id	sid	tid	token	lemma	pos
easat-2091	1	1	edelweiss	edelweiss	PROPN
easat-2091	1	2	applied	apply	VERB
easat-2091	1	3	science	science	NOUN
easat-2091	1	4	and	and	CCONJ
easat-2091	1	5	technology	technology	NOUN
easat-2091	1	6	issn	issn	PROPN
easat-2091	1	7	:	:	PUNCT
easat-2091	1	8	2576	2576	NUM
easat-2091	1	9	-	-	SYM
easat-2091	1	10	8484	8484	NUM
easat-2091	1	11	vol	vol	NOUN
easat-2091	1	12	.	.	PROPN
easat-2091	1	13	8	8	NUM
easat-2091	1	14	,	,	PUNCT
easat-2091	1	15	no	no	INTJ
easat-2091	1	16	.	.	NOUN
easat-2091	1	17	6	6	NUM
easat-2091	1	18	,	,	PUNCT
easat-2091	1	19	394	394	NUM
easat-2091	1	20	-	-	SYM
easat-2091	1	21	400	400	NUM
easat-2091	1	22	2024	2024	NUM
easat-2091	1	23	publisher	publisher	NOUN
easat-2091	1	24	:	:	PUNCT
easat-2091	1	25	learning	learn	VERB
easat-2091	1	26	gate	gate	NOUN
easat-2091	1	27	doi	doi	PROPN
easat-2091	1	28	:	:	PUNCT
easat-2091	1	29	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	1	30	©	©	ADP
easat-2091	1	31	2024	2024	NUM
easat-2091	1	32	by	by	ADP
easat-2091	1	33	the	the	DET
easat-2091	1	34	author	author	NOUN
easat-2091	1	35	;	;	PUNCT
easat-2091	1	36	licensee	licensee	PROPN
easat-2091	1	37	learning	learning	NOUN
easat-2091	1	38	gate	gate	NOUN
easat-2091	1	39	©	©	PROPN
easat-2091	1	40	2024	2024	NUM
easat-2091	1	41	by	by	ADP
easat-2091	1	42	the	the	DET
easat-2091	1	43	author	author	NOUN
easat-2091	1	44	;	;	PUNCT
easat-2091	1	45	licensee	licensee	PROPN
easat-2091	1	46	learning	learn	VERB
easat-2091	1	47	gate	gate	NOUN
easat-2091	1	48	*	*	PUNCT
easat-2091	1	49	correspondence	correspondence	NOUN
easat-2091	1	50	:	:	PUNCT
easat-2091	1	51	o.osman@qu.edu.sa	o.osman@qu.edu.sa	PROPN
easat-2091	1	52	factors	factor	NOUN
easat-2091	1	53	affecting	affect	VERB
easat-2091	1	54	mobility	mobility	NOUN
easat-2091	1	55	of	of	ADP
easat-2091	1	56	harmonic	harmonic	ADJ
easat-2091	1	57	symbols	symbol	NOUN
easat-2091	1	58	osman	osman	PROPN
easat-2091	1	59	abdalla	abdalla	PROPN
easat-2091	1	60	adam	adam	PROPN
easat-2091	1	61	osman1	osman1	PROPN
easat-2091	1	62	*	*	PROPN
easat-2091	1	63	1department	1department	NUM
easat-2091	1	64	of	of	ADP
easat-2091	1	65	mathematics	mathematic	NOUN
easat-2091	1	66	,	,	PUNCT
easat-2091	1	67	college	college	NOUN
easat-2091	1	68	of	of	ADP
easat-2091	1	69	science	science	NOUN
easat-2091	1	70	,	,	PUNCT
easat-2091	1	71	qassim	qassim	PROPN
easat-2091	1	72	university	university	PROPN
easat-2091	1	73	,	,	PUNCT
easat-2091	1	74	buraydah	buraydah	NOUN
easat-2091	1	75	51452	51452	NUM
easat-2091	1	76	,	,	PUNCT
easat-2091	1	77	saudi	saudi	PROPN
easat-2091	1	78	arabia	arabia	PROPN
easat-2091	1	79	;	;	PUNCT
easat-2091	1	80	o.osman@qu.edu.sa	o.osman@qu.edu.sa	PROPN
easat-2091	1	81	(	(	PUNCT
easat-2091	1	82	q.a.a.o	q.a.a.o	PROPN
easat-2091	1	83	.	.	PROPN
easat-2091	1	84	)	)	PUNCT
easat-2091	1	85	.	.	PUNCT
easat-2091	2	1	abstract	abstract	ADV
easat-2091	2	2	:	:	PUNCT
easat-2091	2	3	in	in	ADP
easat-2091	2	4	this	this	DET
easat-2091	2	5	paper	paper	NOUN
easat-2091	2	6	,	,	PUNCT
easat-2091	2	7	we	we	PRON
easat-2091	2	8	have	have	AUX
easat-2091	2	9	illustrated	illustrate	VERB
easat-2091	2	10	the	the	DET
easat-2091	2	11	bergmann	bergmann	PROPN
easat-2091	2	12	domains	domain	NOUN
easat-2091	2	13	and	and	CCONJ
easat-2091	2	14	the	the	DET
easat-2091	2	15	toeplitz	toeplitz	NOUN
easat-2091	2	16	operators	operator	NOUN
easat-2091	2	17	with	with	ADP
easat-2091	2	18	their	their	PRON
easat-2091	2	19	symbiotic	symbiotic	ADJ
easat-2091	2	20	symbols	symbol	NOUN
easat-2091	2	21	in	in	ADP
easat-2091	2	22	some	some	DET
easat-2091	2	23	special	special	ADJ
easat-2091	2	24	domains	domain	NOUN
easat-2091	2	25	in	in	ADP
easat-2091	2	26	order	order	NOUN
easat-2091	2	27	to	to	PART
easat-2091	2	28	clarify	clarify	VERB
easat-2091	2	29	the	the	DET
easat-2091	2	30	properties	property	NOUN
easat-2091	2	31	that	that	PRON
easat-2091	2	32	correspond	correspond	VERB
easat-2091	2	33	to	to	ADP
easat-2091	2	34	the	the	DET
easat-2091	2	35	fixed	fix	VERB
easat-2091	2	36	values	value	NOUN
easat-2091	2	37	.	.	PUNCT
easat-2091	3	1	and	and	CCONJ
easat-2091	3	2	we	we	PRON
easat-2091	3	3	characterize	characterize	VERB
easat-2091	3	4	the	the	DET
easat-2091	3	5	bounded	bounded	ADJ
easat-2091	3	6	harmonic	harmonic	ADJ
easat-2091	3	7	functions	function	NOUN
easat-2091	3	8	for	for	ADP
easat-2091	3	9	which	which	PRON
easat-2091	3	10	the	the	DET
easat-2091	3	11	toeplitz	toeplitz	NOUN
easat-2091	3	12	operators	operator	NOUN
easat-2091	3	13	bergman	bergman	PROPN
easat-2091	3	14	space	space	PROPN
easat-2091	3	15	are	be	AUX
easat-2091	3	16	essentially	essentially	ADV
easat-2091	3	17	commuting	commute	VERB
easat-2091	3	18	.	.	PUNCT
easat-2091	4	1	keywords	keyword	NOUN
easat-2091	4	2	:	:	PUNCT
easat-2091	5	1	bergman	bergman	PROPN
easat-2091	5	2	,	,	PUNCT
easat-2091	5	3	correspond	correspond	NOUN
easat-2091	5	4	,	,	PUNCT
easat-2091	5	5	function	function	NOUN
easat-2091	5	6	,	,	PUNCT
easat-2091	5	7	functions	function	NOUN
easat-2091	5	8	,	,	PUNCT
easat-2091	5	9	harmonic	harmonic	PROPN
easat-2091	5	10	bounded	bound	VERB
easat-2091	5	11	,	,	PUNCT
easat-2091	5	12	invariant	invariant	ADJ
easat-2091	5	13	,	,	PUNCT
easat-2091	5	14	operator	operator	NOUN
easat-2091	5	15	,	,	PUNCT
easat-2091	5	16	spaces	space	NOUN
easat-2091	5	17	,	,	PUNCT
easat-2091	5	18	toeplitz	toeplitz	NOUN
easat-2091	5	19	,	,	PUNCT
easat-2091	5	20	value	value	NOUN
easat-2091	5	21	.	.	PUNCT
easat-2091	6	1	1	1	X
easat-2091	6	2	.	.	X
easat-2091	6	3	introduction	introduction	NOUN
easat-2091	6	4	suppose	suppose	VERB
easat-2091	6	5	that	that	SCONJ
easat-2091	6	6	𝑑𝐴	𝑑𝐴	PRON
easat-2091	6	7	stands	stand	VERB
easat-2091	6	8	for	for	ADP
easat-2091	6	9	the	the	DET
easat-2091	6	10	measure	measure	NOUN
easat-2091	6	11	of	of	ADP
easat-2091	6	12	the	the	DET
easat-2091	6	13	space	space	NOUN
easat-2091	6	14	defined	define	VERB
easat-2091	6	15	on	on	ADP
easat-2091	6	16	the	the	DET
easat-2091	6	17	open	open	ADJ
easat-2091	6	18	unit	unit	NOUN
easat-2091	6	19	disk	disk	NOUN
easat-2091	6	20	in	in	ADP
easat-2091	6	21	𝐷	𝐷	NOUN
easat-2091	6	22	of	of	ADP
easat-2091	6	23	the	the	DET
easat-2091	6	24	complex	complex	ADJ
easat-2091	6	25	plane	plane	NOUN
easat-2091	6	26	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	6	27	,	,	PUNCT
easat-2091	6	28	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	6	29	)	)	PUNCT
easat-2091	6	30	represents	represent	VERB
easat-2091	6	31	the	the	DET
easat-2091	6	32	inner	inner	ADJ
easat-2091	6	33	hilbert	hilbert	NOUN
easat-2091	6	34	space	space	NOUN
easat-2091	6	35	〈	〈	PROPN
easat-2091	6	36	𝑓	𝑓	PROPN
easat-2091	6	37	,	,	PUNCT
easat-2091	6	38	𝑔	𝑔	NOUN
easat-2091	6	39	〉	〉	NOUN
easat-2091	6	40	=	=	SYM
easat-2091	6	41	∫𝐷	∫𝐷	NOUN
easat-2091	6	42	𝑓𝑔	𝑓𝑔	VERB
easat-2091	6	43	𝑑𝐴	𝑑𝐴	NUM
easat-2091	6	44	bergmann	bergmann	PROPN
easat-2091	6	45	space	space	NOUN
easat-2091	6	46	𝐿𝑎	𝐿𝑎	PROPN
easat-2091	6	47	2	2	NUM
easat-2091	6	48	is	be	AUX
easat-2091	6	49	a	a	DET
easat-2091	6	50	set	set	NOUN
easat-2091	6	51	of	of	ADP
easat-2091	6	52	function	function	NOUN
easat-2091	6	53	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	6	54	,	,	PUNCT
easat-2091	6	55	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	6	56	)	)	PUNCT
easat-2091	6	57	which	which	PRON
easat-2091	6	58	properties	property	NOUN
easat-2091	6	59	that	that	PRON
easat-2091	6	60	are	be	AUX
easat-2091	6	61	included	include	VERB
easat-2091	6	62	in	in	ADP
easat-2091	6	63	it	it	PRON
easat-2091	6	64	analytically𝐷.	analytically𝐷.	INTJ
easat-2091	6	65	which	which	PRON
easat-2091	6	66	means	mean	VERB
easat-2091	6	67	that	that	SCONJ
easat-2091	6	68	bergmann	bergmann	PROPN
easat-2091	6	69	's	's	PART
easat-2091	6	70	space	space	NOUN
easat-2091	6	71	𝐿𝑎	𝐿𝑎	PROPN
easat-2091	6	72	2	2	NUM
easat-2091	6	73	it	it	PRON
easat-2091	6	74	is	be	AUX
easat-2091	6	75	a	a	DET
easat-2091	6	76	closed	closed	ADJ
easat-2091	6	77	subspace𝐿2(𝐷	subspace𝐿2(𝐷	NOUN
easat-2091	6	78	,	,	PUNCT
easat-2091	6	79	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	6	80	)	)	PUNCT
easat-2091	6	81	,	,	PUNCT
easat-2091	6	82	and	and	CCONJ
easat-2091	6	83	so	so	ADV
easat-2091	6	84	there	there	PRON
easat-2091	6	85	is	be	VERB
easat-2091	6	86	an	an	DET
easat-2091	6	87	orthogonal	orthogonal	ADJ
easat-2091	6	88	projection	projection	NOUN
easat-2091	6	89	𝑃	𝑃	NOUN
easat-2091	6	90	from	from	ADP
easat-2091	6	91	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	6	92	,	,	PUNCT
easat-2091	6	93	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	6	94	)	)	PUNCT
easat-2091	6	95	to𝐿𝑎	to𝐿𝑎	VERB
easat-2091	6	96	2	2	NUM
easat-2091	6	97	,	,	PUNCT
easat-2091	6	98	[	[	X
easat-2091	6	99	1,6	1,6	NUM
easat-2091	6	100	]	]	PUNCT
easat-2091	6	101	.	.	PUNCT
easat-2091	7	1	for(𝜑	for(𝜑	PROPN
easat-2091	8	1	+	+	NOUN
easat-2091	8	2	1	1	X
easat-2091	8	3	)	)	PUNCT
easat-2091	8	4	∈	∈	PROPN
easat-2091	8	5	𝐿∞(𝐷	𝐿∞(𝐷	NOUN
easat-2091	8	6	,	,	PUNCT
easat-2091	8	7	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	8	8	)	)	PUNCT
easat-2091	8	9	,	,	PUNCT
easat-2091	8	10	the	the	DET
easat-2091	8	11	toeplitz	toeplitz	NOUN
easat-2091	8	12	effects	effect	NOUN
easat-2091	8	13	with	with	ADP
easat-2091	8	14	symbol(𝜑	symbol(𝜑	NOUN
easat-2091	8	15	+	+	PROPN
easat-2091	8	16	1	1	NUM
easat-2091	8	17	)	)	PUNCT
easat-2091	8	18	,	,	PUNCT
easat-2091	8	19	denoted	denote	VERB
easat-2091	8	20	𝑇𝜑+1	𝑇𝜑+1	ADP
easat-2091	8	21	is	be	AUX
easat-2091	8	22	operator	operator	NOUN
easat-2091	8	23	from	from	ADP
easat-2091	8	24	𝐿𝑎	𝐿𝑎	ADP
easat-2091	8	25	2	2	NUM
easat-2091	8	26	to	to	ADP
easat-2091	8	27	𝐿𝑎	𝐿𝑎	PROPN
easat-2091	8	28	2	2	NUM
easat-2091	8	29	knowledge	knowledge	NOUN
easat-2091	8	30	before𝑇𝜑+1𝑓	before𝑇𝜑+1𝑓	NOUN
easat-2091	8	31	=	=	PUNCT
easat-2091	8	32	𝑃{(𝜑	𝑃{(𝜑	X
easat-2091	9	1	+	+	CCONJ
easat-2091	10	1	1)𝑓	1)𝑓	NUM
easat-2091	10	2	}	}	PUNCT
easat-2091	10	3	.	.	PUNCT
easat-2091	11	1	by	by	ADP
easat-2091	11	2	harmonic	harmonic	ADJ
easat-2091	11	3	function	function	NOUN
easat-2091	11	4	,	,	PUNCT
easat-2091	11	5	we	we	PRON
easat-2091	11	6	mean	mean	VERB
easat-2091	11	7	a	a	DET
easat-2091	11	8	function	function	NOUN
easat-2091	11	9	with	with	ADP
easat-2091	11	10	a	a	DET
easat-2091	11	11	complex	complex	ADJ
easat-2091	11	12	value	value	NOUN
easat-2091	11	13	over	over	ADP
easat-2091	11	14	𝐷	𝐷	NOUN
easat-2091	11	15	for	for	ADP
easat-2091	11	16	which	which	PRON
easat-2091	11	17	the	the	DET
easat-2091	11	18	laplacian	laplacian	ADJ
easat-2091	11	19	congruent	congruent	NOUN
easat-2091	11	20	is	be	AUX
easat-2091	11	21	zero	zero	NUM
easat-2091	11	22	.	.	PUNCT
easat-2091	12	1	theorem	theorem	NOUN
easat-2091	12	2	1	1	NUM
easat-2091	12	3	.	.	PUNCT
easat-2091	12	4	suppose	suppose	VERB
easat-2091	12	5	that	that	SCONJ
easat-2091	12	6	(	(	PUNCT
easat-2091	12	7	𝜑	𝜑	X
easat-2091	12	8	+	+	NOUN
easat-2091	12	9	1	1	NUM
easat-2091	12	10	)	)	PUNCT
easat-2091	12	11	and	and	CCONJ
easat-2091	12	12	(	(	PUNCT
easat-2091	12	13	𝜓	𝜓	PROPN
easat-2091	12	14	+	+	NOUN
easat-2091	12	15	1	1	X
easat-2091	12	16	)	)	PUNCT
easat-2091	12	17	definite	definite	ADJ
easat-2091	12	18	harmonic	harmonic	ADJ
easat-2091	12	19	functions	function	NOUN
easat-2091	12	20	on𝐷.	on𝐷.	ADV
easat-2091	12	21	so	so	SCONJ
easat-2091	12	22	that	that	SCONJ
easat-2091	12	23	𝑇𝜑+1𝑇𝜓+1	𝑇𝜑+1𝑇𝜓+1	NOUN
easat-2091	12	24	=	=	SYM
easat-2091	12	25	𝑇𝜓+1𝑇𝜑+1	𝑇𝜓+1𝑇𝜑+1	PROPN
easat-2091	12	26	if	if	SCONJ
easat-2091	12	27	and	and	CCONJ
easat-2091	12	28	only	only	ADV
easat-2091	12	29	if	if	SCONJ
easat-2091	12	30	a(𝜑	a(𝜑	PROPN
easat-2091	12	31	+	+	CCONJ
easat-2091	12	32	1	1	NUM
easat-2091	12	33	)	)	PUNCT
easat-2091	12	34	and	and	CCONJ
easat-2091	12	35	(	(	PUNCT
easat-2091	12	36	𝜓	𝜓	PROPN
easat-2091	12	37	+	+	NOUN
easat-2091	12	38	1	1	NUM
easat-2091	12	39	)	)	PUNCT
easat-2091	12	40	are	be	AUX
easat-2091	12	41	both	both	ADV
easat-2091	12	42	analytic	analytic	ADJ
easat-2091	12	43	on	on	ADP
easat-2091	12	44	𝐷.	𝐷.	PROPN
easat-2091	12	45	b(	b(	VERB
easat-2091	12	46	�	�	PROPN
easat-2091	12	47	̅	̅	NOUN
easat-2091	12	48	�	�	NOUN
easat-2091	12	49	)	)	PUNCT
easat-2091	12	50	and	and	CCONJ
easat-2091	12	51	(	(	PUNCT
easat-2091	12	52	�	�	NOUN
easat-2091	12	53	̅	̅	NOUN
easat-2091	12	54	�	�	NOUN
easat-2091	12	55	)	)	PUNCT
easat-2091	12	56	also	also	ADV
easat-2091	12	57	analytic	analytic	ADJ
easat-2091	12	58	on	on	ADP
easat-2091	12	59	𝐷.	𝐷.	PROPN
easat-2091	12	60	cthere	cthere	NOUN
easat-2091	12	61	are	be	AUX
easat-2091	12	62	constants	constant	NOUN
easat-2091	12	63	𝑎	𝑎	ADP
easat-2091	12	64	,	,	PUNCT
easat-2091	12	65	𝑏	𝑏	PROPN
easat-2091	12	66	∈	∈	PROPN
easat-2091	12	67	𝐶	𝐶	PROPN
easat-2091	12	68	not	not	PART
easat-2091	12	69	both	both	CCONJ
easat-2091	12	70	equal	equal	ADJ
easat-2091	12	71	to	to	ADP
easat-2091	12	72	,	,	PUNCT
easat-2091	12	73	such	such	ADJ
easat-2091	12	74	that	that	SCONJ
easat-2091	12	75	𝑎(𝜑	𝑎(𝜑	PROPN
easat-2091	12	76	+	+	PUNCT
easat-2091	12	77	1	1	NUM
easat-2091	12	78	)	)	PUNCT
easat-2091	12	79	+	+	CCONJ
easat-2091	13	1	𝑏(𝜓	𝑏(𝜓	PROPN
easat-2091	13	2	+	+	SYM
easat-2091	13	3	1	1	NUM
easat-2091	13	4	)	)	PUNCT
easat-2091	13	5	is	be	AUX
easat-2091	13	6	constant	constant	ADJ
easat-2091	13	7	in𝐷.	in𝐷.	SCONJ
easat-2091	13	8	we	we	PRON
easat-2091	13	9	will	will	AUX
easat-2091	13	10	see	see	VERB
easat-2091	13	11	if	if	SCONJ
easat-2091	13	12	the	the	DET
easat-2091	13	13	direction	direction	NOUN
easat-2091	13	14	of	of	ADP
easat-2091	13	15	that	that	DET
easat-2091	13	16	theorem	theorem	NOUN
easat-2091	13	17	is	be	AUX
easat-2091	13	18	trivial	trivial	ADJ
easat-2091	13	19	,	,	PUNCT
easat-2091	13	20	but	but	CCONJ
easat-2091	13	21	proving	prove	VERB
easat-2091	13	22	the	the	DET
easat-2091	13	23	direction	direction	NOUN
easat-2091	13	24	"	"	PUNCT
easat-2091	13	25	only	only	ADV
easat-2091	13	26	if	if	SCONJ
easat-2091	13	27	"	"	PUNCT
easat-2091	13	28	requires	require	VERB
easat-2091	13	29	an	an	DET
easat-2091	13	30	inverse	inverse	NOUN
easat-2091	13	31	of	of	ADP
easat-2091	13	32	the	the	DET
easat-2091	13	33	invariant	invariant	ADJ
easat-2091	13	34	form	form	NOUN
easat-2091	13	35	of	of	ADP
easat-2091	13	36	the	the	DET
easat-2091	13	37	mean	mean	ADJ
easat-2091	13	38	-	-	PUNCT
easat-2091	13	39	value	value	NOUN
easat-2091	13	40	property	property	NOUN
easat-2091	13	41	.	.	PUNCT
easat-2091	14	1	the	the	DET
easat-2091	14	2	clarification	clarification	NOUN
easat-2091	14	3	of	of	ADP
easat-2091	14	4	theorem	theorem	NOUN
easat-2091	14	5	1	1	NUM
easat-2091	14	6	similar	similar	ADJ
easat-2091	14	7	to	to	ADP
easat-2091	14	8	a	a	DET
easat-2091	14	9	similar	similar	ADJ
easat-2091	14	10	result	result	NOUN
easat-2091	14	11	installed	instal	VERB
easat-2091	14	12	in	in	ADP
easat-2091	14	13	[	[	X
easat-2091	14	14	2,7	2,7	NUM
easat-2091	14	15	]	]	PUNCT
easat-2091	14	16	for	for	ADP
easat-2091	14	17	toeplitz	toeplitz	NOUN
easat-2091	14	18	effects	effect	NOUN
easat-2091	14	19	in	in	ADP
easat-2091	14	20	the	the	DET
easat-2091	14	21	symbol	symbol	NOUN
easat-2091	14	22	𝐿∞(𝜕𝐷	𝐿∞(𝜕𝐷	PROPN
easat-2091	14	23	)	)	PUNCT
easat-2091	14	24	acts	act	VERB
easat-2091	14	25	on	on	ADP
easat-2091	14	26	hardy	hardy	ADJ
easat-2091	14	27	space𝐻2(𝜕𝐷	space𝐻2(𝜕𝐷	PROPN
easat-2091	14	28	)	)	PUNCT
easat-2091	14	29	.	.	PUNCT
easat-2091	15	1	brown	brown	PROPN
easat-2091	15	2	and	and	CCONJ
easat-2091	15	3	holmes	holmes	PROPN
easat-2091	15	4	prove	prove	VERB
easat-2091	15	5	these	these	DET
easat-2091	15	6	results	result	NOUN
easat-2091	15	7	through	through	ADP
easat-2091	15	8	the	the	DET
easat-2091	15	9	effect	effect	NOUN
easat-2091	15	10	matrix	matrix	NOUN
easat-2091	15	11	of	of	ADP
easat-2091	15	12	hardy	hardy	ADJ
easat-2091	15	13	spaces	space	NOUN
easat-2091	15	14	in	in	ADP
easat-2091	15	15	bergman	bergman	PROPN
easat-2091	15	16	spaces	space	VERB
easat-2091	15	17	.	.	PUNCT
easat-2091	16	1	𝐿𝑎	𝐿𝑎	ADP
easat-2091	16	2	2	2	NUM
easat-2091	16	3	,	,	PUNCT
easat-2091	16	4	toeplitz	toeplitz	NOUN
easat-2091	16	5	effects	effect	NOUN
easat-2091	16	6	do	do	AUX
easat-2091	16	7	not	not	PART
easat-2091	16	8	have	have	VERB
easat-2091	16	9	good	good	ADJ
easat-2091	16	10	matrices	matrix	NOUN
easat-2091	16	11	,	,	PUNCT
easat-2091	16	12	and	and	CCONJ
easat-2091	16	13	the	the	DET
easat-2091	16	14	techniques	technique	NOUN
easat-2091	16	15	used	use	VERB
easat-2091	16	16	by	by	ADP
easat-2091	16	17	brown	brown	ADJ
easat-2091	16	18	and	and	CCONJ
easat-2091	16	19	holmes	holme	NOUN
easat-2091	16	20	do	do	AUX
easat-2091	16	21	not	not	PART
easat-2091	16	22	seem	seem	VERB
easat-2091	16	23	to	to	PART
easat-2091	16	24	work	work	VERB
easat-2091	16	25	in	in	ADP
easat-2091	16	26	this	this	DET
easat-2091	16	27	context	context	NOUN
easat-2091	16	28	.	.	PUNCT
easat-2091	17	1	thus	thus	ADV
easat-2091	17	2	function	function	VERB
easat-2091	17	3	theory	theory	NOUN
easat-2091	17	4	,	,	PUNCT
easat-2091	17	5	rather	rather	ADV
easat-2091	17	6	than	than	ADP
easat-2091	17	7	matrix	matrix	NOUN
easat-2091	17	8	manipulation	manipulation	NOUN
easat-2091	17	9	,	,	PUNCT
easat-2091	17	10	plays	play	VERB
easat-2091	17	11	a	a	DET
easat-2091	17	12	large	large	ADJ
easat-2091	17	13	role	role	NOUN
easat-2091	17	14	in	in	ADP
easat-2091	17	15	our	our	PRON
easat-2091	17	16	proof	proof	NOUN
easat-2091	17	17	.	.	PUNCT
easat-2091	18	1	a	a	DET
easat-2091	18	2	special	special	ADJ
easat-2091	18	3	case	case	NOUN
easat-2091	18	4	of	of	ADP
easat-2091	18	5	theorem	theorem	NOUN
easat-2091	18	6	1	1	NUM
easat-2091	18	7	was	be	AUX
easat-2091	18	8	proved	prove	VERB
easat-2091	18	9	in	in	ADP
easat-2091	18	10	[	[	PUNCT
easat-2091	18	11	3.8	3.8	NUM
easat-2091	18	12	]	]	PUNCT
easat-2091	18	13	,	,	PUNCT
easat-2091	18	14	using	use	VERB
easat-2091	18	15	function	function	NOUN
easat-2091	18	16	theory	theory	NOUN
easat-2091	18	17	techniques	technique	NOUN
easat-2091	18	18	quite	quite	ADV
easat-2091	18	19	different	different	ADJ
easat-2091	18	20	from	from	ADP
easat-2091	18	21	those	those	PRON
easat-2091	18	22	that	that	PRON
easat-2091	18	23	we	we	PRON
easat-2091	18	24	use	use	VERB
easat-2091	18	25	here	here	ADV
easat-2091	18	26	.	.	PUNCT
easat-2091	19	1	also	also	ADV
easat-2091	19	2	proved	prove	VERB
easat-2091	19	3	a	a	DET
easat-2091	19	4	special	special	ADJ
easat-2091	19	5	case	case	NOUN
easat-2091	19	6	of	of	ADP
easat-2091	19	7	theorem	theorem	NOUN
easat-2091	19	8	1	1	NUM
easat-2091	19	9	;	;	PUNCT
easat-2091	19	10	our	our	PRON
easat-2091	19	11	proof	proof	NOUN
easat-2091	19	12	makes	make	VERB
easat-2091	19	13	use	use	NOUN
easat-2091	19	14	of	of	ADP
easat-2091	19	15	some	some	PRON
easat-2091	19	16	of	of	ADP
easat-2091	19	17	his	his	PRON
easat-2091	19	18	ideas	idea	NOUN
easat-2091	19	19	.	.	PUNCT
easat-2091	20	1	functions	function	NOUN
easat-2091	20	2	in	in	ADP
easat-2091	20	3	𝐿∞(𝜕𝐷	𝐿∞(𝜕𝐷	NOUN
easat-2091	20	4	)	)	PUNCT
easat-2091	20	5	correspond	correspond	ADV
easat-2091	20	6	,	,	PUNCT
easat-2091	20	7	via	via	ADP
easat-2091	20	8	the	the	DET
easat-2091	20	9	poisson	poisson	NOUN
easat-2091	20	10	integral	integral	ADJ
easat-2091	20	11	,	,	PUNCT
easat-2091	20	12	to	to	PART
easat-2091	20	13	bounded	bound	VERB
easat-2091	20	14	harmonic	harmonic	ADJ
easat-2091	20	15	functions	function	NOUN
easat-2091	20	16	on𝐷	on𝐷	PROPN
easat-2091	20	17	,	,	PUNCT
easat-2091	20	18	so	so	SCONJ
easat-2091	20	19	the	the	DET
easat-2091	20	20	restriction	restriction	NOUN
easat-2091	20	21	in	in	ADP
easat-2091	20	22	theorem	theorem	NOUN
easat-2091	20	23	1	1	NUM
easat-2091	20	24	to	to	ADP
easat-2091	20	25	consideration	consideration	NOUN
easat-2091	20	26	only	only	ADV
easat-2091	20	27	of	of	ADP
easat-2091	20	28	toeplitz	toeplitz	NOUN
easat-2091	20	29	effects	effect	NOUN
easat-2091	20	30	with	with	ADP
easat-2091	20	31	harmonic	harmonic	ADJ
easat-2091	20	32	symbols	symbol	NOUN
easat-2091	20	33	is	be	AUX
easat-2091	20	34	natural	natural	ADJ
easat-2091	20	35	.	.	PUNCT
easat-2091	21	1	395	395	NUM
easat-2091	21	2	edelweiss	edelweiss	PROPN
easat-2091	21	3	applied	apply	VERB
easat-2091	21	4	science	science	NOUN
easat-2091	21	5	and	and	CCONJ
easat-2091	21	6	technology	technology	NOUN
easat-2091	21	7	issn	issn	PROPN
easat-2091	21	8	:	:	PUNCT
easat-2091	21	9	2576	2576	NUM
easat-2091	21	10	-	-	SYM
easat-2091	21	11	8484	8484	NUM
easat-2091	21	12	vol	vol	NOUN
easat-2091	21	13	.	.	PROPN
easat-2091	21	14	8	8	NUM
easat-2091	21	15	,	,	PUNCT
easat-2091	21	16	no	no	INTJ
easat-2091	21	17	.	.	NOUN
easat-2091	22	1	6	6	NUM
easat-2091	22	2	:	:	SYM
easat-2091	22	3	394	394	NUM
easat-2091	22	4	-	-	SYM
easat-2091	22	5	400	400	NUM
easat-2091	22	6	,	,	PUNCT
easat-2091	22	7	2024	2024	NUM
easat-2091	22	8	doi	doi	NOUN
easat-2091	22	9	:	:	PUNCT
easat-2091	22	10	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	22	11	©	©	ADP
easat-2091	22	12	2024	2024	NUM
easat-2091	22	13	by	by	ADP
easat-2091	22	14	the	the	DET
easat-2091	22	15	author	author	NOUN
easat-2091	22	16	;	;	PUNCT
easat-2091	22	17	licensee	licensee	PROPN
easat-2091	22	18	learning	learning	NOUN
easat-2091	22	19	gate	gate	VERB
easat-2091	22	20	more	more	ADV
easat-2091	22	21	importantly	importantly	ADV
easat-2091	22	22	,	,	PUNCT
easat-2091	22	23	theorem	theorem	ADJ
easat-2091	22	24	1	1	NUM
easat-2091	22	25	does	do	AUX
easat-2091	22	26	not	not	PART
easat-2091	22	27	hold	hold	VERB
easat-2091	22	28	if	if	SCONJ
easat-2091	22	29	“	"	PUNCT
easat-2091	22	30	we	we	PRON
easat-2091	22	31	can	can	AUX
easat-2091	22	32	replace	replace	VERB
easat-2091	22	33	measurable	measurable	ADJ
easat-2091	22	34	harmonic	harmonic	NOUN
easat-2091	22	35	"	"	PUNCT
easat-2091	22	36	.	.	PUNCT
easat-2091	23	1	for	for	ADP
easat-2091	23	2	example	example	NOUN
easat-2091	23	3	,	,	PUNCT
easat-2091	23	4	paul	paul	PROPN
easat-2091	23	5	bourdon	bourdon	PROPN
easat-2091	23	6	has	have	AUX
easat-2091	23	7	pointed	point	VERB
easat-2091	23	8	out	out	ADP
easat-2091	23	9	to	to	ADP
easat-2091	23	10	us	we	PRON
easat-2091	23	11	that	that	SCONJ
easat-2091	23	12	if	if	SCONJ
easat-2091	23	13	(	(	PUNCT
easat-2091	23	14	𝜑	𝜑	NOUN
easat-2091	23	15	+	+	NOUN
easat-2091	23	16	1	1	NUM
easat-2091	23	17	)	)	PUNCT
easat-2091	23	18	and	and	CCONJ
easat-2091	23	19	(	(	PUNCT
easat-2091	23	20	𝜓	𝜓	PROPN
easat-2091	23	21	+	+	NOUN
easat-2091	23	22	1	1	NUM
easat-2091	23	23	)	)	PUNCT
easat-2091	23	24	are	be	AUX
easat-2091	23	25	any	any	DET
easat-2091	23	26	two	two	NUM
easat-2091	23	27	radial	radial	ADJ
easat-2091	23	28	functions	function	NOUN
easat-2091	23	29	in𝐿∞(𝐷	in𝐿∞(𝐷	NOUN
easat-2091	23	30	,	,	PUNCT
easat-2091	23	31	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	23	32	)	)	PUNCT
easat-2091	23	33	,	,	PUNCT
easat-2091	23	34	then	then	ADV
easat-2091	23	35	𝑇𝜑+1𝑇𝜓+1	𝑇𝜑+1𝑇𝜓+1	VERB
easat-2091	23	36	=	=	SYM
easat-2091	23	37	𝑇𝜓+1𝑇𝜑+1(a	𝑇𝜓+1𝑇𝜑+1(a	PROPN
easat-2091	23	38	function	function	NOUN
easat-2091	23	39	is	be	AUX
easat-2091	23	40	called	call	VERB
easat-2091	23	41	radial	radial	ADJ
easat-2091	23	42	if	if	SCONJ
easat-2091	23	43	its	its	PRON
easat-2091	23	44	value	value	NOUN
easat-2091	23	45	at	at	ADP
easat-2091	23	46	𝑧	𝑧	PROPN
easat-2091	23	47	depends	depend	VERB
easat-2091	23	48	only	only	ADV
easat-2091	23	49	on|𝑧|	on|𝑧|	ADJ
easat-2091	23	50	)	)	PUNCT
easat-2091	23	51	.	.	PUNCT
easat-2091	24	1	thus	thus	ADV
easat-2091	24	2	,	,	PUNCT
easat-2091	24	3	the	the	DET
easat-2091	24	4	following	follow	VERB
easat-2091	24	5	open	open	ADJ
easat-2091	24	6	problem	problem	NOUN
easat-2091	24	7	may	may	AUX
easat-2091	24	8	be	be	AUX
easat-2091	24	9	hard	hard	ADJ
easat-2091	24	10	:	:	PUNCT
easat-2091	24	11	find	find	VERB
easat-2091	24	12	conditions	condition	NOUN
easat-2091	24	13	on	on	ADP
easat-2091	24	14	functions	function	NOUN
easat-2091	24	15	(	(	PUNCT
easat-2091	24	16	𝜑	𝜑	NOUN
easat-2091	24	17	+	+	NOUN
easat-2091	24	18	1	1	NUM
easat-2091	24	19	)	)	PUNCT
easat-2091	24	20	and	and	CCONJ
easat-2091	24	21	(	(	PUNCT
easat-2091	24	22	𝜓	𝜓	PROPN
easat-2091	24	23	+	+	NOUN
easat-2091	24	24	1	1	X
easat-2091	24	25	)	)	PUNCT
easat-2091	24	26	in	in	ADP
easat-2091	24	27	𝐿∞(𝐷	𝐿∞(𝐷	NOUN
easat-2091	24	28	,	,	PUNCT
easat-2091	24	29	𝑑𝐴)that	𝑑𝐴)that	PRON
easat-2091	24	30	are	be	AUX
easat-2091	24	31	necessary	necessary	ADJ
easat-2091	24	32	and	and	CCONJ
easat-2091	24	33	sufficient	sufficient	ADJ
easat-2091	24	34	for	for	ADP
easat-2091	24	35	𝑇𝜑+1to	𝑇𝜑+1to	ADP
easat-2091	24	36	commute	commute	NOUN
easat-2091	24	37	with	with	ADP
easat-2091	24	38	𝑇𝜓+1	𝑇𝜓+1	NOUN
easat-2091	24	39	2	2	NUM
easat-2091	24	40	.	.	PUNCT
easat-2091	25	1	the	the	DET
easat-2091	25	2	constant	constant	ADJ
easat-2091	25	3	property	property	NOUN
easat-2091	25	4	of	of	ADP
easat-2091	25	5	the	the	DET
easat-2091	25	6	argument	argument	NOUN
easat-2091	25	7	value	value	VERB
easat-2091	25	8	a	a	DET
easat-2091	25	9	continuous	continuous	ADJ
easat-2091	25	10	function	function	NOUN
easat-2091	25	11	on	on	ADP
easat-2091	25	12	the	the	DET
easat-2091	25	13	disk	disk	NOUN
easat-2091	25	14	𝐷	𝐷	NOUN
easat-2091	25	15	is	be	AUX
easat-2091	25	16	harmonic	harmonic	ADJ
easat-2091	25	17	if	if	SCONJ
easat-2091	25	18	and	and	CCONJ
easat-2091	25	19	only	only	ADV
easat-2091	25	20	if	if	SCONJ
easat-2091	25	21	it	it	PRON
easat-2091	25	22	has	have	VERB
easat-2091	25	23	the	the	DET
easat-2091	25	24	mean	mean	ADJ
easat-2091	25	25	value	value	NOUN
easat-2091	25	26	property	property	NOUN
easat-2091	25	27	.	.	PUNCT
easat-2091	26	1	we	we	PRON
easat-2091	26	2	characterize	characterize	VERB
easat-2091	26	3	harmonic	harmonic	ADJ
easat-2091	26	4	functions	function	NOUN
easat-2091	26	5	in	in	ADP
easat-2091	26	6	terms	term	NOUN
easat-2091	26	7	of	of	ADP
easat-2091	26	8	an	an	DET
easat-2091	26	9	invariant	invariant	ADJ
easat-2091	26	10	mean	mean	NOUN
easat-2091	26	11	value	value	NOUN
easat-2091	26	12	property	property	NOUN
easat-2091	26	13	.	.	PUNCT
easat-2091	27	1	let	let	AUX
easat-2091	27	2	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NUM
easat-2091	27	3	)	)	PUNCT
easat-2091	27	4	denote	denote	VERB
easat-2091	27	5	the	the	DET
easat-2091	27	6	set	set	NOUN
easat-2091	27	7	of	of	ADP
easat-2091	27	8	analytic	analytic	ADJ
easat-2091	27	9	,	,	PUNCT
easat-2091	27	10	one	one	NUM
easat-2091	27	11	-	-	PUNCT
easat-2091	27	12	to	to	ADP
easat-2091	27	13	-	-	PUNCT
easat-2091	27	14	one	one	NUM
easat-2091	27	15	maps	map	NOUN
easat-2091	27	16	of	of	ADP
easat-2091	27	17	𝐷	𝐷	NOUN
easat-2091	27	18	onto	onto	ADP
easat-2091	27	19	𝐷	𝐷	PROPN
easat-2091	27	20	(	(	PUNCT
easat-2091	27	21	where	where	SCONJ
easat-2091	27	22	𝐴𝑢𝑡	𝐴𝑢𝑡	PROPN
easat-2091	27	23	stands	stand	VERB
easat-2091	27	24	for	for	ADP
easat-2091	27	25	automorphic	automorphic	ADJ
easat-2091	27	26	)	)	PUNCT
easat-2091	27	27	.	.	PUNCT
easat-2091	28	1	a	a	DET
easat-2091	28	2	function	function	NOUN
easat-2091	28	3	ℎ	ℎ	PROPN
easat-2091	28	4	on	on	ADP
easat-2091	28	5	𝐷	𝐷	PROPN
easat-2091	28	6	is	be	AUX
easat-2091	28	7	in	in	ADP
easat-2091	28	8	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	28	9	)	)	PUNCT
easat-2091	29	1	if	if	SCONJ
easat-2091	29	2	and	and	CCONJ
easat-2091	29	3	only	only	ADV
easat-2091	29	4	if	if	SCONJ
easat-2091	29	5	there	there	PRON
easat-2091	29	6	exist	exist	VERB
easat-2091	29	7	𝛼	𝛼	DET
easat-2091	29	8	∈	∈	PROPN
easat-2091	29	9	𝜕𝐷	𝜕𝐷	PROPN
easat-2091	29	10	and	and	CCONJ
easat-2091	29	11	𝛽	𝛽	PROPN
easat-2091	29	12	∈	∈	PROPN
easat-2091	29	13	𝐷	𝐷	NOUN
easat-2091	29	14	such	such	ADJ
easat-2091	29	15	that	that	DET
easat-2091	29	16	ℎ(𝑧	ℎ(𝑧	NOUN
easat-2091	29	17	)	)	PUNCT
easat-2091	30	1	=	=	PUNCT
easat-2091	30	2	𝛼	𝛼	PRON
easat-2091	30	3	𝛽	𝛽	NOUN
easat-2091	30	4	−	−	NOUN
easat-2091	30	5	𝑧	𝑧	PROPN
easat-2091	30	6	1	1	NUM
easat-2091	30	7	−	−	NOUN
easat-2091	30	8	𝛽	𝛽	NOUN
easat-2091	30	9	for	for	ADP
easat-2091	30	10	all	all	DET
easat-2091	30	11	𝑧	𝑧	DET
easat-2091	30	12	∈	∈	PROPN
easat-2091	30	13	𝐷	𝐷	NOUN
easat-2091	30	14	a	a	DET
easat-2091	30	15	function	function	NOUN
easat-2091	30	16	𝑢	𝑢	PROPN
easat-2091	30	17	∈	∈	PROPN
easat-2091	30	18	𝐶(𝐷	𝐶(𝐷	NOUN
easat-2091	30	19	)	)	PUNCT
easat-2091	30	20	is	be	AUX
easat-2091	30	21	said	say	VERB
easat-2091	30	22	to	to	PART
easat-2091	30	23	have	have	VERB
easat-2091	30	24	the	the	DET
easat-2091	30	25	invariant	invariant	ADJ
easat-2091	30	26	mean	mean	NOUN
easat-2091	30	27	value	value	NOUN
easat-2091	30	28	property	property	NOUN
easat-2091	30	29	if	if	SCONJ
easat-2091	30	30	∫	∫	PROPN
easat-2091	30	31	𝑢	𝑢	X
easat-2091	30	32	(	(	PUNCT
easat-2091	30	33	ℎ(𝑟𝑒𝑖𝜃	ℎ(𝑟𝑒𝑖𝜃	PROPN
easat-2091	30	34	)	)	PUNCT
easat-2091	30	35	)	)	PUNCT
easat-2091	30	36	𝑑𝜃	𝑑𝜃	ADP
easat-2091	30	37	2𝜋	2𝜋	NUM
easat-2091	30	38	2𝜋	2𝜋	NOUN
easat-2091	30	39	0	0	PUNCT
easat-2091	30	40	=	=	SYM
easat-2091	30	41	𝑢(ℎ(0	𝑢(ℎ(0	NOUN
easat-2091	30	42	)	)	PUNCT
easat-2091	30	43	)	)	PUNCT
easat-2091	30	44	(	(	PUNCT
easat-2091	30	45	1	1	X
easat-2091	30	46	)	)	PUNCT
easat-2091	30	47	for	for	ADP
easat-2091	30	48	every	every	DET
easat-2091	30	49	ℎ	ℎ	PROPN
easat-2091	30	50	∈	∈	PROPN
easat-2091	30	51	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	30	52	)	)	PUNCT
easat-2091	30	53	and	and	CCONJ
easat-2091	30	54	every	every	DET
easat-2091	30	55	𝑟	𝑟	X
easat-2091	30	56	∈	∈	PROPN
easat-2091	31	1	[	[	X
easat-2091	31	2	0	0	NUM
easat-2091	31	3	,	,	PUNCT
easat-2091	31	4	1	1	NUM
easat-2091	31	5	)	)	PUNCT
easat-2091	31	6	.	.	PUNCT
easat-2091	32	1	here	here	ADV
easat-2091	32	2	"	"	PUNCT
easat-2091	32	3	invariant	invariant	ADJ
easat-2091	32	4	"	"	PUNCT
easat-2091	32	5	refers	refer	VERB
easat-2091	32	6	to	to	ADP
easat-2091	32	7	conformal	conformal	ADJ
easat-2091	32	8	invariance	invariance	NOUN
easat-2091	32	9	,	,	PUNCT
easat-2091	32	10	meaning	mean	VERB
easat-2091	32	11	invariance	invariance	NOUN
easat-2091	32	12	under	under	ADP
easat-2091	32	13	composition	composition	NOUN
easat-2091	32	14	with	with	ADP
easat-2091	32	15	elements	element	NOUN
easat-2091	32	16	of𝐴𝑢𝑡(𝐷	of𝐴𝑢𝑡(𝐷	NOUN
easat-2091	32	17	)	)	PUNCT
easat-2091	32	18	.	.	PUNCT
easat-2091	33	1	if	if	SCONJ
easat-2091	33	2	𝑢	𝑢	NOUN
easat-2091	33	3	is	be	AUX
easat-2091	33	4	harmonic	harmonic	ADJ
easat-2091	33	5	on	on	ADP
easat-2091	33	6	𝐷	𝐷	PROPN
easat-2091	33	7	,	,	PUNCT
easat-2091	33	8	then	then	ADV
easat-2091	33	9	so	so	ADV
easat-2091	33	10	is	be	AUX
easat-2091	33	11	𝑢	𝑢	NOUN
easat-2091	33	12	°	°	ADP
easat-2091	33	13	ℎ	ℎ	NOUN
easat-2091	33	14	for	for	ADP
easat-2091	33	15	everyℎ	everyℎ	ADJ
easat-2091	33	16	∈	∈	PROPN
easat-2091	33	17	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	33	18	)	)	PUNCT
easat-2091	33	19	;	;	PUNCT
easat-2091	33	20	thus	thus	ADV
easat-2091	33	21	harmonic	harmonic	ADJ
easat-2091	33	22	functions	function	NOUN
easat-2091	33	23	have	have	VERB
easat-2091	33	24	the	the	DET
easat-2091	33	25	invariant	invariant	ADJ
easat-2091	33	26	mean	mean	NOUN
easat-2091	33	27	value	value	NOUN
easat-2091	33	28	property	property	NOUN
easat-2091	33	29	.	.	PUNCT
easat-2091	34	1	the	the	DET
easat-2091	34	2	converse	converse	NOUN
easat-2091	34	3	is	be	AUX
easat-2091	34	4	also	also	ADV
easat-2091	34	5	true	true	ADJ
easat-2091	34	6	[	[	X
easat-2091	34	7	4,6	4,6	X
easat-2091	34	8	]	]	X
easat-2091	34	9	,	,	PUNCT
easat-2091	34	10	if	if	SCONJ
easat-2091	34	11	a	a	DET
easat-2091	34	12	function	function	NOUN
easat-2091	34	13	𝑢	𝑢	PROPN
easat-2091	34	14	∈	∈	PROPN
easat-2091	34	15	𝐶(𝐷	𝐶(𝐷	NOUN
easat-2091	34	16	)	)	PUNCT
easat-2091	34	17	has	have	VERB
easat-2091	34	18	the	the	DET
easat-2091	34	19	invariant	invariant	ADJ
easat-2091	34	20	mean	mean	NOUN
easat-2091	34	21	value	value	NOUN
easat-2091	34	22	property	property	NOUN
easat-2091	34	23	,	,	PUNCT
easat-2091	34	24	then	then	ADV
easat-2091	34	25	𝑢	𝑢	NOUN
easat-2091	34	26	is	be	AUX
easat-2091	34	27	harmonic	harmonic	ADJ
easat-2091	34	28	on𝐷.	on𝐷.	ADP
easat-2091	34	29	the	the	DET
easat-2091	34	30	invariant	invariant	ADJ
easat-2091	34	31	mean	mean	NOUN
easat-2091	34	32	value	value	NOUN
easat-2091	34	33	property	property	NOUN
easat-2091	34	34	concerns	concern	NOUN
easat-2091	34	35	averages	average	NOUN
easat-2091	34	36	over	over	ADP
easat-2091	34	37	circles	circle	NOUN
easat-2091	34	38	with	with	ADP
easat-2091	34	39	respect	respect	NOUN
easat-2091	34	40	to	to	ADP
easat-2091	34	41	arc	arc	NOUN
easat-2091	34	42	length	length	NOUN
easat-2091	34	43	measure	measure	NOUN
easat-2091	34	44	.	.	PUNCT
easat-2091	35	1	because	because	SCONJ
easat-2091	35	2	we	we	PRON
easat-2091	35	3	are	be	AUX
easat-2091	35	4	dealing	deal	VERB
easat-2091	35	5	with	with	ADP
easat-2091	35	6	the	the	DET
easat-2091	35	7	bergman	bergman	PROPN
easat-2091	35	8	space	space	NOUN
easat-2091	35	9	,	,	PUNCT
easat-2091	35	10	𝐿𝑎	𝐿𝑎	ADP
easat-2091	35	11	2	2	NUM
easat-2091	35	12	we	we	PRON
easat-2091	35	13	need	need	VERB
easat-2091	35	14	an	an	DET
easat-2091	35	15	invariant	invariant	ADJ
easat-2091	35	16	condition	condition	NOUN
easat-2091	35	17	stated	state	VERB
easat-2091	35	18	in	in	ADP
easat-2091	35	19	terms	term	NOUN
easat-2091	35	20	of	of	ADP
easat-2091	35	21	an	an	DET
easat-2091	35	22	area	area	NOUN
easat-2091	35	23	average	average	NOUN
easat-2091	35	24	over𝐷.	over𝐷.	NOUN
easat-2091	35	25	thus	thus	ADV
easat-2091	35	26	we	we	PRON
easat-2091	35	27	say	say	VERB
easat-2091	35	28	that	that	SCONJ
easat-2091	35	29	a	a	DET
easat-2091	35	30	function	function	NOUN
easat-2091	35	31	𝑢	𝑢	PROPN
easat-2091	35	32	∈	∈	PROPN
easat-2091	35	33	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	35	34	)	)	PUNCT
easat-2091	35	35	∩	∩	PROPN
easat-2091	36	1	𝐿1(𝐷	𝐿1(𝐷	PROPN
easat-2091	36	2	,	,	PUNCT
easat-2091	36	3	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	36	4	)	)	PUNCT
easat-2091	36	5	has	have	VERB
easat-2091	36	6	the	the	DET
easat-2091	36	7	area	area	NOUN
easat-2091	36	8	version	version	NOUN
easat-2091	36	9	of	of	ADP
easat-2091	36	10	the	the	DET
easat-2091	36	11	invariant	invariant	ADJ
easat-2091	36	12	mean	mean	NOUN
easat-2091	36	13	value	value	NOUN
easat-2091	36	14	property	property	NOUN
easat-2091	36	15	if	if	SCONJ
easat-2091	36	16	∫𝐷𝑢οℎ	∫𝐷𝑢οℎ	NOUN
easat-2091	36	17	𝑑𝜃	𝑑𝜃	ADP
easat-2091	36	18	2𝜋	2𝜋	NOUN
easat-2091	36	19	=	=	SYM
easat-2091	36	20	𝑢(ℎ(0	𝑢(ℎ(0	NOUN
easat-2091	36	21	)	)	PUNCT
easat-2091	36	22	)	)	PUNCT
easat-2091	37	1	(	(	PUNCT
easat-2091	37	2	2	2	X
easat-2091	37	3	)	)	PUNCT
easat-2091	37	4	for	for	ADP
easat-2091	37	5	every	every	DET
easat-2091	37	6	ℎ	ℎ	PROPN
easat-2091	37	7	∈	∈	PROPN
easat-2091	37	8	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PROPN
easat-2091	37	9	)	)	PUNCT
easat-2091	37	10	.	.	PUNCT
easat-2091	38	1	if	if	SCONJ
easat-2091	38	2	𝑢	𝑢	NOUN
easat-2091	38	3	is	be	AUX
easat-2091	38	4	in𝐶(𝐷	in𝐶(𝐷	PROPN
easat-2091	38	5	)	)	PUNCT
easat-2091	38	6	∩	∩	PROPN
easat-2091	38	7	𝐿1(𝐷	𝐿1(𝐷	PROPN
easat-2091	38	8	,	,	PUNCT
easat-2091	38	9	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	38	10	)	)	PUNCT
easat-2091	38	11	,	,	PUNCT
easat-2091	38	12	then	then	ADV
easat-2091	38	13	so	so	ADV
easat-2091	38	14	is	be	AUX
easat-2091	38	15	𝑢	𝑢	NOUN
easat-2091	38	16	°	°	ADP
easat-2091	38	17	ℎ	ℎ	NOUN
easat-2091	38	18	for	for	ADP
easat-2091	38	19	everyℎ	everyℎ	ADJ
easat-2091	38	20	∈	∈	PROPN
easat-2091	38	21	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	38	22	)	)	PUNCT
easat-2091	38	23	,	,	PUNCT
easat-2091	38	24	so	so	CCONJ
easat-2091	38	25	the	the	DET
easat-2091	38	26	left	left	ADJ
easat-2091	38	27	-	-	PUNCT
easat-2091	38	28	hand	hand	NOUN
easat-2091	38	29	side	side	NOUN
easat-2091	38	30	of	of	ADP
easat-2091	38	31	the	the	DET
easat-2091	38	32	above	above	ADJ
easat-2091	38	33	equation	equation	NOUN
easat-2091	38	34	makes	make	VERB
easat-2091	38	35	sense	sense	NOUN
easat-2091	38	36	.	.	PUNCT
easat-2091	39	1	note	note	VERB
easat-2091	39	2	that	that	SCONJ
easat-2091	39	3	the	the	DET
easat-2091	39	4	area	area	NOUN
easat-2091	39	5	version	version	NOUN
easat-2091	39	6	of	of	ADP
easat-2091	39	7	the	the	DET
easat-2091	39	8	invariant	invariant	ADJ
easat-2091	39	9	mean	mean	NOUN
easat-2091	39	10	value	value	NOUN
easat-2091	39	11	property	property	NOUN
easat-2091	39	12	deals	deal	NOUN
easat-2091	39	13	with	with	ADP
easat-2091	39	14	integrals	integral	NOUN
easat-2091	39	15	over	over	ADP
easat-2091	39	16	all	all	DET
easat-2091	39	17	of𝐷	of𝐷	PROPN
easat-2091	39	18	,	,	PUNCT
easat-2091	39	19	as	as	SCONJ
easat-2091	39	20	opposed	oppose	VERB
easat-2091	39	21	to	to	ADP
easat-2091	39	22	integrals	integral	NOUN
easat-2091	39	23	over	over	ADP
easat-2091	39	24	𝑟𝐷	𝑟𝐷	NUM
easat-2091	39	25	for𝑟	for𝑟	PROPN
easat-2091	39	26	∈	∈	PROPN
easat-2091	39	27	(	(	PUNCT
easat-2091	39	28	0	0	NUM
easat-2091	39	29	,	,	PUNCT
easat-2091	39	30	1	1	NUM
easat-2091	39	31	)	)	PUNCT
easat-2091	39	32	.	.	PUNCT
easat-2091	40	1	if	if	SCONJ
easat-2091	40	2	𝑢	𝑢	NOUN
easat-2091	40	3	is	be	AUX
easat-2091	40	4	harmonic	harmonic	ADJ
easat-2091	40	5	on	on	ADP
easat-2091	40	6	𝐷	𝐷	PROPN
easat-2091	40	7	and	and	CCONJ
easat-2091	40	8	in𝐿1(𝐷	in𝐿1(𝐷	PROPN
easat-2091	40	9	,	,	PUNCT
easat-2091	40	10	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	40	11	)	)	PUNCT
easat-2091	40	12	,	,	PUNCT
easat-2091	40	13	then	then	ADV
easat-2091	40	14	so	so	ADV
easat-2091	40	15	is	be	AUX
easat-2091	40	16	𝑢	𝑢	NOUN
easat-2091	40	17	°	°	ADP
easat-2091	40	18	ℎ	ℎ	NOUN
easat-2091	40	19	for	for	ADP
easat-2091	40	20	everyℎ	everyℎ	ADJ
easat-2091	40	21	∈	∈	PROPN
easat-2091	40	22	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	40	23	)	)	PUNCT
easat-2091	40	24	.	.	PUNCT
easat-2091	41	1	thus	thus	ADV
easat-2091	41	2	,	,	PUNCT
easat-2091	41	3	by	by	ADP
easat-2091	41	4	the	the	DET
easat-2091	41	5	mean	mean	ADJ
easat-2091	41	6	value	value	NOUN
easat-2091	41	7	property	property	NOUN
easat-2091	41	8	,	,	PUNCT
easat-2091	41	9	harmonic	harmonic	ADJ
easat-2091	41	10	functions	function	NOUN
easat-2091	41	11	have	have	VERB
easat-2091	41	12	the	the	DET
easat-2091	41	13	area	area	NOUN
easat-2091	41	14	version	version	NOUN
easat-2091	41	15	of	of	ADP
easat-2091	41	16	the	the	DET
easat-2091	41	17	invariant	invariant	ADJ
easat-2091	41	18	mean	mean	NOUN
easat-2091	41	19	value	value	NOUN
easat-2091	41	20	property	property	NOUN
easat-2091	41	21	.	.	PUNCT
easat-2091	42	1	whether	whether	SCONJ
easat-2091	42	2	or	or	CCONJ
easat-2091	42	3	not	not	PART
easat-2091	42	4	the	the	DET
easat-2091	42	5	converse	converse	NOUN
easat-2091	42	6	is	be	AUX
easat-2091	42	7	true	true	ADJ
easat-2091	42	8	is	be	AUX
easat-2091	42	9	an	an	DET
easat-2091	42	10	open	open	ADJ
easat-2091	42	11	question	question	NOUN
easat-2091	42	12	.	.	PUNCT
easat-2091	43	1	in	in	ADP
easat-2091	43	2	other	other	ADJ
easat-2091	43	3	words	word	NOUN
easat-2091	43	4	,	,	PUNCT
easat-2091	43	5	if	if	SCONJ
easat-2091	43	6	𝑢	𝑢	PROPN
easat-2091	43	7	∈	∈	PROPN
easat-2091	43	8	𝐶(𝐷	𝐶(𝐷	NOUN
easat-2091	43	9	)	)	PUNCT
easat-2091	43	10	∩	∩	PROPN
easat-2091	43	11	𝐿1(𝐷	𝐿1(𝐷	PROPN
easat-2091	43	12	,	,	PUNCT
easat-2091	43	13	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	43	14	)	)	PUNCT
easat-2091	43	15	has	have	VERB
easat-2091	43	16	the	the	DET
easat-2091	43	17	area	area	NOUN
easat-2091	43	18	version	version	NOUN
easat-2091	43	19	of	of	ADP
easat-2091	43	20	the	the	DET
easat-2091	43	21	invariant	invariant	ADJ
easat-2091	43	22	mean	mean	NOUN
easat-2091	43	23	value	value	NOUN
easat-2091	43	24	property	property	NOUN
easat-2091	43	25	,	,	PUNCT
easat-2091	43	26	must	must	AUX
easat-2091	43	27	𝑢	𝑢	PRON
easat-2091	43	28	be	be	AUX
easat-2091	43	29	harmonic	harmonic	ADJ
easat-2091	43	30	?	?	PUNCT
easat-2091	44	1	this	this	DET
easat-2091	44	2	question	question	NOUN
easat-2091	44	3	has	have	VERB
easat-2091	44	4	an	an	DET
easat-2091	44	5	affirmative	affirmative	ADJ
easat-2091	44	6	answer	answer	NOUN
easat-2091	44	7	if	if	SCONJ
easat-2091	44	8	we	we	PRON
easat-2091	44	9	replace	replace	VERB
easat-2091	44	10	the	the	DET
easat-2091	44	11	hypothesis	hypothesis	NOUN
easat-2091	44	12	that	that	PRON
easat-2091	44	13	𝑢	𝑢	NOUN
easat-2091	44	14	is	be	AUX
easat-2091	44	15	in	in	ADP
easat-2091	44	16	𝐶(𝐷	𝐶(𝐷	NOUN
easat-2091	44	17	)	)	PUNCT
easat-2091	44	18	∩	∩	PROPN
easat-2091	44	19	𝐿1(𝐷	𝐿1(𝐷	PROPN
easat-2091	44	20	,	,	PUNCT
easat-2091	44	21	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	44	22	)	)	PUNCT
easat-2091	44	23	with	with	ADP
easat-2091	44	24	the	the	DET
easat-2091	44	25	stronger	strong	ADJ
easat-2091	44	26	hypothesis	hypothesis	NOUN
easat-2091	44	27	that	that	SCONJ
easat-2091	44	28	𝑢	𝑢	PROPN
easat-2091	44	29	is	be	AUX
easat-2091	44	30	in𝐶(	in𝐶(	PRON
easat-2091	44	31	�	�	NOUN
easat-2091	44	32	̅	̅	NOUN
easat-2091	44	33	�	�	NOUN
easat-2091	44	34	)	)	PUNCT
easat-2091	44	35	;	;	PUNCT
easat-2091	44	36	see	see	VERB
easat-2091	44	37	in	in	ADP
easat-2091	44	38	[	[	X
easat-2091	44	39	4,5	4,5	NUM
easat-2091	44	40	]	]	PUNCT
easat-2091	44	41	.	.	PUNCT
easat-2091	45	1	we	we	PRON
easat-2091	45	2	need	need	VERB
easat-2091	45	3	to	to	PART
easat-2091	45	4	consider	consider	VERB
easat-2091	45	5	functions	function	NOUN
easat-2091	45	6	that	that	PRON
easat-2091	45	7	are	be	AUX
easat-2091	45	8	not	not	PART
easat-2091	45	9	necessarily	necessarily	ADV
easat-2091	45	10	continuous	continuous	ADJ
easat-2091	45	11	on	on	ADP
easat-2091	45	12	the	the	DET
easat-2091	45	13	closed	closed	ADJ
easat-2091	45	14	disk	disk	NOUN
easat-2091	45	15	,	,	PUNCT
easat-2091	45	16	so	so	CCONJ
easat-2091	45	17	the	the	DET
easat-2091	45	18	result	result	NOUN
easat-2091	45	19	mentioned	mention	VERB
easat-2091	45	20	in	in	ADP
easat-2091	45	21	the	the	DET
easat-2091	45	22	last	last	ADJ
easat-2091	45	23	sentence	sentence	NOUN
easat-2091	45	24	will	will	AUX
easat-2091	45	25	not	not	PART
easat-2091	45	26	suffice	suffice	VERB
easat-2091	45	27	.	.	PUNCT
easat-2091	46	1	however	however	ADV
easat-2091	46	2	,	,	PUNCT
easat-2091	46	3	our	our	PRON
easat-2091	46	4	functions	function	NOUN
easat-2091	46	5	do	do	AUX
easat-2091	46	6	have	have	VERB
easat-2091	46	7	the	the	DET
easat-2091	46	8	property	property	NOUN
easat-2091	46	9	that	that	PRON
easat-2091	46	10	their	their	PRON
easat-2091	46	11	radiation	radiation	NOUN
easat-2091	46	12	are	be	AUX
easat-2091	46	13	continuous	continuous	ADJ
easat-2091	46	14	on	on	ADP
easat-2091	46	15	the	the	DET
easat-2091	46	16	closed	closed	ADJ
easat-2091	46	17	disk	disk	NOUN
easat-2091	46	18	,	,	PUNCT
easat-2091	46	19	and	and	CCONJ
easat-2091	46	20	we	we	PRON
easat-2091	46	21	will	will	AUX
easat-2091	46	22	prove	prove	VERB
easat-2091	46	23	that	that	SCONJ
easat-2091	46	24	this	this	DET
easat-2091	46	25	property	property	NOUN
easat-2091	46	26	,	,	PUNCT
easat-2091	46	27	along	along	ADP
easat-2091	46	28	with	with	ADP
easat-2091	46	29	the	the	DET
easat-2091	46	30	area	area	NOUN
easat-2091	46	31	version	version	NOUN
easat-2091	46	32	of	of	ADP
easat-2091	46	33	the	the	DET
easat-2091	46	34	invariant	invariant	ADJ
easat-2091	46	35	mean	mean	NOUN
easat-2091	46	36	value	value	NOUN
easat-2091	46	37	property	property	NOUN
easat-2091	46	38	,	,	PUNCT
easat-2091	46	39	is	be	AUX
easat-2091	46	40	enough	enough	ADJ
easat-2091	46	41	to	to	PART
easat-2091	46	42	imply	imply	VERB
easat-2091	46	43	harmonicity	harmonicity	NOUN
easat-2091	46	44	.	.	PUNCT
easat-2091	47	1	if𝑢	if𝑢	PROPN
easat-2091	47	2	∈	∈	PROPN
easat-2091	47	3	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	47	4	)	)	PUNCT
easat-2091	47	5	,	,	PUNCT
easat-2091	47	6	then	then	ADV
easat-2091	47	7	the	the	DET
easat-2091	47	8	radiation	radiation	NOUN
easat-2091	47	9	of	of	ADP
easat-2091	47	10	𝑢	𝑢	PRON
easat-2091	47	11	denoted𝑅(𝑢	denoted𝑅(𝑢	PROPN
easat-2091	47	12	)	)	PUNCT
easat-2091	47	13	,	,	PUNCT
easat-2091	47	14	is	be	AUX
easat-2091	47	15	the	the	DET
easat-2091	47	16	function	function	NOUN
easat-2091	47	17	on	on	ADP
easat-2091	47	18	𝐷	𝐷	NOUN
easat-2091	47	19	defined	define	VERB
easat-2091	47	20	by	by	ADP
easat-2091	47	21	𝑅(𝑢)(𝑤	𝑅(𝑢)(𝑤	NOUN
easat-2091	47	22	)	)	PUNCT
easat-2091	47	23	=	=	SYM
easat-2091	47	24	∫	∫	PROPN
easat-2091	47	25	𝑢(𝑒𝑖𝜃	𝑢(𝑒𝑖𝜃	PROPN
easat-2091	47	26	)	)	PUNCT
easat-2091	47	27	2𝜋	2𝜋	NOUN
easat-2091	47	28	0	0	NUM
easat-2091	47	29	𝑑𝜃	𝑑𝜃	ADP
easat-2091	47	30	2𝜋	2𝜋	NUM
easat-2091	47	31	(	(	PUNCT
easat-2091	47	32	3	3	NUM
easat-2091	47	33	)	)	PUNCT
easat-2091	47	34	in	in	ADP
easat-2091	47	35	the	the	DET
easat-2091	47	36	following	follow	VERB
easat-2091	47	37	lemma	lemma	PROPN
easat-2091	47	38	,	,	PUNCT
easat-2091	47	39	which	which	PRON
easat-2091	47	40	will	will	AUX
easat-2091	47	41	be	be	AUX
easat-2091	47	42	a	a	DET
easat-2091	47	43	key	key	ADJ
easat-2091	47	44	tool	tool	NOUN
easat-2091	47	45	in	in	ADP
easat-2091	47	46	our	our	PRON
easat-2091	47	47	proof	proof	NOUN
easat-2091	47	48	of	of	ADP
easat-2091	47	49	theorem	theorem	NOUN
easat-2091	47	50	1	1	NUM
easat-2091	47	51	,	,	PUNCT
easat-2091	47	52	the	the	DET
easat-2091	47	53	statement	statement	NOUN
easat-2091	47	54	𝑅(𝑢	𝑅(𝑢	NUM
easat-2091	47	55	°	°	NOUN
easat-2091	47	56	ℎ	ℎ	ADJ
easat-2091	47	57	)	)	PUNCT
easat-2091	47	58	∈	∈	PROPN
easat-2091	47	59	𝐶(	𝐶(	NOUN
easat-2091	47	60	�	�	PROPN
easat-2091	47	61	̅	̅	NOUN
easat-2091	47	62	�	�	NOUN
easat-2091	47	63	)	)	PUNCT
easat-2091	47	64	means	mean	VERB
easat-2091	47	65	𝑅(𝑢	𝑅(𝑢	X
easat-2091	47	66	°	°	NOUN
easat-2091	47	67	ℎ	ℎ	NOUN
easat-2091	47	68	)	)	PUNCT
easat-2091	47	69	can	can	AUX
easat-2091	47	70	be	be	AUX
easat-2091	47	71	extended	extend	VERB
easat-2091	47	72	to	to	ADP
easat-2091	47	73	a	a	DET
easat-2091	47	74	continuous	continuous	ADJ
easat-2091	47	75	complex	complex	ADJ
easat-2091	47	76	valued	value	VERB
easat-2091	47	77	function	function	NOUN
easat-2091	47	78	on	on	ADP
easat-2091	47	79	�	�	PROPN
easat-2091	47	80	̅	̅	NOUN
easat-2091	47	81	�	�	NOUN
easat-2091	47	82	lemma	lemma	PROPN
easat-2091	47	83	2	2	X
easat-2091	47	84	.	.	PUNCT
easat-2091	47	85	suppose	suppose	VERB
easat-2091	47	86	that𝑢	that𝑢	PROPN
easat-2091	47	87	∈	∈	PROPN
easat-2091	47	88	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	47	89	)	)	PUNCT
easat-2091	48	1	∩	∩	PROPN
easat-2091	48	2	𝐿1(𝐷	𝐿1(𝐷	PROPN
easat-2091	48	3	,	,	PUNCT
easat-2091	48	4	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	48	5	)	)	PUNCT
easat-2091	48	6	.	.	PUNCT
easat-2091	49	1	then	then	ADV
easat-2091	49	2	𝑢	𝑢	NOUN
easat-2091	49	3	is	be	AUX
easat-2091	49	4	harmonic	harmonic	ADJ
easat-2091	49	5	on	on	ADP
easat-2091	49	6	𝐷	𝐷	PROPN
easat-2091	49	7	if	if	SCONJ
easat-2091	50	1	and	and	CCONJ
easat-2091	50	2	only	only	ADV
easat-2091	50	3	if	if	SCONJ
easat-2091	50	4	396	396	NUM
easat-2091	50	5	edelweiss	edelweiss	PROPN
easat-2091	50	6	applied	apply	VERB
easat-2091	50	7	science	science	NOUN
easat-2091	50	8	and	and	CCONJ
easat-2091	50	9	technology	technology	NOUN
easat-2091	50	10	issn	issn	PROPN
easat-2091	50	11	:	:	PUNCT
easat-2091	50	12	2576	2576	NUM
easat-2091	50	13	-	-	SYM
easat-2091	50	14	8484	8484	NUM
easat-2091	50	15	vol	vol	NOUN
easat-2091	50	16	.	.	PROPN
easat-2091	51	1	8	8	NUM
easat-2091	51	2	,	,	PUNCT
easat-2091	51	3	no	no	INTJ
easat-2091	51	4	.	.	NOUN
easat-2091	52	1	6	6	NUM
easat-2091	52	2	:	:	SYM
easat-2091	52	3	394	394	NUM
easat-2091	52	4	-	-	SYM
easat-2091	52	5	400	400	NUM
easat-2091	52	6	,	,	PUNCT
easat-2091	52	7	2024	2024	NUM
easat-2091	52	8	doi	doi	NOUN
easat-2091	52	9	:	:	PUNCT
easat-2091	52	10	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	52	11	©	©	ADP
easat-2091	52	12	2024	2024	NUM
easat-2091	52	13	by	by	ADP
easat-2091	52	14	the	the	DET
easat-2091	52	15	author	author	NOUN
easat-2091	52	16	;	;	PUNCT
easat-2091	52	17	licensee	licensee	NOUN
easat-2091	52	18	learning	learning	NOUN
easat-2091	52	19	gate	gate	NOUN
easat-2091	52	20	∫𝐷𝑢οℎ	∫𝐷𝑢οℎ	VERB
easat-2091	52	21	𝑑𝐴	𝑑𝐴	NUM
easat-2091	52	22	𝜋	𝜋	NOUN
easat-2091	52	23	=	=	SYM
easat-2091	52	24	𝑢(ℎ(0	𝑢(ℎ(0	NOUN
easat-2091	52	25	)	)	PUNCT
easat-2091	52	26	)	)	PUNCT
easat-2091	52	27	(	(	PUNCT
easat-2091	52	28	4	4	NUM
easat-2091	52	29	)	)	PUNCT
easat-2091	52	30	and	and	CCONJ
easat-2091	52	31	𝑅(𝑢	𝑅(𝑢	NUM
easat-2091	52	32	°	°	NOUN
easat-2091	52	33	ℎ	ℎ	ADJ
easat-2091	52	34	)	)	PUNCT
easat-2091	52	35	∈	∈	PROPN
easat-2091	52	36	𝐶(	𝐶(	NOUN
easat-2091	52	37	�	�	PROPN
easat-2091	52	38	̅	̅	NOUN
easat-2091	52	39	�	�	NOUN
easat-2091	52	40	)	)	PUNCT
easat-2091	52	41	for	for	ADP
easat-2091	52	42	everyℎ	everyℎ	ADJ
easat-2091	52	43	∈	∈	PROPN
easat-2091	52	44	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	52	45	)	)	PUNCT
easat-2091	52	46	(	(	PUNCT
easat-2091	52	47	5	5	X
easat-2091	52	48	)	)	PUNCT
easat-2091	52	49	proof	proof	NOUN
easat-2091	52	50	.	.	PUNCT
easat-2091	52	51	suppose	suppose	VERB
easat-2091	52	52	that	that	SCONJ
easat-2091	52	53	𝑢	𝑢	NOUN
easat-2091	52	54	is	be	AUX
easat-2091	52	55	harmonic	harmonic	ADJ
easat-2091	52	56	on𝐷.	on𝐷.	ADP
easat-2091	52	57	letℎ	letℎ	ADJ
easat-2091	52	58	∈	∈	PROPN
easat-2091	52	59	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	52	60	)	)	PUNCT
easat-2091	52	61	.	.	PUNCT
easat-2091	53	1	as	as	SCONJ
easat-2091	53	2	we	we	PRON
easat-2091	53	3	discussed	discuss	VERB
easat-2091	53	4	earlier	early	ADV
easat-2091	53	5	,	,	PUNCT
easat-2091	53	6	𝑢	𝑢	NOUN
easat-2091	53	7	°	°	NOUN
easat-2091	53	8	ℎ	ℎ	NOUN
easat-2091	53	9	is	be	AUX
easat-2091	53	10	harmonic	harmonic	ADJ
easat-2091	53	11	and	and	CCONJ
easat-2091	53	12	we	we	PRON
easat-2091	53	13	see	see	VERB
easat-2091	53	14	in	in	ADP
easat-2091	53	15	eq(4	eq(4	PROPN
easat-2091	53	16	)	)	PUNCT
easat-2091	53	17	holds	hold	VERB
easat-2091	53	18	.	.	PUNCT
easat-2091	54	1	the	the	DET
easat-2091	54	2	mean	mean	ADJ
easat-2091	54	3	value	value	NOUN
easat-2091	54	4	property	property	NOUN
easat-2091	54	5	implies	imply	VERB
easat-2091	54	6	that	that	SCONJ
easat-2091	54	7	𝑅(𝑢	𝑅(𝑢	PROPN
easat-2091	54	8	°	°	NOUN
easat-2091	54	9	ℎ	ℎ	NOUN
easat-2091	54	10	)	)	PUNCT
easat-2091	54	11	is	be	AUX
easat-2091	54	12	a	a	DET
easat-2091	54	13	constant	constant	ADJ
easat-2091	54	14	function	function	NOUN
easat-2091	54	15	on𝐷	on𝐷	PROPN
easat-2091	54	16	,	,	PUNCT
easat-2091	54	17	with	with	ADP
easat-2091	54	18	value𝑢(ℎ(0	value𝑢(ℎ(0	NOUN
easat-2091	54	19	)	)	PUNCT
easat-2091	54	20	)	)	PUNCT
easat-2091	54	21	,	,	PUNCT
easat-2091	54	22	we	we	PRON
easat-2091	54	23	see	see	VERB
easat-2091	54	24	in	in	ADP
easat-2091	54	25	eq	eq	NOUN
easat-2091	54	26	(	(	PUNCT
easat-2091	54	27	5	5	NUM
easat-2091	54	28	)	)	PUNCT
easat-2091	54	29	also	also	ADV
easat-2091	54	30	holds	hold	VERB
easat-2091	54	31	.	.	PUNCT
easat-2091	55	1	to	to	PART
easat-2091	55	2	prove	prove	VERB
easat-2091	55	3	the	the	DET
easat-2091	55	4	other	other	ADJ
easat-2091	55	5	direction	direction	NOUN
easat-2091	55	6	,	,	PUNCT
easat-2091	55	7	suppose	suppose	VERB
easat-2091	55	8	that	that	SCONJ
easat-2091	55	9	eq(4	eq(4	PROPN
easat-2091	55	10	)	)	PUNCT
easat-2091	55	11	and	and	CCONJ
easat-2091	55	12	eq(5	eq(5	NOUN
easat-2091	55	13	)	)	PUNCT
easat-2091	55	14	hold	hold	VERB
easat-2091	55	15	.	.	PUNCT
easat-2091	56	1	letℎ	letℎ	PROPN
easat-2091	56	2	∈	∈	PROPN
easat-2091	56	3	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PRON
easat-2091	56	4	)	)	PUNCT
easat-2091	56	5	,	,	PUNCT
easat-2091	57	1	and	and	CCONJ
easat-2091	57	2	let	let	VERB
easat-2091	57	3	𝑣	𝑣	PRON
easat-2091	57	4	∈	∈	PROPN
easat-2091	57	5	𝑅(𝑢	𝑅(𝑢	NUM
easat-2091	57	6	°	°	NOUN
easat-2091	57	7	ℎ	ℎ	NOUN
easat-2091	57	8	)	)	PUNCT
easat-2091	57	9	from	from	ADP
easat-2091	57	10	eq(5	eq(5	NOUN
easat-2091	57	11	)	)	PUNCT
easat-2091	57	12	we	we	PRON
easat-2091	57	13	fine	fine	VERB
easat-2091	57	14	𝑣	𝑣	PRON
easat-2091	57	15	∈	∈	PROPN
easat-2091	57	16	𝐶(	𝐶(	NOUN
easat-2091	57	17	�	�	PROPN
easat-2091	57	18	̅	̅	NOUN
easat-2091	57	19	�	�	NOUN
easat-2091	57	20	)	)	PUNCT
easat-2091	57	21	we	we	PRON
easat-2091	57	22	want	want	VERB
easat-2091	57	23	to	to	PART
easat-2091	57	24	show	show	VERB
easat-2091	57	25	that	that	SCONJ
easat-2091	57	26	𝑣	𝑣	PRON
easat-2091	57	27	has	have	VERB
easat-2091	57	28	the	the	DET
easat-2091	57	29	area	area	NOUN
easat-2091	57	30	version	version	NOUN
easat-2091	57	31	of	of	ADP
easat-2091	57	32	the	the	DET
easat-2091	57	33	invariant	invariant	ADJ
easat-2091	57	34	mean	mean	NOUN
easat-2091	57	35	value	value	NOUN
easat-2091	57	36	property	property	NOUN
easat-2091	57	37	.	.	PUNCT
easat-2091	58	1	to	to	PART
easat-2091	58	2	do	do	VERB
easat-2091	58	3	this	this	PRON
easat-2091	58	4	,	,	PUNCT
easat-2091	58	5	fixg	fixg	NOUN
easat-2091	58	6	∈	∈	PROPN
easat-2091	58	7	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PROPN
easat-2091	58	8	)	)	PUNCT
easat-2091	58	9	.	.	PUNCT
easat-2091	59	1	then	then	ADV
easat-2091	59	2	∫𝐷𝑢	∫𝐷𝑢	PUNCT
easat-2091	59	3	°	°	ADP
easat-2091	59	4	g	g	ADP
easat-2091	59	5	𝑑𝐴	𝑑𝐴	NUM
easat-2091	59	6	𝜋	𝜋	NOUN
easat-2091	59	7	𝑢	𝑢	X
easat-2091	59	8	=	=	X
easat-2091	59	9	∫𝐷ℛ(𝑢	∫𝐷ℛ(𝑢	NOUN
easat-2091	59	10	°	°	NOUN
easat-2091	59	11	ℎ)(g(𝑤	ℎ)(g(𝑤	VERB
easat-2091	59	12	)	)	PUNCT
easat-2091	59	13	)	)	PUNCT
easat-2091	60	1	𝑑𝐴(𝑤	𝑑𝐴(𝑤	PROPN
easat-2091	60	2	)	)	PUNCT
easat-2091	61	1	𝜋	𝜋	NOUN
easat-2091	61	2	=	=	SYM
easat-2091	61	3	∫𝐷	∫𝐷	PROPN
easat-2091	61	4	∫	∫	PROPN
easat-2091	61	5	𝑢	𝑢	PROPN
easat-2091	61	6	(	(	PUNCT
easat-2091	61	7	ℎ(g(𝑤)𝑒𝑖𝜃	ℎ(g(𝑤)𝑒𝑖𝜃	NOUN
easat-2091	61	8	)	)	PUNCT
easat-2091	61	9	)	)	PUNCT
easat-2091	62	1	2𝜋	2𝜋	NOUN
easat-2091	62	2	0	0	NUM
easat-2091	62	3	𝑑𝜃	𝑑𝜃	PROPN
easat-2091	62	4	2𝜋	2𝜋	PROPN
easat-2091	62	5	𝑑𝐴(𝑤	𝑑𝐴(𝑤	PROPN
easat-2091	62	6	)	)	PUNCT
easat-2091	62	7	𝜋	𝜋	NOUN
easat-2091	62	8	(	(	PUNCT
easat-2091	62	9	6	6	NUM
easat-2091	62	10	)	)	PUNCT
easat-2091	62	11	𝜃	𝜃	NOUN
easat-2091	62	12	∈	∈	NOUN
easat-2091	63	1	[	[	X
easat-2091	63	2	0,2𝜋	0,2𝜋	X
easat-2091	63	3	]	]	X
easat-2091	63	4	to	to	PART
easat-2091	63	5	check	check	VERB
easat-2091	63	6	that	that	PRON
easat-2091	63	7	interchanging	interchange	VERB
easat-2091	63	8	the	the	DET
easat-2091	63	9	order	order	NOUN
easat-2091	63	10	of	of	ADP
easat-2091	63	11	integration	integration	NOUN
easat-2091	63	12	in	in	ADP
easat-2091	63	13	the	the	DET
easat-2091	63	14	last	last	ADJ
easat-2091	63	15	integral	integral	NOUN
easat-2091	63	16	is	be	AUX
easat-2091	63	17	valid	valid	ADJ
easat-2091	63	18	,	,	PUNCT
easat-2091	63	19	for	for	ADP
easat-2091	63	20	each	each	PRON
easat-2091	63	21	define	define	NOUN
easat-2091	63	22	𝑓𝜃	𝑓𝜃	ADP
easat-2091	63	23	∈	∈	PROPN
easat-2091	63	24	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	63	25	)	)	PUNCT
easat-2091	63	26	by	by	ADP
easat-2091	63	27	𝑓𝜃(𝑤	𝑓𝜃(𝑤	NUM
easat-2091	63	28	)	)	PUNCT
easat-2091	63	29	=	=	SYM
easat-2091	63	30	ℎ(g(𝑤)𝑒𝑖𝜃	ℎ(g(𝑤)𝑒𝑖𝜃	NOUN
easat-2091	63	31	)	)	PUNCT
easat-2091	63	32	the	the	DET
easat-2091	63	33	inverse	inverse	NOUN
easat-2091	63	34	𝑓𝜃	𝑓𝜃	ADP
easat-2091	63	35	−1	−1	NOUN
easat-2091	63	36	of	of	ADP
easat-2091	63	37	𝑓𝜃	𝑓𝜃	ADV
easat-2091	63	38	is	be	AUX
easat-2091	63	39	also	also	ADV
easat-2091	63	40	an	an	DET
easat-2091	63	41	analytic	analytic	ADJ
easat-2091	63	42	automorphism	automorphism	NOUN
easat-2091	63	43	of𝐷	of𝐷	NOUN
easat-2091	63	44	,	,	PUNCT
easat-2091	63	45	so	so	SCONJ
easat-2091	63	46	there	there	PRON
easat-2091	63	47	exist	exist	VERB
easat-2091	63	48	𝛼	𝛼	DET
easat-2091	63	49	∈	∈	PROPN
easat-2091	63	50	𝜕𝐷	𝜕𝐷	PROPN
easat-2091	63	51	and	and	CCONJ
easat-2091	63	52	𝛽	𝛽	PROPN
easat-2091	63	53	∈	∈	PROPN
easat-2091	63	54	𝐷	𝐷	NOUN
easat-2091	63	55	such	such	ADJ
easat-2091	63	56	that	that	SCONJ
easat-2091	63	57	𝑓𝜃	𝑓𝜃	ADP
easat-2091	63	58	−1(𝑧	−1(𝑧	NOUN
easat-2091	63	59	)	)	PUNCT
easat-2091	63	60	=	=	SYM
easat-2091	63	61	𝛽−𝑍	𝛽−𝑍	NOUN
easat-2091	63	62	1−	1−	NUM
easat-2091	63	63	�	�	NOUN
easat-2091	63	64	̅	̅	NOUN
easat-2091	63	65	�	�	NOUN
easat-2091	63	66	𝑍	𝑍	NOUN
easat-2091	63	67	,	,	PUNCT
easat-2091	63	68	for	for	ADP
easat-2091	63	69	all	all	DET
easat-2091	63	70	𝑍	𝑍	PROPN
easat-2091	63	71	∈	∈	NOUN
easat-2091	63	72	𝐷	𝐷	NOUN
easat-2091	63	73	thus	thus	ADV
easat-2091	63	74	|(𝑓𝜃	|(𝑓𝜃	PROPN
easat-2091	63	75	−1	−1	NOUN
easat-2091	63	76	)	)	PUNCT
easat-2091	63	77	/	/	PUNCT
easat-2091	64	1	(	(	PUNCT
easat-2091	64	2	𝑧)|	𝑧)|	PROPN
easat-2091	64	3	=	=	SYM
easat-2091	64	4	1−|𝛽|2	1−|𝛽|2	NUM
easat-2091	64	5	|1−	|1−	NOUN
easat-2091	64	6	�	�	SYM
easat-2091	64	7	̅	̅	NOUN
easat-2091	64	8	�	�	NOUN
easat-2091	64	9	𝑧|	𝑧|	ADJ
easat-2091	64	10	2	2	NUM
easat-2091	64	11	≤	≤	NOUN
easat-2091	64	12	1+|𝛽|	1+|𝛽|	NUM
easat-2091	64	13	1−|𝛽|	1−|𝛽|	NUM
easat-2091	64	14	,	,	PUNCT
easat-2091	64	15	for	for	ADP
easat-2091	64	16	all	all	DET
easat-2091	64	17	𝑍	𝑍	PROPN
easat-2091	64	18	∈	∈	NOUN
easat-2091	64	19	𝐷	𝐷	NOUN
easat-2091	64	20	note	note	NOUN
easat-2091	64	21	that𝛽	that𝛽	NOUN
easat-2091	64	22	=	=	PUNCT
easat-2091	64	23	𝑓𝜃(0	𝑓𝜃(0	NOUN
easat-2091	64	24	)	)	PUNCT
easat-2091	64	25	=	=	SYM
easat-2091	64	26	ℎ(g(0)𝑒𝑖𝜃	ℎ(g(0)𝑒𝑖𝜃	ADJ
easat-2091	64	27	)	)	PUNCT
easat-2091	64	28	;	;	PUNCT
easat-2091	64	29	we	we	PRON
easat-2091	64	30	are	be	AUX
easat-2091	64	31	thinking	think	VERB
easat-2091	64	32	of	of	ADP
easat-2091	64	33	ℎ	ℎ	PROPN
easat-2091	64	34	and	and	CCONJ
easat-2091	64	35	g	g	NOUN
easat-2091	64	36	as	as	SCONJ
easat-2091	64	37	fixed	fix	VERB
easat-2091	64	38	,	,	PUNCT
easat-2091	64	39	so	so	SCONJ
easat-2091	64	40	the	the	DET
easat-2091	64	41	above	above	ADJ
easat-2091	64	42	inequality	inequality	NOUN
easat-2091	64	43	shows	show	VERB
easat-2091	64	44	there	there	PRON
easat-2091	64	45	is	be	VERB
easat-2091	64	46	a	a	DET
easat-2091	64	47	constant	constant	ADJ
easat-2091	64	48	k	k	NOUN
easat-2091	64	49	such	such	ADJ
easat-2091	64	50	that	that	DET
easat-2091	64	51	|(𝑓𝜃	|(𝑓𝜃	PROPN
easat-2091	64	52	−1	−1	NOUN
easat-2091	64	53	)	)	PUNCT
easat-2091	64	54	/	/	PUNCT
easat-2091	65	1	(	(	PUNCT
easat-2091	65	2	𝑧)|	𝑧)|	PROPN
easat-2091	65	3	≤	≤	PROPN
easat-2091	65	4	𝐾	𝐾	PROPN
easat-2091	65	5	,	,	PUNCT
easat-2091	65	6	for	for	ADP
easat-2091	65	7	all	all	DET
easat-2091	65	8	𝑍	𝑍	PROPN
easat-2091	65	9	∈	∈	NOUN
easat-2091	65	10	𝐷	𝐷	NOUN
easat-2091	65	11	and	and	CCONJ
easat-2091	65	12	𝜃	𝜃	NOUN
easat-2091	65	13	∈	∈	NOUN
easat-2091	66	1	[	[	X
easat-2091	66	2	0,2𝜋	0,2𝜋	X
easat-2091	66	3	]	]	X
easat-2091	66	4	now	now	ADV
easat-2091	66	5	∫	∫	PROPN
easat-2091	66	6	∫𝐷	∫𝐷	PROPN
easat-2091	66	7	|𝑢	|𝑢	PROPN
easat-2091	66	8	(	(	PUNCT
easat-2091	66	9	ℎ(g(𝑤)𝑒	ℎ(g(𝑤)𝑒	PROPN
easat-2091	66	10	𝑖𝜃))|	𝑖𝜃))|	X
easat-2091	66	11	𝑑𝐴(𝑤	𝑑𝐴(𝑤	PROPN
easat-2091	66	12	)	)	PUNCT
easat-2091	67	1	𝜋	𝜋	PRON
easat-2091	67	2	𝑑𝜃	𝑑𝜃	ADP
easat-2091	67	3	2𝜋	2𝜋	NUM
easat-2091	67	4	2𝜋	2𝜋	NOUN
easat-2091	67	5	0	0	NUM
easat-2091	68	1	=	=	SYM
easat-2091	68	2	∫	∫	PROPN
easat-2091	69	1	∫𝐷|𝑢(𝑧)|	∫𝐷|𝑢(𝑧)|	PROPN
easat-2091	69	2	|(𝑓𝜃	|(𝑓𝜃	PROPN
easat-2091	69	3	−1	−1	NOUN
easat-2091	69	4	)	)	PUNCT
easat-2091	69	5	/	/	PUNCT
easat-2091	70	1	(	(	PUNCT
easat-2091	70	2	𝑧)|	𝑧)|	PROPN
easat-2091	70	3	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
easat-2091	70	4	)	)	PUNCT
easat-2091	71	1	𝜋	𝜋	PRON
easat-2091	71	2	𝑑𝜃	𝑑𝜃	ADP
easat-2091	71	3	2𝜋	2𝜋	NUM
easat-2091	71	4	2𝜋	2𝜋	NOUN
easat-2091	71	5	0	0	NUM
easat-2091	71	6	≤	≤	NUM
easat-2091	71	7	𝐾2∫𝐷|𝑢(𝑧)|	𝐾2∫𝐷|𝑢(𝑧)|	PUNCT
easat-2091	71	8	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
easat-2091	71	9	)	)	PUNCT
easat-2091	72	1	𝜋	𝜋	NOUN
easat-2091	72	2	≤	≤	NUM
easat-2091	72	3	∞	∞	NUM
easat-2091	72	4	that	that	PRON
easat-2091	72	5	is	be	AUX
easat-2091	72	6	apply	apply	VERB
easat-2091	72	7	fubini	fubini	NOUN
easat-2091	72	8	's	's	PART
easat-2091	72	9	theorem	theorem	NOUN
easat-2091	72	10	to	to	ADP
easat-2091	72	11	eq(6	eq(6	PROPN
easat-2091	72	12	)	)	PUNCT
easat-2091	72	13	,	,	PUNCT
easat-2091	72	14	getting	get	VERB
easat-2091	72	15	∫𝐷𝑣	∫𝐷𝑣	NOUN
easat-2091	72	16	°	°	ADP
easat-2091	72	17	g	g	ADP
easat-2091	72	18	𝑑𝐴	𝑑𝐴	NUM
easat-2091	72	19	𝜋	𝜋	NOUN
easat-2091	73	1	=	=	NOUN
easat-2091	73	2	∫	∫	PROPN
easat-2091	73	3	∫𝐷𝑢	∫𝐷𝑢	X
easat-2091	73	4	(	(	PUNCT
easat-2091	73	5	ℎ(g(𝑤)𝑒	ℎ(g(𝑤)𝑒	PROPN
easat-2091	73	6	𝑖𝜃	𝑖𝜃	PROPN
easat-2091	73	7	)	)	PUNCT
easat-2091	73	8	)	)	PUNCT
easat-2091	74	1	𝑑𝐴(𝑤	𝑑𝐴(𝑤	X
easat-2091	74	2	)	)	PUNCT
easat-2091	75	1	𝜋	𝜋	PRON
easat-2091	75	2	𝑑𝜃	𝑑𝜃	ADP
easat-2091	75	3	2𝜋	2𝜋	NUM
easat-2091	75	4	2𝜋	2𝜋	NOUN
easat-2091	75	5	0	0	PUNCT
easat-2091	76	1	=	=	SYM
easat-2091	76	2	∫	∫	PROPN
easat-2091	76	3	∫𝐷(𝑣	∫𝐷(𝑣	PROPN
easat-2091	76	4	°	°	PROPN
easat-2091	76	5	𝑓𝜃(𝑤	𝑓𝜃(𝑤	NUM
easat-2091	76	6	)	)	PUNCT
easat-2091	76	7	)	)	PUNCT
easat-2091	77	1	𝑑𝐴(𝑤	𝑑𝐴(𝑤	X
easat-2091	77	2	)	)	PUNCT
easat-2091	78	1	𝜋	𝜋	PRON
easat-2091	78	2	𝑑𝜃	𝑑𝜃	ADP
easat-2091	78	3	2𝜋	2𝜋	NUM
easat-2091	78	4	2𝜋	2𝜋	NOUN
easat-2091	78	5	0	0	NUM
easat-2091	79	1	=	=	SYM
easat-2091	79	2	∫	∫	PROPN
easat-2091	79	3	𝑢(𝑓𝜃(0	𝑢(𝑓𝜃(0	PROPN
easat-2091	79	4	)	)	PUNCT
easat-2091	79	5	)	)	PUNCT
easat-2091	80	1	𝑑𝜃	𝑑𝜃	ADP
easat-2091	80	2	2𝜋	2𝜋	NUM
easat-2091	80	3	2𝜋	2𝜋	NOUN
easat-2091	80	4	0	0	NUM
easat-2091	81	1	=	=	SYM
easat-2091	81	2	∫	∫	PROPN
easat-2091	81	3	𝑢	𝑢	X
easat-2091	81	4	(	(	PUNCT
easat-2091	81	5	ℎ(g(0)𝑒𝑖𝜃	ℎ(g(0)𝑒𝑖𝜃	ADJ
easat-2091	81	6	)	)	PUNCT
easat-2091	81	7	)	)	PUNCT
easat-2091	81	8	𝑑𝜃	𝑑𝜃	ADP
easat-2091	81	9	2𝜋	2𝜋	NUM
easat-2091	81	10	2𝜋	2𝜋	NOUN
easat-2091	81	11	0	0	PUNCT
easat-2091	82	1	=	=	SYM
easat-2091	82	2	ℛ(𝑢	ℛ(𝑢	NUM
easat-2091	82	3	°	°	NUM
easat-2091	82	4	ℎ)(g(0	ℎ)(g(0	NOUN
easat-2091	82	5	)	)	PUNCT
easat-2091	82	6	)	)	PUNCT
easat-2091	83	1	=	=	PUNCT
easat-2091	83	2	𝑣(g(0	𝑣(g(0	X
easat-2091	83	3	)	)	PUNCT
easat-2091	83	4	)	)	PUNCT
easat-2091	83	5	thus	thus	ADV
easat-2091	83	6	𝑣	𝑣	PRON
easat-2091	83	7	is	be	AUX
easat-2091	83	8	a	a	DET
easat-2091	83	9	continuous	continuous	ADJ
easat-2091	83	10	function	function	NOUN
easat-2091	83	11	on	on	ADP
easat-2091	83	12	�	�	PROPN
easat-2091	83	13	̅	̅	NOUN
easat-2091	83	14	�	�	NOUN
easat-2091	83	15	that	that	PRON
easat-2091	83	16	has	have	VERB
easat-2091	83	17	the	the	DET
easat-2091	83	18	area	area	NOUN
easat-2091	83	19	version	version	NOUN
easat-2091	83	20	of	of	ADP
easat-2091	83	21	the	the	DET
easat-2091	83	22	invariant	invariant	ADJ
easat-2091	83	23	mean	mean	NOUN
easat-2091	83	24	value	value	NOUN
easat-2091	83	25	property	property	NOUN
easat-2091	83	26	.	.	PUNCT
easat-2091	84	1	hence	hence	ADV
easat-2091	84	2	𝑣	𝑣	PRON
easat-2091	84	3	is	be	AUX
easat-2091	84	4	harmonic	harmonic	ADJ
easat-2091	84	5	on	on	ADP
easat-2091	84	6	𝐷	𝐷	PROPN
easat-2091	84	7	[	[	X
easat-2091	84	8	4,5	4,5	NUM
easat-2091	84	9	]	]	PUNCT
easat-2091	84	10	.	.	PUNCT
easat-2091	85	1	because	because	SCONJ
easat-2091	85	2	𝑣	𝑣	PRON
easat-2091	85	3	is	be	AUX
easat-2091	85	4	also	also	ADV
easat-2091	85	5	a	a	DET
easat-2091	85	6	radial	radial	ADJ
easat-2091	85	7	function	function	NOUN
easat-2091	85	8	,	,	PUNCT
easat-2091	85	9	the	the	DET
easat-2091	85	10	mean	mean	ADJ
easat-2091	85	11	value	value	NOUN
easat-2091	85	12	property	property	NOUN
easat-2091	85	13	implies	imply	VERB
easat-2091	85	14	that	that	SCONJ
easat-2091	85	15	𝑣	𝑣	PRON
easat-2091	85	16	is	be	AUX
easat-2091	85	17	a	a	DET
easat-2091	85	18	constant	constant	ADJ
easat-2091	85	19	function	function	NOUN
easat-2091	85	20	on	on	ADP
easat-2091	85	21	𝐷	𝐷	PROPN
easat-2091	85	22	,	,	PUNCT
easat-2091	85	23	with	with	ADP
easat-2091	85	24	value	value	NOUN
easat-2091	85	25	𝑣(0	𝑣(0	NUM
easat-2091	85	26	)	)	PUNCT
easat-2091	85	27	.	.	PUNCT
easat-2091	86	1	recall	recall	PROPN
easat-2091	86	2	that𝑣	that𝑣	PROPN
easat-2091	86	3	=	=	SYM
easat-2091	86	4	ℛ(𝑢	ℛ(𝑢	NUM
easat-2091	86	5	°	°	NUM
easat-2091	86	6	ℎ	ℎ	NOUN
easat-2091	86	7	)	)	PUNCT
easat-2091	86	8	,	,	PUNCT
easat-2091	86	9	so	so	ADV
easat-2091	86	10	∫	∫	PROPN
easat-2091	86	11	(	(	PUNCT
easat-2091	86	12	𝑢	𝑢	PROPN
easat-2091	86	13	°	°	PRON
easat-2091	86	14	ℎ)(𝑟𝑒𝑖𝜃	ℎ)(𝑟𝑒𝑖𝜃	X
easat-2091	86	15	)	)	PUNCT
easat-2091	86	16	𝑑𝜃	𝑑𝜃	ADP
easat-2091	86	17	2𝜋	2𝜋	NUM
easat-2091	86	18	2𝜋	2𝜋	NOUN
easat-2091	86	19	0	0	PUNCT
easat-2091	86	20	=	=	SYM
easat-2091	86	21	𝑢(ℎ(0	𝑢(ℎ(0	NOUN
easat-2091	86	22	)	)	PUNCT
easat-2091	86	23	)	)	PUNCT
easat-2091	86	24	for	for	ADP
easat-2091	86	25	every	every	DET
easat-2091	86	26	𝑟	𝑟	PRON
easat-2091	86	27	∈	∈	PROPN
easat-2091	87	1	[	[	X
easat-2091	87	2	0	0	NUM
easat-2091	87	3	,	,	PUNCT
easat-2091	87	4	1	1	NUM
easat-2091	87	5	)	)	PUNCT
easat-2091	87	6	and	and	CCONJ
easat-2091	87	7	for	for	ADP
easat-2091	87	8	each	each	DET
easat-2091	87	9	ℎ	ℎ	ADP
easat-2091	87	10	∈	∈	PROPN
easat-2091	87	11	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PROPN
easat-2091	87	12	)	)	PUNCT
easat-2091	87	13	.	.	PUNCT
easat-2091	88	1	in	in	ADP
easat-2091	88	2	other	other	ADJ
easat-2091	88	3	words	word	NOUN
easat-2091	88	4	,	,	PUNCT
easat-2091	88	5	𝑢	𝑢	PROPN
easat-2091	88	6	has	have	VERB
easat-2091	88	7	the	the	DET
easat-2091	88	8	invariant	invariant	ADJ
easat-2091	88	9	mean	mean	NOUN
easat-2091	88	10	value	value	NOUN
easat-2091	88	11	property	property	NOUN
easat-2091	88	12	.	.	PUNCT
easat-2091	89	1	thus	thus	ADV
easat-2091	89	2	in	in	ADP
easat-2091	89	3	[	[	X
easat-2091	89	4	4	4	NUM
easat-2091	89	5	]	]	PUNCT
easat-2091	89	6	,	,	PUNCT
easat-2091	89	7	𝑢	𝑢	PROPN
easat-2091	89	8	is	be	AUX
easat-2091	89	9	harmonic	harmonic	ADJ
easat-2091	89	10	on	on	ADP
easat-2091	89	11	𝐷.	𝐷.	PROPN
easat-2091	89	12	as	as	SCONJ
easat-2091	89	13	mentioned	mention	VERB
easat-2091	89	14	earlier	early	ADV
easat-2091	89	15	,	,	PUNCT
easat-2091	89	16	it	it	PRON
easat-2091	89	17	is	be	AUX
easat-2091	89	18	unknown	unknown	ADJ
easat-2091	89	19	whether	whether	SCONJ
easat-2091	89	20	lemma	lemma	PROPN
easat-2091	89	21	2	2	NUM
easat-2091	89	22	remains	remain	VERB
easat-2091	89	23	true	true	ADJ
easat-2091	89	24	if	if	SCONJ
easat-2091	89	25	eq(5	eq(5	NOUN
easat-2091	89	26	)	)	PUNCT
easat-2091	89	27	is	be	AUX
easat-2091	89	28	deleted	delete	VERB
easat-2091	89	29	.	.	PUNCT
easat-2091	90	1	we	we	PRON
easat-2091	90	2	believe	believe	VERB
easat-2091	90	3	that	that	SCONJ
easat-2091	90	4	the	the	DET
easat-2091	90	5	following	follow	VERB
easat-2091	90	6	proposition	proposition	NOUN
easat-2091	90	7	,	,	PUNCT
easat-2091	90	8	which	which	PRON
easat-2091	90	9	reduces	reduce	VERB
easat-2091	90	10	this	this	DET
easat-2091	90	11	question	question	NOUN
easat-2091	90	12	to	to	ADP
easat-2091	90	13	a	a	DET
easat-2091	90	14	tempting	tempting	ADJ
easat-2091	90	15	integral	integral	ADJ
easat-2091	90	16	equation	equation	NOUN
easat-2091	90	17	,	,	PUNCT
easat-2091	90	18	is	be	AUX
easat-2091	90	19	the	the	DET
easat-2091	90	20	best	good	ADJ
easat-2091	90	21	way	way	NOUN
easat-2091	90	22	to	to	PART
easat-2091	90	23	attack	attack	VERB
easat-2091	90	24	this	this	DET
easat-2091	90	25	problem	problem	NOUN
easat-2091	90	26	.	.	PUNCT
easat-2091	91	1	patrick	patrick	PROPN
easat-2091	91	2	ahern	ahern	PROPN
easat-2091	91	3	and	and	CCONJ
easat-2091	91	4	walter	walter	PROPN
easat-2091	91	5	rudin	rudin	PROPN
easat-2091	91	6	also	also	ADV
easat-2091	91	7	independently	independently	ADV
easat-2091	91	8	proved	prove	VERB
easat-2091	91	9	lemma	lemma	PROPN
easat-2091	91	10	2	2	NUM
easat-2091	91	11	and	and	CCONJ
easat-2091	91	12	proposition	proposition	NOUN
easat-2091	91	13	3	3	NUM
easat-2091	91	14	at	at	ADP
easat-2091	91	15	about	about	ADV
easat-2091	91	16	the	the	DET
easat-2091	91	17	same	same	ADJ
easat-2091	91	18	time	time	NOUN
easat-2091	91	19	we	we	PRON
easat-2091	91	20	did	do	VERB
easat-2091	91	21	.	.	PUNCT
easat-2091	92	1	proposition	proposition	NOUN
easat-2091	92	2	3	3	NUM
easat-2091	92	3	.	.	PUNCT
easat-2091	92	4	suppose	suppose	VERB
easat-2091	92	5	that	that	SCONJ
easat-2091	92	6	the	the	DET
easat-2091	92	7	constant	constant	ADJ
easat-2091	92	8	functions	function	NOUN
easat-2091	92	9	are	be	AUX
easat-2091	92	10	the	the	DET
easat-2091	92	11	only	only	ADJ
easat-2091	92	12	functions	function	NOUN
easat-2091	92	13	𝑉	𝑉	PROPN
easat-2091	92	14	∈	∈	PROPN
easat-2091	92	15	𝐶([0,1	𝐶([0,1	NOUN
easat-2091	92	16	)	)	PUNCT
easat-2091	92	17	)	)	PUNCT
easat-2091	93	1	∩	∩	PROPN
easat-2091	93	2	𝐿/[0,1	𝐿/[0,1	PROPN
easat-2091	93	3	]	]	X
easat-2091	93	4	such	such	ADJ
easat-2091	93	5	that	that	SCONJ
easat-2091	93	6	397	397	NUM
easat-2091	93	7	edelweiss	edelweiss	PROPN
easat-2091	93	8	applied	apply	VERB
easat-2091	93	9	science	science	NOUN
easat-2091	93	10	and	and	CCONJ
easat-2091	93	11	technology	technology	NOUN
easat-2091	93	12	issn	issn	PROPN
easat-2091	93	13	:	:	PUNCT
easat-2091	93	14	2576	2576	NUM
easat-2091	93	15	-	-	SYM
easat-2091	93	16	8484	8484	NUM
easat-2091	93	17	vol	vol	NOUN
easat-2091	93	18	.	.	PROPN
easat-2091	94	1	8	8	NUM
easat-2091	94	2	,	,	PUNCT
easat-2091	94	3	no	no	INTJ
easat-2091	94	4	.	.	NOUN
easat-2091	95	1	6	6	NUM
easat-2091	95	2	:	:	SYM
easat-2091	95	3	394	394	NUM
easat-2091	95	4	-	-	SYM
easat-2091	95	5	400	400	NUM
easat-2091	95	6	,	,	PUNCT
easat-2091	95	7	2024	2024	NUM
easat-2091	95	8	doi	doi	NOUN
easat-2091	95	9	:	:	PUNCT
easat-2091	95	10	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	95	11	©	©	ADP
easat-2091	95	12	2024	2024	NUM
easat-2091	95	13	by	by	ADP
easat-2091	95	14	the	the	DET
easat-2091	95	15	author	author	NOUN
easat-2091	95	16	;	;	PUNCT
easat-2091	96	1	licensee	licensee	PROPN
easat-2091	96	2	learning	learn	VERB
easat-2091	96	3	gate	gate	NOUN
easat-2091	96	4	𝑣(𝑡	𝑣(𝑡	NOUN
easat-2091	96	5	)	)	PUNCT
easat-2091	96	6	=	=	PUNCT
easat-2091	96	7	(	(	PUNCT
easat-2091	96	8	1	1	NUM
easat-2091	96	9	−	−	PRON
easat-2091	96	10	𝑡)2	𝑡)2	PROPN
easat-2091	96	11	∫	∫	PROPN
easat-2091	96	12	1+𝑡𝑠	1+𝑡𝑠	NUM
easat-2091	96	13	(	(	PUNCT
easat-2091	96	14	1−𝑡𝑠)2	1−𝑡𝑠)2	NUM
easat-2091	96	15	1	1	NUM
easat-2091	96	16	0	0	NUM
easat-2091	96	17	𝑉(𝑠)𝑑𝑠	𝑉(𝑠)𝑑𝑠	NOUN
easat-2091	96	18	,	,	PUNCT
easat-2091	96	19	for	for	ADP
easat-2091	96	20	every	every	DET
easat-2091	96	21	𝑡	𝑡	PROPN
easat-2091	96	22	∈	∈	PROPN
easat-2091	96	23	[	[	X
easat-2091	96	24	0,1	0,1	NUM
easat-2091	96	25	)	)	PUNCT
easat-2091	96	26	(	(	PUNCT
easat-2091	96	27	7	7	X
easat-2091	96	28	)	)	PUNCT
easat-2091	96	29	then	then	ADV
easat-2091	96	30	every	every	DET
easat-2091	96	31	function	function	NOUN
easat-2091	96	32	in	in	ADP
easat-2091	96	33	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	96	34	)	)	PUNCT
easat-2091	96	35	∩	∩	ADJ
easat-2091	96	36	𝐿/(𝐷	𝐿/(𝐷	NOUN
easat-2091	96	37	,	,	PUNCT
easat-2091	96	38	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	96	39	)	)	PUNCT
easat-2091	96	40	having	have	VERB
easat-2091	96	41	the	the	DET
easat-2091	96	42	area	area	NOUN
easat-2091	96	43	version	version	NOUN
easat-2091	96	44	of	of	ADP
easat-2091	96	45	the	the	DET
easat-2091	96	46	invariant	invariant	ADJ
easat-2091	96	47	mean	mean	NOUN
easat-2091	96	48	value	value	NOUN
easat-2091	96	49	property	property	NOUN
easat-2091	96	50	is	be	AUX
easat-2091	96	51	harmonic	harmonic	ADJ
easat-2091	96	52	.	.	PUNCT
easat-2091	97	1	proof	proof	NOUN
easat-2091	97	2	:	:	PUNCT
easat-2091	97	3	first	first	ADV
easat-2091	97	4	,	,	PUNCT
easat-2091	97	5	suppose	suppose	VERB
easat-2091	97	6	that	that	SCONJ
easat-2091	97	7	𝑣	𝑣	PRON
easat-2091	97	8	is	be	AUX
easat-2091	97	9	a	a	DET
easat-2091	97	10	radial	radial	ADJ
easat-2091	97	11	function	function	NOUN
easat-2091	97	12	in	in	ADP
easat-2091	97	13	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	97	14	)	)	PUNCT
easat-2091	97	15	∩	∩	ADJ
easat-2091	97	16	𝐿/(𝐷	𝐿/(𝐷	NOUN
easat-2091	97	17	,	,	PUNCT
easat-2091	97	18	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	97	19	)	)	PUNCT
easat-2091	97	20	having	have	VERB
easat-2091	97	21	the	the	DET
easat-2091	97	22	area	area	NOUN
easat-2091	97	23	version	version	NOUN
easat-2091	97	24	of	of	ADP
easat-2091	97	25	the	the	DET
easat-2091	97	26	invariant	invariant	ADJ
easat-2091	97	27	mean	mean	NOUN
easat-2091	97	28	value	value	NOUN
easat-2091	97	29	property	property	NOUN
easat-2091	97	30	.	.	PUNCT
easat-2091	98	1	we	we	PRON
easat-2091	98	2	will	will	AUX
easat-2091	98	3	show	show	VERB
easat-2091	98	4	that	that	SCONJ
easat-2091	98	5	𝑣	𝑣	PRON
easat-2091	98	6	is	be	AUX
easat-2091	98	7	constant	constant	ADJ
easat-2091	98	8	on	on	ADP
easat-2091	98	9	𝐷.	𝐷.	PROPN
easat-2091	98	10	for	for	ADP
easat-2091	98	11	𝛼	𝛼	NOUN
easat-2091	98	12	∈	∈	PROPN
easat-2091	99	1	[	[	X
easat-2091	99	2	0,1	0,1	NUM
easat-2091	99	3	)	)	PUNCT
easat-2091	99	4	,	,	PUNCT
easat-2091	99	5	let	let	VERB
easat-2091	99	6	ℎ𝛼	ℎ𝛼	PRON
easat-2091	99	7	∈	∈	PROPN
easat-2091	99	8	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	99	9	)	)	PUNCT
easat-2091	99	10	be	be	AUX
easat-2091	99	11	defined	define	VERB
easat-2091	99	12	by	by	ADP
easat-2091	99	13	ℎ𝛼(𝑧	ℎ𝛼(𝑧	ADJ
easat-2091	99	14	)	)	PUNCT
easat-2091	99	15	=	=	SYM
easat-2091	99	16	𝛼−𝑧	𝛼−𝑧	PROPN
easat-2091	99	17	1−𝛼𝑧	1−𝛼𝑧	NUM
easat-2091	99	18	note	note	VERB
easat-2091	99	19	that	that	SCONJ
easat-2091	99	20	ℎ𝛼	ℎ𝛼	PROPN
easat-2091	99	21	is	be	AUX
easat-2091	99	22	its	its	PRON
easat-2091	99	23	own	own	ADJ
easat-2091	99	24	inverse	inverse	NOUN
easat-2091	99	25	under	under	ADP
easat-2091	99	26	composition	composition	NOUN
easat-2091	99	27	.	.	PUNCT
easat-2091	100	1	for	for	ADP
easat-2091	100	2	each	each	DET
easat-2091	100	3	𝛼	𝛼	PROPN
easat-2091	100	4	∈	∈	PROPN
easat-2091	100	5	[	[	X
easat-2091	100	6	0,1	0,1	NUM
easat-2091	100	7	)	)	PUNCT
easat-2091	100	8	we	we	PRON
easat-2091	100	9	have	have	VERB
easat-2091	100	10	𝑣(𝛼	𝑣(𝛼	PROPN
easat-2091	100	11	)	)	PUNCT
easat-2091	100	12	=	=	SYM
easat-2091	101	1	∫𝐷((𝑣	∫𝐷((𝑣	NUM
easat-2091	101	2	°	°	NUM
easat-2091	101	3	ℎ𝛼))(𝑧	ℎ𝛼))(𝑧	NUM
easat-2091	101	4	)	)	PUNCT
easat-2091	101	5	𝑑𝐴(𝑧	𝑑𝐴(𝑧	PROPN
easat-2091	101	6	)	)	PUNCT
easat-2091	101	7	𝜋	𝜋	NOUN
easat-2091	101	8	=	=	PUNCT
easat-2091	101	9	∫𝐷𝑣(𝑤	∫𝐷𝑣(𝑤	NUM
easat-2091	101	10	)	)	PUNCT
easat-2091	102	1	|ℎ𝛼	|ℎ𝛼	NOUN
easat-2091	102	2	/	/	PUNCT
easat-2091	103	1	(	(	PUNCT
easat-2091	103	2	𝑤)|	𝑤)|	NOUN
easat-2091	103	3	2	2	NUM
easat-2091	103	4	𝑑𝐴(𝑧	𝑑𝐴(𝑧	NOUN
easat-2091	103	5	)	)	PUNCT
easat-2091	103	6	𝜋	𝜋	NOUN
easat-2091	103	7	=	=	PUNCT
easat-2091	103	8	(	(	PUNCT
easat-2091	103	9	1	1	NUM
easat-2091	103	10	−	−	PRON
easat-2091	103	11	𝛼2)2	𝛼2)2	NUM
easat-2091	103	12	∫	∫	NOUN
easat-2091	103	13	𝑣(𝑟	𝑣(𝑟	PROPN
easat-2091	103	14	)	)	PUNCT
easat-2091	103	15	1	1	NUM
easat-2091	103	16	0	0	NUM
easat-2091	104	1	𝑟	𝑟	NOUN
easat-2091	104	2	∫	∫	NOUN
easat-2091	104	3	1	1	NUM
easat-2091	104	4	|1−𝛼𝑟𝑒𝑖𝜃|	|1−𝛼𝑟𝑒𝑖𝜃|	NOUN
easat-2091	104	5	4	4	NUM
easat-2091	104	6	2𝜋	2𝜋	NUM
easat-2091	104	7	0	0	NUM
easat-2091	104	8	𝑑𝜃	𝑑𝜃	ADP
easat-2091	104	9	2𝜋	2𝜋	NOUN
easat-2091	104	10	𝑑𝑟	𝑑𝑟	NOUN
easat-2091	104	11	=	=	PUNCT
easat-2091	104	12	(	(	PUNCT
easat-2091	104	13	1	1	NUM
easat-2091	104	14	−	−	NOUN
easat-2091	104	15	𝛼2)2	𝛼2)2	NUM
easat-2091	104	16	∫	∫	PROPN
easat-2091	104	17	𝑟(1+𝑎2𝑟2	𝑟(1+𝑎2𝑟2	PROPN
easat-2091	104	18	)	)	PUNCT
easat-2091	104	19	(	(	PUNCT
easat-2091	104	20	1−𝑎2𝑟2)3	1−𝑎2𝑟2)3	NUM
easat-2091	104	21	1	1	NUM
easat-2091	104	22	0	0	NUM
easat-2091	104	23	𝑣(𝑟)𝑑𝑟	𝑣(𝑟)𝑑𝑟	NOUN
easat-2091	104	24	=	=	PUNCT
easat-2091	104	25	(	(	PUNCT
easat-2091	104	26	1	1	NUM
easat-2091	104	27	−	−	NOUN
easat-2091	104	28	𝛼2)2	𝛼2)2	NUM
easat-2091	104	29	∫	∫	PROPN
easat-2091	104	30	1+𝑎2𝑟2	1+𝑎2𝑟2	NUM
easat-2091	104	31	(	(	PUNCT
easat-2091	104	32	1−𝑎2𝑠)3	1−𝑎2𝑠)3	NUM
easat-2091	104	33	1	1	NUM
easat-2091	104	34	0	0	NUM
easat-2091	104	35	𝑣(√𝑠)𝑑𝑠	𝑣(√𝑠)𝑑𝑠	NOUN
easat-2091	104	36	in	in	ADP
easat-2091	104	37	the	the	DET
easat-2091	104	38	above	above	ADJ
easat-2091	104	39	equation	equation	NOUN
easat-2091	104	40	,	,	PUNCT
easat-2091	104	41	replace	replace	VERB
easat-2091	104	42	𝛼	𝛼	NOUN
easat-2091	104	43	with	with	ADP
easat-2091	104	44	√𝑡	√𝑡	PRON
easat-2091	104	45	and	and	CCONJ
easat-2091	104	46	define	define	VERB
easat-2091	104	47	a	a	DET
easat-2091	104	48	function	function	NOUN
easat-2091	104	49	𝑉	𝑉	PROPN
easat-2091	104	50	on	on	ADP
easat-2091	104	51	[	[	X
easat-2091	104	52	0	0	NUM
easat-2091	104	53	,	,	PUNCT
easat-2091	104	54	1	1	NUM
easat-2091	104	55	)	)	PUNCT
easat-2091	104	56	by𝑉(𝑡	by𝑉(𝑡	PROPN
easat-2091	104	57	)	)	PUNCT
easat-2091	104	58	=	=	PUNCT
easat-2091	104	59	𝑣(√𝑡	𝑣(√𝑡	X
easat-2091	104	60	)	)	PUNCT
easat-2091	104	61	,	,	PUNCT
easat-2091	104	62	transforming	transform	VERB
easat-2091	104	63	the	the	DET
easat-2091	104	64	above	above	ADJ
easat-2091	104	65	equation	equation	NOUN
easat-2091	104	66	into	into	ADP
easat-2091	104	67	eq	eq	NOUN
easat-2091	104	68	(	(	PUNCT
easat-2091	104	69	7	7	NUM
easat-2091	104	70	)	)	PUNCT
easat-2091	104	71	.	.	PUNCT
easat-2091	105	1	hence	hence	ADV
easat-2091	105	2	𝑉	𝑉	PROPN
easat-2091	105	3	is	be	AUX
easat-2091	105	4	constant	constant	ADJ
easat-2091	105	5	on	on	ADP
easat-2091	105	6	[	[	X
easat-2091	105	7	0,1	0,1	NUM
easat-2091	105	8	)	)	PUNCT
easat-2091	105	9	,	,	PUNCT
easat-2091	105	10	and	and	CCONJ
easat-2091	105	11	thus	thus	ADV
easat-2091	105	12	so	so	ADV
easat-2091	105	13	is	be	AUX
easat-2091	105	14	𝑣	𝑣	PRON
easat-2091	105	15	,	,	PUNCT
easat-2091	105	16	as	as	SCONJ
easat-2091	105	17	claimed	claim	VERB
easat-2091	105	18	.	.	PUNCT
easat-2091	106	1	to	to	PART
easat-2091	106	2	complete	complete	VERB
easat-2091	106	3	the	the	DET
easat-2091	106	4	proof	proof	NOUN
easat-2091	106	5	,	,	PUNCT
easat-2091	106	6	now	now	ADV
easat-2091	106	7	suppose	suppose	VERB
easat-2091	106	8	that	that	SCONJ
easat-2091	106	9	𝑢	𝑢	PROPN
easat-2091	106	10	is	be	AUX
easat-2091	106	11	a	a	DET
easat-2091	106	12	function	function	NOUN
easat-2091	106	13	in𝐶(𝐷	in𝐶(𝐷	PROPN
easat-2091	106	14	)	)	PUNCT
easat-2091	106	15	∩	∩	ADJ
easat-2091	106	16	𝐿/(𝐷	𝐿/(𝐷	NOUN
easat-2091	106	17	,	,	PUNCT
easat-2091	106	18	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	106	19	)	)	PUNCT
easat-2091	106	20	,	,	PUNCT
easat-2091	106	21	having	have	VERB
easat-2091	106	22	the	the	DET
easat-2091	106	23	area	area	NOUN
easat-2091	106	24	version	version	NOUN
easat-2091	106	25	of	of	ADP
easat-2091	106	26	the	the	DET
easat-2091	106	27	invariant	invariant	ADJ
easat-2091	106	28	mean	mean	NOUN
easat-2091	106	29	value	value	NOUN
easat-2091	106	30	property	property	NOUN
easat-2091	106	31	.	.	PUNCT
easat-2091	107	1	let	let	VERB
easat-2091	107	2	ℎ	ℎ	PROPN
easat-2091	107	3	∈	∈	PROPN
easat-2091	107	4	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PRON
easat-2091	107	5	)	)	PUNCT
easat-2091	107	6	.	.	PUNCT
easat-2091	108	1	clearly	clearly	ADV
easat-2091	108	2	,	,	PUNCT
easat-2091	108	3	ℛ(𝑢	ℛ(𝑢	PROPN
easat-2091	108	4	°	°	NOUN
easat-2091	108	5	ℎ	ℎ	NOUN
easat-2091	108	6	)	)	PUNCT
easat-2091	108	7	is	be	AUX
easat-2091	108	8	a	a	DET
easat-2091	108	9	radial	radial	ADJ
easat-2091	108	10	function	function	NOUN
easat-2091	108	11	on𝐷	on𝐷	PROPN
easat-2091	108	12	,	,	PUNCT
easat-2091	108	13	and	and	CCONJ
easat-2091	108	14	,	,	PUNCT
easat-2091	108	15	as	as	SCONJ
easat-2091	108	16	shown	show	VERB
easat-2091	108	17	in	in	ADP
easat-2091	108	18	the	the	DET
easat-2091	108	19	proof	proof	NOUN
easat-2091	108	20	of	of	ADP
easat-2091	108	21	lemma	lemma	PROPN
easat-2091	108	22	2	2	NUM
easat-2091	108	23	,	,	PUNCT
easat-2091	108	24	has	have	AUX
easat-2091	108	25	the	the	DET
easat-2091	108	26	area	area	NOUN
easat-2091	108	27	version	version	NOUN
easat-2091	108	28	of	of	ADP
easat-2091	108	29	the	the	DET
easat-2091	108	30	invariant	invariant	ADJ
easat-2091	108	31	mean	mean	NOUN
easat-2091	108	32	value	value	NOUN
easat-2091	108	33	property	property	NOUN
easat-2091	108	34	.	.	PUNCT
easat-2091	109	1	by	by	ADP
easat-2091	109	2	the	the	DET
easat-2091	109	3	above	above	ADJ
easat-2091	109	4	paragraph	paragraph	NOUN
easat-2091	109	5	,	,	PUNCT
easat-2091	109	6	ℛ(𝑢	ℛ(𝑢	NUM
easat-2091	109	7	°	°	NOUN
easat-2091	109	8	ℎ	ℎ	NOUN
easat-2091	109	9	)	)	PUNCT
easat-2091	109	10	is	be	AUX
easat-2091	109	11	constant	constant	ADJ
easat-2091	109	12	on𝐷.	on𝐷.	ADP
easat-2091	109	13	in	in	ADP
easat-2091	109	14	particular,ℛ(𝑢	particular,ℛ(𝑢	PROPN
easat-2091	109	15	°	°	ADP
easat-2091	109	16	ℎ	ℎ	PROPN
easat-2091	109	17	)	)	PUNCT
easat-2091	109	18	∈	∈	PROPN
easat-2091	109	19	𝐶(	𝐶(	NOUN
easat-2091	109	20	�	�	PROPN
easat-2091	109	21	̅	̅	NOUN
easat-2091	109	22	�	�	NOUN
easat-2091	109	23	)	)	PUNCT
easat-2091	109	24	,	,	PUNCT
easat-2091	109	25	and	and	CCONJ
easat-2091	109	26	so	so	ADV
easat-2091	109	27	by	by	ADP
easat-2091	109	28	lemma	lemma	PROPN
easat-2091	109	29	2	2	NUM
easat-2091	109	30	,	,	PUNCT
easat-2091	109	31	𝑢	𝑢	PROPN
easat-2091	109	32	is	be	AUX
easat-2091	109	33	harmonic	harmonic	ADJ
easat-2091	109	34	.	.	PUNCT
easat-2091	110	1	3	3	X
easat-2091	110	2	.	.	X
easat-2091	110	3	the	the	DET
easat-2091	110	4	toeplitz	toeplitz	NOUN
easat-2091	110	5	operators	operator	NOUN
easat-2091	110	6	forℎ	forℎ	VERB
easat-2091	110	7	∈	∈	PROPN
easat-2091	110	8	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	110	9	)	)	PUNCT
easat-2091	110	10	,	,	PUNCT
easat-2091	110	11	define	define	VERB
easat-2091	110	12	an	an	DET
easat-2091	110	13	operator	operator	NOUN
easat-2091	110	14	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	110	15	on	on	ADP
easat-2091	110	16	𝐿𝑎	𝐿𝑎	ADP
easat-2091	110	17	2	2	NUM
easat-2091	110	18	by	by	ADP
easat-2091	110	19	𝑈ℎ𝑓	𝑈ℎ𝑓	PROPN
easat-2091	110	20	=	=	SYM
easat-2091	110	21	(	(	PUNCT
easat-2091	110	22	𝑢	𝑢	NOUN
easat-2091	110	23	°	°	NOUN
easat-2091	110	24	ℎ)ℎ	ℎ)ℎ	NOUN
easat-2091	110	25	a	a	DET
easat-2091	110	26	simple	simple	ADJ
easat-2091	110	27	computation	computation	NOUN
easat-2091	110	28	shows	show	VERB
easat-2091	110	29	that	that	SCONJ
easat-2091	110	30	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	110	31	is	be	AUX
easat-2091	110	32	a	a	DET
easat-2091	110	33	unitary	unitary	ADJ
easat-2091	110	34	operator	operator	NOUN
easat-2091	110	35	from	from	ADP
easat-2091	110	36	𝐿𝑎	𝐿𝑎	ADP
easat-2091	110	37	2	2	NUM
easat-2091	110	38	onto𝐿𝑎	onto𝐿𝑎	NUM
easat-2091	110	39	2	2	NUM
easat-2091	110	40	,	,	PUNCT
easat-2091	110	41	with	with	ADP
easat-2091	110	42	inverse	inverse	NOUN
easat-2091	110	43	𝑈ℎ−1	𝑈ℎ−1	NOUN
easat-2091	110	44	.	.	PUNCT
easat-2091	111	1	in	in	ADP
easat-2091	111	2	the	the	DET
easat-2091	111	3	following	follow	VERB
easat-2091	111	4	lemma	lemma	PROPN
easat-2091	111	5	that	that	DET
easat-2091	111	6	proof	proof	NOUN
easat-2091	111	7	of	of	ADP
easat-2091	111	8	theorem	theorem	NOUN
easat-2091	111	9	1	1	NUM
easat-2091	111	10	.	.	PUNCT
easat-2091	111	11	lemma	lemma	PROPN
easat-2091	111	12	4	4	X
easat-2091	111	13	.	.	PUNCT
easat-2091	111	14	let	let	VERB
easat-2091	111	15	ℎ	ℎ	PROPN
easat-2091	111	16	∈	∈	PROPN
easat-2091	111	17	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	NOUN
easat-2091	111	18	)	)	PUNCT
easat-2091	111	19	and	and	CCONJ
easat-2091	111	20	let(𝜑	let(𝜑	PROPN
easat-2091	111	21	+	+	NOUN
easat-2091	111	22	1	1	X
easat-2091	111	23	)	)	PUNCT
easat-2091	111	24	∈	∈	PROPN
easat-2091	111	25	𝐿∞(𝐷	𝐿∞(𝐷	NOUN
easat-2091	111	26	,	,	PUNCT
easat-2091	111	27	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	111	28	)	)	PUNCT
easat-2091	111	29	.	.	PUNCT
easat-2091	112	1	then	then	ADV
easat-2091	112	2	𝑈ℎ𝑇𝜑	𝑈ℎ𝑇𝜑	PRON
easat-2091	112	3	+	+	CCONJ
easat-2091	112	4	1𝑈ℎ	1𝑈ℎ	NUM
easat-2091	112	5	∗	∗	NOUN
easat-2091	112	6	=	=	SYM
easat-2091	112	7	𝑇(𝜑	𝑇(𝜑	PUNCT
easat-2091	112	8	+	+	NUM
easat-2091	112	9	1)	1)	NUM
easat-2091	112	10	°	°	NOUN
easat-2091	112	11	ℎ	ℎ	NOUN
easat-2091	112	12	proof	proof	NOUN
easat-2091	112	13	.	.	PUNCT
easat-2091	113	1	define	define	VERB
easat-2091	113	2	𝑉ℎ	𝑉ℎ	NOUN
easat-2091	113	3	:	:	PUNCT
easat-2091	113	4	𝐿	𝐿	PROPN
easat-2091	113	5	2(𝐷	2(𝐷	PROPN
easat-2091	113	6	,	,	PUNCT
easat-2091	113	7	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	113	8	)	)	PUNCT
easat-2091	113	9	→	→	SYM
easat-2091	113	10	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	113	11	,	,	PUNCT
easat-2091	113	12	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	113	13	)	)	PUNCT
easat-2091	113	14	by𝑉ℎ𝑓	by𝑉ℎ𝑓	VERB
easat-2091	113	15	=	=	PUNCT
easat-2091	113	16	(	(	PUNCT
easat-2091	113	17	𝑓	𝑓	NOUN
easat-2091	113	18	°	°	NOUN
easat-2091	113	19	ℎ)ℎ	ℎ)ℎ	NOUN
easat-2091	113	20	,	,	PUNCT
easat-2091	113	21	.	.	PUNCT
easat-2091	114	1	then	then	ADV
easat-2091	114	2	𝑉ℎ	𝑉ℎ	PROPN
easat-2091	114	3	is	be	AUX
easat-2091	114	4	a	a	DET
easat-2091	114	5	unitary	unitary	ADJ
easat-2091	114	6	operator	operator	NOUN
easat-2091	114	7	from	from	ADP
easat-2091	114	8	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	114	9	,	,	PUNCT
easat-2091	114	10	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	114	11	)	)	PUNCT
easat-2091	114	12	onto𝐿2(𝐷	onto𝐿2(𝐷	NOUN
easat-2091	114	13	,	,	PUNCT
easat-2091	114	14	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	114	15	)	)	PUNCT
easat-2091	114	16	.	.	PUNCT
easat-2091	115	1	obviously	obviously	ADV
easat-2091	115	2	𝑉ℎ𝐿𝑎	𝑉ℎ𝐿𝑎	PROPN
easat-2091	115	3	2	2	NUM
easat-2091	115	4	=	=	SYM
easat-2091	115	5	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	115	6	.	.	PUNCT
easat-2091	116	1	thus	thus	ADV
easat-2091	116	2	𝑉ℎ	𝑉ℎ	PROPN
easat-2091	116	3	maps	map	VERB
easat-2091	116	4	𝐿𝑎	𝐿𝑎	ADV
easat-2091	116	5	2	2	NUM
easat-2091	116	6	onto𝐿𝑎	onto𝐿𝑎	NUM
easat-2091	116	7	2	2	NUM
easat-2091	116	8	,	,	PUNCT
easat-2091	116	9	so	so	ADV
easat-2091	116	10	𝑃𝑉ℎ	𝑃𝑉ℎ	NOUN
easat-2091	116	11	=	=	SYM
easat-2091	116	12	𝑉ℎ𝑃	𝑉ℎ𝑃	PROPN
easat-2091	116	13	(	(	PUNCT
easat-2091	116	14	8)	8)	NUM
easat-2091	116	15	if	if	SCONJ
easat-2091	116	16	𝑓	𝑓	DET
easat-2091	116	17	∈	∈	NOUN
easat-2091	116	18	𝐿𝑎	𝐿𝑎	ADP
easat-2091	116	19	2	2	NUM
easat-2091	116	20	,	,	PUNCT
easat-2091	116	21	so	so	SCONJ
easat-2091	116	22	that	that	SCONJ
easat-2091	116	23	𝑇(𝜑	𝑇(𝜑	PUNCT
easat-2091	116	24	+	+	NUM
easat-2091	116	25	1)	1)	NUM
easat-2091	116	26	°	°	NUM
easat-2091	116	27	ℎ𝑈ℎ𝑓	ℎ𝑈ℎ𝑓	PROPN
easat-2091	116	28	=	=	PUNCT
easat-2091	116	29	𝑇(𝜑	𝑇(𝜑	PUNCT
easat-2091	116	30	+	+	NUM
easat-2091	116	31	1)	1)	NUM
easat-2091	116	32	°	°	NOUN
easat-2091	116	33	ℎ(𝑓	ℎ(𝑓	NOUN
easat-2091	116	34	°	°	NOUN
easat-2091	116	35	ℎ)ℎ	ℎ)ℎ	NOUN
easat-2091	116	36	,	,	PUNCT
easat-2091	116	37	=	=	SYM
easat-2091	116	38	𝑃[((𝜑	𝑃[((𝜑	PROPN
easat-2091	116	39	+	+	CCONJ
easat-2091	116	40	1)	1)	NUM
easat-2091	116	41	°	°	NOUN
easat-2091	116	42	ℎ)(𝑓	ℎ)(𝑓	NOUN
easat-2091	116	43	°	°	NOUN
easat-2091	116	44	ℎ)ℎ	ℎ)ℎ	NOUN
easat-2091	116	45	,	,	PUNCT
easat-2091	116	46	]	]	PUNCT
easat-2091	116	47	=	=	PUNCT
easat-2091	116	48	𝑃[𝑉ℎ((𝜑	𝑃[𝑉ℎ((𝜑	X
easat-2091	116	49	+	+	CCONJ
easat-2091	116	50	1)𝑓	1)𝑓	NUM
easat-2091	116	51	)	)	PUNCT
easat-2091	116	52	]	]	PUNCT
easat-2091	117	1	=	=	PUNCT
easat-2091	117	2	𝑉ℎ[𝑃((𝜑	𝑉ℎ[𝑃((𝜑	PROPN
easat-2091	117	3	+	+	CCONJ
easat-2091	117	4	1)𝑓	1)𝑓	NUM
easat-2091	117	5	)	)	PUNCT
easat-2091	117	6	]	]	PUNCT
easat-2091	118	1	=	=	PUNCT
easat-2091	118	2	𝑈ℎ𝑇𝜑	𝑈ℎ𝑇𝜑	PROPN
easat-2091	118	3	+	+	CCONJ
easat-2091	118	4	1𝑓	1𝑓	ADJ
easat-2091	118	5	thus	thus	ADV
easat-2091	118	6	𝑇(𝜑	𝑇(𝜑	X
easat-2091	118	7	+	+	NUM
easat-2091	118	8	1)	1)	NUM
easat-2091	118	9	°	°	NOUN
easat-2091	118	10	ℎ𝑈ℎ	ℎ𝑈ℎ	NOUN
easat-2091	118	11	=	=	X
easat-2091	118	12	𝑈ℎ𝑇𝜑	𝑈ℎ𝑇𝜑	X
easat-2091	118	13	+	+	NOUN
easat-2091	118	14	1and	1and	NUM
easat-2091	118	15	because	because	SCONJ
easat-2091	118	16	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	118	17	is	be	AUX
easat-2091	118	18	unitary	unitary	ADJ
easat-2091	118	19	,	,	PUNCT
easat-2091	118	20	this	this	PRON
easat-2091	118	21	implies	imply	VERB
easat-2091	118	22	the	the	DET
easat-2091	118	23	desired	desire	VERB
easat-2091	118	24	result	result	NOUN
easat-2091	118	25	.	.	PUNCT
easat-2091	119	1	let	let	VERB
easat-2091	119	2	𝐻𝜑	𝐻𝜑	PROPN
easat-2091	119	3	+	+	ADJ
easat-2091	119	4	1(𝐷	1(𝐷	X
easat-2091	119	5	)	)	PUNCT
easat-2091	119	6	denote	denote	VERB
easat-2091	119	7	the	the	DET
easat-2091	119	8	usual	usual	ADJ
easat-2091	119	9	is	be	AUX
easat-2091	119	10	the	the	DET
easat-2091	119	11	hardy	hardy	ADJ
easat-2091	119	12	space	space	NOUN
easat-2091	119	13	on	on	ADP
easat-2091	119	14	the	the	DET
easat-2091	119	15	disk	disk	NOUN
easat-2091	119	16	.	.	PUNCT
easat-2091	120	1	it	it	PRON
easat-2091	120	2	is	be	AUX
easat-2091	120	3	well	well	ADV
easat-2091	120	4	known	know	VERB
easat-2091	120	5	that𝐻1(𝐷	that𝐻1(𝐷	NOUN
easat-2091	120	6	)	)	PUNCT
easat-2091	120	7	⊂	⊂	PUNCT
easat-2091	121	1	𝐿𝑎	𝐿𝑎	ADP
easat-2091	121	2	2	2	NUM
easat-2091	121	3	.	.	PUNCT
easat-2091	122	1	in	in	ADP
easat-2091	122	2	the	the	DET
easat-2091	122	3	proof	proof	NOUN
easat-2091	122	4	of	of	ADP
easat-2091	122	5	theorem	theorem	NOUN
easat-2091	122	6	1	1	NUM
easat-2091	122	7	we	we	PRON
easat-2091	122	8	will	will	AUX
easat-2091	122	9	use	use	VERB
easat-2091	122	10	,	,	PUNCT
easat-2091	122	11	without	without	ADP
easat-2091	122	12	comment	comment	NOUN
easat-2091	122	13	,	,	PUNCT
easat-2091	122	14	the	the	DET
easat-2091	122	15	following	follow	VERB
easat-2091	122	16	consequence	consequence	NOUN
easat-2091	122	17	:	:	PUNCT
easat-2091	122	18	if𝑓	if𝑓	ADV
easat-2091	122	19	,	,	PUNCT
easat-2091	122	20	g	g	PROPN
easat-2091	122	21	∈	∈	PROPN
easat-2091	122	22	𝐻2(𝐷	𝐻2(𝐷	PROPN
easat-2091	122	23	)	)	PUNCT
easat-2091	122	24	,	,	PUNCT
easat-2091	122	25	then𝑓	then𝑓	PROPN
easat-2091	122	26	,	,	PUNCT
easat-2091	122	27	g	g	PROPN
easat-2091	122	28	∈	∈	PROPN
easat-2091	123	1	𝐿𝑎	𝐿𝑎	ADP
easat-2091	123	2	2	2	NUM
easat-2091	123	3	,	,	PUNCT
easat-2091	123	4	and	and	CCONJ
easat-2091	123	5	thus	thus	ADV
easat-2091	123	6	𝑓g	𝑓g	PRON
easat-2091	123	7	∈	∈	PROPN
easat-2091	123	8	𝐿2(𝐷	𝐿2(𝐷	PROPN
easat-2091	123	9	,	,	PUNCT
easat-2091	123	10	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	123	11	)	)	PUNCT
easat-2091	123	12	.	.	PUNCT
easat-2091	124	1	we	we	PRON
easat-2091	124	2	have	have	AUX
easat-2091	124	3	now	now	ADV
easat-2091	124	4	assembled	assemble	VERB
easat-2091	124	5	all	all	DET
easat-2091	124	6	the	the	DET
easat-2091	124	7	ingredients	ingredient	NOUN
easat-2091	124	8	needed	need	VERB
easat-2091	124	9	to	to	PART
easat-2091	124	10	prove	prove	VERB
easat-2091	124	11	theorem	theorem	ADJ
easat-2091	124	12	1	1	NUM
easat-2091	124	13	.	.	PUNCT
easat-2091	124	14	proof	proof	NOUN
easat-2091	124	15	of	of	ADP
easat-2091	124	16	theorem	theorem	NOUN
easat-2091	124	17	1	1	NUM
easat-2091	124	18	.	.	PUNCT
easat-2091	125	1	if	if	SCONJ
easat-2091	125	2	we	we	PRON
easat-2091	125	3	begin	begin	VERB
easat-2091	125	4	with	with	ADP
easat-2091	125	5	the	the	DET
easat-2091	125	6	easy	easy	ADJ
easat-2091	125	7	direction	direction	NOUN
easat-2091	125	8	.	.	PUNCT
easat-2091	126	1	first	first	ADV
easat-2091	126	2	suppose	suppose	VERB
easat-2091	126	3	that	that	SCONJ
easat-2091	126	4	(	(	PUNCT
easat-2091	126	5	a	a	X
easat-2091	126	6	)	)	PUNCT
easat-2091	126	7	holds	hold	NOUN
easat-2091	126	8	,	,	PUNCT
easat-2091	126	9	so	so	SCONJ
easat-2091	126	10	that	that	SCONJ
easat-2091	126	11	(	(	PUNCT
easat-2091	126	12	𝜑	𝜑	X
easat-2091	126	13	+	+	NOUN
easat-2091	126	14	1	1	NUM
easat-2091	126	15	)	)	PUNCT
easat-2091	126	16	and	and	CCONJ
easat-2091	126	17	(	(	PUNCT
easat-2091	126	18	𝜓	𝜓	PROPN
easat-2091	126	19	+	+	NOUN
easat-2091	126	20	1	1	NUM
easat-2091	126	21	)	)	PUNCT
easat-2091	126	22	are	be	AUX
easat-2091	126	23	analytic	analytic	ADJ
easat-2091	126	24	on	on	ADP
easat-2091	126	25	𝐷	𝐷	PROPN
easat-2091	126	26	which	which	PRON
easat-2091	126	27	means	mean	VERB
easat-2091	126	28	that	that	SCONJ
easat-2091	126	29	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	126	30	+	+	CCONJ
easat-2091	126	31	1	1	NUM
easat-2091	126	32	and	and	CCONJ
easat-2091	126	33	𝑇𝜓	𝑇𝜓	PROPN
easat-2091	126	34	+	+	CCONJ
easat-2091	126	35	1	1	NUM
easat-2091	126	36	are	be	AUX
easat-2091	126	37	,	,	PUNCT
easat-2091	126	38	respectively	respectively	ADV
easat-2091	126	39	,	,	PUNCT
easat-2091	126	40	the	the	DET
easat-2091	126	41	operators	operator	NOUN
easat-2091	126	42	on	on	ADP
easat-2091	126	43	𝐿𝑎	𝐿𝑎	ADP
easat-2091	126	44	2	2	NUM
easat-2091	126	45	of	of	ADP
easat-2091	126	46	398	398	NUM
easat-2091	126	47	edelweiss	edelweiss	PROPN
easat-2091	126	48	applied	apply	VERB
easat-2091	126	49	science	science	NOUN
easat-2091	126	50	and	and	CCONJ
easat-2091	126	51	technology	technology	NOUN
easat-2091	126	52	issn	issn	PROPN
easat-2091	126	53	:	:	PUNCT
easat-2091	126	54	2576	2576	NUM
easat-2091	126	55	-	-	SYM
easat-2091	126	56	8484	8484	NUM
easat-2091	126	57	vol	vol	NOUN
easat-2091	126	58	.	.	PROPN
easat-2091	126	59	8	8	NUM
easat-2091	126	60	,	,	PUNCT
easat-2091	126	61	no	no	INTJ
easat-2091	126	62	.	.	NOUN
easat-2091	126	63	6	6	NUM
easat-2091	126	64	:	:	SYM
easat-2091	126	65	394	394	NUM
easat-2091	126	66	-	-	SYM
easat-2091	126	67	400	400	NUM
easat-2091	126	68	,	,	PUNCT
easat-2091	126	69	2024	2024	NUM
easat-2091	126	70	doi	doi	NOUN
easat-2091	126	71	:	:	PUNCT
easat-2091	126	72	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	126	73	©	©	ADP
easat-2091	126	74	2024	2024	NUM
easat-2091	126	75	by	by	ADP
easat-2091	126	76	the	the	DET
easat-2091	126	77	author	author	NOUN
easat-2091	126	78	;	;	PUNCT
easat-2091	126	79	licensee	licensee	PROPN
easat-2091	126	80	learning	learn	VERB
easat-2091	126	81	gate	gate	NOUN
easat-2091	126	82	multiplication	multiplication	NOUN
easat-2091	126	83	by	by	ADP
easat-2091	126	84	(	(	PUNCT
easat-2091	126	85	𝜑	𝜑	PROPN
easat-2091	126	86	+	+	NOUN
easat-2091	126	87	1	1	NUM
easat-2091	126	88	)	)	PUNCT
easat-2091	126	89	and	and	CCONJ
easat-2091	126	90	(	(	PUNCT
easat-2091	126	91	𝜓	𝜓	PROPN
easat-2091	126	92	+	+	NOUN
easat-2091	126	93	1	1	NUM
easat-2091	126	94	)	)	PUNCT
easat-2091	126	95	.	.	PUNCT
easat-2091	127	1	so	so	ADV
easat-2091	127	2	that𝑇𝜑	that𝑇𝜑	PRON
easat-2091	127	3	+	+	CCONJ
easat-2091	127	4	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	127	5	+	+	SYM
easat-2091	127	6	1	1	NUM
easat-2091	127	7	=	=	SYM
easat-2091	128	1	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	128	2	+	+	NOUN
easat-2091	128	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	128	4	+	+	CCONJ
easat-2091	128	5	1	1	X
easat-2091	128	6	.	.	PUNCT
easat-2091	128	7	now	now	ADV
easat-2091	128	8	suppose	suppose	VERB
easat-2091	128	9	that	that	SCONJ
easat-2091	128	10	(	(	PUNCT
easat-2091	128	11	b	b	X
easat-2091	128	12	)	)	PUNCT
easat-2091	128	13	holds	hold	VERB
easat-2091	128	14	,	,	PUNCT
easat-2091	128	15	so	so	SCONJ
easat-2091	128	16	that	that	SCONJ
easat-2091	128	17	𝜑	𝜑	PROPN
easat-2091	128	18	+	+	SYM
easat-2091	128	19	1̅̅	1̅̅	NUM
easat-2091	128	20	̅̅	̅̅	PROPN
easat-2091	128	21	̅̅	̅̅	PROPN
easat-2091	128	22	̅̅	̅̅	PROPN
easat-2091	128	23	and	and	CCONJ
easat-2091	128	24	𝜓	𝜓	PROPN
easat-2091	128	25	+	+	PROPN
easat-2091	128	26	1̅̅	1̅̅	NUM
easat-2091	128	27	̅̅	̅̅	PROPN
easat-2091	128	28	̅̅	̅̅	PROPN
easat-2091	128	29	̅̅	̅̅	PROPN
easat-2091	128	30	are	be	AUX
easat-2091	128	31	analytic	analytic	ADJ
easat-2091	128	32	on𝐷.	on𝐷.	ADP
easat-2091	128	33	by	by	ADP
easat-2091	128	34	the	the	DET
easat-2091	128	35	paragraph	paragraph	NOUN
easat-2091	128	36	above	above	ADV
easat-2091	128	37	,	,	PUNCT
easat-2091	128	38	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	128	39	+	+	SYM
easat-2091	128	40	1̅̅	1̅̅	NUM
easat-2091	128	41	̅̅	̅̅	PROPN
easat-2091	128	42	̅̅	̅̅	NOUN
easat-2091	128	43	̅̅	̅̅	PROPN
easat-2091	129	1	𝑇𝜓	𝑇𝜓	PROPN
easat-2091	129	2	+	+	PROPN
easat-2091	129	3	1̅̅	1̅̅	NUM
easat-2091	129	4	̅̅	̅̅	PROPN
easat-2091	129	5	̅̅	̅̅	PROPN
easat-2091	129	6	̅̅	̅̅	PROPN
easat-2091	129	7	=	=	PUNCT
easat-2091	130	1	𝑇𝜓	𝑇𝜓	PROPN
easat-2091	130	2	+	+	CCONJ
easat-2091	130	3	1̅̅	1̅̅	NUM
easat-2091	130	4	̅̅	̅̅	PROPN
easat-2091	130	5	̅̅	̅̅	PROPN
easat-2091	130	6	̅̅	̅̅	PROPN
easat-2091	131	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	131	2	+	+	CCONJ
easat-2091	131	3	1̅̅	1̅̅	NUM
easat-2091	131	4	̅̅	̅̅	PROPN
easat-2091	131	5	̅̅	̅̅	PROPN
easat-2091	131	6	̅̅	̅̅	PROPN
easat-2091	131	7	.	.	PUNCT
easat-2091	132	1	take	take	VERB
easat-2091	132	2	the	the	DET
easat-2091	132	3	adjoin	adjoin	NOUN
easat-2091	132	4	of	of	ADP
easat-2091	132	5	both	both	DET
easat-2091	132	6	sides	side	NOUN
easat-2091	132	7	of	of	ADP
easat-2091	132	8	this	this	DET
easat-2091	132	9	equation	equation	NOUN
easat-2091	132	10	,	,	PUNCT
easat-2091	132	11	and	and	CCONJ
easat-2091	132	12	use	use	VERB
easat-2091	132	13	the	the	DET
easat-2091	132	14	identity	identity	NOUN
easat-2091	132	15	𝑇𝜑+1̅̅	𝑇𝜑+1̅̅	PROPN
easat-2091	132	16	̅̅	̅̅	PROPN
easat-2091	132	17	̅̅	̅̅	PROPN
easat-2091	132	18	∗	∗	NOUN
easat-2091	132	19	=	=	PUNCT
easat-2091	133	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	133	2	+	+	NOUN
easat-2091	133	3	1	1	NUM
easat-2091	133	4	to	to	PART
easat-2091	133	5	conclude	conclude	VERB
easat-2091	133	6	that𝑇𝜑	that𝑇𝜑	PRON
easat-2091	133	7	+	+	CCONJ
easat-2091	133	8	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	133	9	+	+	SYM
easat-2091	133	10	1	1	NUM
easat-2091	133	11	=	=	SYM
easat-2091	134	1	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	134	2	+	+	NOUN
easat-2091	134	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	134	4	+	+	CCONJ
easat-2091	134	5	1	1	X
easat-2091	134	6	.	.	PUNCT
easat-2091	134	7	finally	finally	ADV
easat-2091	134	8	suppose	suppose	VERB
easat-2091	134	9	that	that	SCONJ
easat-2091	134	10	(	(	PUNCT
easat-2091	134	11	c	c	X
easat-2091	134	12	)	)	PUNCT
easat-2091	134	13	holds	hold	NOUN
easat-2091	134	14	,	,	PUNCT
easat-2091	134	15	so	so	SCONJ
easat-2091	134	16	there	there	PRON
easat-2091	134	17	exist	exist	VERB
easat-2091	134	18	constants𝑎	constants𝑎	NOUN
easat-2091	134	19	,	,	PUNCT
easat-2091	134	20	𝑏	𝑏	PROPN
easat-2091	134	21	∈	∈	PROPN
easat-2091	134	22	𝐶	𝐶	PROPN
easat-2091	134	23	,	,	PUNCT
easat-2091	134	24	not	not	PART
easat-2091	134	25	both	both	PRON
easat-2091	134	26	0	0	NUM
easat-2091	134	27	,	,	PUNCT
easat-2091	134	28	such	such	ADJ
easat-2091	134	29	that	that	SCONJ
easat-2091	134	30	𝑎(𝜑	𝑎(𝜑	PROPN
easat-2091	134	31	+	+	PUNCT
easat-2091	134	32	1	1	NUM
easat-2091	134	33	)	)	PUNCT
easat-2091	134	34	+	+	CCONJ
easat-2091	135	1	𝑏(𝜓	𝑏(𝜓	PROPN
easat-2091	135	2	+	+	SYM
easat-2091	135	3	1	1	NUM
easat-2091	135	4	)	)	PUNCT
easat-2091	135	5	is	be	AUX
easat-2091	135	6	constant	constant	ADJ
easat-2091	135	7	on𝐷.	on𝐷.	ADP
easat-2091	135	8	if𝑎	if𝑎	ADP
easat-2091	135	9	≠	≠	PROPN
easat-2091	135	10	0	0	NUM
easat-2091	135	11	,	,	PUNCT
easat-2091	135	12	then	then	ADV
easat-2091	135	13	there	there	PRON
easat-2091	135	14	exist	exist	VERB
easat-2091	135	15	constants	constant	NOUN
easat-2091	135	16	𝛽	𝛽	NOUN
easat-2091	135	17	,	,	PUNCT
easat-2091	135	18	𝛾	𝛾	PROPN
easat-2091	135	19	∈	∈	PROPN
easat-2091	135	20	𝐶	𝐶	PROPN
easat-2091	135	21	such	such	ADJ
easat-2091	135	22	that	that	SCONJ
easat-2091	135	23	𝜑	𝜑	PROPN
easat-2091	135	24	+	+	CCONJ
easat-2091	135	25	1	1	NUM
easat-2091	135	26	=	=	NOUN
easat-2091	135	27	𝛽(𝜓	𝛽(𝜓	NOUN
easat-2091	135	28	+	+	ADJ
easat-2091	135	29	1	1	NUM
easat-2091	135	30	)	)	PUNCT
easat-2091	135	31	+	+	CCONJ
easat-2091	135	32	𝛾	𝛾	PRON
easat-2091	135	33	on𝐷	on𝐷	PROPN
easat-2091	135	34	,	,	PUNCT
easat-2091	135	35	which	which	PRON
easat-2091	135	36	means	mean	VERB
easat-2091	135	37	that	that	SCONJ
easat-2091	136	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	136	2	+	+	CCONJ
easat-2091	136	3	1	1	NUM
easat-2091	136	4	=	=	SYM
easat-2091	136	5	𝛽𝑇𝜓	𝛽𝑇𝜓	NOUN
easat-2091	136	6	+	+	CCONJ
easat-2091	136	7	1	1	NUM
easat-2091	136	8	+	+	NUM
easat-2091	136	9	𝛾𝐼	𝛾𝐼	NOUN
easat-2091	136	10	(	(	PUNCT
easat-2091	136	11	𝐼	𝐼	PROPN
easat-2091	136	12	denotes	denote	VERB
easat-2091	136	13	the	the	DET
easat-2091	136	14	identity	identity	NOUN
easat-2091	136	15	operator	operator	NOUN
easat-2091	136	16	on	on	ADP
easat-2091	136	17	𝐿𝑎	𝐿𝑎	ADP
easat-2091	136	18	2	2	NUM
easat-2091	136	19	)	)	PUNCT
easat-2091	136	20	,	,	PUNCT
easat-2091	136	21	which	which	PRON
easat-2091	136	22	clearly	clearly	ADV
easat-2091	136	23	implies	imply	VERB
easat-2091	136	24	that	that	SCONJ
easat-2091	137	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	137	2	+	+	CCONJ
easat-2091	137	3	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	137	4	+	+	SYM
easat-2091	137	5	1	1	NUM
easat-2091	137	6	=	=	SYM
easat-2091	138	1	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	138	2	+	+	NOUN
easat-2091	138	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	138	4	+	+	CCONJ
easat-2091	138	5	1	1	X
easat-2091	138	6	.	.	PUNCT
easat-2091	139	1	if	if	SCONJ
easat-2091	139	2	𝑏	𝑏	PROPN
easat-2091	139	3	≠	≠	PROPN
easat-2091	139	4	0	0	NUM
easat-2091	139	5	,	,	PUNCT
easat-2091	139	6	we	we	PRON
easat-2091	139	7	conclude	conclude	VERB
easat-2091	139	8	in	in	ADP
easat-2091	139	9	a	a	DET
easat-2091	139	10	similar	similar	ADJ
easat-2091	139	11	fashion	fashion	NOUN
easat-2091	139	12	that	that	PRON
easat-2091	140	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	140	2	+	+	CCONJ
easat-2091	140	3	1	1	NUM
easat-2091	140	4	and	and	CCONJ
easat-2091	140	5	𝑇𝜓	𝑇𝜓	PROPN
easat-2091	140	6	+	+	CCONJ
easat-2091	140	7	1	1	NUM
easat-2091	140	8	commute	commute	NOUN
easat-2091	140	9	.	.	PUNCT
easat-2091	141	1	now	now	ADV
easat-2091	141	2	to	to	PART
easat-2091	141	3	prove	prove	VERB
easat-2091	141	4	the	the	DET
easat-2091	141	5	other	other	ADJ
easat-2091	141	6	direction	direction	NOUN
easat-2091	141	7	of	of	ADP
easat-2091	141	8	theorem	theorem	NOUN
easat-2091	141	9	1	1	NUM
easat-2091	141	10	,	,	PUNCT
easat-2091	141	11	suppose	suppose	VERB
easat-2091	141	12	that	that	SCONJ
easat-2091	141	13	(	(	PUNCT
easat-2091	141	14	𝜑	𝜑	X
easat-2091	141	15	+	+	NOUN
easat-2091	141	16	1	1	NUM
easat-2091	141	17	)	)	PUNCT
easat-2091	141	18	and	and	CCONJ
easat-2091	141	19	(	(	PUNCT
easat-2091	141	20	𝜓	𝜓	PROPN
easat-2091	141	21	+	+	NOUN
easat-2091	141	22	1	1	NUM
easat-2091	141	23	)	)	PUNCT
easat-2091	141	24	are	be	AUX
easat-2091	141	25	bounded	bound	VERB
easat-2091	141	26	harmonic	harmonic	ADJ
easat-2091	141	27	functions	function	NOUN
easat-2091	141	28	on	on	ADP
easat-2091	141	29	𝐷	𝐷	PROPN
easat-2091	141	30	such	such	ADJ
easat-2091	141	31	that𝑇𝜑	that𝑇𝜑	NOUN
easat-2091	142	1	+	+	CCONJ
easat-2091	142	2	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	142	3	+	+	SYM
easat-2091	142	4	1	1	NUM
easat-2091	142	5	=	=	SYM
easat-2091	142	6	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	142	7	+	+	NOUN
easat-2091	142	8	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	142	9	+	+	CCONJ
easat-2091	142	10	1	1	NUM
easat-2091	142	11	.	.	PUNCT
easat-2091	143	1	because	because	SCONJ
easat-2091	143	2	𝜑	𝜑	PROPN
easat-2091	143	3	+	+	CCONJ
easat-2091	143	4	1	1	NUM
easat-2091	143	5	and	and	CCONJ
easat-2091	143	6	𝜓	𝜓	NOUN
easat-2091	143	7	+	+	NOUN
easat-2091	143	8	1	1	NUM
easat-2091	143	9	are	be	AUX
easat-2091	143	10	harmonic	harmonic	ADJ
easat-2091	143	11	on𝐷	on𝐷	PROPN
easat-2091	143	12	,	,	PUNCT
easat-2091	143	13	there	there	PRON
easat-2091	143	14	exist	exist	VERB
easat-2091	143	15	functions	function	NOUN
easat-2091	143	16	𝑓1	𝑓1	ADJ
easat-2091	143	17	,	,	PUNCT
easat-2091	143	18	𝑓2	𝑓2	NOUN
easat-2091	143	19	,	,	PUNCT
easat-2091	143	20	g1	g1	PROPN
easat-2091	143	21	and	and	CCONJ
easat-2091	143	22	g2	g2	PROPN
easat-2091	143	23	are	be	AUX
easat-2091	143	24	analytic	analytic	ADJ
easat-2091	143	25	on	on	ADP
easat-2091	143	26	𝐷	𝐷	PROPN
easat-2091	143	27	such	such	ADJ
easat-2091	143	28	that	that	DET
easat-2091	143	29	t	t	PROPN
easat-2091	143	30	(	(	PUNCT
easat-2091	143	31	𝜑	𝜑	PROPN
easat-2091	144	1	+	+	NOUN
easat-2091	144	2	1	1	NUM
easat-2091	144	3	)	)	PUNCT
easat-2091	144	4	=	=	SYM
easat-2091	144	5	𝑓1	𝑓1	PROPN
easat-2091	144	6	+	+	CCONJ
easat-2091	144	7	𝑓2̅	𝑓2̅	PROPN
easat-2091	144	8	and	and	CCONJ
easat-2091	144	9	(	(	PUNCT
easat-2091	144	10	𝜓	𝜓	PROPN
easat-2091	144	11	+	+	NOUN
easat-2091	144	12	1	1	X
easat-2091	144	13	)	)	PUNCT
easat-2091	144	14	=	=	VERB
easat-2091	144	15	g1	g1	NOUN
easat-2091	144	16	+	+	CCONJ
easat-2091	144	17	g2̅̅	g2̅̅	ADP
easat-2091	144	18	̅	̅	NOUN
easat-2091	144	19	on	on	ADP
easat-2091	144	20	𝐷	𝐷	PROPN
easat-2091	144	21	(	(	PUNCT
easat-2091	144	22	9	9	NUM
easat-2091	144	23	)	)	PUNCT
easat-2091	144	24	because	because	SCONJ
easat-2091	144	25	(	(	PUNCT
easat-2091	144	26	𝜑	𝜑	X
easat-2091	144	27	+	+	NOUN
easat-2091	144	28	1	1	NUM
easat-2091	144	29	)	)	PUNCT
easat-2091	144	30	and	and	CCONJ
easat-2091	144	31	𝜓	𝜓	X
easat-2091	144	32	+	+	NOUN
easat-2091	144	33	1	1	NUM
easat-2091	144	34	are	be	AUX
easat-2091	144	35	bounded	bound	VERB
easat-2091	144	36	on	on	ADP
easat-2091	144	37	𝐷	𝐷	PROPN
easat-2091	144	38	,	,	PUNCT
easat-2091	144	39	the	the	DET
easat-2091	144	40	functions	function	NOUN
easat-2091	144	41	𝑓1	𝑓1	ADJ
easat-2091	144	42	,	,	PUNCT
easat-2091	144	43	𝑓2	𝑓2	NOUN
easat-2091	144	44	,	,	PUNCT
easat-2091	144	45	g1	g1	PROPN
easat-2091	144	46	and	and	CCONJ
easat-2091	144	47	g2	g2	PROPN
easat-2091	144	48	must	must	AUX
easat-2091	144	49	be	be	AUX
easat-2091	144	50	in	in	ADP
easat-2091	144	51	𝐻2(𝐷	𝐻2(𝐷	PROPN
easat-2091	144	52	)	)	PUNCT
easat-2091	144	53	.	.	PUNCT
easat-2091	145	1	this	this	PRON
easat-2091	145	2	means	mean	VERB
easat-2091	145	3	that	that	SCONJ
easat-2091	145	4	the	the	DET
easat-2091	145	5	function	function	NOUN
easat-2091	145	6	is	be	AUX
easat-2091	145	7	constant	constant	ADJ
easat-2091	145	8	at	at	ADP
easat-2091	145	9	𝐷.	𝐷.	PROPN
easat-2091	145	10	so	so	ADV
easat-2091	145	11	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	145	12	+	+	CCONJ
easat-2091	145	13	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	145	14	+	+	SYM
easat-2091	145	15	11	11	NUM
easat-2091	146	1	=	=	SYM
easat-2091	146	2	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	146	3	+	+	NOUN
easat-2091	146	4	1(𝑃(𝜓	1(𝑃(𝜓	NUM
easat-2091	146	5	+	+	CCONJ
easat-2091	146	6	1	1	NUM
easat-2091	146	7	)	)	PUNCT
easat-2091	146	8	)	)	PUNCT
easat-2091	147	1	=	=	PUNCT
easat-2091	148	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	148	2	+	+	NOUN
easat-2091	148	3	1(𝑃(g1	1(𝑃(g1	NUM
easat-2091	148	4	+	+	CCONJ
easat-2091	148	5	g2̅̅	g2̅̅	ADP
easat-2091	148	6	̅	̅	NOUN
easat-2091	148	7	)	)	PUNCT
easat-2091	148	8	)	)	PUNCT
easat-2091	149	1	=	=	PUNCT
easat-2091	150	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	150	2	+	+	NOUN
easat-2091	150	3	1(g1	1(g1	NUM
easat-2091	150	4	+	+	CCONJ
easat-2091	150	5	g2(0)̅̅	g2(0)̅̅	PRON
easat-2091	150	6	̅̅	̅̅	NOUN
easat-2091	150	7	̅̅	̅̅	NOUN
easat-2091	150	8	̅	̅	NOUN
easat-2091	150	9	)	)	PUNCT
easat-2091	150	10	=	=	SYM
easat-2091	150	11	𝑃([𝑓1	𝑃([𝑓1	NOUN
easat-2091	150	12	+	+	CCONJ
easat-2091	150	13	𝑓2̅][g1	𝑓2̅][g1	NOUN
easat-2091	150	14	+	+	CCONJ
easat-2091	150	15	g2(0)̅̅	g2(0)̅̅	PRON
easat-2091	150	16	̅̅	̅̅	NOUN
easat-2091	150	17	̅̅	̅̅	NOUN
easat-2091	150	18	̅	̅	PROPN
easat-2091	150	19	]	]	PUNCT
easat-2091	150	20	)	)	PUNCT
easat-2091	151	1	=	=	PUNCT
easat-2091	152	1	𝑓1g1	𝑓1g1	PROPN
easat-2091	152	2	+	+	CCONJ
easat-2091	153	1	g2(0)̅̅	g2(0)̅̅	DET
easat-2091	153	2	̅̅	̅̅	NOUN
easat-2091	153	3	̅̅	̅̅	NOUN
easat-2091	153	4	̅𝑓1	̅𝑓1	PROPN
easat-2091	154	1	+	+	CCONJ
easat-2091	154	2	𝑃(𝑓2̅g1	𝑃(𝑓2̅g1	NOUN
easat-2091	154	3	)	)	PUNCT
easat-2091	155	1	+	+	CCONJ
easat-2091	156	1	𝑓2(0)̅̅	𝑓2(0)̅̅	ADP
easat-2091	156	2	̅̅	̅̅	PROPN
easat-2091	156	3	̅̅	̅̅	PROPN
easat-2091	156	4	̅g2(0)̅̅	̅g2(0)̅̅	PROPN
easat-2091	156	5	̅̅	̅̅	PROPN
easat-2091	156	6	̅̅	̅̅	PROPN
easat-2091	156	7	̅	̅	PROPN
easat-2091	156	8	=	=	SYM
easat-2091	156	9	〈	〈	PROPN
easat-2091	156	10	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	156	11	+	+	CCONJ
easat-2091	156	12	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	156	13	+	+	SYM
easat-2091	156	14	11,1	11,1	NUM
easat-2091	156	15	〉	〉	NOUN
easat-2091	156	16	=	=	SYM
easat-2091	156	17	〈	〈	PROPN
easat-2091	156	18	𝑓1g1	𝑓1g1	PROPN
easat-2091	156	19	+	+	CCONJ
easat-2091	156	20	g2(0)̅̅	g2(0)̅̅	DET
easat-2091	156	21	̅̅	̅̅	NOUN
easat-2091	156	22	̅̅	̅̅	NOUN
easat-2091	156	23	̅𝑓1	̅𝑓1	PUNCT
easat-2091	157	1	+	+	CCONJ
easat-2091	157	2	𝑓2̅g1	𝑓2̅g1	ADJ
easat-2091	157	3	+	+	CCONJ
easat-2091	157	4	𝑓2(0)̅̅	𝑓2(0)̅̅	ADP
easat-2091	157	5	̅̅	̅̅	PROPN
easat-2091	157	6	̅̅	̅̅	PROPN
easat-2091	157	7	̅g2(0)̅̅	̅g2(0)̅̅	PROPN
easat-2091	157	8	̅̅	̅̅	PROPN
easat-2091	157	9	̅̅	̅̅	PROPN
easat-2091	157	10	̅	̅	PROPN
easat-2091	157	11	,	,	PUNCT
easat-2091	157	12	1	1	NUM
easat-2091	157	13	〉	〉	NOUN
easat-2091	157	14	=	=	PUNCT
easat-2091	157	15	∫𝐷(𝑓1g1	∫𝐷(𝑓1g1	NOUN
easat-2091	158	1	+	+	CCONJ
easat-2091	158	2	g2(0)̅̅	g2(0)̅̅	DET
easat-2091	158	3	̅̅	̅̅	NOUN
easat-2091	158	4	̅̅	̅̅	NOUN
easat-2091	158	5	̅𝑓1	̅𝑓1	PUNCT
easat-2091	159	1	+	+	CCONJ
easat-2091	159	2	𝑓2̅g1	𝑓2̅g1	ADJ
easat-2091	159	3	+	+	CCONJ
easat-2091	159	4	𝑓2(0)̅̅	𝑓2(0)̅̅	ADP
easat-2091	159	5	̅̅	̅̅	PROPN
easat-2091	159	6	̅̅	̅̅	PROPN
easat-2091	159	7	̅g2(0)̅̅	̅g2(0)̅̅	PROPN
easat-2091	159	8	̅̅	̅̅	PROPN
easat-2091	159	9	̅̅	̅̅	PROPN
easat-2091	159	10	̅)𝑑𝐴	̅)𝑑𝐴	NOUN
easat-2091	159	11	=	=	SYM
easat-2091	159	12	𝜋[𝑓2(0)g1(0	𝜋[𝑓2(0)g1(0	SYM
easat-2091	159	13	)	)	PUNCT
easat-2091	159	14	+	+	NUM
easat-2091	159	15	𝑓1(0)g2(0)̅̅	𝑓1(0)g2(0)̅̅	PROPN
easat-2091	159	16	̅̅	̅̅	NOUN
easat-2091	159	17	̅̅	̅̅	PROPN
easat-2091	159	18	̅	̅	NOUN
easat-2091	159	19	+	+	CCONJ
easat-2091	160	1	𝑓2(0)̅̅	𝑓2(0)̅̅	ADP
easat-2091	160	2	̅̅	̅̅	PROPN
easat-2091	160	3	̅̅	̅̅	PROPN
easat-2091	160	4	̅g2(0)̅̅	̅g2(0)̅̅	PROPN
easat-2091	160	5	̅̅	̅̅	PROPN
easat-2091	160	6	̅̅	̅̅	PROPN
easat-2091	160	7	̅	̅	PROPN
easat-2091	160	8	]	]	X
easat-2091	161	1	+	+	CCONJ
easat-2091	161	2	∫𝐷𝑓2̅g1𝑑𝐴	∫𝐷𝑓2̅g1𝑑𝐴	NOUN
easat-2091	161	3	(	(	PUNCT
easat-2091	161	4	10	10	NUM
easat-2091	161	5	)	)	PUNCT
easat-2091	161	6	a	a	DET
easat-2091	161	7	sirnilar	sirnilar	ADJ
easat-2091	161	8	formula	formula	NOUN
easat-2091	161	9	can	can	AUX
easat-2091	161	10	be	be	AUX
easat-2091	161	11	obtained	obtain	VERB
easat-2091	161	12	fo	fo	ADP
easat-2091	161	13	〈	〈	NOUN
easat-2091	161	14	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	161	15	+	+	CCONJ
easat-2091	161	16	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	161	17	+	+	SYM
easat-2091	161	18	11,1	11,1	NUM
easat-2091	161	19	〉	〉	NOUN
easat-2091	161	20	because	because	SCONJ
easat-2091	161	21	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	161	22	+	+	CCONJ
easat-2091	161	23	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	161	24	+	+	SYM
easat-2091	161	25	1	1	NUM
easat-2091	161	26	=	=	SYM
easat-2091	162	1	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	162	2	+	+	NOUN
easat-2091	162	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	162	4	+	+	SYM
easat-2091	162	5	1	1	NUM
easat-2091	162	6	we	we	PRON
easat-2091	162	7	can	can	AUX
easat-2091	162	8	set	set	VERB
easat-2091	162	9	the	the	DET
easat-2091	162	10	right	right	ADJ
easat-2091	162	11	-	-	PUNCT
easat-2091	162	12	hand	hand	NOUN
easat-2091	162	13	side	side	NOUN
easat-2091	162	14	of	of	ADP
easat-2091	162	15	eq	eq	NOUN
easat-2091	162	16	(	(	PUNCT
easat-2091	162	17	10	10	NUM
easat-2091	162	18	)	)	PUNCT
easat-2091	162	19	equal	equal	ADJ
easat-2091	162	20	to	to	ADP
easat-2091	162	21	the	the	DET
easat-2091	162	22	corresponding	correspond	VERB
easat-2091	162	23	formula	formula	NOUN
easat-2091	162	24	for	for	ADP
easat-2091	162	25	〈	〈	NOUN
easat-2091	162	26	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	162	27	+	+	NOUN
easat-2091	162	28	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	162	29	+	+	X
easat-2091	162	30	11,1	11,1	NUM
easat-2091	162	31	〉	〉	NOUN
easat-2091	162	32	,	,	PUNCT
easat-2091	162	33	getting	get	VERB
easat-2091	162	34	∫𝐷(𝑓2̅g1	∫𝐷(𝑓2̅g1	PROPN
easat-2091	162	35	−	−	NUM
easat-2091	162	36	𝑓1g2̅̅	𝑓1g2̅̅	NOUN
easat-2091	162	37	̅	̅	NOUN
easat-2091	162	38	)	)	PUNCT
easat-2091	163	1	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	163	2	𝜋	𝜋	NOUN
easat-2091	163	3	=	=	SYM
easat-2091	163	4	𝑓2(0)̅̅	𝑓2(0)̅̅	ADP
easat-2091	163	5	̅̅	̅̅	PROPN
easat-2091	163	6	̅̅	̅̅	PROPN
easat-2091	163	7	̅g1(0	̅g1(0	PROPN
easat-2091	163	8	)	)	PUNCT
easat-2091	163	9	−	−	PROPN
easat-2091	163	10	𝑓1(0)g2(0)̅̅	𝑓1(0)g2(0)̅̅	PROPN
easat-2091	163	11	̅̅	̅̅	PROPN
easat-2091	163	12	̅̅	̅̅	PROPN
easat-2091	163	13	̅	̅	PROPN
easat-2091	163	14	(	(	PUNCT
easat-2091	163	15	11	11	NUM
easat-2091	163	16	)	)	PUNCT
easat-2091	163	17	let	let	VERB
easat-2091	163	18	ℎ	ℎ	PROPN
easat-2091	163	19	∈	∈	PROPN
easat-2091	163	20	𝐴𝑢𝑡(𝐷	𝐴𝑢𝑡(𝐷	PRON
easat-2091	163	21	)	)	PUNCT
easat-2091	163	22	.	.	PUNCT
easat-2091	164	1	multiplying	multiply	VERB
easat-2091	164	2	both	both	DET
easat-2091	164	3	sides	side	NOUN
easat-2091	164	4	of	of	ADP
easat-2091	164	5	the	the	DET
easat-2091	164	6	equation	equation	NOUN
easat-2091	164	7	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	165	1	+	+	CCONJ
easat-2091	165	2	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	165	3	+	+	SYM
easat-2091	165	4	1	1	NUM
easat-2091	165	5	=	=	SYM
easat-2091	166	1	𝑇𝜓	𝑇𝜓	NOUN
easat-2091	166	2	+	+	NOUN
easat-2091	166	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	166	4	+	+	CCONJ
easat-2091	166	5	1	1	NUM
easat-2091	166	6	on	on	ADP
easat-2091	166	7	the	the	DET
easat-2091	166	8	left	left	NOUN
easat-2091	166	9	and	and	CCONJ
easat-2091	166	10	by	by	ADP
easat-2091	166	11	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	166	12	∗	∗	NOUN
easat-2091	166	13	on	on	ADP
easat-2091	166	14	the	the	DET
easat-2091	166	15	right	right	NOUN
easat-2091	166	16	,	,	PUNCT
easat-2091	166	17	and	and	CCONJ
easat-2091	166	18	recalling	recall	VERB
easat-2091	166	19	that	that	SCONJ
easat-2091	166	20	𝑈ℎ	𝑈ℎ	PROPN
easat-2091	166	21	is	be	AUX
easat-2091	166	22	unitary	unitary	ADJ
easat-2091	166	23	so	so	SCONJ
easat-2091	166	24	that𝑈ℎ	that𝑈ℎ	NUM
easat-2091	166	25	∗𝑈ℎ	∗𝑈ℎ	ADJ
easat-2091	166	26	=	=	SYM
easat-2091	166	27	1	1	NUM
easat-2091	166	28	,	,	PUNCT
easat-2091	166	29	we	we	PRON
easat-2091	166	30	get	get	VERB
easat-2091	166	31	𝑈ℎ𝑇𝜑	𝑈ℎ𝑇𝜑	PRON
easat-2091	167	1	+	+	CCONJ
easat-2091	168	1	1𝑈ℎ	1𝑈ℎ	NUM
easat-2091	168	2	∗𝑈ℎ𝑇𝜓	∗𝑈ℎ𝑇𝜓	NOUN
easat-2091	168	3	+	+	NOUN
easat-2091	168	4	1𝑈ℎ	1𝑈ℎ	NUM
easat-2091	168	5	∗	∗	NOUN
easat-2091	168	6	=	=	SYM
easat-2091	168	7	𝑈ℎ𝑇𝜓	𝑈ℎ𝑇𝜓	NOUN
easat-2091	168	8	+	+	NOUN
easat-2091	168	9	1𝑈ℎ	1𝑈ℎ	NUM
easat-2091	168	10	∗𝑈ℎ𝑇𝜑	∗𝑈ℎ𝑇𝜑	NUM
easat-2091	168	11	+	+	NOUN
easat-2091	168	12	1𝑈ℎ	1𝑈ℎ	NUM
easat-2091	168	13	∗	∗	NOUN
easat-2091	168	14	lemma	lemma	PROPN
easat-2091	168	15	4	4	NUM
easat-2091	168	16	now	now	ADV
easat-2091	168	17	shows	show	VERB
easat-2091	168	18	that	that	SCONJ
easat-2091	168	19	𝑇(𝜑	𝑇(𝜑	PUNCT
easat-2091	169	1	+	+	NUM
easat-2091	169	2	1)∗ℎ𝑇(𝜓	1)∗ℎ𝑇(𝜓	NUM
easat-2091	169	3	+	+	CCONJ
easat-2091	169	4	1)∗ℎ	1)∗ℎ	NUM
easat-2091	169	5	=	=	SYM
easat-2091	170	1	𝑇(𝜓	𝑇(𝜓	NOUN
easat-2091	170	2	+	+	NUM
easat-2091	170	3	1)∗ℎ𝑇(𝜑	1)∗ℎ𝑇(𝜑	NUM
easat-2091	170	4	+	+	X
easat-2091	170	5	1)∗ℎ	1)∗ℎ	NUM
easat-2091	170	6	(	(	PUNCT
easat-2091	170	7	12	12	NUM
easat-2091	170	8	)	)	PUNCT
easat-2091	170	9	composing	compose	VERB
easat-2091	170	10	both	both	DET
easat-2091	170	11	sides	side	NOUN
easat-2091	170	12	of	of	ADP
easat-2091	170	13	the	the	DET
easat-2091	170	14	equations	equation	NOUN
easat-2091	170	15	in	in	ADP
easat-2091	170	16	eq(8	eq(8	NOUN
easat-2091	170	17	)	)	PUNCT
easat-2091	170	18	with	with	ADP
easat-2091	170	19	ℎ	ℎ	PROPN
easat-2091	170	20	expresses	express	VERB
easat-2091	170	21	each	each	PRON
easat-2091	170	22	of	of	ADP
easat-2091	170	23	the	the	DET
easat-2091	170	24	bounded	bounded	ADJ
easat-2091	170	25	harmonic	harmonic	ADJ
easat-2091	170	26	functions	function	NOUN
easat-2091	170	27	(	(	PUNCT
easat-2091	170	28	𝜑	𝜑	NOUN
easat-2091	170	29	+	+	NOUN
easat-2091	170	30	1	1	X
easat-2091	170	31	)	)	PUNCT
easat-2091	170	32	∗	∗	NOUN
easat-2091	170	33	ℎ	ℎ	PROPN
easat-2091	170	34	and	and	CCONJ
easat-2091	170	35	(	(	PUNCT
easat-2091	170	36	𝜓	𝜓	PROPN
easat-2091	170	37	+	+	NOUN
easat-2091	170	38	1	1	X
easat-2091	170	39	)	)	PUNCT
easat-2091	170	40	∗	∗	NOUN
easat-2091	170	41	ℎ	ℎ	PROPN
easat-2091	170	42	as	as	ADP
easat-2091	170	43	the	the	DET
easat-2091	170	44	sum	sum	NOUN
easat-2091	170	45	of	of	ADP
easat-2091	170	46	an	an	DET
easat-2091	170	47	analytic	analytic	ADJ
easat-2091	170	48	function	function	NOUN
easat-2091	170	49	and	and	CCONJ
easat-2091	170	50	a	a	DET
easat-2091	170	51	conjugate	conjugate	ADJ
easat-2091	170	52	analytic	analytic	ADJ
easat-2091	170	53	function	function	NOUN
easat-2091	170	54	:	:	PUNCT
easat-2091	170	55	(	(	PUNCT
easat-2091	170	56	𝜑	𝜑	X
easat-2091	170	57	+	+	NOUN
easat-2091	170	58	1	1	X
easat-2091	170	59	)	)	PUNCT
easat-2091	170	60	∗	∗	NOUN
easat-2091	170	61	ℎ	ℎ	PROPN
easat-2091	170	62	=	=	SYM
easat-2091	170	63	𝑓1	𝑓1	ADJ
easat-2091	170	64	∗	∗	NOUN
easat-2091	170	65	ℎ	ℎ	ADP
easat-2091	170	66	+	+	CCONJ
easat-2091	170	67	𝑓2̅	𝑓2̅	PROPN
easat-2091	170	68	∗	∗	NOUN
easat-2091	170	69	ℎ	ℎ	PROPN
easat-2091	170	70	and	and	CCONJ
easat-2091	170	71	(	(	PUNCT
easat-2091	170	72	𝜓	𝜓	PROPN
easat-2091	170	73	+	+	NOUN
easat-2091	170	74	1	1	X
easat-2091	170	75	)	)	PUNCT
easat-2091	170	76	∗	∗	NOUN
easat-2091	170	77	ℎ	ℎ	NOUN
easat-2091	170	78	=	=	PUNCT
easat-2091	170	79	g1	g1	PROPN
easat-2091	170	80	∗	∗	NOUN
easat-2091	170	81	ℎ	ℎ	PROPN
easat-2091	170	82	+	+	CCONJ
easat-2091	170	83	g2̅̅	g2̅̅	ADP
easat-2091	170	84	̅	̅	NOUN
easat-2091	170	85	∗	∗	NOUN
easat-2091	170	86	ℎ	ℎ	NOUN
easat-2091	170	87	on	on	ADP
easat-2091	170	88	𝐷	𝐷	PROPN
easat-2091	170	89	(	(	PUNCT
easat-2091	170	90	13	13	NUM
easat-2091	170	91	)	)	PUNCT
easat-2091	170	92	from	from	ADP
easat-2091	170	93	eq(11	eq(11	NOUN
easat-2091	170	94	)	)	PUNCT
easat-2091	170	95	was	be	AUX
easat-2091	170	96	derived	derive	VERB
easat-2091	170	97	under	under	ADP
easat-2091	170	98	the	the	DET
easat-2091	170	99	assumption	assumption	NOUN
easat-2091	170	100	that	that	SCONJ
easat-2091	171	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	171	2	+	+	CCONJ
easat-2091	171	3	1𝑇𝜓	1𝑇𝜓	NUM
easat-2091	171	4	+	+	SYM
easat-2091	171	5	1	1	NUM
easat-2091	171	6	=	=	SYM
easat-2091	172	1	𝑇𝜓	𝑇𝜓	PROPN
easat-2091	172	2	+	+	NOUN
easat-2091	172	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	172	4	+	+	CCONJ
easat-2091	172	5	1;thus	1;thus	NUM
easat-2091	172	6	eq(12	eq(12	ADJ
easat-2091	172	7	)	)	PUNCT
easat-2091	172	8	,	,	PUNCT
easat-2091	172	9	combined	combine	VERB
easat-2091	172	10	with	with	ADP
easat-2091	172	11	eq(14	eq(14	NOUN
easat-2091	172	12	)	)	PUNCT
easat-2091	172	13	,	,	PUNCT
easat-2091	172	14	says	say	VERB
easat-2091	172	15	that	that	SCONJ
easat-2091	172	16	eq(11	eq(11	PROPN
easat-2091	172	17	)	)	PUNCT
easat-2091	172	18	is	be	AUX
easat-2091	172	19	still	still	ADV
easat-2091	172	20	valid	valid	ADJ
easat-2091	172	21	when	when	SCONJ
easat-2091	172	22	we	we	PRON
easat-2091	172	23	replace	replace	VERB
easat-2091	172	24	each	each	DET
easat-2091	172	25	function	function	NOUN
easat-2091	172	26	in	in	ADP
easat-2091	172	27	it	it	PRON
easat-2091	172	28	by	by	ADP
easat-2091	172	29	its	its	PRON
easat-2091	172	30	composition	composition	NOUN
easat-2091	172	31	with	with	ADP
easat-2091	172	32	ℎ.	ℎ.	NOUN
easat-2091	172	33	in	in	ADP
easat-2091	172	34	other	other	ADJ
easat-2091	172	35	words	word	NOUN
easat-2091	172	36	,	,	PUNCT
easat-2091	172	37	∫𝐷(𝑓2̅g1	∫𝐷(𝑓2̅g1	NOUN
easat-2091	172	38	−	−	NOUN
easat-2091	172	39	𝑓1g2̅̅	𝑓1g2̅̅	NOUN
easat-2091	172	40	̅	̅	NOUN
easat-2091	172	41	)	)	PUNCT
easat-2091	172	42	∗	∗	NOUN
easat-2091	172	43	ℎ	ℎ	ADP
easat-2091	172	44	𝑑𝐴	𝑑𝐴	PRON
easat-2091	172	45	𝜋	𝜋	X
easat-2091	172	46	=	=	NOUN
easat-2091	172	47	𝑓2̅(ℎ(0))g1(ℎ(0	𝑓2̅(ℎ(0))g1(ℎ(0	NOUN
easat-2091	172	48	)	)	PUNCT
easat-2091	172	49	)	)	PUNCT
easat-2091	173	1	−	−	PROPN
easat-2091	173	2	𝑓1(ℎ(0))g2̅̅	𝑓1(ℎ(0))g2̅̅	VERB
easat-2091	173	3	̅(ℎ(0	̅(ℎ(0	NOUN
easat-2091	173	4	)	)	PUNCT
easat-2091	173	5	)	)	PUNCT
easat-2091	174	1	letting	let	VERB
easat-2091	174	2	𝑢	𝑢	X
easat-2091	174	3	=	=	SYM
easat-2091	174	4	𝑓2̅g1	𝑓2̅g1	ADJ
easat-2091	174	5	−	−	PROPN
easat-2091	174	6	𝑓1g2̅̅	𝑓1g2̅̅	NOUN
easat-2091	174	7	̅	̅	NOUN
easat-2091	174	8	the	the	DET
easat-2091	174	9	equation	equation	NOUN
easat-2091	174	10	above	above	ADV
easat-2091	174	11	becomes	become	VERB
easat-2091	174	12	∫𝐷𝑢	∫𝐷𝑢	NUM
easat-2091	174	13	∗	∗	NOUN
easat-2091	174	14	ℎ	ℎ	ADP
easat-2091	174	15	𝑑𝐴	𝑑𝐴	NOUN
easat-2091	174	16	𝜋	𝜋	NOUN
easat-2091	174	17	=	=	SYM
easat-2091	174	18	𝑢(ℎ(0	𝑢(ℎ(0	NOUN
easat-2091	174	19	)	)	PUNCT
easat-2091	174	20	)	)	PUNCT
easat-2091	174	21	in	in	ADP
easat-2091	174	22	the	the	DET
easat-2091	174	23	other	other	ADJ
easat-2091	174	24	words	word	NOUN
easat-2091	174	25	,	,	PUNCT
easat-2091	174	26	𝑢	𝑢	PROPN
easat-2091	174	27	has	have	VERB
easat-2091	174	28	the	the	DET
easat-2091	174	29	area	area	NOUN
easat-2091	174	30	version	version	NOUN
easat-2091	174	31	of	of	ADP
easat-2091	174	32	the	the	DET
easat-2091	174	33	invariant	invariant	ADJ
easat-2091	174	34	mean	mean	NOUN
easat-2091	174	35	value	value	NOUN
easat-2091	174	36	property	property	NOUN
easat-2091	174	37	.	.	PUNCT
easat-2091	175	1	399	399	NUM
easat-2091	175	2	edelweiss	edelweiss	PROPN
easat-2091	175	3	applied	apply	VERB
easat-2091	175	4	science	science	NOUN
easat-2091	175	5	and	and	CCONJ
easat-2091	175	6	technology	technology	NOUN
easat-2091	175	7	issn	issn	PROPN
easat-2091	175	8	:	:	PUNCT
easat-2091	175	9	2576	2576	NUM
easat-2091	175	10	-	-	SYM
easat-2091	175	11	8484	8484	NUM
easat-2091	175	12	vol	vol	NOUN
easat-2091	175	13	.	.	PROPN
easat-2091	175	14	8	8	NUM
easat-2091	175	15	,	,	PUNCT
easat-2091	175	16	no	no	INTJ
easat-2091	175	17	.	.	NOUN
easat-2091	176	1	6	6	NUM
easat-2091	176	2	:	:	SYM
easat-2091	176	3	394	394	NUM
easat-2091	176	4	-	-	SYM
easat-2091	176	5	400	400	NUM
easat-2091	176	6	,	,	PUNCT
easat-2091	176	7	2024	2024	NUM
easat-2091	176	8	doi	doi	NOUN
easat-2091	176	9	:	:	PUNCT
easat-2091	176	10	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	176	11	©	©	ADP
easat-2091	176	12	2024	2024	NUM
easat-2091	176	13	by	by	ADP
easat-2091	176	14	the	the	DET
easat-2091	176	15	author	author	NOUN
easat-2091	176	16	;	;	PUNCT
easat-2091	176	17	licensee	licensee	PROPN
easat-2091	176	18	learning	learning	NOUN
easat-2091	176	19	gate	gate	NOUN
easat-2091	176	20	we	we	PRON
easat-2091	176	21	can	can	AUX
easat-2091	176	22	want	want	VERB
easat-2091	176	23	to	to	PART
easat-2091	176	24	show	show	VERB
easat-2091	176	25	that	that	SCONJ
easat-2091	176	26	𝑢	𝑢	NOUN
easat-2091	176	27	is	be	AUX
easat-2091	176	28	harmonic	harmonic	ADJ
easat-2091	176	29	on𝐷.	on𝐷.	ADP
easat-2091	176	30	by	by	ADP
easat-2091	176	31	the	the	DET
easat-2091	176	32	above	above	ADJ
easat-2091	176	33	equation	equation	NOUN
easat-2091	176	34	and	and	CCONJ
easat-2091	176	35	lemma	lemma	PROPN
easat-2091	176	36	2	2	NUM
easat-2091	176	37	,	,	PUNCT
easat-2091	176	38	we	we	PRON
easat-2091	176	39	need	need	AUX
easat-2091	176	40	only	only	ADV
easat-2091	176	41	show	show	VERB
easat-2091	176	42	thatℛ(𝑢	thatℛ(𝑢	PROPN
easat-2091	176	43	∗	∗	NOUN
easat-2091	176	44	ℎ	ℎ	PROPN
easat-2091	176	45	)	)	PUNCT
easat-2091	176	46	∈	∈	PROPN
easat-2091	176	47	𝐶(	𝐶(	NOUN
easat-2091	176	48	�	�	PROPN
easat-2091	176	49	̅	̅	NOUN
easat-2091	176	50	�	�	NOUN
easat-2091	176	51	)	)	PUNCT
easat-2091	176	52	.	.	PUNCT
easat-2091	177	1	to	to	PART
easat-2091	177	2	do	do	VERB
easat-2091	177	3	this	this	PRON
easat-2091	177	4	,	,	PUNCT
easat-2091	177	5	represent	represent	VERB
easat-2091	177	6	the	the	DET
easat-2091	177	7	analytic	analytic	ADJ
easat-2091	177	8	functions	function	NOUN
easat-2091	177	9	𝑓2	𝑓2	VERB
easat-2091	177	10	∗	∗	NOUN
easat-2091	177	11	ℎ	ℎ	NOUN
easat-2091	177	12	and	and	CCONJ
easat-2091	177	13	g1	g1	PROPN
easat-2091	177	14	∗	∗	NOUN
easat-2091	177	15	ℎ	ℎ	PROPN
easat-2091	177	16	as	as	ADP
easat-2091	177	17	taylor	taylor	PROPN
easat-2091	177	18	series	series	PROPN
easat-2091	177	19	:	:	PUNCT
easat-2091	177	20	(	(	PUNCT
easat-2091	177	21	𝑓2	𝑓2	NOUN
easat-2091	177	22	∗	∗	NOUN
easat-2091	177	23	ℎ)(𝑧	ℎ)(𝑧	NOUN
easat-2091	177	24	)	)	PUNCT
easat-2091	177	25	=	=	PUNCT
easat-2091	177	26	∑𝛼𝑛𝑍	∑𝛼𝑛𝑍	PRON
easat-2091	177	27	𝑛	𝑛	PRON
easat-2091	177	28	∞	∞	NUM
easat-2091	177	29	𝑛=0	𝑛=0	NOUN
easat-2091	177	30	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-2091	177	31	(	(	PUNCT
easat-2091	177	32	g1	g1	PROPN
easat-2091	177	33	∗	∗	NOUN
easat-2091	177	34	ℎ)(𝑧)∑𝛽𝑛𝑍	ℎ)(𝑧)∑𝛽𝑛𝑍	VERB
easat-2091	177	35	𝑛	𝑛	PRON
easat-2091	177	36	∞	∞	NUM
easat-2091	177	37	𝑛=0	𝑛=0	PROPN
easat-2091	177	38	,	,	PUNCT
easat-2091	177	39	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
easat-2091	177	40	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
easat-2091	177	41	𝑍	𝑍	PROPN
easat-2091	177	42	∈	∈	PROPN
easat-2091	177	43	𝐷	𝐷	NOUN
easat-2091	177	44	because	because	SCONJ
easat-2091	177	45	(	(	PUNCT
easat-2091	177	46	𝜑	𝜑	X
easat-2091	177	47	+	+	NOUN
easat-2091	177	48	1	1	X
easat-2091	177	49	)	)	PUNCT
easat-2091	177	50	∗	∗	NOUN
easat-2091	177	51	ℎ	ℎ	PROPN
easat-2091	177	52	and	and	CCONJ
easat-2091	177	53	(	(	PUNCT
easat-2091	177	54	𝜓	𝜓	PROPN
easat-2091	177	55	+	+	NOUN
easat-2091	177	56	1	1	X
easat-2091	177	57	)	)	PUNCT
easat-2091	177	58	∗	∗	NOUN
easat-2091	177	59	ℎ	ℎ	NOUN
easat-2091	177	60	are	be	AUX
easat-2091	177	61	bounded	bounded	ADJ
easat-2091	177	62	harmonic	harmonic	ADJ
easat-2091	177	63	functions	function	NOUN
easat-2091	177	64	on𝐷	on𝐷	PROPN
easat-2091	177	65	,	,	PUNCT
easat-2091	177	66	eq(13	eq(13	PROPN
easat-2091	177	67	)	)	PUNCT
easat-2091	177	68	implies	imply	VERB
easat-2091	177	69	that	that	SCONJ
easat-2091	177	70	functions	function	VERB
easat-2091	177	71	𝑓2	𝑓2	NOUN
easat-2091	177	72	∗	∗	NOUN
easat-2091	177	73	ℎ	ℎ	NOUN
easat-2091	177	74	and	and	CCONJ
easat-2091	177	75	g1	g1	PROPN
easat-2091	177	76	∗	∗	NOUN
easat-2091	177	77	ℎ	ℎ	NOUN
easat-2091	177	78	are	be	AUX
easat-2091	177	79	in	in	ADP
easat-2091	177	80	𝐻2(𝐷	𝐻2(𝐷	PROPN
easat-2091	177	81	)	)	PUNCT
easat-2091	177	82	𝐻2(𝐷	𝐻2(𝐷	PROPN
easat-2091	177	83	)	)	PUNCT
easat-2091	177	84	,	,	PUNCT
easat-2091	177	85	so	so	SCONJ
easat-2091	177	86	that	that	SCONJ
easat-2091	178	1	∑|𝛼𝑛|	∑|𝛼𝑛|	CCONJ
easat-2091	179	1	2	2	NUM
easat-2091	179	2	∞	∞	NUM
easat-2091	179	3	𝑛=0	𝑛=0	NOUN
easat-2091	179	4	<	<	X
easat-2091	179	5	∞	∞	PROPN
easat-2091	179	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-2091	179	7	∑|𝛽𝑛|	∑|𝛽𝑛|	PROPN
easat-2091	179	8	2	2	NUM
easat-2091	179	9	∞	∞	NUM
easat-2091	179	10	𝑛=0	𝑛=0	X
easat-2091	179	11	<	<	X
easat-2091	179	12	∞	∞	PROPN
easat-2091	179	13	(	(	PUNCT
easat-2091	179	14	14	14	NUM
easat-2091	179	15	)	)	PUNCT
easat-2091	179	16	now	now	ADV
easat-2091	179	17	for	for	ADP
easat-2091	179	18	𝑍	𝑍	PROPN
easat-2091	179	19	∈	∈	NOUN
easat-2091	179	20	𝐷	𝐷	NOUN
easat-2091	179	21	we	we	PRON
easat-2091	179	22	have	have	VERB
easat-2091	179	23	ℛ	ℛ	PROPN
easat-2091	179	24	(	(	PUNCT
easat-2091	179	25	(	(	PUNCT
easat-2091	179	26	𝑓2̅g1	𝑓2̅g1	NOUN
easat-2091	179	27	)	)	PUNCT
easat-2091	179	28	∗	∗	NOUN
easat-2091	179	29	ℎ	ℎ	PROPN
easat-2091	179	30	)	)	PUNCT
easat-2091	179	31	(	(	PUNCT
easat-2091	179	32	𝑧	𝑧	X
easat-2091	179	33	)	)	PUNCT
easat-2091	179	34	=	=	SYM
easat-2091	179	35	∫	∫	PROPN
easat-2091	179	36	(	(	PUNCT
easat-2091	179	37	𝑓2̅	𝑓2̅	PROPN
easat-2091	179	38	∗	∗	VERB
easat-2091	179	39	ℎ)(𝑧𝑒	ℎ)(𝑧𝑒	PROPN
easat-2091	179	40	𝑖𝜃	𝑖𝜃	PROPN
easat-2091	179	41	)	)	PUNCT
easat-2091	179	42	2𝜋	2𝜋	NOUN
easat-2091	179	43	0	0	PUNCT
easat-2091	180	1	(	(	PUNCT
easat-2091	180	2	g1	g1	PROPN
easat-2091	180	3	∗	∗	PROPN
easat-2091	180	4	ℎ)(𝑧𝑒	ℎ)(𝑧𝑒	PROPN
easat-2091	180	5	𝑖𝜃	𝑖𝜃	PROPN
easat-2091	180	6	)	)	PUNCT
easat-2091	180	7	𝑑𝜃	𝑑𝜃	ADP
easat-2091	180	8	2𝜋	2𝜋	NOUN
easat-2091	180	9	=	=	SYM
easat-2091	180	10	∑𝛼𝑛̅̅̅̅	∑𝛼𝑛̅̅̅̅	PROPN
easat-2091	180	11	𝛽𝑛|𝑍|	𝛽𝑛|𝑍|	NUM
easat-2091	181	1	2𝑛	2𝑛	PROPN
easat-2091	181	2	∞	∞	PROPN
easat-2091	182	1	𝑛=0	𝑛=0	PROPN
easat-2091	182	2	the	the	DET
easat-2091	182	3	inequalities	inequality	NOUN
easat-2091	182	4	in	in	ADP
easat-2091	182	5	eq(14	eq(14	NOUN
easat-2091	182	6	)	)	PUNCT
easat-2091	182	7	imply	imply	VERB
easat-2091	182	8	that	that	SCONJ
easat-2091	182	9	∑	∑	PUNCT
easat-2091	182	10	|𝛼𝑛𝛽𝑛|	|𝛼𝑛𝛽𝑛|	NOUN
easat-2091	182	11	∞	∞	NUM
easat-2091	182	12	𝑛=0	𝑛=0	X
easat-2091	182	13	<	<	X
easat-2091	182	14	∞	∞	PROPN
easat-2091	182	15	,	,	PUNCT
easat-2091	182	16	so	so	CCONJ
easat-2091	182	17	the	the	DET
easat-2091	182	18	above	above	ADJ
easat-2091	182	19	formula	formula	NOUN
easat-2091	182	20	for	for	ADP
easat-2091	182	21	ℛ	ℛ	PROPN
easat-2091	182	22	(	(	PUNCT
easat-2091	182	23	(	(	PUNCT
easat-2091	182	24	𝑓2̅g1	𝑓2̅g1	NOUN
easat-2091	182	25	)	)	PUNCT
easat-2091	182	26	∗	∗	NOUN
easat-2091	182	27	ℎ	ℎ	PROPN
easat-2091	182	28	)	)	PUNCT
easat-2091	182	29	∈	∈	PROPN
easat-2091	182	30	𝐶(𝐷	𝐶(𝐷	PROPN
easat-2091	182	31	)	)	PUNCT
easat-2091	182	32	shows	show	VERB
easat-2091	182	33	that	that	SCONJ
easat-2091	182	34	ℛ	ℛ	PROPN
easat-2091	182	35	(	(	PUNCT
easat-2091	182	36	(	(	PUNCT
easat-2091	182	37	𝑓2̅g1	𝑓2̅g1	NOUN
easat-2091	182	38	)	)	PUNCT
easat-2091	182	39	∗	∗	NOUN
easat-2091	182	40	ℎ	ℎ	PROPN
easat-2091	182	41	)	)	PUNCT
easat-2091	182	42	∈	∈	PROPN
easat-2091	182	43	𝐶(	𝐶(	NOUN
easat-2091	182	44	�	�	PROPN
easat-2091	182	45	̅	̅	NOUN
easat-2091	182	46	�	�	NUM
easat-2091	182	47	)	)	PUNCT
easat-2091	182	48	;	;	PUNCT
easat-2091	182	49	similarly	similarly	ADV
easat-2091	182	50	,	,	PUNCT
easat-2091	182	51	we	we	PRON
easat-2091	182	52	get	get	VERB
easat-2091	182	53	that	that	SCONJ
easat-2091	182	54	ℛ((𝑓1g2̅̅	ℛ((𝑓1g2̅̅	NUM
easat-2091	182	55	̅	̅	NOUN
easat-2091	182	56	)	)	PUNCT
easat-2091	182	57	∗	∗	NOUN
easat-2091	182	58	ℎ	ℎ	PROPN
easat-2091	182	59	)	)	PUNCT
easat-2091	182	60	∈	∈	PROPN
easat-2091	182	61	𝐶(	𝐶(	NOUN
easat-2091	182	62	�	�	PROPN
easat-2091	182	63	̅	̅	NOUN
easat-2091	182	64	�	�	NOUN
easat-2091	182	65	)	)	PUNCT
easat-2091	182	66	.	.	PUNCT
easat-2091	183	1	so	so	ADV
easat-2091	183	2	thatℛ(𝑢	thatℛ(𝑢	PROPN
easat-2091	183	3	∗	∗	NOUN
easat-2091	183	4	ℎ	ℎ	PROPN
easat-2091	183	5	)	)	PUNCT
easat-2091	183	6	∈	∈	PROPN
easat-2091	183	7	𝐶(	𝐶(	NOUN
easat-2091	183	8	�	�	PROPN
easat-2091	183	9	̅	̅	NOUN
easat-2091	183	10	�	�	NOUN
easat-2091	183	11	)	)	PUNCT
easat-2091	183	12	,	,	PUNCT
easat-2091	183	13	as	as	SCONJ
easat-2091	183	14	desired	desire	VERB
easat-2091	183	15	.	.	PUNCT
easat-2091	184	1	thus	thus	ADV
easat-2091	184	2	at	at	ADP
easat-2091	184	3	this	this	DET
easat-2091	184	4	stage	stage	NOUN
easat-2091	184	5	of	of	ADP
easat-2091	184	6	the	the	DET
easat-2091	184	7	proof	proof	NOUN
easat-2091	184	8	we	we	PRON
easat-2091	184	9	know	know	VERB
easat-2091	184	10	that	that	SCONJ
easat-2091	184	11	𝑢	𝑢	NOUN
easat-2091	184	12	is	be	AUX
easat-2091	184	13	harmonic	harmonic	ADJ
easat-2091	184	14	.	.	PUNCT
easat-2091	185	1	let	let	VERB
easat-2091	185	2	𝜕	𝜕	PROPN
easat-2091	185	3	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	185	4	and	and	CCONJ
easat-2091	185	5	the	the	DET
easat-2091	185	6	𝜕	𝜕	NOUN
easat-2091	185	7	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	185	8	denote	denote	VERB
easat-2091	185	9	the	the	DET
easat-2091	185	10	usual	usual	ADJ
easat-2091	185	11	operators	operator	NOUN
easat-2091	185	12	defined	define	VERB
easat-2091	185	13	by	by	ADP
easat-2091	185	14	𝜕	𝜕	PROPN
easat-2091	185	15	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	185	16	=	=	SYM
easat-2091	185	17	1	1	NUM
easat-2091	185	18	2	2	NUM
easat-2091	185	19	(	(	PUNCT
easat-2091	185	20	𝜕	𝜕	NOUN
easat-2091	185	21	𝜕𝑥	𝜕𝑥	NOUN
easat-2091	185	22	−	−	PROPN
easat-2091	185	23	𝑖	𝑖	SYM
easat-2091	185	24	𝜕	𝜕	PROPN
easat-2091	185	25	𝜕𝑦	𝜕𝑦	PROPN
easat-2091	185	26	)	)	PUNCT
easat-2091	185	27	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-2091	185	28	𝜕	𝜕	PROPN
easat-2091	185	29	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	185	30	=	=	SYM
easat-2091	185	31	1	1	NUM
easat-2091	185	32	2	2	NUM
easat-2091	185	33	(	(	PUNCT
easat-2091	185	34	𝜕	𝜕	NOUN
easat-2091	185	35	𝜕𝑥	𝜕𝑥	X
easat-2091	185	36	+	+	CCONJ
easat-2091	185	37	𝑖	𝑖	SYM
easat-2091	185	38	𝜕	𝜕	NOUN
easat-2091	185	39	𝜕𝑦	𝜕𝑦	NOUN
easat-2091	185	40	)	)	PUNCT
easat-2091	186	1	if	if	SCONJ
easat-2091	186	2	𝑓	𝑓	PRON
easat-2091	186	3	is	be	AUX
easat-2091	186	4	analytic	analytic	ADJ
easat-2091	186	5	,	,	PUNCT
easat-2091	186	6	then	then	ADV
easat-2091	186	7	the	the	DET
easat-2091	186	8	cauchy	cauchy	PROPN
easat-2091	186	9	-	-	PUNCT
easat-2091	186	10	riemann	riemann	PROPN
easat-2091	186	11	equations	equation	NOUN
easat-2091	186	12	show	show	VERB
easat-2091	186	13	that	that	SCONJ
easat-2091	186	14	𝜕𝑓	𝜕𝑓	ADJ
easat-2091	186	15	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	186	16	=	=	SYM
easat-2091	186	17	𝑓	𝑓	PROPN
easat-2091	186	18	,	,	PUNCT
easat-2091	186	19	𝜕𝑓	𝜕𝑓	ADJ
easat-2091	186	20	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	186	21	=	=	SYM
easat-2091	186	22	0	0	NUM
easat-2091	186	23	,	,	PUNCT
easat-2091	186	24	𝜕𝑓	𝜕𝑓	ADJ
easat-2091	186	25	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	186	26	=	=	SYM
easat-2091	186	27	0	0	NUM
easat-2091	186	28	,	,	PUNCT
easat-2091	186	29	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-2091	186	30	𝜕𝑓	𝜕𝑓	NOUN
easat-2091	186	31	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	186	32	=	=	SYM
easat-2091	186	33	𝑓	𝑓	X
easat-2091	186	34	.	.	PUNCT
easat-2091	187	1	it	it	PRON
easat-2091	187	2	is	be	AUX
easat-2091	187	3	easy	easy	ADJ
easat-2091	187	4	to	to	PART
easat-2091	187	5	check	check	VERB
easat-2091	187	6	that	that	SCONJ
easat-2091	187	7	𝜕	𝜕	PROPN
easat-2091	187	8	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	187	9	and	and	CCONJ
easat-2091	187	10	𝜕	𝜕	PROPN
easat-2091	187	11	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	187	12	obey	obey	VERB
easat-2091	187	13	the	the	DET
easat-2091	187	14	usual	usual	ADJ
easat-2091	187	15	addition	addition	NOUN
easat-2091	187	16	and	and	CCONJ
easat-2091	187	17	multiplication	multiplication	NOUN
easat-2091	187	18	formulas	formula	NOUN
easat-2091	187	19	for	for	ADP
easat-2091	187	20	derivatives	derivative	NOUN
easat-2091	187	21	and	and	CCONJ
easat-2091	187	22	that	that	DET
easat-2091	187	23	𝜕2	𝜕2	NOUN
easat-2091	188	1	𝜕𝑥2	𝜕𝑥2	NOUN
easat-2091	188	2	+	+	CCONJ
easat-2091	188	3	𝜕2	𝜕2	NOUN
easat-2091	188	4	𝜕𝑦2	𝜕𝑦2	ADJ
easat-2091	188	5	=	=	SYM
easat-2091	188	6	4	4	NUM
easat-2091	188	7	𝜕	𝜕	PROPN
easat-2091	188	8	𝜕𝑧	𝜕𝑧	PROPN
easat-2091	188	9	𝜕	𝜕	NOUN
easat-2091	188	10	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	188	11	thus	thus	ADV
easat-2091	188	12	,	,	PUNCT
easat-2091	188	13	because	because	SCONJ
easat-2091	188	14	𝑢	𝑢	NOUN
easat-2091	188	15	is	be	AUX
easat-2091	188	16	harmonic	harmonic	ADJ
easat-2091	188	17	,	,	PUNCT
easat-2091	188	18	we	we	PRON
easat-2091	188	19	have	have	VERB
easat-2091	188	20	0	0	NUM
easat-2091	188	21	=	=	SYM
easat-2091	188	22	4	4	NUM
easat-2091	188	23	𝜕	𝜕	PROPN
easat-2091	188	24	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	188	25	(	(	PUNCT
easat-2091	188	26	𝜕𝑢	𝜕𝑢	NOUN
easat-2091	188	27	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	188	28	)	)	PUNCT
easat-2091	189	1	=	=	SYM
easat-2091	189	2	4	4	NUM
easat-2091	189	3	𝜕	𝜕	NOUN
easat-2091	189	4	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	189	5	|	|	ADV
easat-2091	189	6	𝜕(𝑓2̅g1	𝜕(𝑓2̅g1	NOUN
easat-2091	189	7	−	−	NOUN
easat-2091	189	8	𝑓1g2̅̅	𝑓1g2̅̅	NOUN
easat-2091	189	9	̅	̅	NOUN
easat-2091	189	10	)	)	PUNCT
easat-2091	189	11	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	189	12	|	|	NOUN
easat-2091	189	13	=	=	SYM
easat-2091	189	14	4	4	NUM
easat-2091	189	15	𝜕	𝜕	PROPN
easat-2091	189	16	𝜕𝑧	𝜕𝑧	NOUN
easat-2091	189	17	(	(	PUNCT
easat-2091	189	18	𝑓2̅g1	𝑓2̅g1	X
easat-2091	189	19	,	,	PUNCT
easat-2091	189	20	−	−	PROPN
easat-2091	189	21	f1	f1	NOUN
easat-2091	189	22	,	,	PUNCT
easat-2091	189	23	g2̅̅	g2̅̅	ADP
easat-2091	189	24	̅	̅	NOUN
easat-2091	189	25	)	)	PUNCT
easat-2091	190	1	=	=	SYM
easat-2091	190	2	4(𝑓2̅g1	4(𝑓2̅g1	NOUN
easat-2091	190	3	,	,	PUNCT
easat-2091	190	4	−	−	PROPN
easat-2091	190	5	𝑓1	𝑓1	PROPN
easat-2091	190	6	,	,	PUNCT
easat-2091	190	7	g2̅̅	g2̅̅	ADP
easat-2091	190	8	̅	̅	NOUN
easat-2091	190	9	)	)	PUNCT
easat-2091	190	10	hence	hence	ADV
easat-2091	190	11	𝑓1̅g2	𝑓1̅g2	X
easat-2091	190	12	,	,	PUNCT
easat-2091	190	13	−	−	PROPN
easat-2091	190	14	𝑓2	𝑓2	NOUN
easat-2091	190	15	,	,	PUNCT
easat-2091	190	16	g1̅̅	g1̅̅	NOUN
easat-2091	190	17	̅	̅	NOUN
easat-2091	190	18	(	(	PUNCT
easat-2091	190	19	15	15	NUM
easat-2091	190	20	)	)	PUNCT
easat-2091	190	21	we	we	PRON
easat-2091	190	22	finish	finish	VERB
easat-2091	190	23	the	the	DET
easat-2091	190	24	proof	proof	NOUN
easat-2091	190	25	by	by	ADP
easat-2091	190	26	showing	show	VERB
easat-2091	190	27	that	that	SCONJ
easat-2091	190	28	the	the	DET
easat-2091	190	29	above	above	ADJ
easat-2091	190	30	equation	equation	NOUN
easat-2091	190	31	implies	imply	VERB
easat-2091	190	32	that	that	SCONJ
easat-2091	190	33	(	(	PUNCT
easat-2091	190	34	a	a	X
easat-2091	190	35	)	)	PUNCT
easat-2091	190	36	,	,	PUNCT
easat-2091	190	37	(	(	PUNCT
easat-2091	190	38	b	b	NOUN
easat-2091	190	39	)	)	PUNCT
easat-2091	190	40	,	,	PUNCT
easat-2091	190	41	or	or	CCONJ
easat-2091	190	42	(	(	PUNCT
easat-2091	190	43	c	c	NOUN
easat-2091	190	44	)	)	PUNCT
easat-2091	190	45	holds	hold	VERB
easat-2091	190	46	.	.	PUNCT
easat-2091	191	1	if	if	SCONJ
easat-2091	191	2	g1	g1	PROPN
easat-2091	191	3	,	,	PUNCT
easat-2091	191	4	is	be	AUX
easat-2091	191	5	identically	identically	ADV
easat-2091	191	6	0	0	NUM
easat-2091	192	1	on𝐷	on𝐷	PROPN
easat-2091	192	2	,	,	PUNCT
easat-2091	192	3	then	then	ADV
easat-2091	192	4	eq(l5	eq(l5	NOUN
easat-2091	192	5	)	)	PUNCT
easat-2091	192	6	shows	show	VERB
easat-2091	192	7	that	that	SCONJ
easat-2091	192	8	either	either	CCONJ
easat-2091	192	9	g2	g2	PROPN
easat-2091	192	10	,	,	PUNCT
easat-2091	192	11	is	be	AUX
easat-2091	192	12	identically	identically	ADV
easat-2091	192	13	0	0	NUM
easat-2091	192	14	on	on	ADP
easat-2091	192	15	𝐷	𝐷	PROPN
easat-2091	192	16	,	,	PUNCT
easat-2091	192	17	then	then	ADV
easat-2091	192	18	(	(	PUNCT
easat-2091	192	19	𝜓	𝜓	PROPN
easat-2091	192	20	+	+	NOUN
easat-2091	192	21	1	1	NUM
easat-2091	192	22	)	)	PUNCT
easat-2091	192	23	would	would	AUX
easat-2091	192	24	be	be	AUX
easat-2091	192	25	constant	constant	ADJ
easat-2091	192	26	on	on	ADP
easat-2091	192	27	𝐷	𝐷	PROPN
easat-2091	192	28	and	and	CCONJ
easat-2091	192	29	(	(	PUNCT
easat-2091	192	30	c	c	X
easat-2091	192	31	)	)	PUNCT
easat-2091	192	32	would	would	AUX
easat-2091	192	33	hold	hold	VERB
easat-2091	192	34	or	or	CCONJ
easat-2091	192	35	𝑓1	𝑓1	PROPN
easat-2091	192	36	,	,	PUNCT
easat-2091	192	37	is	be	AUX
easat-2091	192	38	identically	identically	ADV
easat-2091	192	39	0	0	NUM
easat-2091	192	40	on	on	ADP
easat-2091	192	41	𝐷	𝐷	PROPN
easat-2091	192	42	,	,	PUNCT
easat-2091	192	43	so	so	ADV
easat-2091	192	44	both	both	PRON
easat-2091	192	45	(	(	PUNCT
easat-2091	192	46	𝜑	𝜑	PROPN
easat-2091	192	47	+	+	NOUN
easat-2091	192	48	1	1	NUM
easat-2091	192	49	)	)	PUNCT
easat-2091	192	50	and	and	CCONJ
easat-2091	192	51	(	(	PUNCT
easat-2091	192	52	𝜓	𝜓	PROPN
easat-2091	192	53	+	+	NOUN
easat-2091	192	54	1	1	NUM
easat-2091	192	55	)	)	PUNCT
easat-2091	192	56	would	would	AUX
easat-2091	192	57	be	be	AUX
easat-2091	192	58	analytic	analytic	ADJ
easat-2091	192	59	on	on	ADP
easat-2091	192	60	𝐷	𝐷	PROPN
easat-2091	192	61	and	and	CCONJ
easat-2091	192	62	(	(	PUNCT
easat-2091	192	63	b	b	NOUN
easat-2091	192	64	)	)	PUNCT
easat-2091	192	65	would	would	AUX
easat-2091	192	66	hold	hold	VERB
easat-2091	192	67	.	.	PUNCT
easat-2091	193	1	similarly	similarly	ADV
easat-2091	193	2	,	,	PUNCT
easat-2091	193	3	if	if	SCONJ
easat-2091	193	4	g2	g2	PROPN
easat-2091	193	5	,	,	PUNCT
easat-2091	193	6	is	be	AUX
easat-2091	193	7	identically	identically	ADV
easat-2091	193	8	0	0	NUM
easat-2091	194	1	on𝐷	on𝐷	PROPN
easat-2091	194	2	,	,	PUNCT
easat-2091	194	3	then	then	ADV
easat-2091	194	4	eq(l5	eq(l5	NOUN
easat-2091	194	5	)	)	PUNCT
easat-2091	194	6	shows	show	VERB
easat-2091	194	7	that	that	SCONJ
easat-2091	194	8	either	either	CCONJ
easat-2091	194	9	(	(	PUNCT
easat-2091	194	10	c	c	NOUN
easat-2091	194	11	)	)	PUNCT
easat-2091	194	12	or	or	CCONJ
easat-2091	194	13	(	(	PUNCT
easat-2091	194	14	a	a	X
easat-2091	194	15	)	)	PUNCT
easat-2091	194	16	would	would	AUX
easat-2091	194	17	hold	hold	VERB
easat-2091	194	18	.	.	PUNCT
easat-2091	195	1	thus	thus	ADV
easat-2091	195	2	,	,	PUNCT
easat-2091	195	3	we	we	PRON
easat-2091	195	4	may	may	AUX
easat-2091	195	5	assume	assume	VERB
easat-2091	195	6	that	that	SCONJ
easat-2091	195	7	neither	neither	CCONJ
easat-2091	195	8	g1	g1	NOUN
easat-2091	195	9	,	,	PUNCT
easat-2091	195	10	nor	nor	CCONJ
easat-2091	195	11	g2	g2	PROPN
easat-2091	195	12	,	,	PUNCT
easat-2091	195	13	is	be	AUX
easat-2091	195	14	identically	identically	ADV
easat-2091	195	15	0	0	NUM
easat-2091	195	16	on𝐷	on𝐷	PROPN
easat-2091	195	17	,	,	PUNCT
easat-2091	195	18	and	and	CCONJ
easat-2091	195	19	so	so	ADV
easat-2091	195	20	eq	eq	NOUN
easat-2091	195	21	(	(	PUNCT
easat-2091	195	22	15	15	NUM
easat-2091	195	23	)	)	PUNCT
easat-2091	195	24	shows	show	VERB
easat-2091	195	25	that	that	SCONJ
easat-2091	195	26	at	at	ADV
easat-2091	195	27	all	all	DET
easat-2091	195	28	points	point	NOUN
easat-2091	195	29	of	of	ADP
easat-2091	195	30	𝐷	𝐷	NOUN
easat-2091	195	31	except	except	SCONJ
easat-2091	195	32	the	the	DET
easat-2091	195	33	countable	countable	ADJ
easat-2091	195	34	set	set	NOUN
easat-2091	195	35	consisting	consisting	NOUN
easat-2091	195	36	of	of	ADP
easat-2091	195	37	the	the	DET
easat-2091	195	38	zeroes	zero	NOUN
easat-2091	195	39	of	of	ADP
easat-2091	195	40	g1	g1	PROPN
easat-2091	195	41	,	,	PUNCT
easat-2091	195	42	g2	g2	PROPN
easat-2091	195	43	,	,	PUNCT
easat-2091	195	44	.	.	PUNCT
easat-2091	196	1	𝑓1	𝑓1	PROPN
easat-2091	196	2	,	,	PUNCT
easat-2091	196	3	g1	g1	PROPN
easat-2091	196	4	,	,	PUNCT
easat-2091	196	5	=	=	PRON
easat-2091	196	6	[	[	PUNCT
easat-2091	196	7	𝑓2	𝑓2	PROPN
easat-2091	196	8	,	,	PUNCT
easat-2091	196	9	g2	g2	PROPN
easat-2091	196	10	,	,	PUNCT
easat-2091	196	11	]	]	PUNCT
easat-2091	196	12	−	−	PROPN
easat-2091	196	13	the	the	DET
easat-2091	196	14	left	left	ADJ
easat-2091	196	15	-	-	PUNCT
easat-2091	196	16	hand	hand	NOUN
easat-2091	196	17	side	side	NOUN
easat-2091	196	18	of	of	ADP
easat-2091	196	19	the	the	DET
easat-2091	196	20	above	above	ADJ
easat-2091	196	21	equation	equation	NOUN
easat-2091	196	22	is	be	AUX
easat-2091	196	23	an	an	DET
easat-2091	196	24	analytic	analytic	ADJ
easat-2091	196	25	function	function	NOUN
easat-2091	196	26	on	on	ADP
easat-2091	196	27	𝐷	𝐷	PROPN
easat-2091	196	28	with	with	ADP
easat-2091	196	29	the	the	DET
easat-2091	196	30	zeroes	zero	NOUN
easat-2091	196	31	ofg1	ofg1	PROPN
easat-2091	196	32	,	,	PUNCT
easat-2091	196	33	g2	g2	PROPN
easat-2091	196	34	,	,	PUNCT
easat-2091	196	35	,	,	PUNCT
easat-2091	196	36	deleted	delete	VERB
easat-2091	196	37	,	,	PUNCT
easat-2091	196	38	and	and	CCONJ
easat-2091	196	39	the	the	DET
easat-2091	196	40	right	right	ADJ
easat-2091	196	41	hand	hand	NOUN
easat-2091	196	42	side	side	NOUN
easat-2091	196	43	is	be	AUX
easat-2091	196	44	the	the	DET
easat-2091	196	45	complex	complex	ADJ
easat-2091	196	46	conjugate	conjugate	NOUN
easat-2091	196	47	of	of	ADP
easat-2091	196	48	an	an	DET
easat-2091	196	49	analytic	analytic	ADJ
easat-2091	196	50	function	function	NOUN
easat-2091	196	51	on	on	ADP
easat-2091	196	52	the	the	DET
easat-2091	196	53	same	same	ADJ
easat-2091	196	54	domain	domain	NOUN
easat-2091	196	55	,	,	PUNCT
easat-2091	196	56	and	and	CCONJ
easat-2091	196	57	so	so	ADV
easat-2091	196	58	both	both	DET
easat-2091	196	59	sides	side	NOUN
easat-2091	196	60	must	must	AUX
easat-2091	196	61	equal	equal	VERB
easat-2091	196	62	a	a	DET
easat-2091	196	63	constant	constant	ADJ
easat-2091	196	64	𝑐	𝑐	NOUN
easat-2091	196	65	∈	∈	PROPN
easat-2091	196	66	𝐶.	𝐶.	PROPN
easat-2091	196	67	thus𝑓1	thus𝑓1	NOUN
easat-2091	196	68	,	,	PUNCT
easat-2091	196	69	=	=	PUNCT
easat-2091	196	70	𝑐g1	𝑐g1	NOUN
easat-2091	196	71	,	,	PUNCT
easat-2091	196	72	,	,	PUNCT
easat-2091	196	73	and	and	CCONJ
easat-2091	196	74	𝑓2	𝑓2	NOUN
easat-2091	196	75	,	,	PUNCT
easat-2091	196	76	=	=	PUNCT
easat-2091	196	77	𝑐	𝑐	PROPN
easat-2091	196	78	g2	g2	PROPN
easat-2091	196	79	,	,	PUNCT
easat-2091	196	80	on𝐷.	on𝐷.	ADP
easat-2091	196	81	hence	hence	ADV
easat-2091	196	82	𝑓1	𝑓1	ADJ
easat-2091	196	83	−	−	PROPN
easat-2091	196	84	𝑐g1	𝑐g1	NOUN
easat-2091	196	85	and	and	CCONJ
easat-2091	196	86	𝑓	𝑓	DET
easat-2091	196	87	2	2	NUM
easat-2091	196	88	−	−	NOUN
easat-2091	196	89	𝑐g	𝑐g	NOUN
easat-2091	196	90	2	2	NUM
easat-2091	196	91	are	be	AUX
easat-2091	196	92	constant	constant	ADJ
easat-2091	196	93	on	on	ADP
easat-2091	196	94	𝐷	𝐷	PROPN
easat-2091	196	95	,	,	PUNCT
easat-2091	196	96	and	and	CCONJ
easat-2091	196	97	so	so	ADV
easat-2091	196	98	their	their	PRON
easat-2091	196	99	sum	sum	NOUN
easat-2091	196	100	,	,	PUNCT
easat-2091	196	101	which	which	PRON
easat-2091	196	102	equals	equal	VERB
easat-2091	196	103	(	(	PUNCT
easat-2091	196	104	𝜑	𝜑	NOUN
easat-2091	196	105	+	+	NOUN
easat-2091	196	106	1	1	NUM
easat-2091	196	107	)	)	PUNCT
easat-2091	196	108	−	−	NOUN
easat-2091	196	109	𝑐(𝜓	𝑐(𝜓	PRON
easat-2091	196	110	+	+	CCONJ
easat-2091	196	111	1	1	NUM
easat-2091	196	112	)	)	PUNCT
easat-2091	196	113	,	,	PUNCT
easat-2091	196	114	is	be	AUX
easat-2091	196	115	constant	constant	ADJ
easat-2091	196	116	on	on	ADP
easat-2091	196	117	𝐷	𝐷	NOUN
easat-2091	196	118	;	;	PUNCT
easat-2091	196	119	in	in	ADP
easat-2091	196	120	other	other	ADJ
easat-2091	196	121	400	400	NUM
easat-2091	196	122	edelweiss	edelweiss	PROPN
easat-2091	196	123	applied	apply	VERB
easat-2091	196	124	science	science	NOUN
easat-2091	196	125	and	and	CCONJ
easat-2091	196	126	technology	technology	NOUN
easat-2091	196	127	issn	issn	PROPN
easat-2091	196	128	:	:	PUNCT
easat-2091	196	129	2576	2576	NUM
easat-2091	196	130	-	-	SYM
easat-2091	196	131	8484	8484	NUM
easat-2091	196	132	vol	vol	NOUN
easat-2091	196	133	.	.	PROPN
easat-2091	196	134	8	8	NUM
easat-2091	196	135	,	,	PUNCT
easat-2091	196	136	no	no	INTJ
easat-2091	196	137	.	.	NOUN
easat-2091	196	138	6	6	NUM
easat-2091	196	139	:	:	SYM
easat-2091	196	140	394	394	NUM
easat-2091	196	141	-	-	SYM
easat-2091	196	142	400	400	NUM
easat-2091	196	143	,	,	PUNCT
easat-2091	196	144	2024	2024	NUM
easat-2091	196	145	doi	doi	NOUN
easat-2091	196	146	:	:	PUNCT
easat-2091	196	147	10.55214/25768484.v8i6.2091	10.55214/25768484.v8i6.2091	NUM
easat-2091	196	148	©	©	ADP
easat-2091	196	149	2024	2024	NUM
easat-2091	196	150	by	by	ADP
easat-2091	196	151	the	the	DET
easat-2091	196	152	author	author	NOUN
easat-2091	196	153	;	;	PUNCT
easat-2091	196	154	licensee	licensee	PROPN
easat-2091	196	155	learning	learn	VERB
easat-2091	196	156	gate	gate	NOUN
easat-2091	196	157	words	word	NOUN
easat-2091	196	158	,	,	PUNCT
easat-2091	196	159	(	(	PUNCT
easat-2091	196	160	c	c	X
easat-2091	196	161	)	)	PUNCT
easat-2091	196	162	holds	hold	NOUN
easat-2091	196	163	and	and	CCONJ
easat-2091	196	164	the	the	DET
easat-2091	196	165	proof	proof	NOUN
easat-2091	196	166	of	of	ADP
easat-2091	196	167	theorem	theorem	ADJ
easat-2091	196	168	1	1	NUM
easat-2091	196	169	is	be	AUX
easat-2091	196	170	complete	complete	ADJ
easat-2091	196	171	.	.	PUNCT
easat-2091	197	1	recall	recall	VERB
easat-2091	197	2	that	that	SCONJ
easat-2091	197	3	an	an	DET
easat-2091	197	4	operator	operator	NOUN
easat-2091	197	5	is	be	AUX
easat-2091	197	6	called	call	VERB
easat-2091	197	7	normal	normal	ADJ
easat-2091	197	8	if	if	SCONJ
easat-2091	197	9	it	it	PRON
easat-2091	197	10	commutes	commute	VERB
easat-2091	197	11	with	with	ADP
easat-2091	197	12	its	its	PRON
easat-2091	197	13	adjoint	adjoint	NOUN
easat-2091	197	14	.	.	PUNCT
easat-2091	198	1	we	we	PRON
easat-2091	198	2	can	can	AUX
easat-2091	198	3	use	use	VERB
easat-2091	198	4	theorem1	theorem1	NOUN
easat-2091	198	5	to	to	PART
easat-2091	198	6	prove	prove	VERB
easat-2091	198	7	the	the	DET
easat-2091	198	8	following	follow	VERB
easat-2091	198	9	corollary	corollary	NOUN
easat-2091	198	10	,	,	PUNCT
easat-2091	198	11	which	which	PRON
easat-2091	198	12	states	state	VERB
easat-2091	198	13	that	that	SCONJ
easat-2091	198	14	for	for	ADP
easat-2091	198	15	(	(	PUNCT
easat-2091	198	16	𝜑	𝜑	NOUN
easat-2091	198	17	+	+	NOUN
easat-2091	198	18	1	1	X
easat-2091	198	19	)	)	PUNCT
easat-2091	198	20	a	a	DET
easat-2091	198	21	bounded	bound	VERB
easat-2091	198	22	harmonic	harmonic	ADJ
easat-2091	198	23	function	function	NOUN
easat-2091	198	24	on𝐷	on𝐷	PROPN
easat-2091	198	25	,	,	PUNCT
easat-2091	198	26	the	the	DET
easat-2091	198	27	toeplitz	toeplitz	NOUN
easat-2091	198	28	operator	operator	NOUN
easat-2091	198	29	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	198	30	+	+	CCONJ
easat-2091	198	31	1is	1is	ADJ
easat-2091	198	32	normal	normal	ADJ
easat-2091	198	33	only	only	ADV
easat-2091	198	34	in	in	ADP
easat-2091	198	35	the	the	DET
easat-2091	198	36	obvious	obvious	ADJ
easat-2091	198	37	case	case	NOUN
easat-2091	198	38	.	.	PUNCT
easat-2091	199	1	corollary	corollary	ADJ
easat-2091	199	2	5	5	NUM
easat-2091	199	3	.	.	PUNCT
easat-2091	199	4	suppose	suppose	VERB
easat-2091	199	5	that	that	SCONJ
easat-2091	199	6	(	(	PUNCT
easat-2091	199	7	𝜑	𝜑	X
easat-2091	199	8	+	+	NOUN
easat-2091	199	9	1	1	NUM
easat-2091	199	10	)	)	PUNCT
easat-2091	199	11	is	be	AUX
easat-2091	199	12	a	a	DET
easat-2091	199	13	bounded	bounded	ADJ
easat-2091	199	14	harmonic	harmonic	ADJ
easat-2091	199	15	function	function	NOUN
easat-2091	199	16	on𝐷.	on𝐷.	ADP
easat-2091	199	17	then	then	ADV
easat-2091	199	18	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	200	1	+	+	CCONJ
easat-2091	200	2	1	1	NUM
easat-2091	200	3	is	be	AUX
easat-2091	200	4	a	a	DET
easat-2091	200	5	normal	normal	ADJ
easat-2091	200	6	operator	operator	NOUN
easat-2091	200	7	if	if	SCONJ
easat-2091	200	8	and	and	CCONJ
easat-2091	200	9	only	only	ADV
easat-2091	200	10	if	if	SCONJ
easat-2091	200	11	(	(	PUNCT
easat-2091	200	12	𝜑	𝜑	X
easat-2091	200	13	+	+	NUM
easat-2091	200	14	1)(𝐷	1)(𝐷	NUM
easat-2091	200	15	)	)	PUNCT
easat-2091	200	16	lies	lie	VERB
easat-2091	200	17	on	on	ADP
easat-2091	200	18	some	some	DET
easat-2091	200	19	line	line	NOUN
easat-2091	200	20	in𝐶.	in𝐶.	NOUN
easat-2091	200	21	proof	proof	NOUN
easat-2091	200	22	:	:	PUNCT
easat-2091	200	23	first	first	ADV
easat-2091	200	24	,	,	PUNCT
easat-2091	200	25	suppose	suppose	VERB
easat-2091	200	26	that	that	SCONJ
easat-2091	200	27	(	(	PUNCT
easat-2091	200	28	𝜑	𝜑	X
easat-2091	200	29	+	+	NUM
easat-2091	200	30	1)(𝐷	1)(𝐷	NUM
easat-2091	200	31	)	)	PUNCT
easat-2091	200	32	lies	lie	VERB
easat-2091	200	33	on	on	ADP
easat-2091	200	34	some	some	DET
easat-2091	200	35	line	line	NOUN
easat-2091	200	36	in𝐶.	in𝐶.	NOUN
easat-2091	200	37	then	then	ADV
easat-2091	200	38	there	there	PRON
easat-2091	200	39	exist	exist	VERB
easat-2091	200	40	constants𝛼	constants𝛼	NOUN
easat-2091	200	41	,	,	PUNCT
easat-2091	200	42	𝛽	𝛽	PROPN
easat-2091	200	43	∈	∈	PROPN
easat-2091	200	44	𝐶	𝐶	PROPN
easat-2091	200	45	,	,	PUNCT
easat-2091	200	46	with𝛼	with𝛼	NOUN
easat-2091	200	47	≠	≠	PROPN
easat-2091	200	48	0	0	NUM
easat-2091	200	49	,	,	PUNCT
easat-2091	200	50	such	such	ADJ
easat-2091	200	51	that𝛼(𝜑	that𝛼(𝜑	PROPN
easat-2091	200	52	+	+	CCONJ
easat-2091	200	53	1	1	NUM
easat-2091	200	54	)	)	PUNCT
easat-2091	201	1	+	+	X
easat-2091	202	1	𝛽	𝛽	NOUN
easat-2091	202	2	,	,	PUNCT
easat-2091	202	3	is	be	AUX
easat-2091	202	4	real	real	ADV
easat-2091	202	5	valued	value	VERB
easat-2091	202	6	on𝐷.	on𝐷.	ADP
easat-2091	202	7	so	so	SCONJ
easat-2091	202	8	that	that	SCONJ
easat-2091	202	9	𝑇𝛼(𝜑	𝑇𝛼(𝜑	PROPN
easat-2091	202	10	+	+	CCONJ
easat-2091	202	11	1)+𝛽	1)+𝛽	PROPN
easat-2091	202	12	is	be	AUX
easat-2091	202	13	a	a	DET
easat-2091	202	14	self	self	NOUN
easat-2091	202	15	-	-	PUNCT
easat-2091	202	16	adjoint	adjoint	NOUN
easat-2091	202	17	operator	operator	NOUN
easat-2091	202	18	,	,	PUNCT
easat-2091	202	19	and	and	CCONJ
easat-2091	202	20	hence	hence	ADV
easat-2091	202	21	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	202	22	+	+	CCONJ
easat-2091	202	23	1	1	NUM
easat-2091	202	24	which	which	PRON
easat-2091	202	25	equals𝛼−1𝑇𝛼(𝜑	equals𝛼−1𝑇𝛼(𝜑	PROPN
easat-2091	202	26	+	+	CCONJ
easat-2091	202	27	1)+𝛽−𝛽𝐼	1)+𝛽−𝛽𝐼	NUM
easat-2091	202	28	,	,	PUNCT
easat-2091	202	29	is	be	AUX
easat-2091	202	30	a	a	DET
easat-2091	202	31	normal	normal	ADJ
easat-2091	202	32	operator	operator	NOUN
easat-2091	202	33	.	.	PUNCT
easat-2091	203	1	to	to	PART
easat-2091	203	2	prove	prove	VERB
easat-2091	203	3	the	the	DET
easat-2091	203	4	other	other	ADJ
easat-2091	203	5	direction	direction	NOUN
easat-2091	203	6	,	,	PUNCT
easat-2091	203	7	suppose	suppose	VERB
easat-2091	203	8	now	now	ADV
easat-2091	203	9	that	that	SCONJ
easat-2091	203	10	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	203	11	+	+	CCONJ
easat-2091	203	12	1	1	NUM
easat-2091	203	13	is	be	AUX
easat-2091	203	14	a	a	DET
easat-2091	203	15	normal	normal	ADJ
easat-2091	203	16	operator	operator	NOUN
easat-2091	203	17	.	.	PUNCT
easat-2091	204	1	so	so	SCONJ
easat-2091	204	2	that	that	SCONJ
easat-2091	205	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	205	2	+	+	NUM
easat-2091	205	3	1𝑇𝜑	1𝑇𝜑	NUM
easat-2091	205	4	+	+	SYM
easat-2091	205	5	1̅̅	1̅̅	NUM
easat-2091	205	6	̅̅	̅̅	PROPN
easat-2091	205	7	̅̅	̅̅	PROPN
easat-2091	205	8	̅̅	̅̅	PROPN
easat-2091	205	9	=	=	PUNCT
easat-2091	206	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	206	2	+	+	SYM
easat-2091	206	3	1̅̅	1̅̅	NUM
easat-2091	206	4	̅̅	̅̅	PROPN
easat-2091	206	5	̅̅	̅̅	PROPN
easat-2091	206	6	̅̅	̅̅	PROPN
easat-2091	207	1	𝑇𝜑	𝑇𝜑	PROPN
easat-2091	208	1	+	+	CCONJ
easat-2091	208	2	1	1	NUM
easat-2091	208	3	and	and	CCONJ
easat-2091	208	4	so	so	ADV
easat-2091	208	5	theorem	theorem	ADJ
easat-2091	208	6	1	1	NUM
easat-2091	208	7	implies	imply	VERB
easat-2091	208	8	that	that	SCONJ
easat-2091	208	9	(	(	PUNCT
easat-2091	208	10	𝜑	𝜑	X
easat-2091	208	11	+	+	NOUN
easat-2091	208	12	1	1	NUM
easat-2091	208	13	)	)	PUNCT
easat-2091	208	14	and	and	CCONJ
easat-2091	208	15	(	(	PUNCT
easat-2091	208	16	𝜑	𝜑	PROPN
easat-2091	208	17	+	+	SYM
easat-2091	208	18	1̅̅	1̅̅	NUM
easat-2091	208	19	̅̅	̅̅	PROPN
easat-2091	208	20	̅̅	̅̅	PROPN
easat-2091	208	21	̅̅	̅̅	PROPN
easat-2091	208	22	̅	̅	PROPN
easat-2091	208	23	)	)	PUNCT
easat-2091	208	24	are	be	AUX
easat-2091	208	25	both	both	ADV
easat-2091	208	26	analytic	analytic	ADJ
easat-2091	208	27	on	on	ADP
easat-2091	208	28	𝐷	𝐷	PROPN
easat-2091	208	29	(	(	PUNCT
easat-2091	208	30	in	in	ADP
easat-2091	208	31	which	which	DET
easat-2091	208	32	case	case	NOUN
easat-2091	208	33	(	(	PUNCT
easat-2091	208	34	𝜑	𝜑	NOUN
easat-2091	208	35	+	+	NOUN
easat-2091	208	36	1	1	NUM
easat-2091	208	37	)	)	PUNCT
easat-2091	208	38	is	be	AUX
easat-2091	208	39	constant	constant	ADJ
easat-2091	208	40	,	,	PUNCT
easat-2091	208	41	so	so	SCONJ
easat-2091	208	42	we	we	PRON
easat-2091	208	43	are	be	AUX
easat-2091	208	44	done	do	VERB
easat-2091	208	45	)	)	PUNCT
easat-2091	208	46	or	or	CCONJ
easat-2091	208	47	there	there	PRON
easat-2091	208	48	are	be	VERB
easat-2091	208	49	constants𝛼	constants𝛼	NOUN
easat-2091	208	50	,	,	PUNCT
easat-2091	208	51	𝛽	𝛽	PROPN
easat-2091	208	52	∈	∈	PROPN
easat-2091	208	53	𝐶	𝐶	PROPN
easat-2091	208	54	,	,	PUNCT
easat-2091	208	55	not	not	PART
easat-2091	208	56	both	both	PRON
easat-2091	208	57	0	0	NUM
easat-2091	208	58	,	,	PUNCT
easat-2091	208	59	such	such	ADJ
easat-2091	208	60	that	that	SCONJ
easat-2091	208	61	𝑎(𝜑	𝑎(𝜑	PROPN
easat-2091	208	62	+	+	PUNCT
easat-2091	209	1	1	1	NUM
easat-2091	209	2	)	)	PUNCT
easat-2091	209	3	+	+	CCONJ
easat-2091	209	4	𝑏(𝜑	𝑏(𝜑	PROPN
easat-2091	209	5	+	+	SYM
easat-2091	209	6	1̅̅	1̅̅	NUM
easat-2091	209	7	̅̅	̅̅	PROPN
easat-2091	209	8	̅̅	̅̅	PROPN
easat-2091	209	9	̅̅	̅̅	PROPN
easat-2091	209	10	̅	̅	PROPN
easat-2091	209	11	)	)	PUNCT
easat-2091	209	12	(	(	PUNCT
easat-2091	209	13	𝜑	𝜑	X
easat-2091	209	14	+	+	NOUN
easat-2091	209	15	1	1	NUM
easat-2091	209	16	)	)	PUNCT
easat-2091	209	17	is	be	AUX
easat-2091	209	18	constant	constant	ADJ
easat-2091	209	19	on	on	ADP
easat-2091	209	20	𝐷.	𝐷.	PROPN
easat-2091	209	21	the	the	DET
easat-2091	209	22	latter	latter	ADJ
easat-2091	209	23	condition	condition	NOUN
easat-2091	209	24	implies	imply	VERB
easat-2091	209	25	that	that	SCONJ
easat-2091	209	26	(	(	PUNCT
easat-2091	209	27	𝜑	𝜑	X
easat-2091	209	28	+	+	NUM
easat-2091	209	29	1)(𝐷	1)(𝐷	NUM
easat-2091	209	30	)	)	PUNCT
easat-2091	209	31	lies	lie	VERB
easat-2091	209	32	on	on	ADP
easat-2091	209	33	a	a	DET
easat-2091	209	34	line	line	NOUN
easat-2091	209	35	.	.	PUNCT
easat-2091	210	1	copyright	copyright	NOUN
easat-2091	210	2	:	:	PUNCT
easat-2091	210	3	©	©	PROPN
easat-2091	210	4	2024	2024	NUM
easat-2091	210	5	by	by	ADP
easat-2091	210	6	the	the	DET
easat-2091	210	7	authors	author	NOUN
easat-2091	210	8	.	.	PUNCT
easat-2091	211	1	this	this	DET
easat-2091	211	2	article	article	NOUN
easat-2091	211	3	is	be	AUX
easat-2091	211	4	an	an	DET
easat-2091	211	5	open	open	ADJ
easat-2091	211	6	access	access	NOUN
easat-2091	211	7	article	article	NOUN
easat-2091	211	8	distributed	distribute	VERB
easat-2091	211	9	under	under	ADP
easat-2091	211	10	the	the	DET
easat-2091	211	11	terms	term	NOUN
easat-2091	211	12	and	and	CCONJ
easat-2091	211	13	conditions	condition	NOUN
easat-2091	211	14	of	of	ADP
easat-2091	211	15	the	the	DET
easat-2091	211	16	creative	creative	ADJ
easat-2091	211	17	commons	common	NOUN
easat-2091	211	18	attribution	attribution	NOUN
easat-2091	211	19	(	(	PUNCT
easat-2091	211	20	cc	cc	NOUN
easat-2091	211	21	by	by	ADP
easat-2091	211	22	)	)	PUNCT
easat-2091	211	23	license	license	NOUN
easat-2091	211	24	(	(	PUNCT
easat-2091	211	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-2091	211	26	)	)	PUNCT
easat-2091	211	27	.	.	PUNCT
easat-2091	212	1	references	reference	NOUN
easat-2091	212	2	[	[	X
easat-2091	212	3	1	1	X
easat-2091	212	4	]	]	PUNCT
easat-2091	212	5	sheldon	sheldon	PROPN
easat-2091	212	6	axler	axler	PROPN
easat-2091	212	7	*	*	PROPN
easat-2091	212	8	and	and	CCONJ
easat-2091	212	9	zeljko	zeljko	PROPN
easat-2091	212	10	cuckovic	cuckovic	PROPN
easat-2091	212	11	,	,	PUNCT
easat-2091	212	12	commuting	commute	VERB
easat-2091	212	13	toeplitz	toeplitz	NOUN
easat-2091	212	14	operator	operator	NOUN
easat-2091	212	15	with	with	ADP
easat-2091	212	16	harmonic	harmonic	ADJ
easat-2091	212	17	symbols	symbol	NOUN
easat-2091	212	18	,	,	PUNCT
easat-2091	212	19	integral	integral	ADJ
easat-2091	212	20	equations	equation	NOUN
easat-2091	212	21	and	and	CCONJ
easat-2091	212	22	operator	operator	NOUN
easat-2091	212	23	theory	theory	NOUN
easat-2091	212	24	vol	vol	NOUN
easat-2091	212	25	.	.	PUNCT
easat-2091	213	1	14	14	NUM
easat-2091	213	2	(	(	PUNCT
easat-2091	213	3	1991	1991	NUM
easat-2091	213	4	)	)	PUNCT
easat-2091	214	1	[	[	X
easat-2091	214	2	2	2	X
easat-2091	214	3	]	]	X
easat-2091	214	4	arlen	arlen	PROPN
easat-2091	214	5	brown	brown	PROPN
easat-2091	214	6	and	and	CCONJ
easat-2091	214	7	p.	p.	PROPN
easat-2091	214	8	r.	r.	PROPN
easat-2091	214	9	halmas	halmas	PROPN
easat-2091	214	10	,	,	PUNCT
easat-2091	214	11	algebraic	algebraic	ADJ
easat-2091	214	12	properties	property	NOUN
easat-2091	214	13	of	of	ADP
easat-2091	214	14	toeplitz	toeplitz	NOUN
easat-2091	214	15	operators	operator	NOUN
easat-2091	214	16	,	,	PUNCT
easat-2091	214	17	j.	j.	PROPN
easat-2091	214	18	reine	reine	PROPN
easat-2091	214	19	angew	angew	PROPN
easat-2091	214	20	.	.	PUNCT
easat-2091	215	1	math	math	NOUN
easat-2091	215	2	.	.	PUNCT
easat-2091	216	1	213(1964	213(1964	NUM
easat-2091	216	2	)	)	PUNCT
easat-2091	216	3	,	,	PUNCT
easat-2091	216	4	89	89	NUM
easat-2091	216	5	-	-	SYM
easat-2091	216	6	102	102	NUM
easat-2091	216	7	.	.	PUNCT
easat-2091	217	1	[	[	X
easat-2091	217	2	3	3	X
easat-2091	217	3	]	]	X
easat-2091	217	4	sheldon	sheldon	PROPN
easat-2091	217	5	axler	axler	PROPN
easat-2091	217	6	and	and	CCONJ
easat-2091	217	7	pamela	pamela	PROPN
easat-2091	217	8	gorkin	gorkin	PROPN
easat-2091	217	9	,	,	PUNCT
easat-2091	217	10	algebras	algebra	VERB
easat-2091	217	11	on	on	ADP
easat-2091	217	12	the	the	DET
easat-2091	217	13	disk	disk	NOUN
easat-2091	217	14	and	and	CCONJ
easat-2091	217	15	doubly	doubly	ADV
easat-2091	217	16	commuting	commute	VERB
easat-2091	217	17	multiplication	multiplication	NOUN
easat-2091	217	18	operators	operator	NOUN
easat-2091	217	19	,	,	PUNCT
easat-2091	217	20	trans	trans	PROPN
easat-2091	217	21	.	.	PROPN
easat-2091	218	1	amer	amer	PROPN
easat-2091	218	2	.	.	PUNCT
easat-2091	218	3	math	math	PROPN
easat-2091	218	4	.	.	PUNCT
easat-2091	219	1	soc	soc	PROPN
easat-2091	219	2	.	.	PUNCT
easat-2091	220	1	309	309	NUM
easat-2091	220	2	(	(	PUNCT
easat-2091	220	3	1988	1988	NUM
easat-2091	220	4	)	)	PUNCT
easat-2091	220	5	,	,	PUNCT
easat-2091	220	6	711	711	NUM
easat-2091	220	7	-	-	SYM
easat-2091	220	8	723	723	NUM
easat-2091	220	9	.	.	PUNCT
easat-2091	221	1	[	[	X
easat-2091	221	2	4	4	NUM
easat-2091	221	3	]	]	X
easat-2091	221	4	walter	walter	PROPN
easat-2091	221	5	rudin	rudin	PROPN
easat-2091	221	6	,	,	PUNCT
easat-2091	221	7	function	function	NOUN
easat-2091	221	8	theory	theory	NOUN
easat-2091	221	9	on	on	ADP
easat-2091	221	10	the	the	DET
easat-2091	221	11	unit	unit	NOUN
easat-2091	221	12	bail	bail	NOUN
easat-2091	221	13	of	of	ADP
easat-2091	221	14	cn	cn	PROPN
easat-2091	221	15	,	,	PUNCT
easat-2091	221	16	springer	springer	NOUN
easat-2091	221	17	-	-	PUNCT
easat-2091	221	18	verlag	verlag	PROPN
easat-2091	221	19	,	,	PUNCT
easat-2091	221	20	new	new	PROPN
easat-2091	221	21	york	york	PROPN
easat-2091	221	22	1980	1980	NUM
easat-2091	221	23	.	.	PUNCT
easat-2091	222	1	[	[	X
easat-2091	222	2	5	5	X
easat-2091	222	3	]	]	PUNCT
easat-2091	222	4	j.	j.	PROPN
easat-2091	222	5	arazy	arazy	PROPN
easat-2091	222	6	,	,	PUNCT
easat-2091	222	7	s.	s.	PROPN
easat-2091	222	8	d.	d.	PROPN
easat-2091	222	9	fisher	fisher	PROPN
easat-2091	222	10	,	,	PUNCT
easat-2091	222	11	and	and	CCONJ
easat-2091	222	12	j.	j.	PROPN
easat-2091	222	13	peeter	peeter	PROPN
easat-2091	222	14	,	,	PUNCT
easat-2091	222	15	hankel	hankel	NOUN
easat-2091	222	16	operators	operator	NOUN
easat-2091	222	17	on	on	ADP
easat-2091	222	18	weighted	weight	VERB
easat-2091	222	19	bergman	bergman	PROPN
easat-2091	222	20	spaces	space	VERB
easat-2091	222	21	,	,	PUNCT
easat-2091	222	22	american	american	PROPN
easat-2091	222	23	j.	j.	PROPN
easat-2091	222	24	math	math	PROPN
easat-2091	222	25	.	.	PUNCT
easat-2091	223	1	110	110	NUM
easat-2091	223	2	(	(	PUNCT
easat-2091	223	3	1988	1988	NUM
easat-2091	223	4	)	)	PUNCT
easat-2091	223	5	,	,	PUNCT
easat-2091	223	6	989	989	NUM
easat-2091	223	7	-	-	SYM
easat-2091	223	8	1053	1053	NUM
easat-2091	223	9	.	.	PUNCT
easat-2091	224	1	[	[	X
easat-2091	224	2	6	6	NUM
easat-2091	224	3	]	]	SYM
easat-2091	224	4	axler	axler	NOUN
easat-2091	224	5	,	,	PUNCT
easat-2091	224	6	s.	s.	PROPN
easat-2091	224	7	,	,	PUNCT
easat-2091	224	8	the	the	DET
easat-2091	224	9	bergman	bergman	PROPN
easat-2091	224	10	space	space	PROPN
easat-2091	224	11	,	,	PUNCT
easat-2091	224	12	the	the	DET
easat-2091	224	13	bloch	bloch	PROPN
easat-2091	224	14	space	space	NOUN
easat-2091	224	15	,	,	PUNCT
easat-2091	224	16	and	and	CCONJ
easat-2091	224	17	commutators	commutator	NOUN
easat-2091	224	18	of	of	ADP
easat-2091	224	19	multiplication	multiplication	NOUN
easat-2091	224	20	operators	operator	NOUN
easat-2091	224	21	,	,	PUNCT
easat-2091	224	22	duke	duke	PROPN
easat-2091	224	23	j.	j.	PROPN
easat-2091	224	24	math	math	PROPN
easat-2091	224	25	.	.	PUNCT
easat-2091	225	1	53(1986	53(1986	NUM
easat-2091	225	2	)	)	PUNCT
easat-2091	225	3	,	,	PUNCT
easat-2091	225	4	315	315	NUM
easat-2091	225	5	-	-	SYM
easat-2091	225	6	332	332	NUM
easat-2091	225	7	.	.	PUNCT
easat-2091	226	1	[	[	X
easat-2091	226	2	7	7	X
easat-2091	226	3	]	]	X
easat-2091	226	4	zheng	zheng	PROPN
easat-2091	226	5	,	,	PUNCT
easat-2091	226	6	d.	d.	PROPN
easat-2091	226	7	,	,	PUNCT
easat-2091	226	8	hankel	hankel	NOUN
easat-2091	226	9	operators	operator	NOUN
easat-2091	226	10	and	and	CCONJ
easat-2091	226	11	toeplitz	toeplitz	NOUN
easat-2091	226	12	operators	operator	NOUN
easat-2091	226	13	on	on	ADP
easat-2091	226	14	the	the	DET
easat-2091	226	15	bergman	bergman	PROPN
easat-2091	226	16	space	space	PROPN
easat-2091	226	17	,	,	PUNCT
easat-2091	226	18	j.	j.	PROPN
easat-2091	226	19	funct	funct	PROPN
easat-2091	226	20	.	.	PUNCT
easat-2091	227	1	anal	anal	PROPN
easat-2091	227	2	.	.	PUNCT
easat-2091	228	1	83(1989	83(1989	NUM
easat-2091	228	2	)	)	PUNCT
easat-2091	228	3	,	,	PUNCT
easat-2091	229	1	98–120	98–120	NUM
easat-2091	229	2	.	.	PUNCT
easat-2091	230	1	[	[	X
easat-2091	230	2	8	8	NUM
easat-2091	230	3	]	]	SYM
easat-2091	230	4	axler	axler	NOUN
easat-2091	230	5	,	,	PUNCT
easat-2091	230	6	s.	s.	PROPN
easat-2091	230	7	,	,	PUNCT
easat-2091	230	8	chang	chang	PROPN
easat-2091	230	9	,	,	PUNCT
easat-2091	230	10	y.a.s	y.a.s	PROPN
easat-2091	230	11	.	.	PROPN
easat-2091	230	12	and	and	CCONJ
easat-2091	230	13	sarason	sarason	PROPN
easat-2091	230	14	,	,	PUNCT
easat-2091	230	15	d.	d.	PROPN
easat-2091	230	16	,	,	PUNCT
easat-2091	230	17	products	product	NOUN
easat-2091	230	18	of	of	ADP
easat-2091	230	19	toeplitz	toeplitz	NOUN
easat-2091	230	20	operators	operator	NOUN
easat-2091	230	21	,	,	PUNCT
easat-2091	230	22	integral	integral	ADJ
easat-2091	230	23	equations	equation	NOUN
easat-2091	230	24	operator	operator	NOUN
easat-2091	230	25	theory	theory	NOUN
easat-2091	230	26	1(1978	1(1978	NUM
easat-2091	230	27	)	)	PUNCT
easat-2091	230	28	,	,	PUNCT
easat-2091	230	29	285–309	285–309	NUM
easat-2091	230	30	.	.	PUNCT
easat-2091	231	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN