id	sid	tid	token	lemma	pos
cana-3855	1	1	communications	communication	NOUN
cana-3855	1	2	on	on	ADP
cana-3855	1	3	applied	apply	VERB
cana-3855	1	4	nonlinear	nonlinear	ADJ
cana-3855	1	5	analysis	analysis	NOUN
cana-3855	1	6	issn	issn	NOUN
cana-3855	1	7	:	:	PUNCT
cana-3855	1	8	1074	1074	NUM
cana-3855	1	9	-	-	PUNCT
cana-3855	1	10	133x	133x	NUM
cana-3855	1	11	vol	vol	NOUN
cana-3855	1	12	32	32	NUM
cana-3855	1	13	no	no	NOUN
cana-3855	1	14	.	.	PUNCT
cana-3855	2	1	9s	9s	NUM
cana-3855	2	2	(	(	PUNCT
cana-3855	2	3	2025	2025	NUM
cana-3855	2	4	)	)	PUNCT
cana-3855	2	5	277	277	NUM
cana-3855	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	2	7	quasi	quasi	VERB
cana-3855	2	8	d	d	X
cana-3855	2	9	-	-	PUNCT
cana-3855	2	10	limits	limit	NOUN
cana-3855	2	11	of	of	ADP
cana-3855	2	12	bounded	bounded	ADJ
cana-3855	2	13	sequences	sequence	NOUN
cana-3855	2	14	and	and	CCONJ
cana-3855	2	15	regular	regular	ADJ
cana-3855	2	16	matrices	matrix	NOUN
cana-3855	2	17	sakambari	sakambari	ADJ
cana-3855	2	18	mishra1	mishra1	NOUN
cana-3855	2	19	,	,	PUNCT
cana-3855	2	20	balaji	balaji	PROPN
cana-3855	2	21	padhy2	padhy2	PROPN
cana-3855	2	22	*	*	SYM
cana-3855	2	23	1	1	NUM
cana-3855	2	24	department	department	NOUN
cana-3855	2	25	of	of	ADP
cana-3855	2	26	mathematics	mathematic	NOUN
cana-3855	2	27	,	,	PUNCT
cana-3855	2	28	college	college	NOUN
cana-3855	2	29	of	of	ADP
cana-3855	2	30	basic	basic	ADJ
cana-3855	2	31	science	science	NOUN
cana-3855	2	32	and	and	CCONJ
cana-3855	2	33	humanities	humanity	NOUN
cana-3855	2	34	,	,	PUNCT
cana-3855	2	35	odisha	odisha	PROPN
cana-3855	2	36	university	university	PROPN
cana-3855	2	37	of	of	ADP
cana-3855	2	38	agriculture	agriculture	NOUN
cana-3855	2	39	and	and	CCONJ
cana-3855	2	40	technology	technology	NOUN
cana-3855	2	41	,	,	PUNCT
cana-3855	2	42	bhubaneswar	bhubaneswar	NOUN
cana-3855	2	43	751003	751003	NUM
cana-3855	2	44	,	,	PUNCT
cana-3855	2	45	odisha	odisha	PROPN
cana-3855	2	46	,	,	PUNCT
cana-3855	2	47	india	india	PROPN
cana-3855	2	48	.	.	PUNCT
cana-3855	3	1	email	email	NOUN
cana-3855	4	1	i	i	PROPN
cana-3855	4	2	d	d	PROPN
cana-3855	4	3	:	:	PUNCT
cana-3855	4	4	sakambari@ouat.ac.in	sakambari@ouat.ac.in	PUNCT
cana-3855	4	5	(	(	PUNCT
cana-3855	4	6	s.mishra	s.mishra	NOUN
cana-3855	4	7	)	)	PUNCT
cana-3855	4	8	*	*	SYM
cana-3855	4	9	2	2	NUM
cana-3855	4	10	department	department	NOUN
cana-3855	4	11	of	of	ADP
cana-3855	4	12	mathematics	mathematics	PROPN
cana-3855	4	13	,	,	PUNCT
cana-3855	4	14	centurion	centurion	NOUN
cana-3855	4	15	university	university	PROPN
cana-3855	4	16	of	of	ADP
cana-3855	4	17	technology	technology	NOUN
cana-3855	4	18	and	and	CCONJ
cana-3855	4	19	management	management	NOUN
cana-3855	4	20	,	,	PUNCT
cana-3855	4	21	paralakhemundi	paralakhemundi	VERB
cana-3855	4	22	761211	761211	NUM
cana-3855	4	23	,	,	PUNCT
cana-3855	4	24	odisha	odisha	PROPN
cana-3855	4	25	,	,	PUNCT
cana-3855	4	26	india	india	PROPN
cana-3855	4	27	.	.	PUNCT
cana-3855	5	1	e	e	X
cana-3855	5	2	-	-	NOUN
cana-3855	5	3	mail	mail	NOUN
cana-3855	5	4	:	:	PUNCT
cana-3855	5	5	balaji.padhy@cutm.ac.in	balaji.padhy@cutm.ac.in	NUM
cana-3855	5	6	∗corresponding	∗corresponde	VERB
cana-3855	5	7	author	author	NOUN
cana-3855	5	8	article	article	NOUN
cana-3855	5	9	history	history	NOUN
cana-3855	5	10	:	:	PUNCT
cana-3855	5	11	received	receive	VERB
cana-3855	5	12	:	:	PUNCT
cana-3855	5	13	14	14	NUM
cana-3855	5	14	-	-	SYM
cana-3855	5	15	11	11	NUM
cana-3855	5	16	-	-	PUNCT
cana-3855	5	17	2024	2024	NUM
cana-3855	5	18	revised:26	revised:26	PROPN
cana-3855	5	19	-	-	PUNCT
cana-3855	5	20	12	12	NUM
cana-3855	5	21	-	-	PUNCT
cana-3855	5	22	2024	2024	NUM
cana-3855	5	23	accepted:10	accepted:10	PROPN
cana-3855	5	24	-	-	PUNCT
cana-3855	5	25	01	01	NUM
cana-3855	5	26	-	-	PUNCT
cana-3855	5	27	2025	2025	NUM
cana-3855	5	28	abstract	abstract	NOUN
cana-3855	5	29	:	:	PUNCT
cana-3855	5	30	in	in	ADP
cana-3855	5	31	this	this	DET
cana-3855	5	32	paper	paper	NOUN
cana-3855	5	33	,	,	PUNCT
cana-3855	5	34	we	we	PRON
cana-3855	5	35	explore	explore	VERB
cana-3855	5	36	the	the	DET
cana-3855	5	37	concept	concept	NOUN
cana-3855	5	38	of	of	ADP
cana-3855	5	39	quasi	quasi	ADJ
cana-3855	5	40	-	-	ADJ
cana-3855	5	41	d	d	NOUN
cana-3855	5	42	-	-	PUNCT
cana-3855	5	43	limits	limit	NOUN
cana-3855	5	44	for	for	ADP
cana-3855	5	45	bounded	bounded	ADJ
cana-3855	5	46	sequences	sequence	NOUN
cana-3855	5	47	through	through	ADP
cana-3855	5	48	the	the	DET
cana-3855	5	49	nonnegative	nonnegative	ADJ
cana-3855	5	50	regular	regular	ADJ
cana-3855	5	51	matrix	matrix	NOUN
cana-3855	5	52	transformation	transformation	NOUN
cana-3855	5	53	d.	d.	PROPN
cana-3855	5	54	quasi	quasi	PROPN
cana-3855	6	1	d	d	PROPN
cana-3855	6	2	-	-	PUNCT
cana-3855	6	3	limits	limit	NOUN
cana-3855	6	4	extend	extend	VERB
cana-3855	6	5	spaces	space	NOUN
cana-3855	6	6	of	of	ADP
cana-3855	6	7	dlimits	dlimit	NOUN
cana-3855	6	8	notions	notion	NOUN
cana-3855	6	9	,	,	PUNCT
cana-3855	6	10	providing	provide	VERB
cana-3855	6	11	a	a	DET
cana-3855	6	12	framework	framework	NOUN
cana-3855	6	13	to	to	PART
cana-3855	6	14	analyse	analyse	VERB
cana-3855	6	15	the	the	DET
cana-3855	6	16	convergence	convergence	NOUN
cana-3855	6	17	properties	property	NOUN
cana-3855	6	18	of	of	ADP
cana-3855	6	19	sequences	sequence	NOUN
cana-3855	6	20	that	that	PRON
cana-3855	6	21	exhibit	exhibit	VERB
cana-3855	6	22	specific	specific	ADJ
cana-3855	6	23	bounded	bounded	ADJ
cana-3855	6	24	behaviour	behaviour	NOUN
cana-3855	6	25	.	.	PUNCT
cana-3855	7	1	we	we	PRON
cana-3855	7	2	establish	establish	VERB
cana-3855	7	3	criteria	criterion	NOUN
cana-3855	7	4	for	for	ADP
cana-3855	7	5	quasi	quasi	NOUN
cana-3855	7	6	-	-	NOUN
cana-3855	7	7	dlimits	dlimit	NOUN
cana-3855	7	8	,	,	PUNCT
cana-3855	7	9	demonstrate	demonstrate	VERB
cana-3855	7	10	their	their	PRON
cana-3855	7	11	existence	existence	NOUN
cana-3855	7	12	via	via	ADP
cana-3855	7	13	regular	regular	ADJ
cana-3855	7	14	matrix	matrix	NOUN
cana-3855	7	15	transformations	transformation	NOUN
cana-3855	7	16	,	,	PUNCT
cana-3855	7	17	and	and	CCONJ
cana-3855	7	18	apply	apply	VERB
cana-3855	7	19	these	these	DET
cana-3855	7	20	results	result	NOUN
cana-3855	7	21	to	to	ADP
cana-3855	7	22	various	various	ADJ
cana-3855	7	23	mathematical	mathematical	ADJ
cana-3855	7	24	and	and	CCONJ
cana-3855	7	25	applied	applied	ADJ
cana-3855	7	26	contexts	context	NOUN
cana-3855	7	27	.	.	PUNCT
cana-3855	8	1	2010	2010	NUM
cana-3855	8	2	mathematics	mathematic	NOUN
cana-3855	8	3	subject	subject	NOUN
cana-3855	8	4	classification	classification	NOUN
cana-3855	8	5	:	:	PUNCT
cana-3855	8	6	primary	primary	NOUN
cana-3855	8	7	40a05	40a05	X
cana-3855	8	8	.	.	PUNCT
cana-3855	9	1	secondary	secondary	ADJ
cana-3855	9	2	40c05	40c05	NOUN
cana-3855	9	3	.	.	PUNCT
cana-3855	10	1	key	key	ADJ
cana-3855	10	2	words	word	NOUN
cana-3855	10	3	.	.	PUNCT
cana-3855	11	1	linear	linear	ADJ
cana-3855	11	2	functional	functional	ADJ
cana-3855	11	3	,	,	PUNCT
cana-3855	11	4	d	d	NOUN
cana-3855	11	5	-	-	PUNCT
cana-3855	11	6	limit	limit	NOUN
cana-3855	11	7	,	,	PUNCT
cana-3855	11	8	d	d	ADJ
cana-3855	11	9	-	-	PUNCT
cana-3855	11	10	invariant	invariant	ADJ
cana-3855	11	11	,	,	PUNCT
cana-3855	11	12	regular	regular	ADJ
cana-3855	11	13	matrices	matrix	NOUN
cana-3855	11	14	and	and	CCONJ
cana-3855	11	15	quasi	quasi	NOUN
cana-3855	11	16	-	-	NOUN
cana-3855	11	17	dconvergence	dconvergence	NOUN
cana-3855	11	18	.	.	PUNCT
cana-3855	12	1	1	1	X
cana-3855	12	2	.	.	X
cana-3855	12	3	introduction	introduction	NOUN
cana-3855	12	4	:	:	PUNCT
cana-3855	12	5	the	the	DET
cana-3855	12	6	study	study	NOUN
cana-3855	12	7	of	of	ADP
cana-3855	12	8	limits	limit	NOUN
cana-3855	12	9	in	in	ADP
cana-3855	12	10	sequence	sequence	NOUN
cana-3855	12	11	convergence	convergence	NOUN
cana-3855	12	12	has	have	AUX
cana-3855	12	13	been	be	AUX
cana-3855	12	14	a	a	DET
cana-3855	12	15	foundational	foundational	ADJ
cana-3855	12	16	aspect	aspect	NOUN
cana-3855	12	17	of	of	ADP
cana-3855	12	18	analysis	analysis	NOUN
cana-3855	12	19	.	.	PUNCT
cana-3855	13	1	in	in	ADP
cana-3855	13	2	particular	particular	ADJ
cana-3855	13	3	,	,	PUNCT
cana-3855	13	4	bounded	bound	VERB
cana-3855	13	5	sequences	sequence	NOUN
cana-3855	13	6	whose	whose	DET
cana-3855	13	7	terms	term	NOUN
cana-3855	13	8	are	be	AUX
cana-3855	13	9	confined	confine	VERB
cana-3855	13	10	within	within	ADP
cana-3855	13	11	a	a	DET
cana-3855	13	12	specific	specific	ADJ
cana-3855	13	13	range	range	NOUN
cana-3855	13	14	often	often	ADV
cana-3855	13	15	require	require	VERB
cana-3855	13	16	advanced	advanced	ADJ
cana-3855	13	17	techniques	technique	NOUN
cana-3855	13	18	to	to	PART
cana-3855	13	19	elucidate	elucidate	VERB
cana-3855	13	20	their	their	PRON
cana-3855	13	21	limiting	limit	VERB
cana-3855	13	22	behaviour	behaviour	NOUN
cana-3855	13	23	.	.	PUNCT
cana-3855	14	1	quasi	quasi	ADJ
cana-3855	14	2	-	-	ADJ
cana-3855	14	3	d	d	NOUN
cana-3855	14	4	-	-	PUNCT
cana-3855	14	5	limits	limit	NOUN
cana-3855	14	6	offer	offer	VERB
cana-3855	14	7	an	an	DET
cana-3855	14	8	innovative	innovative	ADJ
cana-3855	14	9	approach	approach	NOUN
cana-3855	14	10	,	,	PUNCT
cana-3855	14	11	enabling	enable	VERB
cana-3855	14	12	us	we	PRON
cana-3855	14	13	to	to	PART
cana-3855	14	14	extend	extend	VERB
cana-3855	14	15	conventional	conventional	ADJ
cana-3855	14	16	limit	limit	NOUN
cana-3855	14	17	concepts	concept	NOUN
cana-3855	14	18	.	.	PUNCT
cana-3855	15	1	the	the	DET
cana-3855	15	2	idea	idea	NOUN
cana-3855	15	3	of	of	ADP
cana-3855	15	4	quasi	quasi	NOUN
cana-3855	15	5	almost	almost	ADV
cana-3855	15	6	convergence	convergence	NOUN
cana-3855	15	7	in	in	ADP
cana-3855	15	8	normed	normed	ADJ
cana-3855	15	9	space	space	NOUN
cana-3855	15	10	was	be	AUX
cana-3855	15	11	featured	feature	VERB
cana-3855	15	12	by	by	ADP
cana-3855	15	13	hajdukovic[4	hajdukovic[4	NOUN
cana-3855	15	14	]	]	PUNCT
cana-3855	15	15	.	.	PUNCT
cana-3855	16	1	further	far	ADV
cana-3855	16	2	the	the	DET
cana-3855	16	3	concept	concept	NOUN
cana-3855	16	4	of	of	ADP
cana-3855	16	5	quasi	quasi	NOUN
cana-3855	16	6	banach	banach	NOUN
cana-3855	16	7	limit	limit	NOUN
cana-3855	16	8	is	be	AUX
cana-3855	16	9	introduced	introduce	VERB
cana-3855	16	10	by	by	ADP
cana-3855	16	11	das	das	PROPN
cana-3855	16	12	and	and	CCONJ
cana-3855	16	13	mishra	mishra	PROPN
cana-3855	16	14	in	in	ADP
cana-3855	16	15	[	[	X
cana-3855	16	16	3	3	NUM
cana-3855	16	17	]	]	PUNCT
cana-3855	16	18	and	and	CCONJ
cana-3855	16	19	the	the	DET
cana-3855	16	20	concept	concept	NOUN
cana-3855	16	21	of	of	ADP
cana-3855	16	22	quasi	quasi	ADJ
cana-3855	16	23	-	-	ADJ
cana-3855	16	24	invariant	invariant	ADJ
cana-3855	16	25	limit	limit	NOUN
cana-3855	16	26	is	be	AUX
cana-3855	16	27	recently	recently	ADV
cana-3855	16	28	explained	explain	VERB
cana-3855	16	29	by	by	ADP
cana-3855	16	30	mishra[5	mishra[5	PROPN
cana-3855	16	31	]	]	PUNCT
cana-3855	16	32	in	in	ADP
cana-3855	16	33	the	the	DET
cana-3855	16	34	space	space	NOUN
cana-3855	16	35	of	of	ADP
cana-3855	16	36	real	real	ADJ
cana-3855	16	37	bounded	bound	VERB
cana-3855	16	38	sequences	sequence	NOUN
cana-3855	16	39	which	which	PRON
cana-3855	16	40	yields	yield	VERB
cana-3855	16	41	quasiinvariant	quasiinvariant	ADJ
cana-3855	16	42	convergent	convergent	NOUN
cana-3855	16	43	sequences	sequence	NOUN
cana-3855	16	44	.	.	PUNCT
cana-3855	17	1	this	this	DET
cana-3855	17	2	paper	paper	NOUN
cana-3855	17	3	aims	aim	VERB
cana-3855	17	4	to	to	PART
cana-3855	17	5	investigate	investigate	VERB
cana-3855	17	6	the	the	DET
cana-3855	17	7	quasi	quasi	ADJ
cana-3855	17	8	-	-	ADJ
cana-3855	17	9	d	d	NOUN
cana-3855	17	10	-	-	PUNCT
cana-3855	17	11	limits	limit	NOUN
cana-3855	17	12	of	of	ADP
cana-3855	17	13	bounded	bounded	ADJ
cana-3855	17	14	sequences	sequence	NOUN
cana-3855	17	15	utilizing	utilize	VERB
cana-3855	17	16	matrix	matrix	NOUN
cana-3855	17	17	transformations	transformation	NOUN
cana-3855	17	18	.	.	PUNCT
cana-3855	18	1	we	we	PRON
cana-3855	18	2	first	first	ADV
cana-3855	18	3	introduce	introduce	VERB
cana-3855	18	4	the	the	DET
cana-3855	18	5	necessary	necessary	ADJ
cana-3855	18	6	background	background	NOUN
cana-3855	18	7	on	on	ADP
cana-3855	18	8	quasi	quasi	ADJ
cana-3855	18	9	-	-	ADJ
cana-3855	18	10	d	d	NOUN
cana-3855	18	11	-	-	PUNCT
cana-3855	18	12	limits	limit	NOUN
cana-3855	18	13	and	and	CCONJ
cana-3855	18	14	matrix	matrix	NOUN
cana-3855	18	15	transformations	transformation	NOUN
cana-3855	18	16	,	,	PUNCT
cana-3855	18	17	followed	follow	VERB
cana-3855	18	18	by	by	ADP
cana-3855	18	19	an	an	DET
cana-3855	18	20	exploration	exploration	NOUN
cana-3855	18	21	of	of	ADP
cana-3855	18	22	their	their	PRON
cana-3855	18	23	interrelations	interrelation	NOUN
cana-3855	18	24	.	.	PUNCT
cana-3855	19	1	2	2	X
cana-3855	19	2	.	.	X
cana-3855	19	3	preliminaries	preliminary	NOUN
cana-3855	19	4	:	:	PUNCT
cana-3855	19	5	let	let	VERB
cana-3855	19	6	𝑚	𝑚	PART
cana-3855	19	7	be	be	AUX
cana-3855	19	8	the	the	DET
cana-3855	19	9	space	space	NOUN
cana-3855	19	10	of	of	ADP
cana-3855	19	11	real	real	ADJ
cana-3855	19	12	bounded	bounded	ADJ
cana-3855	19	13	sequences	sequence	NOUN
cana-3855	20	1	𝑥	𝑥	X
cana-3855	20	2	=	=	PRON
cana-3855	20	3	{	{	PUNCT
cana-3855	20	4	𝑥𝑛	𝑥𝑛	AUX
cana-3855	20	5	}	}	PUNCT
cana-3855	20	6	normed	norme	VERB
cana-3855	20	7	by	by	ADP
cana-3855	20	8	‖𝑥‖	‖𝑥‖	PROPN
cana-3855	20	9	=	=	PROPN
cana-3855	20	10	sup	sup	PROPN
cana-3855	20	11	𝑛	𝑛	PROPN
cana-3855	20	12	|𝑥𝑛|	|𝑥𝑛|	PROPN
cana-3855	20	13	.	.	PUNCT
cana-3855	21	1	let	let	VERB
cana-3855	21	2	𝑚∗	𝑚∗	PROPN
cana-3855	21	3	denote	denote	VERB
cana-3855	21	4	the	the	DET
cana-3855	21	5	set	set	NOUN
cana-3855	21	6	of	of	ADP
cana-3855	21	7	all	all	DET
cana-3855	21	8	continuous	continuous	ADJ
cana-3855	21	9	linear	linear	ADJ
cana-3855	21	10	functional	functional	ADJ
cana-3855	21	11	on	on	ADP
cana-3855	21	12	m.	m.	NOUN
cana-3855	21	13	throughout	throughout	ADP
cana-3855	21	14	this	this	DET
cana-3855	21	15	work	work	NOUN
cana-3855	21	16	,	,	PUNCT
cana-3855	21	17	we	we	PRON
cana-3855	21	18	have	have	AUX
cana-3855	21	19	used	use	VERB
cana-3855	21	20	an	an	DET
cana-3855	21	21	infinite	infinite	ADJ
cana-3855	21	22	matrix	matrix	NOUN
cana-3855	21	23	d	d	X
cana-3855	21	24	whose	whose	DET
cana-3855	21	25	scalar	scalar	ADJ
cana-3855	21	26	entries	entry	NOUN
cana-3855	21	27	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	21	28	are	be	AUX
cana-3855	21	29	in	in	ADP
cana-3855	21	30	𝑚.	𝑚.	NOUN
cana-3855	21	31	consider	consider	VERB
cana-3855	21	32	a	a	DET
cana-3855	21	33	sequence	sequence	NOUN
cana-3855	21	34	𝑥	𝑥	NOUN
cana-3855	21	35	in	in	ADP
cana-3855	21	36	𝑚.	𝑚.	NOUN
cana-3855	21	37	let	let	VERB
cana-3855	21	38	𝐷𝑥	𝐷𝑥	PROPN
cana-3855	21	39	be	be	AUX
cana-3855	21	40	the	the	DET
cana-3855	21	41	transformed	transform	VERB
cana-3855	21	42	sequence	sequence	NOUN
cana-3855	21	43	whose	whose	DET
cana-3855	21	44	general	general	ADJ
cana-3855	21	45	term	term	NOUN
cana-3855	21	46	is	be	AUX
cana-3855	21	47	written	write	VERB
cana-3855	21	48	as	as	ADP
cana-3855	21	49	(	(	PUNCT
cana-3855	21	50	𝐷𝑥)𝑛	𝐷𝑥)𝑛	PROPN
cana-3855	21	51	=	=	SYM
cana-3855	21	52	∑	∑	PUNCT
cana-3855	21	53	𝑑𝑛𝑘𝑥𝑘	𝑑𝑛𝑘𝑥𝑘	VERB
cana-3855	21	54	∞	∞	NUM
cana-3855	21	55	𝑘=0	𝑘=0	PROPN
cana-3855	22	1	and	and	CCONJ
cana-3855	22	2	it	it	PRON
cana-3855	22	3	is	be	AUX
cana-3855	22	4	convergent	convergent	ADJ
cana-3855	22	5	for	for	ADP
cana-3855	22	6	each	each	DET
cana-3855	22	7	𝑛	𝑛	PRON
cana-3855	22	8	≥	≥	NOUN
cana-3855	22	9	0	0	NUM
cana-3855	22	10	.	.	PUNCT
cana-3855	23	1	communications	communication	NOUN
cana-3855	23	2	on	on	ADP
cana-3855	23	3	applied	apply	VERB
cana-3855	23	4	nonlinear	nonlinear	ADJ
cana-3855	23	5	analysis	analysis	NOUN
cana-3855	23	6	issn	issn	NOUN
cana-3855	23	7	:	:	PUNCT
cana-3855	23	8	1074	1074	NUM
cana-3855	23	9	-	-	PUNCT
cana-3855	23	10	133x	133x	NUM
cana-3855	23	11	vol	vol	NOUN
cana-3855	23	12	32	32	NUM
cana-3855	23	13	no	no	NOUN
cana-3855	23	14	.	.	PUNCT
cana-3855	24	1	9s	9s	NUM
cana-3855	24	2	(	(	PUNCT
cana-3855	24	3	2025	2025	NUM
cana-3855	24	4	)	)	PUNCT
cana-3855	24	5	278	278	NUM
cana-3855	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	24	7	let	let	VERB
cana-3855	24	8	us	we	PRON
cana-3855	24	9	recall	recall	VERB
cana-3855	24	10	the	the	DET
cana-3855	24	11	definition	definition	NOUN
cana-3855	24	12	of	of	ADP
cana-3855	24	13	conservative	conservative	ADJ
cana-3855	24	14	matrix	matrix	NOUN
cana-3855	24	15	(	(	PUNCT
cana-3855	24	16	due	due	ADP
cana-3855	24	17	to	to	ADP
cana-3855	24	18	stieglitz	stieglitz	PROPN
cana-3855	25	1	[	[	X
cana-3855	25	2	8	8	NUM
cana-3855	25	3	]	]	NUM
cana-3855	25	4	)	)	PUNCT
cana-3855	25	5	.	.	PUNCT
cana-3855	26	1	definition	definition	NOUN
cana-3855	26	2	1	1	NUM
cana-3855	26	3	:	:	PUNCT
cana-3855	26	4	a	a	DET
cana-3855	26	5	matrix	matrix	NOUN
cana-3855	26	6	𝐷	𝐷	NOUN
cana-3855	26	7	=	=	SYM
cana-3855	26	8	(	(	PUNCT
cana-3855	26	9	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	26	10	)	)	PUNCT
cana-3855	26	11	is	be	AUX
cana-3855	26	12	said	say	VERB
cana-3855	26	13	to	to	PART
cana-3855	26	14	be	be	AUX
cana-3855	26	15	conservative	conservative	ADJ
cana-3855	26	16	matrix	matrix	NOUN
cana-3855	26	17	iff	iff	NOUN
cana-3855	26	18	it	it	PRON
cana-3855	26	19	satisfies	satisfy	VERB
cana-3855	26	20	the	the	DET
cana-3855	26	21	following	follow	VERB
cana-3855	26	22	properties	property	NOUN
cana-3855	26	23	:	:	PUNCT
cana-3855	26	24	(	(	PUNCT
cana-3855	26	25	a	a	X
cana-3855	26	26	)	)	PUNCT
cana-3855	26	27	‖𝐷‖	‖𝐷‖	PRON
cana-3855	26	28	<	<	X
cana-3855	26	29	∞	∞	X
cana-3855	26	30	i.e.	i.e.	X
cana-3855	26	31	,	,	PUNCT
cana-3855	26	32	sup	sup	NOUN
cana-3855	26	33	𝑛	𝑛	DET
cana-3855	26	34	∑	∑	ADV
cana-3855	26	35	|𝑑𝑛𝑘|	|𝑑𝑛𝑘|	PROPN
cana-3855	26	36	<	<	X
cana-3855	26	37	∞.∞	∞.∞	PROPN
cana-3855	26	38	𝑘=0	𝑘=0	PROPN
cana-3855	26	39	(	(	PUNCT
cana-3855	26	40	b	b	NOUN
cana-3855	26	41	)	)	PUNCT
cana-3855	26	42	lim	lim	NOUN
cana-3855	26	43	𝑛	𝑛	PRON
cana-3855	26	44	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	26	45	=	=	PUNCT
cana-3855	26	46	𝑑𝑘	𝑑𝑘	ADP
cana-3855	26	47	for	for	ADP
cana-3855	26	48	every	every	DET
cana-3855	26	49	fixed	fix	VERB
cana-3855	26	50	k.	k.	NOUN
cana-3855	26	51	(	(	PUNCT
cana-3855	26	52	c	c	X
cana-3855	26	53	)	)	PUNCT
cana-3855	26	54	lim	lim	NOUN
cana-3855	26	55	𝑛	𝑛	PRON
cana-3855	26	56	∑	∑	PUNCT
cana-3855	26	57	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	26	58	=	=	PUNCT
cana-3855	26	59	𝑑	𝑑	PROPN
cana-3855	26	60	∞	∞	NUM
cana-3855	26	61	𝑘=0	𝑘=0	PROPN
cana-3855	26	62	.	.	PUNCT
cana-3855	27	1	if	if	SCONJ
cana-3855	27	2	𝑑𝑘	𝑑𝑘	ADV
cana-3855	27	3	=	=	SYM
cana-3855	27	4	0	0	NUM
cana-3855	27	5	and	and	CCONJ
cana-3855	27	6	𝑑	𝑑	PROPN
cana-3855	27	7	=	=	ADJ
cana-3855	27	8	1	1	NUM
cana-3855	27	9	,	,	PUNCT
cana-3855	27	10	then	then	ADV
cana-3855	27	11	𝐷	𝐷	PROPN
cana-3855	27	12	is	be	AUX
cana-3855	27	13	called	call	VERB
cana-3855	27	14	regular	regular	ADJ
cana-3855	27	15	(	(	PUNCT
cana-3855	27	16	see	see	VERB
cana-3855	27	17	[	[	X
cana-3855	27	18	9	9	NUM
cana-3855	27	19	]	]	SYM
cana-3855	27	20	p.64	p.64	NOUN
cana-3855	27	21	)	)	PUNCT
cana-3855	27	22	.	.	PUNCT
cana-3855	28	1	let	let	AUX
cana-3855	28	2	𝐷	𝐷	NOUN
cana-3855	28	3	=	=	SYM
cana-3855	28	4	(	(	PUNCT
cana-3855	28	5	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	28	6	)	)	PUNCT
cana-3855	28	7	be	be	AUX
cana-3855	28	8	the	the	DET
cana-3855	28	9	a	a	DET
cana-3855	28	10	fixed	fix	VERB
cana-3855	28	11	regular	regular	ADJ
cana-3855	28	12	matrix	matrix	NOUN
cana-3855	28	13	with	with	ADP
cana-3855	28	14	‖𝐷‖	‖𝐷‖	PROPN
cana-3855	28	15	<	<	X
cana-3855	28	16	1	1	NUM
cana-3855	28	17	(	(	PUNCT
cana-3855	28	18	this	this	PRON
cana-3855	28	19	is	be	AUX
cana-3855	28	20	assumed	assume	VERB
cana-3855	28	21	throughout	throughout	ADP
cana-3855	28	22	the	the	DET
cana-3855	28	23	paper	paper	NOUN
cana-3855	28	24	)	)	PUNCT
cana-3855	28	25	.	.	PUNCT
cana-3855	29	1	definition	definition	NOUN
cana-3855	29	2	2	2	NUM
cana-3855	29	3	:	:	PUNCT
cana-3855	29	4	a	a	DET
cana-3855	29	5	linear	linear	ADJ
cana-3855	29	6	functional	functional	ADJ
cana-3855	29	7	𝜓	𝜓	PROPN
cana-3855	29	8	∈	∈	PROPN
cana-3855	29	9	𝑚∗	𝑚∗	PROPN
cana-3855	29	10	is	be	AUX
cana-3855	29	11	said	say	VERB
cana-3855	29	12	to	to	PART
cana-3855	29	13	be	be	AUX
cana-3855	29	14	an	an	DET
cana-3855	29	15	𝐷-mean	𝐷-mean	PROPN
cana-3855	29	16	or	or	CCONJ
cana-3855	29	17	𝐷-limit	𝐷-limit	PROPN
cana-3855	29	18	if	if	SCONJ
cana-3855	29	19	and	and	CCONJ
cana-3855	29	20	only	only	ADV
cana-3855	29	21	if	if	SCONJ
cana-3855	29	22	the	the	DET
cana-3855	29	23	following	follow	VERB
cana-3855	29	24	properties	property	NOUN
cana-3855	29	25	hold	hold	VERB
cana-3855	29	26	good	good	ADJ
cana-3855	29	27	:	:	PUNCT
cana-3855	29	28	(	(	PUNCT
cana-3855	29	29	i	i	NOUN
cana-3855	29	30	)	)	PUNCT
cana-3855	29	31	for	for	ADP
cana-3855	29	32	𝑥	𝑥	NOUN
cana-3855	29	33	=	=	PRON
cana-3855	29	34	{	{	PUNCT
cana-3855	29	35	𝑥𝑛	𝑥𝑛	NOUN
cana-3855	29	36	}	}	PUNCT
cana-3855	29	37	,	,	PUNCT
cana-3855	29	38	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	29	39	)	)	PUNCT
cana-3855	29	40	≥	≥	NOUN
cana-3855	29	41	0	0	PUNCT
cana-3855	30	1	if	if	SCONJ
cana-3855	30	2	𝑥𝑛	𝑥𝑛	PROPN
cana-3855	30	3	≥	≥	VERB
cana-3855	30	4	0	0	NUM
cana-3855	30	5	for	for	ADP
cana-3855	30	6	all	all	DET
cana-3855	30	7	𝑛.	𝑛.	NOUN
cana-3855	30	8	(	(	PUNCT
cana-3855	30	9	ii	ii	NOUN
cana-3855	30	10	)	)	PUNCT
cana-3855	30	11	𝜓(𝑒	𝜓(𝑒	PROPN
cana-3855	30	12	)	)	PUNCT
cana-3855	30	13	=	=	SYM
cana-3855	30	14	1	1	NUM
cana-3855	30	15	for	for	ADP
cana-3855	30	16	𝑒	𝑒	PROPN
cana-3855	30	17	=	=	SYM
cana-3855	30	18	(	(	PUNCT
cana-3855	30	19	1,1,1	1,1,1	NUM
cana-3855	30	20	,	,	PUNCT
cana-3855	30	21	…	…	PUNCT
cana-3855	30	22	.	.	PUNCT
cana-3855	30	23	)	)	PUNCT
cana-3855	30	24	.	.	PUNCT
cana-3855	31	1	(	(	PUNCT
cana-3855	31	2	iii	iii	X
cana-3855	31	3	)	)	PUNCT
cana-3855	31	4	𝜓(𝐷𝑥	𝜓(𝐷𝑥	NOUN
cana-3855	31	5	)	)	PUNCT
cana-3855	31	6	=	=	PUNCT
cana-3855	32	1	𝜓(𝑥	𝜓(𝑥	X
cana-3855	32	2	)	)	PUNCT
cana-3855	32	3	for	for	SCONJ
cana-3855	32	4	all	all	PRON
cana-3855	32	5	𝑥	𝑥	PRON
cana-3855	32	6	𝜖	𝜖	X
cana-3855	32	7	𝑚	𝑚	X
cana-3855	32	8	.	.	PUNCT
cana-3855	33	1	let	let	VERB
cana-3855	33	2	𝐷	𝐷	NOUN
cana-3855	33	3	=	=	NOUN
cana-3855	33	4	𝐵	𝐵	NOUN
cana-3855	33	5	be	be	VERB
cana-3855	33	6	the	the	DET
cana-3855	33	7	translation	translation	NOUN
cana-3855	33	8	matrix	matrix	NOUN
cana-3855	33	9	i.e.	i.e.	X
cana-3855	33	10	(	(	PUNCT
cana-3855	33	11	𝐵𝑥)𝑛	𝐵𝑥)𝑛	NOUN
cana-3855	33	12	=	=	SYM
cana-3855	34	1	𝑥𝑛+1	𝑥𝑛+1	X
cana-3855	35	1	then	then	ADV
cana-3855	35	2	it	it	PRON
cana-3855	35	3	is	be	AUX
cana-3855	35	4	called	call	VERB
cana-3855	35	5	a	a	DET
cana-3855	35	6	𝐵	𝐵	NOUN
cana-3855	35	7	-limit	-limit	NOUN
cana-3855	35	8	and	and	CCONJ
cana-3855	35	9	is	be	AUX
cana-3855	35	10	often	often	ADV
cana-3855	35	11	called	call	VERB
cana-3855	35	12	as	as	ADP
cana-3855	35	13	a	a	DET
cana-3855	35	14	banach	banach	NOUN
cana-3855	35	15	limit	limit	NOUN
cana-3855	35	16	[	[	X
cana-3855	35	17	1	1	NUM
cana-3855	35	18	]	]	PUNCT
cana-3855	35	19	.	.	PUNCT
cana-3855	36	1	some	some	DET
cana-3855	36	2	inequalities	inequality	NOUN
cana-3855	36	3	are	be	AUX
cana-3855	36	4	shown	show	VERB
cana-3855	36	5	in	in	ADP
cana-3855	36	6	[	[	X
cana-3855	36	7	6	6	NUM
cana-3855	36	8	]	]	PUNCT
cana-3855	36	9	between	between	ADP
cana-3855	36	10	sublinear	sublinear	NOUN
cana-3855	36	11	functionals	functional	NOUN
cana-3855	36	12	emerging	emerge	VERB
cana-3855	36	13	from	from	ADP
cana-3855	36	14	banach	banach	NOUN
cana-3855	36	15	limits	limit	NOUN
cana-3855	36	16	of	of	ADP
cana-3855	36	17	some	some	DET
cana-3855	36	18	sequences	sequence	NOUN
cana-3855	36	19	and	and	CCONJ
cana-3855	36	20	their	their	PRON
cana-3855	36	21	matrix	matrix	NOUN
cana-3855	36	22	transformations	transformation	NOUN
cana-3855	36	23	using	use	VERB
cana-3855	36	24	conservative	conservative	ADJ
cana-3855	36	25	and	and	CCONJ
cana-3855	36	26	regular	regular	ADJ
cana-3855	36	27	matrices	matrix	NOUN
cana-3855	36	28	.	.	PUNCT
cana-3855	37	1	definition	definition	NOUN
cana-3855	37	2	3	3	NUM
cana-3855	37	3	:	:	PUNCT
cana-3855	37	4	a	a	DET
cana-3855	37	5	matrix	matrix	NOUN
cana-3855	37	6	𝐶	𝐶	PROPN
cana-3855	37	7	∈	∈	PROPN
cana-3855	37	8	𝑚	𝑚	NOUN
cana-3855	37	9	is	be	AUX
cana-3855	37	10	called	call	VERB
cana-3855	37	11	a	a	DET
cana-3855	37	12	𝐷-invariant	𝐷-invariant	ADJ
cana-3855	37	13	matrix	matrix	NOUN
cana-3855	37	14	if	if	SCONJ
cana-3855	37	15	𝐶(𝐷	𝐶(𝐷	PROPN
cana-3855	37	16	−	−	PROPN
cana-3855	37	17	𝐼	𝐼	PROPN
cana-3855	37	18	)	)	PUNCT
cana-3855	37	19	behaves	behave	NOUN
cana-3855	37	20	as	as	ADP
cana-3855	37	21	a	a	DET
cana-3855	37	22	zero	zero	NUM
cana-3855	37	23	map	map	NOUN
cana-3855	37	24	and	and	CCONJ
cana-3855	37	25	maps	map	VERB
cana-3855	37	26	every	every	DET
cana-3855	37	27	bounded	bounded	ADJ
cana-3855	37	28	sequence	sequence	NOUN
cana-3855	37	29	to	to	ADP
cana-3855	37	30	null	null	ADJ
cana-3855	37	31	sequence	sequence	NOUN
cana-3855	37	32	.	.	PUNCT
cana-3855	38	1	(	(	PUNCT
cana-3855	38	2	i	i	PRON
cana-3855	38	3	is	be	AUX
cana-3855	38	4	the	the	DET
cana-3855	38	5	identity	identity	NOUN
cana-3855	38	6	matrix	matrix	NOUN
cana-3855	38	7	.	.	PUNCT
cana-3855	38	8	)	)	PUNCT
cana-3855	39	1	the	the	DET
cana-3855	39	2	concept	concept	NOUN
cana-3855	39	3	of	of	ADP
cana-3855	39	4	𝐷-limits	𝐷-limits	PROPN
cana-3855	39	5	were	be	AUX
cana-3855	39	6	first	first	ADV
cana-3855	39	7	introduced	introduce	VERB
cana-3855	39	8	by	by	ADP
cana-3855	39	9	bell[2	bell[2	NOUN
cana-3855	39	10	]	]	PUNCT
cana-3855	39	11	,	,	PUNCT
cana-3855	39	12	where	where	SCONJ
cana-3855	39	13	he	he	PRON
cana-3855	39	14	assumed	assume	VERB
cana-3855	39	15	that	that	SCONJ
cana-3855	39	16	𝐷	𝐷	PROPN
cana-3855	39	17	is	be	AUX
cana-3855	39	18	a	a	DET
cana-3855	39	19	positive	positive	ADJ
cana-3855	39	20	matrix	matrix	NOUN
cana-3855	39	21	such	such	ADJ
cana-3855	39	22	that	that	SCONJ
cana-3855	39	23	‖𝐷‖	‖𝐷‖	PRON
cana-3855	39	24	=	=	NOUN
cana-3855	40	1	1	1	X
cana-3855	40	2	.	.	X
cana-3855	40	3	from	from	ADP
cana-3855	40	4	definition	definition	NOUN
cana-3855	40	5	of	of	ADP
cana-3855	40	6	𝐷-limit	𝐷-limit	PROPN
cana-3855	40	7	,	,	PUNCT
cana-3855	40	8	it	it	PRON
cana-3855	40	9	follows	follow	VERB
cana-3855	40	10	that	that	SCONJ
cana-3855	40	11	𝑥	𝑥	PROPN
cana-3855	40	12	≤	≤	PRON
cana-3855	40	13	𝑦	𝑦	PRON
cana-3855	40	14	⇒	⇒	NOUN
cana-3855	40	15	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	40	16	)	)	PUNCT
cana-3855	40	17	≤	≤	NOUN
cana-3855	40	18	𝜓(𝑦	𝜓(𝑦	NUM
cana-3855	40	19	)	)	PUNCT
cana-3855	40	20	.	.	PUNCT
cana-3855	41	1	and	and	CCONJ
cana-3855	41	2	from	from	ADP
cana-3855	41	3	(	(	PUNCT
cana-3855	41	4	i	i	NOUN
cana-3855	41	5	)	)	PUNCT
cana-3855	41	6	and	and	CCONJ
cana-3855	41	7	(	(	PUNCT
cana-3855	41	8	ii	ii	NOUN
cana-3855	41	9	)	)	PUNCT
cana-3855	41	10	that	that	SCONJ
cana-3855	41	11	‖𝜓‖	‖𝜓‖	PROPN
cana-3855	41	12	=	=	PUNCT
cana-3855	42	1	1	1	X
cana-3855	42	2	.	.	PUNCT
cana-3855	42	3	here	here	ADV
cana-3855	42	4	the	the	DET
cana-3855	42	5	product	product	NOUN
cana-3855	42	6	of	of	ADP
cana-3855	42	7	𝐷	𝐷	PROPN
cana-3855	42	8	with	with	ADP
cana-3855	42	9	itself	itself	PRON
cana-3855	42	10	𝑝	𝑝	PROPN
cana-3855	42	11	times	time	NOUN
cana-3855	42	12	is	be	AUX
cana-3855	42	13	denoted	denote	VERB
cana-3855	42	14	by	by	ADP
cana-3855	42	15	𝐷𝑝.	𝐷𝑝.	PROPN
cana-3855	42	16	let	let	VERB
cana-3855	42	17	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	42	18	be	be	AUX
cana-3855	42	19	the	the	DET
cana-3855	42	20	set	set	NOUN
cana-3855	42	21	of	of	ADP
cana-3855	42	22	all	all	DET
cana-3855	42	23	bounded	bound	VERB
cana-3855	42	24	sequences	sequence	NOUN
cana-3855	42	25	having	have	VERB
cana-3855	42	26	equal	equal	ADJ
cana-3855	42	27	𝐷-means	𝐷-means	PROPN
cana-3855	42	28	.	.	PUNCT
cana-3855	43	1	so	so	ADV
cana-3855	43	2	,	,	PUNCT
cana-3855	43	3	we	we	PRON
cana-3855	43	4	can	can	AUX
cana-3855	43	5	write	write	VERB
cana-3855	43	6	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	43	7	=	=	PRON
cana-3855	43	8	{	{	PUNCT
cana-3855	43	9	𝑥	𝑥	PUNCT
cana-3855	43	10	∈	∈	PROPN
cana-3855	43	11	𝑚	𝑚	NOUN
cana-3855	43	12	:	:	PUNCT
cana-3855	43	13	lim	lim	PROPN
cana-3855	43	14	𝑝	𝑝	PROPN
cana-3855	43	15	ℎ𝑝𝑛(𝑥	ℎ𝑝𝑛(𝑥	PROPN
cana-3855	43	16	)	)	PUNCT
cana-3855	43	17	=	=	SYM
cana-3855	44	1	𝑙	𝑙	NOUN
cana-3855	45	1	uniformly	uniformly	ADV
cana-3855	45	2	in	in	ADP
cana-3855	45	3	𝑛	𝑛	PROPN
cana-3855	45	4	,	,	PUNCT
cana-3855	45	5	𝑙	𝑙	PROPN
cana-3855	45	6	=	=	SYM
cana-3855	45	7	𝐷	𝐷	PROPN
cana-3855	45	8	−	−	PROPN
cana-3855	45	9	𝑙𝑖𝑚𝑥	𝑙𝑖𝑚𝑥	NOUN
cana-3855	45	10	}	}	PUNCT
cana-3855	45	11	where	where	SCONJ
cana-3855	45	12	for	for	ADP
cana-3855	45	13	𝑝	𝑝	PROPN
cana-3855	45	14	≥	≥	NOUN
cana-3855	45	15	0	0	NUM
cana-3855	45	16	,	,	PUNCT
cana-3855	45	17	𝑛	𝑛	PRON
cana-3855	45	18	>	>	X
cana-3855	45	19	0	0	X
cana-3855	45	20	.	.	PUNCT
cana-3855	46	1	ℎ𝑝𝑛(𝑥	ℎ𝑝𝑛(𝑥	NOUN
cana-3855	46	2	)	)	PUNCT
cana-3855	47	1	=	=	PUNCT
cana-3855	47	2	𝑥𝑛+(𝐷𝑥)𝑛+⋯+(𝐷	𝑥𝑛+(𝐷𝑥)𝑛+⋯+(𝐷	NOUN
cana-3855	47	3	𝑝𝑥)𝑛	𝑝𝑥)𝑛	NOUN
cana-3855	47	4	𝑝+1	𝑝+1	NOUN
cana-3855	47	5	.	.	PUNCT
cana-3855	48	1	(	(	PUNCT
cana-3855	48	2	2.1	2.1	NUM
cana-3855	48	3	)	)	PUNCT
cana-3855	48	4	we	we	PRON
cana-3855	48	5	know	know	VERB
cana-3855	48	6	that	that	SCONJ
cana-3855	48	7	a	a	DET
cana-3855	48	8	sublinear	sublinear	NOUN
cana-3855	48	9	functional	functional	ADJ
cana-3855	48	10	p	p	NOUN
cana-3855	48	11	on	on	ADP
cana-3855	48	12	𝑚	𝑚	X
cana-3855	48	13	generates	generate	NOUN
cana-3855	48	14	𝐷-limit	𝐷-limit	PROPN
cana-3855	48	15	if	if	SCONJ
cana-3855	48	16	𝜓	𝜓	PROPN
cana-3855	48	17	∈	∈	PROPN
cana-3855	48	18	𝑚∗and	𝑚∗and	PUNCT
cana-3855	48	19	𝜓	𝜓	X
cana-3855	48	20	<	<	X
cana-3855	48	21	𝑃	𝑃	NOUN
cana-3855	48	22	⇒	⇒	NOUN
cana-3855	48	23	𝜓	𝜓	NOUN
cana-3855	48	24	is	be	AUX
cana-3855	48	25	a	a	DET
cana-3855	48	26	𝐷-limit	𝐷-limit	PROPN
cana-3855	48	27	.	.	PUNCT
cana-3855	49	1	here	here	ADV
cana-3855	49	2	𝜓	𝜓	X
cana-3855	49	3	<	<	X
cana-3855	49	4	𝑃	𝑃	NOUN
cana-3855	49	5	𝑚𝑒𝑎𝑛𝑠	𝑚𝑒𝑎𝑛𝑠	NOUN
cana-3855	49	6	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	49	7	)	)	PUNCT
cana-3855	49	8	≤	≤	NOUN
cana-3855	49	9	𝑃(𝑥	𝑃(𝑥	NUM
cana-3855	49	10	)	)	PUNCT
cana-3855	49	11	for	for	ADP
cana-3855	49	12	all	all	DET
cana-3855	49	13	𝑥	𝑥	PRON
cana-3855	49	14	∈	∈	PROPN
cana-3855	49	15	𝑚.	𝑚.	NOUN
cana-3855	49	16	p	p	NOUN
cana-3855	49	17	is	be	AUX
cana-3855	49	18	said	say	VERB
cana-3855	49	19	to	to	PART
cana-3855	49	20	dominate	dominate	VERB
cana-3855	49	21	𝐷-limit	𝐷-limit	PROPN
cana-3855	49	22	if	if	SCONJ
cana-3855	49	23	every	every	DET
cana-3855	49	24	𝐷-limit	𝐷-limit	PROPN
cana-3855	49	25	communications	communication	NOUN
cana-3855	49	26	on	on	ADP
cana-3855	49	27	applied	apply	VERB
cana-3855	49	28	nonlinear	nonlinear	ADJ
cana-3855	49	29	analysis	analysis	NOUN
cana-3855	49	30	issn	issn	NOUN
cana-3855	49	31	:	:	PUNCT
cana-3855	49	32	1074	1074	NUM
cana-3855	49	33	-	-	PUNCT
cana-3855	49	34	133x	133x	NUM
cana-3855	49	35	vol	vol	NOUN
cana-3855	49	36	32	32	NUM
cana-3855	49	37	no	no	NOUN
cana-3855	49	38	.	.	PUNCT
cana-3855	50	1	9s	9s	NUM
cana-3855	50	2	(	(	PUNCT
cana-3855	50	3	2025	2025	NUM
cana-3855	50	4	)	)	PUNCT
cana-3855	50	5	279	279	NUM
cana-3855	50	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	50	7	𝜓	𝜓	X
cana-3855	50	8	<	<	X
cana-3855	50	9	𝑃.	𝑃.	PROPN
cana-3855	50	10	i.e.	i.e.	X
cana-3855	50	11	𝜓	𝜓	PROPN
cana-3855	50	12	∈	∈	PROPN
cana-3855	50	13	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	50	14	⇒	⇒	VERB
cana-3855	50	15	𝜓	𝜓	PROPN
cana-3855	50	16	<	<	X
cana-3855	50	17	𝑃	𝑃	PROPN
cana-3855	50	18	,	,	PUNCT
cana-3855	50	19	where	where	SCONJ
cana-3855	50	20	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	50	21	is	be	AUX
cana-3855	50	22	the	the	DET
cana-3855	50	23	set	set	NOUN
cana-3855	50	24	of	of	ADP
cana-3855	50	25	all	all	DET
cana-3855	50	26	𝐷-limits	𝐷-limits	PROPN
cana-3855	50	27	.	.	PUNCT
cana-3855	51	1	it	it	PRON
cana-3855	51	2	is	be	AUX
cana-3855	51	3	provided	provide	VERB
cana-3855	51	4	that	that	SCONJ
cana-3855	51	5	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	51	6	is	be	AUX
cana-3855	51	7	a	a	DET
cana-3855	51	8	closed	closed	ADJ
cana-3855	51	9	convex	convex	NOUN
cana-3855	51	10	set	set	NOUN
cana-3855	51	11	.	.	PUNCT
cana-3855	52	1	it	it	PRON
cana-3855	52	2	is	be	AUX
cana-3855	52	3	proved	prove	VERB
cana-3855	52	4	that	that	SCONJ
cana-3855	52	5	𝜓	𝜓	PROPN
cana-3855	52	6	∈	∈	PROPN
cana-3855	52	7	𝑚∗	𝑚∗	PROPN
cana-3855	52	8	is	be	AUX
cana-3855	52	9	a	a	DET
cana-3855	52	10	𝐷-limit	𝐷-limit	PROPN
cana-3855	52	11	if	if	SCONJ
cana-3855	52	12	and	and	CCONJ
cana-3855	52	13	only	only	ADV
cana-3855	52	14	if	if	SCONJ
cana-3855	52	15	𝜓(𝑥	𝜓(𝑥	NOUN
cana-3855	52	16	)	)	PUNCT
cana-3855	52	17	≤	≤	NOUN
cana-3855	52	18	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	52	19	)	)	PUNCT
cana-3855	52	20	,	,	PUNCT
cana-3855	52	21	where	where	SCONJ
cana-3855	52	22	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	52	23	)	)	PUNCT
cana-3855	52	24	is	be	AUX
cana-3855	52	25	a	a	DET
cana-3855	52	26	sublinear	sublinear	NOUN
cana-3855	52	27	functional	functional	ADJ
cana-3855	52	28	on	on	ADP
cana-3855	52	29	𝑚	𝑚	PROPN
cana-3855	52	30	defined	define	VERB
cana-3855	52	31	by	by	ADP
cana-3855	52	32	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	52	33	)	)	PUNCT
cana-3855	52	34	=	=	VERB
cana-3855	52	35	lim	lim	PROPN
cana-3855	52	36	𝑝	𝑝	PROPN
cana-3855	52	37	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-3855	52	38	sup	sup	NOUN
cana-3855	52	39	𝑛	𝑛	DET
cana-3855	52	40	ℎ𝑝𝑛(𝑥	ℎ𝑝𝑛(𝑥	PROPN
cana-3855	52	41	)	)	PUNCT
cana-3855	52	42	.	.	PUNCT
cana-3855	53	1	i.e.	i.e.	X
cana-3855	53	2	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	53	3	)	)	PUNCT
cana-3855	53	4	=	=	VERB
cana-3855	53	5	lim	lim	PROPN
cana-3855	53	6	𝑝	𝑝	PROPN
cana-3855	53	7	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-3855	53	8	sup	sup	NOUN
cana-3855	53	9	𝑛	𝑛	ADV
cana-3855	53	10	1	1	NUM
cana-3855	53	11	𝑝+1	𝑝+1	NOUN
cana-3855	53	12	∑	∑	PUNCT
cana-3855	53	13	(	(	PUNCT
cana-3855	53	14	𝐷𝑘𝑥)𝑛	𝐷𝑘𝑥)𝑛	NOUN
cana-3855	53	15	𝑝	𝑝	PROPN
cana-3855	53	16	𝑘=0	𝑘=0	PROPN
cana-3855	53	17	.	.	PUNCT
cana-3855	54	1	(	(	PUNCT
cana-3855	54	2	2.2	2.2	NUM
cana-3855	54	3	)	)	PUNCT
cana-3855	54	4	we	we	PRON
cana-3855	54	5	now	now	ADV
cana-3855	54	6	take	take	VERB
cana-3855	54	7	the	the	DET
cana-3855	54	8	idea	idea	NOUN
cana-3855	54	9	from	from	ADP
cana-3855	54	10	above	above	ADV
cana-3855	54	11	and	and	CCONJ
cana-3855	54	12	purpose	purpose	VERB
cana-3855	54	13	the	the	DET
cana-3855	54	14	following	follow	VERB
cana-3855	54	15	definition	definition	NOUN
cana-3855	54	16	for	for	ADP
cana-3855	54	17	a	a	DET
cana-3855	54	18	new	new	ADJ
cana-3855	54	19	family	family	NOUN
cana-3855	54	20	of	of	ADP
cana-3855	54	21	functionals	functional	NOUN
cana-3855	54	22	of	of	ADP
cana-3855	54	23	the	the	DET
cana-3855	54	24	kind	kind	NOUN
cana-3855	54	25	of	of	ADP
cana-3855	54	26	𝐷-mean	𝐷-mean	PROPN
cana-3855	54	27	called	call	VERB
cana-3855	54	28	quasi	quasi	PROPN
cana-3855	54	29	𝐷-limit	𝐷-limit	PROPN
cana-3855	54	30	.	.	PUNCT
cana-3855	55	1	definition	definition	NOUN
cana-3855	55	2	4	4	NUM
cana-3855	55	3	:	:	PUNCT
cana-3855	55	4	the	the	DET
cana-3855	55	5	linear	linear	ADJ
cana-3855	55	6	functional	functional	PROPN
cana-3855	55	7	𝜓	𝜓	PROPN
cana-3855	55	8	∈	∈	PROPN
cana-3855	55	9	𝑚∗	𝑚∗	PROPN
cana-3855	55	10	is	be	AUX
cana-3855	55	11	a	a	DET
cana-3855	55	12	quasi	quasi	NOUN
cana-3855	55	13	𝐷-mean	𝐷-mean	ADJ
cana-3855	55	14	or	or	CCONJ
cana-3855	55	15	quasi	quasi	NOUN
cana-3855	55	16	𝐷-limit	𝐷-limit	PROPN
cana-3855	55	17	or	or	CCONJ
cana-3855	55	18	quasi	quasi	ADJ
cana-3855	55	19	matrix	matrix	NOUN
cana-3855	55	20	invariant	invariant	ADJ
cana-3855	55	21	limit	limit	NOUN
cana-3855	55	22	if	if	SCONJ
cana-3855	55	23	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	55	24	)	)	PUNCT
cana-3855	55	25	≤	≤	NUM
cana-3855	55	26	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	55	27	)	)	PUNCT
cana-3855	55	28	,	,	PUNCT
cana-3855	56	1	𝑥	𝑥	PROPN
cana-3855	56	2	∈	∈	PROPN
cana-3855	56	3	𝑚	𝑚	NOUN
cana-3855	56	4	,	,	PUNCT
cana-3855	56	5	where	where	SCONJ
cana-3855	56	6	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	56	7	)	)	PUNCT
cana-3855	56	8	is	be	AUX
cana-3855	56	9	a	a	DET
cana-3855	56	10	sublinear	sublinear	NOUN
cana-3855	56	11	functional	functional	ADJ
cana-3855	56	12	on	on	ADP
cana-3855	56	13	𝑚	𝑚	PROPN
cana-3855	56	14	defined	define	VERB
cana-3855	56	15	by	by	ADP
cana-3855	56	16	.	.	PUNCT
cana-3855	57	1	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	57	2	)	)	PUNCT
cana-3855	58	1	=	=	VERB
cana-3855	58	2	lim	lim	PROPN
cana-3855	58	3	𝑝	𝑝	PROPN
cana-3855	58	4	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-3855	58	5	sup	sup	NOUN
cana-3855	58	6	𝑛	𝑛	ADV
cana-3855	58	7	1	1	NUM
cana-3855	58	8	𝑝+1	𝑝+1	NOUN
cana-3855	58	9	∑	∑	PUNCT
cana-3855	58	10	(	(	PUNCT
cana-3855	58	11	𝐷𝑘𝑥)𝑛𝑝	𝐷𝑘𝑥)𝑛𝑝	PROPN
cana-3855	58	12	𝑝	𝑝	PROPN
cana-3855	58	13	𝑘=0	𝑘=0	PROPN
cana-3855	58	14	.	.	PUNCT
cana-3855	59	1	(	(	PUNCT
cana-3855	59	2	2.3	2.3	NUM
cana-3855	59	3	)	)	PUNCT
cana-3855	59	4	.	.	PUNCT
cana-3855	60	1	quasi	quasi	PROPN
cana-3855	60	2	𝐷-limit	𝐷-limit	PROPN
cana-3855	60	3	need	need	AUX
cana-3855	60	4	not	not	PART
cana-3855	60	5	exist	exist	VERB
cana-3855	60	6	for	for	ADP
cana-3855	60	7	every	every	DET
cana-3855	60	8	regular	regular	ADJ
cana-3855	60	9	matrix	matrix	NOUN
cana-3855	60	10	.	.	PUNCT
cana-3855	61	1	example	example	NOUN
cana-3855	61	2	1	1	NUM
cana-3855	61	3	:	:	PUNCT
cana-3855	61	4	consider	consider	VERB
cana-3855	61	5	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	61	6	=	=	PUNCT
cana-3855	61	7	{	{	PUNCT
cana-3855	61	8	1	1	NUM
cana-3855	61	9	,	,	PUNCT
cana-3855	61	10	𝑖𝑓	𝑖𝑓	ADP
cana-3855	61	11	𝑘	𝑘	X
cana-3855	61	12	=	=	SYM
cana-3855	61	13	2𝑛	2𝑛	PROPN
cana-3855	61	14	0	0	NUM
cana-3855	61	15	,	,	PUNCT
cana-3855	61	16	𝑖𝑓	𝑖𝑓	ADP
cana-3855	61	17	𝑘	𝑘	X
cana-3855	61	18	>	>	X
cana-3855	61	19	2𝑛	2𝑛	PROPN
cana-3855	61	20	1	1	NUM
cana-3855	61	21	𝑛	𝑛	PROPN
cana-3855	61	22	,	,	PUNCT
cana-3855	61	23	𝑖𝑓	𝑖𝑓	ADP
cana-3855	61	24	𝑘	𝑘	PRON
cana-3855	62	1	𝑖𝑠	𝑖𝑠	INTJ
cana-3855	62	2	𝑒𝑣𝑒𝑛	𝑒𝑣𝑒𝑛	ADJ
cana-3855	62	3	1	1	NUM
cana-3855	62	4	≤	≤	NOUN
cana-3855	62	5	𝑘	𝑘	DET
cana-3855	62	6	≤	≤	NOUN
cana-3855	62	7	2𝑛	2𝑛	NUM
cana-3855	62	8	−1	−1	NOUN
cana-3855	62	9	𝑛	𝑛	PROPN
cana-3855	62	10	,	,	PUNCT
cana-3855	62	11	𝑖𝑓	𝑖𝑓	ADP
cana-3855	62	12	𝑘	𝑘	PRON
cana-3855	62	13	𝑖𝑠	𝑖𝑠	NOUN
cana-3855	62	14	𝑜𝑑𝑑	𝑜𝑑𝑑	PROPN
cana-3855	62	15	,	,	PUNCT
cana-3855	62	16	1	1	NUM
cana-3855	62	17	≤	≤	NUM
cana-3855	62	18	𝑘	𝑘	DET
cana-3855	62	19	≤	≤	NOUN
cana-3855	62	20	2𝑛	2𝑛	NOUN
cana-3855	62	21	here	here	ADV
cana-3855	62	22	𝐷	𝐷	PROPN
cana-3855	62	23	is	be	AUX
cana-3855	62	24	regular	regular	ADJ
cana-3855	62	25	.	.	PUNCT
cana-3855	63	1	so	so	ADV
cana-3855	63	2	,	,	PUNCT
cana-3855	63	3	for	for	ADP
cana-3855	63	4	𝑥	𝑥	NOUN
cana-3855	63	5	=	=	SYM
cana-3855	63	6	{	{	PUNCT
cana-3855	63	7	1,0,1,0	1,0,1,0	PROPN
cana-3855	63	8	,	,	PUNCT
cana-3855	63	9	…	…	PUNCT
cana-3855	63	10	.	.	PUNCT
cana-3855	63	11	.	.	PUNCT
cana-3855	64	1	}	}	PUNCT
cana-3855	64	2	,	,	PUNCT
cana-3855	64	3	𝐷𝑥	𝐷𝑥	PROPN
cana-3855	64	4	=	=	PRON
cana-3855	64	5	{	{	PUNCT
cana-3855	64	6	−1,−1,−1,−1	−1,−1,−1,−1	NOUN
cana-3855	64	7	,	,	PUNCT
cana-3855	64	8	…	…	PUNCT
cana-3855	64	9	.	.	PUNCT
cana-3855	64	10	.	.	PUNCT
cana-3855	64	11	}	}	PUNCT
cana-3855	64	12	.	.	PUNCT
cana-3855	65	1	if	if	SCONJ
cana-3855	65	2	𝜓	𝜓	PROPN
cana-3855	65	3	is	be	AUX
cana-3855	65	4	a	a	DET
cana-3855	65	5	quasi	quasi	ADJ
cana-3855	65	6	𝐷-limit	𝐷-limit	PROPN
cana-3855	65	7	then	then	ADV
cana-3855	65	8	0	0	NUM
cana-3855	65	9	≤	≤	NOUN
cana-3855	65	10	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	65	11	)	)	PUNCT
cana-3855	65	12	=	=	SYM
cana-3855	65	13	𝜓(𝐷𝑥	𝜓(𝐷𝑥	X
cana-3855	65	14	)	)	PUNCT
cana-3855	65	15	=	=	SYM
cana-3855	65	16	−1	−1	NOUN
cana-3855	65	17	which	which	PRON
cana-3855	65	18	is	be	AUX
cana-3855	65	19	impossible	impossible	ADJ
cana-3855	65	20	.	.	PUNCT
cana-3855	66	1	so	so	ADV
cana-3855	66	2	quasi	quasi	ADJ
cana-3855	66	3	𝐷-limit	𝐷-limit	PROPN
cana-3855	66	4	does	do	AUX
cana-3855	66	5	not	not	PART
cana-3855	66	6	exist	exist	VERB
cana-3855	66	7	.	.	PUNCT
cana-3855	67	1	we	we	PRON
cana-3855	67	2	now	now	ADV
cana-3855	67	3	propose	propose	VERB
cana-3855	67	4	the	the	DET
cana-3855	67	5	following	follow	VERB
cana-3855	67	6	definitions	definition	NOUN
cana-3855	67	7	for	for	ADP
cana-3855	67	8	new	new	ADJ
cana-3855	67	9	kind	kind	NOUN
cana-3855	67	10	of	of	ADP
cana-3855	67	11	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	67	12	sequences	sequence	NOUN
cana-3855	67	13	,	,	PUNCT
cana-3855	67	14	called	call	VERB
cana-3855	67	15	as	as	ADP
cana-3855	67	16	quasi	quasi	NOUN
cana-3855	67	17	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	67	18	sequence	sequence	NOUN
cana-3855	67	19	.	.	PUNCT
cana-3855	68	1	for	for	ADP
cana-3855	68	2	this	this	PRON
cana-3855	68	3	we	we	PRON
cana-3855	68	4	define	define	VERB
cana-3855	68	5	it	it	PRON
cana-3855	68	6	as	as	SCONJ
cana-3855	68	7	follows	follow	VERB
cana-3855	68	8	definition	definition	NOUN
cana-3855	68	9	5	5	NUM
cana-3855	68	10	:	:	PUNCT
cana-3855	68	11	a	a	DET
cana-3855	68	12	sequence	sequence	NOUN
cana-3855	68	13	𝑥	𝑥	ADP
cana-3855	68	14	∈	∈	NOUN
cana-3855	68	15	𝑚	𝑚	NOUN
cana-3855	68	16	is	be	AUX
cana-3855	68	17	called	call	VERB
cana-3855	68	18	a	a	DET
cana-3855	68	19	quasi	quasi	NOUN
cana-3855	68	20	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	68	21	sequence	sequence	NOUN
cana-3855	68	22	if	if	SCONJ
cana-3855	68	23	−𝑟(−𝑥	−𝑟(−𝑥	VERB
cana-3855	68	24	)	)	PUNCT
cana-3855	68	25	=	=	SYM
cana-3855	68	26	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	68	27	)	)	PUNCT
cana-3855	68	28	.	.	PUNCT
cana-3855	69	1	the	the	DET
cana-3855	69	2	following	follow	VERB
cana-3855	69	3	theorem	theorem	NOUN
cana-3855	69	4	is	be	AUX
cana-3855	69	5	known	know	VERB
cana-3855	69	6	.	.	PUNCT
cana-3855	70	1	theorem	theorem	VERB
cana-3855	70	2	a	a	PRON
cana-3855	70	3	[	[	X
cana-3855	70	4	10	10	NUM
cana-3855	70	5	]	]	X
cana-3855	70	6	:	:	PUNCT
cana-3855	70	7	for	for	SCONJ
cana-3855	70	8	all	all	DET
cana-3855	70	9	𝑥	𝑥	DET
cana-3855	70	10	∈	∈	NOUN
cana-3855	70	11	𝑚	𝑚	NOUN
cana-3855	70	12	,	,	PUNCT
cana-3855	70	13	the	the	DET
cana-3855	70	14	sublinear	sublinear	ADJ
cana-3855	70	15	functional	functional	ADJ
cana-3855	70	16	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	70	17	)	)	PUNCT
cana-3855	70	18	both	both	PRON
cana-3855	70	19	generates	generate	VERB
cana-3855	70	20	and	and	CCONJ
cana-3855	70	21	dominates	dominate	VERB
cana-3855	70	22	𝐷-limit	𝐷-limit	PROPN
cana-3855	70	23	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	70	24	)	)	PUNCT
cana-3855	70	25	if	if	SCONJ
cana-3855	70	26	and	and	CCONJ
cana-3855	70	27	only	only	ADV
cana-3855	70	28	if	if	SCONJ
cana-3855	70	29	𝜓(𝑥	𝜓(𝑥	NOUN
cana-3855	70	30	)	)	PUNCT
cana-3855	70	31	≤	≤	NOUN
cana-3855	70	32	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	70	33	)	)	PUNCT
cana-3855	70	34	.	.	PUNCT
cana-3855	71	1	applying	apply	VERB
cana-3855	71	2	the	the	DET
cana-3855	71	3	above	above	ADJ
cana-3855	71	4	theorem	theorem	NOUN
cana-3855	71	5	,	,	PUNCT
cana-3855	71	6	we	we	PRON
cana-3855	71	7	will	will	AUX
cana-3855	71	8	prove	prove	VERB
cana-3855	71	9	some	some	DET
cana-3855	71	10	new	new	ADJ
cana-3855	71	11	results	result	NOUN
cana-3855	71	12	for	for	ADP
cana-3855	71	13	the	the	DET
cana-3855	71	14	sublinear	sublinear	ADJ
cana-3855	71	15	functional	functional	ADJ
cana-3855	71	16	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	71	17	)	)	PUNCT
cana-3855	71	18	,	,	PUNCT
cana-3855	71	19	where	where	SCONJ
cana-3855	71	20	these	these	PRON
cana-3855	71	21	are	be	AUX
cana-3855	71	22	based	base	VERB
cana-3855	71	23	on	on	ADP
cana-3855	71	24	the	the	DET
cana-3855	71	25	idea	idea	NOUN
cana-3855	71	26	of	of	ADP
cana-3855	71	27	quasi	quasi	ADJ
cana-3855	71	28	banach	banach	NOUN
cana-3855	71	29	limit	limit	NOUN
cana-3855	71	30	defined	define	VERB
cana-3855	71	31	in	in	ADP
cana-3855	71	32	[	[	X
cana-3855	71	33	3	3	NUM
cana-3855	71	34	]	]	PUNCT
cana-3855	71	35	,	,	PUNCT
cana-3855	71	36	quasi	quasi	PROPN
cana-3855	71	37	invariant	invariant	PROPN
cana-3855	71	38	limits	limit	NOUN
cana-3855	71	39	and	and	CCONJ
cana-3855	71	40	their	their	PRON
cana-3855	71	41	convergence	convergence	NOUN
cana-3855	71	42	suggested	suggest	VERB
cana-3855	71	43	by	by	ADP
cana-3855	71	44	mishra[5	mishra[5	PROPN
cana-3855	71	45	]	]	PUNCT
cana-3855	71	46	and	and	CCONJ
cana-3855	71	47	nuray[7	nuray[7	NOUN
cana-3855	71	48	]	]	X
cana-3855	71	49	.	.	PUNCT
cana-3855	72	1	3	3	X
cana-3855	72	2	.	.	X
cana-3855	72	3	main	main	ADJ
cana-3855	72	4	results	result	NOUN
cana-3855	72	5	:	:	PUNCT
cana-3855	72	6	communications	communication	NOUN
cana-3855	72	7	on	on	ADP
cana-3855	72	8	applied	apply	VERB
cana-3855	72	9	nonlinear	nonlinear	ADJ
cana-3855	72	10	analysis	analysis	NOUN
cana-3855	72	11	issn	issn	NOUN
cana-3855	72	12	:	:	PUNCT
cana-3855	72	13	1074	1074	NUM
cana-3855	72	14	-	-	PUNCT
cana-3855	72	15	133x	133x	NUM
cana-3855	72	16	vol	vol	NOUN
cana-3855	72	17	32	32	NUM
cana-3855	72	18	no	no	NOUN
cana-3855	72	19	.	.	PUNCT
cana-3855	73	1	9s	9s	NUM
cana-3855	73	2	(	(	PUNCT
cana-3855	73	3	2025	2025	NUM
cana-3855	73	4	)	)	PUNCT
cana-3855	73	5	280	280	NUM
cana-3855	73	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	73	7	now	now	ADV
cana-3855	73	8	we	we	PRON
cana-3855	73	9	are	be	AUX
cana-3855	73	10	going	go	VERB
cana-3855	73	11	to	to	PART
cana-3855	73	12	prove	prove	VERB
cana-3855	73	13	a	a	DET
cana-3855	73	14	theorem	theorem	NOUN
cana-3855	73	15	on	on	ADP
cana-3855	73	16	existence	existence	NOUN
cana-3855	73	17	o	o	PROPN
cana-3855	73	18	𝐷-limit	𝐷-limit	PROPN
cana-3855	73	19	and	and	CCONJ
cana-3855	73	20	then	then	ADV
cana-3855	73	21	establish	establish	VERB
cana-3855	73	22	some	some	DET
cana-3855	73	23	inequalities	inequality	NOUN
cana-3855	73	24	on	on	ADP
cana-3855	73	25	quasi	quasi	ADJ
cana-3855	73	26	𝐷-limit	𝐷-limit	PROPN
cana-3855	73	27	.	.	PUNCT
cana-3855	74	1	theorem	theorem	NOUN
cana-3855	74	2	1	1	NUM
cana-3855	74	3	:	:	PUNCT
cana-3855	74	4	(	(	PUNCT
cana-3855	74	5	existence	existence	NOUN
cana-3855	74	6	theorem	theorem	VERB
cana-3855	74	7	)	)	PUNCT
cana-3855	74	8	if	if	SCONJ
cana-3855	74	9	there	there	PRON
cana-3855	74	10	is	be	VERB
cana-3855	74	11	a	a	DET
cana-3855	74	12	nonnegative	nonnegative	ADJ
cana-3855	74	13	regular	regular	ADJ
cana-3855	74	14	and	and	CCONJ
cana-3855	74	15	𝐷-invariant	𝐷-invariant	ADJ
cana-3855	74	16	matrix	matrix	NOUN
cana-3855	74	17	then	then	ADV
cana-3855	74	18	𝐷-limits	𝐷-limits	PROPN
cana-3855	74	19	exist	exist	VERB
cana-3855	74	20	or	or	CCONJ
cana-3855	74	21	quasi	quasi	NOUN
cana-3855	74	22	𝐷-limits	𝐷-limits	PROPN
cana-3855	74	23	exist	exist	VERB
cana-3855	74	24	.	.	PUNCT
cana-3855	75	1	proof	proof	NOUN
cana-3855	75	2	:	:	PUNCT
cana-3855	75	3	suppose	suppose	VERB
cana-3855	75	4	𝐶	𝐶	PROPN
cana-3855	75	5	is	be	AUX
cana-3855	75	6	such	such	DET
cana-3855	75	7	a	a	DET
cana-3855	75	8	nonnegative	nonnegative	ADJ
cana-3855	75	9	regular	regular	ADJ
cana-3855	75	10	𝐷-invariant	𝐷-invariant	ADJ
cana-3855	75	11	matrix	matrix	NOUN
cana-3855	75	12	.	.	PUNCT
cana-3855	76	1	define	define	VERB
cana-3855	76	2	a	a	DET
cana-3855	76	3	nonnegative	nonnegative	ADJ
cana-3855	76	4	homogeneous	homogeneous	ADJ
cana-3855	76	5	sublinear	sublinear	NOUN
cana-3855	76	6	functional	functional	PROPN
cana-3855	76	7	𝑝	𝑝	NOUN
cana-3855	76	8	on	on	ADP
cana-3855	76	9	𝑚	𝑚	X
cana-3855	76	10	by	by	ADP
cana-3855	76	11	𝑝(𝑥	𝑝(𝑥	NUM
cana-3855	76	12	)	)	PUNCT
cana-3855	76	13	=	=	PUNCT
cana-3855	76	14	limsup𝐶𝑥	limsup𝐶𝑥	PROPN
cana-3855	76	15	.if	.if	PUNCT
cana-3855	77	1	𝑥	𝑥	PRON
cana-3855	77	2	is	be	AUX
cana-3855	77	3	a	a	DET
cana-3855	77	4	convergent	convergent	NOUN
cana-3855	77	5	sequence	sequence	NOUN
cana-3855	77	6	,	,	PUNCT
cana-3855	77	7	then	then	ADV
cana-3855	77	8	by	by	ADP
cana-3855	77	9	regularity	regularity	NOUN
cana-3855	77	10	of	of	ADP
cana-3855	77	11	𝐶	𝐶	PROPN
cana-3855	77	12	we	we	PRON
cana-3855	77	13	get	get	VERB
cana-3855	77	14	lim	lim	PROPN
cana-3855	77	15	𝑥	𝑥	NOUN
cana-3855	77	16	=	=	PUNCT
cana-3855	77	17	𝑝(𝑥	𝑝(𝑥	PROPN
cana-3855	77	18	)	)	PUNCT
cana-3855	77	19	.	.	PUNCT
cana-3855	78	1	therefore	therefore	ADV
cana-3855	78	2	,	,	PUNCT
cana-3855	78	3	extending	extend	VERB
cana-3855	78	4	the	the	DET
cana-3855	78	5	above	above	ADJ
cana-3855	78	6	limit	limit	NOUN
cana-3855	78	7	to	to	ADP
cana-3855	78	8	𝜓	𝜓	NOUN
cana-3855	78	9	we	we	PRON
cana-3855	78	10	have	have	VERB
cana-3855	78	11	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	78	12	)	)	PUNCT
cana-3855	78	13	≤	≤	NOUN
cana-3855	78	14	𝑝(𝑥	𝑝(𝑥	NOUN
cana-3855	78	15	)	)	PUNCT
cana-3855	78	16	for	for	ADP
cana-3855	78	17	all	all	DET
cana-3855	79	1	𝑥	𝑥	DET
cana-3855	79	2	∈	∈	PROPN
cana-3855	79	3	𝑚.	𝑚.	NOUN
cana-3855	79	4	thus	thus	ADV
cana-3855	79	5	,	,	PUNCT
cana-3855	79	6	−𝑝(−𝑥	−𝑝(−𝑥	NUM
cana-3855	79	7	)	)	PUNCT
cana-3855	79	8	≤	≤	NUM
cana-3855	79	9	−	−	NOUN
cana-3855	79	10	𝜓(−𝑥	𝜓(−𝑥	NOUN
cana-3855	79	11	)	)	PUNCT
cana-3855	79	12	=	=	SYM
cana-3855	79	13	𝜓(𝑥	𝜓(𝑥	X
cana-3855	79	14	)	)	PUNCT
cana-3855	79	15	≤	≤	NOUN
cana-3855	79	16	𝑝(𝑥	𝑝(𝑥	NOUN
cana-3855	79	17	)	)	PUNCT
cana-3855	79	18	for	for	ADP
cana-3855	79	19	all	all	DET
cana-3855	79	20	𝑥	𝑥	DET
cana-3855	79	21	∈	∈	PROPN
cana-3855	79	22	𝑚.	𝑚.	NOUN
cana-3855	79	23	since	since	SCONJ
cana-3855	79	24	𝐶	𝐶	PROPN
cana-3855	79	25	is	be	AUX
cana-3855	79	26	nonnegative	nonnegative	ADJ
cana-3855	79	27	,	,	PUNCT
cana-3855	79	28	we	we	PRON
cana-3855	79	29	have	have	VERB
cana-3855	79	30	0	0	NUM
cana-3855	79	31	≤	≤	NUM
cana-3855	79	32	liminf	liminf	INTJ
cana-3855	80	1	𝐶𝑥	𝐶𝑥	PROPN
cana-3855	80	2	≤	≤	ADJ
cana-3855	80	3	𝜓(𝑥).this	𝜓(𝑥).this	PRON
cana-3855	80	4	implies	imply	VERB
cana-3855	80	5	𝜓	𝜓	NOUN
cana-3855	80	6	is	be	AUX
cana-3855	80	7	nonnegative	nonnegative	ADJ
cana-3855	80	8	.	.	PUNCT
cana-3855	81	1	here	here	ADV
cana-3855	81	2	we	we	PRON
cana-3855	81	3	have	have	VERB
cana-3855	81	4	𝐶(𝐷	𝐶(𝐷	PROPN
cana-3855	81	5	−	−	PROPN
cana-3855	81	6	𝐼)𝑥	𝐼)𝑥	VERB
cana-3855	82	1	=	=	PUNCT
cana-3855	83	1	0.therefore	0.therefore	NUM
cana-3855	83	2	−𝑝(−(𝐷	−𝑝(−(𝐷	NOUN
cana-3855	83	3	−	−	PROPN
cana-3855	83	4	𝐼)𝑥	𝐼)𝑥	NOUN
cana-3855	83	5	)	)	PUNCT
cana-3855	83	6	=	=	SYM
cana-3855	83	7	0	0	PUNCT
cana-3855	84	1	=	=	SYM
cana-3855	84	2	𝑝((𝐷	𝑝((𝐷	PROPN
cana-3855	84	3	−	−	PROPN
cana-3855	84	4	𝐼)𝑥	𝐼)𝑥	NOUN
cana-3855	84	5	)	)	PUNCT
cana-3855	84	6	which	which	PRON
cana-3855	84	7	implies	imply	VERB
cana-3855	84	8	𝜓((𝐷	𝜓((𝐷	PROPN
cana-3855	84	9	−	−	PROPN
cana-3855	84	10	𝐼)𝑥	𝐼)𝑥	NOUN
cana-3855	84	11	)	)	PUNCT
cana-3855	84	12	=	=	SYM
cana-3855	84	13	0	0	X
cana-3855	84	14	.	.	PUNCT
cana-3855	85	1	hence	hence	ADV
cana-3855	85	2	,	,	PUNCT
cana-3855	85	3	𝜓	𝜓	PROPN
cana-3855	85	4	is	be	AUX
cana-3855	85	5	a	a	DET
cana-3855	85	6	𝐷-limit	𝐷-limit	PROPN
cana-3855	85	7	.	.	PUNCT
cana-3855	85	8	considering	consider	VERB
cana-3855	85	9	𝑝(𝑥	𝑝(𝑥	NUM
cana-3855	85	10	)	)	PUNCT
cana-3855	85	11	as	as	ADV
cana-3855	85	12	equal	equal	ADJ
cana-3855	85	13	to	to	ADP
cana-3855	85	14	the	the	DET
cana-3855	85	15	sublinear	sublinear	NOUN
cana-3855	85	16	functional	functional	ADJ
cana-3855	85	17	in	in	ADP
cana-3855	85	18	(	(	PUNCT
cana-3855	85	19	2.3),we	2.3),we	NUM
cana-3855	85	20	can	can	AUX
cana-3855	85	21	say	say	VERB
cana-3855	85	22	that	that	SCONJ
cana-3855	85	23	𝜓	𝜓	PROPN
cana-3855	85	24	is	be	AUX
cana-3855	85	25	a	a	DET
cana-3855	85	26	quasi	quasi	ADJ
cana-3855	85	27	𝐷-limit	𝐷-limit	PROPN
cana-3855	85	28	.	.	PUNCT
cana-3855	85	29	example	example	NOUN
cana-3855	86	1	2	2	NUM
cana-3855	86	2	:	:	PUNCT
cana-3855	86	3	cesaro	cesaro	NOUN
cana-3855	86	4	matrix	matrix	NOUN
cana-3855	86	5	is	be	AUX
cana-3855	86	6	a	a	DET
cana-3855	86	7	nonnegative	nonnegative	ADJ
cana-3855	86	8	regular	regular	ADJ
cana-3855	86	9	𝐵	𝐵	NOUN
cana-3855	86	10	-invariant	-invariant	NOUN
cana-3855	86	11	matrix	matrix	NOUN
cana-3855	86	12	.	.	PUNCT
cana-3855	87	1	so	so	ADV
cana-3855	87	2	,	,	PUNCT
cana-3855	87	3	its	its	PRON
cana-3855	87	4	quasi	quasi	NOUN
cana-3855	87	5	banach	banach	NOUN
cana-3855	87	6	limit[3	limit[3	PROPN
cana-3855	87	7	]	]	PUNCT
cana-3855	87	8	exists	exist	VERB
cana-3855	87	9	.	.	PUNCT
cana-3855	88	1	if	if	SCONJ
cana-3855	88	2	we	we	PRON
cana-3855	88	3	define	define	VERB
cana-3855	88	4	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	88	5	=	=	PUNCT
cana-3855	88	6	{	{	PUNCT
cana-3855	88	7	1	1	NUM
cana-3855	88	8	𝑛	𝑛	NOUN
cana-3855	88	9	,	,	PUNCT
cana-3855	88	10	𝑖𝑓	𝑖𝑓	ADP
cana-3855	88	11	1	1	NUM
cana-3855	88	12	≤	≤	NOUN
cana-3855	88	13	𝑘	𝑘	DET
cana-3855	88	14	≤	≤	NUM
cana-3855	88	15	𝑛	𝑛	DET
cana-3855	88	16	0	0	NUM
cana-3855	88	17	,	,	PUNCT
cana-3855	88	18	𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒	𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒	ADJ
cana-3855	88	19	(	(	PUNCT
cana-3855	88	20	3.1	3.1	NUM
cana-3855	88	21	)	)	PUNCT
cana-3855	88	22	then	then	ADV
cana-3855	88	23	the	the	DET
cana-3855	88	24	matrix	matrix	NOUN
cana-3855	88	25	𝐷	𝐷	NOUN
cana-3855	88	26	=	=	SYM
cana-3855	88	27	(	(	PUNCT
cana-3855	88	28	𝑑𝑛𝑘	𝑑𝑛𝑘	NOUN
cana-3855	88	29	)	)	PUNCT
cana-3855	88	30	is	be	AUX
cana-3855	88	31	a	a	DET
cana-3855	88	32	cesaro	cesaro	ADJ
cana-3855	88	33	matrix	matrix	NOUN
cana-3855	88	34	of	of	ADP
cana-3855	88	35	order	order	NOUN
cana-3855	88	36	one	one	NUM
cana-3855	88	37	which	which	PRON
cana-3855	88	38	is	be	AUX
cana-3855	88	39	a	a	DET
cana-3855	88	40	translation	translation	NOUN
cana-3855	88	41	matrix	matrix	NOUN
cana-3855	88	42	and	and	CCONJ
cana-3855	88	43	its	its	PRON
cana-3855	88	44	quasi	quasi	ADJ
cana-3855	88	45	𝐷-limit	𝐷-limit	PROPN
cana-3855	88	46	[	[	X
cana-3855	88	47	3	3	NUM
cana-3855	88	48	]	]	PUNCT
cana-3855	88	49	or	or	CCONJ
cana-3855	88	50	quasi	quasi	ADJ
cana-3855	88	51	banach	banach	NOUN
cana-3855	88	52	limit	limit	NOUN
cana-3855	88	53	exists	exist	VERB
cana-3855	88	54	as	as	SCONJ
cana-3855	88	55	it	it	PRON
cana-3855	88	56	is	be	AUX
cana-3855	88	57	a	a	DET
cana-3855	88	58	nonnegative	nonnegative	ADJ
cana-3855	88	59	,	,	PUNCT
cana-3855	88	60	regular	regular	ADJ
cana-3855	88	61	and	and	CCONJ
cana-3855	88	62	𝐷(=	𝐷(=	ADJ
cana-3855	88	63	𝐵)invariant	𝐵)invariant	ADJ
cana-3855	88	64	matrix	matrix	NOUN
cana-3855	88	65	.	.	PUNCT
cana-3855	89	1	theorem	theorem	VERB
cana-3855	89	2	2	2	NUM
cana-3855	89	3	:	:	PUNCT
cana-3855	89	4	for	for	SCONJ
cana-3855	89	5	all	all	DET
cana-3855	89	6	𝑥	𝑥	DET
cana-3855	89	7	∈	∈	NOUN
cana-3855	89	8	𝑚	𝑚	NOUN
cana-3855	89	9	,	,	PUNCT
cana-3855	89	10	the	the	DET
cana-3855	89	11	sublinear	sublinear	ADJ
cana-3855	89	12	functional	functional	ADJ
cana-3855	89	13	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	89	14	)	)	PUNCT
cana-3855	89	15	generates	generate	VERB
cana-3855	89	16	𝐷-limit	𝐷-limit	PROPN
cana-3855	89	17	.	.	PUNCT
cana-3855	90	1	proof	proof	NOUN
cana-3855	90	2	:	:	PUNCT
cana-3855	90	3	we	we	PRON
cana-3855	90	4	first	first	ADV
cana-3855	90	5	proceed	proceed	VERB
cana-3855	90	6	to	to	PART
cana-3855	90	7	prove	prove	VERB
cana-3855	90	8	that	that	SCONJ
cana-3855	90	9	the	the	DET
cana-3855	90	10	sublinear	sublinear	NOUN
cana-3855	90	11	functional	functional	ADJ
cana-3855	90	12	𝜓	𝜓	PROPN
cana-3855	90	13	∈	∈	PROPN
cana-3855	90	14	𝑚∗	𝑚∗	PROPN
cana-3855	90	15	generates	generate	VERB
cana-3855	90	16	𝐷-mean	𝐷-mean	PROPN
cana-3855	90	17	.	.	PUNCT
cana-3855	91	1	let	let	VERB
cana-3855	91	2	𝜓	𝜓	PRON
cana-3855	91	3	∈	∈	PROPN
cana-3855	91	4	𝑚∗	𝑚∗	PROPN
cana-3855	91	5	and	and	CCONJ
cana-3855	91	6	satisfies	satisfy	VERB
cana-3855	91	7	the	the	DET
cana-3855	91	8	inequality	inequality	NOUN
cana-3855	91	9	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	91	10	)	)	PUNCT
cana-3855	91	11	≤	≤	NUM
cana-3855	91	12	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	91	13	)	)	PUNCT
cana-3855	91	14	for	for	ADP
cana-3855	91	15	all	all	DET
cana-3855	91	16	𝑥	𝑥	PRON
cana-3855	91	17	∈	∈	PROPN
cana-3855	91	18	𝑚	𝑚	X
cana-3855	91	19	(	(	PUNCT
cana-3855	91	20	3.2	3.2	NUM
cana-3855	91	21	)	)	PUNCT
cana-3855	91	22	so	so	ADV
cana-3855	91	23	by	by	ADP
cana-3855	91	24	linearity	linearity	NOUN
cana-3855	91	25	of	of	ADP
cana-3855	91	26	𝜓(𝑥)and	𝜓(𝑥)and	PROPN
cana-3855	91	27	sub	sub	NOUN
cana-3855	91	28	-	-	NOUN
cana-3855	91	29	linearity	linearity	NOUN
cana-3855	91	30	of	of	ADP
cana-3855	91	31	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	91	32	)	)	PUNCT
cana-3855	91	33	we	we	PRON
cana-3855	91	34	have	have	AUX
cana-3855	91	35	from	from	ADP
cana-3855	91	36	(	(	PUNCT
cana-3855	91	37	3.2	3.2	NUM
cana-3855	91	38	)	)	PUNCT
cana-3855	91	39	,	,	PUNCT
cana-3855	91	40	that	that	PRON
cana-3855	91	41	is	be	AUX
cana-3855	91	42	−𝑟(−𝑥	−𝑟(−𝑥	NOUN
cana-3855	91	43	)	)	PUNCT
cana-3855	91	44	≤	≤	NOUN
cana-3855	91	45	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	91	46	)	)	PUNCT
cana-3855	91	47	≤	≤	NUM
cana-3855	91	48	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	91	49	)	)	PUNCT
cana-3855	91	50	(	(	PUNCT
cana-3855	91	51	3.3	3.3	NUM
cana-3855	91	52	)	)	PUNCT
cana-3855	91	53	this	this	PRON
cana-3855	91	54	indicates	indicate	VERB
cana-3855	91	55	that	that	SCONJ
cana-3855	91	56	for	for	SCONJ
cana-3855	91	57	𝑥	𝑥	PROPN
cana-3855	91	58	≥	≥	NUM
cana-3855	91	59	0	0	NUM
cana-3855	91	60	implies	imply	VERB
cana-3855	91	61	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	91	62	)	)	PUNCT
cana-3855	91	63	≥	≥	NOUN
cana-3855	91	64	0	0	NUM
cana-3855	91	65	and	and	CCONJ
cana-3855	91	66	−𝑟(−𝑥	−𝑟(−𝑥	PROPN
cana-3855	91	67	)	)	PUNCT
cana-3855	91	68	≥	≥	NOUN
cana-3855	91	69	0	0	NUM
cana-3855	91	70	and	and	CCONJ
cana-3855	91	71	so	so	ADV
cana-3855	91	72	from	from	ADP
cana-3855	91	73	(	(	PUNCT
cana-3855	91	74	3.3	3.3	NUM
cana-3855	91	75	)	)	PUNCT
cana-3855	91	76	,	,	PUNCT
cana-3855	91	77	we	we	PRON
cana-3855	91	78	get	get	VERB
cana-3855	91	79	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	91	80	)	)	PUNCT
cana-3855	91	81	≥	≥	NOUN
cana-3855	91	82	0	0	NUM
cana-3855	91	83	,	,	PUNCT
cana-3855	91	84	for	for	ADP
cana-3855	91	85	all	all	DET
cana-3855	91	86	𝑥	𝑥	DET
cana-3855	91	87	≥	≥	NUM
cana-3855	91	88	0	0	NUM
cana-3855	91	89	.	.	PUNCT
cana-3855	92	1	also	also	ADV
cana-3855	92	2	−𝑟(−𝑒	−𝑟(−𝑒	ADJ
cana-3855	92	3	)	)	PUNCT
cana-3855	93	1	=	=	SYM
cana-3855	93	2	𝑟(𝑒	𝑟(𝑒	PROPN
cana-3855	93	3	)	)	PUNCT
cana-3855	93	4	=	=	SYM
cana-3855	94	1	1	1	NUM
cana-3855	94	2	,	,	PUNCT
cana-3855	94	3	where	where	SCONJ
cana-3855	94	4	𝑒	𝑒	NOUN
cana-3855	94	5	=	=	PUNCT
cana-3855	94	6	(	(	PUNCT
cana-3855	94	7	1,1,1,1	1,1,1,1	NUM
cana-3855	94	8	,	,	PUNCT
cana-3855	94	9	…	…	PUNCT
cana-3855	94	10	.	.	PUNCT
cana-3855	95	1	,	,	PUNCT
cana-3855	95	2	1	1	NUM
cana-3855	95	3	)	)	PUNCT
cana-3855	95	4	.	.	PUNCT
cana-3855	96	1	hence	hence	ADV
cana-3855	96	2	from	from	ADP
cana-3855	96	3	(	(	PUNCT
cana-3855	96	4	3.3	3.3	NUM
cana-3855	96	5	)	)	PUNCT
cana-3855	96	6	,	,	PUNCT
cana-3855	96	7	we	we	PRON
cana-3855	96	8	get	get	VERB
cana-3855	96	9	𝜓(𝑒	𝜓(𝑒	NUM
cana-3855	96	10	)	)	PUNCT
cana-3855	96	11	=	=	SYM
cana-3855	96	12	1	1	X
cana-3855	96	13	.	.	X
cana-3855	96	14	communications	communication	NOUN
cana-3855	96	15	on	on	ADP
cana-3855	96	16	applied	apply	VERB
cana-3855	96	17	nonlinear	nonlinear	ADJ
cana-3855	96	18	analysis	analysis	NOUN
cana-3855	96	19	issn	issn	NOUN
cana-3855	96	20	:	:	PUNCT
cana-3855	96	21	1074	1074	NUM
cana-3855	96	22	-	-	PUNCT
cana-3855	96	23	133x	133x	NUM
cana-3855	96	24	vol	vol	NOUN
cana-3855	96	25	32	32	NUM
cana-3855	97	1	no	no	NOUN
cana-3855	97	2	.	.	PUNCT
cana-3855	98	1	9s	9s	NUM
cana-3855	98	2	(	(	PUNCT
cana-3855	98	3	2025	2025	NUM
cana-3855	98	4	)	)	PUNCT
cana-3855	98	5	281	281	NUM
cana-3855	98	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	98	7	taking	take	VERB
cana-3855	98	8	𝐷	𝐷	NOUN
cana-3855	98	9	as	as	ADP
cana-3855	98	10	a	a	DET
cana-3855	98	11	nonnegative	nonnegative	ADJ
cana-3855	98	12	regular	regular	ADJ
cana-3855	98	13	matrix	matrix	NOUN
cana-3855	98	14	,	,	PUNCT
cana-3855	98	15	we	we	PRON
cana-3855	98	16	have	have	VERB
cana-3855	98	17	limsup	limsup	NOUN
cana-3855	98	18	𝑝	𝑝	PROPN
cana-3855	98	19	sup	sup	NOUN
cana-3855	98	20	𝑛	𝑛	PROPN
cana-3855	98	21	1	1	NUM
cana-3855	98	22	𝑝	𝑝	NOUN
cana-3855	99	1	+	+	CCONJ
cana-3855	99	2	1	1	NUM
cana-3855	99	3	∑(𝐷𝑘+1−𝐷𝑘)𝑥𝑛𝑝	∑(𝐷𝑘+1−𝐷𝑘)𝑥𝑛𝑝	NOUN
cana-3855	99	4	𝑝	𝑝	PROPN
cana-3855	99	5	𝑘=0	𝑘=0	PROPN
cana-3855	100	1	=	=	SYM
cana-3855	100	2	limsup	limsup	PROPN
cana-3855	100	3	𝑝	𝑝	PROPN
cana-3855	100	4	sup	sup	NOUN
cana-3855	100	5	𝑛	𝑛	PROPN
cana-3855	100	6	(	(	PUNCT
cana-3855	100	7	𝐷𝑝+1−𝐼)𝑥𝑛𝑝	𝐷𝑝+1−𝐼)𝑥𝑛𝑝	PROPN
cana-3855	100	8	𝑝	𝑝	PROPN
cana-3855	100	9	+	+	CCONJ
cana-3855	100	10	1	1	NUM
cana-3855	100	11	≤	≤	NOUN
cana-3855	100	12	‖𝑥‖	‖𝑥‖	PROPN
cana-3855	100	13	limsup	limsup	PROPN
cana-3855	100	14	𝑝	𝑝	PROPN
cana-3855	100	15	1	1	NUM
cana-3855	100	16	𝑝+1	𝑝+1	NOUN
cana-3855	100	17	=	=	SYM
cana-3855	100	18	0	0	X
cana-3855	100	19	.	.	PUNCT
cana-3855	101	1	this	this	PRON
cana-3855	101	2	shows	show	VERB
cana-3855	101	3	𝑟(𝐷𝑥	𝑟(𝐷𝑥	NOUN
cana-3855	101	4	−	−	PROPN
cana-3855	101	5	𝑥	𝑥	NOUN
cana-3855	101	6	)	)	PUNCT
cana-3855	101	7	=	=	SYM
cana-3855	101	8	0	0	X
cana-3855	101	9	.	.	PUNCT
cana-3855	102	1	similarly	similarly	ADV
cana-3855	102	2	,	,	PUNCT
cana-3855	102	3	we	we	PRON
cana-3855	102	4	also	also	ADV
cana-3855	102	5	obtained	obtain	VERB
cana-3855	102	6	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	102	7	−	−	PUNCT
cana-3855	102	8	𝐷𝑥	𝐷𝑥	PROPN
cana-3855	102	9	)	)	PUNCT
cana-3855	102	10	=	=	NOUN
cana-3855	103	1	0	0	X
cana-3855	103	2	.	.	PUNCT
cana-3855	104	1	hence	hence	ADV
cana-3855	104	2	from	from	ADP
cana-3855	104	3	(	(	PUNCT
cana-3855	104	4	3.3	3.3	NUM
cana-3855	104	5	)	)	PUNCT
cana-3855	104	6	we	we	PRON
cana-3855	104	7	have	have	VERB
cana-3855	104	8	𝜓(𝐷𝑥	𝜓(𝐷𝑥	ADJ
cana-3855	104	9	−	−	PROPN
cana-3855	104	10	𝑥	𝑥	NOUN
cana-3855	104	11	)	)	PUNCT
cana-3855	104	12	=	=	NOUN
cana-3855	105	1	0	0	X
cana-3855	105	2	.	.	PUNCT
cana-3855	106	1	since	since	SCONJ
cana-3855	106	2	𝜓	𝜓	PROPN
cana-3855	106	3	is	be	AUX
cana-3855	106	4	linear	linear	ADJ
cana-3855	106	5	,	,	PUNCT
cana-3855	106	6	𝜓(𝐷𝑥	𝜓(𝐷𝑥	PROPN
cana-3855	106	7	)	)	PUNCT
cana-3855	106	8	=	=	PUNCT
cana-3855	106	9	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	106	10	)	)	PUNCT
cana-3855	106	11	,	,	PUNCT
cana-3855	106	12	𝑥	𝑥	PROPN
cana-3855	106	13	∈	∈	PROPN
cana-3855	106	14	𝑚.	𝑚.	NOUN
cana-3855	106	15	this	this	PRON
cana-3855	106	16	proves	prove	VERB
cana-3855	106	17	that	that	SCONJ
cana-3855	106	18	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	106	19	)	)	PUNCT
cana-3855	106	20	≤	≤	NUM
cana-3855	106	21	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	106	22	)	)	PUNCT
cana-3855	106	23	⇒	⇒	NOUN
cana-3855	106	24	𝜓	𝜓	PROPN
cana-3855	106	25	is	be	AUX
cana-3855	106	26	a	a	DET
cana-3855	106	27	𝐷-limit	𝐷-limit	PROPN
cana-3855	106	28	(	(	PUNCT
cana-3855	106	29	3.4	3.4	NUM
cana-3855	106	30	)	)	PUNCT
cana-3855	106	31	so	so	ADV
cana-3855	106	32	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	106	33	)	)	PUNCT
cana-3855	106	34	also	also	ADV
cana-3855	106	35	generates	generate	VERB
cana-3855	106	36	𝐷	𝐷	NOUN
cana-3855	106	37	–	–	PUNCT
cana-3855	106	38	limit	limit	NOUN
cana-3855	106	39	as	as	SCONJ
cana-3855	106	40	required	require	VERB
cana-3855	106	41	.	.	PUNCT
cana-3855	107	1	note	note	NOUN
cana-3855	107	2	:	:	PUNCT
cana-3855	107	3	from	from	ADP
cana-3855	107	4	above	above	ADP
cana-3855	107	5	we	we	PRON
cana-3855	107	6	say	say	VERB
cana-3855	107	7	that	that	SCONJ
cana-3855	107	8	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	107	9	)	)	PUNCT
cana-3855	107	10	also	also	ADV
cana-3855	107	11	generates	generate	VERB
cana-3855	107	12	quasi	quasi	ADJ
cana-3855	107	13	𝐷	𝐷	PROPN
cana-3855	107	14	-limit	-limit	NOUN
cana-3855	107	15	.	.	PUNCT
cana-3855	108	1	next	next	ADV
cana-3855	108	2	consider	consider	VERB
cana-3855	108	3	𝜓	𝜓	NOUN
cana-3855	108	4	is	be	AUX
cana-3855	108	5	a	a	DET
cana-3855	108	6	quasi	quasi	ADJ
cana-3855	108	7	𝐷-limit	𝐷-limit	PROPN
cana-3855	108	8	.	.	PUNCT
cana-3855	109	1	then	then	ADV
cana-3855	109	2	𝜓(𝑥𝑛𝑝	𝜓(𝑥𝑛𝑝	PROPN
cana-3855	109	3	)	)	PUNCT
cana-3855	109	4	=	=	SYM
cana-3855	109	5	𝜓(𝐷𝑥)𝑛𝑝	𝜓(𝐷𝑥)𝑛𝑝	PROPN
cana-3855	109	6	=	=	PUNCT
cana-3855	109	7	𝜓(𝐷2𝑥)𝑛𝑝	𝜓(𝐷2𝑥)𝑛𝑝	PROPN
cana-3855	109	8	=	=	SYM
cana-3855	109	9	⋯	⋯	NOUN
cana-3855	109	10	…	…	PUNCT
cana-3855	109	11	=	=	PUNCT
cana-3855	109	12	𝜓(𝐷𝑝𝑥)𝑛𝑝	𝜓(𝐷𝑝𝑥)𝑛𝑝	NOUN
cana-3855	109	13	this	this	PRON
cana-3855	109	14	implies	imply	VERB
cana-3855	109	15	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	109	16	)	)	PUNCT
cana-3855	109	17	=	=	SYM
cana-3855	109	18	𝜓	𝜓	X
cana-3855	109	19	(	(	PUNCT
cana-3855	109	20	𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷	𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷	NOUN
cana-3855	109	21	2𝑥)𝑛𝑝+⋯	2𝑥)𝑛𝑝+⋯	NOUN
cana-3855	109	22	…	…	SYM
cana-3855	109	23	+(𝐷	+(𝐷	SYM
cana-3855	109	24	𝑝𝑥)𝑛𝑝	𝑝𝑥)𝑛𝑝	NUM
cana-3855	109	25	𝑝+1	𝑝+1	NOUN
cana-3855	109	26	)	)	PUNCT
cana-3855	109	27	≤	≤	NUM
cana-3855	109	28	limsup	limsup	NOUN
cana-3855	109	29	𝑝	𝑝	PROPN
cana-3855	109	30	sup	sup	NOUN
cana-3855	109	31	𝑛	𝑛	PROPN
cana-3855	109	32	(	(	PUNCT
cana-3855	109	33	𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷	𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷	NUM
cana-3855	109	34	2𝑥)𝑛𝑝+⋯	2𝑥)𝑛𝑝+⋯	NOUN
cana-3855	109	35	…	…	SYM
cana-3855	109	36	+(𝐷	+(𝐷	SYM
cana-3855	109	37	𝑝𝑥)𝑛𝑝	𝑝𝑥)𝑛𝑝	NUM
cana-3855	109	38	𝑝+1	𝑝+1	NOUN
cana-3855	109	39	)	)	PUNCT
cana-3855	109	40	=	=	SYM
cana-3855	109	41	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	109	42	)	)	PUNCT
cana-3855	109	43	i.e.	i.e.	X
cana-3855	109	44	,	,	PUNCT
cana-3855	109	45	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	109	46	)	)	PUNCT
cana-3855	109	47	dominates	dominate	VERB
cana-3855	109	48	quasi	quasi	PROPN
cana-3855	109	49	𝐷	𝐷	PROPN
cana-3855	109	50	–	–	PUNCT
cana-3855	109	51	limit	limit	NOUN
cana-3855	109	52	.	.	PUNCT
cana-3855	110	1	theorem	theorem	NOUN
cana-3855	110	2	3	3	NUM
cana-3855	110	3	:	:	PUNCT
cana-3855	110	4	let	let	VERB
cana-3855	110	5	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	110	6	)	)	PUNCT
cana-3855	110	7	and	and	CCONJ
cana-3855	110	8	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	110	9	)	)	PUNCT
cana-3855	110	10	be	be	VERB
cana-3855	110	11	two	two	NUM
cana-3855	110	12	sublinear	sublinear	NOUN
cana-3855	110	13	functionals	functional	NOUN
cana-3855	110	14	defined	define	VERB
cana-3855	110	15	in	in	ADP
cana-3855	110	16	(	(	PUNCT
cana-3855	110	17	2.2	2.2	NUM
cana-3855	110	18	)	)	PUNCT
cana-3855	110	19	and	and	CCONJ
cana-3855	110	20	(	(	PUNCT
cana-3855	110	21	2.3	2.3	NUM
cana-3855	110	22	)	)	PUNCT
cana-3855	110	23	respectively.then	respectively.then	ADP
cana-3855	110	24	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	110	25	)	)	PUNCT
cana-3855	110	26	≤	≤	NOUN
cana-3855	110	27	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	110	28	)	)	PUNCT
cana-3855	110	29	for	for	ADP
cana-3855	110	30	all	all	DET
cana-3855	110	31	𝑥	𝑥	DET
cana-3855	110	32	∈	∈	NOUN
cana-3855	110	33	𝑚	𝑚	NOUN
cana-3855	110	34	.	.	PUNCT
cana-3855	111	1	proof	proof	NOUN
cana-3855	111	2	:	:	PUNCT
cana-3855	111	3	combining	combine	VERB
cana-3855	111	4	the	the	DET
cana-3855	111	5	given	give	VERB
cana-3855	111	6	condition	condition	NOUN
cana-3855	111	7	of	of	ADP
cana-3855	111	8	(	(	PUNCT
cana-3855	111	9	3.4	3.4	NUM
cana-3855	111	10	)	)	PUNCT
cana-3855	111	11	and	and	CCONJ
cana-3855	111	12	the	the	DET
cana-3855	111	13	inequality	inequality	NOUN
cana-3855	111	14	of	of	ADP
cana-3855	111	15	theorem	theorem	NOUN
cana-3855	111	16	a	a	PRON
cana-3855	111	17	,	,	PUNCT
cana-3855	111	18	we	we	PRON
cana-3855	111	19	will	will	AUX
cana-3855	111	20	get	get	VERB
cana-3855	111	21	that	that	PRON
cana-3855	111	22	for	for	ADP
cana-3855	111	23	all	all	PRON
cana-3855	111	24	𝜓	𝜓	PRON
cana-3855	111	25	∈	∈	PROPN
cana-3855	111	26	𝑚∗.	𝑚∗.	NUM
cana-3855	111	27	𝜓(𝑥	𝜓(𝑥	SYM
cana-3855	111	28	)	)	PUNCT
cana-3855	111	29	≤	≤	NUM
cana-3855	111	30	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	111	31	)	)	PUNCT
cana-3855	111	32	⇒	⇒	NOUN
cana-3855	111	33	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	111	34	)	)	PUNCT
cana-3855	111	35	≤	≤	NOUN
cana-3855	111	36	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	111	37	)	)	PUNCT
cana-3855	111	38	(	(	PUNCT
cana-3855	111	39	3.5	3.5	NUM
cana-3855	111	40	)	)	PUNCT
cana-3855	111	41	claim	claim	NOUN
cana-3855	111	42	:	:	PUNCT
cana-3855	111	43	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	111	44	)	)	PUNCT
cana-3855	111	45	≤	≤	NOUN
cana-3855	111	46	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	111	47	)	)	PUNCT
cana-3855	111	48	for	for	ADP
cana-3855	111	49	all	all	DET
cana-3855	111	50	𝑥	𝑥	DET
cana-3855	111	51	∈	∈	NOUN
cana-3855	111	52	𝑚	𝑚	X
cana-3855	111	53	.	.	PUNCT
cana-3855	112	1	(	(	PUNCT
cana-3855	112	2	3.6	3.6	NUM
cana-3855	112	3	)	)	PUNCT
cana-3855	112	4	suppose	suppose	VERB
cana-3855	112	5	to	to	ADP
cana-3855	112	6	the	the	DET
cana-3855	112	7	contrary	contrary	NOUN
cana-3855	112	8	,	,	PUNCT
cana-3855	112	9	that	that	SCONJ
cana-3855	112	10	(	(	PUNCT
cana-3855	112	11	3.6	3.6	NUM
cana-3855	112	12	)	)	PUNCT
cana-3855	112	13	is	be	AUX
cana-3855	112	14	false	false	ADJ
cana-3855	112	15	.	.	PUNCT
cana-3855	113	1	then	then	ADV
cana-3855	113	2	there	there	PRON
cana-3855	113	3	exists	exist	VERB
cana-3855	113	4	a	a	DET
cana-3855	113	5	sequence	sequence	NOUN
cana-3855	113	6	𝑦	𝑦	NOUN
cana-3855	113	7	∈	∈	NOUN
cana-3855	113	8	𝑚	𝑚	ADP
cana-3855	113	9	such	such	ADJ
cana-3855	113	10	that	that	DET
cana-3855	113	11	𝑟(𝑦	𝑟(𝑦	NUM
cana-3855	113	12	)	)	PUNCT
cana-3855	113	13	>	>	X
cana-3855	113	14	𝑡(𝑦	𝑡(𝑦	NOUN
cana-3855	113	15	)	)	PUNCT
cana-3855	113	16	.	.	PUNCT
cana-3855	114	1	(	(	PUNCT
cana-3855	114	2	3.7	3.7	NUM
cana-3855	114	3	)	)	PUNCT
cana-3855	114	4	communications	communication	NOUN
cana-3855	114	5	on	on	ADP
cana-3855	114	6	applied	apply	VERB
cana-3855	114	7	nonlinear	nonlinear	ADJ
cana-3855	114	8	analysis	analysis	NOUN
cana-3855	114	9	issn	issn	NOUN
cana-3855	114	10	:	:	PUNCT
cana-3855	114	11	1074	1074	NUM
cana-3855	114	12	-	-	PUNCT
cana-3855	114	13	133x	133x	NUM
cana-3855	114	14	vol	vol	NOUN
cana-3855	114	15	32	32	NUM
cana-3855	114	16	no	no	NOUN
cana-3855	114	17	.	.	PUNCT
cana-3855	115	1	9s	9s	NUM
cana-3855	115	2	(	(	PUNCT
cana-3855	115	3	2025	2025	NUM
cana-3855	115	4	)	)	PUNCT
cana-3855	115	5	282	282	NUM
cana-3855	115	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	115	7	since	since	SCONJ
cana-3855	115	8	𝑟	𝑟	PRON
cana-3855	115	9	is	be	AUX
cana-3855	115	10	sublinear	sublinear	ADJ
cana-3855	115	11	,	,	PUNCT
cana-3855	115	12	by	by	ADP
cana-3855	115	13	hahn	hahn	NOUN
cana-3855	115	14	-	-	PUNCT
cana-3855	115	15	banach	banach	NOUN
cana-3855	115	16	theorem	theorem	VERB
cana-3855	115	17	,	,	PUNCT
cana-3855	115	18	there	there	PRON
cana-3855	115	19	exist	exist	VERB
cana-3855	115	20	a	a	DET
cana-3855	115	21	linear	linear	ADJ
cana-3855	115	22	functional	functional	ADJ
cana-3855	115	23	𝑔	𝑔	NOUN
cana-3855	115	24	on	on	ADP
cana-3855	115	25	𝑚	𝑚	ADP
cana-3855	115	26	such	such	ADJ
cana-3855	115	27	that	that	PRON
cana-3855	115	28	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	115	29	)	)	PUNCT
cana-3855	116	1	=	=	SYM
cana-3855	116	2	𝑟(𝑦	𝑟(𝑦	NUM
cana-3855	116	3	)	)	PUNCT
cana-3855	116	4	.	.	PUNCT
cana-3855	117	1	from	from	ADP
cana-3855	117	2	(	(	PUNCT
cana-3855	117	3	3.7	3.7	NUM
cana-3855	117	4	)	)	PUNCT
cana-3855	117	5	,	,	PUNCT
cana-3855	117	6	it	it	PRON
cana-3855	117	7	follows	follow	VERB
cana-3855	117	8	that	that	SCONJ
cana-3855	117	9	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	117	10	)	)	PUNCT
cana-3855	117	11	>	>	X
cana-3855	117	12	𝑡(𝑦	𝑡(𝑦	NOUN
cana-3855	117	13	)	)	PUNCT
cana-3855	117	14	.	.	PUNCT
cana-3855	118	1	however	however	ADV
cana-3855	118	2	,	,	PUNCT
cana-3855	118	3	from	from	ADP
cana-3855	118	4	the	the	DET
cana-3855	118	5	given	give	VERB
cana-3855	118	6	condition	condition	NOUN
cana-3855	118	7	(	(	PUNCT
cana-3855	118	8	3.5	3.5	NUM
cana-3855	118	9	)	)	PUNCT
cana-3855	118	10	,	,	PUNCT
cana-3855	118	11	we	we	PRON
cana-3855	118	12	know	know	VERB
cana-3855	118	13	that	that	SCONJ
cana-3855	118	14	for	for	ADP
cana-3855	118	15	all	all	DET
cana-3855	118	16	𝜓	𝜓	ADP
cana-3855	118	17	∈	∈	PROPN
cana-3855	118	18	𝑚∗	𝑚∗	NOUN
cana-3855	118	19	𝜓(𝑥	𝜓(𝑥	ADV
cana-3855	118	20	)	)	PUNCT
cana-3855	118	21	≤	≤	NUM
cana-3855	118	22	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	118	23	)	)	PUNCT
cana-3855	118	24	⇒	⇒	NOUN
cana-3855	118	25	𝜓(𝑥	𝜓(𝑥	PROPN
cana-3855	118	26	)	)	PUNCT
cana-3855	118	27	≤	≤	NOUN
cana-3855	118	28	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	118	29	)	)	PUNCT
cana-3855	118	30	substituting	substitute	VERB
cana-3855	118	31	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	118	32	)	)	PUNCT
cana-3855	118	33	into	into	ADP
cana-3855	118	34	this	this	DET
cana-3855	118	35	condition	condition	NOUN
cana-3855	118	36	,	,	PUNCT
cana-3855	118	37	we	we	PRON
cana-3855	118	38	have	have	VERB
cana-3855	118	39	:	:	PUNCT
cana-3855	118	40	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	118	41	)	)	PUNCT
cana-3855	118	42	≤	≤	NUM
cana-3855	118	43	𝑡(𝑦	𝑡(𝑦	NOUN
cana-3855	118	44	)	)	PUNCT
cana-3855	118	45	which	which	PRON
cana-3855	118	46	contradicts	contradict	VERB
cana-3855	118	47	our	our	PRON
cana-3855	118	48	earlier	early	ADJ
cana-3855	118	49	assertion	assertion	NOUN
cana-3855	118	50	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	118	51	)	)	PUNCT
cana-3855	118	52	>	>	X
cana-3855	119	1	𝑡(𝑦	𝑡(𝑦	NOUN
cana-3855	119	2	)	)	PUNCT
cana-3855	119	3	thus	thus	ADV
cana-3855	119	4	,	,	PUNCT
cana-3855	119	5	our	our	PRON
cana-3855	119	6	assumption	assumption	NOUN
cana-3855	119	7	that	that	SCONJ
cana-3855	119	8	𝑔(𝑦	𝑔(𝑦	NOUN
cana-3855	119	9	)	)	PUNCT
cana-3855	119	10	>	>	X
cana-3855	119	11	𝑡(𝑦	𝑡(𝑦	NOUN
cana-3855	119	12	)	)	PUNCT
cana-3855	119	13	is	be	AUX
cana-3855	119	14	untenable	untenable	ADJ
cana-3855	119	15	.	.	PUNCT
cana-3855	120	1	therefore	therefore	ADV
cana-3855	120	2	,	,	PUNCT
cana-3855	120	3	we	we	PRON
cana-3855	120	4	conclude	conclude	VERB
cana-3855	120	5	that	that	PRON
cana-3855	120	6	:	:	PUNCT
cana-3855	120	7	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	120	8	)	)	PUNCT
cana-3855	120	9	≤	≤	NOUN
cana-3855	120	10	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	120	11	)	)	PUNCT
cana-3855	120	12	for	for	ADP
cana-3855	120	13	all	all	DET
cana-3855	120	14	𝑥	𝑥	DET
cana-3855	120	15	∈	∈	PROPN
cana-3855	120	16	𝑚.	𝑚.	NOUN
cana-3855	120	17	(	(	PUNCT
cana-3855	120	18	3.8	3.8	NUM
cana-3855	120	19	)	)	PUNCT
cana-3855	120	20	hence	hence	ADV
cana-3855	120	21	proved	prove	VERB
cana-3855	120	22	the	the	DET
cana-3855	120	23	result	result	NOUN
cana-3855	120	24	.	.	PUNCT
cana-3855	121	1	next	next	ADV
cana-3855	121	2	,	,	PUNCT
cana-3855	121	3	we	we	PRON
cana-3855	121	4	are	be	AUX
cana-3855	121	5	going	go	VERB
cana-3855	121	6	to	to	PART
cana-3855	121	7	prove	prove	VERB
cana-3855	121	8	an	an	DET
cana-3855	121	9	important	important	ADJ
cana-3855	121	10	corollary	corollary	NOUN
cana-3855	121	11	of	of	ADP
cana-3855	121	12	above	above	ADP
cana-3855	121	13	theorem	theorem	ADJ
cana-3855	121	14	.	.	PROPN
cana-3855	122	1	from	from	ADP
cana-3855	122	2	the	the	DET
cana-3855	122	3	definition	definition	NOUN
cana-3855	122	4	and	and	CCONJ
cana-3855	122	5	from	from	ADP
cana-3855	122	6	above	above	ADP
cana-3855	122	7	theorem	theorem	NOUN
cana-3855	122	8	,	,	PUNCT
cana-3855	122	9	the	the	DET
cana-3855	122	10	set	set	NOUN
cana-3855	122	11	of	of	ADP
cana-3855	122	12	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	122	13	sequences	sequence	NOUN
cana-3855	122	14	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	122	15	also	also	ADV
cana-3855	122	16	can	can	AUX
cana-3855	122	17	be	be	AUX
cana-3855	122	18	written	write	VERB
cana-3855	122	19	as	as	ADP
cana-3855	122	20	:	:	PUNCT
cana-3855	122	21	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	122	22	=	=	PRON
cana-3855	122	23	{	{	PUNCT
cana-3855	122	24	𝑥	𝑥	X
cana-3855	122	25	∈	∈	PROPN
cana-3855	122	26	𝑚	𝑚	NOUN
cana-3855	122	27	:	:	PUNCT
cana-3855	122	28	−𝑡(−𝑥	−𝑡(−𝑥	NOUN
cana-3855	122	29	)	)	PUNCT
cana-3855	122	30	=	=	SYM
cana-3855	122	31	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	122	32	)	)	PUNCT
cana-3855	122	33	}	}	PUNCT
cana-3855	122	34	.	.	PUNCT
cana-3855	123	1	(	(	PUNCT
cana-3855	123	2	3.9	3.9	NUM
cana-3855	123	3	)	)	PUNCT
cana-3855	123	4	this	this	PRON
cana-3855	123	5	identifies	identify	VERB
cana-3855	123	6	the	the	DET
cana-3855	123	7	structure	structure	NOUN
cana-3855	123	8	of	of	ADP
cana-3855	123	9	sequences	sequence	NOUN
cana-3855	123	10	𝑥	𝑥	PROPN
cana-3855	123	11	for	for	ADP
cana-3855	123	12	which	which	PRON
cana-3855	123	13	−𝑡(−𝑥	−𝑡(−𝑥	PROPN
cana-3855	123	14	)	)	PUNCT
cana-3855	123	15	=	=	SYM
cana-3855	123	16	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	123	17	)	)	PUNCT
cana-3855	123	18	which	which	PRON
cana-3855	123	19	are	be	AUX
cana-3855	123	20	also	also	ADV
cana-3855	123	21	referred	refer	VERB
cana-3855	123	22	to	to	PART
cana-3855	123	23	define	define	VERB
cana-3855	123	24	quasi	quasi	ADJ
cana-3855	123	25	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	123	26	sequences	sequence	NOUN
cana-3855	123	27	.	.	PUNCT
cana-3855	124	1	we	we	PRON
cana-3855	124	2	denote	denote	VERB
cana-3855	124	3	the	the	DET
cana-3855	124	4	set	set	NOUN
cana-3855	124	5	of	of	ADP
cana-3855	124	6	quasi	quasi	NOUN
cana-3855	124	7	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	124	8	sequences	sequence	NOUN
cana-3855	124	9	as	as	ADP
cana-3855	124	10	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	124	11	∗	∗	NOUN
cana-3855	124	12	where	where	SCONJ
cana-3855	124	13	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	124	14	∗	∗	VERB
cana-3855	124	15	=	=	PUNCT
cana-3855	124	16	{	{	PUNCT
cana-3855	124	17	𝑥	𝑥	X
cana-3855	124	18	∈	∈	PROPN
cana-3855	124	19	𝑚	𝑚	NOUN
cana-3855	124	20	:	:	PUNCT
cana-3855	124	21	−𝑟(−𝑥	−𝑟(−𝑥	NOUN
cana-3855	124	22	)	)	PUNCT
cana-3855	124	23	=	=	SYM
cana-3855	124	24	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	124	25	)	)	PUNCT
cana-3855	124	26	}	}	PUNCT
cana-3855	124	27	.	.	PUNCT
cana-3855	125	1	(	(	PUNCT
cana-3855	125	2	3.10	3.10	NUM
cana-3855	125	3	)	)	PUNCT
cana-3855	125	4	then	then	ADV
cana-3855	125	5	we	we	PRON
cana-3855	125	6	now	now	ADV
cana-3855	125	7	prove	prove	VERB
cana-3855	125	8	the	the	DET
cana-3855	125	9	following	follow	VERB
cana-3855	125	10	corollary	corollary	NOUN
cana-3855	125	11	.	.	PUNCT
cana-3855	126	1	corollary	corollary	ADJ
cana-3855	126	2	:	:	PUNCT
cana-3855	126	3	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	126	4	⊆	⊆	NUM
cana-3855	126	5	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	126	6	∗	∗	NOUN
cana-3855	126	7	where	where	SCONJ
cana-3855	126	8	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	126	9	is	be	AUX
cana-3855	126	10	the	the	DET
cana-3855	126	11	set	set	NOUN
cana-3855	126	12	of	of	ADP
cana-3855	126	13	𝐷-convergent	𝐷-convergent	PROPN
cana-3855	126	14	sequences	sequence	NOUN
cana-3855	126	15	,	,	PUNCT
cana-3855	126	16	and	and	CCONJ
cana-3855	126	17	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	126	18	∗is	∗is	PROPN
cana-3855	126	19	the	the	DET
cana-3855	126	20	set	set	NOUN
cana-3855	126	21	of	of	ADP
cana-3855	126	22	quasi	quasi	NOUN
cana-3855	126	23	𝐷-convergent	𝐷-convergent	NOUN
cana-3855	126	24	sequences	sequence	NOUN
cana-3855	126	25	.	.	PUNCT
cana-3855	127	1	proof	proof	NOUN
cana-3855	127	2	:	:	PUNCT
cana-3855	127	3	since	since	SCONJ
cana-3855	127	4	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	127	5	)	)	PUNCT
cana-3855	127	6	and	and	CCONJ
cana-3855	127	7	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	127	8	)	)	PUNCT
cana-3855	127	9	are	be	AUX
cana-3855	127	10	sublinear	sublinear	ADJ
cana-3855	127	11	,	,	PUNCT
cana-3855	127	12	it	it	PRON
cana-3855	127	13	follows	follow	VERB
cana-3855	127	14	from	from	ADP
cana-3855	127	15	above	above	ADP
cana-3855	127	16	theorem	theorem	NOUN
cana-3855	127	17	that	that	SCONJ
cana-3855	127	18	−𝑡(−𝑥	−𝑡(−𝑥	PROPN
cana-3855	127	19	)	)	PUNCT
cana-3855	127	20	≤	≤	NOUN
cana-3855	127	21	−𝑟(−𝑥	−𝑟(−𝑥	ADP
cana-3855	127	22	)	)	PUNCT
cana-3855	127	23	≤	≤	NUM
cana-3855	127	24	𝑟(𝑥	𝑟(𝑥	NOUN
cana-3855	127	25	)	)	PUNCT
cana-3855	127	26	≤	≤	NOUN
cana-3855	127	27	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	127	28	)	)	PUNCT
cana-3855	127	29	for	for	ADP
cana-3855	127	30	all	all	DET
cana-3855	127	31	𝑥	𝑥	PRON
cana-3855	127	32	∈	∈	NOUN
cana-3855	127	33	𝑚.	𝑚.	NOUN
cana-3855	127	34	if	if	SCONJ
cana-3855	127	35	𝑥	𝑥	PROPN
cana-3855	127	36	∈	∈	PROPN
cana-3855	127	37	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	127	38	then	then	ADV
cana-3855	127	39	by	by	ADP
cana-3855	127	40	definition	definition	NOUN
cana-3855	127	41	,	,	PUNCT
cana-3855	127	42	−𝑡(−𝑥	−𝑡(−𝑥	PROPN
cana-3855	127	43	)	)	PUNCT
cana-3855	127	44	=	=	SYM
cana-3855	127	45	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	127	46	)	)	PUNCT
cana-3855	127	47	substituting	substitute	VERB
cana-3855	127	48	this	this	DET
cana-3855	127	49	condition	condition	NOUN
cana-3855	127	50	into	into	ADP
cana-3855	127	51	the	the	DET
cana-3855	127	52	chain	chain	NOUN
cana-3855	127	53	of	of	ADP
cana-3855	127	54	inequalities	inequality	NOUN
cana-3855	127	55	above	above	ADV
cana-3855	127	56	,	,	PUNCT
cana-3855	127	57	we	we	PRON
cana-3855	127	58	have	have	VERB
cana-3855	127	59	:	:	PUNCT
cana-3855	127	60	−𝑡(−𝑥	−𝑡(−𝑥	X
cana-3855	127	61	)	)	PUNCT
cana-3855	127	62	=	=	SYM
cana-3855	127	63	𝑡(𝑥	𝑡(𝑥	PROPN
cana-3855	127	64	)	)	PUNCT
cana-3855	127	65	⇒	⇒	NOUN
cana-3855	127	66	−𝑟(−𝑥	−𝑟(−𝑥	PROPN
cana-3855	127	67	)	)	PUNCT
cana-3855	127	68	=	=	SYM
cana-3855	127	69	𝑟(𝑥	𝑟(𝑥	PROPN
cana-3855	127	70	)	)	PUNCT
cana-3855	127	71	.	.	PUNCT
cana-3855	128	1	this	this	PRON
cana-3855	128	2	implies	imply	VERB
cana-3855	128	3	that	that	SCONJ
cana-3855	128	4	𝑥	𝑥	PROPN
cana-3855	128	5	∈	∈	PROPN
cana-3855	128	6	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	128	7	∗	∗	NOUN
cana-3855	128	8	,	,	PUNCT
cana-3855	128	9	where	where	SCONJ
cana-3855	128	10	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	128	11	∗represents	∗represent	VERB
cana-3855	128	12	the	the	DET
cana-3855	128	13	set	set	NOUN
cana-3855	128	14	of	of	ADP
cana-3855	128	15	quasi	quasi	NOUN
cana-3855	128	16	𝐷-convergent	𝐷-convergent	NOUN
cana-3855	128	17	sequences	sequence	NOUN
cana-3855	128	18	.	.	PUNCT
cana-3855	129	1	communications	communication	NOUN
cana-3855	129	2	on	on	ADP
cana-3855	129	3	applied	apply	VERB
cana-3855	129	4	nonlinear	nonlinear	ADJ
cana-3855	129	5	analysis	analysis	NOUN
cana-3855	129	6	issn	issn	NOUN
cana-3855	129	7	:	:	PUNCT
cana-3855	129	8	1074	1074	NUM
cana-3855	129	9	-	-	PUNCT
cana-3855	129	10	133x	133x	NUM
cana-3855	129	11	vol	vol	NOUN
cana-3855	129	12	32	32	NUM
cana-3855	129	13	no	no	NOUN
cana-3855	129	14	.	.	PUNCT
cana-3855	130	1	9s	9s	NUM
cana-3855	130	2	(	(	PUNCT
cana-3855	130	3	2025	2025	NUM
cana-3855	130	4	)	)	PUNCT
cana-3855	130	5	283	283	NUM
cana-3855	130	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3855	130	7	thus	thus	ADV
cana-3855	130	8	,	,	PUNCT
cana-3855	130	9	every	every	DET
cana-3855	130	10	𝑥	𝑥	PROPN
cana-3855	130	11	∈	∈	PROPN
cana-3855	130	12	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	130	13	(	(	PUNCT
cana-3855	130	14	𝐷-convergent	𝐷-convergent	NOUN
cana-3855	130	15	sequence	sequence	NOUN
cana-3855	130	16	)	)	PUNCT
cana-3855	130	17	is	be	AUX
cana-3855	130	18	also	also	ADV
cana-3855	130	19	an	an	DET
cana-3855	130	20	element	element	NOUN
cana-3855	130	21	of	of	ADP
cana-3855	130	22	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	130	23	∗(quasi	∗(quasi	PUNCT
cana-3855	131	1	𝐷-convergent	𝐷-convergent	NOUN
cana-3855	131	2	sequence	sequence	NOUN
cana-3855	131	3	)	)	PUNCT
cana-3855	131	4	.	.	PUNCT
cana-3855	132	1	therefore	therefore	ADV
cana-3855	132	2	,	,	PUNCT
cana-3855	132	3	we	we	PRON
cana-3855	132	4	conclude	conclude	VERB
cana-3855	132	5	that	that	PRON
cana-3855	132	6	:	:	PUNCT
cana-3855	132	7	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	132	8	⊆	⊆	NUM
cana-3855	132	9	𝑄𝐷	𝑄𝐷	PROPN
cana-3855	132	10	∗	∗	NOUN
cana-3855	132	11	this	this	PRON
cana-3855	132	12	completes	complete	VERB
cana-3855	132	13	the	the	DET
cana-3855	132	14	proof	proof	NOUN
cana-3855	132	15	.	.	PUNCT
cana-3855	133	1	4	4	X
cana-3855	133	2	.	.	X
cana-3855	133	3	conclusion	conclusion	NOUN
cana-3855	133	4	:	:	PUNCT
cana-3855	133	5	this	this	DET
cana-3855	133	6	study	study	NOUN
cana-3855	133	7	presented	present	VERB
cana-3855	133	8	the	the	DET
cana-3855	133	9	quasi	quasi	PROPN
cana-3855	133	10	𝐷-limit	𝐷-limit	PROPN
cana-3855	133	11	for	for	ADP
cana-3855	133	12	bounded	bounded	ADJ
cana-3855	133	13	sequences	sequence	NOUN
cana-3855	133	14	and	and	CCONJ
cana-3855	133	15	demonstrated	demonstrate	VERB
cana-3855	133	16	their	their	PRON
cana-3855	133	17	existence	existence	NOUN
cana-3855	133	18	using	use	VERB
cana-3855	133	19	matrix	matrix	NOUN
cana-3855	133	20	transformations	transformation	NOUN
cana-3855	133	21	.	.	PUNCT
cana-3855	134	1	the	the	DET
cana-3855	134	2	findings	finding	NOUN
cana-3855	134	3	indicate	indicate	VERB
cana-3855	134	4	that	that	DET
cana-3855	134	5	matrix	matrix	NOUN
cana-3855	134	6	transformations	transformation	NOUN
cana-3855	134	7	are	be	AUX
cana-3855	134	8	an	an	DET
cana-3855	134	9	effective	effective	ADJ
cana-3855	134	10	tool	tool	NOUN
cana-3855	134	11	for	for	ADP
cana-3855	134	12	examining	examine	VERB
cana-3855	134	13	the	the	DET
cana-3855	134	14	limiting	limit	VERB
cana-3855	134	15	behaviour	behaviour	NOUN
cana-3855	134	16	of	of	ADP
cana-3855	134	17	various	various	ADJ
cana-3855	134	18	types	type	NOUN
cana-3855	134	19	of	of	ADP
cana-3855	134	20	sequences	sequence	NOUN
cana-3855	134	21	within	within	ADP
cana-3855	134	22	their	their	PRON
cana-3855	134	23	respective	respective	ADJ
cana-3855	134	24	spaces	space	NOUN
cana-3855	134	25	.	.	PUNCT
cana-3855	135	1	furthermore	furthermore	ADV
cana-3855	135	2	,	,	PUNCT
cana-3855	135	3	the	the	DET
cana-3855	135	4	inclusions	inclusion	NOUN
cana-3855	135	5	for	for	ADP
cana-3855	135	6	quasi	quasi	NOUN
cana-3855	135	7	𝐷-limit	𝐷-limit	PROPN
cana-3855	135	8	convergent	convergent	NOUN
cana-3855	135	9	and	and	CCONJ
cana-3855	135	10	𝐷-convergent	𝐷-convergent	NOUN
cana-3855	135	11	sequences	sequence	NOUN
cana-3855	135	12	pave	pave	VERB
cana-3855	135	13	the	the	DET
cana-3855	135	14	way	way	NOUN
cana-3855	135	15	for	for	ADP
cana-3855	135	16	more	more	ADJ
cana-3855	135	17	in	in	ADP
cana-3855	135	18	-	-	PUNCT
cana-3855	135	19	depth	depth	NOUN
cana-3855	135	20	sequence	sequence	NOUN
cana-3855	135	21	analysis	analysis	NOUN
cana-3855	135	22	.	.	PUNCT
cana-3855	136	1	these	these	DET
cana-3855	136	2	findings	finding	NOUN
cana-3855	136	3	have	have	VERB
cana-3855	136	4	major	major	ADJ
cana-3855	136	5	implications	implication	NOUN
cana-3855	136	6	for	for	ADP
cana-3855	136	7	a	a	DET
cana-3855	136	8	variety	variety	NOUN
cana-3855	136	9	of	of	ADP
cana-3855	136	10	mathematical	mathematical	ADJ
cana-3855	136	11	topics	topic	NOUN
cana-3855	136	12	,	,	PUNCT
cana-3855	136	13	including	include	VERB
cana-3855	136	14	functional	functional	ADJ
cana-3855	136	15	analysis	analysis	NOUN
cana-3855	136	16	,	,	PUNCT
cana-3855	136	17	topology	topology	NOUN
cana-3855	136	18	,	,	PUNCT
cana-3855	136	19	and	and	CCONJ
cana-3855	136	20	numerical	numerical	ADJ
cana-3855	136	21	analysis	analysis	NOUN
cana-3855	136	22	,	,	PUNCT
cana-3855	136	23	while	while	SCONJ
cana-3855	136	24	also	also	ADV
cana-3855	136	25	laying	lay	VERB
cana-3855	136	26	the	the	DET
cana-3855	136	27	groundwork	groundwork	NOUN
cana-3855	136	28	for	for	ADP
cana-3855	136	29	future	future	ADJ
cana-3855	136	30	study	study	NOUN
cana-3855	136	31	into	into	ADP
cana-3855	136	32	the	the	DET
cana-3855	136	33	interactions	interaction	NOUN
cana-3855	136	34	between	between	ADP
cana-3855	136	35	sequence	sequence	NOUN
cana-3855	136	36	spaces	space	NOUN
cana-3855	136	37	and	and	CCONJ
cana-3855	136	38	their	their	PRON
cana-3855	136	39	transformations	transformation	NOUN
cana-3855	136	40	.	.	PUNCT
cana-3855	137	1	references	reference	NOUN
cana-3855	137	2	:	:	PUNCT
cana-3855	138	1	[	[	X
cana-3855	138	2	1	1	X
cana-3855	138	3	]	]	PUNCT
cana-3855	138	4	banach	banach	NOUN
cana-3855	138	5	,	,	PUNCT
cana-3855	138	6	s.	s.	PROPN
cana-3855	138	7	"	"	PUNCT
cana-3855	138	8	théorie	théorie	PROPN
cana-3855	138	9	des	des	PROPN
cana-3855	138	10	opérations	opérations	PROPN
cana-3855	138	11	linéaires	linéaires	PROPN
cana-3855	138	12	”	"	PUNCT
cana-3855	138	13	,	,	PUNCT
cana-3855	138	14	chelsea	chelsea	PROPN
cana-3855	138	15	publ	publ	PROPN
cana-3855	138	16	.	.	PUNCT
cana-3855	139	1	co.	co.	PROPN
cana-3855	139	2	,	,	PUNCT
cana-3855	139	3	new	new	PROPN
cana-3855	139	4	york	york	PROPN
cana-3855	139	5	(	(	PUNCT
cana-3855	139	6	1955	1955	NUM
cana-3855	139	7	):	):	PUNCT
cana-3855	139	8	17175	17175	NUM
cana-3855	139	9	.	.	PUNCT
cana-3855	140	1	[	[	X
cana-3855	140	2	2	2	NUM
cana-3855	140	3	]	]	X
cana-3855	140	4	bell	bell	NOUN
cana-3855	140	5	,	,	PUNCT
cana-3855	140	6	howard	howard	PROPN
cana-3855	140	7	t.	t.	PROPN
cana-3855	140	8	"	"	PUNCT
cana-3855	140	9	s	s	NOUN
cana-3855	140	10	-	-	PUNCT
cana-3855	140	11	limits	limit	NOUN
cana-3855	140	12	and	and	CCONJ
cana-3855	140	13	a	a	DET
cana-3855	140	14	-	-	PUNCT
cana-3855	140	15	summability	summability	NOUN
cana-3855	140	16	"	"	PUNCT
cana-3855	140	17	,	,	PUNCT
cana-3855	140	18	proceedings	proceeding	NOUN
cana-3855	140	19	of	of	ADP
cana-3855	140	20	the	the	DET
cana-3855	140	21	american	american	PROPN
cana-3855	140	22	mathematical	mathematical	PROPN
cana-3855	140	23	society	society	NOUN
cana-3855	140	24	(	(	PUNCT
cana-3855	140	25	1976	1976	NUM
cana-3855	140	26	):	):	PUNCT
cana-3855	140	27	49	49	NUM
cana-3855	140	28	-	-	SYM
cana-3855	140	29	53	53	NUM
cana-3855	140	30	.	.	PUNCT
cana-3855	141	1	[	[	X
cana-3855	141	2	3	3	NUM
cana-3855	141	3	]	]	X
cana-3855	141	4	das	das	PROPN
cana-3855	141	5	,	,	PUNCT
cana-3855	141	6	gokulananda	gokulananda	NOUN
cana-3855	141	7	,	,	PUNCT
cana-3855	141	8	mishra	mishra	PROPN
cana-3855	141	9	,	,	PUNCT
cana-3855	141	10	sakambari	sakambari	NOUN
cana-3855	141	11	and	and	CCONJ
cana-3855	141	12	ray	ray	PROPN
cana-3855	141	13	,	,	PUNCT
cana-3855	141	14	braja	braja	PROPN
cana-3855	141	15	kishore	kishore	PROPN
cana-3855	141	16	.	.	PUNCT
cana-3855	142	1	“	"	PUNCT
cana-3855	142	2	quasi	quasi	ADJ
cana-3855	142	3	banach	banach	NOUN
cana-3855	142	4	limits	limit	NOUN
cana-3855	142	5	”	"	PUNCT
cana-3855	142	6	,	,	PUNCT
cana-3855	142	7	journal	journal	NOUN
cana-3855	142	8	of	of	ADP
cana-3855	142	9	odisha	odisha	PROPN
cana-3855	142	10	mathematical	mathematical	ADJ
cana-3855	142	11	society	society	NOUN
cana-3855	142	12	,	,	PUNCT
cana-3855	142	13	v.30,no1(2011	v.30,no1(2011	NUM
cana-3855	142	14	)	)	PUNCT
cana-3855	142	15	,	,	PUNCT
cana-3855	142	16	p.111	p.111	NOUN
cana-3855	142	17	-	-	X
cana-3855	142	18	114	114	NUM
cana-3855	142	19	.	.	PUNCT
cana-3855	143	1	[	[	X
cana-3855	143	2	4	4	NUM
cana-3855	143	3	]	]	SYM
cana-3855	143	4	hajduković	hajduković	NOUN
cana-3855	143	5	,	,	PUNCT
cana-3855	143	6	dimitrije	dimitrije	NOUN
cana-3855	143	7	.	.	PUNCT
cana-3855	144	1	"	"	PUNCT
cana-3855	144	2	quasi	quasi	ADJ
cana-3855	144	3	-	-	ADJ
cana-3855	144	4	almost	almost	ADV
cana-3855	144	5	convergence	convergence	NOUN
cana-3855	144	6	in	in	ADP
cana-3855	144	7	a	a	DET
cana-3855	144	8	normed	normed	ADJ
cana-3855	144	9	space	space	NOUN
cana-3855	144	10	"	"	PUNCT
cana-3855	144	11	,	,	PUNCT
cana-3855	144	12	univ	univ	PROPN
cana-3855	144	13	.	.	PUNCT
cana-3855	145	1	beograd	beograd	PROPN
cana-3855	145	2	.	.	PUNCT
cana-3855	146	1	publikacije	publikacije	NOUN
cana-3855	146	2	elektrotehničkog	elektrotehničkog	PROPN
cana-3855	146	3	fakulteta	fakulteta	PROPN
cana-3855	146	4	.	.	PUNCT
cana-3855	146	5	serija	serija	VERB
cana-3855	146	6	matematika	matematika	PROPN
cana-3855	146	7	,	,	PUNCT
cana-3855	146	8	vol	vol	NOUN
cana-3855	146	9	13	13	NUM
cana-3855	146	10	,	,	PUNCT
cana-3855	146	11	no	no	DET
cana-3855	146	12	3(2002	3(2002	NOUN
cana-3855	146	13	):	):	PUNCT
cana-3855	146	14	36	36	NUM
cana-3855	146	15	-	-	SYM
cana-3855	146	16	41	41	NUM
cana-3855	146	17	.	.	PUNCT
cana-3855	147	1	[	[	X
cana-3855	147	2	5	5	NUM
cana-3855	147	3	]	]	SYM
cana-3855	147	4	mishra	mishra	PROPN
cana-3855	147	5	,	,	PUNCT
cana-3855	147	6	sakambari	sakambari	ADJ
cana-3855	147	7	”	"	PUNCT
cana-3855	147	8	quasi	quasi	NOUN
cana-3855	147	9	invariant	invariant	ADJ
cana-3855	147	10	limits	limit	NOUN
cana-3855	147	11	”	"	PUNCT
cana-3855	147	12	,	,	PUNCT
cana-3855	147	13	journal	journal	NOUN
cana-3855	147	14	of	of	ADP
cana-3855	147	15	orissa	orissa	PROPN
cana-3855	147	16	mathematical	mathematical	PROPN
cana-3855	147	17	society	society	NOUN
cana-3855	147	18	,	,	PUNCT
cana-3855	147	19	v43,no1	v43,no1	PROPN
cana-3855	147	20	-	-	PUNCT
cana-3855	147	21	2(2024	2(2024	NUM
cana-3855	147	22	)	)	PUNCT
cana-3855	147	23	,	,	PUNCT
cana-3855	147	24	p.29	p.29	NOUN
cana-3855	147	25	-	-	SYM
cana-3855	147	26	33	33	NUM
cana-3855	147	27	.	.	PUNCT
cana-3855	148	1	[	[	X
cana-3855	148	2	6	6	NUM
cana-3855	148	3	]	]	SYM
cana-3855	148	4	mishra	mishra	PROPN
cana-3855	148	5	,	,	PUNCT
cana-3855	148	6	sakambari	sakambari	NOUN
cana-3855	148	7	,	,	PUNCT
cana-3855	148	8	das	das	PROPN
cana-3855	148	9	,	,	PUNCT
cana-3855	148	10	g.	g.	PROPN
cana-3855	148	11	and	and	CCONJ
cana-3855	148	12	ray	ray	PROPN
cana-3855	148	13	,	,	PUNCT
cana-3855	148	14	b.k	b.k	PROPN
cana-3855	148	15	.	.	PUNCT
cana-3855	148	16	“	"	PUNCT
cana-3855	148	17	banach	banach	NOUN
cana-3855	148	18	and	and	CCONJ
cana-3855	148	19	knopp	knopp	ADJ
cana-3855	148	20	’s	’s	PART
cana-3855	148	21	core	core	NOUN
cana-3855	148	22	threorems	threorem	NOUN
cana-3855	148	23	and	and	CCONJ
cana-3855	148	24	classes	class	NOUN
cana-3855	148	25	of	of	ADP
cana-3855	148	26	conservative	conservative	ADJ
cana-3855	148	27	matrices	matrix	NOUN
cana-3855	148	28	“	"	PUNCT
cana-3855	148	29	,	,	PUNCT
cana-3855	148	30	azerbaijan	azerbaijan	PROPN
cana-3855	148	31	journal	journal	PROPN
cana-3855	148	32	of	of	ADP
cana-3855	148	33	mathematics	mathematic	NOUN
cana-3855	148	34	,	,	PUNCT
cana-3855	148	35	v.15,no1.(2025	v.15,no1.(2025	PROPN
cana-3855	148	36	)	)	PUNCT
cana-3855	148	37	,	,	PUNCT
cana-3855	148	38	p.11	p.11	PROPN
cana-3855	148	39	-	-	X
cana-3855	148	40	24	24	NUM
cana-3855	148	41	.	.	PUNCT
cana-3855	149	1	[	[	X
cana-3855	149	2	7	7	NUM
cana-3855	149	3	]	]	X
cana-3855	149	4	nuray	nuray	NOUN
cana-3855	149	5	,	,	PUNCT
cana-3855	149	6	fatih	fatih	PROPN
cana-3855	149	7	.	.	PUNCT
cana-3855	150	1	“	"	PUNCT
cana-3855	150	2	quasi	quasi	ADJ
cana-3855	150	3	-	-	ADJ
cana-3855	150	4	invariant	invariant	ADJ
cana-3855	150	5	convergence	convergence	NOUN
cana-3855	150	6	in	in	ADP
cana-3855	150	7	a	a	DET
cana-3855	150	8	normed	normed	ADJ
cana-3855	150	9	space	space	NOUN
cana-3855	150	10	“	"	PUNCT
cana-3855	150	11	,	,	PUNCT
cana-3855	150	12	annals	annal	NOUN
cana-3855	150	13	of	of	ADP
cana-3855	150	14	the	the	DET
cana-3855	150	15	university	university	PROPN
cana-3855	150	16	of	of	ADP
cana-3855	150	17	craiova	craiova	PROPN
cana-3855	150	18	,	,	PUNCT
cana-3855	150	19	mathematics	mathematics	NOUN
cana-3855	150	20	and	and	CCONJ
cana-3855	150	21	computer	computer	NOUN
cana-3855	150	22	science	science	NOUN
cana-3855	150	23	series	series	NOUN
cana-3855	150	24	,	,	PUNCT
cana-3855	150	25	v.41	v.41	ADP
cana-3855	150	26	,	,	PUNCT
cana-3855	150	27	no1(2014	no1(2014	NOUN
cana-3855	150	28	)	)	PUNCT
cana-3855	150	29	,	,	PUNCT
cana-3855	150	30	p.15	p.15	PROPN
cana-3855	150	31	.	.	PUNCT
cana-3855	151	1	[	[	X
cana-3855	151	2	8	8	NUM
cana-3855	151	3	]	]	SYM
cana-3855	151	4	stieglitz	stieglitz	PROPN
cana-3855	151	5	,	,	PUNCT
cana-3855	151	6	michael	michael	PROPN
cana-3855	151	7	.	.	PUNCT
cana-3855	152	1	"	"	PUNCT
cana-3855	152	2	eine	eine	PROPN
cana-3855	152	3	verallgemeinerung	verallgemeinerung	PROPN
cana-3855	152	4	des	des	PROPN
cana-3855	152	5	begriffs	begriffs	PROPN
cana-3855	152	6	der	der	PROPN
cana-3855	152	7	fastkonvergenz",math	fastkonvergenz",math	PROPN
cana-3855	152	8	.	.	PUNCT
cana-3855	153	1	japon	japon	PROPN
cana-3855	153	2	18	18	NUM
cana-3855	153	3	,	,	PUNCT
cana-3855	153	4	no	no	INTJ
cana-3855	153	5	.	.	NOUN
cana-3855	153	6	1	1	NUM
cana-3855	153	7	(	(	PUNCT
cana-3855	153	8	1973	1973	NUM
cana-3855	153	9	):	):	PUNCT
cana-3855	153	10	53	53	NUM
cana-3855	153	11	-	-	SYM
cana-3855	153	12	70	70	NUM
cana-3855	153	13	.	.	PUNCT
cana-3855	154	1	[	[	X
cana-3855	154	2	9	9	NUM
cana-3855	154	3	]	]	SYM
cana-3855	154	4	toeplitz	toeplitz	NOUN
cana-3855	154	5	,	,	PUNCT
cana-3855	154	6	otto	otto	PROPN
cana-3855	154	7	.	.	PUNCT
cana-3855	155	1	"	"	PUNCT
cana-3855	155	2	über	über	PROPN
cana-3855	155	3	allgemeine	allgemeine	PROPN
cana-3855	155	4	lineare	lineare	PROPN
cana-3855	155	5	mittelbildungen	mittelbildungen	NOUN
cana-3855	155	6	.	.	PUNCT
cana-3855	155	7	"	"	PUNCT
cana-3855	156	1	prace	prace	PROPN
cana-3855	156	2	matematycznofizyczne	matematycznofizyczne	PROPN
cana-3855	156	3	22	22	NUM
cana-3855	156	4	,	,	PUNCT
cana-3855	156	5	no	no	INTJ
cana-3855	156	6	.	.	NOUN
cana-3855	156	7	1	1	NUM
cana-3855	156	8	(	(	PUNCT
cana-3855	156	9	1911	1911	NUM
cana-3855	156	10	):	):	PUNCT
cana-3855	156	11	113	113	NUM
cana-3855	156	12	-	-	SYM
cana-3855	156	13	119	119	NUM
cana-3855	156	14	.	.	PUNCT
cana-3855	157	1	[	[	X
cana-3855	157	2	10	10	NUM
cana-3855	157	3	]	]	X
cana-3855	157	4	tripathy	tripathy	PROPN
cana-3855	157	5	,	,	PUNCT
cana-3855	157	6	nandita	nandita	PROPN
cana-3855	157	7	.	.	PUNCT
cana-3855	157	8	:	:	PUNCT
cana-3855	158	1	ph.d	ph.d	PROPN
cana-3855	158	2	thesis	thesis	NOUN
cana-3855	158	3	,	,	PUNCT
cana-3855	158	4	approved	approve	VERB
cana-3855	158	5	by	by	ADP
cana-3855	158	6	utkal	utkal	PROPN
cana-3855	158	7	univ	univ	PROPN
cana-3855	158	8	.	.	PUNCT
cana-3855	159	1	bbsr(orissa),2006	bbsr(orissa),2006	PROPN
cana-3855	159	2	.	.	PUNCT