id	sid	tid	token	lemma	pos
cana-1681	1	1	communications	communication	NOUN
cana-1681	1	2	on	on	ADP
cana-1681	1	3	applied	apply	VERB
cana-1681	1	4	nonlinear	nonlinear	ADJ
cana-1681	1	5	analysis	analysis	NOUN
cana-1681	1	6	issn	issn	NOUN
cana-1681	1	7	:	:	PUNCT
cana-1681	1	8	1074	1074	NUM
cana-1681	1	9	-	-	PUNCT
cana-1681	1	10	133x	133x	NUM
cana-1681	1	11	vol	vol	NOUN
cana-1681	1	12	32	32	NUM
cana-1681	1	13	no	no	NOUN
cana-1681	1	14	.	.	NOUN
cana-1681	1	15	1	1	NUM
cana-1681	1	16	(	(	PUNCT
cana-1681	1	17	2025	2025	NUM
cana-1681	1	18	)	)	PUNCT
cana-1681	1	19	385	385	NUM
cana-1681	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	1	21	𝑳𝟏-convergence	𝑳𝟏-convergence	NOUN
cana-1681	1	22	of	of	ADP
cana-1681	1	23	double	double	ADJ
cana-1681	1	24	fourier	fourier	NOUN
cana-1681	1	25	transform	transform	NOUN
cana-1681	1	26	in	in	ADP
cana-1681	1	27	𝑳𝒑(𝑹	𝑳𝒑(𝑹	PRON
cana-1681	1	28	)	)	PUNCT
cana-1681	1	29	spaces	space	NOUN
cana-1681	1	30	,	,	PUNCT
cana-1681	1	31	𝒑	𝒑	PRON
cana-1681	1	32	≥	≥	NOUN
cana-1681	1	33	𝟏	𝟏	NUM
cana-1681	1	34	sakshi1	sakshi1	PROPN
cana-1681	1	35	,	,	PUNCT
cana-1681	1	36	karanvir	karanvir	PROPN
cana-1681	1	37	singh2	singh2	PUNCT
cana-1681	2	1	sakshi.bfcmt@gmail.com1	sakshi.bfcmt@gmail.com1	PROPN
cana-1681	2	2	,	,	PUNCT
cana-1681	2	3	karanvir@mrsptu.ac.in2	karanvir@mrsptu.ac.in2	PROPN
cana-1681	2	4	1	1	NUM
cana-1681	2	5	department	department	NOUN
cana-1681	2	6	of	of	ADP
cana-1681	2	7	mathematics	mathematic	NOUN
cana-1681	2	8	,	,	PUNCT
cana-1681	2	9	maharaja	maharaja	PROPN
cana-1681	2	10	ranjit	ranjit	PROPN
cana-1681	2	11	singh	singh	PROPN
cana-1681	2	12	,	,	PUNCT
cana-1681	2	13	punjab	punjab	PROPN
cana-1681	2	14	technical	technical	PROPN
cana-1681	2	15	university	university	PROPN
cana-1681	2	16	,	,	PUNCT
cana-1681	2	17	bathinda	bathinda	NOUN
cana-1681	2	18	,	,	PUNCT
cana-1681	2	19	punjab	punjab	PROPN
cana-1681	2	20	,	,	PUNCT
cana-1681	2	21	india	india	PROPN
cana-1681	2	22	.	.	PUNCT
cana-1681	3	1	article	article	PROPN
cana-1681	3	2	history	history	NOUN
cana-1681	3	3	:	:	PUNCT
cana-1681	3	4	received	receive	VERB
cana-1681	3	5	:	:	PUNCT
cana-1681	3	6	18	18	NUM
cana-1681	3	7	-	-	SYM
cana-1681	3	8	07	07	NUM
cana-1681	3	9	-	-	PUNCT
cana-1681	3	10	2024	2024	NUM
cana-1681	3	11	revised	revise	VERB
cana-1681	3	12	:	:	PUNCT
cana-1681	3	13	31	31	NUM
cana-1681	3	14	-	-	SYM
cana-1681	3	15	08	08	NUM
cana-1681	3	16	-	-	PUNCT
cana-1681	3	17	2024	2024	NUM
cana-1681	3	18	accepted	accept	VERB
cana-1681	3	19	:	:	PUNCT
cana-1681	3	20	14	14	NUM
cana-1681	3	21	-	-	SYM
cana-1681	3	22	09	09	NUM
cana-1681	3	23	-	-	PUNCT
cana-1681	3	24	2024	2024	NUM
cana-1681	3	25	abstract	abstract	NOUN
cana-1681	3	26	in	in	ADP
cana-1681	3	27	this	this	DET
cana-1681	3	28	research	research	NOUN
cana-1681	3	29	article	article	NOUN
cana-1681	3	30	,	,	PUNCT
cana-1681	3	31	we	we	PRON
cana-1681	3	32	have	have	AUX
cana-1681	3	33	given	give	VERB
cana-1681	3	34	a	a	DET
cana-1681	3	35	method	method	NOUN
cana-1681	3	36	which	which	PRON
cana-1681	3	37	restrict	restrict	VERB
cana-1681	3	38	the	the	DET
cana-1681	3	39	double	double	ADJ
cana-1681	3	40	fourier	fourier	NOUN
cana-1681	3	41	transform	transform	NOUN
cana-1681	3	42	of	of	ADP
cana-1681	3	43	fϵl^p	fϵl^p	NOUN
cana-1681	3	44	(	(	PUNCT
cana-1681	3	45	r	r	NOUN
cana-1681	3	46	)	)	PUNCT
cana-1681	3	47	spaces	space	NOUN
cana-1681	3	48	,	,	PUNCT
cana-1681	3	49	1≤p≤∞	1≤p≤∞	NOUN
cana-1681	3	50	.	.	PUNCT
cana-1681	4	1	further	far	ADV
cana-1681	4	2	,	,	PUNCT
cana-1681	4	3	we	we	PRON
cana-1681	4	4	have	have	AUX
cana-1681	4	5	discussed	discuss	VERB
cana-1681	4	6	the	the	DET
cana-1681	4	7	convergence	convergence	NOUN
cana-1681	4	8	by	by	ADP
cana-1681	4	9	using	use	VERB
cana-1681	4	10	the	the	DET
cana-1681	4	11	approximate	approximate	ADJ
cana-1681	4	12	identities	identity	NOUN
cana-1681	4	13	.	.	PUNCT
cana-1681	5	1	the	the	DET
cana-1681	5	2	aim	aim	NOUN
cana-1681	5	3	of	of	ADP
cana-1681	5	4	this	this	DET
cana-1681	5	5	paper	paper	NOUN
cana-1681	5	6	is	be	AUX
cana-1681	5	7	to	to	PART
cana-1681	5	8	extend	extend	VERB
cana-1681	5	9	the	the	DET
cana-1681	5	10	results	result	NOUN
cana-1681	5	11	of	of	ADP
cana-1681	5	12	k.	k.	PROPN
cana-1681	5	13	devendra	devendra	PROPN
cana-1681	5	14	and	and	CCONJ
cana-1681	5	15	s.	s.	PROPN
cana-1681	5	16	dimple[3	dimple[3	PROPN
cana-1681	5	17	]	]	PUNCT
cana-1681	5	18	from	from	ADP
cana-1681	5	19	one	one	NUM
cana-1681	5	20	dimensional	dimensional	ADJ
cana-1681	5	21	to	to	ADP
cana-1681	5	22	two	two	NUM
cana-1681	5	23	-	-	PUNCT
cana-1681	5	24	dimensional	dimensional	ADJ
cana-1681	5	25	trigonometric	trigonometric	ADJ
cana-1681	5	26	series	series	NOUN
cana-1681	5	27	.	.	PUNCT
cana-1681	6	1	keywords	keyword	NOUN
cana-1681	6	2	:	:	PUNCT
cana-1681	6	3	schwartz	schwartz	PROPN
cana-1681	6	4	space	space	NOUN
cana-1681	6	5	,	,	PUNCT
cana-1681	6	6	convolution	convolution	NOUN
cana-1681	6	7	operator	operator	NOUN
cana-1681	6	8	,	,	PUNCT
cana-1681	6	9	approximate	approximate	ADJ
cana-1681	6	10	identities	identity	NOUN
cana-1681	6	11	,	,	PUNCT
cana-1681	6	12	l^pconvergence	l^pconvergence	PROPN
cana-1681	6	13	.	.	PROPN
cana-1681	6	14	2020	2020	NUM
cana-1681	6	15	mathematics	mathematic	NOUN
cana-1681	6	16	subject	subject	ADJ
cana-1681	6	17	classification	classification	NOUN
cana-1681	6	18	.	.	PUNCT
cana-1681	7	1	42a20	42a20	NUM
cana-1681	7	2	,	,	PUNCT
cana-1681	7	3	42a32	42a32	NUM
cana-1681	7	4	,	,	PUNCT
cana-1681	7	5	42a38	42a38	NOUN
cana-1681	7	6	.	.	PUNCT
cana-1681	8	1	1	1	X
cana-1681	8	2	.	.	X
cana-1681	8	3	introduction	introduction	NOUN
cana-1681	8	4	:	:	PUNCT
cana-1681	8	5	let	let	VERB
cana-1681	8	6	𝑓𝜖𝐿1(𝑅	𝑓𝜖𝐿1(𝑅	NOUN
cana-1681	8	7	)	)	PUNCT
cana-1681	8	8	.	.	PUNCT
cana-1681	9	1	the	the	DET
cana-1681	9	2	fourier	fourier	NOUN
cana-1681	9	3	transform	transform	NOUN
cana-1681	9	4	of	of	ADP
cana-1681	9	5	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	9	6	,	,	PUNCT
cana-1681	9	7	𝑦	𝑦	NOUN
cana-1681	9	8	)	)	PUNCT
cana-1681	9	9	is	be	AUX
cana-1681	9	10	denoted	denote	VERB
cana-1681	9	11	by	by	ADP
cana-1681	9	12	𝑔(휁	𝑔(휁	PROPN
cana-1681	9	13	,	,	PUNCT
cana-1681	9	14	𝜙	𝜙	X
cana-1681	9	15	)	)	PUNCT
cana-1681	9	16	and	and	CCONJ
cana-1681	9	17	is	be	AUX
cana-1681	9	18	defined	define	VERB
cana-1681	9	19	by	by	ADP
cana-1681	9	20	:	:	PUNCT
cana-1681	9	21	𝑔(휁	𝑔(휁	PROPN
cana-1681	9	22	,	,	PUNCT
cana-1681	9	23	𝜙	𝜙	X
cana-1681	9	24	)	)	PUNCT
cana-1681	9	25	=	=	SYM
cana-1681	9	26	1	1	NUM
cana-1681	9	27	√2𝜋	√2𝜋	DET
cana-1681	9	28	∫	∫	NOUN
cana-1681	9	29	𝑅	𝑅	PROPN
cana-1681	9	30	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	9	31	,	,	PUNCT
cana-1681	9	32	𝑦)𝑒𝑥𝑝−𝜄	𝑦)𝑒𝑥𝑝−𝜄	PROPN
cana-1681	9	33	(	(	PUNCT
cana-1681	9	34	𝑥+𝜙𝑦)𝑑𝑥𝑑𝑦	𝑥+𝜙𝑦)𝑑𝑥𝑑𝑦	X
cana-1681	9	35	,	,	PUNCT
cana-1681	9	36	휁𝜖𝑅	휁𝜖𝑅	PROPN
cana-1681	9	37	if	if	SCONJ
cana-1681	9	38	𝑓	𝑓	PROPN
cana-1681	9	39	,	,	PUNCT
cana-1681	9	40	𝑔𝜖𝐿1(𝑅	𝑔𝜖𝐿1(𝑅	PROPN
cana-1681	9	41	)	)	PUNCT
cana-1681	9	42	,	,	PUNCT
cana-1681	9	43	then	then	ADV
cana-1681	9	44	the	the	DET
cana-1681	9	45	inverse	inverse	ADJ
cana-1681	9	46	fourier	fourier	NOUN
cana-1681	9	47	transform	transform	NOUN
cana-1681	9	48	of	of	ADP
cana-1681	9	49	g	g	NOUN
cana-1681	9	50	is	be	AUX
cana-1681	9	51	defined	define	VERB
cana-1681	9	52	as	as	ADP
cana-1681	9	53	:	:	PUNCT
cana-1681	9	54	:	:	PUNCT
cana-1681	9	55	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	9	56	,	,	PUNCT
cana-1681	9	57	𝑦	𝑦	NOUN
cana-1681	9	58	)	)	PUNCT
cana-1681	9	59	=	=	SYM
cana-1681	10	1	1	1	NUM
cana-1681	10	2	√2𝜋	√2𝜋	PRON
cana-1681	10	3	∫	∫	PROPN
cana-1681	10	4	𝑅	𝑅	PROPN
cana-1681	10	5	𝑔(휁	𝑔(휁	PROPN
cana-1681	10	6	,	,	PUNCT
cana-1681	10	7	𝜙)𝑒𝑥𝑝𝜄	𝜙)𝑒𝑥𝑝𝜄	PROPN
cana-1681	10	8	(	(	PUNCT
cana-1681	10	9	𝑥+𝜙𝑦)𝑑휁𝑑𝜙	𝑥+𝜙𝑦)𝑑휁𝑑𝜙	ADJ
cana-1681	10	10	for	for	ADP
cana-1681	10	11	𝑥𝜖𝑅.	𝑥𝜖𝑅.	X
cana-1681	10	12	“	"	PUNCT
cana-1681	10	13	as	as	SCONJ
cana-1681	10	14	we	we	PRON
cana-1681	10	15	know	know	VERB
cana-1681	10	16	that	that	SCONJ
cana-1681	10	17	several	several	ADJ
cana-1681	10	18	functions	function	NOUN
cana-1681	10	19	such	such	ADJ
cana-1681	10	20	as	as	ADP
cana-1681	10	21	elementary	elementary	ADJ
cana-1681	10	22	constant	constant	ADJ
cana-1681	10	23	functions	function	NOUN
cana-1681	10	24	𝑠𝑖𝑛𝑤𝑡	𝑠𝑖𝑛𝑤𝑡	VERB
cana-1681	10	25	,	,	PUNCT
cana-1681	10	26	𝑐𝑜𝑠𝑤𝑡	𝑐𝑜𝑠𝑤𝑡	ADV
cana-1681	10	27	do	do	AUX
cana-1681	10	28	not	not	PART
cana-1681	10	29	converge	converge	VERB
cana-1681	10	30	in	in	ADP
cana-1681	10	31	𝐿1(𝑅	𝐿1(𝑅	PROPN
cana-1681	10	32	)	)	PUNCT
cana-1681	10	33	and	and	CCONJ
cana-1681	10	34	thus	thus	ADV
cana-1681	10	35	they	they	PRON
cana-1681	10	36	do	do	AUX
cana-1681	10	37	not	not	PART
cana-1681	10	38	have	have	VERB
cana-1681	10	39	fourier	fourier	NOUN
cana-1681	10	40	transforms	transform	NOUN
cana-1681	10	41	.	.	PUNCT
cana-1681	11	1	but	but	CCONJ
cana-1681	11	2	when	when	SCONJ
cana-1681	11	3	these	these	DET
cana-1681	11	4	functions	function	NOUN
cana-1681	11	5	are	be	AUX
cana-1681	11	6	multiplied	multiply	VERB
cana-1681	11	7	by	by	ADP
cana-1681	11	8	characteristic	characteristic	ADJ
cana-1681	11	9	functions	function	NOUN
cana-1681	11	10	,	,	PUNCT
cana-1681	11	11	then	then	ADV
cana-1681	11	12	the	the	DET
cana-1681	11	13	resultiong	resultiong	PROPN
cana-1681	11	14	functions	function	NOUN
cana-1681	11	15	converge	converge	VERB
cana-1681	11	16	in	in	ADP
cana-1681	11	17	𝐿1(𝑅	𝐿1(𝑅	PROPN
cana-1681	11	18	)	)	PUNCT
cana-1681	11	19	and	and	CCONJ
cana-1681	11	20	have	have	VERB
cana-1681	11	21	fourier	fourier	NOUN
cana-1681	11	22	transforms	transform	NOUN
cana-1681	11	23	"	"	PUNCT
cana-1681	11	24	.	.	PUNCT
cana-1681	12	1	as	as	SCONJ
cana-1681	12	2	we	we	PRON
cana-1681	12	3	know	know	VERB
cana-1681	12	4	,	,	PUNCT
cana-1681	12	5	lebesgue	lebesgue	PROPN
cana-1681	12	6	lemma	lemma	PROPN
cana-1681	12	7	states	state	VERB
cana-1681	12	8	that	that	SCONJ
cana-1681	12	9	if	if	SCONJ
cana-1681	12	10	𝑓𝜖𝐿1(𝑅	𝑓𝜖𝐿1(𝑅	PROPN
cana-1681	12	11	)	)	PUNCT
cana-1681	12	12	then	then	ADV
cana-1681	12	13	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	12	14	|	|	ADV
cana-1681	12	15	|→∞	|→∞	NOUN
cana-1681	12	16	|𝑔(휁)|	|𝑔(휁)|	SYM
cana-1681	12	17	=	=	SYM
cana-1681	12	18	0	0	PROPN
cana-1681	12	19	.	.	PUNCT
cana-1681	13	1	from	from	ADP
cana-1681	13	2	which	which	PRON
cana-1681	13	3	it	it	PRON
cana-1681	13	4	follows	follow	VERB
cana-1681	13	5	that	that	SCONJ
cana-1681	13	6	“	"	PUNCT
cana-1681	13	7	fourier	fouri	ADJ
cana-1681	13	8	transform	transform	NOUN
cana-1681	13	9	is	be	AUX
cana-1681	13	10	a	a	DET
cana-1681	13	11	continuous	continuous	ADJ
cana-1681	13	12	linear	linear	NOUN
cana-1681	13	13	operator	operator	NOUN
cana-1681	13	14	from	from	ADP
cana-1681	13	15	𝐿1(𝑅	𝐿1(𝑅	PROPN
cana-1681	13	16	)	)	PUNCT
cana-1681	13	17	into	into	ADP
cana-1681	13	18	𝐶0(𝑅	𝐶0(𝑅	PROPN
cana-1681	13	19	)	)	PUNCT
cana-1681	13	20	,	,	PUNCT
cana-1681	13	21	the	the	DET
cana-1681	13	22	space	space	NOUN
cana-1681	13	23	of	of	ADP
cana-1681	13	24	all	all	DET
cana-1681	13	25	continuous	continuous	ADJ
cana-1681	13	26	functions	function	NOUN
cana-1681	13	27	on	on	ADP
cana-1681	13	28	r	r	NOUN
cana-1681	13	29	which	which	PRON
cana-1681	13	30	decay	decay	VERB
cana-1681	13	31	at	at	ADP
cana-1681	13	32	infinity	infinity	NOUN
cana-1681	13	33	,	,	PUNCT
cana-1681	13	34	i.e.	i.e.	X
cana-1681	13	35	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	13	36	)	)	PUNCT
cana-1681	13	37	→	→	SYM
cana-1681	13	38	0	0	PUNCT
cana-1681	13	39	as	as	SCONJ
cana-1681	13	40	|𝑥|	|𝑥|	PROPN
cana-1681	13	41	→	→	SYM
cana-1681	13	42	∞.	∞.	PROPN
cana-1681	13	43	we	we	PRON
cana-1681	13	44	say	say	VERB
cana-1681	13	45	that	that	SCONJ
cana-1681	13	46	if	if	SCONJ
cana-1681	13	47	𝑓𝜖𝐿1(𝑅	𝑓𝜖𝐿1(𝑅	PROPN
cana-1681	13	48	)	)	PUNCT
cana-1681	13	49	,	,	PUNCT
cana-1681	13	50	it	it	PRON
cana-1681	13	51	is	be	AUX
cana-1681	13	52	not	not	PART
cana-1681	13	53	necessary	necessary	ADJ
cana-1681	13	54	that	that	SCONJ
cana-1681	13	55	g	g	PROPN
cana-1681	13	56	also	also	ADV
cana-1681	13	57	belongs	belong	VERB
cana-1681	13	58	to	to	ADP
cana-1681	13	59	𝐿1(𝑅	𝐿1(𝑅	PROPN
cana-1681	13	60	)	)	PUNCT
cana-1681	13	61	.	.	PUNCT
cana-1681	14	1	in	in	ADP
cana-1681	14	2	the	the	DET
cana-1681	14	3	present	present	ADJ
cana-1681	14	4	article	article	NOUN
cana-1681	14	5	,	,	PUNCT
cana-1681	14	6	we	we	PRON
cana-1681	14	7	provide	provide	VERB
cana-1681	14	8	a	a	DET
cana-1681	14	9	method	method	NOUN
cana-1681	14	10	for	for	ADP
cana-1681	14	11	restricting	restrict	VERB
cana-1681	14	12	fourier	fourier	NOUN
cana-1681	14	13	transform	transform	NOUN
cana-1681	14	14	of	of	ADP
cana-1681	14	15	𝑓𝜖𝐿𝑝(𝑅	𝑓𝜖𝐿𝑝(𝑅	NOUN
cana-1681	14	16	)	)	PUNCT
cana-1681	14	17	spaces	space	NOUN
cana-1681	14	18	using	use	VERB
cana-1681	14	19	the	the	DET
cana-1681	14	20	pointwise	pointwise	ADJ
cana-1681	14	21	convergence	convergence	NOUN
cana-1681	14	22	of	of	ADP
cana-1681	14	23	convolution	convolution	NOUN
cana-1681	14	24	operators	operator	NOUN
cana-1681	14	25	for	for	ADP
cana-1681	14	26	approximate	approximate	ADJ
cana-1681	14	27	identities	identity	NOUN
cana-1681	14	28	.	.	PUNCT
cana-1681	14	29	"	"	PUNCT
cana-1681	15	1	definition	definition	NOUN
cana-1681	15	2	1.1	1.1	NUM
cana-1681	15	3	.	.	PUNCT
cana-1681	16	1	let	let	VERB
cana-1681	16	2	𝜓휀𝐿1(𝑅	𝜓휀𝐿1(𝑅	VERB
cana-1681	16	3	)	)	PUNCT
cana-1681	16	4	suct	suct	NOUN
cana-1681	16	5	that	that	DET
cana-1681	16	6	𝜉(0	𝜉(0	NOUN
cana-1681	16	7	)	)	PUNCT
cana-1681	16	8	=	=	SYM
cana-1681	17	1	1	1	X
cana-1681	17	2	.	.	PUNCT
cana-1681	17	3	then	then	ADV
cana-1681	17	4	𝜓𝜖(𝑥	𝜓𝜖(𝑥	NOUN
cana-1681	17	5	,	,	PUNCT
cana-1681	17	6	𝑦	𝑦	X
cana-1681	17	7	)	)	PUNCT
cana-1681	17	8	=	=	PUNCT
cana-1681	18	1	𝜖−1𝜓	𝜖−1𝜓	NOUN
cana-1681	18	2	(	(	PUNCT
cana-1681	18	3	𝑥	𝑥	X
cana-1681	18	4	𝜖	𝜖	X
cana-1681	18	5	,	,	PUNCT
cana-1681	18	6	𝑦	𝑦	NOUN
cana-1681	18	7	𝜖	𝜖	X
cana-1681	18	8	)	)	PUNCT
cana-1681	18	9	is	be	AUX
cana-1681	18	10	called	call	VERB
cana-1681	18	11	an	an	DET
cana-1681	18	12	approximate	approximate	ADJ
cana-1681	18	13	identity	identity	NOUN
cana-1681	18	14	if	if	SCONJ
cana-1681	18	15	communications	communication	NOUN
cana-1681	18	16	on	on	ADP
cana-1681	18	17	applied	apply	VERB
cana-1681	18	18	nonlinear	nonlinear	ADJ
cana-1681	18	19	analysis	analysis	NOUN
cana-1681	18	20	issn	issn	NOUN
cana-1681	18	21	:	:	PUNCT
cana-1681	18	22	1074	1074	NUM
cana-1681	18	23	-	-	PUNCT
cana-1681	18	24	133x	133x	NUM
cana-1681	18	25	vol	vol	NOUN
cana-1681	18	26	32	32	NUM
cana-1681	18	27	no	no	NOUN
cana-1681	18	28	.	.	NOUN
cana-1681	18	29	1	1	NUM
cana-1681	18	30	(	(	PUNCT
cana-1681	18	31	2025	2025	NUM
cana-1681	18	32	)	)	PUNCT
cana-1681	18	33	386	386	NUM
cana-1681	18	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	18	35	(	(	PUNCT
cana-1681	18	36	i	i	NOUN
cana-1681	18	37	)	)	PUNCT
cana-1681	18	38	∫	∫	PROPN
cana-1681	19	1	𝑅	𝑅	PROPN
cana-1681	19	2	𝜓𝜖(𝑥	𝜓𝜖(𝑥	PROPN
cana-1681	19	3	,	,	PUNCT
cana-1681	19	4	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	X
cana-1681	19	5	=	=	SYM
cana-1681	19	6	1	1	NUM
cana-1681	19	7	(	(	PUNCT
cana-1681	19	8	ii	ii	NOUN
cana-1681	19	9	)	)	PUNCT
cana-1681	19	10	𝑠𝑢𝑝𝜖>0	𝑠𝑢𝑝𝜖>0	PROPN
cana-1681	19	11	∫	∫	PROPN
cana-1681	19	12	𝑅	𝑅	PROPN
cana-1681	19	13	|𝜓𝜖(𝑥	|𝜓𝜖(𝑥	PROPN
cana-1681	19	14	,	,	PUNCT
cana-1681	19	15	𝑦)|𝑑𝑥𝑑𝑦	𝑦)|𝑑𝑥𝑑𝑦	VERB
cana-1681	19	16	<	<	X
cana-1681	19	17	∞	∞	PROPN
cana-1681	19	18	,	,	PUNCT
cana-1681	19	19	(	(	PUNCT
cana-1681	19	20	iii	iii	NOUN
cana-1681	19	21	)	)	PUNCT
cana-1681	19	22	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	19	23	𝜖→0	𝜖→0	X
cana-1681	19	24	∫	∫	PROPN
cana-1681	19	25	|𝑥|>𝛿,|𝑦|>𝛿	|𝑥|>𝛿,|𝑦|>𝛿	PROPN
cana-1681	19	26	|𝜓𝜖(𝑥	|𝜓𝜖(𝑥	PROPN
cana-1681	19	27	,	,	PUNCT
cana-1681	19	28	𝑦)|𝑑𝑥𝑑𝑦	𝑦)|𝑑𝑥𝑑𝑦	ADJ
cana-1681	19	29	=	=	SYM
cana-1681	19	30	0	0	NUM
cana-1681	19	31	,	,	PUNCT
cana-1681	19	32	for	for	ADP
cana-1681	19	33	all	all	PRON
cana-1681	19	34	𝛿	𝛿	PRON
cana-1681	19	35	>	>	X
cana-1681	19	36	0	0	NUM
cana-1681	19	37	"	"	PUNCT
cana-1681	19	38	proof	proof	NOUN
cana-1681	19	39	.	.	PUNCT
cana-1681	20	1	we	we	PRON
cana-1681	20	2	can	can	AUX
cana-1681	20	3	prove	prove	VERB
cana-1681	20	4	properties	property	NOUN
cana-1681	20	5	(	(	PUNCT
cana-1681	20	6	i	i	NOUN
cana-1681	20	7	)	)	PUNCT
cana-1681	20	8	and	and	CCONJ
cana-1681	20	9	(	(	PUNCT
cana-1681	20	10	ii	ii	NOUN
cana-1681	20	11	)	)	PUNCT
cana-1681	20	12	by	by	ADP
cana-1681	20	13	following	follow	VERB
cana-1681	20	14	:	:	PUNCT
cana-1681	20	15	∫	∫	PROPN
cana-1681	20	16	𝑅	𝑅	PROPN
cana-1681	20	17	𝜓𝜖(𝑥	𝜓𝜖(𝑥	PROPN
cana-1681	20	18	,	,	PUNCT
cana-1681	20	19	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	NUM
cana-1681	20	20	=	=	SYM
cana-1681	20	21	∫	∫	PROPN
cana-1681	20	22	𝑅	𝑅	PROPN
cana-1681	20	23	𝜖−1𝜓	𝜖−1𝜓	PROPN
cana-1681	20	24	(	(	PUNCT
cana-1681	20	25	𝑥	𝑥	X
cana-1681	20	26	𝜖	𝜖	X
cana-1681	20	27	,	,	PUNCT
cana-1681	20	28	𝑦	𝑦	NOUN
cana-1681	20	29	𝜖	𝜖	X
cana-1681	20	30	)	)	PUNCT
cana-1681	20	31	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1681	20	32	=	=	SYM
cana-1681	20	33	∫	∫	PROPN
cana-1681	20	34	𝑅	𝑅	PROPN
cana-1681	20	35	𝜓	𝜓	PROPN
cana-1681	20	36	(	(	PUNCT
cana-1681	20	37	𝑥	𝑥	X
cana-1681	20	38	𝜖	𝜖	X
cana-1681	20	39	,	,	PUNCT
cana-1681	20	40	𝑦	𝑦	NOUN
cana-1681	20	41	𝜖	𝜖	X
cana-1681	20	42	)	)	PUNCT
cana-1681	20	43	𝑑	𝑑	PROPN
cana-1681	20	44	(	(	PUNCT
cana-1681	20	45	𝑥	𝑥	X
cana-1681	20	46	𝜖	𝜖	X
cana-1681	20	47	,	,	PUNCT
cana-1681	20	48	𝑦	𝑦	NOUN
cana-1681	20	49	𝜖	𝜖	X
cana-1681	20	50	)	)	PUNCT
cana-1681	20	51	=	=	SYM
cana-1681	20	52	1	1	NUM
cana-1681	20	53	for	for	ADP
cana-1681	20	54	(	(	PUNCT
cana-1681	20	55	iii	iii	NOUN
cana-1681	20	56	)	)	PUNCT
cana-1681	20	57	,	,	PUNCT
cana-1681	20	58	it	it	PRON
cana-1681	20	59	follows	follow	VERB
cana-1681	20	60	that	that	PRON
cana-1681	20	61	:	:	PUNCT
cana-1681	20	62	∫	∫	PROPN
cana-1681	20	63	|𝑥|>𝛿,|𝑦|>𝛿	|𝑥|>𝛿,|𝑦|>𝛿	PROPN
cana-1681	20	64	𝜓𝜖(𝑥	𝜓𝜖(𝑥	PROPN
cana-1681	20	65	,	,	PUNCT
cana-1681	20	66	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	NUM
cana-1681	21	1	=	=	SYM
cana-1681	21	2	∫	∫	PROPN
cana-1681	21	3	|𝑥|>𝛿,|𝑦|>𝛿	|𝑥|>𝛿,|𝑦|>𝛿	PROPN
cana-1681	21	4	1	1	NUM
cana-1681	21	5	𝜖	𝜖	PROPN
cana-1681	21	6	𝜓	𝜓	PROPN
cana-1681	21	7	(	(	PUNCT
cana-1681	21	8	𝑥	𝑥	X
cana-1681	21	9	𝜖	𝜖	X
cana-1681	21	10	,	,	PUNCT
cana-1681	21	11	𝑦	𝑦	NOUN
cana-1681	21	12	𝜖	𝜖	X
cana-1681	21	13	)	)	PUNCT
cana-1681	21	14	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1681	22	1	=	=	SYM
cana-1681	22	2	∫	∫	PROPN
cana-1681	22	3	∞	∞	PROPN
cana-1681	22	4	𝛿	𝛿	PROPN
cana-1681	22	5	1	1	NUM
cana-1681	22	6	𝜖	𝜖	X
cana-1681	22	7	𝜓	𝜓	PROPN
cana-1681	22	8	(	(	PUNCT
cana-1681	22	9	𝑥	𝑥	X
cana-1681	22	10	𝜖	𝜖	X
cana-1681	22	11	,	,	PUNCT
cana-1681	22	12	𝑦	𝑦	NOUN
cana-1681	22	13	𝜖	𝜖	X
cana-1681	22	14	)	)	PUNCT
cana-1681	22	15	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1681	23	1	+	+	CCONJ
cana-1681	23	2	∫	∫	PROPN
cana-1681	23	3	−𝛿	−𝛿	PROPN
cana-1681	23	4	−∞	−∞	ADP
cana-1681	23	5	1	1	NUM
cana-1681	23	6	𝜖	𝜖	PROPN
cana-1681	23	7	𝜓	𝜓	PROPN
cana-1681	23	8	(	(	PUNCT
cana-1681	23	9	𝑥	𝑥	X
cana-1681	23	10	𝜖	𝜖	X
cana-1681	23	11	,	,	PUNCT
cana-1681	23	12	𝑦	𝑦	NOUN
cana-1681	23	13	𝜖	𝜖	NOUN
cana-1681	23	14	)	)	PUNCT
cana-1681	23	15	𝑑𝑥𝑑𝑦𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔	𝑑𝑥𝑑𝑦𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔	NOUN
cana-1681	23	16	𝑧	𝑧	NOUN
cana-1681	23	17	=	=	X
cana-1681	23	18	𝑥	𝑥	X
cana-1681	23	19	𝜖	𝜖	X
cana-1681	23	20	,	,	PUNCT
cana-1681	23	21	𝑡	𝑡	PROPN
cana-1681	23	22	=	=	SYM
cana-1681	23	23	𝑦	𝑦	X
cana-1681	23	24	𝜖	𝜖	X
cana-1681	23	25	,	,	PUNCT
cana-1681	23	26	we	we	PRON
cana-1681	23	27	get	get	VERB
cana-1681	23	28	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	23	29	𝜖→0	𝜖→0	X
cana-1681	23	30	∫	∫	PROPN
cana-1681	23	31	∞	∞	PROPN
cana-1681	23	32	𝛿	𝛿	X
cana-1681	23	33	𝜖	𝜖	X
cana-1681	23	34	𝜓(𝑧	𝜓(𝑧	NOUN
cana-1681	23	35	,	,	PUNCT
cana-1681	23	36	𝑡)𝑑𝑧𝑑𝑡	𝑡)𝑑𝑧𝑑𝑡	X
cana-1681	23	37	+	+	CCONJ
cana-1681	23	38	∫	∫	PROPN
cana-1681	23	39	−	−	PROPN
cana-1681	23	40	𝛿	𝛿	PRON
cana-1681	23	41	𝜖	𝜖	X
cana-1681	23	42	−∞	−∞	ADP
cana-1681	23	43	𝜓(𝑧	𝜓(𝑧	NOUN
cana-1681	23	44	,	,	PUNCT
cana-1681	23	45	𝑡)𝑑𝑧𝑑𝑡	𝑡)𝑑𝑧𝑑𝑡	X
cana-1681	23	46	=	=	SYM
cana-1681	24	1	0	0	X
cana-1681	24	2	.	.	PUNCT
cana-1681	24	3	definition	definition	NOUN
cana-1681	24	4	1.2	1.2	NUM
cana-1681	24	5	.	.	PUNCT
cana-1681	25	1	“	"	PUNCT
cana-1681	25	2	a	a	DET
cana-1681	25	3	sequence	sequence	NOUN
cana-1681	25	4	of	of	ADP
cana-1681	25	5	functions	function	NOUN
cana-1681	25	6	ℎ𝑛𝑛	ℎ𝑛𝑛	ADJ
cana-1681	25	7	𝑁	𝑁	NOUN
cana-1681	25	8	such	such	ADJ
cana-1681	25	9	that	that	PRON
cana-1681	25	10	ℎ𝑛(𝑥	ℎ𝑛(𝑥	NOUN
cana-1681	25	11	,	,	PUNCT
cana-1681	25	12	𝑦	𝑦	NOUN
cana-1681	25	13	)	)	PUNCT
cana-1681	25	14	=	=	SYM
cana-1681	25	15	𝑛ℎ(𝑛𝑥	𝑛ℎ(𝑛𝑥	ADJ
cana-1681	25	16	,	,	PUNCT
cana-1681	25	17	𝑛𝑦	𝑛𝑦	NOUN
cana-1681	25	18	)	)	PUNCT
cana-1681	25	19	where	where	SCONJ
cana-1681	25	20	𝑛	𝑛	PRON
cana-1681	25	21	=	=	SYM
cana-1681	25	22	1	1	NUM
cana-1681	25	23	휀	휀	NOUN
cana-1681	25	24	,	,	PUNCT
cana-1681	25	25	𝑛	𝑛	PROPN
cana-1681	25	26	→	→	SYM
cana-1681	25	27	∞	∞	PROPN
cana-1681	25	28	,	,	PUNCT
cana-1681	25	29	휀	휀	X
cana-1681	25	30	→	→	SYM
cana-1681	25	31	0	0	NUM
cana-1681	25	32	is	be	AUX
cana-1681	25	33	called	call	VERB
cana-1681	25	34	an	an	DET
cana-1681	25	35	approximate	approximate	ADJ
cana-1681	25	36	identity	identity	NOUN
cana-1681	25	37	if	if	SCONJ
cana-1681	25	38	(	(	PUNCT
cana-1681	25	39	i	i	NOUN
cana-1681	25	40	)	)	PUNCT
cana-1681	25	41	∫	∫	PROPN
cana-1681	25	42	𝑅	𝑅	PROPN
cana-1681	25	43	ℎ𝑛(𝑥	ℎ𝑛(𝑥	PROPN
cana-1681	25	44	,	,	PUNCT
cana-1681	25	45	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	X
cana-1681	25	46	=	=	SYM
cana-1681	25	47	1	1	NUM
cana-1681	25	48	for	for	ADP
cana-1681	25	49	all	all	DET
cana-1681	25	50	n	n	CCONJ
cana-1681	25	51	,	,	PUNCT
cana-1681	25	52	(	(	PUNCT
cana-1681	25	53	ii	ii	NOUN
cana-1681	25	54	)	)	PUNCT
cana-1681	25	55	𝑠𝑢𝑝𝑛	𝑠𝑢𝑝𝑛	NOUN
cana-1681	25	56	∫	∫	PROPN
cana-1681	25	57	𝑅	𝑅	PROPN
cana-1681	25	58	ℎ𝑛(𝑥	ℎ𝑛(𝑥	PROPN
cana-1681	25	59	,	,	PUNCT
cana-1681	25	60	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	X
cana-1681	25	61	<	<	X
cana-1681	26	1	+	+	PROPN
cana-1681	26	2	∞	∞	PROPN
cana-1681	26	3	,	,	PUNCT
cana-1681	26	4	(	(	PUNCT
cana-1681	26	5	iii	iii	X
cana-1681	26	6	)	)	PUNCT
cana-1681	26	7	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	26	8	𝑛→∞	𝑛→∞	NUM
cana-1681	26	9	∫	∫	PROPN
cana-1681	26	10	|𝑥|>𝛿	|𝑥|>𝛿	PROPN
cana-1681	26	11	ℎ𝑛(𝑥	ℎ𝑛(𝑥	PROPN
cana-1681	26	12	,	,	PUNCT
cana-1681	26	13	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	PROPN
cana-1681	27	1	=	=	SYM
cana-1681	27	2	0	0	NUM
cana-1681	28	1	for	for	ADP
cana-1681	28	2	every	every	DET
cana-1681	28	3	𝛿	𝛿	PROPN
cana-1681	28	4	>	>	X
cana-1681	28	5	0	0	NUM
cana-1681	28	6	.	.	PUNCT
cana-1681	28	7	"	"	PUNCT
cana-1681	28	8	by	by	ADP
cana-1681	28	9	following	follow	VERB
cana-1681	28	10	the	the	DET
cana-1681	28	11	above	above	ADJ
cana-1681	28	12	definition	definition	NOUN
cana-1681	28	13	,	,	PUNCT
cana-1681	28	14	the	the	DET
cana-1681	28	15	following	follow	VERB
cana-1681	28	16	proposition	proposition	NOUN
cana-1681	28	17	can	can	AUX
cana-1681	28	18	easily	easily	ADV
cana-1681	28	19	prove	prove	VERB
cana-1681	28	20	:	:	PUNCT
cana-1681	28	21	proposition	proposition	NOUN
cana-1681	28	22	1.1	1.1	NUM
cana-1681	28	23	.	.	PUNCT
cana-1681	29	1	“	"	PUNCT
cana-1681	29	2	a	a	DET
cana-1681	29	3	sequence	sequence	NOUN
cana-1681	29	4	of	of	ADP
cana-1681	29	5	functions	function	NOUN
cana-1681	29	6	ℎ𝑛𝑛	ℎ𝑛𝑛	ADJ
cana-1681	29	7	𝑁	𝑁	NOUN
cana-1681	29	8	with	with	ADP
cana-1681	29	9	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	29	10	≥	≥	PROPN
cana-1681	29	11	0	0	NUM
cana-1681	29	12	,	,	PUNCT
cana-1681	29	13	ℎ𝑛(0,0	ℎ𝑛(0,0	NOUN
cana-1681	29	14	)	)	PUNCT
cana-1681	29	15	=	=	SYM
cana-1681	29	16	1	1	NUM
cana-1681	29	17	is	be	AUX
cana-1681	29	18	an	an	DET
cana-1681	29	19	approximate	approximate	ADJ
cana-1681	29	20	identity	identity	NOUN
cana-1681	29	21	if	if	SCONJ
cana-1681	29	22	for	for	ADP
cana-1681	29	23	every	every	DET
cana-1681	29	24	휀	휀	NOUN
cana-1681	29	25	>	>	X
cana-1681	29	26	0	0	PUNCT
cana-1681	29	27	there	there	PRON
cana-1681	29	28	exists	exist	VERB
cana-1681	29	29	𝑛0휀𝑁	𝑛0휀𝑁	ADJ
cana-1681	29	30	so	so	SCONJ
cana-1681	29	31	that	that	SCONJ
cana-1681	29	32	for	for	SCONJ
cana-1681	29	33	all	all	DET
cana-1681	29	34	𝑛	𝑛	DET
cana-1681	29	35	≥	≥	NOUN
cana-1681	29	36	𝑛0	𝑛0	VERB
cana-1681	29	37	we	we	PRON
cana-1681	29	38	have	have	VERB
cana-1681	29	39	∫	∫	PROPN
cana-1681	30	1	−	−	PROPN
cana-1681	30	2	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	30	3	>	>	X
cana-1681	30	4	1	1	NUM
cana-1681	30	5	−	−	NOUN
cana-1681	31	1	휀	휀	X
cana-1681	31	2	.	.	PUNCT
cana-1681	31	3	let	let	VERB
cana-1681	31	4	us	we	PRON
cana-1681	31	5	consider	consider	VERB
cana-1681	31	6	the	the	DET
cana-1681	31	7	class	class	NOUN
cana-1681	31	8	𝑆∗(𝑅	𝑆∗(𝑅	NOUN
cana-1681	31	9	)	)	PUNCT
cana-1681	31	10	of	of	ADP
cana-1681	31	11	𝐶∞-functions	𝐶∞-functions	PROPN
cana-1681	31	12	on	on	ADP
cana-1681	31	13	r	r	NOUN
cana-1681	31	14	which	which	PRON
cana-1681	31	15	are	be	AUX
cana-1681	31	16	rapidly	rapidly	ADV
cana-1681	31	17	decreasing	decrease	VERB
cana-1681	31	18	i.e.	i.e.	X
cana-1681	31	19	schwartz	schwartz	PROPN
cana-1681	31	20	class	class	NOUN
cana-1681	31	21	such	such	DET
cana-1681	31	22	that	that	DET
cana-1681	31	23	𝑆∗(𝑅	𝑆∗(𝑅	NOUN
cana-1681	31	24	)	)	PUNCT
cana-1681	31	25	=	=	SYM
cana-1681	32	1	𝑓	𝑓	X
cana-1681	32	2	:	:	PUNCT
cana-1681	32	3	𝑅	𝑅	PROPN
cana-1681	32	4	→	→	SYM
cana-1681	32	5	𝑅	𝑅	PROPN
cana-1681	32	6	,	,	PUNCT
cana-1681	32	7	𝑠𝑢𝑝𝑥→𝑅(𝑥	𝑠𝑢𝑝𝑥→𝑅(𝑥	NUM
cana-1681	32	8	,	,	PUNCT
cana-1681	32	9	𝑦	𝑦	NOUN
cana-1681	32	10	)	)	PUNCT
cana-1681	32	11	𝑑𝑚	𝑑𝑚	NOUN
cana-1681	32	12	𝑑𝑥𝑚	𝑑𝑥𝑚	NOUN
cana-1681	32	13	𝑑𝑚	𝑑𝑚	NOUN
cana-1681	32	14	𝑑𝑦𝑚	𝑑𝑦𝑚	NOUN
cana-1681	32	15	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	32	16	,	,	PUNCT
cana-1681	32	17	𝑦	𝑦	NOUN
cana-1681	32	18	)	)	PUNCT
cana-1681	32	19	<	<	X
cana-1681	32	20	∞	∞	PROPN
cana-1681	32	21	;	;	PUNCT
cana-1681	32	22	𝑛	𝑛	PROPN
cana-1681	32	23	,	,	PUNCT
cana-1681	32	24	𝑚휀𝑁⋃(0	𝑚휀𝑁⋃(0	NOUN
cana-1681	32	25	)	)	PUNCT
cana-1681	32	26	"	"	PUNCT
cana-1681	32	27	.	.	PUNCT
cana-1681	33	1	we	we	PRON
cana-1681	33	2	know	know	VERB
cana-1681	33	3	that	that	SCONJ
cana-1681	33	4	if	if	SCONJ
cana-1681	33	5	𝑓휀𝑆∗(𝑅	𝑓휀𝑆∗(𝑅	NOUN
cana-1681	33	6	)	)	PUNCT
cana-1681	33	7	,	,	PUNCT
cana-1681	33	8	then	then	ADV
cana-1681	33	9	𝑔휀𝑆∗	𝑔휀𝑆∗	PROPN
cana-1681	33	10	and	and	CCONJ
cana-1681	33	11	“	"	PUNCT
cana-1681	33	12	𝑆∗(𝑅	𝑆∗(𝑅	NOUN
cana-1681	33	13	)	)	PUNCT
cana-1681	33	14	⊂	⊂	NOUN
cana-1681	33	15	𝐿𝑝(𝑅	𝐿𝑝(𝑅	NOUN
cana-1681	33	16	)	)	PUNCT
cana-1681	33	17	"	"	PUNCT
cana-1681	33	18	.	.	PUNCT
cana-1681	34	1	to	to	PART
cana-1681	34	2	prove	prove	VERB
cana-1681	34	3	the	the	DET
cana-1681	34	4	denseness	denseness	NOUN
cana-1681	34	5	of	of	ADP
cana-1681	34	6	𝑆∗(𝑅	𝑆∗(𝑅	NOUN
cana-1681	34	7	)	)	PUNCT
cana-1681	34	8	⊂	⊂	NOUN
cana-1681	34	9	𝐿𝑝(𝑅	𝐿𝑝(𝑅	NOUN
cana-1681	34	10	)	)	PUNCT
cana-1681	34	11	,	,	PUNCT
cana-1681	34	12	we	we	PRON
cana-1681	34	13	have	have	VERB
cana-1681	34	14	휂휀𝑆∗(𝑅	휂휀𝑆∗(𝑅	NOUN
cana-1681	34	15	)	)	PUNCT
cana-1681	34	16	⇒	⇒	PROPN
cana-1681	34	17	|휂(𝑥	|휂(𝑥	PROPN
cana-1681	34	18	,	,	PUNCT
cana-1681	34	19	𝑦)|	𝑦)|	PROPN
cana-1681	34	20	≤	≤	PROPN
cana-1681	34	21	𝑐	𝑐	PROPN
cana-1681	34	22	1+|𝑥𝑦|𝑛.	1+|𝑥𝑦|𝑛.	NUM
cana-1681	34	23	for	for	ADP
cana-1681	34	24	1	1	NUM
cana-1681	34	25	≤	≤	NOUN
cana-1681	34	26	𝑝	𝑝	PROPN
cana-1681	34	27	<	<	X
cana-1681	34	28	∞	∞	PROPN
cana-1681	34	29	,	,	PUNCT
cana-1681	34	30	∫	∫	PROPN
cana-1681	34	31	𝑅	𝑅	PROPN
cana-1681	34	32	|휂(𝑥	|휂(𝑥	PROPN
cana-1681	34	33	,	,	PUNCT
cana-1681	34	34	𝑦)|𝑝𝑑𝑥𝑑𝑦	𝑦)|𝑝𝑑𝑥𝑑𝑦	VERB
cana-1681	34	35	≤	≤	NUM
cana-1681	34	36	∫	∫	PROPN
cana-1681	34	37	𝑅	𝑅	PROPN
cana-1681	34	38	𝑐𝑝	𝑐𝑝	PROPN
cana-1681	34	39	(	(	PUNCT
cana-1681	34	40	1	1	NUM
cana-1681	34	41	+	+	CCONJ
cana-1681	34	42	|𝑥𝑦|𝑛)𝑝	|𝑥𝑦|𝑛)𝑝	PROPN
cana-1681	34	43	<	<	X
cana-1681	34	44	∞𝑤ℎ𝑖𝑐ℎ	∞𝑤ℎ𝑖𝑐ℎ	PROPN
cana-1681	34	45	𝑔𝑖𝑣𝑒𝑠	𝑔𝑖𝑣𝑒𝑠	ADJ
cana-1681	34	46	휂휀𝐿𝑝(𝑅	휂휀𝐿𝑝(𝑅	NOUN
cana-1681	34	47	)	)	PUNCT
cana-1681	34	48	communications	communication	NOUN
cana-1681	34	49	on	on	ADP
cana-1681	34	50	applied	apply	VERB
cana-1681	34	51	nonlinear	nonlinear	ADJ
cana-1681	34	52	analysis	analysis	NOUN
cana-1681	34	53	issn	issn	NOUN
cana-1681	34	54	:	:	PUNCT
cana-1681	34	55	1074	1074	NUM
cana-1681	34	56	-	-	PUNCT
cana-1681	34	57	133x	133x	NUM
cana-1681	34	58	vol	vol	NOUN
cana-1681	34	59	32	32	NUM
cana-1681	35	1	no	no	NOUN
cana-1681	35	2	.	.	NOUN
cana-1681	35	3	1	1	NUM
cana-1681	35	4	(	(	PUNCT
cana-1681	35	5	2025	2025	NUM
cana-1681	35	6	)	)	PUNCT
cana-1681	35	7	387	387	NUM
cana-1681	35	8	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	35	9	.	.	PUNCT
cana-1681	36	1	define	define	VERB
cana-1681	36	2	a	a	DET
cana-1681	36	3	sequence	sequence	NOUN
cana-1681	36	4	휂𝑁	휂𝑁	PUNCT
cana-1681	36	5	such	such	ADJ
cana-1681	36	6	that	that	SCONJ
cana-1681	36	7	휂𝑁(𝑥	휂𝑁(𝑥	PROPN
cana-1681	36	8	,	,	PUNCT
cana-1681	36	9	𝑦	𝑦	NOUN
cana-1681	36	10	)	)	PUNCT
cana-1681	36	11	=	=	SYM
cana-1681	36	12	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	36	13	,	,	PUNCT
cana-1681	36	14	𝑦	𝑦	NOUN
cana-1681	36	15	)	)	PUNCT
cana-1681	36	16	,	,	PUNCT
cana-1681	36	17	if	if	SCONJ
cana-1681	36	18	−𝑁	−𝑁	VERB
cana-1681	36	19	≤	≤	NUM
cana-1681	36	20	𝑥	𝑥	PROPN
cana-1681	36	21	,	,	PUNCT
cana-1681	36	22	𝑦	𝑦	NOUN
cana-1681	36	23	≤	≤	X
cana-1681	36	24	𝑁	𝑁	NOUN
cana-1681	36	25	;	;	PUNCT
cana-1681	36	26	and	and	CCONJ
cana-1681	36	27	otherwise	otherwise	ADV
cana-1681	36	28	it	it	PRON
cana-1681	36	29	will	will	AUX
cana-1681	36	30	become	become	VERB
cana-1681	36	31	0	0	NUM
cana-1681	36	32	.	.	PUNCT
cana-1681	37	1	⇒	⇒	PROPN
cana-1681	37	2	∃휂𝑁휀𝑆(𝑅	∃휂𝑁휀𝑆(𝑅	PROPN
cana-1681	37	3	)	)	PUNCT
cana-1681	37	4	,	,	PUNCT
cana-1681	37	5	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	37	6	)	)	PUNCT
cana-1681	37	7	such	such	ADJ
cana-1681	37	8	that	that	DET
cana-1681	37	9	∫	∫	PROPN
cana-1681	37	10	𝑅	𝑅	PROPN
cana-1681	37	11	|휂𝑁	|휂𝑁	NOUN
cana-1681	37	12	−	−	NOUN
cana-1681	37	13	𝑓|𝑝𝑑𝑥	𝑓|𝑝𝑑𝑥	ADJ
cana-1681	37	14	→	→	SYM
cana-1681	37	15	0	0	NUM
cana-1681	37	16	.	.	PUNCT
cana-1681	38	1	as	as	SCONJ
cana-1681	38	2	𝑁	𝑁	PROPN
cana-1681	38	3	→	→	SYM
cana-1681	38	4	∞.	∞.	PROPN
cana-1681	38	5	“	"	PUNCT
cana-1681	38	6	hence	hence	ADV
cana-1681	38	7	𝑆∗(𝑅	𝑆∗(𝑅	NOUN
cana-1681	38	8	)	)	PUNCT
cana-1681	38	9	is	be	AUX
cana-1681	38	10	dense	dense	ADJ
cana-1681	38	11	in	in	ADP
cana-1681	38	12	𝐿𝑝(𝑅	𝐿𝑝(𝑅	NOUN
cana-1681	38	13	)	)	PUNCT
cana-1681	38	14	.	.	PUNCT
cana-1681	38	15	"	"	PUNCT
cana-1681	39	1	proposition	proposition	NOUN
cana-1681	39	2	1.2	1.2	NUM
cana-1681	39	3	.	.	PUNCT
cana-1681	40	1	let	let	VERB
cana-1681	40	2	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	40	3	=	=	SYM
cana-1681	40	4	𝛼𝑛𝜓𝑛	𝛼𝑛𝜓𝑛	PROPN
cana-1681	40	5	+	+	CCONJ
cana-1681	40	6	(	(	PUNCT
cana-1681	40	7	1	1	NUM
cana-1681	40	8	−	−	NOUN
cana-1681	40	9	𝛼𝑛)𝜎𝑛	𝛼𝑛)𝜎𝑛	NOUN
cana-1681	40	10	,	,	PUNCT
cana-1681	40	11	where	where	SCONJ
cana-1681	40	12	{	{	PUNCT
cana-1681	40	13	𝜓𝑛}𝑛	𝜓𝑛}𝑛	PROPN
cana-1681	40	14	𝑁	𝑁	PROPN
cana-1681	40	15	,	,	PUNCT
cana-1681	40	16	{	{	PUNCT
cana-1681	40	17	𝜎𝑛}𝑛	𝜎𝑛}𝑛	PROPN
cana-1681	40	18	𝑁	𝑁	PROPN
cana-1681	40	19	are	be	AUX
cana-1681	40	20	approximate	approximate	ADJ
cana-1681	40	21	identities	identity	NOUN
cana-1681	40	22	and	and	CCONJ
cana-1681	41	1	0	0	NUM
cana-1681	41	2	≤	≤	NUM
cana-1681	41	3	𝛼𝑛	𝛼𝑛	NOUN
cana-1681	41	4	≤	≤	NUM
cana-1681	41	5	1	1	NUM
cana-1681	41	6	.	.	PUNCT
cana-1681	42	1	“	"	PUNCT
cana-1681	42	2	(	(	PUNCT
cana-1681	42	3	a	a	X
cana-1681	42	4	)	)	PUNCT
cana-1681	42	5	for	for	ADP
cana-1681	42	6	1	1	NUM
cana-1681	42	7	≤	≤	NUM
cana-1681	42	8	𝑝	𝑝	NOUN
cana-1681	42	9	≤	≤	NOUN
cana-1681	43	1	+	+	PROPN
cana-1681	43	2	∞	∞	NUM
cana-1681	43	3	and	and	CCONJ
cana-1681	43	4	every	every	DET
cana-1681	43	5	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	43	6	)	)	PUNCT
cana-1681	43	7	,	,	PUNCT
cana-1681	43	8	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	43	9	𝑛→∞	𝑛→∞	NUM
cana-1681	43	10	(	(	PUNCT
cana-1681	43	11	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	43	12	−	−	PROPN
cana-1681	43	13	𝜓𝑛	𝜓𝑛	NOUN
cana-1681	43	14	)	)	PUNCT
cana-1681	43	15	⋆	⋆	VERB
cana-1681	43	16	𝑓	𝑓	PRON
cana-1681	43	17	→	→	SYM
cana-1681	43	18	0	0	NUM
cana-1681	43	19	and	and	CCONJ
cana-1681	43	20	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	43	21	𝑛→∞	𝑛→∞	NUM
cana-1681	43	22	(	(	PUNCT
cana-1681	43	23	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	43	24	−	−	PROPN
cana-1681	43	25	𝜎𝑛	𝜎𝑛	NOUN
cana-1681	43	26	)	)	PUNCT
cana-1681	43	27	⋆	⋆	VERB
cana-1681	43	28	𝑓	𝑓	PRON
cana-1681	43	29	→	→	SYM
cana-1681	43	30	0	0	NUM
cana-1681	43	31	.	.	PUNCT
cana-1681	44	1	(	(	PUNCT
cana-1681	44	2	b	b	X
cana-1681	44	3	)	)	PUNCT
cana-1681	44	4	for	for	ADP
cana-1681	44	5	every	every	DET
cana-1681	44	6	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	NOUN
cana-1681	44	7	)	)	PUNCT
cana-1681	44	8	,	,	PUNCT
cana-1681	44	9	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	44	10	𝑛→∞	𝑛→∞	NUM
cana-1681	44	11	(	(	PUNCT
cana-1681	44	12	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	44	13	−	−	PROPN
cana-1681	44	14	𝜓𝑛	𝜓𝑛	NOUN
cana-1681	44	15	)	)	PUNCT
cana-1681	44	16	⋆	⋆	VERB
cana-1681	44	17	𝑓	𝑓	PRON
cana-1681	44	18	→	→	SYM
cana-1681	44	19	0	0	NUM
cana-1681	44	20	a.e	a.e	PROPN
cana-1681	44	21	..	..	PUNCT
cana-1681	44	22	(	(	PUNCT
cana-1681	44	23	c	c	X
cana-1681	44	24	)	)	PUNCT
cana-1681	44	25	for	for	ADP
cana-1681	44	26	1	1	NUM
cana-1681	44	27	≤	≤	NUM
cana-1681	44	28	𝑝	𝑝	NOUN
cana-1681	44	29	≤	≤	NOUN
cana-1681	45	1	+	+	PROPN
cana-1681	46	1	∞	∞	PROPN
cana-1681	46	2	,	,	PUNCT
cana-1681	46	3	if	if	SCONJ
cana-1681	46	4	∑	∑	ADP
cana-1681	46	5	𝑛	𝑛	PROPN
cana-1681	46	6	(	(	PUNCT
cana-1681	46	7	1	1	NUM
cana-1681	46	8	−	−	PROPN
cana-1681	46	9	𝛼𝑛)𝑝	𝛼𝑛)𝑝	NUM
cana-1681	46	10	<	<	X
cana-1681	46	11	+	+	PROPN
cana-1681	46	12	∞	∞	PROPN
cana-1681	46	13	,	,	PUNCT
cana-1681	46	14	then	then	ADV
cana-1681	46	15	for	for	ADP
cana-1681	46	16	every	every	DET
cana-1681	46	17	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	46	18	)	)	PUNCT
cana-1681	46	19	,	,	PUNCT
cana-1681	46	20	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1681	46	21	𝑛→∞	𝑛→∞	NUM
cana-1681	46	22	(	(	PUNCT
cana-1681	46	23	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	46	24	−	−	PROPN
cana-1681	46	25	𝜓𝑛	𝜓𝑛	NOUN
cana-1681	46	26	)	)	PUNCT
cana-1681	46	27	⋆	⋆	VERB
cana-1681	46	28	𝑓	𝑓	PRON
cana-1681	46	29	→	→	SYM
cana-1681	46	30	0	0	NUM
cana-1681	46	31	"	"	PUNCT
cana-1681	46	32	a.e	a.e	PROPN
cana-1681	46	33	.	.	PROPN
cana-1681	46	34	proof	proof	NOUN
cana-1681	46	35	.	.	PUNCT
cana-1681	47	1	(	(	PUNCT
cana-1681	47	2	a	a	X
cana-1681	47	3	)	)	PUNCT
cana-1681	47	4	“	"	PUNCT
cana-1681	47	5	if	if	SCONJ
cana-1681	47	6	1	1	NUM
cana-1681	47	7	≤	≤	NUM
cana-1681	47	8	𝑝	𝑝	NOUN
cana-1681	47	9	≤	≤	NOUN
cana-1681	48	1	+	+	PROPN
cana-1681	48	2	∞	∞	NUM
cana-1681	48	3	and	and	CCONJ
cana-1681	48	4	every	every	DET
cana-1681	48	5	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	48	6	)	)	PUNCT
cana-1681	48	7	.	.	PUNCT
cana-1681	49	1	using	use	VERB
cana-1681	49	2	minkowski	minkowski	PROPN
cana-1681	49	3	’s	’s	PART
cana-1681	49	4	inequality	inequality	NOUN
cana-1681	49	5	,	,	PUNCT
cana-1681	49	6	||(ℎ𝑛	||(ℎ𝑛	NOUN
cana-1681	49	7	−	−	NOUN
cana-1681	49	8	𝜓𝑛	𝜓𝑛	X
cana-1681	49	9	)	)	PUNCT
cana-1681	49	10	⋆	⋆	VERB
cana-1681	49	11	𝑓||	𝑓||	X
cana-1681	49	12	𝑝	𝑝	NOUN
cana-1681	49	13	≤	≤	NOUN
cana-1681	49	14	(	(	PUNCT
cana-1681	49	15	1	1	NUM
cana-1681	49	16	−	−	PROPN
cana-1681	49	17	𝛼𝑛	𝛼𝑛	PROPN
cana-1681	49	18	)	)	PUNCT
cana-1681	49	19	(	(	PUNCT
cana-1681	49	20	||𝜎𝑛	||𝜎𝑛	PROPN
cana-1681	49	21	⋆	⋆	VERB
cana-1681	49	22	𝑓	𝑓	DET
cana-1681	49	23	−	−	PUNCT
cana-1681	49	24	𝑓||	𝑓||	PROPN
cana-1681	49	25	𝑝	𝑝	NOUN
cana-1681	50	1	+	+	NUM
cana-1681	50	2	||𝜓𝑛	||𝜓𝑛	ADJ
cana-1681	50	3	⋆	⋆	VERB
cana-1681	50	4	𝑓	𝑓	PRON
cana-1681	50	5	−	−	PUNCT
cana-1681	50	6	𝑓||	𝑓||	PROPN
cana-1681	50	7	𝑝	𝑝	NOUN
cana-1681	50	8	)	)	PUNCT
cana-1681	50	9	now	now	ADV
cana-1681	50	10	,	,	PUNCT
cana-1681	50	11	as	as	SCONJ
cana-1681	50	12	proved	prove	VERB
cana-1681	50	13	by	by	ADP
cana-1681	50	14	singh	singh	PROPN
cana-1681	50	15	d.	d.	PROPN
cana-1681	50	16	and	and	CCONJ
cana-1681	50	17	singh	singh	PROPN
cana-1681	50	18	d.	d.	PROPN
cana-1681	51	1	[	[	X
cana-1681	51	2	2	2	X
cana-1681	51	3	]	]	PUNCT
cana-1681	51	4	,	,	PUNCT
cana-1681	51	5	we	we	PRON
cana-1681	51	6	have	have	AUX
cana-1681	51	7	if	if	SCONJ
cana-1681	51	8	ℎ𝑛(𝑥	ℎ𝑛(𝑥	PROPN
cana-1681	51	9	)	)	PUNCT
cana-1681	51	10	is	be	AUX
cana-1681	51	11	an	an	DET
cana-1681	51	12	approximate	approximate	ADJ
cana-1681	51	13	identity	identity	NOUN
cana-1681	51	14	and	and	CCONJ
cana-1681	51	15	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	51	16	)	)	PUNCT
cana-1681	51	17	,	,	PUNCT
cana-1681	51	18	then	then	ADV
cana-1681	51	19	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	51	20	⋆	⋆	VERB
cana-1681	51	21	𝑓	𝑓	PRON
cana-1681	51	22	→	→	SYM
cana-1681	51	23	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	51	24	)	)	PUNCT
cana-1681	51	25	.	.	PUNCT
cana-1681	52	1	so	so	ADV
cana-1681	52	2	,	,	PUNCT
cana-1681	52	3	by	by	ADP
cana-1681	52	4	using	use	VERB
cana-1681	52	5	the	the	DET
cana-1681	52	6	above	above	NOUN
cana-1681	52	7	,	,	PUNCT
cana-1681	52	8	we	we	PRON
cana-1681	52	9	obtain	obtain	VERB
cana-1681	52	10	||(ℎ𝑛	||(ℎ𝑛	PROPN
cana-1681	52	11	−	−	NOUN
cana-1681	52	12	𝜎𝑛	𝜎𝑛	NOUN
cana-1681	52	13	)	)	PUNCT
cana-1681	52	14	⋆	⋆	VERB
cana-1681	52	15	𝑓||	𝑓||	X
cana-1681	52	16	𝑝	𝑝	PROPN
cana-1681	52	17	→	→	SYM
cana-1681	52	18	0	0	NUM
cana-1681	52	19	.	.	PUNCT
cana-1681	52	20	"	"	PUNCT
cana-1681	53	1	(	(	PUNCT
cana-1681	53	2	b	b	NOUN
cana-1681	53	3	)	)	PUNCT
cana-1681	53	4	“	"	PUNCT
cana-1681	53	5	for	for	ADP
cana-1681	53	6	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	PROPN
cana-1681	53	7	)	)	PUNCT
cana-1681	53	8	,	,	PUNCT
cana-1681	53	9	|(ℎ𝑛	|(ℎ𝑛	NOUN
cana-1681	53	10	−	−	NOUN
cana-1681	53	11	𝜓𝑛	𝜓𝑛	X
cana-1681	53	12	)	)	PUNCT
cana-1681	53	13	⋆	⋆	PUNCT
cana-1681	53	14	𝑓|	𝑓|	PROPN
cana-1681	53	15	≤	≤	NUM
cana-1681	53	16	||(ℎ𝑛	||(ℎ𝑛	NOUN
cana-1681	53	17	−	−	NOUN
cana-1681	53	18	𝜓𝑛	𝜓𝑛	X
cana-1681	53	19	)	)	PUNCT
cana-1681	53	20	⋆	⋆	PUNCT
cana-1681	53	21	𝑓||	𝑓||	PRON
cana-1681	53	22	→	→	SYM
cana-1681	53	23	0	0	NUM
cana-1681	53	24	by	by	ADP
cana-1681	53	25	part	part	NOUN
cana-1681	53	26	(	(	PUNCT
cana-1681	53	27	a	a	NOUN
cana-1681	53	28	)	)	PUNCT
cana-1681	53	29	.	.	PUNCT
cana-1681	53	30	"	"	PUNCT
cana-1681	54	1	(	(	PUNCT
cana-1681	54	2	c	c	NOUN
cana-1681	54	3	)	)	PUNCT
cana-1681	54	4	“	"	PUNCT
cana-1681	54	5	for	for	ADP
cana-1681	54	6	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	54	7	)	)	PUNCT
cana-1681	54	8	,	,	PUNCT
cana-1681	54	9	∫	∫	PROPN
cana-1681	54	10	𝑅	𝑅	PROPN
cana-1681	54	11	∑	∑	PROPN
cana-1681	54	12	𝑛	𝑛	PROPN
cana-1681	54	13	(	(	PUNCT
cana-1681	54	14	1	1	NUM
cana-1681	54	15	−	−	NOUN
cana-1681	54	16	𝛼𝑛)𝑝|𝜎𝑛	𝛼𝑛)𝑝|𝜎𝑛	NOUN
cana-1681	54	17	⋆	⋆	VERB
cana-1681	54	18	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	54	19	,	,	PUNCT
cana-1681	54	20	𝑦)|𝑝𝑑𝑥𝑑𝑦	𝑦)|𝑝𝑑𝑥𝑑𝑦	VERB
cana-1681	54	21	=	=	PUNCT
cana-1681	54	22	∑	∑	PUNCT
cana-1681	54	23	𝑛	𝑛	PROPN
cana-1681	54	24	||	||	NOUN
cana-1681	55	1	(	(	PUNCT
cana-1681	55	2	1	1	NUM
cana-1681	55	3	−	−	NOUN
cana-1681	55	4	𝛼𝑛)𝜎𝑛	𝛼𝑛)𝜎𝑛	NUM
cana-1681	55	5	⋆	⋆	VERB
cana-1681	55	6	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	55	7	,	,	PUNCT
cana-1681	55	8	𝑦)||𝑝	𝑦)||𝑝	PROPN
cana-1681	55	9	𝑝	𝑝	NOUN
cana-1681	55	10	≤	≤	PROPN
cana-1681	55	11	∑	∑	PUNCT
cana-1681	55	12	𝑛	𝑛	PROPN
cana-1681	55	13	(	(	PUNCT
cana-1681	55	14	1	1	NUM
cana-1681	55	15	−	−	PROPN
cana-1681	55	16	𝜎𝑛)𝑝||𝑓||𝑝	𝜎𝑛)𝑝||𝑓||𝑝	PROPN
cana-1681	55	17	𝑝	𝑝	PROPN
cana-1681	55	18	<	<	X
cana-1681	55	19	+	+	PROPN
cana-1681	55	20	∞.	∞.	PROPN
cana-1681	55	21	"	"	PUNCT
cana-1681	55	22	then	then	ADV
cana-1681	55	23	(	(	PUNCT
cana-1681	55	24	1	1	NUM
cana-1681	55	25	−	−	NOUN
cana-1681	55	26	𝛼𝑛)𝜎𝑛	𝛼𝑛)𝜎𝑛	PUNCT
cana-1681	55	27	⋆	⋆	VERB
cana-1681	55	28	𝑓	𝑓	PRON
cana-1681	55	29	→	→	SYM
cana-1681	55	30	0	0	NUM
cana-1681	55	31	a.e	a.e	PROPN
cana-1681	55	32	.	.	PROPN
cana-1681	55	33	.	.	PUNCT
cana-1681	56	1	similarly	similarly	ADV
cana-1681	56	2	(	(	PUNCT
cana-1681	56	3	𝛼𝑛	𝛼𝑛	INTJ
cana-1681	56	4	−	−	PROPN
cana-1681	56	5	1)𝜓𝑛	1)𝜓𝑛	PROPN
cana-1681	56	6	⋆	⋆	VERB
cana-1681	56	7	𝑓	𝑓	PRON
cana-1681	56	8	→	→	SYM
cana-1681	56	9	0	0	NUM
cana-1681	56	10	a.e	a.e	PROPN
cana-1681	56	11	.	.	PROPN
cana-1681	56	12	definition	definition	NOUN
cana-1681	56	13	1.3	1.3	NUM
cana-1681	56	14	.	.	PUNCT
cana-1681	57	1	“	"	PUNCT
cana-1681	57	2	an	an	DET
cana-1681	57	3	approximate	approximate	ADJ
cana-1681	57	4	identity	identity	NOUN
cana-1681	57	5	{	{	PUNCT
cana-1681	57	6	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	57	7	}	}	PUNCT
cana-1681	57	8	is	be	AUX
cana-1681	57	9	called	call	VERB
cana-1681	57	10	𝐿𝑝-good	𝐿𝑝-good	PROPN
cana-1681	57	11	if	if	SCONJ
cana-1681	57	12	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	57	13	⋆	⋆	VERB
cana-1681	57	14	𝑓	𝑓	PROPN
cana-1681	57	15	→	→	PUNCT
cana-1681	57	16	𝑓	𝑓	DET
cana-1681	57	17	a.e	a.e	PROPN
cana-1681	57	18	.	.	PROPN
cana-1681	57	19	for	for	ADP
cana-1681	57	20	all	all	DET
cana-1681	57	21	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	57	22	)	)	PUNCT
cana-1681	57	23	,	,	PUNCT
cana-1681	57	24	and	and	CCONJ
cana-1681	57	25	it	it	PRON
cana-1681	57	26	is	be	AUX
cana-1681	57	27	called	call	VERB
cana-1681	57	28	good	good	ADJ
cana-1681	57	29	if	if	SCONJ
cana-1681	57	30	it	it	PRON
cana-1681	57	31	is	be	AUX
cana-1681	57	32	𝐿𝑝-good	𝐿𝑝-good	PROPN
cana-1681	57	33	for	for	ADP
cana-1681	57	34	every	every	DET
cana-1681	57	35	1	1	NUM
cana-1681	57	36	≤	≤	NUM
cana-1681	57	37	𝑝	𝑝	NOUN
cana-1681	57	38	≤	≤	NOUN
cana-1681	58	1	+	+	CCONJ
cana-1681	58	2	∞.	∞.	PROPN
cana-1681	58	3	an	an	DET
cana-1681	58	4	approximate	approximate	ADJ
cana-1681	58	5	identity	identity	NOUN
cana-1681	58	6	{	{	PUNCT
cana-1681	58	7	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	58	8	}	}	PUNCT
cana-1681	58	9	is	be	AUX
cana-1681	58	10	called	call	VERB
cana-1681	58	11	𝐿𝑝	𝐿𝑝	ADV
cana-1681	58	12	-bad	-bad	PUNCT
cana-1681	58	13	if	if	SCONJ
cana-1681	58	14	therte	therte	NOUN
cana-1681	58	15	exists	exist	VERB
cana-1681	58	16	𝑓휀𝐿𝑝(𝑅	𝑓휀𝐿𝑝(𝑅	NOUN
cana-1681	58	17	)	)	PUNCT
cana-1681	58	18	such	such	ADJ
cana-1681	58	19	that	that	SCONJ
cana-1681	58	20	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	58	21	⋆	⋆	VERB
cana-1681	58	22	𝑓	𝑓	PRON
cana-1681	58	23	not	not	PART
cana-1681	58	24	approachable	approachable	ADJ
cana-1681	58	25	to	to	ADP
cana-1681	58	26	𝑓	𝑓	PRON
cana-1681	58	27	on	on	ADP
cana-1681	58	28	a	a	DET
cana-1681	58	29	set	set	NOUN
cana-1681	58	30	of	of	ADP
cana-1681	58	31	positive	positive	ADJ
cana-1681	58	32	measure	measure	NOUN
cana-1681	58	33	.	.	PUNCT
cana-1681	59	1	definition	definition	NOUN
cana-1681	59	2	1.4	1.4	NUM
cana-1681	59	3	.	.	PUNCT
cana-1681	60	1	let	let	VERB
cana-1681	60	2	{	{	PUNCT
cana-1681	60	3	𝜓𝑛}𝑛	𝜓𝑛}𝑛	PROPN
cana-1681	60	4	𝑁	𝑁	PROPN
cana-1681	60	5	and	and	CCONJ
cana-1681	60	6	{	{	PUNCT
cana-1681	60	7	𝜎}𝑛	𝜎}𝑛	PROPN
cana-1681	60	8	𝑁	𝑁	PROPN
cana-1681	60	9	be	be	VERB
cana-1681	60	10	approximate	approximate	ADJ
cana-1681	60	11	identities	identity	NOUN
cana-1681	60	12	,	,	PUNCT
cana-1681	60	13	𝛼𝑛	𝛼𝑛	INTJ
cana-1681	60	14	be	be	AUX
cana-1681	60	15	a	a	DET
cana-1681	60	16	sequence	sequence	NOUN
cana-1681	60	17	of	of	ADP
cana-1681	60	18	real	real	ADJ
cana-1681	60	19	numbers	number	NOUN
cana-1681	60	20	with	with	ADP
cana-1681	60	21	0	0	NUM
cana-1681	60	22	≤	≤	NUM
cana-1681	60	23	𝛼𝑛	𝛼𝑛	NOUN
cana-1681	60	24	≤	≤	NUM
cana-1681	60	25	1	1	NUM
cana-1681	60	26	and	and	CCONJ
cana-1681	60	27	𝛼𝑛	𝛼𝑛	PROPN
cana-1681	60	28	→	→	SYM
cana-1681	60	29	1	1	X
cana-1681	60	30	.	.	X
cana-1681	61	1	we	we	PRON
cana-1681	61	2	call	call	VERB
cana-1681	61	3	preturbed	preturbe	VERB
cana-1681	61	4	approximate	approximate	ADJ
cana-1681	61	5	identities	identity	NOUN
cana-1681	61	6	any	any	DET
cana-1681	61	7	approximate	approximate	ADJ
cana-1681	61	8	identity	identity	NOUN
cana-1681	61	9	{	{	PUNCT
cana-1681	61	10	ℎ𝑛}𝑛	ℎ𝑛}𝑛	PROPN
cana-1681	61	11	𝑁	𝑁	PROPN
cana-1681	61	12	of	of	ADP
cana-1681	61	13	the	the	DET
cana-1681	61	14	form	form	NOUN
cana-1681	61	15	ℎ𝑛𝜓𝑛	ℎ𝑛𝜓𝑛	VERB
cana-1681	61	16	+	+	CCONJ
cana-1681	61	17	(	(	PUNCT
cana-1681	61	18	1	1	NUM
cana-1681	61	19	−	−	NOUN
cana-1681	61	20	𝛼𝑛)𝜎𝑛.	𝛼𝑛)𝜎𝑛.	NOUN
cana-1681	61	21	"	"	PUNCT
cana-1681	61	22	2	2	NUM
cana-1681	61	23	.	.	PUNCT
cana-1681	61	24	main	main	ADJ
cana-1681	61	25	results	result	NOUN
cana-1681	61	26	.	.	PUNCT
cana-1681	62	1	theorem	theorem	VERB
cana-1681	62	2	2.1	2.1	NUM
cana-1681	62	3	.	.	PUNCT
cana-1681	63	1	(	(	PUNCT
cana-1681	63	2	i	i	NOUN
cana-1681	63	3	)	)	PUNCT
cana-1681	63	4	“	"	PUNCT
cana-1681	63	5	given	give	VERB
cana-1681	63	6	any	any	DET
cana-1681	63	7	good	good	ADJ
cana-1681	63	8	approximate	approximate	ADJ
cana-1681	63	9	identity	identity	NOUN
cana-1681	63	10	{	{	PUNCT
cana-1681	63	11	𝜓𝑛}𝑛	𝜓𝑛}𝑛	PROPN
cana-1681	63	12	𝑁	𝑁	PROPN
cana-1681	63	13	there	there	PRON
cana-1681	63	14	exists	exist	VERB
cana-1681	63	15	a	a	DET
cana-1681	63	16	perturbed	perturb	VERB
cana-1681	63	17	approximate	approximate	ADJ
cana-1681	63	18	identity	identity	NOUN
cana-1681	63	19	{	{	PUNCT
cana-1681	63	20	ℎ𝑛}𝑛휀𝑁	ℎ𝑛}𝑛휀𝑁	VERB
cana-1681	63	21	such	such	ADJ
cana-1681	63	22	that	that	PRON
cana-1681	63	23	𝑓휀𝐿𝑞(𝑅	𝑓휀𝐿𝑞(𝑅	NOUN
cana-1681	63	24	)	)	PUNCT
cana-1681	63	25	(	(	PUNCT
cana-1681	63	26	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	63	27	⋆	⋆	VERB
cana-1681	63	28	𝑓)(휁	𝑓)(휁	NOUN
cana-1681	63	29	,	,	PUNCT
cana-1681	63	30	𝜙	𝜙	NOUN
cana-1681	63	31	)	)	PUNCT
cana-1681	63	32	=	=	SYM
cana-1681	64	1	ℎ	ℎ	PROPN
cana-1681	64	2	�	�	PROPN
cana-1681	64	3	̂	̂	SYM
cana-1681	64	4	�	�	NOUN
cana-1681	64	5	(휁	(휁	VERB
cana-1681	64	6	,	,	PUNCT
cana-1681	64	7	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	8	,	,	PUNCT
cana-1681	64	9	𝜙	𝜙	NOUN
cana-1681	64	10	)	)	PUNCT
cana-1681	64	11	(	(	PUNCT
cana-1681	64	12	ℎ	ℎ	X
cana-1681	64	13	�	�	PROPN
cana-1681	64	14	̂	̂	SYM
cana-1681	64	15	�	�	NOUN
cana-1681	64	16	(휁	(휁	VERB
cana-1681	64	17	,	,	PUNCT
cana-1681	64	18	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	19	,	,	PUNCT
cana-1681	64	20	𝜙	𝜙	NOUN
cana-1681	64	21	)	)	PUNCT
cana-1681	64	22	)	)	PUNCT
cana-1681	64	23	→	→	PUNCT
cana-1681	64	24	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	64	25	,	,	PUNCT
cana-1681	64	26	𝑦)1	𝑦)1	NOUN
cana-1681	64	27	≤	≤	PUNCT
cana-1681	64	28	𝑞	𝑞	X
cana-1681	64	29	<	<	X
cana-1681	64	30	𝑝	𝑝	PROPN
cana-1681	64	31	(	(	PUNCT
cana-1681	64	32	ii	ii	NOUN
cana-1681	64	33	)	)	PUNCT
cana-1681	64	34	(	(	PUNCT
cana-1681	64	35	ℎ	ℎ	X
cana-1681	64	36	�	�	PROPN
cana-1681	64	37	̂	̂	SYM
cana-1681	64	38	�	�	NOUN
cana-1681	64	39	(휁	(휁	VERB
cana-1681	64	40	,	,	PUNCT
cana-1681	64	41	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	42	,	,	PUNCT
cana-1681	64	43	𝜙	𝜙	NOUN
cana-1681	64	44	)	)	PUNCT
cana-1681	64	45	)	)	PUNCT
cana-1681	64	46	→	→	PUNCT
cana-1681	64	47	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	64	48	,	,	PUNCT
cana-1681	64	49	𝑦	𝑦	NOUN
cana-1681	64	50	)	)	PUNCT
cana-1681	64	51	for	for	ADP
cana-1681	64	52	𝑞	𝑞	PROPN
cana-1681	64	53	>	>	X
cana-1681	64	54	𝑝	𝑝	PROPN
cana-1681	64	55	and	and	CCONJ
cana-1681	64	56	(	(	PUNCT
cana-1681	64	57	ℎ	ℎ	PROPN
cana-1681	64	58	�	�	PROPN
cana-1681	64	59	̂	̂	SYM
cana-1681	64	60	�	�	NOUN
cana-1681	64	61	(휁	(휁	VERB
cana-1681	64	62	,	,	PUNCT
cana-1681	64	63	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	64	,	,	PUNCT
cana-1681	64	65	𝜙	𝜙	NOUN
cana-1681	64	66	)	)	PUNCT
cana-1681	64	67	)	)	PUNCT
cana-1681	64	68	not	not	PART
cana-1681	64	69	approaches	approach	VERB
cana-1681	64	70	to𝑓(𝑥	to𝑓(𝑥	ADP
cana-1681	64	71	,	,	PUNCT
cana-1681	64	72	𝑦	𝑦	NOUN
cana-1681	64	73	)	)	PUNCT
cana-1681	64	74	for	for	ADP
cana-1681	64	75	1	1	NUM
cana-1681	64	76	≤	≤	NUM
cana-1681	64	77	𝑞	𝑞	PROPN
cana-1681	64	78	≤	≤	PROPN
cana-1681	64	79	𝑝.	𝑝.	NOUN
cana-1681	64	80	communications	communication	NOUN
cana-1681	64	81	on	on	ADP
cana-1681	64	82	applied	apply	VERB
cana-1681	64	83	nonlinear	nonlinear	ADJ
cana-1681	64	84	analysis	analysis	NOUN
cana-1681	64	85	issn	issn	NOUN
cana-1681	64	86	:	:	PUNCT
cana-1681	64	87	1074	1074	NUM
cana-1681	64	88	-	-	PUNCT
cana-1681	64	89	133x	133x	NUM
cana-1681	64	90	vol	vol	NOUN
cana-1681	64	91	32	32	NUM
cana-1681	64	92	no	no	NOUN
cana-1681	64	93	.	.	NOUN
cana-1681	64	94	1	1	NUM
cana-1681	64	95	(	(	PUNCT
cana-1681	64	96	2025	2025	NUM
cana-1681	64	97	)	)	PUNCT
cana-1681	64	98	388	388	NUM
cana-1681	64	99	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	64	100	(	(	PUNCT
cana-1681	64	101	iii	iii	NOUN
cana-1681	64	102	)	)	PUNCT
cana-1681	64	103	(	(	PUNCT
cana-1681	64	104	ℎ	ℎ	X
cana-1681	64	105	�	�	PROPN
cana-1681	64	106	̂	̂	SYM
cana-1681	64	107	�	�	NOUN
cana-1681	64	108	(휁	(휁	VERB
cana-1681	64	109	,	,	PUNCT
cana-1681	64	110	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	111	,	,	PUNCT
cana-1681	64	112	𝜙	𝜙	NOUN
cana-1681	64	113	)	)	PUNCT
cana-1681	64	114	)	)	PUNCT
cana-1681	64	115	→	→	PUNCT
cana-1681	64	116	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	64	117	,	,	PUNCT
cana-1681	64	118	𝑦	𝑦	NOUN
cana-1681	64	119	)	)	PUNCT
cana-1681	64	120	for	for	ADP
cana-1681	64	121	𝑞	𝑞	X
cana-1681	64	122	=	=	SYM
cana-1681	64	123	∞	∞	PROPN
cana-1681	64	124	(	(	PUNCT
cana-1681	64	125	ℎ	ℎ	PROPN
cana-1681	64	126	�	�	PROPN
cana-1681	64	127	̂	̂	SYM
cana-1681	64	128	�	�	NOUN
cana-1681	64	129	(휁	(휁	VERB
cana-1681	64	130	,	,	PUNCT
cana-1681	64	131	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	64	132	,	,	PUNCT
cana-1681	64	133	𝜙	𝜙	NOUN
cana-1681	64	134	)	)	PUNCT
cana-1681	64	135	not	not	PART
cana-1681	64	136	approachable	approachable	ADJ
cana-1681	64	137	to	to	ADP
cana-1681	64	138	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	64	139	,	,	PUNCT
cana-1681	64	140	𝑦	𝑦	NOUN
cana-1681	64	141	)	)	PUNCT
cana-1681	64	142	for	for	ADP
cana-1681	64	143	1	1	NUM
cana-1681	64	144	≤	≤	NUM
cana-1681	64	145	𝑞	𝑞	X
cana-1681	64	146	<	<	X
cana-1681	64	147	∞.	∞.	PROPN
cana-1681	64	148	"	"	PUNCT
cana-1681	64	149	proof	proof	NOUN
cana-1681	64	150	.	.	PUNCT
cana-1681	65	1	(	(	PUNCT
cana-1681	65	2	i	i	NOUN
cana-1681	65	3	)	)	PUNCT
cana-1681	65	4	let	let	VERB
cana-1681	65	5	𝑔𝑛(𝑥	𝑔𝑛(𝑥	NOUN
cana-1681	65	6	)	)	PUNCT
cana-1681	65	7	=	=	SYM
cana-1681	65	8	1	1	NUM
cana-1681	65	9	√2𝜋	√2𝜋	DET
cana-1681	65	10	∫	∫	PROPN
cana-1681	65	11	𝑅	𝑅	PROPN
cana-1681	65	12	𝑒𝜄(𝑥	𝑒𝜄(𝑥	NOUN
cana-1681	65	13	+	+	PROPN
cana-1681	65	14	𝑦𝜙)ℎ	𝑦𝜙)ℎ	PROPN
cana-1681	65	15	�	�	PROPN
cana-1681	65	16	̂	̂	SYM
cana-1681	65	17	�	�	NOUN
cana-1681	65	18	(휁	(휁	VERB
cana-1681	65	19	,	,	PUNCT
cana-1681	65	20	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	65	21	,	,	PUNCT
cana-1681	65	22	𝜙)𝑑휁𝑑𝜙	𝜙)𝑑휁𝑑𝜙	ADV
cana-1681	65	23	=	=	SYM
cana-1681	65	24	1	1	NUM
cana-1681	65	25	√2𝜋	√2𝜋	DET
cana-1681	65	26	∫	∫	PROPN
cana-1681	65	27	𝑅	𝑅	PROPN
cana-1681	65	28	𝑒𝜄(𝑥	𝑒𝜄(𝑥	NOUN
cana-1681	65	29	+	+	PROPN
cana-1681	65	30	𝑦𝜙)ℎ	𝑦𝜙)ℎ	PROPN
cana-1681	65	31	�	�	PROPN
cana-1681	65	32	̂	̂	SYM
cana-1681	65	33	�	�	NOUN
cana-1681	65	34	(휁	(휁	NOUN
cana-1681	65	35	,	,	PUNCT
cana-1681	65	36	𝜙	𝜙	NOUN
cana-1681	65	37	)	)	PUNCT
cana-1681	65	38	∫	∫	PROPN
cana-1681	65	39	𝑅	𝑅	PROPN
cana-1681	65	40	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	65	41	,	,	PUNCT
cana-1681	65	42	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	X
cana-1681	65	43	=	=	SYM
cana-1681	65	44	1	1	NUM
cana-1681	65	45	√2𝜋	√2𝜋	PRON
cana-1681	65	46	∫	∫	PROPN
cana-1681	65	47	𝑅	𝑅	PROPN
cana-1681	65	48	ℎ𝑛(𝑥	ℎ𝑛(𝑥	PRON
cana-1681	65	49	+	+	NUM
cana-1681	65	50	𝑦)𝑓(𝑥	𝑦)𝑓(𝑥	NOUN
cana-1681	65	51	,	,	PUNCT
cana-1681	65	52	𝑦)𝑑𝑥𝑑𝑦𝑜𝑟	𝑦)𝑑𝑥𝑑𝑦𝑜𝑟	NOUN
cana-1681	65	53	=	=	SYM
cana-1681	65	54	(	(	PUNCT
cana-1681	65	55	ℎ𝑛	ℎ𝑛	INTJ
cana-1681	65	56	⋆	⋆	VERB
cana-1681	65	57	𝑓)(𝑥	𝑓)(𝑥	PROPN
cana-1681	65	58	,	,	PUNCT
cana-1681	65	59	𝑦)ℎ	𝑦)ℎ	SYM
cana-1681	65	60	�	�	NOUN
cana-1681	65	61	̂	̂	SYM
cana-1681	65	62	�	�	NOUN
cana-1681	65	63	(휁	(휁	VERB
cana-1681	65	64	,	,	PUNCT
cana-1681	65	65	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	65	66	,	,	PUNCT
cana-1681	65	67	𝜙	𝜙	NOUN
cana-1681	65	68	)	)	PUNCT
cana-1681	65	69	=	=	SYM
cana-1681	66	1	1	1	NUM
cana-1681	66	2	√2𝜋	√2𝜋	PRON
cana-1681	66	3	∫	∫	PROPN
cana-1681	66	4	𝑅	𝑅	PROPN
cana-1681	66	5	𝑒𝜄(𝑥	𝑒𝜄(𝑥	NOUN
cana-1681	66	6	+	+	PROPN
cana-1681	66	7	𝑦𝜙)ℎ	𝑦𝜙)ℎ	PROPN
cana-1681	66	8	�	�	PROPN
cana-1681	66	9	̂	̂	SYM
cana-1681	66	10	�	�	NOUN
cana-1681	66	11	(휁	(휁	VERB
cana-1681	66	12	,	,	PUNCT
cana-1681	66	13	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	66	14	,	,	PUNCT
cana-1681	66	15	𝜙)𝑑휁𝑑𝜙	𝜙)𝑑휁𝑑𝜙	ADV
cana-1681	66	16	=	=	SYM
cana-1681	66	17	(	(	PUNCT
cana-1681	66	18	ℎ𝑛	ℎ𝑛	INTJ
cana-1681	66	19	⋆	⋆	VERB
cana-1681	66	20	𝑓)(𝑥	𝑓)(𝑥	PROPN
cana-1681	66	21	,	,	PUNCT
cana-1681	66	22	𝑦)“𝐹𝑖𝑥	𝑦)“𝐹𝑖𝑥	PROPN
cana-1681	66	23	𝑞	𝑞	PROPN
cana-1681	66	24	≥	≥	PROPN
cana-1681	66	25	𝑝	𝑝	PROPN
cana-1681	66	26	and	and	CCONJ
cana-1681	66	27	taking	take	VERB
cana-1681	66	28	1	1	NUM
cana-1681	66	29	−	−	PROPN
cana-1681	66	30	𝛼𝑛	𝛼𝑛	NOUN
cana-1681	66	31	=	=	SYM
cana-1681	66	32	1	1	NUM
cana-1681	66	33	(	(	PUNCT
cana-1681	66	34	𝑛𝑙𝑜𝑔2𝑛)1/𝑝	𝑛𝑙𝑜𝑔2𝑛)1/𝑝	NOUN
cana-1681	66	35	.	.	PUNCT
cana-1681	67	1	since	since	SCONJ
cana-1681	67	2	∑	∑	PROPN
cana-1681	67	3	𝑛	𝑛	PROPN
cana-1681	67	4	(	(	PUNCT
cana-1681	67	5	1	1	NUM
cana-1681	67	6	−	−	NOUN
cana-1681	67	7	𝛼𝑛)𝑞	𝛼𝑛)𝑞	ADV
cana-1681	67	8	<	<	X
cana-1681	67	9	+	+	NOUN
cana-1681	67	10	∞	∞	NUM
cana-1681	67	11	and	and	CCONJ
cana-1681	67	12	𝜓𝑛	𝜓𝑛	PRON
cana-1681	67	13	is	be	AUX
cana-1681	67	14	an	an	DET
cana-1681	67	15	𝐿𝑞-good	𝐿𝑞-good	NOUN
cana-1681	67	16	approximate	approximate	ADJ
cana-1681	67	17	identity	identity	NOUN
cana-1681	67	18	,	,	PUNCT
cana-1681	67	19	using	use	VERB
cana-1681	67	20	proposition	proposition	NOUN
cana-1681	67	21	1.4	1.4	NUM
cana-1681	67	22	.	.	PUNCT
cana-1681	68	1	we	we	PRON
cana-1681	68	2	obtain	obtain	VERB
cana-1681	68	3	that	that	PRON
cana-1681	68	4	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	68	5	is	be	AUX
cana-1681	68	6	also	also	ADV
cana-1681	68	7	an	an	DET
cana-1681	68	8	𝐿𝑞-good	𝐿𝑞-good	ADJ
cana-1681	68	9	approximate	approximate	ADJ
cana-1681	68	10	identity	identity	NOUN
cana-1681	68	11	.	.	PUNCT
cana-1681	68	12	"	"	PUNCT
cana-1681	69	1	hence	hence	ADV
cana-1681	69	2	for	for	ADP
cana-1681	69	3	𝑞	𝑞	PROPN
cana-1681	69	4	≥	≥	X
cana-1681	69	5	𝑝,(ℎ𝑛	𝑝,(ℎ𝑛	NOUN
cana-1681	69	6	⋆	⋆	PUNCT
cana-1681	69	7	𝑓)(𝑥	𝑓)(𝑥	PROPN
cana-1681	69	8	,	,	PUNCT
cana-1681	69	9	𝑦	𝑦	NOUN
cana-1681	69	10	)	)	PUNCT
cana-1681	69	11	→	→	SYM
cana-1681	69	12	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	69	13	,	,	PUNCT
cana-1681	69	14	𝑦	𝑦	NOUN
cana-1681	69	15	)	)	PUNCT
cana-1681	69	16	now	now	ADV
cana-1681	69	17	,	,	PUNCT
cana-1681	69	18	we	we	PRON
cana-1681	69	19	have	have	VERB
cana-1681	69	20	to	to	PART
cana-1681	69	21	prove	prove	VERB
cana-1681	69	22	that	that	SCONJ
cana-1681	69	23	for	for	ADP
cana-1681	69	24	each	each	DET
cana-1681	69	25	1	1	NUM
cana-1681	69	26	≤	≤	NOUN
cana-1681	69	27	𝑞	𝑞	X
cana-1681	69	28	<	<	X
cana-1681	69	29	𝑝.	𝑝.	PROPN
cana-1681	69	30	there	there	PRON
cana-1681	69	31	exists	exist	VERB
cana-1681	69	32	𝑓𝑞휀𝐿𝑞(𝑅	𝑓𝑞휀𝐿𝑞(𝑅	PROPN
cana-1681	69	33	)	)	PUNCT
cana-1681	69	34	so	so	SCONJ
cana-1681	69	35	that	that	SCONJ
cana-1681	69	36	“	"	PUNCT
cana-1681	69	37	𝑙𝑖𝑚𝑠𝑢𝑝𝑘|𝑥|𝑘	𝑙𝑖𝑚𝑠𝑢𝑝𝑘|𝑥|𝑘	VERB
cana-1681	69	38	𝑑𝑘	𝑑𝑘	ADP
cana-1681	69	39	𝑑𝑥𝑘	𝑑𝑥𝑘	NOUN
cana-1681	69	40	(	(	PUNCT
cana-1681	69	41	ℎ𝑘	ℎ𝑘	PROPN
cana-1681	69	42	⋆	⋆	VERB
cana-1681	69	43	𝑓𝑞	𝑓𝑞	INTJ
cana-1681	69	44	→	→	SYM
cana-1681	69	45	∞	∞	NUM
cana-1681	69	46	)	)	PUNCT
cana-1681	69	47	"	"	PUNCT
cana-1681	69	48	on	on	ADP
cana-1681	69	49	a	a	DET
cana-1681	69	50	set	set	NOUN
cana-1681	69	51	of	of	ADP
cana-1681	69	52	positive	positive	ADJ
cana-1681	69	53	measure	measure	NOUN
cana-1681	69	54	.	.	PUNCT
cana-1681	70	1	set	set	NOUN
cana-1681	70	2	,	,	PUNCT
cana-1681	70	3	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NUM
cana-1681	70	4	,	,	PUNCT
cana-1681	70	5	𝑦	𝑦	X
cana-1681	70	6	)	)	PUNCT
cana-1681	70	7	=	=	SYM
cana-1681	70	8	1	1	NUM
cana-1681	70	9	(	(	PUNCT
cana-1681	70	10	𝑥𝑦𝑙𝑜𝑔2(𝑥/2,𝑦/2	𝑥𝑦𝑙𝑜𝑔2(𝑥/2,𝑦/2	NOUN
cana-1681	70	11	)	)	PUNCT
cana-1681	70	12	)	)	PUNCT
cana-1681	70	13	1/𝑞	1/𝑞	NUM
cana-1681	70	14	𝜒[0,1](𝑥)휀𝐿𝑞(𝑅	𝜒[0,1](𝑥)휀𝐿𝑞(𝑅	NOUN
cana-1681	70	15	)	)	PUNCT
cana-1681	70	16	.	.	PUNCT
cana-1681	71	1	take	take	VERB
cana-1681	71	2	𝑟𝑛	𝑟𝑛	ADV
cana-1681	71	3	=	=	SYM
cana-1681	71	4	1	1	NUM
cana-1681	71	5	𝑛1	𝑛1	NOUN
cana-1681	71	6	+	+	PROPN
cana-1681	71	7	1/𝑝(𝑙𝑜𝑔𝑛)2/𝑝	1/𝑝(𝑙𝑜𝑔𝑛)2/𝑝	PROPN
cana-1681	71	8	,	,	PUNCT
cana-1681	71	9	𝑎𝑛	𝑎𝑛	NOUN
cana-1681	71	10	=	=	PUNCT
cana-1681	71	11	𝑟𝑛	𝑟𝑛	ADP
cana-1681	71	12	1	1	NUM
cana-1681	71	13	𝑝+1	𝑝+1	NOUN
cana-1681	71	14	=	=	SYM
cana-1681	71	15	1	1	NUM
cana-1681	71	16	𝑛1/𝑝	𝑛1/𝑝	NOUN
cana-1681	71	17	(	(	PUNCT
cana-1681	71	18	𝑙𝑜𝑔𝑛	𝑙𝑜𝑔𝑛	SYM
cana-1681	71	19	)	)	PUNCT
cana-1681	71	20	2	2	NUM
cana-1681	71	21	𝑝(𝑝+1	𝑝(𝑝+1	NOUN
cana-1681	71	22	)	)	PUNCT
cana-1681	71	23	,	,	PUNCT
cana-1681	71	24	“	"	PUNCT
cana-1681	71	25	𝐽𝑛	𝐽𝑛	NOUN
cana-1681	71	26	=	=	SYM
cana-1681	72	1	[	[	X
cana-1681	72	2	𝑎𝑛	𝑎𝑛	PRON
cana-1681	72	3	−	−	NOUN
cana-1681	72	4	𝑟𝑛	𝑟𝑛	ADP
cana-1681	72	5	,	,	PUNCT
cana-1681	72	6	𝑎𝑛	𝑎𝑛	PROPN
cana-1681	72	7	+	+	X
cana-1681	73	1	𝑟𝑛	𝑟𝑛	ADP
cana-1681	73	2	]	]	X
cana-1681	73	3	and	and	CCONJ
cana-1681	73	4	𝑈𝑛	𝑈𝑛	PROPN
cana-1681	73	5	=	=	PUNCT
cana-1681	74	1	[	[	X
cana-1681	74	2	−𝑎𝑛	−𝑎𝑛	NOUN
cana-1681	74	3	+	+	CCONJ
cana-1681	74	4	𝑟𝑛	𝑟𝑛	ADJ
cana-1681	74	5	,	,	PUNCT
cana-1681	74	6	−𝑎𝑛+1	−𝑎𝑛+1	PROPN
cana-1681	74	7	+	+	CCONJ
cana-1681	74	8	𝑟𝑛+1	𝑟𝑛+1	NUM
cana-1681	74	9	]	]	PUNCT
cana-1681	74	10	,	,	PUNCT
cana-1681	74	11	"	"	PUNCT
cana-1681	74	12	for	for	ADP
cana-1681	74	13	sufficiently	sufficiently	ADV
cana-1681	74	14	large	large	ADJ
cana-1681	74	15	n	n	NOUN
cana-1681	74	16	and	and	CCONJ
cana-1681	74	17	for	for	ADP
cana-1681	74	18	all	all	PRON
cana-1681	74	19	𝑘	𝑘	DET
cana-1681	74	20	≥	≥	NOUN
cana-1681	74	21	𝑛	𝑛	NOUN
cana-1681	74	22	,	,	PUNCT
cana-1681	74	23	𝑥휀𝑈𝑘.	𝑥휀𝑈𝑘.	SCONJ
cana-1681	74	24	ℎ𝑘	ℎ𝑘	PRON
cana-1681	74	25	⋆	⋆	VERB
cana-1681	74	26	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	74	27	,	,	PUNCT
cana-1681	74	28	𝑦	𝑦	X
cana-1681	74	29	)	)	PUNCT
cana-1681	74	30	≥	≥	NOUN
cana-1681	74	31	(	(	PUNCT
cana-1681	74	32	1	1	NUM
cana-1681	74	33	−	−	NOUN
cana-1681	74	34	𝛼𝑘)𝜎𝑘	𝛼𝑘)𝜎𝑘	X
cana-1681	74	35	⋆	⋆	VERB
cana-1681	74	36	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	74	37	)	)	PUNCT
cana-1681	74	38	≥	≥	NOUN
cana-1681	74	39	1	1	NUM
cana-1681	74	40	(	(	PUNCT
cana-1681	74	41	𝑘𝑙𝑜𝑔2(𝑘	𝑘𝑙𝑜𝑔2(𝑘	NOUN
cana-1681	74	42	)	)	PUNCT
cana-1681	74	43	)	)	PUNCT
cana-1681	75	1	1/𝑝	1/𝑝	NUM
cana-1681	75	2	∫	∫	PROPN
cana-1681	75	3	−𝐽𝑘	−𝐽𝑘	PROPN
cana-1681	75	4	𝜎𝑘(𝑥	𝜎𝑘(𝑥	NOUN
cana-1681	75	5	,	,	PUNCT
cana-1681	75	6	𝑦)𝑓𝑞(𝑥	𝑦)𝑓𝑞(𝑥	VERB
cana-1681	75	7	−	−	PROPN
cana-1681	75	8	𝑦)𝑑𝑥𝑑𝑦𝑁𝑜𝑤	𝑦)𝑑𝑥𝑑𝑦𝑁𝑜𝑤	ADJ
cana-1681	75	9	,	,	PUNCT
cana-1681	75	10	𝑤𝑒	𝑤𝑒	PROPN
cana-1681	75	11	ℎ𝑎𝑣𝑒	ℎ𝑎𝑣𝑒	NOUN
cana-1681	75	12	,	,	PUNCT
cana-1681	75	13	ℎ𝑘	ℎ𝑘	PROPN
cana-1681	75	14	⋆	⋆	VERB
cana-1681	75	15	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	75	16	,	,	PUNCT
cana-1681	75	17	𝑦	𝑦	NOUN
cana-1681	75	18	)	)	PUNCT
cana-1681	75	19	≥	≥	NOUN
cana-1681	76	1	𝑓𝑞(𝐶𝑟𝑘	𝑓𝑞(𝐶𝑟𝑘	PROPN
cana-1681	76	2	(	(	PUNCT
cana-1681	76	3	𝑙𝑜𝑔𝑘)2/𝑝+1	𝑙𝑜𝑔𝑘)2/𝑝+1	NUM
cana-1681	76	4	)	)	PUNCT
cana-1681	76	5	(	(	PUNCT
cana-1681	76	6	𝑘𝑙𝑜𝑔2𝑘)1/𝑝	𝑘𝑙𝑜𝑔2𝑘)1/𝑝	NOUN
cana-1681	76	7	∫	∫	PROPN
cana-1681	76	8	−𝐽𝑘	−𝐽𝑘	PROPN
cana-1681	76	9	𝜎𝑘(𝑥	𝜎𝑘(𝑥	PROPN
cana-1681	76	10	,	,	PUNCT
cana-1681	76	11	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	PROPN
cana-1681	76	12	or	or	CCONJ
cana-1681	76	13	,	,	PUNCT
cana-1681	76	14	𝑓𝑞(𝐶𝑟𝑘	𝑓𝑞(𝐶𝑟𝑘	PROPN
cana-1681	76	15	(	(	PUNCT
cana-1681	76	16	𝑙𝑜𝑔𝑘)2/𝑝+1	𝑙𝑜𝑔𝑘)2/𝑝+1	NUM
cana-1681	76	17	)	)	PUNCT
cana-1681	76	18	=	=	SYM
cana-1681	76	19	𝑘1/𝑞+1/𝑝𝑞(𝑙𝑜𝑔𝑘	𝑘1/𝑞+1/𝑝𝑞(𝑙𝑜𝑔𝑘	PROPN
cana-1681	76	20	)	)	PUNCT
cana-1681	76	21	2	2	NUM
cana-1681	76	22	𝑝𝑞(𝑝+1	𝑝𝑞(𝑝+1	NOUN
cana-1681	76	23	)	)	PUNCT
cana-1681	76	24	𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1	𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1	NOUN
cana-1681	76	25	)	)	PUNCT
cana-1681	76	26	)	)	PUNCT
cana-1681	76	27	)	)	PUNCT
cana-1681	77	1	2/𝑞	2/𝑞	NUM
cana-1681	77	2	then	then	ADV
cana-1681	77	3	,	,	PUNCT
cana-1681	77	4	ℎ𝑘	ℎ𝑘	PROPN
cana-1681	77	5	⋆	⋆	VERB
cana-1681	77	6	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	77	7	,	,	PUNCT
cana-1681	77	8	𝑦	𝑦	X
cana-1681	77	9	)	)	PUNCT
cana-1681	77	10	≥	≥	PRON
cana-1681	78	1	𝐶𝑘	𝐶𝑘	PROPN
cana-1681	78	2	1	1	NUM
cana-1681	78	3	𝑞	𝑞	NOUN
cana-1681	78	4	−	−	PROPN
cana-1681	78	5	1	1	NUM
cana-1681	78	6	𝑝	𝑝	PROPN
cana-1681	78	7	+	+	CCONJ
cana-1681	78	8	1	1	NUM
cana-1681	78	9	𝑝𝑞𝐻𝑞(𝑘	𝑝𝑞𝐻𝑞(𝑘	NOUN
cana-1681	78	10	)	)	PUNCT
cana-1681	78	11	>	>	X
cana-1681	79	1	𝑘𝛿	𝑘𝛿	ADP
cana-1681	79	2	≥	≥	NOUN
cana-1681	79	3	𝑛𝛿	𝑛𝛿	PROPN
cana-1681	79	4	)	)	PUNCT
cana-1681	79	5	,	,	PUNCT
cana-1681	79	6	where	where	SCONJ
cana-1681	79	7	,	,	PUNCT
cana-1681	79	8	𝐻𝑞(𝑘	𝐻𝑞(𝑘	NOUN
cana-1681	79	9	)	)	PUNCT
cana-1681	79	10	=	=	SYM
cana-1681	79	11	(	(	PUNCT
cana-1681	79	12	𝑙𝑜𝑔𝑘	𝑙𝑜𝑔𝑘	PROPN
cana-1681	79	13	)	)	PUNCT
cana-1681	79	14	2	2	NUM
cana-1681	79	15	𝑝𝑞(𝑝+1	𝑝𝑞(𝑝+1	NOUN
cana-1681	79	16	)	)	PUNCT
cana-1681	79	17	−	−	ADP
cana-1681	79	18	2	2	NUM
cana-1681	79	19	𝑝	𝑝	NOUN
cana-1681	79	20	𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1	𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1	NOUN
cana-1681	79	21	)	)	PUNCT
cana-1681	79	22	)	)	PUNCT
cana-1681	79	23	2/𝑞	2/𝑞	NUM
cana-1681	79	24	communications	communication	NOUN
cana-1681	79	25	on	on	ADP
cana-1681	79	26	applied	apply	VERB
cana-1681	79	27	nonlinear	nonlinear	ADJ
cana-1681	79	28	analysis	analysis	NOUN
cana-1681	79	29	issn	issn	NOUN
cana-1681	79	30	:	:	PUNCT
cana-1681	79	31	1074	1074	NUM
cana-1681	79	32	-	-	PUNCT
cana-1681	79	33	133x	133x	NUM
cana-1681	79	34	vol	vol	NOUN
cana-1681	79	35	32	32	NUM
cana-1681	79	36	no	no	NOUN
cana-1681	79	37	.	.	NOUN
cana-1681	79	38	1	1	NUM
cana-1681	79	39	(	(	PUNCT
cana-1681	79	40	2025	2025	NUM
cana-1681	79	41	)	)	PUNCT
cana-1681	79	42	389	389	NUM
cana-1681	79	43	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	79	44	and	and	CCONJ
cana-1681	79	45	0	0	NUM
cana-1681	79	46	<	<	X
cana-1681	79	47	𝛿	𝛿	X
cana-1681	79	48	<	<	X
cana-1681	79	49	1/𝑞	1/𝑞	NUM
cana-1681	79	50	−	−	PROPN
cana-1681	80	1	1/𝑝	1/𝑝	NUM
cana-1681	81	1	+	+	SYM
cana-1681	82	1	1/𝑝𝑞.	1/𝑝𝑞.	VERB
cana-1681	82	2	so	so	ADV
cana-1681	82	3	,	,	PUNCT
cana-1681	82	4	𝑑𝑘	𝑑𝑘	ADV
cana-1681	82	5	𝑑𝑥𝑘	𝑑𝑥𝑘	VERB
cana-1681	82	6	𝑑𝑘	𝑑𝑘	ADV
cana-1681	82	7	𝑑𝑦𝑘	𝑑𝑦𝑘	NOUN
cana-1681	82	8	(	(	PUNCT
cana-1681	82	9	ℎ𝑘	ℎ𝑘	NOUN
cana-1681	82	10	⋆	⋆	VERB
cana-1681	82	11	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	82	12	,	,	PUNCT
cana-1681	82	13	𝑦	𝑦	NOUN
cana-1681	82	14	)	)	PUNCT
cana-1681	82	15	)	)	PUNCT
cana-1681	83	1	≥	≥	X
cana-1681	84	1	𝐶	𝐶	PROPN
cana-1681	84	2	𝑑𝑘	𝑑𝑘	ADJ
cana-1681	84	3	𝑑𝑥𝑘	𝑑𝑥𝑘	NOUN
cana-1681	84	4	𝑑𝑘	𝑑𝑘	ADV
cana-1681	84	5	𝑑𝑦𝑘	𝑑𝑦𝑘	ADJ
cana-1681	84	6	(	(	PUNCT
cana-1681	84	7	𝑘1/𝑞−1/𝑝+1/𝑝𝑞𝐻𝑞(𝑘	𝑘1/𝑞−1/𝑝+1/𝑝𝑞𝐻𝑞(𝑘	NOUN
cana-1681	84	8	)	)	PUNCT
cana-1681	84	9	)	)	PUNCT
cana-1681	84	10	or	or	CCONJ
cana-1681	84	11	,	,	PUNCT
cana-1681	84	12	|𝑥𝑦|𝑛	|𝑥𝑦|𝑛	PROPN
cana-1681	84	13	𝑑𝑛	𝑑𝑛	AUX
cana-1681	84	14	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
cana-1681	84	15	𝑑𝑛	𝑑𝑛	X
cana-1681	84	16	𝑑𝑦𝑛	𝑑𝑦𝑛	INTJ
cana-1681	84	17	(	(	PUNCT
cana-1681	84	18	ℎ𝑘	ℎ𝑘	NOUN
cana-1681	84	19	⋆	⋆	VERB
cana-1681	84	20	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	84	21	,	,	PUNCT
cana-1681	84	22	𝑦	𝑦	NOUN
cana-1681	84	23	)	)	PUNCT
cana-1681	84	24	)	)	PUNCT
cana-1681	84	25	≥	≥	PROPN
cana-1681	85	1	|𝑥𝑦|𝑛	|𝑥𝑦|𝑛	NOUN
cana-1681	85	2	∫	∫	PROPN
cana-1681	85	3	−𝐽𝑘	−𝐽𝑘	PROPN
cana-1681	85	4	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	85	5	−	−	PROPN
cana-1681	85	6	𝑦	𝑦	X
cana-1681	85	7	)	)	PUNCT
cana-1681	85	8	𝑑𝑛	𝑑𝑛	X
cana-1681	85	9	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
cana-1681	85	10	𝑑𝑛	𝑑𝑛	X
cana-1681	85	11	𝑑𝑦𝑛	𝑑𝑦𝑛	PROPN
cana-1681	85	12	𝜎𝑘(𝑥	𝜎𝑘(𝑥	NUM
cana-1681	85	13	,	,	PUNCT
cana-1681	85	14	𝑦)𝑑𝑥𝑑𝑦	𝑦)𝑑𝑥𝑑𝑦	PROPN
cana-1681	85	15	for	for	ADP
cana-1681	85	16	𝑘	𝑘	DET
cana-1681	85	17	≥	≥	NOUN
cana-1681	85	18	𝑛	𝑛	NOUN
cana-1681	85	19	,	,	PUNCT
cana-1681	86	1	|𝑥𝑦|𝑘	|𝑥𝑦|𝑘	PRON
cana-1681	86	2	𝑑𝑘	𝑑𝑘	ADV
cana-1681	86	3	𝑑𝑥𝑘	𝑑𝑥𝑘	VERB
cana-1681	86	4	𝑑𝑘	𝑑𝑘	ADV
cana-1681	86	5	𝑑𝑦𝑘	𝑑𝑦𝑘	NOUN
cana-1681	86	6	(	(	PUNCT
cana-1681	86	7	ℎ𝑘	ℎ𝑘	NOUN
cana-1681	86	8	⋆	⋆	VERB
cana-1681	86	9	𝑓𝑞(𝑥	𝑓𝑞(𝑥	NOUN
cana-1681	86	10	,	,	PUNCT
cana-1681	86	11	𝑦	𝑦	NOUN
cana-1681	86	12	)	)	PUNCT
cana-1681	86	13	)	)	PUNCT
cana-1681	86	14	≥	≥	PROPN
cana-1681	87	1	|𝑥𝑦|𝑛	|𝑥𝑦|𝑛	NOUN
cana-1681	87	2	𝑑𝑛	𝑑𝑛	X
cana-1681	87	3	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
cana-1681	87	4	𝑛𝛿	𝑛𝛿	ADP
cana-1681	87	5	≥	≥	PROPN
cana-1681	87	6	|𝑥𝑦|𝑛	|𝑥𝑦|𝑛	NOUN
cana-1681	87	7	𝑑𝑛	𝑑𝑛	AUX
cana-1681	87	8	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
cana-1681	87	9	𝑑𝑛	𝑑𝑛	X
cana-1681	87	10	𝑑𝑦𝑛	𝑑𝑦𝑛	PROPN
cana-1681	87	11	(	(	PUNCT
cana-1681	87	12	1	1	NUM
cana-1681	87	13	(	(	PUNCT
cana-1681	87	14	𝑥−𝑦)𝑝𝛿	𝑥−𝑦)𝑝𝛿	NOUN
cana-1681	87	15	)	)	PUNCT
cana-1681	87	16	=	=	SYM
cana-1681	87	17	|𝑥𝑦|𝑛(−1)𝑛(𝑝𝛿+𝑛−1	|𝑥𝑦|𝑛(−1)𝑛(𝑝𝛿+𝑛−1	NOUN
cana-1681	87	18	)	)	PUNCT
cana-1681	87	19	!	!	PUNCT
cana-1681	88	1	(	(	PUNCT
cana-1681	88	2	𝑝𝛿)!(𝑥−𝑦)𝑝𝛿+𝑛	𝑝𝛿)!(𝑥−𝑦)𝑝𝛿+𝑛	NOUN
cana-1681	88	3	≥	≥	PROPN
cana-1681	88	4	|𝑥𝑦|𝑛	|𝑥𝑦|𝑛	NOUN
cana-1681	88	5	(	(	PUNCT
cana-1681	88	6	−1)𝑛(𝑝𝛿+𝑛−1	−1)𝑛(𝑝𝛿+𝑛−1	NOUN
cana-1681	88	7	)	)	PUNCT
cana-1681	88	8	!	!	PUNCT
cana-1681	89	1	(	(	PUNCT
cana-1681	89	2	𝑝𝛿)!𝐶𝑟𝑛(𝑙𝑜𝑔𝑛)2/𝑝+1(𝑙𝑜𝑔𝑛)2𝛿/𝑝+1	𝑝𝛿)!𝐶𝑟𝑛(𝑙𝑜𝑔𝑛)2/𝑝+1(𝑙𝑜𝑔𝑛)2𝛿/𝑝+1	NOUN
cana-1681	89	3	→	→	SYM
cana-1681	89	4	∞	∞	PROPN
cana-1681	89	5	as	as	ADP
cana-1681	89	6	𝑛	𝑛	PROPN
cana-1681	89	7	→	→	SYM
cana-1681	89	8	∞.	∞.	PROPN
cana-1681	89	9	“	"	PUNCT
cana-1681	89	10	in	in	ADP
cana-1681	89	11	view	view	NOUN
cana-1681	89	12	of	of	ADP
cana-1681	89	13	sawyer	sawyer	PROPN
cana-1681	89	14	’s	’s	PART
cana-1681	89	15	principle	principle	NOUN
cana-1681	89	16	[	[	X
cana-1681	89	17	4	4	X
cana-1681	89	18	]	]	PUNCT
cana-1681	89	19	,	,	PUNCT
cana-1681	89	20	there	there	PRON
cana-1681	89	21	exists	exist	VERB
cana-1681	89	22	a	a	DET
cana-1681	89	23	functions	function	NOUN
cana-1681	89	24	𝑓휀𝐿𝑞([0,1	𝑓휀𝐿𝑞([0,1	ADV
cana-1681	89	25	)	)	PUNCT
cana-1681	89	26	)	)	PUNCT
cana-1681	90	1	⊆	⊆	NUM
cana-1681	90	2	𝐿𝑞(𝑅	𝐿𝑞(𝑅	NOUN
cana-1681	90	3	)	)	PUNCT
cana-1681	90	4	such	such	ADJ
cana-1681	90	5	that	that	SCONJ
cana-1681	90	6	𝑙𝑖𝑚𝑠𝑢𝑝𝑛|𝑥𝑦|𝑛	𝑙𝑖𝑚𝑠𝑢𝑝𝑛|𝑥𝑦|𝑛	PUNCT
cana-1681	90	7	𝑑𝑛	𝑑𝑛	X
cana-1681	90	8	𝑑𝑥𝑛	𝑑𝑥𝑛	VERB
cana-1681	90	9	𝑑𝑛	𝑑𝑛	X
cana-1681	90	10	𝑑𝑦𝑛	𝑑𝑦𝑛	PROPN
cana-1681	90	11	(	(	PUNCT
cana-1681	90	12	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	90	13	⋆	⋆	VERB
cana-1681	90	14	𝑓	𝑓	NOUN
cana-1681	90	15	)	)	PUNCT
cana-1681	90	16	→	→	SYM
cana-1681	90	17	∞	∞	NUM
cana-1681	90	18	a.e	a.e	PROPN
cana-1681	90	19	.	.	PROPN
cana-1681	90	20	on	on	ADP
cana-1681	90	21	a	a	DET
cana-1681	90	22	set	set	NOUN
cana-1681	90	23	of	of	ADP
cana-1681	90	24	positive	positive	ADJ
cana-1681	90	25	measure	measure	NOUN
cana-1681	90	26	in	in	ADP
cana-1681	90	27	r.	r.	PROPN
cana-1681	90	28	it	it	PRON
cana-1681	90	29	follows	follow	VERB
cana-1681	90	30	that	that	SCONJ
cana-1681	90	31	(	(	PUNCT
cana-1681	90	32	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	90	33	⋆	⋆	X
cana-1681	90	34	𝑓	𝑓	X
cana-1681	90	35	)	)	PUNCT
cana-1681	90	36	not	not	PART
cana-1681	90	37	belongs	belong	VERB
cana-1681	90	38	to	to	ADP
cana-1681	90	39	𝑆(𝑅	𝑆(𝑅	ADV
cana-1681	90	40	)	)	PUNCT
cana-1681	90	41	or	or	CCONJ
cana-1681	90	42	ℎ𝑛	ℎ𝑛	ADV
cana-1681	90	43	⋆	⋆	VERB
cana-1681	90	44	𝑓not	𝑓not	ADV
cana-1681	90	45	approachable	approachable	ADJ
cana-1681	90	46	to	to	ADP
cana-1681	90	47	𝑓	𝑓	DET
cana-1681	90	48	or	or	CCONJ
cana-1681	90	49	ℎ	ℎ	PRON
cana-1681	90	50	�	�	PROPN
cana-1681	90	51	̂	̂	VERB
cana-1681	90	52	�	�	NOUN
cana-1681	90	53	(휁	(휁	VERB
cana-1681	90	54	,	,	PUNCT
cana-1681	90	55	𝜙)𝑓(휁	𝜙)𝑓(휁	ADJ
cana-1681	90	56	,	,	PUNCT
cana-1681	90	57	𝜙	𝜙	NOUN
cana-1681	90	58	)	)	PUNCT
cana-1681	90	59	not	not	PART
cana-1681	90	60	approachable	approachable	ADJ
cana-1681	90	61	to	to	ADP
cana-1681	90	62	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	90	63	,	,	PUNCT
cana-1681	90	64	𝑦	𝑦	NOUN
cana-1681	90	65	)	)	PUNCT
cana-1681	90	66	for	for	ADP
cana-1681	90	67	1	1	NUM
cana-1681	90	68	≤	≤	NOUN
cana-1681	90	69	𝑞	𝑞	X
cana-1681	90	70	<	<	X
cana-1681	90	71	𝑝.	𝑝.	PROPN
cana-1681	90	72	let	let	VERB
cana-1681	90	73	𝑝𝑛	𝑝𝑛	PART
cana-1681	90	74	be	be	AUX
cana-1681	90	75	a	a	DET
cana-1681	90	76	decreasing	decrease	VERB
cana-1681	90	77	sequence	sequence	NOUN
cana-1681	90	78	of	of	ADP
cana-1681	90	79	real	real	ADJ
cana-1681	90	80	numbers	number	NOUN
cana-1681	90	81	such	such	ADJ
cana-1681	90	82	that	that	DET
cana-1681	90	83	𝑝1	𝑝1	NOUN
cana-1681	90	84	>	>	X
cana-1681	90	85	𝑝2	𝑝2	PROPN
cana-1681	90	86	>	>	PUNCT
cana-1681	90	87	.	.	PUNCT
cana-1681	90	88	.	.	PUNCT
cana-1681	90	89	.	.	PUNCT
cana-1681	90	90	.	.	PUNCT
cana-1681	90	91	.	.	PUNCT
cana-1681	91	1	𝑝𝑛	𝑝𝑛	INTJ
cana-1681	91	2	>	>	PUNCT
cana-1681	91	3	.	.	PUNCT
cana-1681	91	4	.	.	PUNCT
cana-1681	91	5	.	.	PUNCT
cana-1681	91	6	.	.	PUNCT
cana-1681	91	7	.	.	PUNCT
cana-1681	91	8	.	.	PUNCT
cana-1681	92	1	𝑝.	𝑝.	VERB
cana-1681	92	2	for	for	ADP
cana-1681	92	3	each	each	DET
cana-1681	92	4	𝑝𝑖	𝑝𝑖	NOUN
cana-1681	92	5	we	we	PRON
cana-1681	92	6	can	can	AUX
cana-1681	92	7	construct	construct	VERB
cana-1681	92	8	a	a	DET
cana-1681	92	9	perturbation	perturbation	NOUN
cana-1681	92	10	{	{	PUNCT
cana-1681	92	11	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	92	12	𝑖	𝑖	PROPN
cana-1681	92	13	}	}	PUNCT
cana-1681	92	14	𝑛	𝑛	PRON
cana-1681	92	15	of	of	ADP
cana-1681	92	16	{	{	PUNCT
cana-1681	92	17	𝜓𝑛	𝜓𝑛	NOUN
cana-1681	92	18	}	}	PUNCT
cana-1681	92	19	that	that	PRON
cana-1681	92	20	is	be	AUX
cana-1681	92	21	𝐿𝑞-good	𝐿𝑞-good	PUNCT
cana-1681	92	22	for	for	ADP
cana-1681	92	23	𝑞	𝑞	PROPN
cana-1681	92	24	≥	≥	PROPN
cana-1681	92	25	𝑝𝑖	𝑝𝑖	PROPN
cana-1681	92	26	,	,	PUNCT
cana-1681	92	27	and	and	CCONJ
cana-1681	92	28	𝐿𝑞-bad	𝐿𝑞-bad	NOUN
cana-1681	92	29	for	for	ADP
cana-1681	92	30	1	1	NUM
cana-1681	92	31	≥	≥	NOUN
cana-1681	92	32	𝑞	𝑞	X
cana-1681	92	33	<	<	X
cana-1681	92	34	𝑝𝑖.	𝑝𝑖.	X
cana-1681	92	35	consider	consider	VERB
cana-1681	92	36	a	a	DET
cana-1681	92	37	sequence	sequence	NOUN
cana-1681	92	38	of	of	ADP
cana-1681	92	39	blocks	block	NOUN
cana-1681	92	40	{	{	PUNCT
cana-1681	92	41	𝑇𝑘}𝑘	𝑇𝑘}𝑘	NOUN
cana-1681	92	42	𝑁	𝑁	PROPN
cana-1681	92	43	,	,	PUNCT
cana-1681	92	44	where	where	SCONJ
cana-1681	92	45	𝑇𝑘	𝑇𝑘	PROPN
cana-1681	92	46	=	=	PRON
cana-1681	92	47	{	{	PUNCT
cana-1681	92	48	ℎ𝑛𝑘−1	ℎ𝑛𝑘−1	PROPN
cana-1681	92	49	+	+	NOUN
cana-1681	92	50	1	1	NUM
cana-1681	92	51	,	,	PUNCT
cana-1681	92	52	.	.	PUNCT
cana-1681	92	53	.	.	PUNCT
cana-1681	92	54	.	.	PUNCT
cana-1681	92	55	.	.	PUNCT
cana-1681	93	1	.	.	PUNCT
cana-1681	94	1	,	,	PUNCT
cana-1681	94	2	ℎ𝑛𝑘	ℎ𝑛𝑘	PROPN
cana-1681	94	3	𝑘	𝑘	X
cana-1681	94	4	}	}	PUNCT
cana-1681	94	5	and	and	CCONJ
cana-1681	94	6	{	{	PUNCT
cana-1681	94	7	𝑛𝑘	𝑛𝑘	NOUN
cana-1681	94	8	}	}	PUNCT
cana-1681	94	9	is	be	AUX
cana-1681	94	10	a	a	DET
cana-1681	94	11	sequence	sequence	NOUN
cana-1681	94	12	of	of	ADP
cana-1681	94	13	positive	positive	ADJ
cana-1681	94	14	integers	integer	NOUN
cana-1681	94	15	increasing	increase	VERB
cana-1681	94	16	to	to	ADP
cana-1681	94	17	infinity	infinity	NOUN
cana-1681	94	18	.	.	PUNCT
cana-1681	95	1	let	let	VERB
cana-1681	95	2	𝑆𝑘	𝑆𝑘	PROPN
cana-1681	95	3	=	=	PUNCT
cana-1681	95	4	{	{	PUNCT
cana-1681	95	5	𝑛𝑘−1	𝑛𝑘−1	PROPN
cana-1681	95	6	+	+	PROPN
cana-1681	95	7	1	1	NUM
cana-1681	95	8	,	,	PUNCT
cana-1681	95	9	.	.	PUNCT
cana-1681	95	10	.	.	PUNCT
cana-1681	95	11	.	.	PUNCT
cana-1681	96	1	.	.	PUNCT
cana-1681	97	1	,	,	PUNCT
cana-1681	97	2	𝑛𝑘	𝑛𝑘	NOUN
cana-1681	97	3	}	}	PUNCT
cana-1681	97	4	.	.	PUNCT
cana-1681	98	1	and	and	CCONJ
cana-1681	98	2	let	let	VERB
cana-1681	98	3	{	{	PUNCT
cana-1681	98	4	ℎ𝑛}𝑛	ℎ𝑛}𝑛	PROPN
cana-1681	98	5	=	=	SYM
cana-1681	98	6	𝑈𝑘𝑇𝑘.	𝑈𝑘𝑇𝑘.	PROPN
cana-1681	98	7	now	now	ADV
cana-1681	98	8	,	,	PUNCT
cana-1681	98	9	fix	fix	VERB
cana-1681	98	10	𝑞	𝑞	X
cana-1681	98	11	>	>	X
cana-1681	98	12	𝑝.	𝑝.	PROPN
cana-1681	98	13	there	there	PRON
cana-1681	98	14	exists	exist	VERB
cana-1681	98	15	𝑛0휀𝑁	𝑛0휀𝑁	ADJ
cana-1681	98	16	so	so	SCONJ
cana-1681	98	17	that	that	SCONJ
cana-1681	98	18	for	for	ADP
cana-1681	98	19	all	all	DET
cana-1681	98	20	𝑛	𝑛	PRON
cana-1681	98	21	>	>	X
cana-1681	98	22	𝑛0	𝑛0	NOUN
cana-1681	98	23	we	we	PRON
cana-1681	98	24	have	have	AUX
cana-1681	98	25	𝑝𝑛	𝑝𝑛	ADP
cana-1681	98	26	<	<	X
cana-1681	98	27	𝑞	𝑞	X
cana-1681	98	28	"	"	PUNCT
cana-1681	98	29	.	.	PUNCT
cana-1681	99	1	∑	∑	PUNCT
cana-1681	100	1	∞	∞	NUM
cana-1681	100	2	𝑘=𝑛0	𝑘=𝑛0	PROPN
cana-1681	100	3	∑	∑	PUNCT
cana-1681	100	4	𝑛	𝑛	DET
cana-1681	100	5	𝑆𝑘	𝑆𝑘	PROPN
cana-1681	100	6	(	(	PUNCT
cana-1681	100	7	1	1	NUM
cana-1681	100	8	−	−	PROPN
cana-1681	100	9	𝛼𝑛	𝛼𝑛	NOUN
cana-1681	100	10	𝑘)𝑞	𝑘)𝑞	NOUN
cana-1681	100	11	≤	≤	NUM
cana-1681	100	12	∑	∑	PUNCT
cana-1681	100	13	∞	∞	NUM
cana-1681	100	14	𝑘=𝑛0	𝑘=𝑛0	NOUN
cana-1681	100	15	∑	∑	PUNCT
cana-1681	100	16	𝑛	𝑛	DET
cana-1681	100	17	𝑆𝑘	𝑆𝑘	PROPN
cana-1681	100	18	1	1	NUM
cana-1681	100	19	(	(	PUNCT
cana-1681	100	20	𝑛𝑙𝑜𝑔2𝑛	𝑛𝑙𝑜𝑔2𝑛	PROPN
cana-1681	100	21	)	)	PUNCT
cana-1681	100	22	𝑞/𝑝𝑛0	𝑞/𝑝𝑛0	ADV
cana-1681	100	23	≤	≤	X
cana-1681	100	24	∑	∑	PUNCT
cana-1681	100	25	𝑛	𝑛	PROPN
cana-1681	100	26	1	1	NUM
cana-1681	100	27	(	(	PUNCT
cana-1681	100	28	𝑛𝑙𝑜𝑔2𝑛)𝑞/𝑝𝑛0	𝑛𝑙𝑜𝑔2𝑛)𝑞/𝑝𝑛0	X
cana-1681	100	29	<	<	X
cana-1681	100	30	∞.	∞.	PROPN
cana-1681	100	31	using	use	VERB
cana-1681	100	32	proposition	proposition	NOUN
cana-1681	100	33	1.2(c	1.2(c	NUM
cana-1681	100	34	)	)	PUNCT
cana-1681	100	35	,	,	PUNCT
cana-1681	100	36	we	we	PRON
cana-1681	100	37	get	get	VERB
cana-1681	100	38	ℎ𝑛	ℎ𝑛	INTJ
cana-1681	100	39	⋆	⋆	VERB
cana-1681	100	40	𝑓	𝑓	PROPN
cana-1681	100	41	→	→	PUNCT
cana-1681	100	42	𝑓	𝑓	PROPN
cana-1681	100	43	for	for	ADP
cana-1681	100	44	𝑓휀𝐿𝑞(𝑅	𝑓휀𝐿𝑞(𝑅	NOUN
cana-1681	100	45	)	)	PUNCT
cana-1681	100	46	,	,	PUNCT
cana-1681	100	47	𝑞	𝑞	X
cana-1681	100	48	>	>	X
cana-1681	100	49	𝑝	𝑝	PROPN
cana-1681	100	50	,	,	PUNCT
cana-1681	100	51	or	or	CCONJ
cana-1681	100	52	ℎ	ℎ	PART
cana-1681	100	53	�	�	NOUN
cana-1681	100	54	̂	̂	VERB
cana-1681	100	55	�	�	NOUN
cana-1681	100	56	(휁	(휁	VERB
cana-1681	100	57	,	,	PUNCT
cana-1681	100	58	𝜓)𝑓(휁	𝜓)𝑓(휁	NOUN
cana-1681	100	59	,	,	PUNCT
cana-1681	100	60	𝜓	𝜓	NOUN
cana-1681	100	61	)	)	PUNCT
cana-1681	100	62	→	→	SYM
cana-1681	100	63	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	100	64	,	,	PUNCT
cana-1681	100	65	𝑦	𝑦	NOUN
cana-1681	100	66	)	)	PUNCT
cana-1681	100	67	for	for	ADP
cana-1681	100	68	𝑞	𝑞	PROPN
cana-1681	100	69	>	>	X
cana-1681	100	70	𝑝.	𝑝.	PROPN
cana-1681	100	71	now	now	ADV
cana-1681	100	72	consider	consider	VERB
cana-1681	100	73	a	a	DET
cana-1681	100	74	sequence	sequence	NOUN
cana-1681	100	75	“	"	PUNCT
cana-1681	100	76	𝐶𝑖	𝐶𝑖	VERB
cana-1681	100	77	𝑁	𝑁	PROPN
cana-1681	100	78	→	→	SYM
cana-1681	100	79	∞	∞	PROPN
cana-1681	100	80	as	as	ADP
cana-1681	100	81	𝑖	𝑖	NOUN
cana-1681	100	82	→	→	SYM
cana-1681	100	83	∞.	∞.	PROPN
cana-1681	100	84	since	since	SCONJ
cana-1681	100	85	{	{	PUNCT
cana-1681	100	86	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	100	87	𝑖	𝑖	PROPN
cana-1681	100	88	}	}	PUNCT
cana-1681	100	89	𝑛	𝑛	PRON
cana-1681	100	90	is	be	AUX
cana-1681	100	91	𝐿𝑞-bad	𝐿𝑞-bad	NOUN
cana-1681	100	92	for	for	ADP
cana-1681	100	93	all	all	PRON
cana-1681	100	94	𝑞	𝑞	X
cana-1681	100	95	<	<	X
cana-1681	100	96	𝑝𝑖	𝑝𝑖	PROPN
cana-1681	100	97	,	,	PUNCT
cana-1681	100	98	it	it	PRON
cana-1681	100	99	is	be	AUX
cana-1681	100	100	also	also	ADV
cana-1681	100	101	𝐿𝑝-bad	𝐿𝑝-bad	PROPN
cana-1681	100	102	.	.	PUNCT
cana-1681	101	1	these	these	PRON
cana-1681	101	2	exists	exist	VERB
cana-1681	101	3	𝑓𝑖휀𝐿𝑝([0,1	𝑓𝑖휀𝐿𝑝([0,1	ADV
cana-1681	101	4	)	)	PUNCT
cana-1681	101	5	)	)	PUNCT
cana-1681	101	6	and	and	CCONJ
cana-1681	101	7	𝜆𝑖	𝜆𝑖	VERB
cana-1681	101	8	𝑁	𝑁	NOUN
cana-1681	101	9	>	>	X
cana-1681	101	10	0	0	NUM
cana-1681	102	1	such	such	ADJ
cana-1681	102	2	that	that	SCONJ
cana-1681	102	3	||𝑠𝑢𝑝𝑛>𝑛𝑖−1	||𝑠𝑢𝑝𝑛>𝑛𝑖−1	NOUN
cana-1681	102	4	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	102	5	𝑖	𝑖	PROPN
cana-1681	102	6	⋆	⋆	VERB
cana-1681	102	7	𝑓𝑖(𝑥	𝑓𝑖(𝑥	NUM
cana-1681	102	8	,	,	PUNCT
cana-1681	102	9	𝑦	𝑦	NOUN
cana-1681	102	10	)	)	PUNCT
cana-1681	102	11	∥	∥	NUM
cana-1681	102	12	>	>	X
cana-1681	102	13	∫	∫	PROPN
cana-1681	102	14	||ℎ𝑛	||ℎ𝑛	NOUN
cana-1681	102	15	𝑖	𝑖	SYM
cana-1681	102	16	⋆	⋆	VERB
cana-1681	102	17	𝑓𝑖(𝑥	𝑓𝑖(𝑥	NUM
cana-1681	102	18	,	,	PUNCT
cana-1681	102	19	𝑦	𝑦	X
cana-1681	102	20	)	)	PUNCT
cana-1681	103	1	∥𝑝	∥𝑝	PROPN
cana-1681	103	2	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	X
cana-1681	103	3	>	>	X
cana-1681	103	4	𝐶𝑁||𝑓𝑖(𝑥𝑦	𝐶𝑁||𝑓𝑖(𝑥𝑦	NOUN
cana-1681	103	5	−	−	PROPN
cana-1681	103	6	𝜆𝑖	𝜆𝑖	NOUN
cana-1681	103	7	𝑁)||𝑝	𝑁)||𝑝	PROPN
cana-1681	103	8	𝑝	𝑝	NOUN
cana-1681	103	9	=	=	SYM
cana-1681	103	10	2𝐶𝑖	2𝐶𝑖	NUM
cana-1681	103	11	𝑁	𝑁	NOUN
cana-1681	103	12	,	,	PUNCT
cana-1681	103	13	[	[	X
cana-1681	103	14	||𝑓𝑖(𝑥𝑦	||𝑓𝑖(𝑥𝑦	NOUN
cana-1681	103	15	−	−	NOUN
cana-1681	103	16	𝜆𝑖	𝜆𝑖	NOUN
cana-1681	103	17	𝑁)||𝑝	𝑁)||𝑝	X
cana-1681	103	18	=	=	SYM
cana-1681	103	19	21−𝑖	21−𝑖	NUM
cana-1681	103	20	,	,	PUNCT
cana-1681	103	21	𝐶𝑁	𝐶𝑁	PROPN
cana-1681	103	22	=	=	SYM
cana-1681	103	23	2(𝑖−1)𝑝+1𝐶𝑖	2(𝑖−1)𝑝+1𝐶𝑖	NUM
cana-1681	103	24	𝑁	𝑁	PROPN
cana-1681	103	25	]	]	PUNCT
cana-1681	103	26	.	.	PUNCT
cana-1681	103	27	"	"	PUNCT
cana-1681	104	1	it	it	PRON
cana-1681	104	2	follows	follow	VERB
cana-1681	104	3	that	that	SCONJ
cana-1681	104	4	there	there	PRON
cana-1681	104	5	exists	exist	VERB
cana-1681	104	6	𝑛𝑖	𝑛𝑖	PRON
cana-1681	104	7	>	>	X
cana-1681	104	8	𝑛𝑖−1	𝑛𝑖−1	NOUN
cana-1681	104	9	,	,	PUNCT
cana-1681	104	10	so	so	SCONJ
cana-1681	104	11	that	that	SCONJ
cana-1681	104	12	||𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖	||𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖	PROPN
cana-1681	104	13	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	104	14	𝑖	𝑖	PRON
cana-1681	104	15	⋆	⋆	VERB
cana-1681	104	16	𝑓𝑖	𝑓𝑖	PROPN
cana-1681	104	17	∥	∥	PRON
cana-1681	104	18	>	>	X
cana-1681	104	19	𝐶𝑖	𝐶𝑖	PROPN
cana-1681	104	20	𝑁.	𝑁.	PROPN
cana-1681	104	21	set	set	VERB
cana-1681	104	22	𝑓	𝑓	PROPN
cana-1681	104	23	=	=	SYM
cana-1681	104	24	∑	∑	PROPN
cana-1681	104	25	𝑖	𝑖	SYM
cana-1681	104	26	𝑓𝑖	𝑓𝑖	PROPN
cana-1681	104	27	,	,	PUNCT
cana-1681	104	28	then	then	ADV
cana-1681	104	29	||𝑓||𝑝	||𝑓||𝑝	PROPN
cana-1681	104	30	≤	≤	PROPN
cana-1681	104	31	∑	∑	PUNCT
cana-1681	105	1	𝑖	𝑖	SYM
cana-1681	105	2	||𝑓𝑖||𝑝	||𝑓𝑖||𝑝	NOUN
cana-1681	105	3	≤	≤	ADJ
cana-1681	105	4	2	2	NUM
cana-1681	105	5	.	.	PUNCT
cana-1681	105	6	“	"	PUNCT
cana-1681	105	7	suppose	suppose	VERB
cana-1681	105	8	that	that	SCONJ
cana-1681	105	9	{	{	PUNCT
cana-1681	105	10	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	105	11	}	}	PUNCT
cana-1681	105	12	satisfies	satisfy	VERB
cana-1681	105	13	a	a	DET
cana-1681	105	14	weak	weak	ADJ
cana-1681	105	15	(	(	PUNCT
cana-1681	105	16	𝑝	𝑝	NOUN
cana-1681	105	17	,	,	PUNCT
cana-1681	105	18	𝑝	𝑝	NOUN
cana-1681	105	19	)	)	PUNCT
cana-1681	105	20	inequality	inequality	NOUN
cana-1681	105	21	in	in	ADP
cana-1681	105	22	𝐿𝑝([0,1	𝐿𝑝([0,1	PROPN
cana-1681	105	23	)	)	PUNCT
cana-1681	105	24	)	)	PUNCT
cana-1681	105	25	,	,	PUNCT
cana-1681	105	26	we	we	PRON
cana-1681	105	27	know	know	VERB
cana-1681	105	28	that	that	SCONJ
cana-1681	105	29	if	if	SCONJ
cana-1681	105	30	𝜇	𝜇	ADP
cana-1681	105	31	be	be	AUX
cana-1681	105	32	a	a	DET
cana-1681	105	33	finite	finite	ADJ
cana-1681	105	34	positive	positive	ADJ
cana-1681	105	35	borel	borel	NOUN
cana-1681	105	36	measure	measure	NOUN
cana-1681	105	37	,	,	PUNCT
cana-1681	105	38	then	then	ADV
cana-1681	105	39	there	there	PRON
cana-1681	105	40	exists	exist	VERB
cana-1681	105	41	a	a	DET
cana-1681	105	42	sequence	sequence	NOUN
cana-1681	105	43	𝜇𝑛	𝜇𝑛	NOUN
cana-1681	105	44	of	of	ADP
cana-1681	105	45	atomic	atomic	ADJ
cana-1681	105	46	measure	measure	NOUN
cana-1681	105	47	that	that	PRON
cana-1681	105	48	converges	converge	VERB
cana-1681	105	49	to	to	ADP
cana-1681	105	50	𝜇	𝜇	ADV
cana-1681	105	51	weakly	weakly	ADJ
cana-1681	106	1	or	or	CCONJ
cana-1681	106	2	if	if	SCONJ
cana-1681	106	3	f	f	PROPN
cana-1681	106	4	has	have	VERB
cana-1681	106	5	compact	compact	ADJ
cana-1681	106	6	support	support	NOUN
cana-1681	106	7	then	then	ADV
cana-1681	106	8	∫	∫	PROPN
cana-1681	106	9	𝑅	𝑅	PROPN
cana-1681	106	10	𝑑𝜇𝑛𝑓(𝑥	𝑑𝜇𝑛𝑓(𝑥	PROPN
cana-1681	106	11	,	,	PUNCT
cana-1681	106	12	𝑦	𝑦	NOUN
cana-1681	106	13	)	)	PUNCT
cana-1681	106	14	→	→	SYM
cana-1681	106	15	∫	∫	PROPN
cana-1681	106	16	𝑅	𝑅	PROPN
cana-1681	106	17	𝑓(𝑥	𝑓(𝑥	PROPN
cana-1681	106	18	,	,	PUNCT
cana-1681	106	19	𝑦)𝑑𝜇𝑑𝛾	𝑦)𝑑𝜇𝑑𝛾	VERB
cana-1681	106	20	where	where	SCONJ
cana-1681	106	21	,	,	PUNCT
cana-1681	106	22	𝜇𝑛	𝜇𝑛	INTJ
cana-1681	106	23	→	→	SYM
cana-1681	106	24	𝜇	𝜇	X
cana-1681	106	25	and	and	CCONJ
cana-1681	106	26	𝛾𝑛	𝛾𝑛	INTJ
cana-1681	106	27	→	→	SYM
cana-1681	106	28	𝛾	𝛾	NOUN
cana-1681	106	29	,	,	PUNCT
cana-1681	106	30	weakly	weakly	ADJ
cana-1681	106	31	.	.	PUNCT
cana-1681	107	1	if	if	SCONJ
cana-1681	107	2	𝑓휀𝐿1(𝑅	𝑓휀𝐿1(𝑅	NOUN
cana-1681	107	3	)	)	PUNCT
cana-1681	107	4	,	,	PUNCT
cana-1681	107	5	𝑑𝜇𝑑𝛾	𝑑𝜇𝑑𝛾	NOUN
cana-1681	107	6	=	=	SYM
cana-1681	107	7	|𝑓(𝑥	|𝑓(𝑥	PROPN
cana-1681	107	8	,	,	PUNCT
cana-1681	107	9	𝑦)|𝑑𝑥𝑑𝑦	𝑦)|𝑑𝑥𝑑𝑦	NOUN
cana-1681	107	10	is	be	AUX
cana-1681	107	11	a	a	DET
cana-1681	107	12	finite	finite	PROPN
cana-1681	107	13	borel	borel	NOUN
cana-1681	107	14	measure	measure	NOUN
cana-1681	107	15	,	,	PUNCT
cana-1681	107	16	so	so	ADV
cana-1681	107	17	we	we	PRON
cana-1681	107	18	can	can	AUX
cana-1681	107	19	find	find	VERB
cana-1681	107	20	𝛾𝑛𝜇𝑛	𝛾𝑛𝜇𝑛	NOUN
cana-1681	107	21	=	=	PUNCT
cana-1681	107	22	∑	∑	PUNCT
cana-1681	107	23	𝑁	𝑁	PROPN
cana-1681	107	24	𝑖=1	𝑖=1	PROPN
cana-1681	107	25	𝐶𝑖	𝐶𝑖	PROPN
cana-1681	107	26	𝑁𝛿𝜆𝑖	𝑁𝛿𝜆𝑖	PROPN
cana-1681	107	27	𝑁	𝑁	PROPN
cana-1681	107	28	→	→	SYM
cana-1681	107	29	𝜇𝛾	𝜇𝛾	X
cana-1681	107	30	weakly	weakly	ADJ
cana-1681	107	31	.	.	PUNCT
cana-1681	108	1	communications	communication	NOUN
cana-1681	108	2	on	on	ADP
cana-1681	108	3	applied	apply	VERB
cana-1681	108	4	nonlinear	nonlinear	ADJ
cana-1681	108	5	analysis	analysis	NOUN
cana-1681	108	6	issn	issn	NOUN
cana-1681	108	7	:	:	PUNCT
cana-1681	108	8	1074	1074	NUM
cana-1681	108	9	-	-	PUNCT
cana-1681	108	10	133x	133x	NUM
cana-1681	108	11	vol	vol	NOUN
cana-1681	108	12	32	32	NUM
cana-1681	108	13	no	no	NOUN
cana-1681	108	14	.	.	NOUN
cana-1681	108	15	1	1	NUM
cana-1681	108	16	(	(	PUNCT
cana-1681	108	17	2025	2025	NUM
cana-1681	108	18	)	)	PUNCT
cana-1681	108	19	390	390	NUM
cana-1681	108	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	108	21	consider	consider	VERB
cana-1681	108	22	|{𝑠𝑢𝑝𝑛(ℎ𝑛	|{𝑠𝑢𝑝𝑛(ℎ𝑛	NOUN
cana-1681	108	23	𝑖	𝑖	AUX
cana-1681	108	24	⋆	⋆	VERB
cana-1681	108	25	𝑓)}|	𝑓)}|	PUNCT
cana-1681	109	1	=	=	SYM
cana-1681	109	2	∫	∫	PROPN
cana-1681	110	1	−𝐽𝑘	−𝐽𝑘	INTJ
cana-1681	110	2	|ℎ𝑛	|ℎ𝑛	NOUN
cana-1681	110	3	𝑖	𝑖	SYM
cana-1681	110	4	(	(	PUNCT
cana-1681	110	5	𝑥	𝑥	NOUN
cana-1681	110	6	,	,	PUNCT
cana-1681	110	7	𝑦)𝑓(𝑥	𝑦)𝑓(𝑥	VERB
cana-1681	110	8	−	−	PROPN
cana-1681	110	9	𝑦)|	𝑦)|	PROPN
cana-1681	110	10	𝑝	𝑝	PROPN
cana-1681	110	11	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1681	110	12	≤	≤	NUM
cana-1681	110	13	∫	∫	PROPN
cana-1681	111	1	−𝐽𝑘	−𝐽𝑘	INTJ
cana-1681	112	1	|ℎ𝑛	|ℎ𝑛	NOUN
cana-1681	112	2	𝑖	𝑖	SYM
cana-1681	112	3	(	(	PUNCT
cana-1681	112	4	𝑥	𝑥	NOUN
cana-1681	112	5	,	,	PUNCT
cana-1681	112	6	𝑦)𝑑𝜇𝑛(𝑥	𝑦)𝑑𝜇𝑛(𝑥	VERB
cana-1681	112	7	−	−	PROPN
cana-1681	112	8	𝑦)|	𝑦)|	PROPN
cana-1681	112	9	𝑝	𝑝	PROPN
cana-1681	112	10	𝑑𝑥𝑑𝑦	𝑑𝑥𝑑𝑦	PROPN
cana-1681	112	11	≤	≤	NUM
cana-1681	112	12	||	||	NOUN
cana-1681	113	1	∑	∑	PUNCT
cana-1681	113	2	𝑁	𝑁	PROPN
cana-1681	113	3	𝑖=1	𝑖=1	PROPN
cana-1681	113	4	𝑓(𝑥𝑦	𝑓(𝑥𝑦	NOUN
cana-1681	113	5	−	−	NUM
cana-1681	113	6	𝜆𝑖	𝜆𝑖	NOUN
cana-1681	114	1	𝑁)𝐶𝑖	𝑁)𝐶𝑖	PROPN
cana-1681	114	2	𝑁||𝑝	𝑁||𝑝	PROPN
cana-1681	114	3	𝑝	𝑝	ADP
cana-1681	114	4	∑	∑	ADV
cana-1681	114	5	𝑁	𝑁	PROPN
cana-1681	114	6	𝑖=1	𝑖=1	PROPN
cana-1681	114	7	𝐶𝑖	𝐶𝑖	PROPN
cana-1681	114	8	𝑁||𝑓(𝑥𝑦	𝑁||𝑓(𝑥𝑦	NOUN
cana-1681	114	9	−	−	PROPN
cana-1681	114	10	𝜆𝑖	𝜆𝑖	PROPN
cana-1681	114	11	𝑁)||𝑝	𝑁)||𝑝	PROPN
cana-1681	114	12	𝑝	𝑝	NOUN
cana-1681	114	13	≤	≤	NUM
cana-1681	114	14	𝐶0	𝐶0	NOUN
cana-1681	114	15	𝑁||𝑓||𝑝	𝑁||𝑓||𝑝	NOUN
cana-1681	114	16	𝑝	𝑝	PROPN
cana-1681	114	17	=	=	SYM
cana-1681	114	18	2𝑝𝐶0	2𝑝𝐶0	NUM
cana-1681	114	19	𝑁.	𝑁.	PROPN
cana-1681	114	20	"	"	PUNCT
cana-1681	114	21	(	(	PUNCT
cana-1681	114	22	1	1	NUM
cana-1681	114	23	)	)	PUNCT
cana-1681	114	24	on	on	ADP
cana-1681	114	25	the	the	DET
cana-1681	114	26	other	other	ADJ
cana-1681	114	27	hand	hand	NOUN
cana-1681	114	28	|{𝑠𝑢𝑝𝑛(ℎ𝑛	|{𝑠𝑢𝑝𝑛(ℎ𝑛	NOUN
cana-1681	114	29	⋆	⋆	VERB
cana-1681	114	30	𝑓)}|	𝑓)}|	ADP
cana-1681	114	31	≤	≤	NOUN
cana-1681	114	32	|{𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖	|{𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖	NOUN
cana-1681	114	33	(	(	PUNCT
cana-1681	114	34	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	114	35	𝑖	𝑖	PRON
cana-1681	114	36	⋆	⋆	VERB
cana-1681	114	37	𝑓(𝑖))}|	𝑓(𝑖))}|	NOUN
cana-1681	114	38	>	>	X
cana-1681	114	39	𝐶𝑖	𝐶𝑖	PROPN
cana-1681	114	40	𝑁	𝑁	PROPN
cana-1681	114	41	(	(	PUNCT
cana-1681	114	42	2	2	NUM
cana-1681	114	43	)	)	PUNCT
cana-1681	114	44	combining	combine	VERB
cana-1681	114	45	equations	equation	NOUN
cana-1681	114	46	,	,	PUNCT
cana-1681	114	47	we	we	PRON
cana-1681	114	48	get	get	VERB
cana-1681	114	49	𝐶0	𝐶0	NOUN
cana-1681	114	50	𝑁	𝑁	PROPN
cana-1681	114	51	>	>	X
cana-1681	114	52	𝐶𝑖	𝐶𝑖	PROPN
cana-1681	114	53	𝑁𝐵𝑢𝑡	𝑁𝐵𝑢𝑡	PUNCT
cana-1681	115	1	𝐶𝑖	𝐶𝑖	VERB
cana-1681	115	2	𝑁	𝑁	PROPN
cana-1681	115	3	→	→	SYM
cana-1681	115	4	∞	∞	PROPN
cana-1681	115	5	as	as	ADP
cana-1681	115	6	𝑖	𝑖	NOUN
cana-1681	115	7	→	→	PUNCT
cana-1681	115	8	+	+	NOUN
cana-1681	115	9	∞.	∞.	PROPN
cana-1681	115	10	hence	hence	ADV
cana-1681	116	1	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	116	2	⋆	⋆	VERB
cana-1681	116	3	𝑓	𝑓	PRON
cana-1681	116	4	not	not	PART
cana-1681	116	5	approachable	approachable	ADJ
cana-1681	116	6	to	to	ADP
cana-1681	116	7	f	f	PROPN
cana-1681	116	8	in	in	ADP
cana-1681	116	9	𝐿𝑝([0,1	𝐿𝑝([0,1	PROPN
cana-1681	116	10	)	)	PUNCT
cana-1681	116	11	)	)	PUNCT
cana-1681	116	12	.	.	PUNCT
cana-1681	117	1	since	since	SCONJ
cana-1681	117	2	the	the	DET
cana-1681	117	3	spaces	space	NOUN
cana-1681	117	4	𝐿𝑞([0,1	𝐿𝑞([0,1	NOUN
cana-1681	117	5	)	)	PUNCT
cana-1681	117	6	)	)	PUNCT
cana-1681	117	7	are	be	AUX
cana-1681	117	8	nested	nest	VERB
cana-1681	117	9	,	,	PUNCT
cana-1681	117	10	{	{	PUNCT
cana-1681	117	11	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	117	12	}	}	PUNCT
cana-1681	117	13	is	be	AUX
cana-1681	117	14	𝐿𝑞([0,1))-bad	𝐿𝑞([0,1))-bad	VERB
cana-1681	117	15	for	for	ADP
cana-1681	117	16	all	all	DET
cana-1681	117	17	1	1	NUM
cana-1681	117	18	≤	≤	NUM
cana-1681	117	19	𝑞	𝑞	X
cana-1681	117	20	≤	≤	PROPN
cana-1681	117	21	𝑝.	𝑝.	NOUN
cana-1681	117	22	therefore	therefore	ADV
cana-1681	117	23	,	,	PUNCT
cana-1681	117	24	such	such	DET
cana-1681	117	25	a	a	DET
cana-1681	117	26	choice	choice	NOUN
cana-1681	117	27	of	of	ADP
cana-1681	117	28	{	{	PUNCT
cana-1681	117	29	𝑛𝑘	𝑛𝑘	NOUN
cana-1681	117	30	}	}	PUNCT
cana-1681	117	31	makes	make	VERB
cana-1681	117	32	{	{	PUNCT
cana-1681	117	33	ℎ𝑛	ℎ𝑛	NOUN
cana-1681	117	34	}	}	PUNCT
cana-1681	117	35	𝐿𝑞(𝑅)-bad	𝐿𝑞(𝑅)-bad	VERB
cana-1681	117	36	for	for	ADP
cana-1681	117	37	all	all	DET
cana-1681	117	38	1	1	NUM
cana-1681	117	39	≤	≤	NUM
cana-1681	117	40	𝑞	𝑞	X
cana-1681	117	41	≤	≤	PROPN
cana-1681	117	42	𝑝.	𝑝.	NOUN
cana-1681	117	43	this	this	PRON
cana-1681	117	44	implies	imply	VERB
cana-1681	117	45	that	that	SCONJ
cana-1681	117	46	ℎ	ℎ	PROPN
cana-1681	117	47	�	�	PROPN
cana-1681	117	48	̂	̂	SYM
cana-1681	117	49	�	�	NOUN
cana-1681	117	50	(휁	(휁	VERB
cana-1681	117	51	,	,	PUNCT
cana-1681	117	52	𝜓)𝑓(휁	𝜓)𝑓(휁	NOUN
cana-1681	117	53	,	,	PUNCT
cana-1681	117	54	𝜓	𝜓	NOUN
cana-1681	117	55	)	)	PUNCT
cana-1681	117	56	not	not	PART
cana-1681	117	57	approachable	approachable	ADJ
cana-1681	117	58	to	to	ADP
cana-1681	117	59	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	117	60	,	,	PUNCT
cana-1681	117	61	𝑦	𝑦	NOUN
cana-1681	117	62	)	)	PUNCT
cana-1681	117	63	for	for	ADP
cana-1681	117	64	1	1	NUM
cana-1681	117	65	≤	≤	NUM
cana-1681	117	66	𝑞	𝑞	PROPN
cana-1681	117	67	≤	≤	PROPN
cana-1681	117	68	𝑝.	𝑝.	NOUN
cana-1681	117	69	(	(	PUNCT
cana-1681	117	70	iii	iii	NOUN
cana-1681	117	71	)	)	PUNCT
cana-1681	117	72	“	"	PUNCT
cana-1681	117	73	let	let	VERB
cana-1681	117	74	{	{	PUNCT
cana-1681	117	75	𝜓𝑛}𝑛	𝜓𝑛}𝑛	PROPN
cana-1681	117	76	𝑁	𝑁	PROPN
cana-1681	117	77	be	be	VERB
cana-1681	117	78	a	a	DET
cana-1681	117	79	good	good	ADJ
cana-1681	117	80	approximate	approximate	ADJ
cana-1681	117	81	identity	identity	NOUN
cana-1681	117	82	and	and	CCONJ
cana-1681	117	83	let	let	VERB
cana-1681	117	84	{	{	PUNCT
cana-1681	117	85	휁𝑛}𝑛	휁𝑛}𝑛	X
cana-1681	117	86	𝑁	𝑁	PROPN
cana-1681	117	87	be	be	VERB
cana-1681	117	88	any	any	DET
cana-1681	117	89	approximate	approximate	ADJ
cana-1681	117	90	identity	identity	NOUN
cana-1681	117	91	.	.	PUNCT
cana-1681	118	1	let	let	AUX
cana-1681	118	2	{	{	PUNCT
cana-1681	118	3	𝑝𝑛	𝑝𝑛	PART
cana-1681	118	4	}	}	PUNCT
cana-1681	118	5	be	be	AUX
cana-1681	118	6	a	a	DET
cana-1681	118	7	sequence	sequence	NOUN
cana-1681	118	8	of	of	ADP
cana-1681	118	9	real	real	ADJ
cana-1681	118	10	numbers	number	NOUN
cana-1681	118	11	satisfying	satisfy	VERB
cana-1681	118	12	1	1	NUM
cana-1681	118	13	≤	≤	NOUN
cana-1681	118	14	𝑝1	𝑝1	NOUN
cana-1681	118	15	<	<	X
cana-1681	118	16	𝑝2	𝑝2	PROPN
cana-1681	118	17	<	<	X
cana-1681	118	18	.	.	PUNCT
cana-1681	118	19	.	.	PUNCT
cana-1681	118	20	.	.	PUNCT
cana-1681	118	21	.	.	PUNCT
cana-1681	118	22	.	.	PUNCT
cana-1681	119	1	.	.	PUNCT
cana-1681	120	1	<	<	X
cana-1681	120	2	𝑝𝑛	𝑝𝑛	INTJ
cana-1681	120	3	→	→	SYM
cana-1681	120	4	∞	∞	PROPN
cana-1681	120	5	consider	consider	VERB
cana-1681	120	6	the	the	DET
cana-1681	120	7	blocks	block	NOUN
cana-1681	120	8	{	{	PUNCT
cana-1681	120	9	𝑇𝑘	𝑇𝑘	PROPN
cana-1681	120	10	}	}	PUNCT
cana-1681	120	11	,	,	PUNCT
cana-1681	120	12	where	where	SCONJ
cana-1681	120	13	each	each	DET
cana-1681	120	14	block	block	NOUN
cana-1681	120	15	𝑇𝑘	𝑇𝑘	PROPN
cana-1681	120	16	is	be	AUX
cana-1681	120	17	related	relate	VERB
cana-1681	120	18	to	to	ADP
cana-1681	120	19	𝑝𝑖	𝑝𝑖	NOUN
cana-1681	120	20	,	,	PUNCT
cana-1681	120	21	for	for	ADP
cana-1681	120	22	𝑖휀𝑆𝑛	𝑖휀𝑆𝑛	NOUN
cana-1681	120	23	,	,	PUNCT
cana-1681	120	24	let	let	VERB
cana-1681	120	25	ℎ𝑖	ℎ𝑖	NOUN
cana-1681	120	26	=	=	PUNCT
cana-1681	120	27	𝛼𝑖	𝛼𝑖	INTJ
cana-1681	120	28	𝑘𝜓𝑖	𝑘𝜓𝑖	NOUN
cana-1681	120	29	𝑘	𝑘	X
cana-1681	121	1	+	+	CCONJ
cana-1681	121	2	(	(	PUNCT
cana-1681	121	3	1	1	NUM
cana-1681	121	4	−	−	NUM
cana-1681	121	5	𝛼𝑖	𝛼𝑖	INTJ
cana-1681	121	6	𝑘)𝜎𝑖	𝑘)𝜎𝑖	PROPN
cana-1681	121	7	𝑘	𝑘	PROPN
cana-1681	121	8	.	.	PUNCT
cana-1681	122	1	choose	choose	VERB
cana-1681	122	2	𝑛𝑘	𝑛𝑘	ADP
cana-1681	122	3	such	such	ADJ
cana-1681	122	4	that	that	SCONJ
cana-1681	122	5	𝛼𝑖	𝛼𝑖	ADP
cana-1681	122	6	𝑘	𝑘	X
cana-1681	122	7	→	→	SYM
cana-1681	122	8	1	1	NUM
cana-1681	122	9	.	.	PUNCT
cana-1681	123	1	then	then	ADV
cana-1681	123	2	since	since	SCONJ
cana-1681	123	3	{	{	PUNCT
cana-1681	123	4	𝜓𝑛	𝜓𝑛	NOUN
cana-1681	123	5	}	}	PUNCT
cana-1681	123	6	is	be	AUX
cana-1681	123	7	𝐿∞	𝐿∞	NOUN
cana-1681	123	8	good	good	ADJ
cana-1681	123	9	,	,	PUNCT
cana-1681	123	10	𝜓𝑛	𝜓𝑛	VERB
cana-1681	123	11	⋆	⋆	X
cana-1681	123	12	𝑓	𝑓	PROPN
cana-1681	123	13	→	→	SYM
cana-1681	123	14	𝑓a.e	𝑓a.e	PROPN
cana-1681	123	15	.	.	PUNCT
cana-1681	124	1	for	for	ADP
cana-1681	124	2	all	all	DET
cana-1681	124	3	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	NOUN
cana-1681	124	4	)	)	PUNCT
cana-1681	124	5	,	,	PUNCT
cana-1681	124	6	and	and	CCONJ
cana-1681	124	7	,	,	PUNCT
cana-1681	124	8	𝛼𝑖	𝛼𝑖	INTJ
cana-1681	124	9	𝑘𝜓𝑖	𝑘𝜓𝑖	VERB
cana-1681	124	10	𝑘	𝑘	X
cana-1681	124	11	⋆	⋆	VERB
cana-1681	124	12	𝑓	𝑓	PROPN
cana-1681	124	13	→	→	PUNCT
cana-1681	124	14	𝑓	𝑓	DET
cana-1681	124	15	a.e	a.e	PROPN
cana-1681	124	16	.	.	PROPN
cana-1681	124	17	for	for	ADP
cana-1681	124	18	all	all	DET
cana-1681	124	19	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	NOUN
cana-1681	124	20	)	)	PUNCT
cana-1681	124	21	since	since	SCONJ
cana-1681	124	22	,	,	PUNCT
cana-1681	124	23	𝜎𝑖	𝜎𝑖	X
cana-1681	124	24	𝑘	𝑘	PRON
cana-1681	124	25	⋆	⋆	PUNCT
cana-1681	124	26	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	124	27	)	)	PUNCT
cana-1681	124	28	≤	≤	NOUN
cana-1681	124	29	||𝑓||∞.	||𝑓||∞.	PROPN
cana-1681	124	30	(	(	PUNCT
cana-1681	124	31	1	1	NUM
cana-1681	124	32	−	−	NUM
cana-1681	124	33	𝛼𝑖	𝛼𝑖	INTJ
cana-1681	124	34	𝑘)𝜎𝑖	𝑘)𝜎𝑖	PROPN
cana-1681	124	35	𝑘	𝑘	X
cana-1681	124	36	⋆	⋆	VERB
cana-1681	124	37	𝑓	𝑓	PROPN
cana-1681	124	38	→	→	SYM
cana-1681	124	39	0	0	NUM
cana-1681	124	40	a.e	a.e	PROPN
cana-1681	124	41	.	.	PROPN
cana-1681	124	42	for	for	ADP
cana-1681	124	43	all	all	DET
cana-1681	124	44	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	NOUN
cana-1681	124	45	)	)	PUNCT
cana-1681	124	46	.	.	PUNCT
cana-1681	125	1	it	it	PRON
cana-1681	125	2	follows	follow	VERB
cana-1681	125	3	that	that	SCONJ
cana-1681	125	4	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	125	5	⋆	⋆	VERB
cana-1681	125	6	𝑓	𝑓	PRON
cana-1681	125	7	→	→	SYM
cana-1681	125	8	0	0	NUM
cana-1681	125	9	a.e	a.e	PROPN
cana-1681	125	10	.	.	PROPN
cana-1681	125	11	for	for	ADP
cana-1681	125	12	all	all	DET
cana-1681	125	13	𝑓휀𝐿∞(𝑅	𝑓휀𝐿∞(𝑅	NOUN
cana-1681	125	14	)	)	PUNCT
cana-1681	125	15	.	.	PUNCT
cana-1681	126	1	this	this	PRON
cana-1681	126	2	implies	imply	VERB
cana-1681	126	3	that	that	SCONJ
cana-1681	126	4	(	(	PUNCT
cana-1681	126	5	ℎ	ℎ	X
cana-1681	126	6	�	�	PROPN
cana-1681	126	7	̂	̂	SYM
cana-1681	126	8	�	�	NOUN
cana-1681	126	9	(휁	(휁	VERB
cana-1681	126	10	,	,	PUNCT
cana-1681	126	11	𝜓)𝑓(휁	𝜓)𝑓(휁	NOUN
cana-1681	126	12	,	,	PUNCT
cana-1681	126	13	𝜓	𝜓	NOUN
cana-1681	126	14	)	)	PUNCT
cana-1681	126	15	)	)	PUNCT
cana-1681	126	16	→	→	PUNCT
cana-1681	126	17	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	126	18	,	,	PUNCT
cana-1681	126	19	𝑦	𝑦	NOUN
cana-1681	126	20	)	)	PUNCT
cana-1681	126	21	for	for	ADP
cana-1681	126	22	𝑞	𝑞	PROPN
cana-1681	126	23	=	=	PROPN
cana-1681	126	24	∞.	∞.	PROPN
cana-1681	126	25	the	the	DET
cana-1681	126	26	approximate	approximate	ADJ
cana-1681	126	27	identity	identity	NOUN
cana-1681	126	28	{	{	PUNCT
cana-1681	126	29	ℎ𝑛	ℎ𝑛	INTJ
cana-1681	126	30	𝑘}𝑛	𝑘}𝑛	PROPN
cana-1681	126	31	is	be	AUX
cana-1681	126	32	𝐿𝑝𝑚-bad	𝐿𝑝𝑚-bad	NOUN
cana-1681	126	33	for	for	ADP
cana-1681	126	34	every	every	DET
cana-1681	126	35	𝑚휀{1	𝑚휀{1	NOUN
cana-1681	126	36	,	,	PUNCT
cana-1681	126	37	.	.	PUNCT
cana-1681	126	38	.	.	PUNCT
cana-1681	126	39	.	.	PUNCT
cana-1681	127	1	.	.	PUNCT
cana-1681	128	1	,	,	PUNCT
cana-1681	128	2	𝑘	𝑘	X
cana-1681	128	3	}	}	PUNCT
cana-1681	128	4	,	,	PUNCT
cana-1681	128	5	since	since	SCONJ
cana-1681	128	6	it	it	PRON
cana-1681	128	7	is	be	AUX
cana-1681	128	8	𝐿𝑞-bad	𝐿𝑞-bad	NOUN
cana-1681	128	9	for	for	ADP
cana-1681	128	10	every	every	DET
cana-1681	128	11	1	1	NUM
cana-1681	128	12	≤	≤	NUM
cana-1681	128	13	𝑞	𝑞	PROPN
cana-1681	128	14	≤	≤	ADJ
cana-1681	128	15	𝑝𝑘.	𝑝𝑘.	NOUN
cana-1681	128	16	there	there	PRON
cana-1681	128	17	exists	exist	VERB
cana-1681	128	18	𝑓𝑚	𝑓𝑚	VERB
cana-1681	128	19	𝑘휀𝐿𝑝𝑚([0,1	𝑘휀𝐿𝑝𝑚([0,1	NOUN
cana-1681	128	20	)	)	PUNCT
cana-1681	128	21	)	)	PUNCT
cana-1681	128	22	with	with	ADP
cana-1681	128	23	||𝑓𝑚	||𝑓𝑚	NOUN
cana-1681	128	24	𝑘(𝑥𝑦	𝑘(𝑥𝑦	NOUN
cana-1681	128	25	−	−	NOUN
cana-1681	128	26	𝜆𝑚	𝜆𝑚	NOUN
cana-1681	128	27	𝑘(𝑁	𝑘(𝑁	PROPN
cana-1681	128	28	)	)	PUNCT
cana-1681	128	29	||	||	PUNCT
cana-1681	129	1	=	=	PUNCT
cana-1681	129	2	2−𝑘	2−𝑘	NUM
cana-1681	129	3	,	,	PUNCT
cana-1681	129	4	𝜆𝑚	𝜆𝑚	ADP
cana-1681	129	5	𝑘(𝑁	𝑘(𝑁	PROPN
cana-1681	129	6	)	)	PUNCT
cana-1681	129	7	>	>	X
cana-1681	129	8	0	0	PUNCT
cana-1681	130	1	and	and	CCONJ
cana-1681	130	2	𝑛𝑚	𝑛𝑚	ADP
cana-1681	130	3	𝑘	𝑘	X
cana-1681	130	4	>	>	X
cana-1681	130	5	𝑚𝑘−1	𝑚𝑘−1	PROPN
cana-1681	130	6	so	so	SCONJ
cana-1681	130	7	that	that	SCONJ
cana-1681	130	8	|{𝑠𝑢𝑝𝑛𝑘−1	|{𝑠𝑢𝑝𝑛𝑘−1	NOUN
cana-1681	130	9	<	<	X
cana-1681	130	10	𝑛	𝑛	X
cana-1681	130	11	<	<	X
cana-1681	130	12	𝑛𝑚	𝑛𝑚	X
cana-1681	130	13	𝑘	𝑘	PROPN
cana-1681	130	14	(	(	PUNCT
cana-1681	130	15	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	130	16	𝑘	𝑘	PROPN
cana-1681	130	17	⋆	⋆	NOUN
cana-1681	130	18	𝑓𝑚	𝑓𝑚	PROPN
cana-1681	130	19	𝑘)}|	𝑘)}|	X
cana-1681	130	20	>	>	X
cana-1681	130	21	𝐶𝑁||𝑓𝑚	𝐶𝑁||𝑓𝑚	NOUN
cana-1681	130	22	𝑘(𝑥𝑦	𝑘(𝑥𝑦	X
cana-1681	130	23	−	−	PROPN
cana-1681	130	24	𝜆𝑚	𝜆𝑚	NOUN
cana-1681	130	25	𝑘(𝑁	𝑘(𝑁	NOUN
cana-1681	130	26	)	)	PUNCT
cana-1681	130	27	||𝑝𝑚	||𝑝𝑚	NOUN
cana-1681	130	28	𝑝𝑚	𝑝𝑚	NOUN
cana-1681	130	29	=	=	PUNCT
cana-1681	130	30	𝐶𝑘	𝐶𝑘	PROPN
cana-1681	130	31	𝑁	𝑁	PROPN
cana-1681	130	32	2𝑘𝑝𝑚	2𝑘𝑝𝑚	NOUN
cana-1681	130	33	let	let	VERB
cana-1681	130	34	𝑓	𝑓	PRON
cana-1681	130	35	=	=	PUNCT
cana-1681	130	36	∑	∑	PUNCT
cana-1681	130	37	𝑘≥𝑘0	𝑘≥𝑘0	PROPN
cana-1681	130	38	𝑓𝑘0	𝑓𝑘0	NOUN
cana-1681	130	39	𝑘	𝑘	PRON
cana-1681	130	40	,	,	PUNCT
cana-1681	130	41	then	then	ADV
cana-1681	130	42	||𝑓||𝑝𝑘0	||𝑓||𝑝𝑘0	ADP
cana-1681	130	43	<	<	X
cana-1681	130	44	2	2	NUM
cana-1681	130	45	.	.	PUNCT
cana-1681	131	1	so	so	ADV
cana-1681	131	2	,	,	PUNCT
cana-1681	131	3	|{𝑠𝑢𝑝𝑛(ℎ𝑛	|{𝑠𝑢𝑝𝑛(ℎ𝑛	PROPN
cana-1681	131	4	⋆	⋆	VERB
cana-1681	131	5	𝑓)}|	𝑓)}|	ADP
cana-1681	131	6	≤	≤	NOUN
cana-1681	131	7	𝐶0||𝑓||𝑝𝑘0	𝐶0||𝑓||𝑝𝑘0	ADP
cana-1681	131	8	𝑝𝑘0	𝑝𝑘0	NOUN
cana-1681	131	9	≤	≤	NOUN
cana-1681	131	10	2𝑝𝑘0	2𝑝𝑘0	NUM
cana-1681	131	11	𝐶0	𝐶0	NOUN
cana-1681	131	12	𝑁.	𝑁.	PROPN
cana-1681	131	13	(	(	PUNCT
cana-1681	131	14	3	3	NUM
cana-1681	131	15	)	)	PUNCT
cana-1681	131	16	hence	hence	ADV
cana-1681	131	17	,	,	PUNCT
cana-1681	131	18	|{𝑠𝑢𝑝𝑛(ℎ𝑛	|{𝑠𝑢𝑝𝑛(ℎ𝑛	PROPN
cana-1681	131	19	⋆	⋆	VERB
cana-1681	131	20	𝑓)}|	𝑓)}|	ADP
cana-1681	131	21	≥	≥	NOUN
cana-1681	131	22	|{𝑠𝑢𝑝𝑛𝑘−1	|{𝑠𝑢𝑝𝑛𝑘−1	X
cana-1681	131	23	<	<	X
cana-1681	131	24	𝑛	𝑛	X
cana-1681	131	25	<	<	X
cana-1681	131	26	𝑛𝑚	𝑛𝑚	X
cana-1681	131	27	𝑘	𝑘	PROPN
cana-1681	131	28	(	(	PUNCT
cana-1681	131	29	ℎ𝑛	ℎ𝑛	PROPN
cana-1681	131	30	𝑘	𝑘	X
cana-1681	131	31	⋆	⋆	X
cana-1681	131	32	𝑓𝑘0	𝑓𝑘0	PROPN
cana-1681	131	33	𝑘	𝑘	X
cana-1681	131	34	)	)	PUNCT
cana-1681	131	35	}	}	PUNCT
cana-1681	131	36	|	|	ADV
cana-1681	131	37	>	>	X
cana-1681	131	38	𝐶𝑘	𝐶𝑘	VERB
cana-1681	131	39	𝑁	𝑁	PROPN
cana-1681	131	40	2	2	NUM
cana-1681	131	41	𝑘𝑝𝑘0	𝑘𝑝𝑘0	NOUN
cana-1681	131	42	"	"	PUNCT
cana-1681	131	43	(	(	PUNCT
cana-1681	131	44	4	4	X
cana-1681	131	45	)	)	PUNCT
cana-1681	131	46	communications	communication	NOUN
cana-1681	131	47	on	on	ADP
cana-1681	131	48	applied	apply	VERB
cana-1681	131	49	nonlinear	nonlinear	ADJ
cana-1681	131	50	analysis	analysis	NOUN
cana-1681	131	51	issn	issn	NOUN
cana-1681	131	52	:	:	PUNCT
cana-1681	131	53	1074	1074	NUM
cana-1681	131	54	-	-	PUNCT
cana-1681	131	55	133x	133x	NUM
cana-1681	131	56	vol	vol	NOUN
cana-1681	131	57	32	32	NUM
cana-1681	131	58	no	no	NOUN
cana-1681	131	59	.	.	NOUN
cana-1681	131	60	1	1	NUM
cana-1681	131	61	(	(	PUNCT
cana-1681	131	62	2025	2025	NUM
cana-1681	131	63	)	)	PUNCT
cana-1681	131	64	391	391	NUM
cana-1681	131	65	https://internationalpubls.com	https://internationalpubls.com	X
cana-1681	131	66	using	use	VERB
cana-1681	131	67	the	the	DET
cana-1681	131	68	equations	equation	NOUN
cana-1681	131	69	,	,	PUNCT
cana-1681	131	70	we	we	PRON
cana-1681	131	71	get	get	VERB
cana-1681	131	72	𝐶0	𝐶0	NOUN
cana-1681	131	73	𝑁	𝑁	PROPN
cana-1681	131	74	>	>	SYM
cana-1681	131	75	𝐶𝑘	𝐶𝑘	PROPN
cana-1681	131	76	𝑁	𝑁	PROPN
cana-1681	131	77	2𝑘𝑝𝑘0	2𝑘𝑝𝑘0	PROPN
cana-1681	131	78	(	(	PUNCT
cana-1681	131	79	𝑘+1	𝑘+1	NOUN
cana-1681	131	80	)	)	PUNCT
cana-1681	131	81	→	→	PUNCT
cana-1681	132	1	+	+	NUM
cana-1681	132	2	∞	∞	NOUN
cana-1681	132	3	thus	thus	ADV
cana-1681	132	4	we	we	PRON
cana-1681	132	5	conclude	conclude	VERB
cana-1681	132	6	that	that	SCONJ
cana-1681	132	7	ℎ	ℎ	PROPN
cana-1681	132	8	�	�	PROPN
cana-1681	132	9	̂	̂	SYM
cana-1681	132	10	�	�	NOUN
cana-1681	132	11	(휁	(휁	VERB
cana-1681	132	12	,	,	PUNCT
cana-1681	132	13	𝜓)𝑓(휁	𝜓)𝑓(휁	NOUN
cana-1681	132	14	,	,	PUNCT
cana-1681	132	15	𝜓	𝜓	NOUN
cana-1681	132	16	)	)	PUNCT
cana-1681	132	17	not	not	PART
cana-1681	132	18	approachable	approachable	ADJ
cana-1681	132	19	to	to	ADP
cana-1681	132	20	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1681	132	21	,	,	PUNCT
cana-1681	132	22	𝑦	𝑦	NOUN
cana-1681	132	23	)	)	PUNCT
cana-1681	132	24	for	for	ADP
cana-1681	132	25	1	1	NUM
cana-1681	132	26	≤	≤	NUM
cana-1681	132	27	𝑞	𝑞	X
cana-1681	132	28	≤	≤	NOUN
cana-1681	132	29	∞.	∞.	PROPN
cana-1681	132	30	hence	hence	ADV
cana-1681	132	31	,	,	PUNCT
cana-1681	132	32	the	the	DET
cana-1681	132	33	proof	proof	NOUN
cana-1681	132	34	is	be	AUX
cana-1681	132	35	completed	complete	VERB
cana-1681	132	36	.	.	PUNCT
cana-1681	133	1	references	reference	NOUN
cana-1681	133	2	:	:	PUNCT
cana-1681	134	1	[	[	X
cana-1681	134	2	1	1	NUM
cana-1681	134	3	]	]	X
cana-1681	134	4	a.bellow	a.bellow	ADJ
cana-1681	134	5	,	,	PUNCT
cana-1681	134	6	perturbation	perturbation	NOUN
cana-1681	134	7	of	of	ADP
cana-1681	134	8	a	a	DET
cana-1681	134	9	sequence	sequence	NOUN
cana-1681	134	10	,	,	PUNCT
cana-1681	134	11	advances	advance	NOUN
cana-1681	134	12	in	in	ADP
cana-1681	134	13	mathematics	mathematic	NOUN
cana-1681	134	14	,	,	PUNCT
cana-1681	134	15	78(1989	78(1989	NUM
cana-1681	134	16	)	)	PUNCT
cana-1681	134	17	,	,	PUNCT
cana-1681	134	18	131	131	NUM
cana-1681	134	19	-	-	SYM
cana-1681	134	20	139	139	NUM
cana-1681	134	21	.	.	PUNCT
cana-1681	135	1	[	[	X
cana-1681	135	2	2	2	X
cana-1681	135	3	]	]	PUNCT
cana-1681	135	4	k.	k.	PROPN
cana-1681	135	5	devendra	devendra	PROPN
cana-1681	135	6	and	and	CCONJ
cana-1681	135	7	s.	s.	PROPN
cana-1681	135	8	dimple	dimple	PROPN
cana-1681	135	9	,	,	PUNCT
cana-1681	135	10	fourier	fourier	NOUN
cana-1681	135	11	transform	transform	NOUN
cana-1681	135	12	in	in	ADP
cana-1681	135	13	𝐿𝑝(𝑅	𝐿𝑝(𝑅	NOUN
cana-1681	135	14	)	)	PUNCT
cana-1681	135	15	spaces	space	NOUN
cana-1681	135	16	,	,	PUNCT
cana-1681	135	17	𝑝	𝑝	PRON
cana-1681	135	18	≥	≥	NOUN
cana-1681	135	19	1	1	NUM
cana-1681	135	20	,	,	PUNCT
cana-1681	135	21	3(2011	3(2011	NUM
cana-1681	135	22	)	)	PUNCT
cana-1681	135	23	,	,	PUNCT
cana-1681	135	24	14	14	NUM
cana-1681	135	25	-	-	SYM
cana-1681	135	26	25	25	NUM
cana-1681	135	27	.	.	PUNCT
cana-1681	136	1	[	[	X
cana-1681	136	2	3	3	X
cana-1681	136	3	]	]	PUNCT
cana-1681	136	4	k.	k.	PROPN
cana-1681	136	5	reinhold	reinhold	PROPN
cana-1681	136	6	-	-	PUNCT
cana-1681	136	7	larsson	larsson	PROPN
cana-1681	136	8	,	,	PUNCT
cana-1681	136	9	discrepancy	discrepancy	NOUN
cana-1681	136	10	of	of	ADP
cana-1681	136	11	behaviour	behaviour	NOUN
cana-1681	136	12	of	of	ADP
cana-1681	136	13	perturbed	perturb	VERB
cana-1681	136	14	sequences	sequence	NOUN
cana-1681	136	15	in	in	ADP
cana-1681	136	16	𝐿𝑝-spaces	𝐿𝑝-space	NOUN
cana-1681	136	17	,	,	PUNCT
cana-1681	136	18	proc	proc	NOUN
cana-1681	136	19	.	.	PUNCT
cana-1681	137	1	amer	amer	PROPN
cana-1681	137	2	.	.	PUNCT
cana-1681	137	3	math	math	PROPN
cana-1681	137	4	.	.	PUNCT
cana-1681	138	1	soc	soc	PROPN
cana-1681	138	2	.	.	PUNCT
cana-1681	138	3	,	,	PUNCT
cana-1681	138	4	120(1994	120(1994	NUM
cana-1681	138	5	)	)	PUNCT
cana-1681	138	6	,	,	PUNCT
cana-1681	138	7	865	865	NUM
cana-1681	138	8	-	-	SYM
cana-1681	138	9	874	874	NUM
cana-1681	138	10	.	.	PUNCT
cana-1681	139	1	[	[	X
cana-1681	139	2	4	4	X
cana-1681	139	3	]	]	PUNCT
cana-1681	139	4	s.	s.	PROPN
cana-1681	139	5	sawyer	sawyer	PROPN
cana-1681	139	6	,	,	PUNCT
cana-1681	139	7	maximal	maximal	ADJ
cana-1681	139	8	inequalities	inequality	NOUN
cana-1681	139	9	of	of	ADP
cana-1681	139	10	weak	weak	ADJ
cana-1681	139	11	type	type	NOUN
cana-1681	139	12	,	,	PUNCT
cana-1681	139	13	ann	ann	PROPN
cana-1681	139	14	.	.	PROPN
cana-1681	139	15	of	of	ADP
cana-1681	139	16	math	math	NOUN
cana-1681	139	17	,	,	PUNCT
cana-1681	139	18	84(2)(1966	84(2)(1966	NUM
cana-1681	139	19	)	)	PUNCT
cana-1681	139	20	,	,	PUNCT
cana-1681	139	21	157	157	NUM
cana-1681	139	22	-	-	SYM
cana-1681	139	23	174	174	NUM
cana-1681	139	24	.	.	PUNCT