
asir-3636	344	11	−	−	PROPN
asir-3636	344	12	𝜖)𝑒	𝜖)𝑒	X
asir-3636	344	13	𝑖𝑡2	𝑖𝑡2	X
asir-3636	344	14	)	)	PUNCT
asir-3636	344	15	]	]	PUNCT
asir-3636	344	16	2	2	NUM
asir-3636	344	17	𝑗	𝑗	SYM
asir-3636	344	18	≤	≤	NUM
asir-3636	344	19	64	64	NUM
asir-3636	344	20	(	(	PUNCT
asir-3636	344	21	1	1	NUM
asir-3636	344	22	−	−	NOUN
asir-3636	344	23	𝜇	𝜇	X
asir-3636	344	24	(	(	PUNCT
asir-3636	344	25	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	344	26	2	2	NUM
asir-3636	344	27	)	)	PUNCT
asir-3636	344	28	𝜀(1+𝜖	𝜀(1+𝜖	NOUN
asir-3636	344	29	)	)	PUNCT
asir-3636	344	30	log2	log2	PROPN
asir-3636	344	31	(	(	PUNCT
asir-3636	344	32	𝑑((1	𝑑((1	PROPN
asir-3636	344	33	−	−	PROPN
asir-3636	344	34	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	344	35	2	2	NUM
asir-3636	344	36	)	)	PUNCT
asir-3636	344	37	)	)	PUNCT
asir-3636	344	38	≤	≤	NUM
asir-3636	344	39	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	344	40	.	.	PUNCT
asir-3636	345	1	it	it	PRON
asir-3636	345	2	is	be	AUX
asir-3636	345	3	clear	clear	ADJ
asir-3636	345	4	that	that	SCONJ
asir-3636	345	5	𝜖	𝜖	PROPN
asir-3636	345	6	≤	≤	ADV
asir-3636	345	7	1	1	NUM
asir-3636	345	8	−	−	NOUN
asir-3636	345	9	𝜇	𝜇	X
asir-3636	345	10	(	(	PUNCT
asir-3636	345	11	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	345	12	2	2	NUM
asir-3636	345	13	≤	≤	NOUN
asir-3636	345	14	𝑑((1	𝑑((1	PROPN
asir-3636	345	15	−	−	PROPN
asir-3636	345	16	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	345	17	2	2	NUM
asir-3636	345	18	)	)	PUNCT
asir-3636	345	19	≤	≤	NOUN
asir-3636	345	20	1	1	NUM
asir-3636	345	21	2	2	NUM
asir-3636	345	22	and	and	CCONJ
asir-3636	345	23	so	so	ADV
asir-3636	345	24	1	1	NUM
asir-3636	345	25	2	2	NUM
asir-3636	345	26	≤	≤	NOUN
asir-3636	345	27	𝑑((1	𝑑((1	PROPN
asir-3636	345	28	−	−	PROPN
asir-3636	345	29	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	345	30	2	2	NUM
asir-3636	345	31	)	)	PUNCT
asir-3636	345	32	≤	≤	NOUN
asir-3636	345	33	(	(	PUNCT
asir-3636	345	34	1	1	NUM
asir-3636	345	35	−	−	PROPN
asir-3636	345	36	𝜖	𝜖	NOUN
asir-3636	345	37	)	)	PUNCT
asir-3636	345	38	.	.	PUNCT
asir-3636	346	1	we	we	PRON
asir-3636	346	2	have	have	VERB
asir-3636	346	3	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	346	4	applied	apply	VERB
asir-3636	346	5	science	science	NOUN
asir-3636	346	6	and	and	CCONJ
asir-3636	346	7	innovative	innovative	ADJ
asir-3636	346	8	research	research	NOUN
asir-3636	346	9	vol	vol	NOUN
asir-3636	346	10	.	.	PROPN
asir-3636	347	1	5	5	NUM
asir-3636	347	2	,	,	PUNCT
asir-3636	347	3	no	no	INTJ
asir-3636	347	4	.	.	NOUN
asir-3636	347	5	1	1	NUM
asir-3636	347	6	,	,	PUNCT
asir-3636	347	7	2021	2021	NUM
asir-3636	347	8	36	36	NUM
asir-3636	347	9	published	publish	VERB
asir-3636	347	10	by	by	ADP
asir-3636	347	11	scholink	scholink	PROPN
asir-3636	347	12	inc	inc	PROPN
asir-3636	347	13	.	.	PROPN
asir-3636	347	14	∫	∫	PROPN
asir-3636	348	1	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	348	2	2	2	NUM
asir-3636	348	3	(	(	PUNCT
asir-3636	348	4	(	(	PUNCT
asir-3636	348	5	1	1	NUM
asir-3636	348	6	−	−	PROPN
asir-3636	348	7	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	8	2	2	NUM
asir-3636	348	9	)	)	PUNCT
asir-3636	348	10	−	−	ADP
asir-3636	348	11	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	348	12	2	2	NUM
asir-3636	348	13	(	(	PUNCT
asir-3636	348	14	𝜇	𝜇	X
asir-3636	348	15	(	(	PUNCT
asir-3636	348	16	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	348	17	2(1	2(1	NUM
asir-3636	348	18	−	−	NOUN
asir-3636	348	19	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	20	2	2	NUM
asir-3636	348	21	)	)	PUNCT
asir-3636	348	22	|	|	ADV
asir-3636	348	23	𝑗	𝑗	NOUN
asir-3636	348	24	2(1+𝜖	2(1+𝜖	NUM
asir-3636	348	25	)	)	PUNCT
asir-3636	348	26	𝑎𝛾	𝑎𝛾	ADP
asir-3636	348	27	2((1	2((1	NUM
asir-3636	348	28	−	−	PROPN
asir-3636	348	29	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	30	2	2	NUM
asir-3636	348	31	)	)	PUNCT
asir-3636	348	32	(	(	PUNCT
asir-3636	348	33	1	1	NUM
asir-3636	348	34	−	−	PROPN
asir-3636	348	35	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	348	36	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	PROPN
asir-3636	348	37	𝟐𝟐	𝟐𝟐	PROPN
asir-3636	348	38	≤	≤	NOUN
asir-3636	348	39	𝐶1+𝜖∫	𝐶1+𝜖∫	ADP
asir-3636	348	40	∑	∑	PUNCT
asir-3636	348	41	|𝑓𝑗	|𝑓𝑗	SYM
asir-3636	348	42	2	2	NUM
asir-3636	348	43	(	(	PUNCT
asir-3636	348	44	(	(	PUNCT
asir-3636	348	45	1	1	NUM
asir-3636	348	46	−	−	PROPN
asir-3636	348	47	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	48	2	2	NUM
asir-3636	348	49	)	)	PUNCT
asir-3636	348	50	−	−	ADP
asir-3636	348	51	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	348	52	2	2	NUM
asir-3636	348	53	(	(	PUNCT
asir-3636	348	54	𝜇	𝜇	X
asir-3636	348	55	(	(	PUNCT
asir-3636	348	56	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	348	57	2(1	2(1	NUM
asir-3636	348	58	−	−	NOUN
asir-3636	348	59	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	60	2	2	NUM
asir-3636	348	61	)	)	PUNCT
asir-3636	348	62	|	|	ADV
asir-3636	348	63	𝜖+2	𝜖+2	NUM
asir-3636	348	64	(	(	PUNCT
asir-3636	348	65	1	1	NUM
asir-3636	348	66	−	−	NOUN
asir-3636	348	67	𝜇	𝜇	X
asir-3636	348	68	(	(	PUNCT
asir-3636	348	69	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	348	70	2	2	NUM
asir-3636	348	71	)	)	PUNCT
asir-3636	348	72	2	2	NUM
asir-3636	348	73	(	(	PUNCT
asir-3636	348	74	1	1	NUM
asir-3636	348	75	𝑗	𝑗	NOUN
asir-3636	348	76	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	348	77	𝟐𝟐	𝟐𝟐	NUM
asir-3636	348	78	−	−	PROPN
asir-3636	348	79	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	348	80	≤	≤	NUM
asir-3636	348	81	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	348	82	𝜖1−𝜀(1+𝜖	𝜖1−𝜀(1+𝜖	ADV
asir-3636	348	83	)	)	PUNCT
asir-3636	348	84	∫	∫	PROPN
asir-3636	348	85	(	(	PUNCT
asir-3636	348	86	∫	∫	PROPN
asir-3636	348	87	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	348	88	2	2	NUM
asir-3636	348	89	)	)	PUNCT
asir-3636	348	90	′	′	NOUN
asir-3636	348	91	(	(	PUNCT
asir-3636	348	92	(	(	PUNCT
asir-3636	348	93	1	1	NUM
asir-3636	348	94	2	2	NUM
asir-3636	348	95	+	+	NOUN
asir-3636	348	96	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	348	97	2	2	NUM
asir-3636	348	98	)	)	PUNCT
asir-3636	349	1	|	|	ADV
asir-3636	349	2	𝑗	𝑗	INTJ
asir-3636	349	3	2	2	NUM
asir-3636	349	4	𝑑	𝑑	NOUN
asir-3636	349	5	(	(	PUNCT
asir-3636	349	6	1	1	NUM
asir-3636	349	7	2	2	NUM
asir-3636	349	8	+	+	NOUN
asir-3636	349	9	𝜖	𝜖	X
asir-3636	349	10	)	)	PUNCT
asir-3636	349	11	(	(	PUNCT
asir-3636	349	12	1−𝜖	1−𝜖	NUM
asir-3636	349	13	)	)	PUNCT
asir-3636	349	14	𝜇	𝜇	X
asir-3636	349	15	(	(	PUNCT
asir-3636	349	16	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	349	17	2(1−𝜖	2(1−𝜖	NUM
asir-3636	349	18	)	)	PUNCT
asir-3636	349	19	)	)	PUNCT
asir-3636	350	1	(	(	PUNCT
asir-3636	350	2	1	1	NUM
asir-3636	350	3	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	350	4	𝟐𝟐	𝟐𝟐	NUM
asir-3636	350	5	−	−	PROPN
asir-3636	350	6	𝜖)𝑑𝑡2	𝜖)𝑑𝑡2	PROPN
asir-3636	350	7	≤	≤	NUM
asir-3636	350	8	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	350	9	𝜖1−𝜀(1+𝜖	𝜖1−𝜀(1+𝜖	ADV
asir-3636	350	10	)	)	PUNCT
asir-3636	350	11	∫	∫	PROPN
asir-3636	350	12	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	350	13	2	2	NUM
asir-3636	350	14	)	)	PUNCT
asir-3636	350	15	′	′	NOUN
asir-3636	350	16	(	(	PUNCT
asir-3636	350	17	(	(	PUNCT
asir-3636	350	18	1	1	NUM
asir-3636	350	19	2	2	NUM
asir-3636	350	20	+	+	NOUN
asir-3636	350	21	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	350	22	2	2	NUM
asir-3636	350	23	)	)	PUNCT
asir-3636	351	1	|	|	ADV
asir-3636	351	2	𝑗	𝑗	INTJ
asir-3636	351	3	2	2	NUM
asir-3636	351	4	(	(	PUNCT
asir-3636	351	5	1	1	NUM
asir-3636	351	6	2	2	NUM
asir-3636	351	7	+	+	NOUN
asir-3636	351	8	𝜖	𝜖	X
asir-3636	351	9	)	)	PUNCT
asir-3636	351	10	𝑑	𝑑	NOUN
asir-3636	351	11	(	(	PUNCT
asir-3636	351	12	1	1	NUM
asir-3636	351	13	2	2	NUM
asir-3636	351	14	+	+	NOUN
asir-3636	351	15	𝜖	𝜖	NOUN
asir-3636	351	16	)	)	PUNCT
asir-3636	351	17	𝑑𝑡2	𝑑𝑡2	NOUN
asir-3636	351	18	(	(	PUNCT
asir-3636	351	19	1	1	NUM
asir-3636	351	20	2+𝜖	2+𝜖	NUM
asir-3636	351	21	)	)	PUNCT
asir-3636	351	22	(	(	PUNCT
asir-3636	351	23	1−𝜖	1−𝜖	NUM
asir-3636	351	24	)	)	PUNCT
asir-3636	351	25	≤	≤	NUM
asir-3636	351	26	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	351	27	𝜖1−𝜀(1+𝜖	𝜖1−𝜀(1+𝜖	ADV
asir-3636	351	28	)	)	PUNCT
asir-3636	351	29	∫	∫	PROPN
asir-3636	351	30	∑|(𝑓𝑗	∑|(𝑓𝑗	PROPN
asir-3636	351	31	2	2	NUM
asir-3636	351	32	)	)	PUNCT
asir-3636	351	33	′(𝑧	′(𝑧	NOUN
asir-3636	352	1	−	−	PROPN
asir-3636	352	2	𝜖)|	𝜖)|	ADV
asir-3636	352	3	2	2	NUM
asir-3636	352	4	𝑗	𝑗	NOUN
asir-3636	352	5	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	352	6	−	−	NUM
asir-3636	352	7	𝜖	𝜖	NOUN
asir-3636	352	8	)	)	PUNCT
asir-3636	352	9	,	,	PUNCT
asir-3636	352	10	(	(	PUNCT
asir-3636	352	11	1	1	NUM
asir-3636	352	12	2+𝜖	2+𝜖	NUM
asir-3636	352	13	)	)	PUNCT
asir-3636	352	14	(	(	PUNCT
asir-3636	352	15	1−𝜖	1−𝜖	NUM
asir-3636	352	16	)	)	PUNCT
asir-3636	352	17	where	where	SCONJ
asir-3636	352	18	𝑆(1−𝜖	𝑆(1−𝜖	ADJ
asir-3636	352	19	)	)	PUNCT
asir-3636	352	20	≔	≔	NOUN
asir-3636	352	21	{	{	PUNCT
asir-3636	352	22	(	(	PUNCT
asir-3636	352	23	𝑧	𝑧	PROPN
asir-3636	352	24	−	−	PROPN
asir-3636	352	25	𝜖	𝜖	NOUN
asir-3636	352	26	)	)	PUNCT
asir-3636	352	27	∈	∈	PROPN
asir-3636	352	28	𝔻	𝔻	PROPN
asir-3636	352	29	∶	∶	NOUN
asir-3636	352	30	0	0	NUM
asir-3636	352	31	≤	≤	NUM
asir-3636	352	32	|𝑧	|𝑧	NOUN
asir-3636	352	33	−	−	PROPN
asir-3636	352	34	𝜖|	𝜖|	PROPN
asir-3636	352	35	≤	≤	NOUN
asir-3636	352	36	(	(	PUNCT
asir-3636	352	37	1	1	NUM
asir-3636	352	38	−	−	PROPN
asir-3636	352	39	𝜖	𝜖	NOUN
asir-3636	352	40	)	)	PUNCT
asir-3636	352	41	and	and	CCONJ
asir-3636	352	42	𝑧	𝑧	ADP
asir-3636	352	43	−	−	PROPN
asir-3636	352	44	𝜖	𝜖	PROPN
asir-3636	352	45	|𝑧	|𝑧	NOUN
asir-3636	352	46	−	−	PROPN
asir-3636	352	47	𝜖|	𝜖|	PROPN
asir-3636	352	48	∈	∈	PROPN
asir-3636	352	49	𝛾	𝛾	NOUN
asir-3636	352	50	}	}	PUNCT
asir-3636	352	51	.	.	PUNCT
asir-3636	353	1	the	the	DET
asir-3636	353	2	proof	proof	NOUN
asir-3636	353	3	is	be	AUX
asir-3636	353	4	therefore	therefore	ADV
asir-3636	353	5	completed	complete	VERB
asir-3636	353	6	.	.	PUNCT
asir-3636	354	1	the	the	DET
asir-3636	354	2	last	last	ADJ
asir-3636	354	3	result	result	NOUN
asir-3636	354	4	that	that	SCONJ
asir-3636	354	5	we	we	PRON
asir-3636	354	6	need	need	VERB
asir-3636	354	7	before	before	ADP
asir-3636	354	8	giving	give	VERB
asir-3636	354	9	the	the	DET
asir-3636	354	10	proof	proof	NOUN
asir-3636	354	11	of	of	ADP
asir-3636	354	12	theorem	theorem	NOUN
asir-3636	354	13	(	(	PUNCT
asir-3636	354	14	2.1	2.1	NUM
asir-3636	354	15	)	)	PUNCT
asir-3636	354	16	is	be	AUX
asir-3636	354	17	the	the	DET
asir-3636	354	18	following	follow	VERB
asir-3636	354	19	one	one	NUM
asir-3636	354	20	(	(	PUNCT
asir-3636	354	21	see	see	VERB
asir-3636	354	22	brahim	brahim	PROPN
asir-3636	354	23	bouya	bouya	PROPN
asir-3636	354	24	,	,	PUNCT
asir-3636	354	25	2008	2008	NUM
asir-3636	354	26	)	)	PUNCT
asir-3636	354	27	.	.	PUNCT
asir-3636	355	1	lemma	lemma	PROPN
asir-3636	355	2	(	(	PUNCT
asir-3636	355	3	4.8	4.8	NUM
asir-3636	355	4	):	):	PUNCT
asir-3636	355	5	∫	∫	PROPN
asir-3636	356	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	356	2	2	2	NUM
asir-3636	356	3	(	(	PUNCT
asir-3636	356	4	𝑧)|	𝑧)|	ADV
asir-3636	356	5	𝑗	𝑗	ADJ
asir-3636	356	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	356	7	)	)	PUNCT
asir-3636	356	8	𝑎𝛾	𝑎𝛾	ADP
asir-3636	356	9	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	356	10	)	)	PUNCT
asir-3636	356	11	∆𝜸	∆𝜸	PROPN
asir-3636	356	12	𝟐𝟐	𝟐𝟐	NUM
asir-3636	356	13	≤∑𝐶1+𝜖‖(𝑓𝑗	≤∑𝐶1+𝜖‖(𝑓𝑗	SYM
asir-3636	356	14	2	2	NUM
asir-3636	356	15	)	)	PUNCT
asir-3636	356	16	′‖	′‖	X
asir-3636	356	17	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	356	18	)	)	PUNCT
asir-3636	356	19	2	2	NUM
asir-3636	356	20	+	+	CCONJ
asir-3636	356	21	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	356	22	)	)	PUNCT
asir-3636	356	23	𝑗	𝑗	INTJ
asir-3636	356	24	.	.	PUNCT
asir-3636	357	1	proof	proof	NOUN
asir-3636	357	2	:	:	PUNCT
asir-3636	357	3	using	use	VERB
asir-3636	357	4	(	(	PUNCT
asir-3636	357	5	19	19	NUM
asir-3636	357	6	)	)	PUNCT
asir-3636	357	7	and	and	CCONJ
asir-3636	357	8	lemmas	lemmas	PROPN
asir-3636	357	9	(	(	PUNCT
asir-3636	357	10	4.6	4.6	NUM
asir-3636	357	11	)	)	PUNCT
asir-3636	357	12	and	and	CCONJ
asir-3636	357	13	(	(	PUNCT
asir-3636	357	14	4.7	4.7	NUM
asir-3636	357	15	)	)	PUNCT
asir-3636	357	16	,	,	PUNCT
asir-3636	357	17	we	we	PRON
asir-3636	357	18	find	find	VERB
asir-3636	357	19	that	that	SCONJ
asir-3636	357	20	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	357	21	applied	apply	VERB
asir-3636	357	22	science	science	NOUN
asir-3636	357	23	and	and	CCONJ
asir-3636	357	24	innovative	innovative	ADJ
asir-3636	357	25	research	research	NOUN
asir-3636	357	26	vol	vol	NOUN
asir-3636	357	27	.	.	PROPN
asir-3636	358	1	5	5	NUM
asir-3636	358	2	,	,	PUNCT
asir-3636	358	3	no	no	INTJ
asir-3636	358	4	.	.	NOUN
asir-3636	358	5	1	1	NUM
asir-3636	358	6	,	,	PUNCT
asir-3636	358	7	2021	2021	NUM
asir-3636	358	8	37	37	NUM
asir-3636	358	9	published	publish	VERB
asir-3636	358	10	by	by	ADP
asir-3636	358	11	scholink	scholink	PROPN
asir-3636	358	12	inc	inc	PROPN
asir-3636	358	13	.	.	PROPN
asir-3636	358	14	∫	∫	PROPN
asir-3636	359	1	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	359	2	2	2	NUM
asir-3636	359	3	(	(	PUNCT
asir-3636	359	4	𝑧)|	𝑧)|	ADV
asir-3636	359	5	𝑗	𝑗	ADJ
asir-3636	359	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	359	7	)	)	PUNCT
asir-3636	359	8	𝑎𝛾	𝑎𝛾	ADP
asir-3636	359	9	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	359	10	)	)	PUNCT
asir-3636	359	11	∆𝜸	∆𝜸	PROPN
asir-3636	359	12	𝟐𝟐	𝟐𝟐	NUM
asir-3636	359	13	=	=	SYM
asir-3636	359	14	1	1	NUM
asir-3636	359	15	𝜋	𝜋	NOUN
asir-3636	359	16	∫	∫	PROPN
asir-3636	359	17	(	(	PUNCT
asir-3636	359	18	∫	∫	PROPN
asir-3636	359	19	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	359	20	2	2	NUM
asir-3636	359	21	(	(	PUNCT
asir-3636	359	22	(	(	PUNCT
asir-3636	359	23	1	1	NUM
asir-3636	359	24	−	−	PROPN
asir-3636	359	25	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	359	26	2	2	NUM
asir-3636	359	27	)	)	PUNCT
asir-3636	360	1	|	|	ADV
asir-3636	360	2	𝑗	𝑗	NOUN
asir-3636	360	3	2(1+𝜖	2(1+𝜖	NUM
asir-3636	360	4	)	)	PUNCT
asir-3636	360	5	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	360	6	𝟐𝟐	𝟐𝟐	VERB
asir-3636	360	7	𝑎𝛾	𝑎𝛾	VERB
asir-3636	360	8	2((1	2((1	NUM
asir-3636	360	9	−	−	PROPN
asir-3636	361	1	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	361	2	2	2	NUM
asir-3636	361	3	)	)	PUNCT
asir-3636	361	4	(	(	PUNCT
asir-3636	361	5	1	1	NUM
asir-3636	361	6	−	−	NOUN
asir-3636	361	7	𝜖)𝑑𝑡2)𝑑(1	𝜖)𝑑𝑡2)𝑑(1	NOUN
asir-3636	361	8	1	1	NUM
asir-3636	361	9	0	0	NUM
asir-3636	361	10	−	−	NUM
asir-3636	361	11	𝜖	𝜖	NOUN
asir-3636	361	12	)	)	PUNCT
asir-3636	361	13	≤	≤	NOUN
asir-3636	361	14	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	361	15	)	)	PUNCT
asir-3636	362	1	+	+	CCONJ
asir-3636	362	2	2(2𝜖+1)∫	2(2𝜖+1)∫	NUM
asir-3636	362	3	(	(	PUNCT
asir-3636	362	4	∫	∫	PROPN
asir-3636	362	5	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	362	6	2	2	NUM
asir-3636	362	7	(	(	PUNCT
asir-3636	362	8	(	(	PUNCT
asir-3636	362	9	1	1	NUM
asir-3636	362	10	−	−	PROPN
asir-3636	362	11	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	362	12	2	2	NUM
asir-3636	362	13	)	)	PUNCT
asir-3636	362	14	𝑗	𝑗	PROPN
asir-3636	362	15	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	362	16	𝟐𝟐	𝟐𝟐	NUM
asir-3636	362	17	1	1	NUM
asir-3636	362	18	0	0	NUM
asir-3636	362	19	−	−	NOUN
asir-3636	362	20	𝑓𝑗	𝑓𝑗	ADJ
asir-3636	362	21	2	2	NUM
asir-3636	362	22	(	(	PUNCT
asir-3636	362	23	𝜇	𝜇	X
asir-3636	362	24	(	(	PUNCT
asir-3636	362	25	1−𝜖)𝑒𝑖𝑡	1−𝜖)𝑒𝑖𝑡	NUM
asir-3636	362	26	2(1	2(1	NUM
asir-3636	362	27	−	−	NOUN
asir-3636	362	28	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	362	29	2	2	NUM
asir-3636	362	30	)	)	PUNCT
asir-3636	362	31	|	|	ADV
asir-3636	362	32	2(1+𝜖	2(1+𝜖	NUM
asir-3636	362	33	)	)	PUNCT
asir-3636	362	34	𝑎𝛾	𝑎𝛾	ADP
asir-3636	362	35	2((1	2((1	NUM
asir-3636	362	36	−	−	PROPN
asir-3636	363	1	𝜖)𝑒𝑖𝑡	𝜖)𝑒𝑖𝑡	PROPN
asir-3636	363	2	2	2	NUM
asir-3636	363	3	)	)	PUNCT
asir-3636	363	4	(	(	PUNCT
asir-3636	363	5	1	1	NUM
asir-3636	363	6	−	−	PROPN
asir-3636	363	7	𝜖)𝑑𝑡2)𝑑(1	𝜖)𝑑𝑡2)𝑑(1	NOUN
asir-3636	363	8	−	−	PROPN
asir-3636	363	9	𝜖	𝜖	NOUN
asir-3636	363	10	)	)	PUNCT
asir-3636	363	11	≤	≤	NOUN
asir-3636	363	12	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	363	13	)	)	PUNCT
asir-3636	364	1	+	+	ADP
asir-3636	364	2	∑𝐶1+𝜖‖(𝑓𝑗	∑𝐶1+𝜖‖(𝑓𝑗	X
asir-3636	364	3	2	2	NUM
asir-3636	364	4	)	)	PUNCT
asir-3636	364	5	′‖	′‖	X
asir-3636	364	6	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	364	7	)	)	PUNCT
asir-3636	364	8	2	2	NUM
asir-3636	364	9	𝑗	𝑗	NOUN
asir-3636	364	10	.	.	PUNCT
asir-3636	365	1	this	this	PRON
asir-3636	365	2	completes	complete	VERB
asir-3636	365	3	the	the	DET
asir-3636	365	4	proof	proof	NOUN
asir-3636	365	5	of	of	ADP
asir-3636	365	6	the	the	DET
asir-3636	365	7	lemma	lemma	PROPN
asir-3636	365	8	.	.	PUNCT
asir-3636	365	9	conclusion	conclusion	NOUN
asir-3636	365	10	.	.	PUNCT
asir-3636	366	1	now	now	ADV
asir-3636	366	2	,	,	PUNCT
asir-3636	366	3	according	accord	VERB
asir-3636	366	4	to	to	ADP
asir-3636	366	5	(	(	PUNCT
asir-3636	366	6	18	18	NUM
asir-3636	366	7	)	)	PUNCT
asir-3636	366	8	and	and	CCONJ
asir-3636	366	9	lemmas	lemmas	PROPN
asir-3636	366	10	(	(	PUNCT
asir-3636	366	11	4.4	4.4	NUM
asir-3636	366	12	)	)	PUNCT
asir-3636	366	13	,	,	PUNCT
asir-3636	366	14	(	(	PUNCT
asir-3636	366	15	4.5	4.5	NUM
asir-3636	366	16	)	)	PUNCT
asir-3636	366	17	and	and	CCONJ
asir-3636	366	18	(	(	PUNCT
asir-3636	366	19	4.8	4.8	NUM
asir-3636	366	20	)	)	PUNCT
asir-3636	366	21	,	,	PUNCT
asir-3636	366	22	we	we	PRON
asir-3636	366	23	obtain	obtain	VERB
asir-3636	366	24	∫	∫	NOUN
asir-3636	367	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	367	2	2	2	NUM
asir-3636	367	3	(	(	PUNCT
asir-3636	367	4	𝑧)|	𝑧)|	ADV
asir-3636	367	5	𝑗	𝑗	ADJ
asir-3636	367	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	367	7	)	)	PUNCT
asir-3636	367	8	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	367	9	2	2	NUM
asir-3636	367	10	)	)	PUNCT
asir-3636	367	11	γ	γ	NOUN
asir-3636	367	12	)	)	PUNCT
asir-3636	367	13	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	367	14	2	2	NUM
asir-3636	367	15	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	367	16	)	)	PUNCT
asir-3636	367	17	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	367	18	𝟐𝟐	𝟐𝟐	NUM
asir-3636	367	19	≤	≤	NUM
asir-3636	367	20	2∑‖(𝑓𝑗	2∑‖(𝑓𝑗	NUM
asir-3636	367	21	2	2	NUM
asir-3636	367	22	)	)	PUNCT
asir-3636	367	23	′‖	′‖	X
asir-3636	367	24	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	367	25	)	)	PUNCT
asir-3636	367	26	2	2	NUM
asir-3636	367	27	𝑗	𝑗	NOUN
asir-3636	367	28	+	+	NUM
asir-3636	367	29	8∫	8∫	NUM
asir-3636	367	30	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	367	31	2	2	NUM
asir-3636	367	32	(	(	PUNCT
asir-3636	367	33	𝑧)|	𝑧)|	NOUN
asir-3636	367	34	2(1+𝜖	2(1+𝜖	NUM
asir-3636	367	35	)	)	PUNCT
asir-3636	367	36	𝑎𝛾	𝑎𝛾	ADP
asir-3636	367	37	2(𝑧)𝑑𝐴(𝑧	2(𝑧)𝑑𝐴(𝑧	NUM
asir-3636	367	38	)	)	PUNCT
asir-3636	368	1	𝑗	𝑗	PRON
asir-3636	368	2	𝛾(1−𝜖)⋂∆𝜸	𝛾(1−𝜖)⋂∆𝜸	NOUN
asir-3636	368	3	𝟐𝟐	𝟐𝟐	NUM
asir-3636	368	4	≤∑𝐶1+𝜖‖(𝑓𝑗	≤∑𝐶1+𝜖‖(𝑓𝑗	SYM
asir-3636	368	5	2	2	NUM
asir-3636	368	6	)	)	PUNCT
asir-3636	368	7	′‖	′‖	X
asir-3636	368	8	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	368	9	)	)	PUNCT
asir-3636	368	10	2	2	NUM
asir-3636	368	11	𝑗	𝑗	NOUN
asir-3636	368	12	+	+	CCONJ
asir-3636	368	13	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	368	14	)	)	PUNCT
asir-3636	368	15	.	.	PUNCT
asir-3636	369	1	combining	combine	VERB
asir-3636	369	2	this	this	PRON
asir-3636	369	3	with	with	ADP
asir-3636	369	4	lemma	lemma	PROPN
asir-3636	369	5	(	(	PUNCT
asir-3636	369	6	4.3	4.3	NUM
asir-3636	369	7	)	)	PUNCT
asir-3636	369	8	,	,	PUNCT
asir-3636	369	9	we	we	PRON
asir-3636	369	10	deduce	deduce	VERB
asir-3636	369	11	that	that	SCONJ
asir-3636	369	12	∫	∫	NOUN
asir-3636	370	1	∑|𝑓𝑗	∑|𝑓𝑗	ADJ
asir-3636	370	2	2	2	NUM
asir-3636	370	3	(	(	PUNCT
asir-3636	370	4	𝑧)|	𝑧)|	ADV
asir-3636	370	5	𝑗	𝑗	ADJ
asir-3636	370	6	2(1+𝜖	2(1+𝜖	NUM
asir-3636	370	7	)	)	PUNCT
asir-3636	370	8	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	370	9	2	2	NUM
asir-3636	370	10	)	)	PUNCT
asir-3636	370	11	γ	γ	NOUN
asir-3636	370	12	)	)	PUNCT
asir-3636	370	13	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	370	14	2	2	NUM
asir-3636	370	15	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	370	16	)	)	PUNCT
asir-3636	370	17	∆𝛾	∆𝛾	PROPN
asir-3636	370	18	≤∑𝐶1+𝜖‖(𝑓𝑗	≤∑𝐶1+𝜖‖(𝑓𝑗	PROPN
asir-3636	370	19	2	2	NUM
asir-3636	370	20	)	)	PUNCT
asir-3636	370	21	′‖	′‖	X
asir-3636	370	22	𝐿2(∆𝛾	𝐿2(∆𝛾	NUM
asir-3636	370	23	)	)	PUNCT
asir-3636	370	24	2	2	NUM
asir-3636	370	25	𝑗	𝑗	NOUN
asir-3636	370	26	+	+	CCONJ
asir-3636	370	27	𝐶𝐴(∆𝛾	𝐶𝐴(∆𝛾	NOUN
asir-3636	370	28	)	)	PUNCT
asir-3636	370	29	.	.	PUNCT
asir-3636	371	1	hence	hence	ADV
asir-3636	371	2	∫	∫	X
asir-3636	371	3	𝔻	𝔻	ADJ
asir-3636	371	4	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	371	5	2	2	NUM
asir-3636	371	6	(	(	PUNCT
asir-3636	371	7	𝑧)|	𝑧)|	ADV
asir-3636	371	8	𝑗	𝑗	ADJ
asir-3636	371	9	2(1+𝜖	2(1+𝜖	NUM
asir-3636	371	10	)	)	PUNCT
asir-3636	371	11	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	371	12	2	2	NUM
asir-3636	371	13	)	)	PUNCT
asir-3636	371	14	γ	γ	NOUN
asir-3636	371	15	)	)	PUNCT
asir-3636	371	16	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	371	17	2	2	NUM
asir-3636	371	18	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	371	19	)	)	PUNCT
asir-3636	371	20	=	=	SYM
asir-3636	372	1	∑∫	∑∫	PROPN
asir-3636	372	2	∆𝛾𝑛	∆𝛾𝑛	ADJ
asir-3636	372	3	∑|𝑓𝑗	∑|𝑓𝑗	PROPN
asir-3636	372	4	2	2	NUM
asir-3636	372	5	(	(	PUNCT
asir-3636	372	6	𝑧)|	𝑧)|	ADV
asir-3636	372	7	𝑗	𝑗	ADJ
asir-3636	372	8	2(1+𝜖	2(1+𝜖	NUM
asir-3636	372	9	)	)	PUNCT
asir-3636	372	10	|((𝑓𝑗	|((𝑓𝑗	PROPN
asir-3636	372	11	2	2	NUM
asir-3636	372	12	)	)	PUNCT
asir-3636	372	13	γ	γ	NOUN
asir-3636	372	14	)	)	PUNCT
asir-3636	372	15	′(𝑧)|	′(𝑧)|	ADJ
asir-3636	372	16	2	2	NUM
asir-3636	372	17	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NUM
asir-3636	372	18	)	)	PUNCT
asir-3636	372	19	∞	∞	PROPN
asir-3636	372	20	𝑛=1	𝑛=1	NOUN
asir-3636	372	21	≤∑𝐶1+𝜖∑‖(𝑓𝑗	≤∑𝐶1+𝜖∑‖(𝑓𝑗	X
asir-3636	372	22	2	2	NUM
asir-3636	372	23	)	)	PUNCT
asir-3636	372	24	′‖	′‖	X
asir-3636	372	25	𝐿2(∆𝛾𝑛	𝐿2(∆𝛾𝑛	SYM
asir-3636	372	26	)	)	PUNCT
asir-3636	372	27	2	2	NUM
asir-3636	372	28	∞	∞	NUM
asir-3636	372	29	𝑛=1𝑗	𝑛=1𝑗	NOUN
asir-3636	372	30	+	+	CCONJ
asir-3636	372	31	𝐶∑𝐴(∆𝛾𝑛	𝐶∑𝐴(∆𝛾𝑛	NOUN
asir-3636	372	32	)	)	PUNCT
asir-3636	373	1	∞	∞	PROPN
asir-3636	373	2	𝑛=1	𝑛=1	NOUN
asir-3636	373	3	≤	≤	NUM
asir-3636	373	4	𝐶1+𝜖	𝐶1+𝜖	NOUN
asir-3636	373	5	.	.	PUNCT
asir-3636	374	1	this	this	PRON
asir-3636	374	2	completes	complete	VERB
asir-3636	374	3	the	the	DET
asir-3636	374	4	proof	proof	NOUN
asir-3636	374	5	of	of	ADP
asir-3636	374	6	theorem	theorem	NOUN
asir-3636	374	7	(	(	PUNCT
asir-3636	374	8	2.1	2.1	NUM
asir-3636	374	9	)	)	PUNCT
asir-3636	374	10	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-3636	374	11	applied	apply	VERB
asir-3636	374	12	science	science	NOUN
asir-3636	374	13	and	and	CCONJ
asir-3636	374	14	innovative	innovative	ADJ
asir-3636	374	15	research	research	NOUN
asir-3636	374	16	vol	vol	NOUN
asir-3636	374	17	.	.	PROPN
asir-3636	375	1	5	5	NUM
asir-3636	375	2	,	,	PUNCT
asir-3636	375	3	no	no	INTJ
asir-3636	375	4	.	.	NOUN
asir-3636	375	5	1	1	NUM
asir-3636	375	6	,	,	PUNCT
asir-3636	375	7	2021	2021	NUM
asir-3636	375	8	38	38	NUM
asir-3636	375	9	published	publish	VERB
asir-3636	375	10	by	by	ADP
asir-3636	375	11	scholink	scholink	PROPN
asir-3636	375	12	inc	inc	PROPN
asir-3636	375	13	.	.	PROPN
asir-3636	375	14	references	reference	NOUN
asir-3636	375	15	bouya	bouya	PROPN
asir-3636	375	16	,	,	PUNCT
asir-3636	375	17	b.	b.	PROPN
asir-3636	375	18	(	(	PUNCT
asir-3636	375	19	2006	2006	NUM
asir-3636	375	20	)	)	PUNCT
asir-3636	375	21	.	.	PUNCT
asir-3636	376	1	i	i	PRON
asir-3636	376	2	d	d	PROPN
asir-3636	376	3	éaux	éaux	PROPN
asir-3636	376	4	ferm	ferm	PROPN
asir-3636	376	5	és	és	PROPN
asir-3636	376	6	de	de	X
asir-3636	376	7	certaines	certaines	X
asir-3636	376	8	alg`ebres	alg`ebre	NOUN
asir-3636	376	9	de	de	ADP
asir-3636	376	10	fonctions	fonction	NOUN
asir-3636	376	11	analytiques	analytique	NOUN
asir-3636	376	12	.	.	PUNCT
asir-3636	377	1	c.	c.	PROPN
asir-3636	377	2	r.	r.	PROPN
asir-3636	377	3	math	math	PROPN
asir-3636	377	4	.	.	PUNCT
asir-3636	378	1	acad	acad	PROPN
asir-3636	378	2	.	.	PUNCT
asir-3636	379	1	sci	sci	PROPN
asir-3636	379	2	.	.	PROPN
asir-3636	379	3	paris	paris	PROPN
asir-3636	379	4	,	,	PUNCT
asir-3636	379	5	343(4	343(4	NUM
asir-3636	379	6	)	)	PUNCT
asir-3636	379	7	,	,	PUNCT
asir-3636	379	8	235	235	NUM
asir-3636	379	9	-	-	SYM
asir-3636	379	10	238	238	NUM
asir-3636	379	11	.	.	PUNCT
asir-3636	380	1	https://doi.org/10.1016/j.crma.2006.06.021	https://doi.org/10.1016/j.crma.2006.06.021	PROPN
asir-3636	380	2	brahim	brahim	PROPN
asir-3636	380	3	bouya	bouya	PROPN
asir-3636	380	4	.	.	PUNCT
asir-3636	381	1	(	(	PUNCT
asir-3636	381	2	2008	2008	NUM
asir-3636	381	3	)	)	PUNCT
asir-3636	381	4	.	.	PUNCT
asir-3636	382	1	closed	close	VERB
asir-3636	382	2	ideals	ideal	NOUN
asir-3636	382	3	in	in	ADP
asir-3636	382	4	some	some	DET
asir-3636	382	5	algebras	algebra	NOUN
asir-3636	382	6	of	of	ADP
asir-3636	382	7	analytic	analytic	ADJ
asir-3636	382	8	function	function	NOUN
asir-3636	382	9	.	.	PUNCT
asir-3636	383	1	https://doi.org/10.4153/cjm-2009-014-5	https://doi.org/10.4153/cjm-2009-014-5	PROPN
asir-3636	383	2	carleson	carleson	PROPN
asir-3636	383	3	,	,	PUNCT
asir-3636	383	4	l.	l.	PROPN
asir-3636	383	5	(	(	PUNCT
asir-3636	383	6	1960	1960	NUM
asir-3636	383	7	)	)	PUNCT
asir-3636	383	8	.	.	PUNCT
asir-3636	384	1	a	a	DET
asir-3636	384	2	representation	representation	NOUN
asir-3636	384	3	formula	formula	NOUN
asir-3636	384	4	in	in	ADP
asir-3636	384	5	the	the	DET
asir-3636	384	6	dirichlet	dirichlet	PROPN
asir-3636	384	7	space	space	NOUN
asir-3636	384	8	.	.	PUNCT
asir-3636	385	1	math	math	NOUN
asir-3636	385	2	.	.	PUNCT
asir-3636	386	1	z.	z.	PROPN
asir-3636	386	2	,	,	PUNCT
asir-3636	386	3	73	73	NUM
asir-3636	386	4	,	,	PUNCT
asir-3636	386	5	190	190	NUM
asir-3636	386	6	-	-	SYM
asir-3636	386	7	196	196	NUM
asir-3636	386	8	.	.	PUNCT
asir-3636	387	1	https://doi.org/10.1007/bf01162477	https://doi.org/10.1007/bf01162477	X
asir-3636	387	2	duren	duren	PROPN
asir-3636	387	3	,	,	PUNCT
asir-3636	387	4	p.	p.	PROPN
asir-3636	387	5	l.	l.	PROPN
asir-3636	387	6	(	(	PUNCT
asir-3636	387	7	1970	1970	NUM
asir-3636	387	8	)	)	PUNCT
asir-3636	387	9	.	.	PUNCT
asir-3636	388	1	theory	theory	NOUN
asir-3636	388	2	of	of	ADP
asir-3636	388	3	hp	hp	ADJ
asir-3636	388	4	spaces	space	NOUN
asir-3636	388	5	.	.	PUNCT
asir-3636	389	1	academic	academic	ADJ
asir-3636	389	2	press	press	NOUN
asir-3636	389	3	,	,	PUNCT
asir-3636	389	4	new	new	PROPN
asir-3636	389	5	york	york	PROPN
asir-3636	389	6	.	.	PUNCT
asir-3636	390	1	el	el	PROPN
asir-3636	390	2	-	-	PUNCT
asir-3636	390	3	fallah	fallah	ADJ
asir-3636	390	4	,	,	PUNCT
asir-3636	390	5	o.	o.	PROPN
asir-3636	390	6	,	,	PUNCT
asir-3636	390	7	kellay	kellay	PROPN
asir-3636	390	8	,	,	PUNCT
asir-3636	390	9	k.	k.	PROPN
asir-3636	390	10	,	,	PUNCT
asir-3636	390	11	&	&	CCONJ
asir-3636	390	12	ransford	ransford	PROPN
asir-3636	390	13	,	,	PUNCT
asir-3636	390	14	t.	t.	PROPN
asir-3636	390	15	(	(	PUNCT
asir-3636	390	16	2006	2006	NUM
asir-3636	390	17	)	)	PUNCT
asir-3636	390	18	cyclicity	cyclicity	NOUN
asir-3636	390	19	in	in	ADP
asir-3636	390	20	the	the	DET
asir-3636	390	21	dirichlet	dirichlet	PROPN
asir-3636	390	22	space	space	NOUN
asir-3636	390	23	.	.	PUNCT
asir-3636	391	1	ark	ark	PROPN
asir-3636	391	2	.	.	PROPN
asir-3636	391	3	mat	mat	PROPN
asir-3636	391	4	.	.	PROPN
asir-3636	391	5	,	,	PUNCT
asir-3636	391	6	44(1	44(1	PROPN
asir-3636	391	7	)	)	PUNCT
asir-3636	391	8	,	,	PUNCT
asir-3636	391	9	61	61	NUM
asir-3636	391	10	-	-	SYM
asir-3636	391	11	86	86	NUM
asir-3636	391	12	.	.	PUNCT
asir-3636	392	1	https://doi.org/10.1007/s11512-005-0008-z	https://doi.org/10.1007/s11512-005-0008-z	PROPN
asir-3636	392	2	esterle	esterle	PROPN
asir-3636	392	3	,	,	PUNCT
asir-3636	392	4	j.	j.	PROPN
asir-3636	392	5	,	,	PUNCT
asir-3636	392	6	strouse	strouse	PROPN
asir-3636	392	7	,	,	PUNCT
asir-3636	392	8	e.	e.	PROPN
asir-3636	392	9	,	,	PUNCT
asir-3636	392	10	&	&	CCONJ
asir-3636	392	11	zouakia	zouakia	PROPN
asir-3636	392	12	,	,	PUNCT
asir-3636	392	13	f.	f.	PROPN
asir-3636	392	14	(	(	PUNCT
asir-3636	392	15	1994	1994	NUM
asir-3636	392	16	)	)	PUNCT
asir-3636	392	17	.	.	PUNCT
asir-3636	393	1	closed	close	VERB
asir-3636	393	2	ideal	ideal	NOUN
asir-3636	393	3	of	of	ADP
asir-3636	393	4	a+	a+	PUNCT
asir-3636	393	5	and	and	CCONJ
asir-3636	393	6	the	the	DET
asir-3636	393	7	cantor	cantor	PROPN
asir-3636	393	8	set	set	PROPN
asir-3636	393	9	.	.	PUNCT
asir-3636	394	1	j.	j.	PROPN
asir-3636	394	2	reine	reine	PROPN
asir-3636	394	3	angew	angew	PROPN
asir-3636	394	4	.	.	PUNCT
asir-3636	395	1	math	math	NOUN
asir-3636	395	2	.	.	PUNCT
asir-3636	396	1	,	,	PUNCT
asir-3636	396	2	449	449	NUM
asir-3636	396	3	,	,	PUNCT
asir-3636	396	4	65	65	NUM
asir-3636	396	5	-	-	SYM
asir-3636	396	6	79	79	NUM
asir-3636	396	7	.	.	PUNCT
asir-3636	396	8	https://doi.org/10.1515/crll.1994.449.65	https://doi.org/10.1515/crll.1994.449.65	PROPN
asir-3636	396	9	hedenmalm	hedenmalm	PROPN
asir-3636	396	10	,	,	PUNCT
asir-3636	396	11	h.	h.	PROPN
asir-3636	396	12	(	(	PUNCT
asir-3636	396	13	1990	1990	NUM
asir-3636	396	14	)	)	PUNCT
asir-3636	396	15	.	.	PUNCT
asir-3636	397	1	shields	shield	NOUN
asir-3636	397	2	,	,	PUNCT
asir-3636	397	3	invariant	invariant	ADJ
asir-3636	397	4	subspaces	subspace	NOUN
asir-3636	397	5	in	in	ADP
asir-3636	397	6	banach	banach	NOUN
asir-3636	397	7	spaces	space	NOUN
asir-3636	397	8	of	of	ADP
asir-3636	397	9	analytic	analytic	ADJ
asir-3636	397	10	functions	function	NOUN
asir-3636	397	11	.	.	PUNCT
asir-3636	398	1	mich	mich	PROPN
asir-3636	398	2	.	.	PUNCT
asir-3636	398	3	math	math	PROPN
asir-3636	398	4	.	.	PUNCT
asir-3636	399	1	j.	j.	PROPN
asir-3636	399	2	,	,	PUNCT
asir-3636	399	3	37	37	NUM
asir-3636	399	4	,	,	PUNCT
asir-3636	399	5	91	91	NUM
asir-3636	399	6	-	-	SYM
asir-3636	399	7	104	104	NUM
asir-3636	399	8	.	.	PUNCT
asir-3636	399	9	https://doi.org/10.1307/mmj/1029004068	https://doi.org/10.1307/mmj/1029004068	PROPN
asir-3636	399	10	hoffman	hoffman	PROPN
asir-3636	399	11	,	,	PUNCT
asir-3636	399	12	k.	k.	PROPN
asir-3636	399	13	(	(	PUNCT
asir-3636	399	14	1988	1988	NUM
asir-3636	399	15	)	)	PUNCT
asir-3636	399	16	.	.	PUNCT
asir-3636	400	1	banach	banach	NOUN
asir-3636	400	2	spaces	space	NOUN
asir-3636	400	3	of	of	ADP
asir-3636	400	4	analytic	analytic	ADJ
asir-3636	400	5	functions	function	NOUN
asir-3636	400	6	.	.	PUNCT
asir-3636	401	1	dover	dover	PROPN
asir-3636	401	2	publications	publications	PROPN
asir-3636	401	3	inc	inc	PROPN
asir-3636	401	4	.	.	PROPN
asir-3636	401	5	,	,	PUNCT
asir-3636	401	6	new	new	PROPN
asir-3636	401	7	york	york	PROPN
asir-3636	401	8	.	.	PUNCT
asir-3636	402	1	reprint	reprint	NOUN
asir-3636	402	2	of	of	ADP
asir-3636	402	3	the	the	DET
asir-3636	402	4	1962	1962	NUM
asir-3636	402	5	original	original	NOUN
asir-3636	402	6	.	.	PUNCT
asir-3636	403	1	korenblum	korenblum	PROPN
asir-3636	403	2	,	,	PUNCT
asir-3636	403	3	b.	b.	PROPN
asir-3636	403	4	i.	i.	PROPN
asir-3636	403	5	(	(	PUNCT
asir-3636	403	6	1972	1972	NUM
asir-3636	403	7	)	)	PUNCT
asir-3636	403	8	invariant	invariant	ADJ
asir-3636	403	9	subspaces	subspace	NOUN
asir-3636	403	10	of	of	ADP
asir-3636	403	11	the	the	DET
asir-3636	403	12	shift	shift	NOUN
asir-3636	403	13	operator	operator	NOUN
asir-3636	403	14	in	in	ADP
asir-3636	403	15	a	a	DET
asir-3636	403	16	weighted	weighted	ADJ
asir-3636	403	17	hilbert	hilbert	NOUN
asir-3636	403	18	space	space	NOUN
asir-3636	403	19	.	.	PUNCT
asir-3636	404	1	mat	mat	PROPN
asir-3636	404	2	.	.	PUNCT
asir-3636	404	3	sb	sb	PROPN
asir-3636	404	4	.	.	PROPN
asir-3636	404	5	,	,	PUNCT
asir-3636	404	6	89(131	89(131	PROPN
asir-3636	404	7	)	)	PUNCT
asir-3636	404	8	,	,	PUNCT
asir-3636	404	9	110	110	NUM
asir-3636	404	10	-	-	SYM
asir-3636	404	11	138	138	NUM
asir-3636	404	12	.	.	PUNCT
asir-3636	405	1	https://doi.org/10.1070/sm1972v018n01abeh001617	https://doi.org/10.1070/sm1972v018n01abeh001617	PROPN
asir-3636	405	2	matheson	matheson	PROPN
asir-3636	405	3	,	,	PUNCT
asir-3636	405	4	a.	a.	NOUN
asir-3636	405	5	(	(	PUNCT
asir-3636	405	6	1978	1978	NUM
asir-3636	405	7	)	)	PUNCT
asir-3636	405	8	.	.	PUNCT
asir-3636	406	1	approximation	approximation	NOUN
asir-3636	406	2	of	of	ADP
asir-3636	406	3	analytic	analytic	ADJ
asir-3636	406	4	functions	function	NOUN
asir-3636	406	5	satisfying	satisfy	VERB
asir-3636	406	6	a	a	DET
asir-3636	406	7	lipschitz	lipschitz	NOUN
asir-3636	406	8	condition	condition	NOUN
asir-3636	406	9	.	.	PUNCT
asir-3636	407	1	mich	mich	PROPN
asir-3636	407	2	.	.	PUNCT
asir-3636	407	3	math	math	PROPN
asir-3636	407	4	.	.	PUNCT
asir-3636	408	1	j.	j.	PROPN
asir-3636	408	2	,	,	PUNCT
asir-3636	408	3	25(3	25(3	NUM
asir-3636	408	4	)	)	PUNCT
asir-3636	408	5	,	,	PUNCT
asir-3636	408	6	289	289	NUM
asir-3636	408	7	-	-	SYM
asir-3636	408	8	298	298	NUM
asir-3636	408	9	.	.	PUNCT
asir-3636	408	10	https://doi.org/10.1307/mmj/1029002111	https://doi.org/10.1307/mmj/1029002111	PROPN
asir-3636	408	11	rudin	rudin	PROPN
asir-3636	408	12	,	,	PUNCT
asir-3636	408	13	w.	w.	PROPN
asir-3636	408	14	(	(	PUNCT
asir-3636	408	15	1974	1974	NUM
asir-3636	408	16	)	)	PUNCT
asir-3636	408	17	.	.	PUNCT
asir-3636	409	1	real	real	ADJ
asir-3636	409	2	and	and	CCONJ
asir-3636	409	3	complex	complex	ADJ
asir-3636	409	4	analysis	analysis	NOUN
asir-3636	409	5	(	(	PUNCT
asir-3636	409	6	2nd	2nd	ADJ
asir-3636	409	7	ed	ed	NOUN
asir-3636	409	8	.	.	PUNCT
asir-3636	409	9	)	)	PUNCT
asir-3636	409	10	.	.	PUNCT
asir-3636	410	1	mcgraw	mcgraw	PROPN
asir-3636	410	2	-	-	PUNCT
asir-3636	410	3	hill	hill	NOUN
asir-3636	410	4	series	series	NOUN
asir-3636	410	5	in	in	ADP
asir-3636	410	6	higher	high	ADJ
asir-3636	410	7	mathematics	mathematic	NOUN
asir-3636	410	8	,	,	PUNCT
asir-3636	410	9	mcgraw	mcgraw	PROPN
asir-3636	410	10	-	-	PUNCT
asir-3636	410	11	hill	hill	PROPN
asir-3636	410	12	book	book	PROPN
asir-3636	410	13	co.	co.	PROPN
asir-3636	410	14	,	,	PUNCT
asir-3636	410	15	new	new	PROPN
asir-3636	410	16	york	york	PROPN
asir-3636	410	17	.	.	PUNCT
asir-3636	411	1	shamoyan	shamoyan	PROPN
asir-3636	411	2	,	,	PUNCT
asir-3636	411	3	f.	f.	PROPN
asir-3636	411	4	a.	a.	PROPN
asir-3636	411	5	(	(	PUNCT
asir-3636	411	6	1994	1994	NUM
asir-3636	411	7	)	)	PUNCT
asir-3636	411	8	.	.	PUNCT
asir-3636	412	1	closed	close	VERB
asir-3636	412	2	ideals	ideal	NOUN
asir-3636	412	3	in	in	ADP
asir-3636	412	4	algebras	algebra	NOUN
asir-3636	412	5	of	of	ADP
asir-3636	412	6	functions	function	NOUN
asir-3636	412	7	that	that	PRON
asir-3636	412	8	are	be	AUX
asir-3636	412	9	analytic	analytic	ADJ
asir-3636	412	10	in	in	ADP
asir-3636	412	11	the	the	DET
asir-3636	412	12	disk	disk	NOUN
asir-3636	412	13	and	and	CCONJ
asir-3636	412	14	smooth	smooth	VERB
asir-3636	412	15	up	up	ADP
asir-3636	412	16	to	to	ADP
asir-3636	412	17	its	its	PRON
asir-3636	412	18	boundary	boundary	NOUN
asir-3636	412	19	.	.	PUNCT
asir-3636	413	1	mat	mat	PROPN
asir-3636	413	2	.	.	PUNCT
asir-3636	413	3	sb	sb	PROPN
asir-3636	413	4	.	.	PROPN
asir-3636	413	5	,	,	PUNCT
asir-3636	413	6	79(2	79(2	NUM
asir-3636	413	7	)	)	PUNCT
asir-3636	413	8	,	,	PUNCT
asir-3636	413	9	425	425	NUM
asir-3636	413	10	-	-	SYM
asir-3636	413	11	445	445	NUM
asir-3636	413	12	.	.	PUNCT
asir-3636	414	1	https://doi.org/10.1070/sm1994v079n02abeh003508	https://doi.org/10.1070/sm1994v079n02abeh003508	ADJ
asir-3636	414	2	shirokov	shirokov	NOUN
asir-3636	414	3	,	,	PUNCT
asir-3636	414	4	n.	n.	NOUN
asir-3636	414	5	a.	a.	NOUN
asir-3636	414	6	(	(	PUNCT
asir-3636	414	7	1982	1982	NUM
asir-3636	414	8	)	)	PUNCT
asir-3636	414	9	.	.	PUNCT
asir-3636	415	1	closed	close	VERB
asir-3636	415	2	ideals	ideal	NOUN
asir-3636	415	3	of	of	ADP
asir-3636	415	4	algebras	algebra	NOUN
asir-3636	415	5	of	of	ADP
asir-3636	415	6	b	b	PROPN
asir-3636	415	7	_	_	PRON
asir-3636	415	8	pq	pq	NOUN
asir-3636	415	9	-	-	PUNCT
asir-3636	415	10	type	type	NOUN
asir-3636	415	11	,	,	PUNCT
asir-3636	415	12	(	(	PUNCT
asir-3636	415	13	russian	russian	ADJ
asir-3636	415	14	)	)	PUNCT
asir-3636	415	15	izv	izv	PROPN
asir-3636	415	16	.	.	PROPN
asir-3636	415	17	akad	akad	PROPN
asir-3636	415	18	.	.	PUNCT
asir-3636	416	1	nauk	nauk	PROPN
asir-3636	416	2	.	.	PROPN
asir-3636	416	3	sssr	sssr	PROPN
asir-3636	416	4	,	,	PUNCT
asir-3636	416	5	mat	mat	NOUN
asir-3636	416	6	.	.	PROPN
asir-3636	416	7	,	,	PUNCT
asir-3636	416	8	46(6	46(6	NOUN
asir-3636	416	9	)	)	PUNCT
asir-3636	416	10	,	,	PUNCT
asir-3636	416	11	1316	1316	NUM
asir-3636	416	12	-	-	SYM
asir-3636	416	13	1333	1333	NUM
asir-3636	416	14	.	.	PUNCT
asir-3636	417	1	shirokov	shirokov	PROPN
asir-3636	417	2	,	,	PUNCT
asir-3636	417	3	n.	n.	NOUN
asir-3636	417	4	a.	a.	PROPN
asir-3636	417	5	(	(	PUNCT
asir-3636	417	6	1988	1988	NUM
asir-3636	417	7	)	)	PUNCT
asir-3636	417	8	.	.	PUNCT
asir-3636	418	1	analytic	analytic	ADJ
asir-3636	418	2	functions	function	NOUN
asir-3636	418	3	smooth	smooth	VERB
asir-3636	418	4	up	up	ADP
asir-3636	418	5	to	to	ADP
asir-3636	418	6	the	the	DET
asir-3636	418	7	boundary	boundary	NOUN
asir-3636	418	8	,	,	PUNCT
asir-3636	418	9	lecture	lecture	NOUN
asir-3636	418	10	notes	note	NOUN
asir-3636	418	11	in	in	ADP
asir-3636	418	12	mathematics	mathematic	NOUN
asir-3636	418	13	,	,	PUNCT
asir-3636	418	14	1312	1312	NUM
asir-3636	418	15	.	.	PUNCT
asir-3636	419	1	springer	springer	NOUN
asir-3636	419	2	-	-	PUNCT
asir-3636	419	3	verlag	verlag	PROPN
asir-3636	419	4	,	,	PUNCT
asir-3636	419	5	berlin	berlin	PROPN
asir-3636	419	6	.	.	PUNCT
asir-3636	420	1	taylor	taylor	PROPN
asir-3636	420	2	,	,	PUNCT
asir-3636	420	3	b.	b.	PROPN
asir-3636	420	4	a.	a.	PROPN
asir-3636	420	5	,	,	PUNCT
asir-3636	420	6	&	&	CCONJ
asir-3636	420	7	williams	williams	PROPN
asir-3636	420	8	,	,	PUNCT
asir-3636	420	9	d.	d.	PROPN
asir-3636	420	10	l.	l.	PROPN
asir-3636	420	11	(	(	PUNCT
asir-3636	420	12	1970	1970	NUM
asir-3636	420	13	)	)	PUNCT
asir-3636	420	14	ideals	ideal	NOUN
asir-3636	420	15	in	in	ADP
asir-3636	420	16	rings	ring	NOUN
asir-3636	420	17	of	of	ADP
asir-3636	420	18	analytic	analytic	ADJ
asir-3636	420	19	functions	function	NOUN
asir-3636	420	20	with	with	ADP
asir-3636	420	21	smooth	smooth	ADJ
asir-3636	420	22	boundary	boundary	ADJ
asir-3636	420	23	values	value	NOUN
asir-3636	420	24	.	.	PUNCT
asir-3636	421	1	can	can	AUX
asir-3636	421	2	.	.	PUNCT
asir-3636	422	1	j.	j.	PROPN
asir-3636	422	2	math	math	PROPN
asir-3636	422	3	.	.	PUNCT
asir-3636	422	4	,	,	PUNCT
asir-3636	422	5	22	22	NUM
asir-3636	422	6	,	,	PUNCT
asir-3636	422	7	1266	1266	NUM
asir-3636	422	8	-	-	SYM
asir-3636	422	9	1283	1283	NUM
asir-3636	422	10	.	.	PUNCT
asir-3636	423	1	https://doi.org/10.4153/cjm-1970-143-x	https://doi.org/10.4153/cjm-1970-143-x	NOUN