id	sid	tid	token	lemma	pos
cana-5853	1	1	matrix	matrix	NOUN
cana-5853	1	2	transformations	transformation	NOUN
cana-5853	1	3	on	on	ADP
cana-5853	1	4	modulated	modulate	VERB
cana-5853	1	5	orlicz	orlicz	ADJ
cana-5853	1	6	-	-	PUNCT
cana-5853	1	7	type	type	NOUN
cana-5853	1	8	sequence	sequence	NOUN
cana-5853	1	9	spaces	space	VERB
cana-5853	1	10	sanskriti∗	sanskriti∗	PROPN
cana-5853	1	11	dr	dr	PROPN
cana-5853	1	12	.	.	PROPN
cana-5853	1	13	h.	h.	PROPN
cana-5853	1	14	c.	c.	PROPN
cana-5853	1	15	jha†	jha†	PROPN
cana-5853	1	16	abstract	abstract	ADP
cana-5853	1	17	this	this	DET
cana-5853	1	18	paper	paper	NOUN
cana-5853	1	19	investigates	investigate	VERB
cana-5853	1	20	the	the	DET
cana-5853	1	21	boundedness	boundedness	NOUN
cana-5853	1	22	,	,	PUNCT
cana-5853	1	23	compactness	compactness	NOUN
cana-5853	1	24	,	,	PUNCT
cana-5853	1	25	and	and	CCONJ
cana-5853	1	26	spectral	spectral	ADJ
cana-5853	1	27	properties	property	NOUN
cana-5853	1	28	of	of	ADP
cana-5853	1	29	matrix	matrix	NOUN
cana-5853	1	30	transformations	transformation	NOUN
cana-5853	1	31	acting	act	VERB
cana-5853	1	32	on	on	ADP
cana-5853	1	33	modulated	modulate	VERB
cana-5853	1	34	orlicz	orlicz	ADJ
cana-5853	1	35	-	-	PUNCT
cana-5853	1	36	type	type	NOUN
cana-5853	1	37	sequence	sequence	NOUN
cana-5853	1	38	spaces	space	VERB
cana-5853	1	39	.	.	PUNCT
cana-5853	2	1	by	by	ADP
cana-5853	2	2	extending	extend	VERB
cana-5853	2	3	classical	classical	ADJ
cana-5853	2	4	summability	summability	NOUN
cana-5853	2	5	and	and	CCONJ
cana-5853	2	6	operator	operator	NOUN
cana-5853	2	7	theory	theory	NOUN
cana-5853	2	8	to	to	ADP
cana-5853	2	9	these	these	DET
cana-5853	2	10	generalized	generalized	ADJ
cana-5853	2	11	spaces	space	NOUN
cana-5853	2	12	,	,	PUNCT
cana-5853	2	13	we	we	PRON
cana-5853	2	14	develop	develop	VERB
cana-5853	2	15	criteria	criterion	NOUN
cana-5853	2	16	for	for	ADP
cana-5853	2	17	diagonal	diagonal	ADJ
cana-5853	2	18	,	,	PUNCT
cana-5853	2	19	triangular	triangular	NOUN
cana-5853	2	20	,	,	PUNCT
cana-5853	2	21	and	and	CCONJ
cana-5853	2	22	cesàro	cesàro	ADJ
cana-5853	2	23	-	-	PUNCT
cana-5853	2	24	type	type	NOUN
cana-5853	2	25	matrices	matrix	NOUN
cana-5853	2	26	.	.	PUNCT
cana-5853	3	1	applications	application	NOUN
cana-5853	3	2	to	to	PART
cana-5853	3	3	discrete	discrete	VERB
cana-5853	3	4	operator	operator	NOUN
cana-5853	3	5	theory	theory	NOUN
cana-5853	3	6	are	be	AUX
cana-5853	3	7	also	also	ADV
cana-5853	3	8	discussed	discuss	VERB
cana-5853	3	9	.	.	PUNCT
cana-5853	4	1	1	1	NUM
cana-5853	4	2	introduction	introduction	NOUN
cana-5853	4	3	the	the	DET
cana-5853	4	4	study	study	NOUN
cana-5853	4	5	of	of	ADP
cana-5853	4	6	sequence	sequence	NOUN
cana-5853	4	7	spaces	space	NOUN
cana-5853	4	8	has	have	AUX
cana-5853	4	9	long	long	ADV
cana-5853	4	10	held	hold	VERB
cana-5853	4	11	a	a	DET
cana-5853	4	12	central	central	ADJ
cana-5853	4	13	place	place	NOUN
cana-5853	4	14	in	in	ADP
cana-5853	4	15	functional	functional	ADJ
cana-5853	4	16	analysis	analysis	NOUN
cana-5853	4	17	,	,	PUNCT
cana-5853	4	18	operator	operator	NOUN
cana-5853	4	19	theory	theory	NOUN
cana-5853	4	20	,	,	PUNCT
cana-5853	4	21	and	and	CCONJ
cana-5853	4	22	summability	summability	NOUN
cana-5853	4	23	theory	theory	NOUN
cana-5853	4	24	.	.	PUNCT
cana-5853	5	1	classical	classical	ADJ
cana-5853	5	2	spaces	space	NOUN
cana-5853	5	3	such	such	ADJ
cana-5853	5	4	as	as	ADP
cana-5853	5	5	ℓp	ℓp	NOUN
cana-5853	5	6	,	,	PUNCT
cana-5853	5	7	c0	c0	NOUN
cana-5853	5	8	,	,	PUNCT
cana-5853	5	9	and	and	CCONJ
cana-5853	5	10	ℓ∞	ℓ∞	NOUN
cana-5853	5	11	provide	provide	VERB
cana-5853	5	12	the	the	DET
cana-5853	5	13	basic	basic	ADJ
cana-5853	5	14	framework	framework	NOUN
cana-5853	5	15	for	for	ADP
cana-5853	5	16	understanding	understand	VERB
cana-5853	5	17	convergence	convergence	NOUN
cana-5853	5	18	,	,	PUNCT
cana-5853	5	19	boundedness	boundedness	NOUN
cana-5853	5	20	,	,	PUNCT
cana-5853	5	21	and	and	CCONJ
cana-5853	5	22	operator	operator	NOUN
cana-5853	5	23	behavior	behavior	NOUN
cana-5853	5	24	in	in	ADP
cana-5853	5	25	infinitedimensional	infinitedimensional	ADJ
cana-5853	5	26	settings	setting	NOUN
cana-5853	5	27	.	.	PUNCT
cana-5853	6	1	these	these	DET
cana-5853	6	2	spaces	space	NOUN
cana-5853	6	3	offer	offer	VERB
cana-5853	6	4	clean	clean	ADJ
cana-5853	6	5	,	,	PUNCT
cana-5853	6	6	well	well	ADV
cana-5853	6	7	-	-	PUNCT
cana-5853	6	8	understood	understand	VERB
cana-5853	6	9	duality	duality	NOUN
cana-5853	6	10	theory	theory	NOUN
cana-5853	6	11	,	,	PUNCT
cana-5853	6	12	basis	basis	NOUN
cana-5853	6	13	properties	property	NOUN
cana-5853	6	14	,	,	PUNCT
cana-5853	6	15	and	and	CCONJ
cana-5853	6	16	a	a	DET
cana-5853	6	17	robust	robust	ADJ
cana-5853	6	18	operator	operator	NOUN
cana-5853	6	19	calculus	calculus	NOUN
cana-5853	6	20	that	that	PRON
cana-5853	6	21	have	have	AUX
cana-5853	6	22	been	be	AUX
cana-5853	6	23	applied	apply	VERB
cana-5853	6	24	in	in	ADP
cana-5853	6	25	approximation	approximation	NOUN
cana-5853	6	26	theory	theory	NOUN
cana-5853	6	27	,	,	PUNCT
cana-5853	6	28	fourier	fourier	ADJ
cana-5853	6	29	analysis	analysis	NOUN
cana-5853	6	30	,	,	PUNCT
cana-5853	6	31	and	and	CCONJ
cana-5853	6	32	numerical	numerical	ADJ
cana-5853	6	33	methods	method	NOUN
cana-5853	6	34	.	.	PUNCT
cana-5853	7	1	in	in	ADP
cana-5853	7	2	the	the	DET
cana-5853	7	3	early	early	ADJ
cana-5853	7	4	20th	20th	ADJ
cana-5853	7	5	century	century	NOUN
cana-5853	7	6	,	,	PUNCT
cana-5853	7	7	researchers	researcher	NOUN
cana-5853	7	8	recognized	recognize	VERB
cana-5853	7	9	the	the	DET
cana-5853	7	10	limitations	limitation	NOUN
cana-5853	7	11	of	of	ADP
cana-5853	7	12	these	these	DET
cana-5853	7	13	classical	classical	ADJ
cana-5853	7	14	spaces	space	NOUN
cana-5853	7	15	in	in	ADP
cana-5853	7	16	modeling	model	VERB
cana-5853	7	17	sequences	sequence	NOUN
cana-5853	7	18	whose	whose	DET
cana-5853	7	19	entries	entry	NOUN
cana-5853	7	20	may	may	AUX
cana-5853	7	21	exhibit	exhibit	VERB
cana-5853	7	22	varying	vary	VERB
cana-5853	7	23	growth	growth	NOUN
cana-5853	7	24	or	or	CCONJ
cana-5853	7	25	decay	decay	NOUN
cana-5853	7	26	rates	rate	NOUN
cana-5853	7	27	.	.	PUNCT
cana-5853	8	1	to	to	PART
cana-5853	8	2	address	address	VERB
cana-5853	8	3	this	this	PRON
cana-5853	8	4	,	,	PUNCT
cana-5853	8	5	mathematicians	mathematician	NOUN
cana-5853	8	6	introduced	introduce	VERB
cana-5853	8	7	orlicz	orlicz	ADJ
cana-5853	8	8	sequence	sequence	NOUN
cana-5853	8	9	spaces	space	NOUN
cana-5853	8	10	,	,	PUNCT
cana-5853	8	11	generalizing	generalize	VERB
cana-5853	8	12	ℓp	ℓp	ADJ
cana-5853	8	13	spaces	space	NOUN
cana-5853	8	14	by	by	ADP
cana-5853	8	15	replacing	replace	VERB
cana-5853	8	16	the	the	DET
cana-5853	8	17	fixed	fix	VERB
cana-5853	8	18	power	power	NOUN
cana-5853	8	19	function	function	NOUN
cana-5853	8	20	with	with	ADP
cana-5853	8	21	a	a	DET
cana-5853	8	22	convex	convex	NOUN
cana-5853	8	23	,	,	PUNCT
cana-5853	8	24	increasing	increase	VERB
cana-5853	8	25	orlicz	orlicz	ADJ
cana-5853	8	26	function	function	NOUN
cana-5853	8	27	.	.	PUNCT
cana-5853	9	1	these	these	DET
cana-5853	9	2	spaces	space	NOUN
cana-5853	9	3	allowed	allow	VERB
cana-5853	9	4	for	for	ADP
cana-5853	9	5	greater	great	ADJ
cana-5853	9	6	flexibility	flexibility	NOUN
cana-5853	9	7	,	,	PUNCT
cana-5853	9	8	capturing	capture	VERB
cana-5853	9	9	behaviors	behavior	NOUN
cana-5853	9	10	that	that	PRON
cana-5853	9	11	lie	lie	VERB
cana-5853	9	12	outside	outside	ADP
cana-5853	9	13	the	the	DET
cana-5853	9	14	scope	scope	NOUN
cana-5853	9	15	of	of	ADP
cana-5853	9	16	power	power	NOUN
cana-5853	9	17	growth	growth	NOUN
cana-5853	9	18	and	and	CCONJ
cana-5853	9	19	enabling	enable	VERB
cana-5853	9	20	finer	fine	ADJ
cana-5853	9	21	analysis	analysis	NOUN
cana-5853	9	22	of	of	ADP
cana-5853	9	23	convergence	convergence	NOUN
cana-5853	9	24	and	and	CCONJ
cana-5853	9	25	summability	summability	NOUN
cana-5853	9	26	.	.	PUNCT
cana-5853	10	1	the	the	DET
cana-5853	10	2	duality	duality	NOUN
cana-5853	10	3	theory	theory	NOUN
cana-5853	10	4	of	of	ADP
cana-5853	10	5	orlicz	orlicz	PROPN
cana-5853	10	6	spaces	space	NOUN
cana-5853	10	7	,	,	PUNCT
cana-5853	10	8	relying	rely	VERB
cana-5853	10	9	on	on	ADP
cana-5853	10	10	complementary	complementary	ADJ
cana-5853	10	11	functions	function	NOUN
cana-5853	10	12	and	and	CCONJ
cana-5853	10	13	young	young	ADJ
cana-5853	10	14	’s	’s	PART
cana-5853	10	15	inequality	inequality	NOUN
cana-5853	10	16	,	,	PUNCT
cana-5853	10	17	became	become	VERB
cana-5853	10	18	a	a	DET
cana-5853	10	19	standard	standard	ADJ
cana-5853	10	20	tool	tool	NOUN
cana-5853	10	21	in	in	ADP
cana-5853	10	22	functional	functional	ADJ
cana-5853	10	23	analysis	analysis	NOUN
cana-5853	10	24	.	.	PUNCT
cana-5853	11	1	yet	yet	CCONJ
cana-5853	11	2	even	even	ADV
cana-5853	11	3	orlicz	orlicz	ADJ
cana-5853	11	4	sequence	sequence	NOUN
cana-5853	11	5	spaces	space	NOUN
cana-5853	11	6	impose	impose	VERB
cana-5853	11	7	a	a	DET
cana-5853	11	8	certain	certain	ADJ
cana-5853	11	9	uniformity	uniformity	NOUN
cana-5853	11	10	:	:	PUNCT
cana-5853	11	11	the	the	DET
cana-5853	11	12	same	same	ADJ
cana-5853	11	13	orlicz	orlicz	NOUN
cana-5853	11	14	function	function	NOUN
cana-5853	11	15	governs	govern	VERB
cana-5853	11	16	the	the	DET
cana-5853	11	17	growth	growth	NOUN
cana-5853	11	18	condition	condition	NOUN
cana-5853	11	19	at	at	ADP
cana-5853	11	20	every	every	DET
cana-5853	11	21	coordinate	coordinate	NOUN
cana-5853	11	22	.	.	PUNCT
cana-5853	12	1	this	this	DET
cana-5853	12	2	assumption	assumption	NOUN
cana-5853	12	3	can	can	AUX
cana-5853	12	4	be	be	AUX
cana-5853	12	5	too	too	ADV
cana-5853	12	6	restrictive	restrictive	ADJ
cana-5853	12	7	in	in	ADP
cana-5853	12	8	real	real	ADJ
cana-5853	12	9	-	-	PUNCT
cana-5853	12	10	world	world	NOUN
cana-5853	12	11	applications	application	NOUN
cana-5853	12	12	where	where	SCONJ
cana-5853	12	13	the	the	DET
cana-5853	12	14	importance	importance	NOUN
cana-5853	12	15	,	,	PUNCT
cana-5853	12	16	weight	weight	NOUN
cana-5853	12	17	,	,	PUNCT
cana-5853	12	18	or	or	CCONJ
cana-5853	12	19	variability	variability	NOUN
cana-5853	12	20	of	of	ADP
cana-5853	12	21	sequence	sequence	NOUN
cana-5853	12	22	entries	entry	NOUN
cana-5853	12	23	may	may	AUX
cana-5853	12	24	depend	depend	VERB
cana-5853	12	25	on	on	ADP
cana-5853	12	26	their	their	PRON
cana-5853	12	27	position	position	NOUN
cana-5853	12	28	.	.	PUNCT
cana-5853	13	1	for	for	ADP
cana-5853	13	2	example	example	NOUN
cana-5853	13	3	,	,	PUNCT
cana-5853	13	4	in	in	ADP
cana-5853	13	5	signal	signal	ADJ
cana-5853	13	6	processing	processing	NOUN
cana-5853	13	7	,	,	PUNCT
cana-5853	13	8	higher	high	ADJ
cana-5853	13	9	-	-	PUNCT
cana-5853	13	10	frequency	frequency	NOUN
cana-5853	13	11	components	component	NOUN
cana-5853	13	12	may	may	AUX
cana-5853	13	13	be	be	AUX
cana-5853	13	14	penalized	penalize	VERB
cana-5853	13	15	more	more	ADV
cana-5853	13	16	heavily	heavily	ADV
cana-5853	13	17	to	to	PART
cana-5853	13	18	enforce	enforce	VERB
cana-5853	13	19	smoothness	smoothness	NOUN
cana-5853	13	20	;	;	PUNCT
cana-5853	13	21	in	in	ADP
cana-5853	13	22	numerical	numerical	ADJ
cana-5853	13	23	methods	method	NOUN
cana-5853	13	24	,	,	PUNCT
cana-5853	13	25	discretizations	discretization	NOUN
cana-5853	13	26	∗research	∗research	NUM
cana-5853	13	27	scholar	scholar	NOUN
cana-5853	13	28	,	,	PUNCT
cana-5853	13	29	university	university	NOUN
cana-5853	13	30	department	department	NOUN
cana-5853	13	31	of	of	ADP
cana-5853	13	32	mathematics	mathematics	PROPN
cana-5853	13	33	,	,	PUNCT
cana-5853	13	34	lalit	lalit	PROPN
cana-5853	13	35	narayan	narayan	PROPN
cana-5853	13	36	mithila	mithila	PROPN
cana-5853	13	37	university	university	PROPN
cana-5853	13	38	,	,	PUNCT
cana-5853	13	39	darbhanga	darbhanga	NOUN
cana-5853	13	40	,	,	PUNCT
cana-5853	13	41	bihar	bihar	PROPN
cana-5853	13	42	.	.	PUNCT
cana-5853	13	43	†retired	†retire	VERB
cana-5853	13	44	professor	professor	NOUN
cana-5853	13	45	and	and	CCONJ
cana-5853	13	46	head	head	NOUN
cana-5853	13	47	,	,	PUNCT
cana-5853	13	48	university	university	NOUN
cana-5853	13	49	department	department	NOUN
cana-5853	13	50	of	of	ADP
cana-5853	13	51	mathematics	mathematics	PROPN
cana-5853	13	52	,	,	PUNCT
cana-5853	13	53	lalit	lalit	PROPN
cana-5853	13	54	narayan	narayan	PROPN
cana-5853	13	55	mithila	mithila	PROPN
cana-5853	13	56	university	university	PROPN
cana-5853	13	57	,	,	PUNCT
cana-5853	13	58	darbhanga	darbhanga	NOUN
cana-5853	13	59	,	,	PUNCT
cana-5853	13	60	bihar	bihar	NOUN
cana-5853	13	61	.	.	PUNCT
cana-5853	14	1	communications	communication	NOUN
cana-5853	14	2	on	on	ADP
cana-5853	14	3	applied	apply	VERB
cana-5853	14	4	nonlinear	nonlinear	ADJ
cana-5853	14	5	analysis	analysis	NOUN
cana-5853	14	6	issn	issn	NOUN
cana-5853	14	7	:	:	PUNCT
cana-5853	14	8	1074	1074	NUM
cana-5853	14	9	-	-	PUNCT
cana-5853	14	10	133x	133x	NUM
cana-5853	14	11	vol	vol	NOUN
cana-5853	14	12	31	31	NUM
cana-5853	14	13	no	no	NOUN
cana-5853	14	14	.	.	NOUN
cana-5853	14	15	2	2	NUM
cana-5853	14	16	(	(	PUNCT
cana-5853	14	17	2024	2024	NUM
cana-5853	14	18	)	)	PUNCT
cana-5853	15	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	15	2	461	461	NUM
cana-5853	15	3	on	on	ADP
cana-5853	15	4	non	non	ADJ
cana-5853	15	5	-	-	ADJ
cana-5853	15	6	uniform	uniform	ADJ
cana-5853	15	7	grids	grid	NOUN
cana-5853	15	8	naturally	naturally	ADV
cana-5853	15	9	lead	lead	VERB
cana-5853	15	10	to	to	ADP
cana-5853	15	11	non	non	ADJ
cana-5853	15	12	-	-	ADJ
cana-5853	15	13	uniform	uniform	ADJ
cana-5853	15	14	weighting	weighting	NOUN
cana-5853	15	15	.	.	PUNCT
cana-5853	16	1	in	in	ADP
cana-5853	16	2	such	such	ADJ
cana-5853	16	3	contexts	context	NOUN
cana-5853	16	4	,	,	PUNCT
cana-5853	16	5	a	a	DET
cana-5853	16	6	more	more	ADV
cana-5853	16	7	refined	refined	ADJ
cana-5853	16	8	model	model	NOUN
cana-5853	16	9	is	be	AUX
cana-5853	16	10	needed	need	VERB
cana-5853	16	11	.	.	PUNCT
cana-5853	17	1	modulated	modulate	VERB
cana-5853	17	2	orlicz	orlicz	ADJ
cana-5853	17	3	-	-	PUNCT
cana-5853	17	4	type	type	NOUN
cana-5853	17	5	sequence	sequence	NOUN
cana-5853	17	6	spaces	space	NOUN
cana-5853	17	7	offer	offer	VERB
cana-5853	17	8	this	this	DET
cana-5853	17	9	refinement	refinement	NOUN
cana-5853	17	10	.	.	PUNCT
cana-5853	18	1	instead	instead	ADV
cana-5853	18	2	of	of	ADP
cana-5853	18	3	a	a	DET
cana-5853	18	4	single	single	ADJ
cana-5853	18	5	orlicz	orlicz	NOUN
cana-5853	18	6	function	function	NOUN
cana-5853	18	7	applied	apply	VERB
cana-5853	18	8	uniformly	uniformly	ADV
cana-5853	18	9	,	,	PUNCT
cana-5853	18	10	these	these	DET
cana-5853	18	11	spaces	space	NOUN
cana-5853	18	12	use	use	VERB
cana-5853	18	13	an	an	DET
cana-5853	18	14	index	index	NOUN
cana-5853	18	15	-	-	PUNCT
cana-5853	18	16	dependent	dependent	ADJ
cana-5853	18	17	family	family	NOUN
cana-5853	18	18	m(n	m(n	PROPN
cana-5853	18	19	,	,	PUNCT
cana-5853	18	20	t	t	PROPN
cana-5853	18	21	)	)	PUNCT
cana-5853	18	22	that	that	PRON
cana-5853	18	23	allows	allow	VERB
cana-5853	18	24	the	the	DET
cana-5853	18	25	growth	growth	NOUN
cana-5853	18	26	condition	condition	NOUN
cana-5853	18	27	to	to	PART
cana-5853	18	28	vary	vary	VERB
cana-5853	18	29	with	with	ADP
cana-5853	18	30	n.	n.	NOUN
cana-5853	18	31	this	this	DET
cana-5853	18	32	generalization	generalization	NOUN
cana-5853	18	33	opens	open	VERB
cana-5853	18	34	the	the	DET
cana-5853	18	35	door	door	NOUN
cana-5853	18	36	to	to	ADP
cana-5853	18	37	analyzing	analyze	VERB
cana-5853	18	38	sequences	sequence	NOUN
cana-5853	18	39	with	with	ADP
cana-5853	18	40	spatially	spatially	ADV
cana-5853	18	41	inhomogeneous	inhomogeneous	ADJ
cana-5853	18	42	behavior	behavior	NOUN
cana-5853	18	43	,	,	PUNCT
cana-5853	18	44	adaptive	adaptive	ADJ
cana-5853	18	45	approximation	approximation	NOUN
cana-5853	18	46	schemes	scheme	NOUN
cana-5853	18	47	,	,	PUNCT
cana-5853	18	48	and	and	CCONJ
cana-5853	18	49	variable	variable	ADJ
cana-5853	18	50	-	-	PUNCT
cana-5853	18	51	exponent	exponent	NOUN
cana-5853	18	52	models	model	NOUN
cana-5853	18	53	.	.	PUNCT
cana-5853	19	1	it	it	PRON
cana-5853	19	2	also	also	ADV
cana-5853	19	3	brings	bring	VERB
cana-5853	19	4	new	new	ADJ
cana-5853	19	5	mathematical	mathematical	ADJ
cana-5853	19	6	challenges	challenge	NOUN
cana-5853	19	7	:	:	PUNCT
cana-5853	20	1	completeness	completeness	NOUN
cana-5853	20	2	,	,	PUNCT
cana-5853	20	3	duality	duality	NOUN
cana-5853	20	4	,	,	PUNCT
cana-5853	20	5	operator	operator	NOUN
cana-5853	20	6	boundedness	boundedness	NOUN
cana-5853	20	7	,	,	PUNCT
cana-5853	20	8	and	and	CCONJ
cana-5853	20	9	compactness	compactness	NOUN
cana-5853	20	10	criteria	criterion	NOUN
cana-5853	20	11	all	all	PRON
cana-5853	20	12	require	require	VERB
cana-5853	20	13	careful	careful	ADJ
cana-5853	20	14	generalization	generalization	NOUN
cana-5853	20	15	.	.	PUNCT
cana-5853	21	1	the	the	DET
cana-5853	21	2	primary	primary	ADJ
cana-5853	21	3	objective	objective	NOUN
cana-5853	21	4	of	of	ADP
cana-5853	21	5	this	this	DET
cana-5853	21	6	paper	paper	NOUN
cana-5853	21	7	is	be	AUX
cana-5853	21	8	to	to	PART
cana-5853	21	9	systematically	systematically	ADV
cana-5853	21	10	develop	develop	VERB
cana-5853	21	11	the	the	DET
cana-5853	21	12	theory	theory	NOUN
cana-5853	21	13	of	of	ADP
cana-5853	21	14	matrix	matrix	NOUN
cana-5853	21	15	transformations	transformation	NOUN
cana-5853	21	16	acting	act	VERB
cana-5853	21	17	on	on	ADP
cana-5853	21	18	modulated	modulate	VERB
cana-5853	21	19	orlicz	orlicz	ADJ
cana-5853	21	20	-	-	PUNCT
cana-5853	21	21	type	type	NOUN
cana-5853	21	22	sequence	sequence	NOUN
cana-5853	21	23	spaces	space	VERB
cana-5853	21	24	.	.	PUNCT
cana-5853	22	1	we	we	PRON
cana-5853	22	2	aim	aim	VERB
cana-5853	22	3	to	to	PART
cana-5853	22	4	:	:	PUNCT
cana-5853	22	5	•	•	ADP
cana-5853	22	6	define	define	VERB
cana-5853	22	7	these	these	DET
cana-5853	22	8	spaces	space	NOUN
cana-5853	22	9	rigorously	rigorously	ADV
cana-5853	22	10	and	and	CCONJ
cana-5853	22	11	explore	explore	VERB
cana-5853	22	12	their	their	PRON
cana-5853	22	13	foundational	foundational	ADJ
cana-5853	22	14	properties	property	NOUN
cana-5853	22	15	.	.	PUNCT
cana-5853	23	1	•	•	NUM
cana-5853	23	2	establish	establish	VERB
cana-5853	23	3	criteria	criterion	NOUN
cana-5853	23	4	for	for	ADP
cana-5853	23	5	the	the	DET
cana-5853	23	6	boundedness	boundedness	NOUN
cana-5853	23	7	and	and	CCONJ
cana-5853	23	8	compactness	compactness	NOUN
cana-5853	23	9	of	of	ADP
cana-5853	23	10	matrix	matrix	NOUN
cana-5853	23	11	operators	operator	NOUN
cana-5853	23	12	,	,	PUNCT
cana-5853	23	13	extending	extend	VERB
cana-5853	23	14	classical	classical	ADJ
cana-5853	23	15	results	result	NOUN
cana-5853	23	16	from	from	ADP
cana-5853	23	17	ℓp	ℓp	ADJ
cana-5853	23	18	and	and	CCONJ
cana-5853	23	19	orlicz	orlicz	ADJ
cana-5853	23	20	spaces	space	NOUN
cana-5853	23	21	.	.	PUNCT
cana-5853	24	1	•	•	NUM
cana-5853	24	2	characterize	characterize	VERB
cana-5853	24	3	special	special	ADJ
cana-5853	24	4	classes	class	NOUN
cana-5853	24	5	of	of	ADP
cana-5853	24	6	matrices	matrix	NOUN
cana-5853	24	7	such	such	ADJ
cana-5853	24	8	as	as	ADP
cana-5853	24	9	diagonal	diagonal	ADJ
cana-5853	24	10	,	,	PUNCT
cana-5853	24	11	triangular	triangular	NOUN
cana-5853	24	12	,	,	PUNCT
cana-5853	24	13	and	and	CCONJ
cana-5853	24	14	cesàro	cesàro	ADJ
cana-5853	24	15	-	-	PUNCT
cana-5853	24	16	type	type	NOUN
cana-5853	24	17	operators	operator	NOUN
cana-5853	24	18	within	within	ADP
cana-5853	24	19	this	this	DET
cana-5853	24	20	modular	modular	ADJ
cana-5853	24	21	framework	framework	NOUN
cana-5853	24	22	.	.	PUNCT
cana-5853	25	1	•	•	NUM
cana-5853	25	2	analyze	analyze	VERB
cana-5853	25	3	the	the	DET
cana-5853	25	4	spectral	spectral	ADJ
cana-5853	25	5	properties	property	NOUN
cana-5853	25	6	of	of	ADP
cana-5853	25	7	such	such	ADJ
cana-5853	25	8	operators	operator	NOUN
cana-5853	25	9	,	,	PUNCT
cana-5853	25	10	particularly	particularly	ADV
cana-5853	25	11	in	in	ADP
cana-5853	25	12	the	the	DET
cana-5853	25	13	context	context	NOUN
cana-5853	25	14	of	of	ADP
cana-5853	25	15	compactness	compactness	NOUN
cana-5853	25	16	.	.	PUNCT
cana-5853	26	1	•	•	NUM
cana-5853	26	2	discuss	discuss	VERB
cana-5853	26	3	potential	potential	ADJ
cana-5853	26	4	applications	application	NOUN
cana-5853	26	5	to	to	ADP
cana-5853	26	6	summability	summability	NOUN
cana-5853	26	7	theory	theory	NOUN
cana-5853	26	8	and	and	CCONJ
cana-5853	26	9	discrete	discrete	ADJ
cana-5853	26	10	operator	operator	NOUN
cana-5853	26	11	theory	theory	NOUN
cana-5853	26	12	,	,	PUNCT
cana-5853	26	13	demonstrating	demonstrate	VERB
cana-5853	26	14	how	how	SCONJ
cana-5853	26	15	these	these	DET
cana-5853	26	16	abstract	abstract	ADJ
cana-5853	26	17	results	result	NOUN
cana-5853	26	18	can	can	AUX
cana-5853	26	19	be	be	AUX
cana-5853	26	20	used	use	VERB
cana-5853	26	21	in	in	ADP
cana-5853	26	22	concrete	concrete	ADJ
cana-5853	26	23	analytic	analytic	ADJ
cana-5853	26	24	settings	setting	NOUN
cana-5853	26	25	.	.	PUNCT
cana-5853	27	1	by	by	ADP
cana-5853	27	2	pursuing	pursue	VERB
cana-5853	27	3	these	these	DET
cana-5853	27	4	goals	goal	NOUN
cana-5853	27	5	,	,	PUNCT
cana-5853	27	6	the	the	DET
cana-5853	27	7	paper	paper	NOUN
cana-5853	27	8	seeks	seek	VERB
cana-5853	27	9	not	not	PART
cana-5853	27	10	only	only	ADV
cana-5853	27	11	to	to	PART
cana-5853	27	12	generalize	generalize	VERB
cana-5853	27	13	existing	exist	VERB
cana-5853	27	14	results	result	NOUN
cana-5853	27	15	to	to	ADP
cana-5853	27	16	a	a	DET
cana-5853	27	17	richer	rich	ADJ
cana-5853	27	18	class	class	NOUN
cana-5853	27	19	of	of	ADP
cana-5853	27	20	sequence	sequence	NOUN
cana-5853	27	21	spaces	space	NOUN
cana-5853	27	22	but	but	CCONJ
cana-5853	27	23	also	also	ADV
cana-5853	27	24	to	to	PART
cana-5853	27	25	provide	provide	VERB
cana-5853	27	26	a	a	DET
cana-5853	27	27	framework	framework	NOUN
cana-5853	27	28	for	for	ADP
cana-5853	27	29	further	further	ADJ
cana-5853	27	30	study	study	NOUN
cana-5853	27	31	in	in	ADP
cana-5853	27	32	operator	operator	NOUN
cana-5853	27	33	theory	theory	NOUN
cana-5853	27	34	,	,	PUNCT
cana-5853	27	35	approximation	approximation	NOUN
cana-5853	27	36	methods	method	NOUN
cana-5853	27	37	,	,	PUNCT
cana-5853	27	38	and	and	CCONJ
cana-5853	27	39	applied	apply	VERB
cana-5853	27	40	analysis	analysis	NOUN
cana-5853	27	41	.	.	PUNCT
cana-5853	28	1	our	our	PRON
cana-5853	28	2	approach	approach	NOUN
cana-5853	28	3	emphasizes	emphasize	VERB
cana-5853	28	4	both	both	DET
cana-5853	28	5	theoretical	theoretical	ADJ
cana-5853	28	6	rigor	rigor	NOUN
cana-5853	28	7	and	and	CCONJ
cana-5853	28	8	practical	practical	ADJ
cana-5853	28	9	relevance	relevance	NOUN
cana-5853	28	10	,	,	PUNCT
cana-5853	28	11	ensuring	ensure	VERB
cana-5853	28	12	that	that	SCONJ
cana-5853	28	13	the	the	DET
cana-5853	28	14	results	result	NOUN
cana-5853	28	15	can	can	AUX
cana-5853	28	16	serve	serve	VERB
cana-5853	28	17	as	as	ADP
cana-5853	28	18	a	a	DET
cana-5853	28	19	foundation	foundation	NOUN
cana-5853	28	20	for	for	ADP
cana-5853	28	21	future	future	ADJ
cana-5853	28	22	research	research	NOUN
cana-5853	28	23	and	and	CCONJ
cana-5853	28	24	applications	application	NOUN
cana-5853	28	25	in	in	ADP
cana-5853	28	26	mathematical	mathematical	ADJ
cana-5853	28	27	analysis	analysis	NOUN
cana-5853	28	28	and	and	CCONJ
cana-5853	28	29	beyond	beyond	ADP
cana-5853	28	30	.	.	NOUN
cana-5853	28	31	2	2	NUM
cana-5853	28	32	preliminaries	preliminary	NOUN
cana-5853	28	33	in	in	ADP
cana-5853	28	34	this	this	DET
cana-5853	28	35	section	section	NOUN
cana-5853	28	36	,	,	PUNCT
cana-5853	28	37	we	we	PRON
cana-5853	28	38	establish	establish	VERB
cana-5853	28	39	the	the	DET
cana-5853	28	40	fundamental	fundamental	ADJ
cana-5853	28	41	definitions	definition	NOUN
cana-5853	28	42	and	and	CCONJ
cana-5853	28	43	notation	notation	NOUN
cana-5853	28	44	necessary	necessary	ADJ
cana-5853	28	45	for	for	ADP
cana-5853	28	46	our	our	PRON
cana-5853	28	47	study	study	NOUN
cana-5853	28	48	of	of	ADP
cana-5853	28	49	modulated	modulate	VERB
cana-5853	28	50	orlicz	orlicz	ADJ
cana-5853	28	51	-	-	PUNCT
cana-5853	28	52	type	type	NOUN
cana-5853	28	53	sequence	sequence	NOUN
cana-5853	28	54	spaces	space	VERB
cana-5853	28	55	.	.	PUNCT
cana-5853	29	1	we	we	PRON
cana-5853	29	2	begin	begin	VERB
cana-5853	29	3	by	by	ADP
cana-5853	29	4	defining	define	VERB
cana-5853	29	5	the	the	DET
cana-5853	29	6	modular	modular	ADJ
cana-5853	29	7	functions	function	NOUN
cana-5853	29	8	that	that	PRON
cana-5853	29	9	govern	govern	VERB
cana-5853	29	10	the	the	DET
cana-5853	29	11	growth	growth	NOUN
cana-5853	29	12	conditions	condition	NOUN
cana-5853	29	13	in	in	ADP
cana-5853	29	14	these	these	DET
cana-5853	29	15	spaces	space	NOUN
cana-5853	29	16	,	,	PUNCT
cana-5853	29	17	then	then	ADV
cana-5853	29	18	introduce	introduce	VERB
cana-5853	29	19	the	the	DET
cana-5853	29	20	spaces	space	NOUN
cana-5853	29	21	themselves	themselves	PRON
cana-5853	29	22	,	,	PUNCT
cana-5853	29	23	their	their	PRON
cana-5853	29	24	associated	associated	ADJ
cana-5853	29	25	norms	norm	NOUN
cana-5853	29	26	(	(	PUNCT
cana-5853	29	27	or	or	CCONJ
cana-5853	29	28	modulars	modular	NOUN
cana-5853	29	29	)	)	PUNCT
cana-5853	29	30	,	,	PUNCT
cana-5853	29	31	and	and	CCONJ
cana-5853	29	32	the	the	DET
cana-5853	29	33	concept	concept	NOUN
cana-5853	29	34	of	of	ADP
cana-5853	29	35	complementary	complementary	ADJ
cana-5853	29	36	modular	modular	ADJ
cana-5853	29	37	functions	function	NOUN
cana-5853	29	38	.	.	PUNCT
cana-5853	30	1	finally	finally	ADV
cana-5853	30	2	,	,	PUNCT
cana-5853	30	3	we	we	PRON
cana-5853	30	4	illustrate	illustrate	VERB
cana-5853	30	5	these	these	DET
cana-5853	30	6	ideas	idea	NOUN
cana-5853	30	7	with	with	ADP
cana-5853	30	8	classical	classical	ADJ
cana-5853	30	9	examples	example	NOUN
cana-5853	30	10	that	that	PRON
cana-5853	30	11	fit	fit	VERB
cana-5853	30	12	within	within	ADP
cana-5853	30	13	this	this	DET
cana-5853	30	14	general	general	ADJ
cana-5853	30	15	framework	framework	NOUN
cana-5853	30	16	.	.	PUNCT
cana-5853	31	1	2.1	2.1	NUM
cana-5853	31	2	modular	modular	ADJ
cana-5853	31	3	functions	function	NOUN
cana-5853	31	4	m(n	m(n	PROPN
cana-5853	31	5	,	,	PUNCT
cana-5853	31	6	t	t	PROPN
cana-5853	31	7	)	)	PUNCT
cana-5853	31	8	a	a	DET
cana-5853	31	9	central	central	ADJ
cana-5853	31	10	feature	feature	NOUN
cana-5853	31	11	of	of	ADP
cana-5853	31	12	modulated	modulate	VERB
cana-5853	31	13	orlicz	orlicz	ADJ
cana-5853	31	14	-	-	PUNCT
cana-5853	31	15	type	type	NOUN
cana-5853	31	16	sequence	sequence	NOUN
cana-5853	31	17	spaces	space	NOUN
cana-5853	31	18	is	be	AUX
cana-5853	31	19	the	the	DET
cana-5853	31	20	use	use	NOUN
cana-5853	31	21	of	of	ADP
cana-5853	31	22	index	index	NOUN
cana-5853	31	23	-	-	PUNCT
cana-5853	31	24	dependent	dependent	ADJ
cana-5853	31	25	modular	modular	ADJ
cana-5853	31	26	functions	function	NOUN
cana-5853	31	27	.	.	PUNCT
cana-5853	32	1	formally	formally	ADV
cana-5853	32	2	,	,	PUNCT
cana-5853	32	3	let	let	VERB
cana-5853	32	4	m	m	PRON
cana-5853	32	5	:	:	PUNCT
cana-5853	32	6	n	n	PROPN
cana-5853	32	7	×	×	NOUN
cana-5853	32	8	f	f	X
cana-5853	32	9	→	→	PUNCT
cana-5853	33	1	[	[	X
cana-5853	33	2	0,∞	0,∞	NOUN
cana-5853	33	3	)	)	PUNCT
cana-5853	33	4	,	,	PUNCT
cana-5853	33	5	where	where	SCONJ
cana-5853	33	6	f	f	PROPN
cana-5853	33	7	is	be	AUX
cana-5853	33	8	either	either	PRON
cana-5853	33	9	r	r	NOUN
cana-5853	33	10	or	or	CCONJ
cana-5853	33	11	c.	c.	NOUN
cana-5853	33	12	for	for	ADP
cana-5853	33	13	each	each	DET
cana-5853	33	14	fixed	fix	VERB
cana-5853	33	15	n	n	PROPN
cana-5853	33	16	∈	∈	PROPN
cana-5853	33	17	n	n	CCONJ
cana-5853	33	18	,	,	PUNCT
cana-5853	33	19	the	the	DET
cana-5853	33	20	function	function	NOUN
cana-5853	33	21	m(n	m(n	PROPN
cana-5853	33	22	,	,	PUNCT
cana-5853	33	23	·	·	PUNCT
cana-5853	33	24	)	)	PUNCT
cana-5853	33	25	is	be	AUX
cana-5853	33	26	assumed	assume	VERB
cana-5853	33	27	to	to	PART
cana-5853	33	28	satisfy	satisfy	VERB
cana-5853	33	29	:	:	PUNCT
cana-5853	33	30	•	•	NUM
cana-5853	33	31	m(n	m(n	PROPN
cana-5853	33	32	,	,	PUNCT
cana-5853	33	33	0	0	NUM
cana-5853	33	34	)	)	PUNCT
cana-5853	33	35	=	=	SYM
cana-5853	34	1	0	0	X
cana-5853	34	2	.	.	PUNCT
cana-5853	35	1	communications	communication	NOUN
cana-5853	35	2	on	on	ADP
cana-5853	35	3	applied	apply	VERB
cana-5853	35	4	nonlinear	nonlinear	ADJ
cana-5853	35	5	analysis	analysis	NOUN
cana-5853	35	6	issn	issn	NOUN
cana-5853	35	7	:	:	PUNCT
cana-5853	35	8	1074	1074	NUM
cana-5853	35	9	-	-	PUNCT
cana-5853	35	10	133x	133x	NUM
cana-5853	35	11	vol	vol	NOUN
cana-5853	35	12	31	31	NUM
cana-5853	35	13	no	no	NOUN
cana-5853	35	14	.	.	NOUN
cana-5853	35	15	2	2	NUM
cana-5853	35	16	(	(	PUNCT
cana-5853	35	17	2024	2024	NUM
cana-5853	35	18	)	)	PUNCT
cana-5853	35	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	36	1	462	462	NUM
cana-5853	36	2	•	•	NUM
cana-5853	36	3	m(n	m(n	PROPN
cana-5853	36	4	,	,	PUNCT
cana-5853	36	5	t	t	PROPN
cana-5853	36	6	)	)	PUNCT
cana-5853	36	7	is	be	AUX
cana-5853	36	8	continuous	continuous	ADJ
cana-5853	36	9	in	in	ADP
cana-5853	36	10	t.	t.	PROPN
cana-5853	36	11	•	•	NOUN
cana-5853	36	12	m(n	m(n	PROPN
cana-5853	36	13	,	,	PUNCT
cana-5853	36	14	t	t	PROPN
cana-5853	36	15	)	)	PUNCT
cana-5853	36	16	is	be	AUX
cana-5853	36	17	even	even	ADV
cana-5853	36	18	and	and	CCONJ
cana-5853	36	19	convex	convex	VERB
cana-5853	36	20	in	in	ADP
cana-5853	36	21	t.	t.	PROPN
cana-5853	36	22	•	•	PROPN
cana-5853	36	23	m(n	m(n	PROPN
cana-5853	36	24	,	,	PUNCT
cana-5853	36	25	t	t	PROPN
cana-5853	36	26	)	)	PUNCT
cana-5853	36	27	is	be	AUX
cana-5853	36	28	increasing	increase	VERB
cana-5853	36	29	for	for	ADP
cana-5853	36	30	t	t	PROPN
cana-5853	36	31	≥	≥	NOUN
cana-5853	36	32	0	0	NUM
cana-5853	36	33	.	.	PUNCT
cana-5853	37	1	additionally	additionally	ADV
cana-5853	37	2	,	,	PUNCT
cana-5853	37	3	to	to	PART
cana-5853	37	4	ensure	ensure	VERB
cana-5853	37	5	desirable	desirable	ADJ
cana-5853	37	6	analytic	analytic	ADJ
cana-5853	37	7	properties	property	NOUN
cana-5853	37	8	(	(	PUNCT
cana-5853	37	9	such	such	ADJ
cana-5853	37	10	as	as	ADP
cana-5853	37	11	completeness	completeness	NOUN
cana-5853	37	12	of	of	ADP
cana-5853	37	13	the	the	DET
cana-5853	37	14	associated	associated	ADJ
cana-5853	37	15	space	space	NOUN
cana-5853	37	16	)	)	PUNCT
cana-5853	37	17	,	,	PUNCT
cana-5853	37	18	we	we	PRON
cana-5853	37	19	often	often	ADV
cana-5853	37	20	impose	impose	VERB
cana-5853	37	21	a	a	DET
cana-5853	37	22	∆2	∆2	NOUN
cana-5853	37	23	-	-	PUNCT
cana-5853	37	24	type	type	NOUN
cana-5853	37	25	condition	condition	NOUN
cana-5853	37	26	:	:	PUNCT
cana-5853	37	27	there	there	PRON
cana-5853	37	28	exists	exist	VERB
cana-5853	37	29	a	a	DET
cana-5853	37	30	constant	constant	ADJ
cana-5853	37	31	k	k	X
cana-5853	37	32	>	>	X
cana-5853	37	33	0	0	NUM
cana-5853	37	34	such	such	ADJ
cana-5853	37	35	that	that	PRON
cana-5853	37	36	for	for	ADP
cana-5853	37	37	all	all	DET
cana-5853	37	38	n	n	PRON
cana-5853	37	39	and	and	CCONJ
cana-5853	37	40	all	all	DET
cana-5853	37	41	t	t	PROPN
cana-5853	37	42	,	,	PUNCT
cana-5853	37	43	m(n	m(n	PROPN
cana-5853	37	44	,	,	PUNCT
cana-5853	37	45	2	2	NUM
cana-5853	37	46	t	t	NOUN
cana-5853	37	47	)	)	PUNCT
cana-5853	37	48	≤	≤	NOUN
cana-5853	37	49	km(n	km(n	X
cana-5853	37	50	,	,	PUNCT
cana-5853	37	51	t	t	PROPN
cana-5853	37	52	)	)	PUNCT
cana-5853	38	1	+	+	ADP
cana-5853	38	2	k.	k.	NOUN
cana-5853	38	3	this	this	DET
cana-5853	38	4	condition	condition	NOUN
cana-5853	38	5	controls	control	VERB
cana-5853	38	6	the	the	DET
cana-5853	38	7	growth	growth	NOUN
cana-5853	38	8	of	of	ADP
cana-5853	38	9	m	m	PRON
cana-5853	38	10	and	and	CCONJ
cana-5853	38	11	ensures	ensure	VERB
cana-5853	38	12	modular	modular	ADJ
cana-5853	38	13	convergence	convergence	NOUN
cana-5853	38	14	is	be	AUX
cana-5853	38	15	compatible	compatible	ADJ
cana-5853	38	16	with	with	ADP
cana-5853	38	17	the	the	DET
cana-5853	38	18	vector	vector	NOUN
cana-5853	38	19	space	space	NOUN
cana-5853	38	20	structure	structure	NOUN
cana-5853	38	21	.	.	PUNCT
cana-5853	39	1	2.2	2.2	NUM
cana-5853	39	2	the	the	DET
cana-5853	39	3	space	space	NOUN
cana-5853	39	4	xm	xm	PROPN
cana-5853	39	5	and	and	CCONJ
cana-5853	39	6	its	its	PRON
cana-5853	39	7	norm	norm	NOUN
cana-5853	39	8	/	/	SYM
cana-5853	39	9	modular	modular	NOUN
cana-5853	39	10	given	give	VERB
cana-5853	39	11	such	such	DET
cana-5853	39	12	a	a	DET
cana-5853	39	13	family	family	NOUN
cana-5853	39	14	of	of	ADP
cana-5853	39	15	modular	modular	ADJ
cana-5853	39	16	functions	function	NOUN
cana-5853	39	17	m(n	m(n	PROPN
cana-5853	39	18	,	,	PUNCT
cana-5853	39	19	t	t	PROPN
cana-5853	39	20	)	)	PUNCT
cana-5853	39	21	,	,	PUNCT
cana-5853	39	22	we	we	PRON
cana-5853	39	23	define	define	VERB
cana-5853	39	24	the	the	DET
cana-5853	39	25	modulated	modulate	VERB
cana-5853	39	26	orlicz	orlicz	ADJ
cana-5853	39	27	-	-	PUNCT
cana-5853	39	28	type	type	NOUN
cana-5853	39	29	sequence	sequence	NOUN
cana-5853	39	30	space	space	NOUN
cana-5853	39	31	xm	xm	PROPN
cana-5853	40	1	=	=	PUNCT
cana-5853	40	2	{	{	PUNCT
cana-5853	40	3	x	x	PUNCT
cana-5853	40	4	=	=	SYM
cana-5853	40	5	(	(	PUNCT
cana-5853	40	6	xn	xn	X
cana-5853	40	7	)	)	PUNCT
cana-5853	40	8	∈	∈	PROPN
cana-5853	40	9	fn	fn	NOUN
cana-5853	40	10	:	:	PUNCT
cana-5853	40	11	ρm(x	ρm(x	NUM
cana-5853	40	12	)	)	PUNCT
cana-5853	40	13	:	:	PUNCT
cana-5853	41	1	=	=	NOUN
cana-5853	41	2	∞∑	∞∑	NUM
cana-5853	41	3	n=1	n=1	ADP
cana-5853	41	4	m(n	m(n	PROPN
cana-5853	41	5	,	,	PUNCT
cana-5853	41	6	xn	xn	NUM
cana-5853	41	7	)	)	PUNCT
cana-5853	41	8	<	<	X
cana-5853	41	9	∞	∞	PROPN
cana-5853	41	10	}	}	PUNCT
cana-5853	41	11	.	.	PUNCT
cana-5853	42	1	the	the	DET
cana-5853	42	2	quantity	quantity	NOUN
cana-5853	42	3	ρm(x	ρm(x	NOUN
cana-5853	42	4	)	)	PUNCT
cana-5853	42	5	is	be	AUX
cana-5853	42	6	called	call	VERB
cana-5853	42	7	the	the	DET
cana-5853	42	8	modular	modular	NOUN
cana-5853	42	9	of	of	ADP
cana-5853	42	10	x.	x.	NOUN
cana-5853	42	11	under	under	ADP
cana-5853	42	12	mild	mild	ADJ
cana-5853	42	13	conditions	condition	NOUN
cana-5853	42	14	on	on	ADP
cana-5853	42	15	m	m	PROPN
cana-5853	42	16	(	(	PUNCT
cana-5853	42	17	including	include	VERB
cana-5853	42	18	convexity	convexity	NOUN
cana-5853	42	19	and	and	CCONJ
cana-5853	42	20	∆2	∆2	NOUN
cana-5853	42	21	-	-	PUNCT
cana-5853	42	22	type	type	NOUN
cana-5853	42	23	growth	growth	NOUN
cana-5853	42	24	)	)	PUNCT
cana-5853	42	25	,	,	PUNCT
cana-5853	42	26	ρm	ρm	PROPN
cana-5853	42	27	behaves	behave	VERB
cana-5853	42	28	analogously	analogously	ADV
cana-5853	42	29	to	to	ADP
cana-5853	42	30	a	a	DET
cana-5853	42	31	norm	norm	NOUN
cana-5853	42	32	and	and	CCONJ
cana-5853	42	33	can	can	AUX
cana-5853	42	34	often	often	ADV
cana-5853	42	35	be	be	AUX
cana-5853	42	36	used	use	VERB
cana-5853	42	37	to	to	PART
cana-5853	42	38	define	define	VERB
cana-5853	42	39	an	an	DET
cana-5853	42	40	equivalent	equivalent	ADJ
cana-5853	42	41	norm	norm	NOUN
cana-5853	42	42	on	on	ADP
cana-5853	42	43	xm	xm	PROPN
cana-5853	42	44	.	.	PUNCT
cana-5853	43	1	in	in	ADP
cana-5853	43	2	many	many	ADJ
cana-5853	43	3	treatments	treatment	NOUN
cana-5853	43	4	,	,	PUNCT
cana-5853	43	5	one	one	NUM
cana-5853	43	6	introduces	introduce	VERB
cana-5853	43	7	the	the	DET
cana-5853	43	8	luxemburg	luxemburg	PROPN
cana-5853	43	9	norm	norm	NOUN
cana-5853	43	10	:	:	PUNCT
cana-5853	44	1	∥x∥m	∥x∥m	X
cana-5853	44	2	=	=	X
cana-5853	44	3	inf	inf	NOUN
cana-5853	44	4	{	{	PUNCT
cana-5853	44	5	λ	λ	X
cana-5853	44	6	>	>	X
cana-5853	44	7	0	0	NUM
cana-5853	44	8	:	:	PUNCT
cana-5853	44	9	ρm	ρm	INTJ
cana-5853	44	10	(	(	PUNCT
cana-5853	44	11	x	x	PART
cana-5853	44	12	λ	λ	PROPN
cana-5853	44	13	)	)	PUNCT
cana-5853	44	14	≤	≤	NUM
cana-5853	44	15	1	1	NUM
cana-5853	44	16	}	}	PUNCT
cana-5853	44	17	.	.	PUNCT
cana-5853	45	1	this	this	DET
cana-5853	45	2	norm	norm	NOUN
cana-5853	45	3	turns	turn	VERB
cana-5853	45	4	xm	xm	PROPN
cana-5853	45	5	into	into	ADP
cana-5853	45	6	a	a	DET
cana-5853	45	7	banach	banach	NOUN
cana-5853	45	8	space	space	NOUN
cana-5853	45	9	under	under	ADP
cana-5853	45	10	appropriate	appropriate	ADJ
cana-5853	45	11	conditions	condition	NOUN
cana-5853	45	12	,	,	PUNCT
cana-5853	45	13	ensuring	ensure	VERB
cana-5853	45	14	the	the	DET
cana-5853	45	15	applicability	applicability	NOUN
cana-5853	45	16	of	of	ADP
cana-5853	45	17	standard	standard	ADJ
cana-5853	45	18	tools	tool	NOUN
cana-5853	45	19	of	of	ADP
cana-5853	45	20	functional	functional	ADJ
cana-5853	45	21	analysis	analysis	NOUN
cana-5853	45	22	.	.	PUNCT
cana-5853	46	1	2.3	2.3	NUM
cana-5853	46	2	complementary	complementary	ADJ
cana-5853	46	3	modular	modular	ADJ
cana-5853	46	4	functions	function	NOUN
cana-5853	46	5	m∗(n	m∗(n	PROPN
cana-5853	46	6	,	,	PUNCT
cana-5853	46	7	y	y	PROPN
cana-5853	46	8	)	)	PUNCT
cana-5853	46	9	a	a	DET
cana-5853	46	10	crucial	crucial	ADJ
cana-5853	46	11	concept	concept	NOUN
cana-5853	46	12	in	in	ADP
cana-5853	46	13	duality	duality	NOUN
cana-5853	46	14	theory	theory	NOUN
cana-5853	46	15	for	for	ADP
cana-5853	46	16	modular	modular	ADJ
cana-5853	46	17	spaces	space	NOUN
cana-5853	46	18	is	be	AUX
cana-5853	46	19	the	the	DET
cana-5853	46	20	notion	notion	NOUN
cana-5853	46	21	of	of	ADP
cana-5853	46	22	the	the	DET
cana-5853	46	23	complementary	complementary	ADJ
cana-5853	46	24	modular	modular	ADJ
cana-5853	46	25	function	function	NOUN
cana-5853	46	26	,	,	PUNCT
cana-5853	46	27	generalizing	generalize	VERB
cana-5853	46	28	the	the	DET
cana-5853	46	29	legendre	legendre	PROPN
cana-5853	46	30	-	-	PUNCT
cana-5853	46	31	fenchel	fenchel	PROPN
cana-5853	46	32	transform	transform	NOUN
cana-5853	46	33	.	.	PUNCT
cana-5853	47	1	for	for	ADP
cana-5853	47	2	each	each	DET
cana-5853	47	3	n	n	PRON
cana-5853	47	4	∈	∈	PROPN
cana-5853	47	5	n	n	CCONJ
cana-5853	47	6	,	,	PUNCT
cana-5853	47	7	define	define	VERB
cana-5853	47	8	m∗(n	m∗(n	PROPN
cana-5853	47	9	,	,	PUNCT
cana-5853	47	10	y	y	NOUN
cana-5853	47	11	)	)	PUNCT
cana-5853	47	12	=	=	SYM
cana-5853	47	13	sup	sup	NOUN
cana-5853	47	14	t∈f	t∈f	X
cana-5853	47	15	{	{	PUNCT
cana-5853	47	16	|ty|	|ty|	PROPN
cana-5853	47	17	−m(n	−m(n	PROPN
cana-5853	47	18	,	,	PUNCT
cana-5853	47	19	t	t	PROPN
cana-5853	47	20	)	)	PUNCT
cana-5853	47	21	}	}	PUNCT
cana-5853	47	22	.	.	PUNCT
cana-5853	48	1	the	the	DET
cana-5853	48	2	function	function	NOUN
cana-5853	48	3	m∗(n	m∗(n	PROPN
cana-5853	48	4	,	,	PUNCT
cana-5853	48	5	·	·	PUNCT
cana-5853	48	6	)	)	PUNCT
cana-5853	48	7	inherits	inherit	VERB
cana-5853	48	8	convexity	convexity	NOUN
cana-5853	48	9	and	and	CCONJ
cana-5853	48	10	lower	low	ADJ
cana-5853	48	11	semicontinuity	semicontinuity	NOUN
cana-5853	48	12	properties	property	NOUN
cana-5853	48	13	,	,	PUNCT
cana-5853	48	14	and	and	CCONJ
cana-5853	48	15	serves	serve	VERB
cana-5853	48	16	to	to	PART
cana-5853	48	17	characterize	characterize	VERB
cana-5853	48	18	bounded	bound	VERB
cana-5853	48	19	linear	linear	ADJ
cana-5853	48	20	functionals	functional	NOUN
cana-5853	48	21	on	on	ADP
cana-5853	48	22	xm	xm	PROPN
cana-5853	48	23	.	.	PUNCT
cana-5853	49	1	specifically	specifically	ADV
cana-5853	49	2	,	,	PUNCT
cana-5853	49	3	if	if	SCONJ
cana-5853	49	4	y	y	PROPN
cana-5853	49	5	=	=	SYM
cana-5853	49	6	(	(	PUNCT
cana-5853	49	7	yn	yn	INTJ
cana-5853	49	8	)	)	PUNCT
cana-5853	49	9	∈	∈	PROPN
cana-5853	49	10	fn	fn	NOUN
cana-5853	49	11	satisfies	satisfie	NOUN
cana-5853	49	12	∞∑	∞∑	NUM
cana-5853	49	13	n=1	n=1	PROPN
cana-5853	49	14	m∗(n	m∗(n	PROPN
cana-5853	49	15	,	,	PUNCT
cana-5853	49	16	yn	yn	PROPN
cana-5853	49	17	)	)	PUNCT
cana-5853	49	18	<	<	X
cana-5853	50	1	∞	∞	PROPN
cana-5853	50	2	,	,	PUNCT
cana-5853	50	3	then	then	ADV
cana-5853	50	4	the	the	DET
cana-5853	50	5	functional	functional	ADJ
cana-5853	50	6	ly(x	ly(x	NOUN
cana-5853	50	7	)	)	PUNCT
cana-5853	50	8	=	=	NOUN
cana-5853	51	1	∞∑	∞∑	NUM
cana-5853	51	2	n=1	n=1	PROPN
cana-5853	51	3	xnyn	xnyn	PROPN
cana-5853	51	4	is	be	AUX
cana-5853	51	5	well	well	ADV
cana-5853	51	6	-	-	PUNCT
cana-5853	51	7	defined	define	VERB
cana-5853	51	8	and	and	CCONJ
cana-5853	51	9	bounded	bound	VERB
cana-5853	51	10	on	on	ADP
cana-5853	51	11	xm	xm	PROPN
cana-5853	51	12	.	.	PUNCT
cana-5853	52	1	this	this	DET
cana-5853	52	2	pairing	pairing	NOUN
cana-5853	52	3	between	between	ADP
cana-5853	52	4	x	x	PROPN
cana-5853	52	5	and	and	CCONJ
cana-5853	52	6	y	y	PROPN
cana-5853	52	7	underpins	underpin	VERB
cana-5853	52	8	the	the	DET
cana-5853	52	9	duality	duality	NOUN
cana-5853	52	10	theory	theory	NOUN
cana-5853	52	11	of	of	ADP
cana-5853	52	12	xm	xm	PROPN
cana-5853	52	13	spaces	space	NOUN
cana-5853	52	14	,	,	PUNCT
cana-5853	52	15	generalizing	generalize	VERB
cana-5853	52	16	the	the	DET
cana-5853	52	17	well	well	ADV
cana-5853	52	18	-	-	PUNCT
cana-5853	52	19	known	know	VERB
cana-5853	52	20	relation	relation	NOUN
cana-5853	52	21	between	between	ADP
cana-5853	52	22	ℓp	ℓp	NOUN
cana-5853	52	23	and	and	CCONJ
cana-5853	52	24	ℓq	ℓq	PROPN
cana-5853	52	25	spaces	space	NOUN
cana-5853	52	26	.	.	PUNCT
cana-5853	53	1	communications	communication	NOUN
cana-5853	53	2	on	on	ADP
cana-5853	53	3	applied	apply	VERB
cana-5853	53	4	nonlinear	nonlinear	ADJ
cana-5853	53	5	analysis	analysis	NOUN
cana-5853	53	6	issn	issn	NOUN
cana-5853	53	7	:	:	PUNCT
cana-5853	53	8	1074	1074	NUM
cana-5853	53	9	-	-	PUNCT
cana-5853	53	10	133x	133x	NUM
cana-5853	53	11	vol	vol	NOUN
cana-5853	53	12	31	31	NUM
cana-5853	53	13	no	no	NOUN
cana-5853	53	14	.	.	NOUN
cana-5853	53	15	2	2	NUM
cana-5853	53	16	(	(	PUNCT
cana-5853	53	17	2024	2024	NUM
cana-5853	53	18	)	)	PUNCT
cana-5853	53	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	53	20	463	463	NUM
cana-5853	53	21	2.4	2.4	NUM
cana-5853	53	22	examples	example	NOUN
cana-5853	53	23	to	to	PART
cana-5853	53	24	ground	ground	VERB
cana-5853	53	25	these	these	DET
cana-5853	53	26	abstract	abstract	ADJ
cana-5853	53	27	definitions	definition	NOUN
cana-5853	53	28	,	,	PUNCT
cana-5853	53	29	we	we	PRON
cana-5853	53	30	present	present	VERB
cana-5853	53	31	two	two	NUM
cana-5853	53	32	important	important	ADJ
cana-5853	53	33	special	special	ADJ
cana-5853	53	34	cases	case	NOUN
cana-5853	53	35	that	that	PRON
cana-5853	53	36	illustrate	illustrate	VERB
cana-5853	53	37	how	how	SCONJ
cana-5853	53	38	classical	classical	ADJ
cana-5853	53	39	sequence	sequence	NOUN
cana-5853	53	40	spaces	space	NOUN
cana-5853	53	41	fit	fit	ADJ
cana-5853	53	42	into	into	ADP
cana-5853	53	43	this	this	DET
cana-5853	53	44	modular	modular	ADJ
cana-5853	53	45	framework	framework	NOUN
cana-5853	53	46	.	.	PUNCT
cana-5853	54	1	example	example	NOUN
cana-5853	55	1	1	1	NUM
cana-5853	55	2	:	:	PUNCT
cana-5853	55	3	ℓp	ℓp	ADJ
cana-5853	55	4	spaces	space	NOUN
cana-5853	55	5	let	let	VERB
cana-5853	55	6	1	1	NUM
cana-5853	55	7	≤	≤	NOUN
cana-5853	56	1	p	p	NOUN
cana-5853	56	2	<	<	AUX
cana-5853	56	3	∞.	∞.	PROPN
cana-5853	56	4	define	define	VERB
cana-5853	56	5	m(n	m(n	PROPN
cana-5853	56	6	,	,	PUNCT
cana-5853	56	7	t	t	PROPN
cana-5853	56	8	)	)	PUNCT
cana-5853	56	9	=	=	PROPN
cana-5853	56	10	|t|p	|t|p	NOUN
cana-5853	56	11	p	p	NOUN
cana-5853	56	12	.	.	PUNCT
cana-5853	57	1	then	then	ADV
cana-5853	57	2	xm	xm	PROPN
cana-5853	58	1	=	=	PUNCT
cana-5853	58	2	{	{	PUNCT
cana-5853	58	3	x	x	PUNCT
cana-5853	58	4	∈	∈	NOUN
cana-5853	58	5	fn	fn	NOUN
cana-5853	58	6	:	:	PUNCT
cana-5853	58	7	∞∑	∞∑	NUM
cana-5853	58	8	n=1	n=1	PUNCT
cana-5853	58	9	|xn|p	|xn|p	NOUN
cana-5853	58	10	p	p	X
cana-5853	58	11	<	<	X
cana-5853	58	12	∞	∞	NUM
cana-5853	58	13	}	}	PUNCT
cana-5853	58	14	=	=	SYM
cana-5853	58	15	ℓp	ℓp	NOUN
cana-5853	58	16	.	.	PUNCT
cana-5853	59	1	the	the	DET
cana-5853	59	2	complementary	complementary	ADJ
cana-5853	59	3	function	function	NOUN
cana-5853	59	4	is	be	AUX
cana-5853	59	5	m∗(n	m∗(n	PROPN
cana-5853	59	6	,	,	PUNCT
cana-5853	59	7	y	y	NOUN
cana-5853	59	8	)	)	PUNCT
cana-5853	60	1	=	=	SYM
cana-5853	60	2	|y|q	|y|q	ADJ
cana-5853	60	3	q	q	X
cana-5853	60	4	,	,	PUNCT
cana-5853	60	5	where	where	SCONJ
cana-5853	60	6	1	1	NUM
cana-5853	60	7	p	p	NOUN
cana-5853	60	8	+	+	NOUN
cana-5853	60	9	1	1	NUM
cana-5853	60	10	q	q	NOUN
cana-5853	60	11	=	=	SYM
cana-5853	60	12	1	1	NUM
cana-5853	60	13	,	,	PUNCT
cana-5853	60	14	yielding	yield	VERB
cana-5853	60	15	the	the	DET
cana-5853	60	16	classical	classical	ADJ
cana-5853	60	17	duality	duality	NOUN
cana-5853	60	18	ℓp	ℓp	NOUN
cana-5853	60	19	∼=	∼=	PROPN
cana-5853	60	20	(	(	PUNCT
cana-5853	60	21	ℓq)∗.	ℓq)∗.	NOUN
cana-5853	60	22	example	example	NOUN
cana-5853	60	23	2	2	NUM
cana-5853	60	24	:	:	PUNCT
cana-5853	60	25	weighted	weight	VERB
cana-5853	60	26	orlicz	orlicz	ADJ
cana-5853	60	27	spaces	space	NOUN
cana-5853	60	28	consider	consider	VERB
cana-5853	60	29	a	a	DET
cana-5853	60	30	weight	weight	NOUN
cana-5853	60	31	sequence	sequence	NOUN
cana-5853	60	32	(	(	PUNCT
cana-5853	60	33	ωn)n∈n	ωn)n∈n	NUM
cana-5853	60	34	with	with	ADP
cana-5853	60	35	ωn	ωn	ADP
cana-5853	60	36	>	>	X
cana-5853	60	37	0	0	NUM
cana-5853	60	38	,	,	PUNCT
cana-5853	60	39	and	and	CCONJ
cana-5853	60	40	let	let	VERB
cana-5853	60	41	φ	φ	NOUN
cana-5853	60	42	:	:	PUNCT
cana-5853	61	1	[	[	X
cana-5853	61	2	0,∞	0,∞	NUM
cana-5853	61	3	)	)	PUNCT
cana-5853	61	4	→	→	PUNCT
cana-5853	62	1	[	[	X
cana-5853	62	2	0,∞	0,∞	X
cana-5853	62	3	)	)	PUNCT
cana-5853	62	4	be	be	VERB
cana-5853	62	5	an	an	DET
cana-5853	62	6	orlicz	orlicz	ADJ
cana-5853	62	7	function	function	NOUN
cana-5853	62	8	(	(	PUNCT
cana-5853	62	9	convex	convex	PROPN
cana-5853	62	10	,	,	PUNCT
cana-5853	62	11	increasing	increase	VERB
cana-5853	62	12	,	,	PUNCT
cana-5853	62	13	with	with	ADP
cana-5853	62	14	φ(0	φ(0	ADJ
cana-5853	62	15	)	)	PUNCT
cana-5853	62	16	=	=	SYM
cana-5853	62	17	0	0	NUM
cana-5853	62	18	)	)	PUNCT
cana-5853	62	19	.	.	PUNCT
cana-5853	63	1	define	define	VERB
cana-5853	63	2	m(n	m(n	PROPN
cana-5853	63	3	,	,	PUNCT
cana-5853	63	4	t	t	PROPN
cana-5853	63	5	)	)	PUNCT
cana-5853	63	6	=	=	NOUN
cana-5853	63	7	ωn	ωn	ADP
cana-5853	63	8	φ(|t|	φ(|t|	PROPN
cana-5853	63	9	)	)	PUNCT
cana-5853	63	10	.	.	PUNCT
cana-5853	64	1	then	then	ADV
cana-5853	64	2	xm	xm	PROPN
cana-5853	64	3	=	=	PUNCT
cana-5853	65	1	{	{	PUNCT
cana-5853	65	2	x	x	PUNCT
cana-5853	65	3	∈	∈	NOUN
cana-5853	65	4	fn	fn	NOUN
cana-5853	65	5	:	:	PUNCT
cana-5853	66	1	∞∑	∞∑	NUM
cana-5853	66	2	n=1	n=1	NUM
cana-5853	66	3	ωn	ωn	ADP
cana-5853	66	4	φ(|xn|	φ(|xn|	NUM
cana-5853	66	5	)	)	PUNCT
cana-5853	66	6	<	<	X
cana-5853	66	7	∞	∞	PROPN
cana-5853	66	8	}	}	PUNCT
cana-5853	66	9	is	be	AUX
cana-5853	66	10	the	the	DET
cana-5853	66	11	weighted	weight	VERB
cana-5853	66	12	orlicz	orlicz	ADJ
cana-5853	66	13	sequence	sequence	NOUN
cana-5853	66	14	space	space	NOUN
cana-5853	66	15	.	.	PUNCT
cana-5853	67	1	the	the	DET
cana-5853	67	2	complementary	complementary	ADJ
cana-5853	67	3	function	function	NOUN
cana-5853	67	4	is	be	AUX
cana-5853	67	5	given	give	VERB
cana-5853	67	6	by	by	ADP
cana-5853	67	7	m∗(n	m∗(n	PROPN
cana-5853	67	8	,	,	PUNCT
cana-5853	67	9	y	y	NOUN
cana-5853	67	10	)	)	PUNCT
cana-5853	67	11	=	=	PUNCT
cana-5853	67	12	ωn	ωn	PROPN
cana-5853	67	13	φ	φ	PROPN
cana-5853	67	14	∗(|y|	∗(|y|	PROPN
cana-5853	67	15	)	)	PUNCT
cana-5853	67	16	,	,	PUNCT
cana-5853	67	17	where	where	SCONJ
cana-5853	67	18	φ∗	φ∗	NOUN
cana-5853	67	19	is	be	AUX
cana-5853	67	20	the	the	DET
cana-5853	67	21	standard	standard	ADJ
cana-5853	67	22	complementary	complementary	ADJ
cana-5853	67	23	orlicz	orlicz	NOUN
cana-5853	67	24	function	function	NOUN
cana-5853	67	25	,	,	PUNCT
cana-5853	67	26	ensuring	ensure	VERB
cana-5853	67	27	duality	duality	NOUN
cana-5853	67	28	relations	relation	NOUN
cana-5853	67	29	similar	similar	ADJ
cana-5853	67	30	to	to	ADP
cana-5853	67	31	the	the	DET
cana-5853	67	32	unweighted	unweighted	ADJ
cana-5853	67	33	case	case	NOUN
cana-5853	67	34	but	but	CCONJ
cana-5853	67	35	modulated	modulate	VERB
cana-5853	67	36	by	by	ADP
cana-5853	67	37	the	the	DET
cana-5853	67	38	weights	weight	NOUN
cana-5853	67	39	.	.	PUNCT
cana-5853	68	1	these	these	DET
cana-5853	68	2	examples	example	NOUN
cana-5853	68	3	demonstrate	demonstrate	VERB
cana-5853	68	4	that	that	SCONJ
cana-5853	68	5	the	the	DET
cana-5853	68	6	framework	framework	NOUN
cana-5853	68	7	of	of	ADP
cana-5853	68	8	modulated	modulate	VERB
cana-5853	68	9	orlicz	orlicz	ADJ
cana-5853	68	10	-	-	PUNCT
cana-5853	68	11	type	type	NOUN
cana-5853	68	12	sequence	sequence	NOUN
cana-5853	68	13	spaces	space	NOUN
cana-5853	68	14	encompasses	encompass	VERB
cana-5853	68	15	many	many	ADJ
cana-5853	68	16	classical	classical	ADJ
cana-5853	68	17	spaces	space	NOUN
cana-5853	68	18	while	while	SCONJ
cana-5853	68	19	allowing	allow	VERB
cana-5853	68	20	for	for	ADP
cana-5853	68	21	greater	great	ADJ
cana-5853	68	22	flexibility	flexibility	NOUN
cana-5853	68	23	through	through	ADP
cana-5853	68	24	the	the	DET
cana-5853	68	25	choice	choice	NOUN
cana-5853	68	26	of	of	ADP
cana-5853	68	27	index	index	NOUN
cana-5853	68	28	-	-	PUNCT
cana-5853	68	29	dependent	dependent	ADJ
cana-5853	68	30	modular	modular	ADJ
cana-5853	68	31	functions	function	NOUN
cana-5853	68	32	.	.	PUNCT
cana-5853	69	1	this	this	DET
cana-5853	69	2	flexibility	flexibility	NOUN
cana-5853	69	3	is	be	AUX
cana-5853	69	4	the	the	DET
cana-5853	69	5	foundation	foundation	NOUN
cana-5853	69	6	for	for	ADP
cana-5853	69	7	the	the	DET
cana-5853	69	8	operator	operator	NOUN
cana-5853	69	9	-	-	PUNCT
cana-5853	69	10	theoretic	theoretic	NOUN
cana-5853	69	11	investigations	investigation	NOUN
cana-5853	69	12	developed	develop	VERB
cana-5853	69	13	in	in	ADP
cana-5853	69	14	the	the	DET
cana-5853	69	15	remainder	remainder	NOUN
cana-5853	69	16	of	of	ADP
cana-5853	69	17	this	this	DET
cana-5853	69	18	paper	paper	NOUN
cana-5853	69	19	.	.	PUNCT
cana-5853	70	1	3	3	NUM
cana-5853	70	2	bounded	bound	VERB
cana-5853	70	3	linear	linear	PROPN
cana-5853	70	4	operators	operator	NOUN
cana-5853	70	5	on	on	ADP
cana-5853	70	6	xm	xm	PROPN
cana-5853	70	7	having	having	AUX
cana-5853	70	8	established	establish	VERB
cana-5853	70	9	the	the	DET
cana-5853	70	10	foundational	foundational	ADJ
cana-5853	70	11	structure	structure	NOUN
cana-5853	70	12	of	of	ADP
cana-5853	70	13	modulated	modulate	VERB
cana-5853	70	14	orlicz	orlicz	ADJ
cana-5853	70	15	-	-	PUNCT
cana-5853	70	16	type	type	NOUN
cana-5853	70	17	sequence	sequence	NOUN
cana-5853	70	18	spaces	space	VERB
cana-5853	70	19	xm	xm	PROPN
cana-5853	70	20	,	,	PUNCT
cana-5853	70	21	we	we	PRON
cana-5853	70	22	now	now	ADV
cana-5853	70	23	turn	turn	VERB
cana-5853	70	24	to	to	ADP
cana-5853	70	25	the	the	DET
cana-5853	70	26	study	study	NOUN
cana-5853	70	27	of	of	ADP
cana-5853	70	28	bounded	bounded	ADJ
cana-5853	70	29	linear	linear	PROPN
cana-5853	70	30	operators	operator	NOUN
cana-5853	70	31	acting	act	VERB
cana-5853	70	32	on	on	ADP
cana-5853	70	33	these	these	DET
cana-5853	70	34	spaces	space	NOUN
cana-5853	70	35	.	.	PUNCT
cana-5853	71	1	this	this	DET
cana-5853	71	2	section	section	NOUN
cana-5853	71	3	defines	define	VERB
cana-5853	71	4	such	such	ADJ
cana-5853	71	5	operators	operator	NOUN
cana-5853	71	6	,	,	PUNCT
cana-5853	71	7	describes	describe	VERB
cana-5853	71	8	the	the	DET
cana-5853	71	9	role	role	NOUN
cana-5853	71	10	of	of	ADP
cana-5853	71	11	infinite	infinite	ADJ
cana-5853	71	12	matrices	matrix	NOUN
cana-5853	71	13	as	as	ADP
cana-5853	71	14	concrete	concrete	ADJ
cana-5853	71	15	realizations	realization	NOUN
cana-5853	71	16	,	,	PUNCT
cana-5853	71	17	establishes	establish	VERB
cana-5853	71	18	general	general	ADJ
cana-5853	71	19	criteria	criterion	NOUN
cana-5853	71	20	for	for	ADP
cana-5853	71	21	boundedness	boundedness	NOUN
cana-5853	71	22	,	,	PUNCT
cana-5853	71	23	and	and	CCONJ
cana-5853	71	24	explains	explain	VERB
cana-5853	71	25	how	how	SCONJ
cana-5853	71	26	young	young	ADJ
cana-5853	71	27	-	-	PUNCT
cana-5853	71	28	type	type	NOUN
cana-5853	71	29	inequalities	inequality	NOUN
cana-5853	71	30	in	in	ADP
cana-5853	71	31	the	the	DET
cana-5853	71	32	modular	modular	ADJ
cana-5853	71	33	setting	setting	NOUN
cana-5853	71	34	provide	provide	VERB
cana-5853	71	35	powerful	powerful	ADJ
cana-5853	71	36	analytical	analytical	ADJ
cana-5853	71	37	tools	tool	NOUN
cana-5853	71	38	.	.	PUNCT
cana-5853	72	1	communications	communication	NOUN
cana-5853	72	2	on	on	ADP
cana-5853	72	3	applied	apply	VERB
cana-5853	72	4	nonlinear	nonlinear	ADJ
cana-5853	72	5	analysis	analysis	NOUN
cana-5853	72	6	issn	issn	NOUN
cana-5853	72	7	:	:	PUNCT
cana-5853	72	8	1074	1074	NUM
cana-5853	72	9	-	-	PUNCT
cana-5853	72	10	133x	133x	NUM
cana-5853	72	11	vol	vol	NOUN
cana-5853	72	12	31	31	NUM
cana-5853	72	13	no	no	NOUN
cana-5853	72	14	.	.	NOUN
cana-5853	72	15	2	2	NUM
cana-5853	72	16	(	(	PUNCT
cana-5853	72	17	2024	2024	NUM
cana-5853	72	18	)	)	PUNCT
cana-5853	73	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	73	2	464	464	NUM
cana-5853	73	3	3.1	3.1	NUM
cana-5853	73	4	definition	definition	NOUN
cana-5853	73	5	and	and	CCONJ
cana-5853	73	6	general	general	ADJ
cana-5853	73	7	criteria	criterion	NOUN
cana-5853	73	8	let	let	VERB
cana-5853	73	9	xm	xm	PROPN
cana-5853	73	10	be	be	AUX
cana-5853	73	11	a	a	DET
cana-5853	73	12	modulated	modulate	VERB
cana-5853	73	13	orlicz	orlicz	ADJ
cana-5853	73	14	-	-	PUNCT
cana-5853	73	15	type	type	NOUN
cana-5853	73	16	sequence	sequence	NOUN
cana-5853	73	17	space	space	NOUN
cana-5853	73	18	over	over	ADP
cana-5853	73	19	the	the	DET
cana-5853	73	20	field	field	NOUN
cana-5853	73	21	f.	f.	PROPN
cana-5853	74	1	a	a	DET
cana-5853	74	2	mapping	mapping	NOUN
cana-5853	74	3	t	t	NOUN
cana-5853	74	4	:	:	PUNCT
cana-5853	74	5	xm	xm	PROPN
cana-5853	74	6	→	→	SYM
cana-5853	74	7	xm	xm	PROPN
cana-5853	74	8	is	be	AUX
cana-5853	74	9	called	call	VERB
cana-5853	74	10	a	a	DET
cana-5853	74	11	bounded	bounded	ADJ
cana-5853	74	12	linear	linear	ADJ
cana-5853	74	13	operator	operator	NOUN
cana-5853	74	14	if	if	SCONJ
cana-5853	74	15	:	:	PUNCT
cana-5853	74	16	1	1	X
cana-5853	74	17	.	.	X
cana-5853	74	18	t	t	PROPN
cana-5853	74	19	is	be	AUX
cana-5853	74	20	linear	linear	ADJ
cana-5853	74	21	:	:	PUNCT
cana-5853	74	22	for	for	ADP
cana-5853	74	23	all	all	DET
cana-5853	74	24	x	x	NOUN
cana-5853	74	25	,	,	PUNCT
cana-5853	74	26	y	y	PROPN
cana-5853	74	27	∈	∈	PROPN
cana-5853	74	28	xm	xm	PROPN
cana-5853	74	29	and	and	CCONJ
cana-5853	74	30	scalars	scalar	VERB
cana-5853	74	31	α	α	PROPN
cana-5853	74	32	,	,	PUNCT
cana-5853	75	1	β	β	X
cana-5853	75	2	∈	∈	PROPN
cana-5853	75	3	f	f	X
cana-5853	75	4	,	,	PUNCT
cana-5853	75	5	t	t	PROPN
cana-5853	75	6	(	(	PUNCT
cana-5853	75	7	αx+	αx+	VERB
cana-5853	75	8	βy	βy	ADP
cana-5853	75	9	)	)	PUNCT
cana-5853	76	1	=	=	SYM
cana-5853	76	2	αt	αt	PROPN
cana-5853	76	3	(	(	PUNCT
cana-5853	76	4	x	x	X
cana-5853	76	5	)	)	PUNCT
cana-5853	76	6	+	+	CCONJ
cana-5853	76	7	βt	βt	PRON
cana-5853	76	8	(	(	PUNCT
cana-5853	76	9	y	y	NOUN
cana-5853	76	10	)	)	PUNCT
cana-5853	76	11	.	.	PUNCT
cana-5853	77	1	2	2	X
cana-5853	77	2	.	.	X
cana-5853	77	3	t	t	PROPN
cana-5853	77	4	is	be	AUX
cana-5853	77	5	bounded	bound	VERB
cana-5853	77	6	:	:	PUNCT
cana-5853	77	7	there	there	PRON
cana-5853	77	8	exists	exist	VERB
cana-5853	77	9	c	c	NOUN
cana-5853	77	10	>	>	X
cana-5853	77	11	0	0	NUM
cana-5853	78	1	such	such	ADJ
cana-5853	78	2	that	that	PRON
cana-5853	78	3	for	for	ADP
cana-5853	78	4	all	all	DET
cana-5853	78	5	x	x	SYM
cana-5853	78	6	∈	∈	PROPN
cana-5853	78	7	xm	xm	PROPN
cana-5853	78	8	,	,	PUNCT
cana-5853	78	9	∥t	∥t	PROPN
cana-5853	78	10	(	(	PUNCT
cana-5853	78	11	x)∥m	x)∥m	PROPN
cana-5853	78	12	≤	≤	PROPN
cana-5853	78	13	c∥x∥m	c∥x∥m	NOUN
cana-5853	78	14	.	.	PUNCT
cana-5853	79	1	boundedness	boundedness	PROPN
cana-5853	79	2	ensures	ensure	VERB
cana-5853	79	3	continuity	continuity	NOUN
cana-5853	79	4	and	and	CCONJ
cana-5853	79	5	guarantees	guarantee	NOUN
cana-5853	79	6	that	that	SCONJ
cana-5853	79	7	t	t	PROPN
cana-5853	79	8	respects	respect	VERB
cana-5853	79	9	the	the	DET
cana-5853	79	10	topological	topological	ADJ
cana-5853	79	11	structure	structure	NOUN
cana-5853	79	12	of	of	ADP
cana-5853	79	13	xm	xm	PROPN
cana-5853	79	14	,	,	PUNCT
cana-5853	79	15	allowing	allow	VERB
cana-5853	79	16	the	the	DET
cana-5853	79	17	use	use	NOUN
cana-5853	79	18	of	of	ADP
cana-5853	79	19	standard	standard	ADJ
cana-5853	79	20	results	result	NOUN
cana-5853	79	21	such	such	ADJ
cana-5853	79	22	as	as	ADP
cana-5853	79	23	the	the	DET
cana-5853	79	24	uniform	uniform	PROPN
cana-5853	79	25	boundedness	boundedness	PROPN
cana-5853	79	26	principle	principle	NOUN
cana-5853	79	27	and	and	CCONJ
cana-5853	79	28	the	the	DET
cana-5853	79	29	open	open	ADJ
cana-5853	79	30	mapping	mapping	NOUN
cana-5853	79	31	theorem	theorem	VERB
cana-5853	79	32	.	.	PROPN
cana-5853	80	1	3.2	3.2	NUM
cana-5853	80	2	matrix	matrix	NOUN
cana-5853	80	3	transformations	transformation	NOUN
cana-5853	80	4	as	as	ADP
cana-5853	80	5	operators	operator	NOUN
cana-5853	80	6	a	a	DET
cana-5853	80	7	large	large	ADJ
cana-5853	80	8	and	and	CCONJ
cana-5853	80	9	important	important	ADJ
cana-5853	80	10	class	class	NOUN
cana-5853	80	11	of	of	ADP
cana-5853	80	12	linear	linear	PROPN
cana-5853	80	13	operators	operator	NOUN
cana-5853	80	14	on	on	ADP
cana-5853	80	15	sequence	sequence	NOUN
cana-5853	80	16	spaces	space	NOUN
cana-5853	80	17	can	can	AUX
cana-5853	80	18	be	be	AUX
cana-5853	80	19	represented	represent	VERB
cana-5853	80	20	by	by	ADP
cana-5853	80	21	infinite	infinite	ADJ
cana-5853	80	22	matrices	matrix	NOUN
cana-5853	80	23	.	.	PUNCT
cana-5853	81	1	let	let	VERB
cana-5853	81	2	a	a	DET
cana-5853	81	3	=	=	SYM
cana-5853	81	4	(	(	PUNCT
cana-5853	81	5	ank	ank	PROPN
cana-5853	81	6	)	)	PUNCT
cana-5853	81	7	be	be	VERB
cana-5853	81	8	a	a	DET
cana-5853	81	9	double	double	ADJ
cana-5853	81	10	sequence	sequence	NOUN
cana-5853	81	11	of	of	ADP
cana-5853	81	12	scalars	scalar	NOUN
cana-5853	81	13	in	in	ADP
cana-5853	81	14	f.	f.	PROPN
cana-5853	81	15	define	define	VERB
cana-5853	81	16	the	the	DET
cana-5853	81	17	formal	formal	ADJ
cana-5853	81	18	matrix	matrix	NOUN
cana-5853	81	19	transformation	transformation	NOUN
cana-5853	81	20	a	a	DET
cana-5853	81	21	acting	act	VERB
cana-5853	81	22	on	on	ADP
cana-5853	81	23	x	x	X
cana-5853	81	24	=	=	SYM
cana-5853	81	25	(	(	PUNCT
cana-5853	81	26	xk	xk	NOUN
cana-5853	81	27	)	)	PUNCT
cana-5853	81	28	by	by	ADP
cana-5853	81	29	:	:	PUNCT
cana-5853	81	30	(	(	PUNCT
cana-5853	81	31	ax)n	ax)n	PROPN
cana-5853	81	32	=	=	PROPN
cana-5853	81	33	∞∑	∞∑	NUM
cana-5853	81	34	k=1	k=1	ADV
cana-5853	81	35	ankxk	ankxk	ADJ
cana-5853	81	36	.	.	PUNCT
cana-5853	82	1	for	for	SCONJ
cana-5853	82	2	this	this	DET
cana-5853	82	3	formal	formal	ADJ
cana-5853	82	4	sum	sum	NOUN
cana-5853	82	5	to	to	PART
cana-5853	82	6	define	define	VERB
cana-5853	82	7	a	a	DET
cana-5853	82	8	well	well	ADV
cana-5853	82	9	-	-	PUNCT
cana-5853	82	10	defined	define	VERB
cana-5853	82	11	element	element	NOUN
cana-5853	82	12	of	of	ADP
cana-5853	82	13	xm	xm	PROPN
cana-5853	82	14	,	,	PUNCT
cana-5853	82	15	the	the	DET
cana-5853	82	16	series	series	NOUN
cana-5853	82	17	must	must	AUX
cana-5853	82	18	converge	converge	VERB
cana-5853	82	19	for	for	ADP
cana-5853	82	20	each	each	DET
cana-5853	82	21	n	n	CCONJ
cana-5853	82	22	,	,	PUNCT
cana-5853	82	23	and	and	CCONJ
cana-5853	82	24	the	the	DET
cana-5853	82	25	resulting	result	VERB
cana-5853	82	26	sequence	sequence	NOUN
cana-5853	82	27	must	must	AUX
cana-5853	82	28	belong	belong	VERB
cana-5853	82	29	to	to	ADP
cana-5853	82	30	xm	xm	PROPN
cana-5853	82	31	.	.	PUNCT
cana-5853	83	1	that	that	PRON
cana-5853	83	2	is	be	AUX
cana-5853	83	3	:	:	PUNCT
cana-5853	83	4	ρm(ax	ρm(ax	PROPN
cana-5853	83	5	)	)	PUNCT
cana-5853	84	1	=	=	PUNCT
cana-5853	85	1	∞∑	∞∑	NUM
cana-5853	85	2	n=1	n=1	NUM
cana-5853	85	3	m	m	PROPN
cana-5853	85	4	(	(	PUNCT
cana-5853	85	5	n	n	CCONJ
cana-5853	85	6	,	,	PUNCT
cana-5853	85	7	∞∑	∞∑	NUM
cana-5853	85	8	k=1	k=1	ADV
cana-5853	85	9	ankxk	ankxk	ADJ
cana-5853	85	10	)	)	PUNCT
cana-5853	85	11	<	<	X
cana-5853	86	1	∞.	∞.	PROPN
cana-5853	86	2	not	not	PART
cana-5853	86	3	all	all	DET
cana-5853	86	4	infinite	infinite	ADJ
cana-5853	86	5	matrices	matrix	NOUN
cana-5853	86	6	define	define	VERB
cana-5853	86	7	bounded	bounded	ADJ
cana-5853	86	8	operators	operator	NOUN
cana-5853	86	9	on	on	ADP
cana-5853	86	10	xm	xm	PROPN
cana-5853	86	11	.	.	PUNCT
cana-5853	87	1	establishing	establish	VERB
cana-5853	87	2	criteria	criterion	NOUN
cana-5853	87	3	under	under	ADP
cana-5853	87	4	which	which	PRON
cana-5853	87	5	a	a	DET
cana-5853	87	6	yields	yield	NOUN
cana-5853	87	7	a	a	DET
cana-5853	87	8	bounded	bounded	ADJ
cana-5853	87	9	linear	linear	ADJ
cana-5853	87	10	operator	operator	NOUN
cana-5853	87	11	is	be	AUX
cana-5853	87	12	therefore	therefore	ADV
cana-5853	87	13	a	a	DET
cana-5853	87	14	fundamental	fundamental	ADJ
cana-5853	87	15	problem	problem	NOUN
cana-5853	87	16	in	in	ADP
cana-5853	87	17	this	this	DET
cana-5853	87	18	theory	theory	NOUN
cana-5853	87	19	.	.	PUNCT
cana-5853	88	1	3.3	3.3	NUM
cana-5853	88	2	conditions	condition	NOUN
cana-5853	88	3	for	for	ADP
cana-5853	88	4	boundedness	boundedness	NOUN
cana-5853	88	5	a	a	DET
cana-5853	88	6	general	general	ADJ
cana-5853	88	7	approach	approach	NOUN
cana-5853	88	8	to	to	ADP
cana-5853	88	9	boundedness	boundedness	PROPN
cana-5853	88	10	uses	use	VERB
cana-5853	88	11	modular	modular	ADJ
cana-5853	88	12	inequalities	inequality	NOUN
cana-5853	88	13	.	.	PUNCT
cana-5853	89	1	suppose	suppose	VERB
cana-5853	90	1	that	that	SCONJ
cana-5853	90	2	for	for	ADP
cana-5853	90	3	all	all	DET
cana-5853	90	4	x	x	SYM
cana-5853	90	5	∈	∈	PROPN
cana-5853	90	6	xm	xm	PROPN
cana-5853	90	7	,	,	PUNCT
cana-5853	90	8	∞∑	∞∑	NUM
cana-5853	90	9	n=1	n=1	NOUN
cana-5853	90	10	m	m	PROPN
cana-5853	90	11	(	(	PUNCT
cana-5853	90	12	n	n	CCONJ
cana-5853	90	13	,	,	PUNCT
cana-5853	90	14	∞∑	∞∑	NUM
cana-5853	90	15	k=1	k=1	ADJ
cana-5853	90	16	ankxk	ankxk	ADJ
cana-5853	90	17	)	)	PUNCT
cana-5853	90	18	≤	≤	NUM
cana-5853	91	1	c	c	X
cana-5853	91	2	∞∑	∞∑	NUM
cana-5853	91	3	k=1	k=1	PROPN
cana-5853	91	4	m(k	m(k	PROPN
cana-5853	91	5	,	,	PUNCT
cana-5853	91	6	xk	xk	PROPN
cana-5853	91	7	)	)	PUNCT
cana-5853	91	8	,	,	PUNCT
cana-5853	91	9	for	for	ADP
cana-5853	91	10	some	some	DET
cana-5853	91	11	constant	constant	ADJ
cana-5853	91	12	c	c	NOUN
cana-5853	91	13	>	>	X
cana-5853	91	14	0	0	X
cana-5853	91	15	.	.	PUNCT
cana-5853	92	1	then	then	ADV
cana-5853	92	2	a	a	PRON
cana-5853	92	3	is	be	AUX
cana-5853	92	4	a	a	DET
cana-5853	92	5	bounded	bounded	ADJ
cana-5853	92	6	linear	linear	ADJ
cana-5853	92	7	operator	operator	NOUN
cana-5853	92	8	from	from	ADP
cana-5853	92	9	xm	xm	PROPN
cana-5853	92	10	into	into	ADP
cana-5853	92	11	itself	itself	PRON
cana-5853	92	12	,	,	PUNCT
cana-5853	92	13	with	with	ADP
cana-5853	92	14	operator	operator	NOUN
cana-5853	92	15	norm	norm	NOUN
cana-5853	92	16	controlled	control	VERB
cana-5853	92	17	by	by	ADP
cana-5853	92	18	c.	c.	PROPN
cana-5853	92	19	in	in	ADP
cana-5853	92	20	practice	practice	NOUN
cana-5853	92	21	,	,	PUNCT
cana-5853	92	22	sufficient	sufficient	ADJ
cana-5853	92	23	conditions	condition	NOUN
cana-5853	92	24	for	for	ADP
cana-5853	92	25	boundedness	boundedness	NOUN
cana-5853	92	26	often	often	ADV
cana-5853	92	27	arise	arise	VERB
cana-5853	92	28	from	from	ADP
cana-5853	92	29	more	more	ADJ
cana-5853	92	30	concrete	concrete	ADJ
cana-5853	92	31	estimates	estimate	NOUN
cana-5853	92	32	:	:	PUNCT
cana-5853	92	33	communications	communication	NOUN
cana-5853	92	34	on	on	ADP
cana-5853	92	35	applied	apply	VERB
cana-5853	92	36	nonlinear	nonlinear	ADJ
cana-5853	92	37	analysis	analysis	NOUN
cana-5853	92	38	issn	issn	NOUN
cana-5853	92	39	:	:	PUNCT
cana-5853	92	40	1074	1074	NUM
cana-5853	92	41	-	-	PUNCT
cana-5853	92	42	133x	133x	NUM
cana-5853	92	43	vol	vol	NOUN
cana-5853	92	44	31	31	NUM
cana-5853	92	45	no	no	NOUN
cana-5853	92	46	.	.	NOUN
cana-5853	92	47	2	2	NUM
cana-5853	92	48	(	(	PUNCT
cana-5853	92	49	2024	2024	NUM
cana-5853	92	50	)	)	PUNCT
cana-5853	92	51	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	92	52	465	465	NUM
cana-5853	92	53	|ankxk|	|ankxk|	NUM
cana-5853	92	54	≤	≤	PUNCT
cana-5853	92	55	ηnkm(k	ηnkm(k	PROPN
cana-5853	92	56	,	,	PUNCT
cana-5853	92	57	xk	xk	ADJ
cana-5853	92	58	)	)	PUNCT
cana-5853	92	59	+	+	CCONJ
cana-5853	92	60	θnk	θnk	ADJ
cana-5853	92	61	,	,	PUNCT
cana-5853	92	62	for	for	ADP
cana-5853	92	63	non	non	ADJ
cana-5853	92	64	-	-	ADJ
cana-5853	92	65	negative	negative	ADJ
cana-5853	92	66	sequences	sequence	NOUN
cana-5853	92	67	ηnk	ηnk	NOUN
cana-5853	92	68	,	,	PUNCT
cana-5853	92	69	θnk	θnk	NOUN
cana-5853	92	70	satisfying	satisfy	VERB
cana-5853	92	71	suitable	suitable	ADJ
cana-5853	92	72	summability	summability	NOUN
cana-5853	92	73	conditions	condition	NOUN
cana-5853	92	74	.	.	PUNCT
cana-5853	93	1	summing	sum	VERB
cana-5853	93	2	over	over	ADP
cana-5853	93	3	k	k	PROPN
cana-5853	93	4	and	and	CCONJ
cana-5853	93	5	using	use	VERB
cana-5853	93	6	convexity	convexity	NOUN
cana-5853	93	7	of	of	ADP
cana-5853	93	8	m(n	m(n	PROPN
cana-5853	93	9	,	,	PUNCT
cana-5853	93	10	·	·	PUNCT
cana-5853	93	11	)	)	PUNCT
cana-5853	93	12	allows	allow	VERB
cana-5853	93	13	estimation	estimation	NOUN
cana-5853	93	14	of	of	ADP
cana-5853	93	15	m(n	m(n	PROPN
cana-5853	93	16	,	,	PUNCT
cana-5853	93	17	(	(	PUNCT
cana-5853	93	18	ax)n	ax)n	PROPN
cana-5853	93	19	)	)	PUNCT
cana-5853	93	20	in	in	ADP
cana-5853	93	21	terms	term	NOUN
cana-5853	93	22	of	of	ADP
cana-5853	93	23	the	the	DET
cana-5853	93	24	modular	modular	NOUN
cana-5853	93	25	of	of	ADP
cana-5853	93	26	x.	x.	NOUN
cana-5853	93	27	diagonal	diagonal	ADJ
cana-5853	93	28	and	and	CCONJ
cana-5853	93	29	triangular	triangular	NOUN
cana-5853	93	30	matrices	matrix	NOUN
cana-5853	93	31	.	.	PUNCT
cana-5853	94	1	for	for	ADP
cana-5853	94	2	diagonal	diagonal	ADJ
cana-5853	94	3	matrices	matrix	NOUN
cana-5853	94	4	d	d	X
cana-5853	94	5	=	=	SYM
cana-5853	94	6	diag(λn	diag(λn	PROPN
cana-5853	94	7	)	)	PUNCT
cana-5853	94	8	,	,	PUNCT
cana-5853	94	9	boundedness	boundedness	PROPN
cana-5853	94	10	requires	require	VERB
cana-5853	94	11	control	control	NOUN
cana-5853	94	12	over	over	ADP
cana-5853	94	13	m(n	m(n	PROPN
cana-5853	94	14	,	,	PUNCT
cana-5853	94	15	λnxn	λnxn	NOUN
cana-5853	94	16	)	)	PUNCT
cana-5853	94	17	.	.	PUNCT
cana-5853	95	1	using	use	VERB
cana-5853	95	2	properties	property	NOUN
cana-5853	95	3	of	of	ADP
cana-5853	95	4	m(n	m(n	PROPN
cana-5853	95	5	,	,	PUNCT
cana-5853	95	6	·	·	PUNCT
cana-5853	95	7	)	)	PUNCT
cana-5853	95	8	(	(	PUNCT
cana-5853	95	9	like	like	ADP
cana-5853	95	10	∆2	∆2	PROPN
cana-5853	95	11	conditions	condition	NOUN
cana-5853	95	12	)	)	PUNCT
cana-5853	95	13	,	,	PUNCT
cana-5853	95	14	one	one	PRON
cana-5853	95	15	can	can	AUX
cana-5853	95	16	derive	derive	VERB
cana-5853	95	17	simple	simple	ADJ
cana-5853	95	18	criteria	criterion	NOUN
cana-5853	95	19	:	:	PUNCT
cana-5853	95	20	m(n	m(n	PROPN
cana-5853	95	21	,	,	PUNCT
cana-5853	95	22	λnt	λnt	PROPN
cana-5853	95	23	)	)	PUNCT
cana-5853	95	24	≤	≤	NOUN
cana-5853	95	25	cnm(n	cnm(n	PROPN
cana-5853	95	26	,	,	PUNCT
cana-5853	95	27	t	t	PROPN
cana-5853	95	28	)	)	PUNCT
cana-5853	96	1	+	+	CCONJ
cana-5853	97	1	cn	cn	PROPN
cana-5853	97	2	.	.	PUNCT
cana-5853	97	3	similarly	similarly	ADV
cana-5853	97	4	,	,	PUNCT
cana-5853	97	5	for	for	ADP
cana-5853	97	6	lower	low	ADJ
cana-5853	97	7	triangular	triangular	NOUN
cana-5853	97	8	matrices	matrix	NOUN
cana-5853	97	9	,	,	PUNCT
cana-5853	97	10	tail	tail	NOUN
cana-5853	97	11	conditions	condition	NOUN
cana-5853	97	12	and	and	CCONJ
cana-5853	97	13	growth	growth	NOUN
cana-5853	97	14	estimates	estimate	NOUN
cana-5853	97	15	ensure	ensure	VERB
cana-5853	97	16	boundedness	boundedness	PROPN
cana-5853	97	17	.	.	PUNCT
cana-5853	98	1	this	this	DET
cana-5853	98	2	analysis	analysis	NOUN
cana-5853	98	3	generalizes	generalize	VERB
cana-5853	98	4	classical	classical	ADJ
cana-5853	98	5	results	result	NOUN
cana-5853	98	6	from	from	ADP
cana-5853	98	7	ℓp	ℓp	ADJ
cana-5853	98	8	spaces	space	NOUN
cana-5853	98	9	and	and	CCONJ
cana-5853	98	10	orlicz	orlicz	NOUN
cana-5853	98	11	spaces	space	NOUN
cana-5853	98	12	to	to	ADP
cana-5853	98	13	the	the	DET
cana-5853	98	14	modulated	modulate	VERB
cana-5853	98	15	setting	setting	NOUN
cana-5853	98	16	.	.	PUNCT
cana-5853	99	1	3.4	3.4	NUM
cana-5853	99	2	young	young	ADJ
cana-5853	99	3	-	-	PUNCT
cana-5853	99	4	type	type	NOUN
cana-5853	99	5	inequalities	inequality	NOUN
cana-5853	99	6	in	in	ADP
cana-5853	99	7	the	the	DET
cana-5853	99	8	modular	modular	NOUN
cana-5853	99	9	setting	set	VERB
cana-5853	99	10	a	a	DET
cana-5853	99	11	crucial	crucial	ADJ
cana-5853	99	12	tool	tool	NOUN
cana-5853	99	13	in	in	ADP
cana-5853	99	14	these	these	DET
cana-5853	99	15	boundedness	boundedness	NOUN
cana-5853	99	16	proofs	proof	NOUN
cana-5853	99	17	is	be	AUX
cana-5853	99	18	the	the	DET
cana-5853	99	19	modular	modular	ADJ
cana-5853	99	20	analog	analog	NOUN
cana-5853	99	21	of	of	ADP
cana-5853	99	22	young	young	PROPN
cana-5853	99	23	’s	’s	PART
cana-5853	99	24	inequality	inequality	NOUN
cana-5853	99	25	.	.	PUNCT
cana-5853	100	1	recall	recall	VERB
cana-5853	100	2	that	that	PRON
cana-5853	100	3	for	for	ADP
cana-5853	100	4	each	each	DET
cana-5853	100	5	n	n	CCONJ
cana-5853	100	6	,	,	PUNCT
cana-5853	100	7	the	the	DET
cana-5853	100	8	complementary	complementary	ADJ
cana-5853	100	9	modular	modular	ADJ
cana-5853	100	10	function	function	NOUN
cana-5853	100	11	is	be	AUX
cana-5853	100	12	defined	define	VERB
cana-5853	100	13	as	as	ADP
cana-5853	100	14	:	:	PUNCT
cana-5853	100	15	m∗(n	m∗(n	PROPN
cana-5853	100	16	,	,	PUNCT
cana-5853	100	17	y	y	NOUN
cana-5853	100	18	)	)	PUNCT
cana-5853	100	19	=	=	SYM
cana-5853	100	20	sup	sup	NOUN
cana-5853	100	21	t∈f	t∈f	X
cana-5853	100	22	{	{	PUNCT
cana-5853	100	23	|ty|	|ty|	PROPN
cana-5853	100	24	−m(n	−m(n	PROPN
cana-5853	100	25	,	,	PUNCT
cana-5853	100	26	t	t	PROPN
cana-5853	100	27	)	)	PUNCT
cana-5853	100	28	}	}	PUNCT
cana-5853	100	29	.	.	PUNCT
cana-5853	101	1	young	young	PROPN
cana-5853	101	2	’s	’s	PART
cana-5853	101	3	inequality	inequality	NOUN
cana-5853	101	4	then	then	ADV
cana-5853	101	5	states	state	VERB
cana-5853	101	6	:	:	PUNCT
cana-5853	101	7	|ty|	|ty|	ADV
cana-5853	101	8	≤	≤	ADJ
cana-5853	101	9	m(n	m(n	NOUN
cana-5853	101	10	,	,	PUNCT
cana-5853	101	11	t	t	PROPN
cana-5853	101	12	)	)	PUNCT
cana-5853	102	1	+	+	NOUN
cana-5853	102	2	m∗(n	m∗(n	PROPN
cana-5853	102	3	,	,	PUNCT
cana-5853	102	4	y	y	NOUN
cana-5853	102	5	)	)	PUNCT
cana-5853	102	6	.	.	PUNCT
cana-5853	103	1	this	this	DET
cana-5853	103	2	inequality	inequality	NOUN
cana-5853	103	3	provides	provide	VERB
cana-5853	103	4	an	an	DET
cana-5853	103	5	upper	upper	ADJ
cana-5853	103	6	bound	bind	VERB
cana-5853	103	7	for	for	ADP
cana-5853	103	8	the	the	DET
cana-5853	103	9	bilinear	bilinear	NOUN
cana-5853	103	10	form	form	NOUN
cana-5853	103	11	ty	ty	INTJ
cana-5853	103	12	in	in	ADP
cana-5853	103	13	terms	term	NOUN
cana-5853	103	14	of	of	ADP
cana-5853	103	15	the	the	DET
cana-5853	103	16	modular	modular	ADJ
cana-5853	103	17	functions	function	NOUN
cana-5853	103	18	.	.	PUNCT
cana-5853	104	1	it	it	PRON
cana-5853	104	2	is	be	AUX
cana-5853	104	3	indispensable	indispensable	ADJ
cana-5853	104	4	when	when	SCONJ
cana-5853	104	5	analyzing	analyze	VERB
cana-5853	104	6	matrix	matrix	NOUN
cana-5853	104	7	operators	operator	NOUN
cana-5853	104	8	,	,	PUNCT
cana-5853	104	9	as	as	SCONJ
cana-5853	104	10	it	it	PRON
cana-5853	104	11	controls	control	VERB
cana-5853	104	12	terms	term	NOUN
cana-5853	104	13	like	like	ADP
cana-5853	104	14	ankxk	ankxk	NOUN
cana-5853	104	15	:	:	PUNCT
cana-5853	104	16	|ankxk|	|ankxk|	NUM
cana-5853	104	17	≤	≤	PROPN
cana-5853	104	18	m(k	m(k	PROPN
cana-5853	104	19	,	,	PUNCT
cana-5853	104	20	xk	xk	PROPN
cana-5853	104	21	)	)	PUNCT
cana-5853	104	22	+	+	PROPN
cana-5853	104	23	m∗(k	m∗(k	PROPN
cana-5853	104	24	,	,	PUNCT
cana-5853	104	25	ank	ank	PROPN
cana-5853	104	26	)	)	PUNCT
cana-5853	104	27	.	.	PUNCT
cana-5853	105	1	summing	sum	VERB
cana-5853	105	2	over	over	ADP
cana-5853	105	3	k	k	PROPN
cana-5853	105	4	and	and	CCONJ
cana-5853	105	5	applying	apply	VERB
cana-5853	105	6	convexity	convexity	NOUN
cana-5853	105	7	yields	yield	NOUN
cana-5853	105	8	:	:	PUNCT
cana-5853	105	9	∞∑	∞∑	NUM
cana-5853	105	10	k=1	k=1	ADP
cana-5853	105	11	|ankxk|	|ankxk|	VERB
cana-5853	105	12	≤	≤	NOUN
cana-5853	105	13	∞∑	∞∑	NUM
cana-5853	105	14	k=1	k=1	PROPN
cana-5853	105	15	m(k	m(k	PROPN
cana-5853	105	16	,	,	PUNCT
cana-5853	105	17	xk	xk	PROPN
cana-5853	105	18	)	)	PUNCT
cana-5853	105	19	+	+	CCONJ
cana-5853	105	20	∞∑	∞∑	NUM
cana-5853	105	21	k=1	k=1	ADP
cana-5853	105	22	m∗(k	m∗(k	PROPN
cana-5853	105	23	,	,	PUNCT
cana-5853	105	24	ank	ank	PROPN
cana-5853	105	25	)	)	PUNCT
cana-5853	105	26	.	.	PUNCT
cana-5853	106	1	thus	thus	ADV
cana-5853	106	2	,	,	PUNCT
cana-5853	106	3	boundedness	boundedness	NOUN
cana-5853	106	4	of	of	ADP
cana-5853	106	5	a	a	PRON
cana-5853	106	6	can	can	AUX
cana-5853	106	7	be	be	AUX
cana-5853	106	8	ensured	ensure	VERB
cana-5853	106	9	if	if	SCONJ
cana-5853	106	10	:	:	PUNCT
cana-5853	106	11	∞∑	∞∑	NUM
cana-5853	106	12	n=1	n=1	NUM
cana-5853	106	13	∞∑	∞∑	NUM
cana-5853	106	14	k=1	k=1	ADP
cana-5853	106	15	m∗(k	m∗(k	PROPN
cana-5853	106	16	,	,	PUNCT
cana-5853	106	17	ank	ank	PROPN
cana-5853	106	18	)	)	PUNCT
cana-5853	106	19	<	<	X
cana-5853	106	20	∞	∞	PROPN
cana-5853	106	21	,	,	PUNCT
cana-5853	106	22	together	together	ADV
cana-5853	106	23	with	with	ADP
cana-5853	106	24	control	control	NOUN
cana-5853	106	25	over	over	ADP
cana-5853	106	26	the	the	DET
cana-5853	106	27	modular	modular	NOUN
cana-5853	106	28	of	of	ADP
cana-5853	106	29	x.	x.	NOUN
cana-5853	106	30	this	this	DET
cana-5853	106	31	approach	approach	NOUN
cana-5853	106	32	generalizes	generalize	VERB
cana-5853	106	33	classical	classical	ADJ
cana-5853	106	34	schurtype	schurtype	NOUN
cana-5853	106	35	tests	test	NOUN
cana-5853	106	36	and	and	CCONJ
cana-5853	106	37	summability	summability	NOUN
cana-5853	106	38	criteria	criterion	NOUN
cana-5853	106	39	,	,	PUNCT
cana-5853	106	40	providing	provide	VERB
cana-5853	106	41	a	a	DET
cana-5853	106	42	flexible	flexible	ADJ
cana-5853	106	43	framework	framework	NOUN
cana-5853	106	44	for	for	ADP
cana-5853	106	45	verifying	verifying	NOUN
cana-5853	106	46	boundedness	boundedness	NOUN
cana-5853	106	47	in	in	ADP
cana-5853	106	48	modulated	modulate	VERB
cana-5853	106	49	orlicz	orlicz	ADJ
cana-5853	106	50	-	-	PUNCT
cana-5853	106	51	type	type	NOUN
cana-5853	106	52	sequence	sequence	NOUN
cana-5853	106	53	spaces	space	VERB
cana-5853	106	54	.	.	PUNCT
cana-5853	107	1	summary	summary	NOUN
cana-5853	107	2	.	.	PUNCT
cana-5853	108	1	through	through	ADP
cana-5853	108	2	these	these	DET
cana-5853	108	3	definitions	definition	NOUN
cana-5853	108	4	and	and	CCONJ
cana-5853	108	5	criteria	criterion	NOUN
cana-5853	108	6	,	,	PUNCT
cana-5853	108	7	this	this	DET
cana-5853	108	8	section	section	NOUN
cana-5853	108	9	has	have	AUX
cana-5853	108	10	established	establish	VERB
cana-5853	108	11	the	the	DET
cana-5853	108	12	theoretical	theoretical	ADJ
cana-5853	108	13	basis	basis	NOUN
cana-5853	108	14	for	for	ADP
cana-5853	108	15	analyzing	analyze	VERB
cana-5853	108	16	infinite	infinite	ADJ
cana-5853	108	17	matrices	matrix	NOUN
cana-5853	108	18	as	as	ADP
cana-5853	108	19	bounded	bound	VERB
cana-5853	108	20	linear	linear	PROPN
cana-5853	108	21	operators	operator	NOUN
cana-5853	108	22	on	on	ADP
cana-5853	108	23	xm	xm	PROPN
cana-5853	108	24	.	.	PUNCT
cana-5853	109	1	the	the	DET
cana-5853	109	2	modular	modular	ADJ
cana-5853	109	3	communications	communication	NOUN
cana-5853	109	4	on	on	ADP
cana-5853	109	5	applied	apply	VERB
cana-5853	109	6	nonlinear	nonlinear	ADJ
cana-5853	109	7	analysis	analysis	NOUN
cana-5853	109	8	issn	issn	NOUN
cana-5853	109	9	:	:	PUNCT
cana-5853	109	10	1074	1074	NUM
cana-5853	109	11	-	-	PUNCT
cana-5853	109	12	133x	133x	NUM
cana-5853	109	13	vol	vol	NOUN
cana-5853	109	14	31	31	NUM
cana-5853	109	15	no	no	NOUN
cana-5853	109	16	.	.	NOUN
cana-5853	109	17	2	2	NUM
cana-5853	109	18	(	(	PUNCT
cana-5853	109	19	2024	2024	NUM
cana-5853	109	20	)	)	PUNCT
cana-5853	109	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	109	22	466	466	NUM
cana-5853	109	23	inequalities	inequality	NOUN
cana-5853	109	24	,	,	PUNCT
cana-5853	109	25	particularly	particularly	ADV
cana-5853	109	26	young	young	ADJ
cana-5853	109	27	’s	’s	PART
cana-5853	109	28	inequality	inequality	NOUN
cana-5853	109	29	adapted	adapt	VERB
cana-5853	109	30	to	to	ADP
cana-5853	109	31	the	the	DET
cana-5853	109	32	index	index	NOUN
cana-5853	109	33	-	-	PUNCT
cana-5853	109	34	dependent	dependent	ADJ
cana-5853	109	35	setting	setting	NOUN
cana-5853	109	36	,	,	PUNCT
cana-5853	109	37	serve	serve	VERB
cana-5853	109	38	as	as	ADP
cana-5853	109	39	essential	essential	ADJ
cana-5853	109	40	tools	tool	NOUN
cana-5853	109	41	for	for	ADP
cana-5853	109	42	proving	prove	VERB
cana-5853	109	43	boundedness	boundedness	NOUN
cana-5853	109	44	results	result	NOUN
cana-5853	109	45	,	,	PUNCT
cana-5853	109	46	which	which	PRON
cana-5853	109	47	will	will	AUX
cana-5853	109	48	be	be	AUX
cana-5853	109	49	systematically	systematically	ADV
cana-5853	109	50	developed	develop	VERB
cana-5853	109	51	for	for	ADP
cana-5853	109	52	specific	specific	ADJ
cana-5853	109	53	matrix	matrix	NOUN
cana-5853	109	54	classes	class	NOUN
cana-5853	109	55	in	in	ADP
cana-5853	109	56	subsequent	subsequent	ADJ
cana-5853	109	57	sections	section	NOUN
cana-5853	109	58	.	.	PUNCT
cana-5853	110	1	4	4	NUM
cana-5853	110	2	diagonal	diagonal	ADJ
cana-5853	110	3	and	and	CCONJ
cana-5853	110	4	triangular	triangular	NOUN
cana-5853	110	5	matrices	matrice	VERB
cana-5853	110	6	4.1	4.1	NUM
cana-5853	110	7	boundedness	boundedness	NOUN
cana-5853	110	8	criteria	criterion	NOUN
cana-5853	110	9	a	a	DET
cana-5853	110	10	central	central	ADJ
cana-5853	110	11	question	question	NOUN
cana-5853	110	12	in	in	ADP
cana-5853	110	13	the	the	DET
cana-5853	110	14	study	study	NOUN
cana-5853	110	15	of	of	ADP
cana-5853	110	16	matrix	matrix	NOUN
cana-5853	110	17	transformations	transformation	NOUN
cana-5853	110	18	on	on	ADP
cana-5853	110	19	modulated	modulate	VERB
cana-5853	110	20	orlicz	orlicz	ADJ
cana-5853	110	21	-	-	PUNCT
cana-5853	110	22	type	type	NOUN
cana-5853	110	23	sequence	sequence	NOUN
cana-5853	110	24	spaces	space	VERB
cana-5853	110	25	xm	xm	PROPN
cana-5853	110	26	is	be	AUX
cana-5853	110	27	determining	determine	VERB
cana-5853	110	28	when	when	SCONJ
cana-5853	110	29	an	an	DET
cana-5853	110	30	infinite	infinite	ADJ
cana-5853	110	31	matrix	matrix	NOUN
cana-5853	110	32	a	a	DET
cana-5853	110	33	=	=	SYM
cana-5853	110	34	(	(	PUNCT
cana-5853	110	35	ank	ank	PROPN
cana-5853	110	36	)	)	PUNCT
cana-5853	110	37	defines	define	VERB
cana-5853	110	38	a	a	DET
cana-5853	110	39	bounded	bounded	ADJ
cana-5853	110	40	linear	linear	ADJ
cana-5853	110	41	operator	operator	NOUN
cana-5853	110	42	from	from	ADP
cana-5853	110	43	xm	xm	PROPN
cana-5853	110	44	into	into	ADP
cana-5853	110	45	itself	itself	PRON
cana-5853	110	46	.	.	PUNCT
cana-5853	111	1	in	in	ADP
cana-5853	111	2	this	this	DET
cana-5853	111	3	subsection	subsection	NOUN
cana-5853	111	4	,	,	PUNCT
cana-5853	111	5	we	we	PRON
cana-5853	111	6	present	present	VERB
cana-5853	111	7	general	general	ADJ
cana-5853	111	8	sufficient	sufficient	ADJ
cana-5853	111	9	and	and	CCONJ
cana-5853	111	10	necessary	necessary	ADJ
cana-5853	111	11	conditions	condition	NOUN
cana-5853	111	12	for	for	ADP
cana-5853	111	13	boundedness	boundedness	NOUN
cana-5853	111	14	,	,	PUNCT
cana-5853	111	15	followed	follow	VERB
cana-5853	111	16	by	by	ADP
cana-5853	111	17	illustrative	illustrative	ADJ
cana-5853	111	18	examples	example	NOUN
cana-5853	111	19	and	and	CCONJ
cana-5853	111	20	counterexamples	counterexample	NOUN
cana-5853	111	21	to	to	PART
cana-5853	111	22	clarify	clarify	VERB
cana-5853	111	23	the	the	DET
cana-5853	111	24	theory	theory	NOUN
cana-5853	111	25	.	.	PUNCT
cana-5853	112	1	sufficient	sufficient	ADJ
cana-5853	112	2	conditions	condition	NOUN
cana-5853	112	3	let	let	VERB
cana-5853	112	4	xm	xm	PROPN
cana-5853	112	5	be	be	AUX
cana-5853	112	6	defined	define	VERB
cana-5853	112	7	via	via	ADP
cana-5853	112	8	a	a	DET
cana-5853	112	9	family	family	NOUN
cana-5853	112	10	of	of	ADP
cana-5853	112	11	modular	modular	ADJ
cana-5853	112	12	functions	function	NOUN
cana-5853	112	13	m(n	m(n	PROPN
cana-5853	112	14	,	,	PUNCT
cana-5853	112	15	t	t	NOUN
cana-5853	112	16	)	)	PUNCT
cana-5853	112	17	satisfying	satisfy	VERB
cana-5853	112	18	standard	standard	ADJ
cana-5853	112	19	conditions	condition	NOUN
cana-5853	112	20	(	(	PUNCT
cana-5853	112	21	e.g.	e.g.	ADV
cana-5853	112	22	,	,	PUNCT
cana-5853	112	23	convexity	convexity	NOUN
cana-5853	112	24	,	,	PUNCT
cana-5853	112	25	continuity	continuity	NOUN
cana-5853	112	26	,	,	PUNCT
cana-5853	112	27	∆2	∆2	NOUN
cana-5853	112	28	-	-	PUNCT
cana-5853	112	29	type	type	NOUN
cana-5853	112	30	growth	growth	NOUN
cana-5853	112	31	)	)	PUNCT
cana-5853	112	32	.	.	PUNCT
cana-5853	113	1	for	for	ADP
cana-5853	113	2	a	a	DET
cana-5853	113	3	matrix	matrix	NOUN
cana-5853	113	4	a	a	DET
cana-5853	113	5	=	=	SYM
cana-5853	113	6	(	(	PUNCT
cana-5853	113	7	ank	ank	PROPN
cana-5853	113	8	)	)	PUNCT
cana-5853	113	9	,	,	PUNCT
cana-5853	113	10	define	define	VERB
cana-5853	113	11	the	the	DET
cana-5853	113	12	formal	formal	ADJ
cana-5853	113	13	action	action	NOUN
cana-5853	113	14	(	(	PUNCT
cana-5853	113	15	ax)n	ax)n	PROPN
cana-5853	113	16	=	=	PROPN
cana-5853	114	1	∞∑	∞∑	NUM
cana-5853	114	2	k=1	k=1	ADV
cana-5853	114	3	ankxk	ankxk	ADJ
cana-5853	114	4	.	.	PUNCT
cana-5853	115	1	a	a	DET
cana-5853	115	2	typical	typical	ADJ
cana-5853	115	3	sufficient	sufficient	ADJ
cana-5853	115	4	condition	condition	NOUN
cana-5853	115	5	for	for	SCONJ
cana-5853	115	6	a	a	PRON
cana-5853	115	7	to	to	PART
cana-5853	115	8	be	be	AUX
cana-5853	115	9	a	a	DET
cana-5853	115	10	bounded	bounded	ADJ
cana-5853	115	11	operator	operator	NOUN
cana-5853	115	12	on	on	ADP
cana-5853	115	13	xm	xm	PROPN
cana-5853	115	14	is	be	AUX
cana-5853	115	15	the	the	DET
cana-5853	115	16	existence	existence	NOUN
cana-5853	115	17	of	of	ADP
cana-5853	115	18	a	a	DET
cana-5853	115	19	constant	constant	ADJ
cana-5853	115	20	c	c	NOUN
cana-5853	115	21	>	>	X
cana-5853	115	22	0	0	NUM
cana-5853	116	1	such	such	ADJ
cana-5853	116	2	that	that	SCONJ
cana-5853	116	3	∞∑	∞∑	NUM
cana-5853	116	4	n=1	n=1	NOUN
cana-5853	116	5	m	m	PROPN
cana-5853	116	6	(	(	PUNCT
cana-5853	116	7	n	n	CCONJ
cana-5853	116	8	,	,	PUNCT
cana-5853	116	9	∞∑	∞∑	NUM
cana-5853	116	10	k=1	k=1	ADJ
cana-5853	116	11	ankxk	ankxk	ADJ
cana-5853	116	12	)	)	PUNCT
cana-5853	116	13	≤	≤	NUM
cana-5853	116	14	c	c	X
cana-5853	116	15	∞∑	∞∑	NUM
cana-5853	116	16	k=1	k=1	PROPN
cana-5853	116	17	m(k	m(k	PROPN
cana-5853	116	18	,	,	PUNCT
cana-5853	116	19	xk	xk	PROPN
cana-5853	116	20	)	)	PUNCT
cana-5853	116	21	for	for	ADP
cana-5853	116	22	all	all	DET
cana-5853	116	23	x	x	SYM
cana-5853	116	24	∈	∈	PROPN
cana-5853	116	25	xm	xm	PROPN
cana-5853	116	26	.	.	PUNCT
cana-5853	117	1	this	this	DET
cana-5853	117	2	inequality	inequality	NOUN
cana-5853	117	3	ensures	ensure	VERB
cana-5853	117	4	that	that	SCONJ
cana-5853	117	5	the	the	DET
cana-5853	117	6	modular	modular	NOUN
cana-5853	117	7	of	of	ADP
cana-5853	117	8	ax	ax	NOUN
cana-5853	117	9	is	be	AUX
cana-5853	117	10	controlled	control	VERB
cana-5853	117	11	by	by	ADP
cana-5853	117	12	the	the	DET
cana-5853	117	13	modular	modular	NOUN
cana-5853	117	14	of	of	ADP
cana-5853	117	15	x	x	X
cana-5853	117	16	,	,	PUNCT
cana-5853	117	17	directly	directly	ADV
cana-5853	117	18	yielding	yield	VERB
cana-5853	117	19	∥ax∥m	∥ax∥m	X
cana-5853	117	20	≤	≤	NUM
cana-5853	117	21	c	c	NOUN
cana-5853	117	22	′∥x∥m	′∥x∥m	NOUN
cana-5853	117	23	for	for	ADP
cana-5853	117	24	an	an	DET
cana-5853	117	25	equivalent	equivalent	ADJ
cana-5853	117	26	norm	norm	NOUN
cana-5853	117	27	.	.	PUNCT
cana-5853	118	1	a	a	DET
cana-5853	118	2	more	more	ADV
cana-5853	118	3	tractable	tractable	ADJ
cana-5853	118	4	sufficient	sufficient	ADJ
cana-5853	118	5	condition	condition	NOUN
cana-5853	118	6	uses	use	VERB
cana-5853	118	7	modular	modular	ADJ
cana-5853	118	8	analogs	analog	NOUN
cana-5853	118	9	of	of	ADP
cana-5853	118	10	schur	schur	PROPN
cana-5853	118	11	’s	’s	PART
cana-5853	118	12	test	test	NOUN
cana-5853	118	13	.	.	PUNCT
cana-5853	119	1	if	if	SCONJ
cana-5853	119	2	there	there	PRON
cana-5853	119	3	exist	exist	VERB
cana-5853	119	4	non	non	ADJ
cana-5853	119	5	-	-	ADJ
cana-5853	119	6	negative	negative	ADJ
cana-5853	119	7	sequences	sequence	NOUN
cana-5853	119	8	(	(	PUNCT
cana-5853	119	9	un	un	PROPN
cana-5853	119	10	)	)	PUNCT
cana-5853	119	11	,	,	PUNCT
cana-5853	119	12	(	(	PUNCT
cana-5853	119	13	vk	vk	NOUN
cana-5853	119	14	)	)	PUNCT
cana-5853	119	15	with	with	ADP
cana-5853	119	16	∞∑	∞∑	NUM
cana-5853	119	17	n=1	n=1	PROPN
cana-5853	119	18	un	un	PROPN
cana-5853	119	19	<	<	X
cana-5853	119	20	∞	∞	PROPN
cana-5853	119	21	,	,	PUNCT
cana-5853	119	22	∞∑	∞∑	NUM
cana-5853	119	23	k=1	k=1	ADJ
cana-5853	119	24	vk	vk	X
cana-5853	119	25	<	<	X
cana-5853	119	26	∞	∞	PROPN
cana-5853	119	27	,	,	PUNCT
cana-5853	119	28	and	and	CCONJ
cana-5853	119	29	for	for	ADP
cana-5853	119	30	all	all	DET
cana-5853	119	31	n	n	CCONJ
cana-5853	119	32	,	,	PUNCT
cana-5853	119	33	k	k	NOUN
cana-5853	119	34	,	,	PUNCT
cana-5853	119	35	|ank|	|ank|	ADJ
cana-5853	119	36	≤	≤	NUM
cana-5853	119	37	ηnk	ηnk	NOUN
cana-5853	119	38	,	,	PUNCT
cana-5853	119	39	m(n	m(n	PROPN
cana-5853	119	40	,	,	PUNCT
cana-5853	119	41	ηnkt	ηnkt	VERB
cana-5853	119	42	)	)	PUNCT
cana-5853	119	43	≤	≤	NOUN
cana-5853	119	44	un	un	PROPN
cana-5853	120	1	+	+	CCONJ
cana-5853	120	2	vk	vk	PROPN
cana-5853	120	3	+	+	ADJ
cana-5853	120	4	m(k	m(k	PROPN
cana-5853	120	5	,	,	PUNCT
cana-5853	120	6	t	t	PROPN
cana-5853	120	7	)	)	PUNCT
cana-5853	120	8	,	,	PUNCT
cana-5853	120	9	then	then	ADV
cana-5853	120	10	summing	sum	VERB
cana-5853	120	11	over	over	ADP
cana-5853	120	12	n	n	NOUN
cana-5853	120	13	and	and	CCONJ
cana-5853	120	14	k	k	PROPN
cana-5853	120	15	shows	show	VERB
cana-5853	120	16	a	a	PRON
cana-5853	120	17	is	be	AUX
cana-5853	120	18	bounded	bound	VERB
cana-5853	120	19	.	.	PUNCT
cana-5853	121	1	these	these	DET
cana-5853	121	2	conditions	condition	NOUN
cana-5853	121	3	generalize	generalize	VERB
cana-5853	121	4	classical	classical	ADJ
cana-5853	121	5	results	result	NOUN
cana-5853	121	6	for	for	ADP
cana-5853	121	7	ℓp	ℓp	ADJ
cana-5853	121	8	spaces	space	NOUN
cana-5853	121	9	and	and	CCONJ
cana-5853	121	10	orlicz	orlicz	ADJ
cana-5853	121	11	spaces	space	NOUN
cana-5853	121	12	,	,	PUNCT
cana-5853	121	13	where	where	SCONJ
cana-5853	121	14	simple	simple	ADJ
cana-5853	121	15	weighted	weight	VERB
cana-5853	121	16	inequalities	inequality	NOUN
cana-5853	121	17	suffice	suffice	VERB
cana-5853	121	18	.	.	PUNCT
cana-5853	122	1	communications	communication	NOUN
cana-5853	122	2	on	on	ADP
cana-5853	122	3	applied	apply	VERB
cana-5853	122	4	nonlinear	nonlinear	ADJ
cana-5853	122	5	analysis	analysis	NOUN
cana-5853	122	6	issn	issn	NOUN
cana-5853	122	7	:	:	PUNCT
cana-5853	122	8	1074	1074	NUM
cana-5853	122	9	-	-	PUNCT
cana-5853	122	10	133x	133x	NUM
cana-5853	122	11	vol	vol	NOUN
cana-5853	122	12	31	31	NUM
cana-5853	122	13	no	no	NOUN
cana-5853	122	14	.	.	NOUN
cana-5853	122	15	2	2	NUM
cana-5853	122	16	(	(	PUNCT
cana-5853	122	17	2024	2024	NUM
cana-5853	122	18	)	)	PUNCT
cana-5853	123	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	123	2	467	467	NUM
cana-5853	123	3	necessary	necessary	ADJ
cana-5853	123	4	conditions	condition	NOUN
cana-5853	123	5	necessary	necessary	ADJ
cana-5853	123	6	conditions	condition	NOUN
cana-5853	123	7	for	for	ADP
cana-5853	123	8	boundedness	boundedness	NOUN
cana-5853	123	9	are	be	AUX
cana-5853	123	10	often	often	ADV
cana-5853	123	11	expressed	express	VERB
cana-5853	123	12	in	in	ADP
cana-5853	123	13	terms	term	NOUN
cana-5853	123	14	of	of	ADP
cana-5853	123	15	the	the	DET
cana-5853	123	16	behavior	behavior	NOUN
cana-5853	123	17	of	of	ADP
cana-5853	123	18	a	a	PRON
cana-5853	123	19	on	on	ADP
cana-5853	123	20	unit	unit	NOUN
cana-5853	123	21	vectors	vector	NOUN
cana-5853	123	22	.	.	PUNCT
cana-5853	124	1	for	for	ADP
cana-5853	124	2	e(m	e(m	PROPN
cana-5853	124	3	)	)	PUNCT
cana-5853	124	4	the	the	DET
cana-5853	124	5	sequence	sequence	NOUN
cana-5853	124	6	with	with	ADP
cana-5853	124	7	1	1	NUM
cana-5853	124	8	in	in	ADP
cana-5853	124	9	position	position	NOUN
cana-5853	124	10	m	m	NOUN
cana-5853	124	11	and	and	CCONJ
cana-5853	124	12	0	0	NUM
cana-5853	124	13	elsewhere	elsewhere	ADV
cana-5853	124	14	,	,	PUNCT
cana-5853	124	15	boundedness	boundedness	NOUN
cana-5853	124	16	of	of	ADP
cana-5853	124	17	a	a	DET
cana-5853	124	18	implies	implie	NOUN
cana-5853	124	19	:	:	PUNCT
cana-5853	124	20	∥ae(m)∥m	∥ae(m)∥m	PROPN
cana-5853	124	21	≤	≤	PROPN
cana-5853	124	22	c∥e(m)∥m	c∥e(m)∥m	PROPN
cana-5853	124	23	.	.	PUNCT
cana-5853	125	1	but	but	CCONJ
cana-5853	125	2	(	(	PUNCT
cana-5853	125	3	ae(m))n	ae(m))n	PRON
cana-5853	125	4	=	=	SYM
cana-5853	125	5	anm	anm	PROPN
cana-5853	125	6	,	,	PUNCT
cana-5853	125	7	so	so	SCONJ
cana-5853	125	8	∞∑	∞∑	NUM
cana-5853	125	9	n=1	n=1	ADP
cana-5853	125	10	m(n	m(n	PROPN
cana-5853	125	11	,	,	PUNCT
cana-5853	125	12	anm	anm	PROPN
cana-5853	125	13	)	)	PUNCT
cana-5853	125	14	≤	≤	NOUN
cana-5853	125	15	cm(m	cm(m	PUNCT
cana-5853	125	16	,	,	PUNCT
cana-5853	125	17	1	1	NUM
cana-5853	125	18	)	)	PUNCT
cana-5853	125	19	.	.	PUNCT
cana-5853	126	1	thus	thus	ADV
cana-5853	126	2	,	,	PUNCT
cana-5853	126	3	a	a	DET
cana-5853	126	4	necessary	necessary	ADJ
cana-5853	126	5	condition	condition	NOUN
cana-5853	126	6	is	be	AUX
cana-5853	126	7	that	that	SCONJ
cana-5853	126	8	the	the	DET
cana-5853	126	9	columns	column	NOUN
cana-5853	126	10	(	(	PUNCT
cana-5853	126	11	anm	anm	INTJ
cana-5853	126	12	)	)	PUNCT
cana-5853	126	13	lie	lie	NOUN
cana-5853	126	14	in	in	ADP
cana-5853	126	15	xm	xm	PROPN
cana-5853	126	16	with	with	ADP
cana-5853	126	17	modular	modular	ADJ
cana-5853	126	18	sums	sum	NOUN
cana-5853	126	19	uniformly	uniformly	ADV
cana-5853	126	20	bounded	bound	VERB
cana-5853	126	21	relative	relative	ADJ
cana-5853	126	22	to	to	ADP
cana-5853	126	23	m(m	m(m	NOUN
cana-5853	126	24	,	,	PUNCT
cana-5853	126	25	1	1	NUM
cana-5853	126	26	)	)	PUNCT
cana-5853	126	27	.	.	PUNCT
cana-5853	127	1	this	this	DET
cana-5853	127	2	condition	condition	NOUN
cana-5853	127	3	ensures	ensure	VERB
cana-5853	127	4	that	that	SCONJ
cana-5853	127	5	a	a	PRON
cana-5853	127	6	can	can	AUX
cana-5853	127	7	not	not	PART
cana-5853	127	8	”	"	PUNCT
cana-5853	127	9	blow	blow	VERB
cana-5853	127	10	up	up	ADP
cana-5853	127	11	”	"	PUNCT
cana-5853	127	12	single	single	ADJ
cana-5853	127	13	coefficients	coefficient	NOUN
cana-5853	127	14	disproportionately	disproportionately	ADV
cana-5853	127	15	,	,	PUNCT
cana-5853	127	16	reflecting	reflect	VERB
cana-5853	127	17	the	the	DET
cana-5853	127	18	modular	modular	NOUN
cana-5853	127	19	’s	’s	PART
cana-5853	127	20	control	control	NOUN
cana-5853	127	21	of	of	ADP
cana-5853	127	22	local	local	ADJ
cana-5853	127	23	growth	growth	NOUN
cana-5853	127	24	.	.	PUNCT
cana-5853	128	1	examples	example	NOUN
cana-5853	128	2	diagonal	diagonal	ADJ
cana-5853	128	3	operators	operator	NOUN
cana-5853	128	4	.	.	PUNCT
cana-5853	129	1	let	let	VERB
cana-5853	129	2	a	a	DET
cana-5853	129	3	=	=	X
cana-5853	129	4	diag(λn	diag(λn	PROPN
cana-5853	129	5	)	)	PUNCT
cana-5853	129	6	,	,	PUNCT
cana-5853	129	7	so	so	CCONJ
cana-5853	129	8	ank	ank	PROPN
cana-5853	129	9	=	=	PUNCT
cana-5853	129	10	λnδnk	λnδnk	ADJ
cana-5853	129	11	.	.	PUNCT
cana-5853	130	1	then	then	ADV
cana-5853	130	2	(	(	PUNCT
cana-5853	130	3	ax)n	ax)n	PROPN
cana-5853	130	4	=	=	SYM
cana-5853	130	5	λnxn	λnxn	ADJ
cana-5853	130	6	.	.	PUNCT
cana-5853	131	1	boundedness	boundedness	PROPN
cana-5853	131	2	requires	require	VERB
cana-5853	131	3	:	:	PUNCT
cana-5853	131	4	∞∑	∞∑	NUM
cana-5853	131	5	n=1	n=1	ADP
cana-5853	131	6	m(n	m(n	PROPN
cana-5853	131	7	,	,	PUNCT
cana-5853	131	8	λnxn	λnxn	ADJ
cana-5853	131	9	)	)	PUNCT
cana-5853	131	10	≤	≤	NOUN
cana-5853	132	1	c	c	VERB
cana-5853	132	2	∞∑	∞∑	NUM
cana-5853	132	3	n=1	n=1	PROPN
cana-5853	132	4	m(n	m(n	PROPN
cana-5853	132	5	,	,	PUNCT
cana-5853	132	6	xn	xn	PROPN
cana-5853	132	7	)	)	PUNCT
cana-5853	132	8	.	.	PUNCT
cana-5853	133	1	if	if	SCONJ
cana-5853	133	2	m(n	m(n	PROPN
cana-5853	133	3	,	,	PUNCT
cana-5853	133	4	·	·	PUNCT
cana-5853	133	5	)	)	PUNCT
cana-5853	133	6	satisfies	satisfy	VERB
cana-5853	133	7	m(n	m(n	PROPN
cana-5853	133	8	,	,	PUNCT
cana-5853	133	9	λnt	λnt	PROPN
cana-5853	133	10	)	)	PUNCT
cana-5853	133	11	≤	≤	NOUN
cana-5853	134	1	cnm(n	cnm(n	PROPN
cana-5853	134	2	,	,	PUNCT
cana-5853	134	3	t	t	PROPN
cana-5853	134	4	)	)	PUNCT
cana-5853	135	1	+	+	X
cana-5853	135	2	cn	cn	ADJ
cana-5853	135	3	,	,	PUNCT
cana-5853	135	4	uniformly	uniformly	ADV
cana-5853	135	5	over	over	ADP
cana-5853	135	6	n	n	CCONJ
cana-5853	135	7	,	,	PUNCT
cana-5853	135	8	then	then	ADV
cana-5853	135	9	a	a	PRON
cana-5853	135	10	is	be	AUX
cana-5853	135	11	bounded	bound	VERB
cana-5853	135	12	.	.	PUNCT
cana-5853	136	1	for	for	ADP
cana-5853	136	2	example	example	NOUN
cana-5853	136	3	,	,	PUNCT
cana-5853	136	4	if	if	SCONJ
cana-5853	136	5	m(n	m(n	PROPN
cana-5853	136	6	,	,	PUNCT
cana-5853	136	7	t	t	PROPN
cana-5853	136	8	)	)	PUNCT
cana-5853	136	9	=	=	SYM
cana-5853	136	10	ωn|t|p	ωn|t|p	NOUN
cana-5853	136	11	/	/	SYM
cana-5853	136	12	p	p	X
cana-5853	136	13	,	,	PUNCT
cana-5853	136	14	then	then	ADV
cana-5853	136	15	|λn|pωn	|λn|pωn	ADJ
cana-5853	136	16	≤	≤	NUM
cana-5853	136	17	cωn	cωn	NOUN
cana-5853	136	18	implies	imply	VERB
cana-5853	136	19	|λn|p	|λn|p	PROPN
cana-5853	136	20	≤	≤	NUM
cana-5853	136	21	c	c	NOUN
cana-5853	136	22	for	for	ADP
cana-5853	136	23	all	all	DET
cana-5853	136	24	n.	n.	NOUN
cana-5853	136	25	triangular	triangular	NOUN
cana-5853	136	26	matrices	matrix	NOUN
cana-5853	136	27	.	.	PUNCT
cana-5853	137	1	consider	consider	VERB
cana-5853	137	2	lower	low	ADJ
cana-5853	137	3	triangular	triangular	NOUN
cana-5853	137	4	matrices	matrix	NOUN
cana-5853	137	5	ank	ank	PROPN
cana-5853	137	6	=	=	PROPN
cana-5853	137	7	0	0	PROPN
cana-5853	137	8	for	for	SCONJ
cana-5853	137	9	k	k	PROPN
cana-5853	137	10	>	>	X
cana-5853	137	11	n.	n.	PROPN
cana-5853	137	12	sufficient	sufficient	ADJ
cana-5853	137	13	boundedness	boundedness	NOUN
cana-5853	137	14	conditions	condition	NOUN
cana-5853	137	15	follow	follow	VERB
cana-5853	137	16	by	by	ADP
cana-5853	137	17	controlling	control	VERB
cana-5853	137	18	the	the	DET
cana-5853	137	19	cumulative	cumulative	ADJ
cana-5853	137	20	growth	growth	NOUN
cana-5853	137	21	:	:	PUNCT
cana-5853	138	1	n∑	n∑	INTJ
cana-5853	138	2	k=1	k=1	PROPN
cana-5853	138	3	|ankxk|	|ankxk|	NOUN
cana-5853	138	4	≤	≤	PUNCT
cana-5853	139	1	n∑	n∑	X
cana-5853	139	2	k=1	k=1	PROPN
cana-5853	139	3	ηnkm(k	ηnkm(k	PROPN
cana-5853	139	4	,	,	PUNCT
cana-5853	139	5	xk	xk	PROPN
cana-5853	139	6	)	)	PUNCT
cana-5853	139	7	+	+	CCONJ
cana-5853	139	8	θnk	θnk	NOUN
cana-5853	139	9	.	.	PUNCT
cana-5853	140	1	if	if	SCONJ
cana-5853	140	2	∞∑	∞∑	NUM
cana-5853	140	3	n=1	n=1	NUM
cana-5853	140	4	un	un	PROPN
cana-5853	140	5	<	<	X
cana-5853	140	6	∞	∞	PROPN
cana-5853	140	7	where	where	SCONJ
cana-5853	140	8	un	un	PROPN
cana-5853	140	9	=	=	PROPN
cana-5853	140	10	n∑	n∑	PROPN
cana-5853	140	11	k=1	k=1	PROPN
cana-5853	140	12	ηnk	ηnk	NOUN
cana-5853	140	13	,	,	PUNCT
cana-5853	140	14	and	and	CCONJ
cana-5853	140	15	the	the	DET
cana-5853	140	16	modular	modular	ADJ
cana-5853	140	17	satisfies	satisfie	NOUN
cana-5853	140	18	convexity	convexity	NOUN
cana-5853	140	19	and	and	CCONJ
cana-5853	140	20	∆2	∆2	PROPN
cana-5853	140	21	growth	growth	NOUN
cana-5853	140	22	,	,	PUNCT
cana-5853	140	23	then	then	ADV
cana-5853	140	24	a	a	PRON
cana-5853	140	25	is	be	AUX
cana-5853	140	26	bounded	bound	VERB
cana-5853	140	27	.	.	PUNCT
cana-5853	141	1	communications	communication	NOUN
cana-5853	141	2	on	on	ADP
cana-5853	141	3	applied	apply	VERB
cana-5853	141	4	nonlinear	nonlinear	ADJ
cana-5853	141	5	analysis	analysis	NOUN
cana-5853	141	6	issn	issn	NOUN
cana-5853	141	7	:	:	PUNCT
cana-5853	141	8	1074	1074	NUM
cana-5853	141	9	-	-	PUNCT
cana-5853	141	10	133x	133x	NUM
cana-5853	141	11	vol	vol	NOUN
cana-5853	141	12	31	31	NUM
cana-5853	141	13	no	no	NOUN
cana-5853	141	14	.	.	NOUN
cana-5853	141	15	2	2	NUM
cana-5853	141	16	(	(	PUNCT
cana-5853	141	17	2024	2024	NUM
cana-5853	141	18	)	)	PUNCT
cana-5853	141	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	141	20	468	468	NUM
cana-5853	141	21	counterexamples	counterexample	VERB
cana-5853	141	22	unbounded	unbounded	ADJ
cana-5853	141	23	diagonal	diagonal	ADJ
cana-5853	141	24	scaling	scaling	NOUN
cana-5853	141	25	.	.	PUNCT
cana-5853	142	1	suppose	suppose	VERB
cana-5853	142	2	a	a	DET
cana-5853	142	3	=	=	SYM
cana-5853	142	4	diag(λn	diag(λn	PROPN
cana-5853	142	5	)	)	PUNCT
cana-5853	142	6	with	with	ADP
cana-5853	142	7	λn	λn	NOUN
cana-5853	142	8	→	→	SYM
cana-5853	142	9	∞.	∞.	PROPN
cana-5853	142	10	even	even	ADV
cana-5853	142	11	if	if	SCONJ
cana-5853	142	12	x	x	PROPN
cana-5853	142	13	∈	∈	PROPN
cana-5853	142	14	xm	xm	PROPN
cana-5853	142	15	,	,	PUNCT
cana-5853	142	16	(	(	PUNCT
cana-5853	142	17	λnxn	λnxn	ADJ
cana-5853	142	18	)	)	PUNCT
cana-5853	142	19	may	may	AUX
cana-5853	142	20	not	not	PART
cana-5853	142	21	belong	belong	VERB
cana-5853	142	22	to	to	ADP
cana-5853	142	23	xm	xm	PROPN
cana-5853	142	24	if	if	SCONJ
cana-5853	142	25	m(n	m(n	NOUN
cana-5853	142	26	,	,	PUNCT
cana-5853	142	27	λnxn	λnxn	ADJ
cana-5853	142	28	)	)	PUNCT
cana-5853	142	29	grows	grow	VERB
cana-5853	142	30	too	too	ADV
cana-5853	142	31	fast	fast	ADV
cana-5853	142	32	.	.	PUNCT
cana-5853	143	1	for	for	ADP
cana-5853	143	2	instance	instance	NOUN
cana-5853	143	3	,	,	PUNCT
cana-5853	143	4	in	in	ADP
cana-5853	143	5	ℓp	ℓp	NOUN
cana-5853	143	6	with	with	ADP
cana-5853	143	7	m(n	m(n	PROPN
cana-5853	143	8	,	,	PUNCT
cana-5853	143	9	t	t	PROPN
cana-5853	143	10	)	)	PUNCT
cana-5853	143	11	=	=	PUNCT
cana-5853	143	12	|t|p	|t|p	NOUN
cana-5853	143	13	/	/	SYM
cana-5853	143	14	p	p	X
cana-5853	143	15	,	,	PUNCT
cana-5853	143	16	taking	take	VERB
cana-5853	143	17	λn	λn	PRON
cana-5853	143	18	→	→	SYM
cana-5853	143	19	∞	∞	NUM
cana-5853	143	20	breaks	break	NOUN
cana-5853	143	21	boundedness	boundedness	NOUN
cana-5853	143	22	immediately	immediately	ADV
cana-5853	143	23	.	.	PUNCT
cana-5853	144	1	highly	highly	ADV
cana-5853	144	2	oscillating	oscillate	VERB
cana-5853	144	3	off	off	ADJ
cana-5853	144	4	-	-	PUNCT
cana-5853	144	5	diagonal	diagonal	ADJ
cana-5853	144	6	terms	term	NOUN
cana-5853	144	7	.	.	PUNCT
cana-5853	145	1	consider	consider	VERB
cana-5853	145	2	matrices	matrix	NOUN
cana-5853	145	3	with	with	ADP
cana-5853	145	4	large	large	ADJ
cana-5853	145	5	off	off	ADJ
cana-5853	145	6	-	-	PUNCT
cana-5853	145	7	diagonal	diagonal	ADJ
cana-5853	145	8	entries	entry	NOUN
cana-5853	145	9	that	that	PRON
cana-5853	145	10	do	do	AUX
cana-5853	145	11	not	not	PART
cana-5853	145	12	decay	decay	VERB
cana-5853	145	13	suitably	suitably	ADV
cana-5853	145	14	.	.	PUNCT
cana-5853	146	1	if	if	SCONJ
cana-5853	146	2	∞∑	∞∑	NUM
cana-5853	146	3	k=1	k=1	PROPN
cana-5853	146	4	m(k	m(k	PROPN
cana-5853	146	5	,	,	PUNCT
cana-5853	146	6	xk	xk	PROPN
cana-5853	146	7	)	)	PUNCT
cana-5853	146	8	<	<	X
cana-5853	146	9	∞	∞	PROPN
cana-5853	146	10	but	but	CCONJ
cana-5853	146	11	∞∑	∞∑	NUM
cana-5853	146	12	n=1	n=1	PROPN
cana-5853	146	13	m	m	PROPN
cana-5853	146	14	(	(	PUNCT
cana-5853	146	15	n	n	CCONJ
cana-5853	146	16	,	,	PUNCT
cana-5853	146	17	∞∑	∞∑	NUM
cana-5853	146	18	k=1	k=1	ADJ
cana-5853	146	19	ankxk	ankxk	ADJ
cana-5853	146	20	)	)	PUNCT
cana-5853	147	1	=	=	SYM
cana-5853	147	2	∞	∞	PROPN
cana-5853	147	3	,	,	PUNCT
cana-5853	147	4	boundedness	boundedness	NOUN
cana-5853	147	5	fails	fail	VERB
cana-5853	147	6	.	.	PUNCT
cana-5853	148	1	such	such	ADJ
cana-5853	148	2	matrices	matrix	NOUN
cana-5853	148	3	might	might	AUX
cana-5853	148	4	map	map	VERB
cana-5853	148	5	sparse	sparse	ADJ
cana-5853	148	6	sequences	sequence	NOUN
cana-5853	148	7	to	to	ADP
cana-5853	148	8	dense	dense	ADJ
cana-5853	148	9	,	,	PUNCT
cana-5853	148	10	unbounded	unbounded	ADJ
cana-5853	148	11	images	image	NOUN
cana-5853	148	12	,	,	PUNCT
cana-5853	148	13	violating	violate	VERB
cana-5853	148	14	modular	modular	ADJ
cana-5853	148	15	control	control	NOUN
cana-5853	148	16	.	.	PUNCT
cana-5853	149	1	summary	summary	NOUN
cana-5853	149	2	.	.	PUNCT
cana-5853	150	1	the	the	DET
cana-5853	150	2	boundedness	boundedness	NOUN
cana-5853	150	3	of	of	ADP
cana-5853	150	4	matrix	matrix	NOUN
cana-5853	150	5	operators	operator	NOUN
cana-5853	150	6	on	on	ADP
cana-5853	150	7	xm	xm	PROPN
cana-5853	150	8	spaces	space	NOUN
cana-5853	150	9	relies	rely	VERB
cana-5853	150	10	on	on	ADP
cana-5853	150	11	delicate	delicate	ADJ
cana-5853	150	12	balancing	balancing	NOUN
cana-5853	150	13	of	of	ADP
cana-5853	150	14	entrywise	entrywise	ADJ
cana-5853	150	15	growth	growth	NOUN
cana-5853	150	16	through	through	ADP
cana-5853	150	17	modular	modular	ADJ
cana-5853	150	18	functions	function	NOUN
cana-5853	150	19	.	.	PUNCT
cana-5853	151	1	sufficient	sufficient	ADJ
cana-5853	151	2	conditions	condition	NOUN
cana-5853	151	3	often	often	ADV
cana-5853	151	4	exploit	exploit	VERB
cana-5853	151	5	modular	modular	ADJ
cana-5853	151	6	inequalities	inequality	NOUN
cana-5853	151	7	and	and	CCONJ
cana-5853	151	8	convexity	convexity	NOUN
cana-5853	151	9	,	,	PUNCT
cana-5853	151	10	while	while	SCONJ
cana-5853	151	11	necessary	necessary	ADJ
cana-5853	151	12	conditions	condition	NOUN
cana-5853	151	13	ensure	ensure	VERB
cana-5853	151	14	that	that	SCONJ
cana-5853	151	15	matrix	matrix	NOUN
cana-5853	151	16	columns	column	NOUN
cana-5853	151	17	remain	remain	VERB
cana-5853	151	18	controlled	control	VERB
cana-5853	151	19	in	in	ADP
cana-5853	151	20	the	the	DET
cana-5853	151	21	modular	modular	ADJ
cana-5853	151	22	sum	sum	NOUN
cana-5853	151	23	.	.	PUNCT
cana-5853	152	1	by	by	ADP
cana-5853	152	2	examining	examine	VERB
cana-5853	152	3	diagonal	diagonal	ADJ
cana-5853	152	4	,	,	PUNCT
cana-5853	152	5	triangular	triangular	NOUN
cana-5853	152	6	,	,	PUNCT
cana-5853	152	7	and	and	CCONJ
cana-5853	152	8	general	general	ADJ
cana-5853	152	9	matrices	matrix	NOUN
cana-5853	152	10	,	,	PUNCT
cana-5853	152	11	one	one	PRON
cana-5853	152	12	sees	see	VERB
cana-5853	152	13	both	both	CCONJ
cana-5853	152	14	the	the	DET
cana-5853	152	15	richness	richness	NOUN
cana-5853	152	16	and	and	CCONJ
cana-5853	152	17	the	the	DET
cana-5853	152	18	challenges	challenge	NOUN
cana-5853	152	19	of	of	ADP
cana-5853	152	20	operator	operator	NOUN
cana-5853	152	21	theory	theory	NOUN
cana-5853	152	22	in	in	ADP
cana-5853	152	23	this	this	DET
cana-5853	152	24	flexible	flexible	ADJ
cana-5853	152	25	modular	modular	ADJ
cana-5853	152	26	framework	framework	NOUN
cana-5853	152	27	.	.	PUNCT
cana-5853	153	1	4.2	4.2	NUM
cana-5853	153	2	compactness	compactness	NOUN
cana-5853	153	3	characterizations	characterization	NOUN
cana-5853	153	4	beyond	beyond	ADP
cana-5853	153	5	boundedness	boundedness	NOUN
cana-5853	153	6	,	,	PUNCT
cana-5853	153	7	the	the	DET
cana-5853	153	8	compactness	compactness	NOUN
cana-5853	153	9	of	of	ADP
cana-5853	153	10	matrix	matrix	NOUN
cana-5853	153	11	operators	operator	NOUN
cana-5853	153	12	on	on	ADP
cana-5853	153	13	modulated	modulate	VERB
cana-5853	153	14	orlicz	orlicz	ADJ
cana-5853	153	15	-	-	PUNCT
cana-5853	153	16	type	type	NOUN
cana-5853	153	17	sequence	sequence	NOUN
cana-5853	153	18	spaces	space	VERB
cana-5853	153	19	xm	xm	PROPN
cana-5853	153	20	is	be	AUX
cana-5853	153	21	a	a	DET
cana-5853	153	22	central	central	ADJ
cana-5853	153	23	question	question	NOUN
cana-5853	153	24	in	in	ADP
cana-5853	153	25	operator	operator	NOUN
cana-5853	153	26	theory	theory	NOUN
cana-5853	153	27	.	.	PUNCT
cana-5853	154	1	compact	compact	ADJ
cana-5853	154	2	operators	operator	NOUN
cana-5853	154	3	have	have	VERB
cana-5853	154	4	wellunderstood	wellunderstood	NOUN
cana-5853	154	5	spectral	spectral	ADJ
cana-5853	154	6	properties	property	NOUN
cana-5853	154	7	,	,	PUNCT
cana-5853	154	8	and	and	CCONJ
cana-5853	154	9	their	their	PRON
cana-5853	154	10	study	study	NOUN
cana-5853	154	11	is	be	AUX
cana-5853	154	12	critical	critical	ADJ
cana-5853	154	13	in	in	ADP
cana-5853	154	14	approximation	approximation	NOUN
cana-5853	154	15	theory	theory	NOUN
cana-5853	154	16	,	,	PUNCT
cana-5853	154	17	spectral	spectral	ADJ
cana-5853	154	18	theory	theory	NOUN
cana-5853	154	19	,	,	PUNCT
cana-5853	154	20	and	and	CCONJ
cana-5853	154	21	summability	summability	NOUN
cana-5853	154	22	methods	method	NOUN
cana-5853	154	23	.	.	PUNCT
cana-5853	155	1	in	in	ADP
cana-5853	155	2	this	this	DET
cana-5853	155	3	subsection	subsection	NOUN
cana-5853	155	4	,	,	PUNCT
cana-5853	155	5	we	we	PRON
cana-5853	155	6	provide	provide	VERB
cana-5853	155	7	general	general	ADJ
cana-5853	155	8	criteria	criterion	NOUN
cana-5853	155	9	for	for	ADP
cana-5853	155	10	compactness	compactness	NOUN
cana-5853	155	11	in	in	ADP
cana-5853	155	12	xm	xm	PROPN
cana-5853	155	13	spaces	space	NOUN
cana-5853	155	14	,	,	PUNCT
cana-5853	155	15	with	with	ADP
cana-5853	155	16	particular	particular	ADJ
cana-5853	155	17	emphasis	emphasis	NOUN
cana-5853	155	18	on	on	ADP
cana-5853	155	19	tail	tail	NOUN
cana-5853	155	20	conditions	condition	NOUN
cana-5853	155	21	and	and	CCONJ
cana-5853	155	22	modular	modular	ADJ
cana-5853	155	23	domination	domination	NOUN
cana-5853	155	24	estimates	estimate	NOUN
cana-5853	155	25	.	.	PUNCT
cana-5853	156	1	tail	tail	NOUN
cana-5853	156	2	conditions	condition	NOUN
cana-5853	156	3	a	a	DET
cana-5853	156	4	classical	classical	ADJ
cana-5853	156	5	approach	approach	NOUN
cana-5853	156	6	to	to	ADP
cana-5853	156	7	characterizing	characterize	VERB
cana-5853	156	8	compactness	compactness	NOUN
cana-5853	156	9	in	in	ADP
cana-5853	156	10	sequence	sequence	NOUN
cana-5853	156	11	spaces	space	NOUN
cana-5853	156	12	involves	involve	VERB
cana-5853	156	13	tail	tail	NOUN
cana-5853	156	14	estimates	estimate	NOUN
cana-5853	156	15	.	.	PUNCT
cana-5853	157	1	intuitively	intuitively	ADV
cana-5853	157	2	,	,	PUNCT
cana-5853	157	3	a	a	DET
cana-5853	157	4	bounded	bound	VERB
cana-5853	157	5	operator	operator	NOUN
cana-5853	157	6	a	a	PRON
cana-5853	157	7	on	on	ADP
cana-5853	157	8	xm	xm	PROPN
cana-5853	157	9	is	be	AUX
cana-5853	157	10	compact	compact	ADJ
cana-5853	157	11	if	if	SCONJ
cana-5853	157	12	it	it	PRON
cana-5853	157	13	maps	map	VERB
cana-5853	157	14	bounded	bound	VERB
cana-5853	157	15	sets	set	NOUN
cana-5853	157	16	into	into	ADP
cana-5853	157	17	subsets	subset	NOUN
cana-5853	157	18	whose	whose	DET
cana-5853	157	19	”	"	PUNCT
cana-5853	157	20	tails	tail	NOUN
cana-5853	157	21	”	"	PUNCT
cana-5853	157	22	become	become	VERB
cana-5853	157	23	small	small	ADJ
cana-5853	157	24	uniformly	uniformly	ADV
cana-5853	157	25	.	.	PUNCT
cana-5853	158	1	formally	formally	ADV
cana-5853	158	2	,	,	PUNCT
cana-5853	158	3	let	let	VERB
cana-5853	158	4	b	b	PROPN
cana-5853	158	5	⊂	⊂	PROPN
cana-5853	158	6	xm	xm	PROPN
cana-5853	158	7	be	be	AUX
cana-5853	158	8	the	the	DET
cana-5853	158	9	unit	unit	NOUN
cana-5853	158	10	ball	ball	NOUN
cana-5853	158	11	.	.	PUNCT
cana-5853	159	1	for	for	SCONJ
cana-5853	159	2	a	a	PRON
cana-5853	159	3	to	to	PART
cana-5853	159	4	be	be	AUX
cana-5853	159	5	compact	compact	ADJ
cana-5853	159	6	,	,	PUNCT
cana-5853	159	7	it	it	PRON
cana-5853	159	8	is	be	AUX
cana-5853	159	9	necessary	necessary	ADJ
cana-5853	159	10	and	and	CCONJ
cana-5853	159	11	sufficient	sufficient	ADJ
cana-5853	159	12	(	(	PUNCT
cana-5853	159	13	in	in	ADP
cana-5853	159	14	many	many	ADJ
cana-5853	159	15	settings	setting	NOUN
cana-5853	159	16	)	)	PUNCT
cana-5853	159	17	that	that	SCONJ
cana-5853	159	18	for	for	ADP
cana-5853	159	19	every	every	DET
cana-5853	159	20	ϵ	ϵ	PROPN
cana-5853	159	21	>	>	X
cana-5853	159	22	0	0	NUM
cana-5853	159	23	,	,	PUNCT
cana-5853	159	24	there	there	PRON
cana-5853	159	25	exists	exist	VERB
cana-5853	159	26	n	n	PRON
cana-5853	159	27	∈	∈	PROPN
cana-5853	159	28	n	n	PRON
cana-5853	159	29	such	such	ADJ
cana-5853	159	30	that	that	PRON
cana-5853	159	31	:	:	PUNCT
cana-5853	159	32	sup	sup	NOUN
cana-5853	159	33	x∈b	x∈b	NOUN
cana-5853	160	1	∞∑	∞∑	NUM
cana-5853	160	2	n	n	CCONJ
cana-5853	160	3	=	=	SYM
cana-5853	160	4	n+1	n+1	PROPN
cana-5853	160	5	m	m	PROPN
cana-5853	160	6	(	(	PUNCT
cana-5853	160	7	n	n	X
cana-5853	160	8	,	,	PUNCT
cana-5853	160	9	(	(	PUNCT
cana-5853	160	10	ax)n	ax)n	PROPN
cana-5853	160	11	)	)	PUNCT
cana-5853	160	12	<	<	X
cana-5853	160	13	ϵ.	ϵ.	NOUN
cana-5853	160	14	communications	communication	NOUN
cana-5853	160	15	on	on	ADP
cana-5853	160	16	applied	apply	VERB
cana-5853	160	17	nonlinear	nonlinear	ADJ
cana-5853	160	18	analysis	analysis	NOUN
cana-5853	160	19	issn	issn	NOUN
cana-5853	160	20	:	:	PUNCT
cana-5853	160	21	1074	1074	NUM
cana-5853	160	22	-	-	PUNCT
cana-5853	160	23	133x	133x	NUM
cana-5853	160	24	vol	vol	NOUN
cana-5853	160	25	31	31	NUM
cana-5853	160	26	no	no	NOUN
cana-5853	160	27	.	.	NOUN
cana-5853	160	28	2	2	NUM
cana-5853	160	29	(	(	PUNCT
cana-5853	160	30	2024	2024	NUM
cana-5853	160	31	)	)	PUNCT
cana-5853	160	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	160	33	469	469	NUM
cana-5853	160	34	this	this	DET
cana-5853	160	35	condition	condition	NOUN
cana-5853	160	36	ensures	ensure	VERB
cana-5853	160	37	that	that	SCONJ
cana-5853	160	38	the	the	DET
cana-5853	160	39	image	image	NOUN
cana-5853	160	40	of	of	ADP
cana-5853	160	41	b	b	NOUN
cana-5853	160	42	under	under	ADP
cana-5853	160	43	a	a	PRON
cana-5853	160	44	has	have	VERB
cana-5853	160	45	uniformly	uniformly	ADV
cana-5853	160	46	small	small	ADJ
cana-5853	160	47	tail	tail	NOUN
cana-5853	160	48	in	in	ADP
cana-5853	160	49	the	the	DET
cana-5853	160	50	modular	modular	ADJ
cana-5853	160	51	sense	sense	NOUN
cana-5853	160	52	.	.	PUNCT
cana-5853	161	1	it	it	PRON
cana-5853	161	2	prevents	prevent	VERB
cana-5853	161	3	the	the	DET
cana-5853	161	4	operator	operator	NOUN
cana-5853	161	5	from	from	ADP
cana-5853	161	6	”	"	PUNCT
cana-5853	161	7	spreading	spread	VERB
cana-5853	161	8	”	"	PUNCT
cana-5853	161	9	mass	mass	NOUN
cana-5853	161	10	into	into	ADP
cana-5853	161	11	higher	high	ADJ
cana-5853	161	12	indices	index	NOUN
cana-5853	161	13	in	in	ADP
cana-5853	161	14	an	an	DET
cana-5853	161	15	uncontrolled	uncontrolled	ADJ
cana-5853	161	16	way	way	NOUN
cana-5853	161	17	.	.	PUNCT
cana-5853	162	1	for	for	ADP
cana-5853	162	2	matrices	matrix	NOUN
cana-5853	162	3	a	a	DET
cana-5853	162	4	=	=	SYM
cana-5853	162	5	(	(	PUNCT
cana-5853	162	6	ank	ank	PROPN
cana-5853	162	7	)	)	PUNCT
cana-5853	162	8	,	,	PUNCT
cana-5853	162	9	this	this	PRON
cana-5853	162	10	translates	translate	VERB
cana-5853	162	11	to	to	ADP
cana-5853	162	12	controlling	control	VERB
cana-5853	162	13	:	:	PUNCT
cana-5853	162	14	∞∑	∞∑	NUM
cana-5853	162	15	n	n	CCONJ
cana-5853	162	16	=	=	SYM
cana-5853	162	17	n+1	n+1	PROPN
cana-5853	162	18	m	m	PROPN
cana-5853	162	19	(	(	PUNCT
cana-5853	162	20	n	n	CCONJ
cana-5853	162	21	,	,	PUNCT
cana-5853	162	22	∞∑	∞∑	NUM
cana-5853	162	23	k=1	k=1	ADV
cana-5853	162	24	ankxk	ankxk	ADJ
cana-5853	162	25	)	)	PUNCT
cana-5853	162	26	.	.	PUNCT
cana-5853	163	1	one	one	NUM
cana-5853	163	2	sufficient	sufficient	ADJ
cana-5853	163	3	strategy	strategy	NOUN
cana-5853	163	4	is	be	AUX
cana-5853	163	5	to	to	PART
cana-5853	163	6	impose	impose	VERB
cana-5853	163	7	decay	decay	NOUN
cana-5853	163	8	on	on	ADP
cana-5853	163	9	the	the	DET
cana-5853	163	10	matrix	matrix	NOUN
cana-5853	163	11	rows	row	NOUN
cana-5853	163	12	:	:	PUNCT
cana-5853	163	13	∞∑	∞∑	NUM
cana-5853	163	14	k=1	k=1	ADP
cana-5853	163	15	|ank|	|ank|	PROPN
cana-5853	163	16	→	→	SYM
cana-5853	163	17	0	0	NUM
cana-5853	163	18	as	as	ADP
cana-5853	163	19	n	n	NUM
cana-5853	163	20	→	→	SYM
cana-5853	163	21	∞	∞	PROPN
cana-5853	163	22	,	,	PUNCT
cana-5853	163	23	together	together	ADV
cana-5853	163	24	with	with	ADP
cana-5853	163	25	uniform	uniform	ADJ
cana-5853	163	26	modular	modular	ADJ
cana-5853	163	27	estimates	estimate	NOUN
cana-5853	163	28	ensuring	ensure	VERB
cana-5853	163	29	that	that	SCONJ
cana-5853	163	30	the	the	DET
cana-5853	163	31	sums	sum	NOUN
cana-5853	163	32	remain	remain	AUX
cana-5853	163	33	controlled	control	VERB
cana-5853	163	34	by	by	ADP
cana-5853	163	35	the	the	DET
cana-5853	163	36	modular	modular	NOUN
cana-5853	163	37	of	of	ADP
cana-5853	163	38	x.	x.	NOUN
cana-5853	163	39	this	this	DET
cana-5853	163	40	approach	approach	NOUN
cana-5853	163	41	generalizes	generalize	VERB
cana-5853	163	42	the	the	DET
cana-5853	163	43	classical	classical	ADJ
cana-5853	163	44	compactness	compactness	NOUN
cana-5853	163	45	conditions	condition	NOUN
cana-5853	163	46	known	know	VERB
cana-5853	163	47	for	for	ADP
cana-5853	163	48	ℓp	ℓp	ADJ
cana-5853	163	49	spaces	space	NOUN
cana-5853	163	50	and	and	CCONJ
cana-5853	163	51	orlicz	orlicz	ADJ
cana-5853	163	52	spaces	space	NOUN
cana-5853	163	53	.	.	PUNCT
cana-5853	164	1	modular	modular	ADJ
cana-5853	164	2	domination	domination	NOUN
cana-5853	164	3	another	another	DET
cana-5853	164	4	powerful	powerful	ADJ
cana-5853	164	5	method	method	NOUN
cana-5853	164	6	for	for	ADP
cana-5853	164	7	proving	prove	VERB
cana-5853	164	8	compactness	compactness	NOUN
cana-5853	164	9	involves	involve	VERB
cana-5853	164	10	modular	modular	ADJ
cana-5853	164	11	domination	domination	NOUN
cana-5853	164	12	inequalities	inequality	NOUN
cana-5853	164	13	.	.	PUNCT
cana-5853	165	1	this	this	DET
cana-5853	165	2	approach	approach	NOUN
cana-5853	165	3	relies	rely	VERB
cana-5853	165	4	on	on	ADP
cana-5853	165	5	comparing	compare	VERB
cana-5853	165	6	a	a	PRON
cana-5853	165	7	to	to	ADP
cana-5853	165	8	operators	operator	NOUN
cana-5853	165	9	that	that	PRON
cana-5853	165	10	are	be	AUX
cana-5853	165	11	already	already	ADV
cana-5853	165	12	known	know	VERB
cana-5853	165	13	to	to	PART
cana-5853	165	14	be	be	AUX
cana-5853	165	15	compact	compact	ADJ
cana-5853	165	16	,	,	PUNCT
cana-5853	165	17	often	often	ADV
cana-5853	165	18	via	via	ADP
cana-5853	165	19	modular	modular	ADJ
cana-5853	165	20	inequalities	inequality	NOUN
cana-5853	165	21	.	.	PUNCT
cana-5853	166	1	suppose	suppose	VERB
cana-5853	166	2	there	there	PRON
cana-5853	166	3	exists	exist	VERB
cana-5853	166	4	a	a	DET
cana-5853	166	5	sequence	sequence	NOUN
cana-5853	166	6	(	(	PUNCT
cana-5853	166	7	θn	θn	NOUN
cana-5853	166	8	)	)	PUNCT
cana-5853	166	9	with	with	ADP
cana-5853	166	10	θn	θn	NOUN
cana-5853	166	11	→	→	SYM
cana-5853	166	12	0	0	PROPN
cana-5853	166	13	as	as	ADP
cana-5853	166	14	n	n	NUM
cana-5853	166	15	→	→	SYM
cana-5853	166	16	∞	∞	PROPN
cana-5853	166	17	,	,	PUNCT
cana-5853	166	18	such	such	ADJ
cana-5853	166	19	that	that	PRON
cana-5853	166	20	for	for	ADP
cana-5853	166	21	all	all	DET
cana-5853	166	22	x	x	SYM
cana-5853	166	23	∈	∈	PROPN
cana-5853	166	24	xm	xm	PROPN
cana-5853	166	25	,	,	PUNCT
cana-5853	166	26	m	m	PROPN
cana-5853	166	27	(	(	PUNCT
cana-5853	166	28	n	n	CCONJ
cana-5853	166	29	,	,	PUNCT
cana-5853	166	30	(	(	PUNCT
cana-5853	166	31	ax)n	ax)n	PROPN
cana-5853	166	32	)	)	PUNCT
cana-5853	166	33	≤	≤	NOUN
cana-5853	166	34	θn	θn	ADP
cana-5853	166	35	∞∑	∞∑	PROPN
cana-5853	166	36	k=1	k=1	X
cana-5853	166	37	m(k	m(k	PROPN
cana-5853	166	38	,	,	PUNCT
cana-5853	166	39	xk	xk	PROPN
cana-5853	166	40	)	)	PUNCT
cana-5853	166	41	+	+	CCONJ
cana-5853	167	1	ϕn	ϕn	INTJ
cana-5853	167	2	,	,	PUNCT
cana-5853	167	3	where	where	SCONJ
cana-5853	167	4	(	(	PUNCT
cana-5853	167	5	ϕn	ϕn	NOUN
cana-5853	167	6	)	)	PUNCT
cana-5853	167	7	is	be	AUX
cana-5853	167	8	a	a	DET
cana-5853	167	9	summable	summable	ADJ
cana-5853	167	10	sequence	sequence	NOUN
cana-5853	167	11	independent	independent	ADJ
cana-5853	167	12	of	of	ADP
cana-5853	167	13	x.	x.	NOUN
cana-5853	167	14	then	then	ADV
cana-5853	167	15	summing	sum	VERB
cana-5853	167	16	over	over	ADP
cana-5853	167	17	n	n	PRON
cana-5853	167	18	yields	yield	NOUN
cana-5853	167	19	:	:	PUNCT
cana-5853	167	20	∞∑	∞∑	NUM
cana-5853	167	21	n=1	n=1	PROPN
cana-5853	167	22	m	m	PROPN
cana-5853	167	23	(	(	PUNCT
cana-5853	167	24	n	n	CCONJ
cana-5853	167	25	,	,	PUNCT
cana-5853	167	26	(	(	PUNCT
cana-5853	167	27	ax)n	ax)n	PROPN
cana-5853	167	28	)	)	PUNCT
cana-5853	167	29	≤	≤	NOUN
cana-5853	167	30	(	(	PUNCT
cana-5853	167	31	∞∑	∞∑	NUM
cana-5853	167	32	n=1	n=1	PROPN
cana-5853	167	33	θn	θn	NOUN
cana-5853	167	34	)	)	PUNCT
cana-5853	167	35	ρm(x	ρm(x	NUM
cana-5853	167	36	)	)	PUNCT
cana-5853	168	1	+	+	CCONJ
cana-5853	168	2	∞∑	∞∑	NUM
cana-5853	168	3	n=1	n=1	ADJ
cana-5853	168	4	ϕn	ϕn	PROPN
cana-5853	168	5	.	.	PUNCT
cana-5853	169	1	since	since	SCONJ
cana-5853	169	2	θn	θn	INTJ
cana-5853	169	3	→	→	SYM
cana-5853	169	4	0	0	NUM
cana-5853	169	5	,	,	PUNCT
cana-5853	169	6	for	for	ADP
cana-5853	169	7	large	large	ADJ
cana-5853	169	8	n	n	CCONJ
cana-5853	169	9	the	the	DET
cana-5853	169	10	tail	tail	NOUN
cana-5853	169	11	sum	sum	NOUN
cana-5853	169	12	∑∞	∑∞	NOUN
cana-5853	169	13	n	n	CCONJ
cana-5853	169	14	=	=	SYM
cana-5853	169	15	n+1	n+1	PROPN
cana-5853	169	16	θn	θn	NOUN
cana-5853	169	17	can	can	AUX
cana-5853	169	18	be	be	AUX
cana-5853	169	19	made	make	VERB
cana-5853	169	20	arbitrarily	arbitrarily	ADV
cana-5853	169	21	small	small	ADJ
cana-5853	169	22	.	.	PUNCT
cana-5853	170	1	this	this	DET
cana-5853	170	2	yields	yield	NOUN
cana-5853	170	3	,	,	PUNCT
cana-5853	170	4	for	for	ADP
cana-5853	170	5	the	the	DET
cana-5853	170	6	unit	unit	NOUN
cana-5853	170	7	ball	ball	PROPN
cana-5853	170	8	b	b	PROPN
cana-5853	170	9	,	,	PUNCT
cana-5853	170	10	sup	sup	NOUN
cana-5853	170	11	x∈b	x∈b	NOUN
cana-5853	171	1	∞∑	∞∑	NUM
cana-5853	171	2	n	n	CCONJ
cana-5853	171	3	=	=	SYM
cana-5853	171	4	n+1	n+1	PROPN
cana-5853	171	5	m	m	PROPN
cana-5853	171	6	(	(	PUNCT
cana-5853	171	7	n	n	X
cana-5853	171	8	,	,	PUNCT
cana-5853	171	9	(	(	PUNCT
cana-5853	171	10	ax)n	ax)n	PROPN
cana-5853	171	11	)	)	PUNCT
cana-5853	171	12	<	<	X
cana-5853	171	13	ϵ	ϵ	X
cana-5853	171	14	for	for	ADP
cana-5853	171	15	n	n	X
cana-5853	171	16	large	large	ADJ
cana-5853	171	17	enough	enough	ADV
cana-5853	171	18	,	,	PUNCT
cana-5853	171	19	proving	prove	VERB
cana-5853	171	20	compactness	compactness	NOUN
cana-5853	171	21	.	.	PUNCT
cana-5853	172	1	such	such	ADJ
cana-5853	172	2	domination	domination	NOUN
cana-5853	172	3	conditions	condition	NOUN
cana-5853	172	4	are	be	AUX
cana-5853	172	5	modular	modular	ADJ
cana-5853	172	6	generalizations	generalization	NOUN
cana-5853	172	7	of	of	ADP
cana-5853	172	8	classical	classical	ADJ
cana-5853	172	9	operator	operator	NOUN
cana-5853	172	10	ideal	ideal	NOUN
cana-5853	172	11	techniques	technique	NOUN
cana-5853	172	12	,	,	PUNCT
cana-5853	172	13	where	where	SCONJ
cana-5853	172	14	an	an	DET
cana-5853	172	15	operator	operator	NOUN
cana-5853	172	16	is	be	AUX
cana-5853	172	17	dominated	dominate	VERB
cana-5853	172	18	(	(	PUNCT
cana-5853	172	19	in	in	ADP
cana-5853	172	20	norm	norm	NOUN
cana-5853	172	21	or	or	CCONJ
cana-5853	172	22	modular	modular	ADJ
cana-5853	172	23	sense	sense	NOUN
cana-5853	172	24	)	)	PUNCT
cana-5853	172	25	by	by	ADP
cana-5853	172	26	a	a	DET
cana-5853	172	27	compact	compact	ADJ
cana-5853	172	28	one	one	NUM
cana-5853	172	29	.	.	PUNCT
cana-5853	173	1	examples	example	NOUN
cana-5853	173	2	diagonal	diagonal	ADJ
cana-5853	173	3	operators	operator	NOUN
cana-5853	173	4	.	.	PUNCT
cana-5853	174	1	for	for	ADP
cana-5853	174	2	a	a	DET
cana-5853	174	3	=	=	SYM
cana-5853	174	4	diag(λn	diag(λn	PROPN
cana-5853	174	5	)	)	PUNCT
cana-5853	174	6	,	,	PUNCT
cana-5853	174	7	boundedness	boundedness	PROPN
cana-5853	174	8	requires	require	VERB
cana-5853	174	9	control	control	NOUN
cana-5853	174	10	over	over	ADP
cana-5853	174	11	m(n	m(n	PROPN
cana-5853	174	12	,	,	PUNCT
cana-5853	174	13	λnt	λnt	PROPN
cana-5853	174	14	)	)	PUNCT
cana-5853	174	15	.	.	PUNCT
cana-5853	175	1	compactness	compactness	NOUN
cana-5853	175	2	requires	require	VERB
cana-5853	175	3	additionally	additionally	ADV
cana-5853	175	4	:	:	PUNCT
cana-5853	175	5	communications	communication	NOUN
cana-5853	175	6	on	on	ADP
cana-5853	175	7	applied	apply	VERB
cana-5853	175	8	nonlinear	nonlinear	ADJ
cana-5853	175	9	analysis	analysis	NOUN
cana-5853	175	10	issn	issn	NOUN
cana-5853	175	11	:	:	PUNCT
cana-5853	175	12	1074	1074	NUM
cana-5853	175	13	-	-	PUNCT
cana-5853	175	14	133x	133x	NUM
cana-5853	175	15	vol	vol	NOUN
cana-5853	175	16	31	31	NUM
cana-5853	175	17	no	no	NOUN
cana-5853	175	18	.	.	NOUN
cana-5853	175	19	2	2	NUM
cana-5853	175	20	(	(	PUNCT
cana-5853	175	21	2024	2024	NUM
cana-5853	175	22	)	)	PUNCT
cana-5853	175	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	175	24	470	470	NUM
cana-5853	175	25	λn	λn	NOUN
cana-5853	175	26	→	→	SYM
cana-5853	175	27	0	0	PUNCT
cana-5853	175	28	as	as	ADP
cana-5853	175	29	n	n	PROPN
cana-5853	175	30	→	→	SYM
cana-5853	175	31	∞.	∞.	PROPN
cana-5853	175	32	indeed	indeed	ADV
cana-5853	175	33	,	,	PUNCT
cana-5853	175	34	for	for	ADP
cana-5853	175	35	any	any	DET
cana-5853	175	36	bounded	bounded	ADJ
cana-5853	175	37	sequence	sequence	NOUN
cana-5853	175	38	x	x	PUNCT
cana-5853	175	39	in	in	ADP
cana-5853	175	40	xm	xm	PROPN
cana-5853	175	41	,	,	PUNCT
cana-5853	175	42	the	the	DET
cana-5853	175	43	tail	tail	NOUN
cana-5853	175	44	sum	sum	NOUN
cana-5853	175	45	:	:	PUNCT
cana-5853	175	46	∞∑	∞∑	NUM
cana-5853	175	47	n	n	CCONJ
cana-5853	175	48	=	=	SYM
cana-5853	175	49	n+1	n+1	PRON
cana-5853	175	50	m(n	m(n	PROPN
cana-5853	175	51	,	,	PUNCT
cana-5853	175	52	λnxn	λnxn	ADJ
cana-5853	175	53	)	)	PUNCT
cana-5853	175	54	can	can	AUX
cana-5853	175	55	be	be	AUX
cana-5853	175	56	made	make	VERB
cana-5853	175	57	small	small	ADJ
cana-5853	175	58	uniformly	uniformly	ADV
cana-5853	175	59	if	if	SCONJ
cana-5853	175	60	λn	λn	NOUN
cana-5853	175	61	decays	decay	VERB
cana-5853	175	62	to	to	ADP
cana-5853	175	63	zero	zero	NUM
cana-5853	175	64	and	and	CCONJ
cana-5853	175	65	m	m	VERB
cana-5853	175	66	satisfies	satisfy	VERB
cana-5853	175	67	appropriate	appropriate	ADJ
cana-5853	175	68	growth	growth	NOUN
cana-5853	175	69	conditions	condition	NOUN
cana-5853	175	70	.	.	PUNCT
cana-5853	176	1	triangular	triangular	NOUN
cana-5853	176	2	matrices	matrix	NOUN
cana-5853	176	3	.	.	PUNCT
cana-5853	177	1	for	for	ADP
cana-5853	177	2	lower	low	ADJ
cana-5853	177	3	-	-	PUNCT
cana-5853	177	4	triangular	triangular	NOUN
cana-5853	177	5	matrices	matrix	NOUN
cana-5853	177	6	a	a	DET
cana-5853	177	7	=	=	SYM
cana-5853	177	8	(	(	PUNCT
cana-5853	177	9	ank	ank	PROPN
cana-5853	177	10	)	)	PUNCT
cana-5853	177	11	with	with	ADP
cana-5853	177	12	ank	ank	PROPN
cana-5853	177	13	=	=	PROPN
cana-5853	177	14	0	0	PROPN
cana-5853	177	15	for	for	ADP
cana-5853	177	16	k	k	PROPN
cana-5853	177	17	>	>	PUNCT
cana-5853	177	18	n	n	CCONJ
cana-5853	177	19	,	,	PUNCT
cana-5853	177	20	tail	tail	NOUN
cana-5853	177	21	conditions	condition	NOUN
cana-5853	177	22	involve	involve	VERB
cana-5853	177	23	:	:	PUNCT
cana-5853	177	24	n∑	n∑	INTJ
cana-5853	177	25	k=1	k=1	PUNCT
cana-5853	178	1	|ank|	|ank|	PROPN
cana-5853	178	2	→	→	SYM
cana-5853	178	3	0	0	NUM
cana-5853	178	4	as	as	ADP
cana-5853	178	5	n	n	NOUN
cana-5853	178	6	→	→	SYM
cana-5853	178	7	∞.	∞.	PROPN
cana-5853	178	8	together	together	ADV
cana-5853	178	9	with	with	ADP
cana-5853	178	10	modular	modular	ADJ
cana-5853	178	11	inequalities	inequality	NOUN
cana-5853	178	12	,	,	PUNCT
cana-5853	178	13	this	this	PRON
cana-5853	178	14	ensures	ensure	VERB
cana-5853	178	15	the	the	DET
cana-5853	178	16	images	image	NOUN
cana-5853	178	17	of	of	ADP
cana-5853	178	18	bounded	bounded	ADJ
cana-5853	178	19	sets	set	NOUN
cana-5853	178	20	have	have	AUX
cana-5853	178	21	vanishing	vanish	VERB
cana-5853	178	22	tails	tail	NOUN
cana-5853	178	23	,	,	PUNCT
cana-5853	178	24	yielding	yield	VERB
cana-5853	178	25	compactness	compactness	NOUN
cana-5853	178	26	.	.	PUNCT
cana-5853	179	1	cesàro	cesàro	ADJ
cana-5853	179	2	-	-	PUNCT
cana-5853	179	3	type	type	NOUN
cana-5853	179	4	matrices	matrix	NOUN
cana-5853	179	5	.	.	PUNCT
cana-5853	180	1	cesàro	cesàro	ADJ
cana-5853	180	2	-	-	PUNCT
cana-5853	180	3	type	type	NOUN
cana-5853	180	4	averaging	averaging	NOUN
cana-5853	180	5	matrices	matrix	NOUN
cana-5853	180	6	often	often	ADV
cana-5853	180	7	satisfy	satisfy	VERB
cana-5853	180	8	modular	modular	ADJ
cana-5853	180	9	domination	domination	NOUN
cana-5853	180	10	naturally	naturally	ADV
cana-5853	180	11	,	,	PUNCT
cana-5853	180	12	as	as	SCONJ
cana-5853	180	13	their	their	PRON
cana-5853	180	14	entries	entry	NOUN
cana-5853	180	15	decay	decay	VERB
cana-5853	180	16	with	with	ADP
cana-5853	180	17	n	n	CCONJ
cana-5853	180	18	:	:	PUNCT
cana-5853	180	19	ank	ank	PROPN
cana-5853	180	20	=	=	PROPN
cana-5853	180	21	1	1	NUM
cana-5853	180	22	n	n	NOUN
cana-5853	180	23	(	(	PUNCT
cana-5853	180	24	k	k	PROPN
cana-5853	180	25	≤	≤	PROPN
cana-5853	180	26	n	n	CCONJ
cana-5853	180	27	)	)	PUNCT
cana-5853	180	28	.	.	PUNCT
cana-5853	181	1	in	in	ADP
cana-5853	181	2	such	such	ADJ
cana-5853	181	3	cases	case	NOUN
cana-5853	181	4	,	,	PUNCT
cana-5853	181	5	the	the	DET
cana-5853	181	6	tail	tail	NOUN
cana-5853	181	7	estimates	estimate	NOUN
cana-5853	181	8	can	can	AUX
cana-5853	181	9	be	be	AUX
cana-5853	181	10	explicitly	explicitly	ADV
cana-5853	181	11	calculated	calculate	VERB
cana-5853	181	12	to	to	PART
cana-5853	181	13	show	show	VERB
cana-5853	181	14	uniform	uniform	ADJ
cana-5853	181	15	modular	modular	ADJ
cana-5853	181	16	smallness	smallness	NOUN
cana-5853	181	17	on	on	ADP
cana-5853	181	18	bounded	bounded	ADJ
cana-5853	181	19	sets	set	NOUN
cana-5853	181	20	.	.	PUNCT
cana-5853	182	1	summary	summary	NOUN
cana-5853	182	2	compactness	compactness	NOUN
cana-5853	182	3	characterizations	characterization	NOUN
cana-5853	182	4	in	in	ADP
cana-5853	182	5	xm	xm	PROPN
cana-5853	182	6	spaces	space	NOUN
cana-5853	182	7	thus	thus	ADV
cana-5853	182	8	rely	rely	VERB
cana-5853	182	9	on	on	ADP
cana-5853	182	10	controlling	control	VERB
cana-5853	182	11	the	the	DET
cana-5853	182	12	modular	modular	NOUN
cana-5853	182	13	of	of	ADP
cana-5853	182	14	the	the	DET
cana-5853	182	15	tails	tail	NOUN
cana-5853	182	16	of	of	ADP
cana-5853	182	17	operator	operator	NOUN
cana-5853	182	18	images	image	NOUN
cana-5853	182	19	and	and	CCONJ
cana-5853	182	20	establishing	establish	VERB
cana-5853	182	21	domination	domination	NOUN
cana-5853	182	22	inequalities	inequality	NOUN
cana-5853	182	23	that	that	PRON
cana-5853	182	24	ensure	ensure	VERB
cana-5853	182	25	decay	decay	NOUN
cana-5853	182	26	.	.	PUNCT
cana-5853	183	1	these	these	DET
cana-5853	183	2	criteria	criterion	NOUN
cana-5853	183	3	generalize	generalize	VERB
cana-5853	183	4	classical	classical	ADJ
cana-5853	183	5	results	result	NOUN
cana-5853	183	6	for	for	ADP
cana-5853	183	7	ℓp	ℓp	ADJ
cana-5853	183	8	and	and	CCONJ
cana-5853	183	9	orlicz	orlicz	ADJ
cana-5853	183	10	spaces	space	NOUN
cana-5853	183	11	while	while	SCONJ
cana-5853	183	12	leveraging	leverage	VERB
cana-5853	183	13	the	the	DET
cana-5853	183	14	flexibility	flexibility	NOUN
cana-5853	183	15	of	of	ADP
cana-5853	183	16	index	index	NOUN
cana-5853	183	17	-	-	PUNCT
cana-5853	183	18	dependent	dependent	ADJ
cana-5853	183	19	modular	modular	ADJ
cana-5853	183	20	functions	function	NOUN
cana-5853	183	21	.	.	PUNCT
cana-5853	184	1	they	they	PRON
cana-5853	184	2	form	form	VERB
cana-5853	184	3	the	the	DET
cana-5853	184	4	foundation	foundation	NOUN
cana-5853	184	5	for	for	ADP
cana-5853	184	6	studying	study	VERB
cana-5853	184	7	spectral	spectral	ADJ
cana-5853	184	8	theory	theory	NOUN
cana-5853	184	9	,	,	PUNCT
cana-5853	184	10	approximation	approximation	NOUN
cana-5853	184	11	methods	method	NOUN
cana-5853	184	12	,	,	PUNCT
cana-5853	184	13	and	and	CCONJ
cana-5853	184	14	summability	summability	NOUN
cana-5853	184	15	techniques	technique	NOUN
cana-5853	184	16	in	in	ADP
cana-5853	184	17	these	these	DET
cana-5853	184	18	generalized	generalized	ADJ
cana-5853	184	19	sequence	sequence	NOUN
cana-5853	184	20	spaces	space	NOUN
cana-5853	184	21	.	.	PUNCT
cana-5853	185	1	5	5	NUM
cana-5853	185	2	cesàro	cesàro	NOUN
cana-5853	185	3	-	-	PUNCT
cana-5853	185	4	type	type	NOUN
cana-5853	185	5	and	and	CCONJ
cana-5853	185	6	summability	summability	NOUN
cana-5853	185	7	matrices	matrix	NOUN
cana-5853	185	8	in	in	ADP
cana-5853	185	9	this	this	DET
cana-5853	185	10	section	section	NOUN
cana-5853	185	11	,	,	PUNCT
cana-5853	185	12	we	we	PRON
cana-5853	185	13	focus	focus	VERB
cana-5853	185	14	on	on	ADP
cana-5853	185	15	an	an	DET
cana-5853	185	16	important	important	ADJ
cana-5853	185	17	class	class	NOUN
cana-5853	185	18	of	of	ADP
cana-5853	185	19	operators	operator	NOUN
cana-5853	185	20	on	on	ADP
cana-5853	185	21	sequence	sequence	NOUN
cana-5853	185	22	spaces	space	NOUN
cana-5853	185	23	:	:	PUNCT
cana-5853	185	24	cesàrotype	cesàrotype	NOUN
cana-5853	185	25	and	and	CCONJ
cana-5853	185	26	related	relate	VERB
cana-5853	185	27	summability	summability	NOUN
cana-5853	185	28	matrices	matrix	NOUN
cana-5853	185	29	.	.	PUNCT
cana-5853	186	1	classical	classical	ADJ
cana-5853	186	2	cesàro	cesàro	ADJ
cana-5853	186	3	matrices	matrix	NOUN
cana-5853	186	4	play	play	VERB
cana-5853	186	5	a	a	DET
cana-5853	186	6	central	central	ADJ
cana-5853	186	7	role	role	NOUN
cana-5853	186	8	in	in	ADP
cana-5853	186	9	summability	summability	NOUN
cana-5853	186	10	theory	theory	NOUN
cana-5853	186	11	,	,	PUNCT
cana-5853	186	12	fourier	fourier	ADJ
cana-5853	186	13	analysis	analysis	NOUN
cana-5853	186	14	,	,	PUNCT
cana-5853	186	15	and	and	CCONJ
cana-5853	186	16	approximation	approximation	NOUN
cana-5853	186	17	theory	theory	NOUN
cana-5853	186	18	.	.	PUNCT
cana-5853	187	1	our	our	PRON
cana-5853	187	2	aim	aim	NOUN
cana-5853	187	3	is	be	AUX
cana-5853	187	4	to	to	PART
cana-5853	187	5	generalize	generalize	VERB
cana-5853	187	6	their	their	PRON
cana-5853	187	7	study	study	NOUN
cana-5853	187	8	to	to	PART
cana-5853	187	9	modulated	modulate	VERB
cana-5853	187	10	orlicz	orlicz	ADJ
cana-5853	187	11	-	-	PUNCT
cana-5853	187	12	type	type	NOUN
cana-5853	187	13	sequence	sequence	NOUN
cana-5853	187	14	spaces	space	VERB
cana-5853	187	15	xm	xm	PROPN
cana-5853	187	16	,	,	PUNCT
cana-5853	187	17	examining	examine	VERB
cana-5853	187	18	both	both	PRON
cana-5853	187	19	boundedness	boundedness	NOUN
cana-5853	187	20	and	and	CCONJ
cana-5853	187	21	compactness	compactness	NOUN
cana-5853	187	22	criteria	criterion	NOUN
cana-5853	187	23	within	within	ADP
cana-5853	187	24	this	this	DET
cana-5853	187	25	flexible	flexible	ADJ
cana-5853	187	26	modular	modular	ADJ
cana-5853	187	27	framework	framework	NOUN
cana-5853	187	28	.	.	PUNCT
cana-5853	188	1	communications	communication	NOUN
cana-5853	188	2	on	on	ADP
cana-5853	188	3	applied	apply	VERB
cana-5853	188	4	nonlinear	nonlinear	ADJ
cana-5853	188	5	analysis	analysis	NOUN
cana-5853	188	6	issn	issn	NOUN
cana-5853	188	7	:	:	PUNCT
cana-5853	188	8	1074	1074	NUM
cana-5853	188	9	-	-	PUNCT
cana-5853	188	10	133x	133x	NUM
cana-5853	188	11	vol	vol	NOUN
cana-5853	188	12	31	31	NUM
cana-5853	188	13	no	no	NOUN
cana-5853	188	14	.	.	NOUN
cana-5853	188	15	2	2	NUM
cana-5853	188	16	(	(	PUNCT
cana-5853	188	17	2024	2024	NUM
cana-5853	188	18	)	)	PUNCT
cana-5853	189	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	189	2	471	471	NUM
cana-5853	189	3	5.1	5.1	NUM
cana-5853	189	4	generalizations	generalization	NOUN
cana-5853	189	5	of	of	ADP
cana-5853	189	6	classical	classical	ADJ
cana-5853	189	7	cesàro	cesàro	NOUN
cana-5853	189	8	matrices	matrix	NOUN
cana-5853	189	9	the	the	DET
cana-5853	189	10	classical	classical	ADJ
cana-5853	189	11	cesàro	cesàro	NOUN
cana-5853	189	12	matrix	matrix	NOUN
cana-5853	189	13	c	c	NOUN
cana-5853	190	1	=	=	SYM
cana-5853	190	2	(	(	PUNCT
cana-5853	190	3	cnk	cnk	PROPN
cana-5853	190	4	)	)	PUNCT
cana-5853	190	5	is	be	AUX
cana-5853	190	6	defined	define	VERB
cana-5853	190	7	by	by	ADP
cana-5853	190	8	:	:	PUNCT
cana-5853	190	9	cnk	cnk	PROPN
cana-5853	190	10	=	=	SYM
cana-5853	190	11	{	{	PUNCT
cana-5853	190	12	1	1	NUM
cana-5853	190	13	n	n	NOUN
cana-5853	190	14	if	if	SCONJ
cana-5853	190	15	1	1	NUM
cana-5853	190	16	≤	≤	NUM
cana-5853	190	17	k	k	NOUN
cana-5853	190	18	≤	≤	PROPN
cana-5853	190	19	n	n	CCONJ
cana-5853	190	20	,	,	PUNCT
cana-5853	190	21	0	0	PUNCT
cana-5853	191	1	if	if	SCONJ
cana-5853	191	2	k	k	PROPN
cana-5853	191	3	>	>	X
cana-5853	191	4	n.	n.	NOUN
cana-5853	191	5	its	its	PRON
cana-5853	191	6	action	action	NOUN
cana-5853	191	7	on	on	ADP
cana-5853	191	8	a	a	DET
cana-5853	191	9	sequence	sequence	NOUN
cana-5853	191	10	x	x	PUNCT
cana-5853	191	11	=	=	SYM
cana-5853	191	12	(	(	PUNCT
cana-5853	191	13	xk	xk	NOUN
cana-5853	191	14	)	)	PUNCT
cana-5853	191	15	yields	yield	VERB
cana-5853	191	16	the	the	DET
cana-5853	191	17	sequence	sequence	NOUN
cana-5853	191	18	of	of	ADP
cana-5853	191	19	arithmetic	arithmetic	ADJ
cana-5853	191	20	means	mean	NOUN
cana-5853	191	21	:	:	PUNCT
cana-5853	191	22	(	(	PUNCT
cana-5853	191	23	cx)n	cx)n	PROPN
cana-5853	191	24	=	=	SYM
cana-5853	191	25	1	1	NUM
cana-5853	191	26	n	n	NUM
cana-5853	191	27	n∑	n∑	NOUN
cana-5853	191	28	k=1	k=1	X
cana-5853	192	1	xk	xk	PROPN
cana-5853	192	2	.	.	PUNCT
cana-5853	192	3	generalizations	generalization	NOUN
cana-5853	192	4	of	of	ADP
cana-5853	192	5	cesàro	cesàro	ADJ
cana-5853	192	6	matrices	matrix	NOUN
cana-5853	192	7	allow	allow	VERB
cana-5853	192	8	more	more	ADV
cana-5853	192	9	flexible	flexible	ADJ
cana-5853	192	10	averaging	averaging	NOUN
cana-5853	192	11	schemes	scheme	NOUN
cana-5853	192	12	.	.	PUNCT
cana-5853	193	1	for	for	ADP
cana-5853	193	2	instance	instance	NOUN
cana-5853	193	3	,	,	PUNCT
cana-5853	193	4	one	one	PRON
cana-5853	193	5	can	can	AUX
cana-5853	193	6	define	define	VERB
cana-5853	193	7	weighted	weight	VERB
cana-5853	193	8	cesàro	cesàro	ADJ
cana-5853	193	9	matrices	matrix	NOUN
cana-5853	193	10	cw	cw	NOUN
cana-5853	193	11	=	=	SYM
cana-5853	193	12	(	(	PUNCT
cana-5853	193	13	cnk	cnk	PROPN
cana-5853	193	14	)	)	PUNCT
cana-5853	193	15	by	by	ADP
cana-5853	193	16	:	:	PUNCT
cana-5853	193	17	cnk	cnk	PROPN
cana-5853	194	1	=	=	SYM
cana-5853	194	2	{	{	PUNCT
cana-5853	194	3	wk	wk	INTJ
cana-5853	194	4	wn	wn	PROPN
cana-5853	194	5	1	1	NUM
cana-5853	194	6	≤	≤	PROPN
cana-5853	194	7	k	k	PROPN
cana-5853	194	8	≤	≤	PROPN
cana-5853	194	9	n	n	CCONJ
cana-5853	194	10	,	,	PUNCT
cana-5853	194	11	0	0	PUNCT
cana-5853	194	12	k	k	PROPN
cana-5853	194	13	>	>	PUNCT
cana-5853	194	14	n	n	CCONJ
cana-5853	194	15	,	,	PUNCT
cana-5853	194	16	where	where	SCONJ
cana-5853	194	17	wk	wk	INTJ
cana-5853	194	18	>	>	X
cana-5853	194	19	0	0	NUM
cana-5853	194	20	are	be	AUX
cana-5853	194	21	weights	weight	NOUN
cana-5853	194	22	and	and	CCONJ
cana-5853	194	23	wn	wn	PROPN
cana-5853	194	24	=	=	PUNCT
cana-5853	194	25	∑n	∑n	PROPN
cana-5853	194	26	k=1wk	k=1wk	X
cana-5853	194	27	.	.	PUNCT
cana-5853	195	1	these	these	DET
cana-5853	195	2	matrices	matrix	NOUN
cana-5853	195	3	preserve	preserve	VERB
cana-5853	195	4	the	the	DET
cana-5853	195	5	averaging	averaging	NOUN
cana-5853	195	6	character	character	NOUN
cana-5853	195	7	while	while	SCONJ
cana-5853	195	8	adapting	adapt	VERB
cana-5853	195	9	to	to	ADP
cana-5853	195	10	non	non	ADJ
cana-5853	195	11	-	-	ADJ
cana-5853	195	12	uniform	uniform	ADJ
cana-5853	195	13	contexts	contexts	NOUN
cana-5853	195	14	.	.	PUNCT
cana-5853	196	1	in	in	ADP
cana-5853	196	2	modulated	modulate	VERB
cana-5853	196	3	orlicz	orlicz	ADJ
cana-5853	196	4	-	-	PUNCT
cana-5853	196	5	type	type	NOUN
cana-5853	196	6	sequence	sequence	NOUN
cana-5853	196	7	spaces	space	NOUN
cana-5853	196	8	,	,	PUNCT
cana-5853	196	9	such	such	ADJ
cana-5853	196	10	matrices	matrix	NOUN
cana-5853	196	11	naturally	naturally	ADV
cana-5853	196	12	arise	arise	VERB
cana-5853	196	13	in	in	ADP
cana-5853	196	14	models	model	NOUN
cana-5853	196	15	where	where	SCONJ
cana-5853	196	16	local	local	ADJ
cana-5853	196	17	smoothing	smoothing	NOUN
cana-5853	196	18	or	or	CCONJ
cana-5853	196	19	regularization	regularization	NOUN
cana-5853	196	20	is	be	AUX
cana-5853	196	21	applied	apply	VERB
cana-5853	196	22	with	with	ADP
cana-5853	196	23	position	position	NOUN
cana-5853	196	24	-	-	PUNCT
cana-5853	196	25	dependent	dependent	ADJ
cana-5853	196	26	penalties	penalty	NOUN
cana-5853	196	27	.	.	PUNCT
cana-5853	197	1	the	the	DET
cana-5853	197	2	challenge	challenge	NOUN
cana-5853	197	3	lies	lie	VERB
cana-5853	197	4	in	in	ADP
cana-5853	197	5	determining	determine	VERB
cana-5853	197	6	conditions	condition	NOUN
cana-5853	197	7	under	under	ADP
cana-5853	197	8	which	which	PRON
cana-5853	197	9	these	these	DET
cana-5853	197	10	matrices	matrix	NOUN
cana-5853	197	11	define	define	VERB
cana-5853	197	12	bounded	bounded	ADJ
cana-5853	197	13	(	(	PUNCT
cana-5853	197	14	or	or	CCONJ
cana-5853	197	15	compact	compact	ADJ
cana-5853	197	16	)	)	PUNCT
cana-5853	197	17	operators	operator	NOUN
cana-5853	197	18	on	on	ADP
cana-5853	197	19	xm	xm	PROPN
cana-5853	197	20	.	.	PUNCT
cana-5853	198	1	5.2	5.2	NUM
cana-5853	198	2	boundedness	boundedness	NOUN
cana-5853	198	3	in	in	ADP
cana-5853	198	4	xm	xm	PROPN
cana-5853	198	5	to	to	PART
cana-5853	198	6	analyze	analyze	VERB
cana-5853	198	7	boundedness	boundedness	NOUN
cana-5853	198	8	,	,	PUNCT
cana-5853	198	9	consider	consider	VERB
cana-5853	198	10	a	a	DET
cana-5853	198	11	=	=	SYM
cana-5853	198	12	(	(	PUNCT
cana-5853	198	13	ank	ank	PROPN
cana-5853	198	14	)	)	PUNCT
cana-5853	198	15	of	of	ADP
cana-5853	198	16	cesàro	cesàro	NOUN
cana-5853	198	17	-	-	PUNCT
cana-5853	198	18	type	type	NOUN
cana-5853	198	19	form	form	NOUN
cana-5853	198	20	:	:	PUNCT
cana-5853	198	21	ank	ank	PROPN
cana-5853	198	22	=	=	X
cana-5853	198	23	{	{	PUNCT
cana-5853	198	24	αnk	αnk	INTJ
cana-5853	198	25	n	n	ADV
cana-5853	198	26	1	1	NUM
cana-5853	198	27	≤	≤	NUM
cana-5853	198	28	k	k	NOUN
cana-5853	198	29	≤	≤	PROPN
cana-5853	198	30	n	n	CCONJ
cana-5853	198	31	,	,	PUNCT
cana-5853	198	32	0	0	PUNCT
cana-5853	199	1	k	k	PROPN
cana-5853	199	2	>	>	PUNCT
cana-5853	199	3	n	n	CCONJ
cana-5853	199	4	,	,	PUNCT
cana-5853	199	5	where	where	SCONJ
cana-5853	199	6	αnk	αnk	PROPN
cana-5853	199	7	are	be	AUX
cana-5853	199	8	bounded	bound	VERB
cana-5853	199	9	and	and	CCONJ
cana-5853	199	10	possibly	possibly	ADV
cana-5853	199	11	vary	vary	VERB
cana-5853	199	12	with	with	ADP
cana-5853	199	13	n	n	PRON
cana-5853	199	14	and	and	CCONJ
cana-5853	199	15	k.	k.	PROPN
cana-5853	199	16	let	let	VERB
cana-5853	199	17	x	x	SYM
cana-5853	199	18	∈	∈	PROPN
cana-5853	199	19	xm	xm	PROPN
cana-5853	199	20	.	.	PUNCT
cana-5853	200	1	then	then	ADV
cana-5853	200	2	:	:	PUNCT
cana-5853	200	3	(	(	PUNCT
cana-5853	200	4	ax)n	ax)n	PROPN
cana-5853	200	5	=	=	SYM
cana-5853	200	6	1	1	NUM
cana-5853	200	7	n	n	NUM
cana-5853	200	8	n∑	n∑	NOUN
cana-5853	200	9	k=1	k=1	PROPN
cana-5853	200	10	αnkxk	αnkxk	VERB
cana-5853	200	11	.	.	PUNCT
cana-5853	201	1	applying	apply	VERB
cana-5853	201	2	the	the	DET
cana-5853	201	3	convexity	convexity	NOUN
cana-5853	201	4	of	of	ADP
cana-5853	201	5	m(n	m(n	PROPN
cana-5853	201	6	,	,	PUNCT
cana-5853	201	7	·	·	PUNCT
cana-5853	201	8	)	)	PUNCT
cana-5853	201	9	and	and	CCONJ
cana-5853	201	10	jensen	jensen	PROPN
cana-5853	201	11	’s	’s	PART
cana-5853	201	12	inequality	inequality	NOUN
cana-5853	201	13	(	(	PUNCT
cana-5853	201	14	which	which	PRON
cana-5853	201	15	holds	hold	VERB
cana-5853	201	16	for	for	ADP
cana-5853	201	17	convex	convex	NOUN
cana-5853	201	18	modulars	modular	NOUN
cana-5853	201	19	)	)	PUNCT
cana-5853	201	20	,	,	PUNCT
cana-5853	201	21	we	we	PRON
cana-5853	201	22	obtain	obtain	VERB
cana-5853	201	23	:	:	PUNCT
cana-5853	201	24	m	m	VERB
cana-5853	201	25	(	(	PUNCT
cana-5853	201	26	n	n	CCONJ
cana-5853	201	27	,	,	PUNCT
cana-5853	201	28	(	(	PUNCT
cana-5853	201	29	ax)n	ax)n	PROPN
cana-5853	201	30	)	)	PUNCT
cana-5853	201	31	≤	≤	NOUN
cana-5853	201	32	1	1	NUM
cana-5853	201	33	n	n	NUM
cana-5853	201	34	n∑	n∑	NOUN
cana-5853	201	35	k=1	k=1	NOUN
cana-5853	202	1	m	m	VERB
cana-5853	202	2	(	(	PUNCT
cana-5853	202	3	n	n	CCONJ
cana-5853	202	4	,	,	PUNCT
cana-5853	202	5	αnkxk	αnkxk	VERB
cana-5853	202	6	)	)	PUNCT
cana-5853	202	7	.	.	PUNCT
cana-5853	203	1	summing	sum	VERB
cana-5853	203	2	over	over	ADP
cana-5853	203	3	n	n	PRON
cana-5853	203	4	yields	yield	NOUN
cana-5853	203	5	:	:	PUNCT
cana-5853	203	6	∞∑	∞∑	NUM
cana-5853	203	7	n=1	n=1	PROPN
cana-5853	203	8	m	m	PROPN
cana-5853	203	9	(	(	PUNCT
cana-5853	203	10	n	n	CCONJ
cana-5853	203	11	,	,	PUNCT
cana-5853	203	12	(	(	PUNCT
cana-5853	203	13	ax)n	ax)n	PROPN
cana-5853	203	14	)	)	PUNCT
cana-5853	203	15	≤	≤	NOUN
cana-5853	204	1	∞∑	∞∑	NUM
cana-5853	204	2	n=1	n=1	ADP
cana-5853	204	3	1	1	NUM
cana-5853	204	4	n	n	NUM
cana-5853	204	5	n∑	n∑	NOUN
cana-5853	204	6	k=1	k=1	NOUN
cana-5853	204	7	m	m	VERB
cana-5853	204	8	(	(	PUNCT
cana-5853	204	9	n	n	CCONJ
cana-5853	204	10	,	,	PUNCT
cana-5853	204	11	αnkxk	αnkxk	VERB
cana-5853	204	12	)	)	PUNCT
cana-5853	204	13	.	.	PUNCT
cana-5853	205	1	communications	communication	NOUN
cana-5853	205	2	on	on	ADP
cana-5853	205	3	applied	apply	VERB
cana-5853	205	4	nonlinear	nonlinear	ADJ
cana-5853	205	5	analysis	analysis	NOUN
cana-5853	205	6	issn	issn	NOUN
cana-5853	205	7	:	:	PUNCT
cana-5853	205	8	1074	1074	NUM
cana-5853	205	9	-	-	PUNCT
cana-5853	205	10	133x	133x	NUM
cana-5853	205	11	vol	vol	NOUN
cana-5853	205	12	31	31	NUM
cana-5853	205	13	no	no	NOUN
cana-5853	205	14	.	.	NOUN
cana-5853	205	15	2	2	NUM
cana-5853	205	16	(	(	PUNCT
cana-5853	205	17	2024	2024	NUM
cana-5853	205	18	)	)	PUNCT
cana-5853	206	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	206	2	472	472	NUM
cana-5853	206	3	to	to	PART
cana-5853	206	4	ensure	ensure	VERB
cana-5853	206	5	boundedness	boundedness	NOUN
cana-5853	206	6	of	of	ADP
cana-5853	206	7	a	a	PRON
cana-5853	206	8	,	,	PUNCT
cana-5853	206	9	it	it	PRON
cana-5853	206	10	suffices	suffice	VERB
cana-5853	206	11	that	that	SCONJ
cana-5853	206	12	there	there	PRON
cana-5853	206	13	exists	exist	VERB
cana-5853	206	14	c	c	NOUN
cana-5853	206	15	>	>	X
cana-5853	206	16	0	0	NUM
cana-5853	206	17	such	such	ADJ
cana-5853	206	18	that	that	PRON
cana-5853	206	19	for	for	ADP
cana-5853	206	20	all	all	DET
cana-5853	206	21	n	n	CCONJ
cana-5853	206	22	,	,	PUNCT
cana-5853	206	23	k	k	PROPN
cana-5853	206	24	,	,	PUNCT
cana-5853	206	25	m	m	VERB
cana-5853	206	26	(	(	PUNCT
cana-5853	206	27	n	n	CCONJ
cana-5853	206	28	,	,	PUNCT
cana-5853	206	29	αnkt	αnkt	NOUN
cana-5853	206	30	)	)	PUNCT
cana-5853	206	31	≤	≤	NOUN
cana-5853	206	32	cm(k	cm(k	NOUN
cana-5853	206	33	,	,	PUNCT
cana-5853	206	34	t	t	PROPN
cana-5853	206	35	)	)	PUNCT
cana-5853	206	36	+	+	CCONJ
cana-5853	206	37	c.	c.	NOUN
cana-5853	206	38	under	under	ADP
cana-5853	206	39	this	this	DET
cana-5853	206	40	condition	condition	NOUN
cana-5853	206	41	,	,	PUNCT
cana-5853	206	42	we	we	PRON
cana-5853	206	43	get	get	VERB
cana-5853	206	44	:	:	PUNCT
cana-5853	206	45	∞∑	∞∑	NUM
cana-5853	206	46	n=1	n=1	PROPN
cana-5853	206	47	m	m	PROPN
cana-5853	206	48	(	(	PUNCT
cana-5853	206	49	n	n	CCONJ
cana-5853	206	50	,	,	PUNCT
cana-5853	206	51	(	(	PUNCT
cana-5853	206	52	ax)n	ax)n	PROPN
cana-5853	206	53	)	)	PUNCT
cana-5853	206	54	≤	≤	NUM
cana-5853	207	1	c	c	VERB
cana-5853	207	2	∞∑	∞∑	NUM
cana-5853	207	3	k=1	k=1	PROPN
cana-5853	207	4	m(k	m(k	PROPN
cana-5853	207	5	,	,	PUNCT
cana-5853	207	6	xk	xk	PROPN
cana-5853	207	7	)	)	PUNCT
cana-5853	207	8	+	+	NUM
cana-5853	207	9	c	c	NOUN
cana-5853	207	10	′	′	NOUN
cana-5853	207	11	,	,	PUNCT
cana-5853	207	12	for	for	ADP
cana-5853	207	13	all	all	DET
cana-5853	207	14	x	x	SYM
cana-5853	207	15	∈	∈	PROPN
cana-5853	207	16	xm	xm	PROPN
cana-5853	207	17	.	.	PUNCT
cana-5853	208	1	therefore	therefore	ADV
cana-5853	208	2	,	,	PUNCT
cana-5853	208	3	a	a	PRON
cana-5853	208	4	is	be	AUX
cana-5853	208	5	bounded	bound	VERB
cana-5853	208	6	on	on	ADP
cana-5853	208	7	xm	xm	PROPN
cana-5853	208	8	.	.	PUNCT
cana-5853	209	1	this	this	DET
cana-5853	209	2	argument	argument	NOUN
cana-5853	209	3	generalizes	generalize	VERB
cana-5853	209	4	the	the	DET
cana-5853	209	5	classical	classical	ADJ
cana-5853	209	6	boundedness	boundedness	NOUN
cana-5853	209	7	of	of	ADP
cana-5853	209	8	cesàro	cesàro	PROPN
cana-5853	209	9	operators	operator	NOUN
cana-5853	209	10	in	in	ADP
cana-5853	209	11	ℓp	ℓp	ADJ
cana-5853	209	12	spaces	space	NOUN
cana-5853	209	13	,	,	PUNCT
cana-5853	209	14	where	where	SCONJ
cana-5853	209	15	power	power	NOUN
cana-5853	209	16	-	-	PUNCT
cana-5853	209	17	type	type	NOUN
cana-5853	209	18	modular	modular	ADJ
cana-5853	209	19	functions	function	NOUN
cana-5853	209	20	yield	yield	VERB
cana-5853	209	21	standard	standard	ADJ
cana-5853	209	22	estimates	estimate	NOUN
cana-5853	209	23	.	.	PUNCT
cana-5853	210	1	5.3	5.3	NUM
cana-5853	210	2	compactness	compactness	NOUN
cana-5853	210	3	analysis	analysis	NOUN
cana-5853	210	4	compactness	compactness	NOUN
cana-5853	210	5	of	of	ADP
cana-5853	210	6	cesàro	cesàro	ADJ
cana-5853	210	7	-	-	PUNCT
cana-5853	210	8	type	type	NOUN
cana-5853	210	9	matrices	matrix	NOUN
cana-5853	210	10	on	on	ADP
cana-5853	210	11	xm	xm	PROPN
cana-5853	210	12	typically	typically	ADV
cana-5853	210	13	requires	require	VERB
cana-5853	210	14	additional	additional	ADJ
cana-5853	210	15	decay	decay	NOUN
cana-5853	210	16	conditions	condition	NOUN
cana-5853	210	17	to	to	PART
cana-5853	210	18	ensure	ensure	VERB
cana-5853	210	19	images	image	NOUN
cana-5853	210	20	of	of	ADP
cana-5853	210	21	bounded	bounded	ADJ
cana-5853	210	22	sets	set	NOUN
cana-5853	210	23	have	have	AUX
cana-5853	210	24	uniformly	uniformly	ADV
cana-5853	210	25	vanishing	vanish	VERB
cana-5853	210	26	tails	tail	NOUN
cana-5853	210	27	.	.	PUNCT
cana-5853	211	1	consider	consider	VERB
cana-5853	211	2	the	the	DET
cana-5853	211	3	image	image	NOUN
cana-5853	211	4	of	of	ADP
cana-5853	211	5	the	the	DET
cana-5853	211	6	unit	unit	NOUN
cana-5853	211	7	ball	ball	PROPN
cana-5853	211	8	b	b	PROPN
cana-5853	211	9	of	of	ADP
cana-5853	211	10	xm	xm	PROPN
cana-5853	211	11	under	under	ADP
cana-5853	211	12	a.	a.	NOUN
cana-5853	211	13	we	we	PRON
cana-5853	211	14	analyze	analyze	VERB
cana-5853	211	15	:	:	PUNCT
cana-5853	211	16	∞∑	∞∑	NUM
cana-5853	211	17	n	n	CCONJ
cana-5853	211	18	=	=	SYM
cana-5853	211	19	n+1	n+1	PROPN
cana-5853	211	20	m	m	PROPN
cana-5853	211	21	(	(	PUNCT
cana-5853	211	22	n	n	X
cana-5853	211	23	,	,	PUNCT
cana-5853	211	24	(	(	PUNCT
cana-5853	211	25	ax)n	ax)n	PROPN
cana-5853	211	26	)	)	PUNCT
cana-5853	211	27	.	.	PUNCT
cana-5853	212	1	given	give	VERB
cana-5853	212	2	the	the	DET
cana-5853	212	3	cesàro	cesàro	NOUN
cana-5853	212	4	-	-	PUNCT
cana-5853	212	5	type	type	NOUN
cana-5853	212	6	structure	structure	NOUN
cana-5853	212	7	,	,	PUNCT
cana-5853	212	8	we	we	PRON
cana-5853	212	9	have	have	VERB
cana-5853	212	10	:	:	PUNCT
cana-5853	212	11	(	(	PUNCT
cana-5853	212	12	ax)n	ax)n	PROPN
cana-5853	212	13	=	=	SYM
cana-5853	212	14	1	1	NUM
cana-5853	212	15	n	n	NUM
cana-5853	212	16	n∑	n∑	NOUN
cana-5853	212	17	k=1	k=1	PROPN
cana-5853	212	18	αnkxk	αnkxk	PROPN
cana-5853	212	19	,	,	PUNCT
cana-5853	212	20	where	where	SCONJ
cana-5853	212	21	αnk	αnk	PROPN
cana-5853	212	22	are	be	AUX
cana-5853	212	23	bounded	bound	VERB
cana-5853	212	24	.	.	PUNCT
cana-5853	213	1	for	for	ADP
cana-5853	213	2	large	large	ADJ
cana-5853	213	3	n	n	CCONJ
cana-5853	213	4	,	,	PUNCT
cana-5853	213	5	the	the	DET
cana-5853	213	6	term	term	NOUN
cana-5853	213	7	1	1	NUM
cana-5853	213	8	/	/	SYM
cana-5853	213	9	n	n	PRON
cana-5853	213	10	decays	decay	VERB
cana-5853	213	11	to	to	ADP
cana-5853	213	12	zero	zero	NUM
cana-5853	213	13	.	.	PUNCT
cana-5853	214	1	additionally	additionally	ADV
cana-5853	214	2	,	,	PUNCT
cana-5853	214	3	if	if	SCONJ
cana-5853	214	4	αnk	αnk	NOUN
cana-5853	214	5	remain	remain	VERB
cana-5853	214	6	uniformly	uniformly	ADV
cana-5853	214	7	bounded	bound	VERB
cana-5853	214	8	,	,	PUNCT
cana-5853	214	9	then	then	ADV
cana-5853	214	10	for	for	ADP
cana-5853	214	11	all	all	DET
cana-5853	214	12	x	x	SYM
cana-5853	214	13	∈	∈	PROPN
cana-5853	214	14	b	b	PROPN
cana-5853	214	15	,	,	PUNCT
cana-5853	214	16	1	1	NUM
cana-5853	214	17	n	n	NUM
cana-5853	214	18	n∑	n∑	NOUN
cana-5853	214	19	k=1	k=1	NOUN
cana-5853	214	20	|αnkxk|	|αnkxk|	NUM
cana-5853	214	21	→	→	SYM
cana-5853	214	22	0	0	NUM
cana-5853	214	23	as	as	ADP
cana-5853	214	24	n	n	NUM
cana-5853	214	25	→	→	SYM
cana-5853	214	26	∞	∞	PROPN
cana-5853	214	27	,	,	PUNCT
cana-5853	214	28	since	since	SCONJ
cana-5853	214	29	xk	xk	PROPN
cana-5853	214	30	are	be	AUX
cana-5853	214	31	controlled	control	VERB
cana-5853	214	32	in	in	ADP
cana-5853	214	33	modular	modular	ADJ
cana-5853	214	34	sum	sum	NOUN
cana-5853	214	35	and	and	CCONJ
cana-5853	214	36	the	the	DET
cana-5853	214	37	weights	weight	NOUN
cana-5853	214	38	1	1	NUM
cana-5853	214	39	/	/	SYM
cana-5853	214	40	n	n	NOUN
cana-5853	214	41	diminish	diminish	NOUN
cana-5853	214	42	.	.	PUNCT
cana-5853	215	1	by	by	ADP
cana-5853	215	2	modular	modular	ADJ
cana-5853	215	3	convexity	convexity	NOUN
cana-5853	215	4	:	:	PUNCT
cana-5853	215	5	m	m	PROPN
cana-5853	215	6	(	(	PUNCT
cana-5853	215	7	n	n	CCONJ
cana-5853	215	8	,	,	PUNCT
cana-5853	215	9	(	(	PUNCT
cana-5853	215	10	ax)n	ax)n	PROPN
cana-5853	215	11	)	)	PUNCT
cana-5853	215	12	≤	≤	NOUN
cana-5853	215	13	1	1	NUM
cana-5853	215	14	n	n	NUM
cana-5853	215	15	n∑	n∑	NOUN
cana-5853	215	16	k=1	k=1	NOUN
cana-5853	215	17	m	m	VERB
cana-5853	215	18	(	(	PUNCT
cana-5853	215	19	n	n	CCONJ
cana-5853	215	20	,	,	PUNCT
cana-5853	215	21	αnkxk	αnkxk	VERB
cana-5853	215	22	)	)	PUNCT
cana-5853	215	23	.	.	PUNCT
cana-5853	216	1	for	for	ADP
cana-5853	216	2	n	n	CCONJ
cana-5853	216	3	large	large	ADJ
cana-5853	216	4	,	,	PUNCT
cana-5853	216	5	1	1	NUM
cana-5853	216	6	/	/	SYM
cana-5853	216	7	n	n	NOUN
cana-5853	216	8	enforces	enforce	VERB
cana-5853	216	9	that	that	SCONJ
cana-5853	216	10	these	these	DET
cana-5853	216	11	modular	modular	ADJ
cana-5853	216	12	sums	sum	NOUN
cana-5853	216	13	become	become	VERB
cana-5853	216	14	arbitrarily	arbitrarily	ADV
cana-5853	216	15	small	small	ADJ
cana-5853	216	16	uniformly	uniformly	ADV
cana-5853	216	17	over	over	ADP
cana-5853	216	18	x	x	PUNCT
cana-5853	216	19	∈	∈	PROPN
cana-5853	216	20	b	b	NOUN
cana-5853	216	21	,	,	PUNCT
cana-5853	216	22	provided	provide	VERB
cana-5853	216	23	the	the	DET
cana-5853	216	24	modular	modular	ADJ
cana-5853	216	25	growth	growth	NOUN
cana-5853	216	26	is	be	AUX
cana-5853	216	27	controlled	control	VERB
cana-5853	216	28	and	and	CCONJ
cana-5853	216	29	satisfies	satisfy	VERB
cana-5853	216	30	∆2	∆2	NOUN
cana-5853	216	31	-	-	PUNCT
cana-5853	216	32	type	type	NOUN
cana-5853	216	33	conditions	condition	NOUN
cana-5853	216	34	.	.	PUNCT
cana-5853	217	1	consequently	consequently	ADV
cana-5853	217	2	:	:	PUNCT
cana-5853	217	3	sup	sup	NOUN
cana-5853	217	4	x∈b	x∈b	NOUN
cana-5853	218	1	∞∑	∞∑	NUM
cana-5853	218	2	n	n	CCONJ
cana-5853	218	3	=	=	SYM
cana-5853	218	4	n+1	n+1	PROPN
cana-5853	218	5	m	m	PROPN
cana-5853	218	6	(	(	PUNCT
cana-5853	218	7	n	n	X
cana-5853	218	8	,	,	PUNCT
cana-5853	218	9	(	(	PUNCT
cana-5853	218	10	ax)n	ax)n	PROPN
cana-5853	218	11	)	)	PUNCT
cana-5853	218	12	<	<	X
cana-5853	218	13	ϵ	ϵ	X
cana-5853	218	14	for	for	ADP
cana-5853	218	15	sufficiently	sufficiently	ADV
cana-5853	218	16	large	large	ADJ
cana-5853	218	17	n	n	NOUN
cana-5853	218	18	,	,	PUNCT
cana-5853	218	19	proving	prove	VERB
cana-5853	218	20	compactness	compactness	NOUN
cana-5853	218	21	.	.	PUNCT
cana-5853	219	1	communications	communication	NOUN
cana-5853	219	2	on	on	ADP
cana-5853	219	3	applied	apply	VERB
cana-5853	219	4	nonlinear	nonlinear	ADJ
cana-5853	219	5	analysis	analysis	NOUN
cana-5853	219	6	issn	issn	NOUN
cana-5853	219	7	:	:	PUNCT
cana-5853	219	8	1074	1074	NUM
cana-5853	219	9	-	-	PUNCT
cana-5853	219	10	133x	133x	NUM
cana-5853	219	11	vol	vol	NOUN
cana-5853	219	12	31	31	NUM
cana-5853	219	13	no	no	NOUN
cana-5853	219	14	.	.	NOUN
cana-5853	219	15	2	2	NUM
cana-5853	219	16	(	(	PUNCT
cana-5853	219	17	2024	2024	NUM
cana-5853	219	18	)	)	PUNCT
cana-5853	219	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	219	20	473	473	NUM
cana-5853	219	21	example	example	NOUN
cana-5853	219	22	:	:	PUNCT
cana-5853	219	23	classical	classical	ADJ
cana-5853	219	24	cesàro	cesàro	NOUN
cana-5853	219	25	matrix	matrix	NOUN
cana-5853	219	26	in	in	ADP
cana-5853	219	27	ℓp	ℓp	NOUN
cana-5853	219	28	when	when	SCONJ
cana-5853	219	29	m(n	m(n	PROPN
cana-5853	219	30	,	,	PUNCT
cana-5853	219	31	t	t	PROPN
cana-5853	219	32	)	)	PUNCT
cana-5853	219	33	=	=	PUNCT
cana-5853	220	1	|t|p	|t|p	NOUN
cana-5853	220	2	/	/	SYM
cana-5853	220	3	p	p	X
cana-5853	220	4	,	,	PUNCT
cana-5853	220	5	the	the	DET
cana-5853	220	6	argument	argument	NOUN
cana-5853	220	7	reduces	reduce	VERB
cana-5853	220	8	to	to	ADP
cana-5853	220	9	the	the	DET
cana-5853	220	10	well	well	ADV
cana-5853	220	11	-	-	PUNCT
cana-5853	220	12	known	know	VERB
cana-5853	220	13	fact	fact	NOUN
cana-5853	220	14	that	that	SCONJ
cana-5853	220	15	the	the	DET
cana-5853	220	16	cesàro	cesàro	NOUN
cana-5853	220	17	operator	operator	NOUN
cana-5853	220	18	is	be	AUX
cana-5853	220	19	bounded	bound	VERB
cana-5853	220	20	on	on	ADP
cana-5853	220	21	ℓp	ℓp	NOUN
cana-5853	220	22	for	for	ADP
cana-5853	220	23	1	1	NUM
cana-5853	220	24	<	<	X
cana-5853	220	25	p	p	X
cana-5853	220	26	<	<	X
cana-5853	220	27	∞	∞	PROPN
cana-5853	220	28	and	and	CCONJ
cana-5853	220	29	is	be	AUX
cana-5853	220	30	compact	compact	ADJ
cana-5853	220	31	because	because	SCONJ
cana-5853	220	32	1	1	NUM
cana-5853	220	33	/	/	SYM
cana-5853	220	34	n	n	CCONJ
cana-5853	220	35	→	→	SYM
cana-5853	220	36	0	0	NUM
cana-5853	220	37	ensures	ensure	VERB
cana-5853	220	38	tail	tail	NOUN
cana-5853	220	39	smallness	smallness	NOUN
cana-5853	220	40	.	.	PUNCT
cana-5853	221	1	example	example	NOUN
cana-5853	221	2	:	:	PUNCT
cana-5853	221	3	weighted	weight	VERB
cana-5853	221	4	cesàro	cesàro	PROPN
cana-5853	221	5	in	in	ADP
cana-5853	221	6	xm	xm	PROPN
cana-5853	221	7	for	for	ADP
cana-5853	221	8	weighted	weight	VERB
cana-5853	221	9	cesàro	cesàro	ADJ
cana-5853	221	10	matrices	matrix	NOUN
cana-5853	221	11	with	with	ADP
cana-5853	221	12	decaying	decay	VERB
cana-5853	221	13	weights	weight	NOUN
cana-5853	221	14	wk	wk	PROPN
cana-5853	221	15	,	,	PUNCT
cana-5853	221	16	provided	provide	VERB
cana-5853	221	17	wn	wn	PROPN
cana-5853	221	18	grows	grow	VERB
cana-5853	221	19	sufficiently	sufficiently	ADV
cana-5853	221	20	to	to	PART
cana-5853	221	21	ensure	ensure	VERB
cana-5853	221	22	1	1	NUM
cana-5853	221	23	/	/	SYM
cana-5853	221	24	wn	wn	PROPN
cana-5853	221	25	→	→	SYM
cana-5853	221	26	0	0	NUM
cana-5853	221	27	,	,	PUNCT
cana-5853	221	28	similar	similar	ADJ
cana-5853	221	29	estimates	estimate	NOUN
cana-5853	221	30	yield	yield	VERB
cana-5853	221	31	compactness	compactness	NOUN
cana-5853	221	32	on	on	ADP
cana-5853	221	33	xm	xm	PROPN
cana-5853	221	34	spaces	space	NOUN
cana-5853	221	35	.	.	PUNCT
cana-5853	222	1	summary	summary	VERB
cana-5853	222	2	cesàro	cesàro	NOUN
cana-5853	222	3	-	-	PUNCT
cana-5853	222	4	type	type	NOUN
cana-5853	222	5	and	and	CCONJ
cana-5853	222	6	summability	summability	NOUN
cana-5853	222	7	matrices	matrix	NOUN
cana-5853	222	8	offer	offer	VERB
cana-5853	222	9	natural	natural	ADJ
cana-5853	222	10	,	,	PUNCT
cana-5853	222	11	concrete	concrete	ADJ
cana-5853	222	12	examples	example	NOUN
cana-5853	222	13	of	of	ADP
cana-5853	222	14	operators	operator	NOUN
cana-5853	222	15	on	on	ADP
cana-5853	222	16	xm	xm	PROPN
cana-5853	222	17	.	.	PUNCT
cana-5853	223	1	by	by	ADP
cana-5853	223	2	leveraging	leverage	VERB
cana-5853	223	3	convexity	convexity	NOUN
cana-5853	223	4	and	and	CCONJ
cana-5853	223	5	modular	modular	ADJ
cana-5853	223	6	inequalities	inequality	NOUN
cana-5853	223	7	,	,	PUNCT
cana-5853	223	8	we	we	PRON
cana-5853	223	9	obtain	obtain	VERB
cana-5853	223	10	clear	clear	ADJ
cana-5853	223	11	boundedness	boundedness	NOUN
cana-5853	223	12	criteria	criterion	NOUN
cana-5853	223	13	via	via	ADP
cana-5853	223	14	control	control	NOUN
cana-5853	223	15	over	over	ADP
cana-5853	223	16	matrix	matrix	NOUN
cana-5853	223	17	weights	weight	NOUN
cana-5853	223	18	.	.	PUNCT
cana-5853	224	1	compactness	compactness	NOUN
cana-5853	224	2	emerges	emerge	VERB
cana-5853	224	3	through	through	ADP
cana-5853	224	4	tail	tail	NOUN
cana-5853	224	5	decay	decay	NOUN
cana-5853	224	6	properties	property	NOUN
cana-5853	224	7	,	,	PUNCT
cana-5853	224	8	with	with	ADP
cana-5853	224	9	the	the	DET
cana-5853	224	10	1	1	NUM
cana-5853	224	11	/	/	SYM
cana-5853	224	12	n	n	NOUN
cana-5853	224	13	averaging	average	VERB
cana-5853	224	14	enforcing	enforce	VERB
cana-5853	224	15	vanishing	vanish	VERB
cana-5853	224	16	modular	modular	ADJ
cana-5853	224	17	sums	sum	NOUN
cana-5853	224	18	in	in	ADP
cana-5853	224	19	high	high	ADJ
cana-5853	224	20	indices	index	NOUN
cana-5853	224	21	.	.	PUNCT
cana-5853	225	1	these	these	DET
cana-5853	225	2	analyses	analysis	NOUN
cana-5853	225	3	generalize	generalize	VERB
cana-5853	225	4	classical	classical	ADJ
cana-5853	225	5	results	result	NOUN
cana-5853	225	6	from	from	ADP
cana-5853	225	7	ℓp	ℓp	ADJ
cana-5853	225	8	and	and	CCONJ
cana-5853	225	9	orlicz	orlicz	ADJ
cana-5853	225	10	spaces	space	NOUN
cana-5853	225	11	,	,	PUNCT
cana-5853	225	12	demonstrating	demonstrate	VERB
cana-5853	225	13	the	the	DET
cana-5853	225	14	strength	strength	NOUN
cana-5853	225	15	and	and	CCONJ
cana-5853	225	16	flexibility	flexibility	NOUN
cana-5853	225	17	of	of	ADP
cana-5853	225	18	the	the	DET
cana-5853	225	19	modular	modular	ADJ
cana-5853	225	20	framework	framework	NOUN
cana-5853	225	21	for	for	ADP
cana-5853	225	22	operator	operator	NOUN
cana-5853	225	23	theory	theory	NOUN
cana-5853	225	24	on	on	ADP
cana-5853	225	25	sequence	sequence	NOUN
cana-5853	225	26	spaces	space	NOUN
cana-5853	225	27	.	.	PUNCT
cana-5853	226	1	6	6	NUM
cana-5853	226	2	spectral	spectral	ADJ
cana-5853	226	3	properties	property	NOUN
cana-5853	226	4	of	of	ADP
cana-5853	226	5	matrix	matrix	NOUN
cana-5853	226	6	operators	operator	NOUN
cana-5853	226	7	in	in	ADP
cana-5853	226	8	addition	addition	NOUN
cana-5853	226	9	to	to	ADP
cana-5853	226	10	boundedness	boundedness	NOUN
cana-5853	226	11	and	and	CCONJ
cana-5853	226	12	compactness	compactness	NOUN
cana-5853	226	13	,	,	PUNCT
cana-5853	226	14	understanding	understand	VERB
cana-5853	226	15	the	the	DET
cana-5853	226	16	spectral	spectral	ADJ
cana-5853	226	17	properties	property	NOUN
cana-5853	226	18	of	of	ADP
cana-5853	226	19	matrix	matrix	NOUN
cana-5853	226	20	operators	operator	NOUN
cana-5853	226	21	on	on	ADP
cana-5853	226	22	modulated	modulate	VERB
cana-5853	226	23	orlicz	orlicz	ADJ
cana-5853	226	24	-	-	PUNCT
cana-5853	226	25	type	type	NOUN
cana-5853	226	26	sequence	sequence	NOUN
cana-5853	226	27	spaces	space	VERB
cana-5853	226	28	xm	xm	PROPN
cana-5853	226	29	is	be	AUX
cana-5853	226	30	crucial	crucial	ADJ
cana-5853	226	31	for	for	ADP
cana-5853	226	32	operator	operator	NOUN
cana-5853	226	33	theory	theory	NOUN
cana-5853	226	34	.	.	PUNCT
cana-5853	227	1	spectral	spectral	ADJ
cana-5853	227	2	theory	theory	NOUN
cana-5853	227	3	describes	describe	VERB
cana-5853	227	4	the	the	DET
cana-5853	227	5	set	set	NOUN
cana-5853	227	6	of	of	ADP
cana-5853	227	7	scalars	scalar	NOUN
cana-5853	227	8	λ	λ	PROPN
cana-5853	227	9	for	for	ADP
cana-5853	227	10	which	which	PRON
cana-5853	227	11	(	(	PUNCT
cana-5853	227	12	a−λi	a−λi	NOUN
cana-5853	227	13	)	)	PUNCT
cana-5853	227	14	fails	fail	VERB
cana-5853	227	15	to	to	PART
cana-5853	227	16	be	be	AUX
cana-5853	227	17	invertible	invertible	ADJ
cana-5853	227	18	,	,	PUNCT
cana-5853	227	19	informing	inform	VERB
cana-5853	227	20	stability	stability	NOUN
cana-5853	227	21	analysis	analysis	NOUN
cana-5853	227	22	,	,	PUNCT
cana-5853	227	23	iterative	iterative	NOUN
cana-5853	227	24	methods	method	NOUN
cana-5853	227	25	,	,	PUNCT
cana-5853	227	26	and	and	CCONJ
cana-5853	227	27	functional	functional	ADJ
cana-5853	227	28	calculus	calculus	NOUN
cana-5853	227	29	in	in	ADP
cana-5853	227	30	infinite	infinite	ADJ
cana-5853	227	31	dimensions	dimension	NOUN
cana-5853	227	32	.	.	PUNCT
cana-5853	228	1	in	in	ADP
cana-5853	228	2	this	this	DET
cana-5853	228	3	section	section	NOUN
cana-5853	228	4	,	,	PUNCT
cana-5853	228	5	we	we	PRON
cana-5853	228	6	discuss	discuss	VERB
cana-5853	228	7	general	general	ADJ
cana-5853	228	8	aspects	aspect	NOUN
cana-5853	228	9	of	of	ADP
cana-5853	228	10	spectral	spectral	ADJ
cana-5853	228	11	theory	theory	NOUN
cana-5853	228	12	in	in	ADP
cana-5853	228	13	sequence	sequence	NOUN
cana-5853	228	14	spaces	space	NOUN
cana-5853	228	15	,	,	PUNCT
cana-5853	228	16	special	special	ADJ
cana-5853	228	17	results	result	NOUN
cana-5853	228	18	for	for	ADP
cana-5853	228	19	compact	compact	ADJ
cana-5853	228	20	operators	operator	NOUN
cana-5853	228	21	on	on	ADP
cana-5853	228	22	xm	xm	PROPN
cana-5853	228	23	,	,	PUNCT
cana-5853	228	24	and	and	CCONJ
cana-5853	228	25	detailed	detailed	ADJ
cana-5853	228	26	analysis	analysis	NOUN
cana-5853	228	27	of	of	ADP
cana-5853	228	28	diagonal	diagonal	ADJ
cana-5853	228	29	matrices	matrix	NOUN
cana-5853	228	30	as	as	ADP
cana-5853	228	31	prototypical	prototypical	ADJ
cana-5853	228	32	examples	example	NOUN
cana-5853	228	33	.	.	PUNCT
cana-5853	229	1	6.1	6.1	NUM
cana-5853	229	2	spectral	spectral	ADJ
cana-5853	229	3	theory	theory	NOUN
cana-5853	229	4	in	in	ADP
cana-5853	229	5	sequence	sequence	NOUN
cana-5853	229	6	spaces	space	NOUN
cana-5853	229	7	let	let	VERB
cana-5853	229	8	a	a	DET
cana-5853	229	9	:	:	PUNCT
cana-5853	229	10	xm	xm	PROPN
cana-5853	229	11	→	→	SYM
cana-5853	229	12	xm	xm	PROPN
cana-5853	229	13	be	be	AUX
cana-5853	229	14	a	a	DET
cana-5853	229	15	bounded	bounded	ADJ
cana-5853	229	16	linear	linear	ADJ
cana-5853	229	17	operator	operator	NOUN
cana-5853	229	18	.	.	PUNCT
cana-5853	230	1	the	the	DET
cana-5853	230	2	spectrum	spectrum	NOUN
cana-5853	230	3	of	of	ADP
cana-5853	230	4	a	a	PRON
cana-5853	230	5	,	,	PUNCT
cana-5853	230	6	denoted	denote	VERB
cana-5853	230	7	σ(a	σ(a	PROPN
cana-5853	230	8	)	)	PUNCT
cana-5853	230	9	,	,	PUNCT
cana-5853	230	10	is	be	AUX
cana-5853	230	11	defined	define	VERB
cana-5853	230	12	as	as	ADP
cana-5853	230	13	:	:	PUNCT
cana-5853	230	14	σ(a	σ(a	NUM
cana-5853	230	15	)	)	PUNCT
cana-5853	230	16	=	=	PRON
cana-5853	231	1	{	{	PUNCT
cana-5853	231	2	λ	λ	X
cana-5853	231	3	∈	∈	NOUN
cana-5853	231	4	f	f	X
cana-5853	231	5	:	:	PUNCT
cana-5853	231	6	(	(	PUNCT
cana-5853	231	7	a−	a−	PROPN
cana-5853	231	8	λi	λi	PART
cana-5853	231	9	)	)	PUNCT
cana-5853	231	10	is	be	AUX
cana-5853	231	11	not	not	PART
cana-5853	231	12	invertible	invertible	ADJ
cana-5853	231	13	}	}	PUNCT
cana-5853	231	14	.	.	PUNCT
cana-5853	232	1	standard	standard	ADJ
cana-5853	232	2	operator	operator	NOUN
cana-5853	232	3	theory	theory	NOUN
cana-5853	232	4	partitions	partition	VERB
cana-5853	232	5	the	the	DET
cana-5853	232	6	spectrum	spectrum	NOUN
cana-5853	232	7	into	into	ADP
cana-5853	232	8	:	:	PUNCT
cana-5853	232	9	•	•	ADP
cana-5853	232	10	the	the	DET
cana-5853	232	11	point	point	NOUN
cana-5853	232	12	spectrum	spectrum	NOUN
cana-5853	232	13	(	(	PUNCT
cana-5853	232	14	eigenvalues	eigenvalue	NOUN
cana-5853	232	15	):	):	PUNCT
cana-5853	232	16	σp(a	σp(a	NOUN
cana-5853	232	17	)	)	PUNCT
cana-5853	232	18	=	=	SYM
cana-5853	232	19	{	{	PUNCT
cana-5853	232	20	λ	λ	X
cana-5853	232	21	:	:	PUNCT
cana-5853	232	22	∃	∃	PROPN
cana-5853	232	23	x	x	PUNCT
cana-5853	232	24	̸=	̸=	PROPN
cana-5853	232	25	0	0	NUM
cana-5853	232	26	,	,	PUNCT
cana-5853	232	27	ax	ax	NOUN
cana-5853	232	28	=	=	PUNCT
cana-5853	232	29	λx	λx	PROPN
cana-5853	232	30	}	}	PUNCT
cana-5853	232	31	.	.	PUNCT
cana-5853	233	1	•	•	NOUN
cana-5853	233	2	the	the	DET
cana-5853	233	3	continuous	continuous	ADJ
cana-5853	233	4	spectrum	spectrum	NOUN
cana-5853	233	5	:	:	PUNCT
cana-5853	233	6	where	where	SCONJ
cana-5853	233	7	(	(	PUNCT
cana-5853	233	8	a	a	DET
cana-5853	233	9	−	−	NOUN
cana-5853	233	10	λi	λi	NUM
cana-5853	233	11	)	)	PUNCT
cana-5853	233	12	is	be	AUX
cana-5853	233	13	injective	injective	ADJ
cana-5853	233	14	with	with	ADP
cana-5853	233	15	dense	dense	ADJ
cana-5853	233	16	range	range	NOUN
cana-5853	233	17	but	but	CCONJ
cana-5853	233	18	not	not	PART
cana-5853	233	19	surjective	surjective	ADJ
cana-5853	233	20	.	.	PUNCT
cana-5853	233	21	•	•	NUM
cana-5853	234	1	the	the	DET
cana-5853	234	2	residual	residual	ADJ
cana-5853	234	3	spectrum	spectrum	NOUN
cana-5853	234	4	:	:	PUNCT
cana-5853	234	5	where	where	SCONJ
cana-5853	234	6	(	(	PUNCT
cana-5853	234	7	a−	a−	PROPN
cana-5853	234	8	λi	λi	PART
cana-5853	234	9	)	)	PUNCT
cana-5853	234	10	is	be	AUX
cana-5853	234	11	injective	injective	ADJ
cana-5853	234	12	but	but	CCONJ
cana-5853	234	13	has	have	VERB
cana-5853	234	14	non	non	ADJ
cana-5853	234	15	-	-	ADJ
cana-5853	234	16	dense	dense	ADJ
cana-5853	234	17	range	range	NOUN
cana-5853	234	18	.	.	PUNCT
cana-5853	235	1	in	in	ADP
cana-5853	235	2	sequence	sequence	NOUN
cana-5853	235	3	spaces	space	NOUN
cana-5853	235	4	like	like	ADP
cana-5853	235	5	xm	xm	PROPN
cana-5853	235	6	,	,	PUNCT
cana-5853	235	7	these	these	DET
cana-5853	235	8	notions	notion	NOUN
cana-5853	235	9	behave	behave	VERB
cana-5853	235	10	analogously	analogously	ADV
cana-5853	235	11	to	to	ADP
cana-5853	235	12	classical	classical	ADJ
cana-5853	235	13	ℓp	ℓp	ADJ
cana-5853	235	14	settings	setting	NOUN
cana-5853	235	15	,	,	PUNCT
cana-5853	235	16	but	but	CCONJ
cana-5853	235	17	the	the	DET
cana-5853	235	18	index	index	NOUN
cana-5853	235	19	-	-	PUNCT
cana-5853	235	20	dependent	dependent	ADJ
cana-5853	235	21	modular	modular	ADJ
cana-5853	235	22	structure	structure	NOUN
cana-5853	235	23	requires	require	VERB
cana-5853	235	24	verifying	verify	VERB
cana-5853	235	25	conditions	condition	NOUN
cana-5853	235	26	with	with	ADP
cana-5853	235	27	care	care	NOUN
cana-5853	235	28	.	.	PUNCT
cana-5853	236	1	key	key	ADJ
cana-5853	236	2	general	general	ADJ
cana-5853	236	3	facts	fact	NOUN
cana-5853	236	4	include	include	VERB
cana-5853	236	5	:	:	PUNCT
cana-5853	236	6	communications	communication	NOUN
cana-5853	236	7	on	on	ADP
cana-5853	236	8	applied	apply	VERB
cana-5853	236	9	nonlinear	nonlinear	ADJ
cana-5853	236	10	analysis	analysis	NOUN
cana-5853	236	11	issn	issn	NOUN
cana-5853	236	12	:	:	PUNCT
cana-5853	236	13	1074	1074	NUM
cana-5853	236	14	-	-	PUNCT
cana-5853	236	15	133x	133x	NUM
cana-5853	236	16	vol	vol	NOUN
cana-5853	236	17	31	31	NUM
cana-5853	236	18	no	no	NOUN
cana-5853	236	19	.	.	NOUN
cana-5853	236	20	2	2	NUM
cana-5853	236	21	(	(	PUNCT
cana-5853	236	22	2024	2024	NUM
cana-5853	236	23	)	)	PUNCT
cana-5853	236	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	236	25	474	474	NUM
cana-5853	236	26	σ(a	σ(a	PROPN
cana-5853	236	27	)	)	PUNCT
cana-5853	236	28	is	be	AUX
cana-5853	236	29	nonempty	nonempty	ADJ
cana-5853	236	30	,	,	PUNCT
cana-5853	236	31	compact	compact	ADJ
cana-5853	236	32	,	,	PUNCT
cana-5853	236	33	and	and	CCONJ
cana-5853	236	34	contained	contain	VERB
cana-5853	236	35	in	in	ADP
cana-5853	236	36	{	{	PUNCT
cana-5853	236	37	λ	λ	X
cana-5853	236	38	:	:	PUNCT
cana-5853	236	39	|λ|	|λ|	PROPN
cana-5853	236	40	≤	≤	NUM
cana-5853	236	41	∥a∥	∥a∥	NOUN
cana-5853	236	42	}	}	PUNCT
cana-5853	236	43	.	.	PUNCT
cana-5853	237	1	this	this	PRON
cana-5853	237	2	remains	remain	VERB
cana-5853	237	3	valid	valid	ADJ
cana-5853	237	4	in	in	ADP
cana-5853	237	5	xm	xm	PROPN
cana-5853	237	6	since	since	SCONJ
cana-5853	237	7	bounded	bound	VERB
cana-5853	237	8	linear	linear	PROPN
cana-5853	237	9	operators	operator	NOUN
cana-5853	237	10	on	on	ADP
cana-5853	237	11	banach	banach	NOUN
cana-5853	237	12	spaces	space	NOUN
cana-5853	237	13	share	share	VERB
cana-5853	237	14	these	these	DET
cana-5853	237	15	spectral	spectral	ADJ
cana-5853	237	16	properties	property	NOUN
cana-5853	237	17	.	.	PUNCT
cana-5853	238	1	6.2	6.2	NUM
cana-5853	238	2	compact	compact	ADJ
cana-5853	238	3	operator	operator	NOUN
cana-5853	238	4	spectra	spectra	NOUN
cana-5853	238	5	in	in	ADP
cana-5853	238	6	xm	xm	PROPN
cana-5853	238	7	a	a	DET
cana-5853	238	8	particularly	particularly	ADV
cana-5853	238	9	tractable	tractable	ADJ
cana-5853	238	10	class	class	NOUN
cana-5853	238	11	of	of	ADP
cana-5853	238	12	operators	operator	NOUN
cana-5853	238	13	are	be	AUX
cana-5853	238	14	compact	compact	ADJ
cana-5853	238	15	operators	operator	NOUN
cana-5853	238	16	,	,	PUNCT
cana-5853	238	17	which	which	PRON
cana-5853	238	18	map	map	VERB
cana-5853	238	19	bounded	bounded	ADJ
cana-5853	238	20	sets	set	NOUN
cana-5853	238	21	into	into	ADP
cana-5853	238	22	relatively	relatively	ADV
cana-5853	238	23	compact	compact	ADJ
cana-5853	238	24	sets	set	NOUN
cana-5853	238	25	.	.	PUNCT
cana-5853	239	1	recall	recall	VERB
cana-5853	239	2	that	that	SCONJ
cana-5853	239	3	in	in	ADP
cana-5853	239	4	any	any	DET
cana-5853	239	5	infinite	infinite	ADJ
cana-5853	239	6	-	-	PUNCT
cana-5853	239	7	dimensional	dimensional	ADJ
cana-5853	239	8	banach	banach	NOUN
cana-5853	239	9	space	space	NOUN
cana-5853	239	10	:	:	PUNCT
cana-5853	239	11	σess(a	σess(a	NOUN
cana-5853	239	12	)	)	PUNCT
cana-5853	239	13	=	=	PUNCT
cana-5853	239	14	{	{	PUNCT
cana-5853	239	15	0	0	NUM
cana-5853	239	16	}	}	PUNCT
cana-5853	239	17	if	if	SCONJ
cana-5853	239	18	a	a	PRON
cana-5853	239	19	is	be	AUX
cana-5853	239	20	compact	compact	ADJ
cana-5853	239	21	.	.	PUNCT
cana-5853	240	1	hence	hence	ADV
cana-5853	240	2	,	,	PUNCT
cana-5853	240	3	the	the	DET
cana-5853	240	4	spectrum	spectrum	NOUN
cana-5853	240	5	of	of	ADP
cana-5853	240	6	a	a	DET
cana-5853	240	7	compact	compact	ADJ
cana-5853	240	8	operator	operator	NOUN
cana-5853	240	9	on	on	ADP
cana-5853	240	10	xm	xm	PROPN
cana-5853	240	11	consists	consist	VERB
cana-5853	240	12	of	of	ADP
cana-5853	240	13	:	:	PUNCT
cana-5853	240	14	σ(a	σ(a	NUM
cana-5853	240	15	)	)	PUNCT
cana-5853	240	16	=	=	PRON
cana-5853	241	1	{	{	PUNCT
cana-5853	241	2	0	0	NUM
cana-5853	241	3	}	}	PUNCT
cana-5853	241	4	∪	∪	X
cana-5853	241	5	{	{	PUNCT
cana-5853	241	6	λj	λj	NOUN
cana-5853	241	7	}	}	PUNCT
cana-5853	241	8	,	,	PUNCT
cana-5853	241	9	where	where	SCONJ
cana-5853	241	10	the	the	DET
cana-5853	241	11	non	non	ADJ
cana-5853	241	12	-	-	ADJ
cana-5853	241	13	zero	zero	NUM
cana-5853	241	14	λj	λj	NOUN
cana-5853	241	15	form	form	NOUN
cana-5853	241	16	at	at	ADP
cana-5853	241	17	most	most	ADV
cana-5853	241	18	a	a	DET
cana-5853	241	19	countable	countable	ADJ
cana-5853	241	20	set	set	NOUN
cana-5853	241	21	with	with	ADP
cana-5853	241	22	|λj|	|λj|	PROPN
cana-5853	241	23	→	→	SYM
cana-5853	241	24	0	0	NUM
cana-5853	241	25	.	.	PUNCT
cana-5853	242	1	each	each	DET
cana-5853	242	2	non	non	ADJ
cana-5853	242	3	-	-	ADJ
cana-5853	242	4	zero	zero	NUM
cana-5853	242	5	eigenvalue	eigenvalue	NOUN
cana-5853	242	6	has	have	AUX
cana-5853	242	7	finite	finite	VERB
cana-5853	242	8	algebraic	algebraic	ADJ
cana-5853	242	9	multiplicity	multiplicity	NOUN
cana-5853	242	10	.	.	PUNCT
cana-5853	243	1	this	this	DET
cana-5853	243	2	result	result	NOUN
cana-5853	243	3	holds	hold	VERB
cana-5853	243	4	in	in	ADP
cana-5853	243	5	xm	xm	PROPN
cana-5853	243	6	under	under	ADP
cana-5853	243	7	standard	standard	ADJ
cana-5853	243	8	completeness	completeness	NOUN
cana-5853	243	9	and	and	CCONJ
cana-5853	243	10	modular	modular	ADJ
cana-5853	243	11	convexity	convexity	NOUN
cana-5853	243	12	assumptions	assumption	NOUN
cana-5853	243	13	.	.	PUNCT
cana-5853	244	1	the	the	DET
cana-5853	244	2	proof	proof	ADJ
cana-5853	244	3	strategy	strategy	NOUN
cana-5853	244	4	mirrors	mirror	VERB
cana-5853	244	5	classical	classical	ADJ
cana-5853	244	6	functional	functional	ADJ
cana-5853	244	7	analysis	analysis	NOUN
cana-5853	244	8	:	:	PUNCT
cana-5853	244	9	•	•	ADP
cana-5853	244	10	use	use	VERB
cana-5853	244	11	the	the	DET
cana-5853	244	12	fact	fact	NOUN
cana-5853	244	13	that	that	SCONJ
cana-5853	244	14	a	a	PRON
cana-5853	244	15	is	be	AUX
cana-5853	244	16	compact	compact	ADJ
cana-5853	244	17	=	=	NOUN
cana-5853	244	18	⇒	⇒	NOUN
cana-5853	244	19	a−	a−	PROPN
cana-5853	244	20	λi	λi	SCONJ
cana-5853	244	21	is	be	AUX
cana-5853	244	22	fredholm	fredholm	NOUN
cana-5853	244	23	of	of	ADP
cana-5853	244	24	index	index	NOUN
cana-5853	244	25	0	0	NUM
cana-5853	244	26	for	for	ADP
cana-5853	244	27	λ	λ	PROPN
cana-5853	244	28	̸=	̸=	PROPN
cana-5853	244	29	0	0	NUM
cana-5853	244	30	.	.	NOUN
cana-5853	244	31	•	•	NUM
cana-5853	244	32	apply	apply	VERB
cana-5853	244	33	riesz	riesz	NOUN
cana-5853	244	34	-	-	PUNCT
cana-5853	244	35	schauder	schauder	NOUN
cana-5853	244	36	theory	theory	NOUN
cana-5853	244	37	to	to	PART
cana-5853	244	38	conclude	conclude	VERB
cana-5853	244	39	spectral	spectral	ADJ
cana-5853	244	40	properties	property	NOUN
cana-5853	244	41	.	.	PUNCT
cana-5853	245	1	compactness	compactness	NOUN
cana-5853	245	2	criteria	criterion	NOUN
cana-5853	245	3	established	establish	VERB
cana-5853	245	4	earlier	early	ADV
cana-5853	245	5	(	(	PUNCT
cana-5853	245	6	tail	tail	NOUN
cana-5853	245	7	conditions	condition	NOUN
cana-5853	245	8	,	,	PUNCT
cana-5853	245	9	modular	modular	ADJ
cana-5853	245	10	domination	domination	NOUN
cana-5853	245	11	)	)	PUNCT
cana-5853	245	12	therefore	therefore	ADV
cana-5853	245	13	directly	directly	ADV
cana-5853	245	14	lead	lead	VERB
cana-5853	245	15	to	to	ADP
cana-5853	245	16	spectral	spectral	ADJ
cana-5853	245	17	structure	structure	NOUN
cana-5853	245	18	results	result	NOUN
cana-5853	245	19	for	for	ADP
cana-5853	245	20	many	many	ADJ
cana-5853	245	21	matrix	matrix	NOUN
cana-5853	245	22	classes	class	NOUN
cana-5853	245	23	.	.	PUNCT
cana-5853	246	1	6.3	6.3	NUM
cana-5853	246	2	diagonal	diagonal	ADJ
cana-5853	246	3	operators	operator	NOUN
cana-5853	246	4	and	and	CCONJ
cana-5853	246	5	eigenvalue	eigenvalue	VERB
cana-5853	246	6	analysis	analysis	NOUN
cana-5853	246	7	diagonal	diagonal	ADJ
cana-5853	246	8	operators	operator	NOUN
cana-5853	246	9	provide	provide	VERB
cana-5853	246	10	an	an	DET
cana-5853	246	11	instructive	instructive	ADJ
cana-5853	246	12	special	special	ADJ
cana-5853	246	13	case	case	NOUN
cana-5853	246	14	.	.	PUNCT
cana-5853	247	1	let	let	VERB
cana-5853	247	2	:	:	PUNCT
cana-5853	247	3	a	a	DET
cana-5853	247	4	=	=	X
cana-5853	247	5	diag(λn	diag(λn	PROPN
cana-5853	247	6	)	)	PUNCT
cana-5853	247	7	,	,	PUNCT
cana-5853	247	8	(	(	PUNCT
cana-5853	247	9	ax)n	ax)n	PROPN
cana-5853	247	10	=	=	SYM
cana-5853	247	11	λnxn	λnxn	PROPN
cana-5853	247	12	.	.	PUNCT
cana-5853	248	1	here	here	ADV
cana-5853	248	2	,	,	PUNCT
cana-5853	248	3	the	the	DET
cana-5853	248	4	spectral	spectral	ADJ
cana-5853	248	5	analysis	analysis	NOUN
cana-5853	248	6	is	be	AUX
cana-5853	248	7	particularly	particularly	ADV
cana-5853	248	8	transparent	transparent	ADJ
cana-5853	248	9	.	.	PUNCT
cana-5853	249	1	for	for	ADP
cana-5853	249	2	any	any	DET
cana-5853	249	3	x	x	SYM
cana-5853	249	4	∈	∈	PROPN
cana-5853	249	5	xm	xm	PROPN
cana-5853	249	6	:	:	PUNCT
cana-5853	249	7	ax	ax	NOUN
cana-5853	249	8	=	=	PUNCT
cana-5853	249	9	λx	λx	X
cana-5853	249	10	⇐	⇐	PROPN
cana-5853	249	11	⇒	⇒	PROPN
cana-5853	249	12	∀n	∀n	PROPN
cana-5853	249	13	,	,	PUNCT
cana-5853	249	14	λnxn	λnxn	NOUN
cana-5853	249	15	=	=	SYM
cana-5853	249	16	λxn	λxn	PROPN
cana-5853	249	17	.	.	PUNCT
cana-5853	250	1	eigenvalues	eigenvalue	VERB
cana-5853	250	2	arise	arise	VERB
cana-5853	250	3	as	as	ADP
cana-5853	250	4	:	:	PUNCT
cana-5853	250	5	λ	λ	X
cana-5853	250	6	=	=	PRON
cana-5853	250	7	λn	λn	PROPN
cana-5853	250	8	for	for	ADP
cana-5853	250	9	some	some	DET
cana-5853	250	10	n	n	CCONJ
cana-5853	250	11	,	,	PUNCT
cana-5853	250	12	with	with	ADP
cana-5853	250	13	eigenvectors	eigenvector	NOUN
cana-5853	250	14	e(n	e(n	PROPN
cana-5853	250	15	)	)	PUNCT
cana-5853	250	16	(	(	PUNCT
cana-5853	250	17	the	the	DET
cana-5853	250	18	unit	unit	NOUN
cana-5853	250	19	vector	vector	NOUN
cana-5853	250	20	with	with	ADP
cana-5853	250	21	1	1	NUM
cana-5853	250	22	at	at	ADP
cana-5853	250	23	position	position	NOUN
cana-5853	250	24	n	n	CCONJ
cana-5853	250	25	)	)	PUNCT
cana-5853	250	26	.	.	PUNCT
cana-5853	251	1	thus	thus	ADV
cana-5853	251	2	:	:	PUNCT
cana-5853	251	3	σp(a	σp(a	X
cana-5853	251	4	)	)	PUNCT
cana-5853	251	5	=	=	SYM
cana-5853	251	6	{	{	PUNCT
cana-5853	251	7	λn	λn	NOUN
cana-5853	251	8	:	:	PUNCT
cana-5853	251	9	n	n	CCONJ
cana-5853	251	10	∈	∈	PROPN
cana-5853	251	11	n	n	CCONJ
cana-5853	251	12	}	}	PUNCT
cana-5853	251	13	.	.	PUNCT
cana-5853	252	1	the	the	DET
cana-5853	252	2	full	full	ADJ
cana-5853	252	3	spectrum	spectrum	NOUN
cana-5853	252	4	is	be	AUX
cana-5853	252	5	:	:	PUNCT
cana-5853	252	6	communications	communication	NOUN
cana-5853	252	7	on	on	ADP
cana-5853	252	8	applied	apply	VERB
cana-5853	252	9	nonlinear	nonlinear	ADJ
cana-5853	252	10	analysis	analysis	NOUN
cana-5853	252	11	issn	issn	NOUN
cana-5853	252	12	:	:	PUNCT
cana-5853	252	13	1074	1074	NUM
cana-5853	252	14	-	-	PUNCT
cana-5853	252	15	133x	133x	NUM
cana-5853	252	16	vol	vol	NOUN
cana-5853	252	17	31	31	NUM
cana-5853	252	18	no	no	NOUN
cana-5853	252	19	.	.	NOUN
cana-5853	252	20	2	2	NUM
cana-5853	252	21	(	(	PUNCT
cana-5853	252	22	2024	2024	NUM
cana-5853	252	23	)	)	PUNCT
cana-5853	252	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	252	25	475	475	NUM
cana-5853	252	26	σ(a	σ(a	PROPN
cana-5853	252	27	)	)	PUNCT
cana-5853	252	28	=	=	PRON
cana-5853	253	1	{	{	PUNCT
cana-5853	253	2	λn	λn	NOUN
cana-5853	253	3	:	:	PUNCT
cana-5853	253	4	n	n	CCONJ
cana-5853	253	5	∈	∈	PROPN
cana-5853	253	6	n	n	CCONJ
cana-5853	253	7	}	}	PUNCT
cana-5853	253	8	.	.	PUNCT
cana-5853	254	1	compactness	compactness	NOUN
cana-5853	254	2	of	of	ADP
cana-5853	254	3	a	a	PRON
cana-5853	254	4	is	be	AUX
cana-5853	254	5	equivalent	equivalent	ADJ
cana-5853	254	6	to	to	ADP
cana-5853	254	7	λn	λn	PROPN
cana-5853	254	8	→	→	SYM
cana-5853	254	9	0	0	NUM
cana-5853	254	10	as	as	ADP
cana-5853	254	11	n	n	PROPN
cana-5853	254	12	→	→	SYM
cana-5853	254	13	∞.	∞.	PROPN
cana-5853	254	14	indeed	indeed	ADV
cana-5853	254	15	:	:	PUNCT
cana-5853	254	16	•	•	INTJ
cana-5853	254	17	if	if	SCONJ
cana-5853	254	18	λn	λn	PROPN
cana-5853	254	19	→	→	SYM
cana-5853	254	20	0	0	NUM
cana-5853	254	21	,	,	PUNCT
cana-5853	254	22	then	then	ADV
cana-5853	254	23	a	a	DET
cana-5853	254	24	maps	map	NOUN
cana-5853	254	25	bounded	bound	VERB
cana-5853	254	26	sequences	sequence	NOUN
cana-5853	254	27	to	to	ADP
cana-5853	254	28	sequences	sequence	NOUN
cana-5853	254	29	with	with	ADP
cana-5853	254	30	vanishing	vanish	VERB
cana-5853	254	31	tails	tail	NOUN
cana-5853	254	32	,	,	PUNCT
cana-5853	254	33	ensuring	ensure	VERB
cana-5853	254	34	compactness	compactness	NOUN
cana-5853	254	35	via	via	ADP
cana-5853	254	36	modular	modular	ADJ
cana-5853	254	37	tail	tail	NOUN
cana-5853	254	38	control	control	NOUN
cana-5853	254	39	.	.	PUNCT
cana-5853	255	1	•	•	NUM
cana-5853	255	2	conversely	conversely	ADV
cana-5853	255	3	,	,	PUNCT
cana-5853	255	4	if	if	SCONJ
cana-5853	255	5	a	a	PRON
cana-5853	255	6	is	be	AUX
cana-5853	255	7	compact	compact	ADJ
cana-5853	255	8	,	,	PUNCT
cana-5853	255	9	any	any	DET
cana-5853	255	10	bounded	bounded	ADJ
cana-5853	255	11	sequence	sequence	NOUN
cana-5853	255	12	of	of	ADP
cana-5853	255	13	unit	unit	NOUN
cana-5853	255	14	vectors	vector	NOUN
cana-5853	255	15	e(n	e(n	NOUN
cana-5853	255	16	)	)	PUNCT
cana-5853	255	17	must	must	AUX
cana-5853	255	18	have	have	VERB
cana-5853	255	19	images	image	NOUN
cana-5853	255	20	ae(n	ae(n	NOUN
cana-5853	255	21	)	)	PUNCT
cana-5853	255	22	converging	converge	VERB
cana-5853	255	23	to	to	ADP
cana-5853	255	24	0	0	NUM
cana-5853	255	25	in	in	ADP
cana-5853	255	26	xm	xm	PROPN
cana-5853	255	27	,	,	PUNCT
cana-5853	255	28	forcing	force	VERB
cana-5853	255	29	λn	λn	NOUN
cana-5853	255	30	→	→	SYM
cana-5853	255	31	0	0	NUM
cana-5853	255	32	.	.	PUNCT
cana-5853	256	1	hence	hence	ADV
cana-5853	256	2	,	,	PUNCT
cana-5853	256	3	for	for	ADP
cana-5853	256	4	diagonal	diagonal	ADJ
cana-5853	256	5	operators	operator	NOUN
cana-5853	256	6	in	in	ADP
cana-5853	256	7	xm	xm	PROPN
cana-5853	256	8	,	,	PUNCT
cana-5853	256	9	the	the	DET
cana-5853	256	10	spectral	spectral	ADJ
cana-5853	256	11	characterization	characterization	NOUN
cana-5853	256	12	aligns	align	VERB
cana-5853	256	13	with	with	ADP
cana-5853	256	14	classical	classical	ADJ
cana-5853	256	15	ℓp	ℓp	ADJ
cana-5853	256	16	results	result	NOUN
cana-5853	256	17	while	while	SCONJ
cana-5853	256	18	generalizing	generalize	VERB
cana-5853	256	19	to	to	ADP
cana-5853	256	20	the	the	DET
cana-5853	256	21	index	index	NOUN
cana-5853	256	22	-	-	PUNCT
cana-5853	256	23	modulated	modulate	VERB
cana-5853	256	24	modular	modular	ADJ
cana-5853	256	25	context	context	NOUN
cana-5853	256	26	.	.	PUNCT
cana-5853	257	1	example	example	NOUN
cana-5853	257	2	.	.	PUNCT
cana-5853	258	1	if	if	SCONJ
cana-5853	258	2	m(n	m(n	PROPN
cana-5853	258	3	,	,	PUNCT
cana-5853	258	4	t	t	PROPN
cana-5853	258	5	)	)	PUNCT
cana-5853	258	6	=	=	SYM
cana-5853	258	7	ωn|t|p	ωn|t|p	NOUN
cana-5853	258	8	p	p	NOUN
cana-5853	258	9	,	,	PUNCT
cana-5853	258	10	then	then	ADV
cana-5853	258	11	xm	xm	PROPN
cana-5853	258	12	=	=	SYM
cana-5853	258	13	ℓp(ω	ℓp(ω	X
cana-5853	258	14	)	)	PUNCT
cana-5853	258	15	,	,	PUNCT
cana-5853	258	16	the	the	DET
cana-5853	258	17	weighted	weight	VERB
cana-5853	258	18	ℓp	ℓp	ADJ
cana-5853	258	19	space	space	NOUN
cana-5853	258	20	.	.	PUNCT
cana-5853	259	1	for	for	ADP
cana-5853	259	2	a	a	DET
cana-5853	259	3	=	=	SYM
cana-5853	259	4	diag(λn	diag(λn	PROPN
cana-5853	259	5	):	):	PUNCT
cana-5853	259	6	∥ax∥pxm	∥ax∥pxm	ADJ
cana-5853	259	7	=	=	SYM
cana-5853	259	8	∞∑	∞∑	NUM
cana-5853	259	9	n=1	n=1	PROPN
cana-5853	259	10	ωn|λnxn|p	ωn|λnxn|p	NOUN
cana-5853	259	11	.	.	PUNCT
cana-5853	259	12	boundedness	boundedness	PROPN
cana-5853	259	13	requires	require	VERB
cana-5853	259	14	supn	supn	NOUN
cana-5853	259	15	|λn|	|λn|	PROPN
cana-5853	259	16	<	<	X
cana-5853	259	17	∞.	∞.	PROPN
cana-5853	259	18	compactness	compactness	NOUN
cana-5853	259	19	requires	require	VERB
cana-5853	259	20	λn	λn	PROPN
cana-5853	259	21	→	→	SYM
cana-5853	259	22	0	0	NUM
cana-5853	259	23	,	,	PUNCT
cana-5853	259	24	yielding	yield	VERB
cana-5853	259	25	:	:	PUNCT
cana-5853	260	1	σ(a	σ(a	NUM
cana-5853	260	2	)	)	PUNCT
cana-5853	260	3	=	=	PRON
cana-5853	260	4	{	{	PUNCT
cana-5853	260	5	λn	λn	NOUN
cana-5853	260	6	}	}	PUNCT
cana-5853	260	7	with	with	ADP
cana-5853	260	8	0	0	NUM
cana-5853	260	9	as	as	ADP
cana-5853	260	10	the	the	DET
cana-5853	260	11	only	only	ADJ
cana-5853	260	12	possible	possible	ADJ
cana-5853	260	13	accumulation	accumulation	NOUN
cana-5853	260	14	point	point	NOUN
cana-5853	260	15	.	.	PUNCT
cana-5853	261	1	general	general	ADJ
cana-5853	261	2	matrix	matrix	NOUN
cana-5853	261	3	operators	operator	NOUN
cana-5853	261	4	.	.	PUNCT
cana-5853	262	1	for	for	ADP
cana-5853	262	2	more	more	ADV
cana-5853	262	3	general	general	ADJ
cana-5853	262	4	matrices	matrix	NOUN
cana-5853	262	5	a	a	DET
cana-5853	262	6	=	=	SYM
cana-5853	262	7	(	(	PUNCT
cana-5853	262	8	ank	ank	PROPN
cana-5853	262	9	)	)	PUNCT
cana-5853	262	10	,	,	PUNCT
cana-5853	262	11	spectral	spectral	ADJ
cana-5853	262	12	analysis	analysis	NOUN
cana-5853	262	13	is	be	AUX
cana-5853	262	14	subtler	subtle	ADJ
cana-5853	262	15	.	.	PUNCT
cana-5853	263	1	however	however	ADV
cana-5853	263	2	,	,	PUNCT
cana-5853	263	3	if	if	SCONJ
cana-5853	263	4	a	a	PRON
cana-5853	263	5	is	be	AUX
cana-5853	263	6	compact	compact	ADJ
cana-5853	263	7	on	on	ADP
cana-5853	263	8	xm	xm	PROPN
cana-5853	263	9	(	(	PUNCT
cana-5853	263	10	e.g.	e.g.	ADV
cana-5853	263	11	,	,	PUNCT
cana-5853	263	12	satisfying	satisfy	VERB
cana-5853	263	13	modular	modular	ADJ
cana-5853	263	14	domination	domination	NOUN
cana-5853	263	15	with	with	ADP
cana-5853	263	16	decaying	decay	VERB
cana-5853	263	17	tails	tail	NOUN
cana-5853	263	18	)	)	PUNCT
cana-5853	263	19	,	,	PUNCT
cana-5853	263	20	then	then	ADV
cana-5853	263	21	its	its	PRON
cana-5853	263	22	spectrum	spectrum	NOUN
cana-5853	263	23	is	be	AUX
cana-5853	263	24	discrete	discrete	ADJ
cana-5853	263	25	outside	outside	ADP
cana-5853	263	26	of	of	ADP
cana-5853	263	27	0	0	NUM
cana-5853	263	28	,	,	PUNCT
cana-5853	263	29	with	with	ADP
cana-5853	263	30	eigenvalues	eigenvalue	NOUN
cana-5853	263	31	converging	converge	VERB
cana-5853	263	32	to	to	ADP
cana-5853	263	33	0	0	NUM
cana-5853	263	34	.	.	PUNCT
cana-5853	264	1	this	this	DET
cana-5853	264	2	structure	structure	NOUN
cana-5853	264	3	enables	enable	VERB
cana-5853	264	4	applying	apply	VERB
cana-5853	264	5	spectral	spectral	ADJ
cana-5853	264	6	approximation	approximation	NOUN
cana-5853	264	7	,	,	PUNCT
cana-5853	264	8	regularization	regularization	NOUN
cana-5853	264	9	methods	method	NOUN
cana-5853	264	10	,	,	PUNCT
cana-5853	264	11	and	and	CCONJ
cana-5853	264	12	functional	functional	ADJ
cana-5853	264	13	calculi	calculi	NOUN
cana-5853	264	14	to	to	PART
cana-5853	264	15	solve	solve	VERB
cana-5853	264	16	operator	operator	NOUN
cana-5853	264	17	equations	equation	NOUN
cana-5853	264	18	in	in	ADP
cana-5853	264	19	xm	xm	PROPN
cana-5853	264	20	.	.	PUNCT
cana-5853	265	1	summary	summary	VERB
cana-5853	265	2	spectral	spectral	ADJ
cana-5853	265	3	theory	theory	NOUN
cana-5853	265	4	for	for	ADP
cana-5853	265	5	matrix	matrix	NOUN
cana-5853	265	6	operators	operator	NOUN
cana-5853	265	7	onxm	onxm	NOUN
cana-5853	265	8	thus	thus	ADV
cana-5853	265	9	combines	combine	VERB
cana-5853	265	10	classical	classical	ADJ
cana-5853	265	11	operator	operator	NOUN
cana-5853	265	12	-	-	PUNCT
cana-5853	265	13	theoretic	theoretic	NOUN
cana-5853	265	14	results	result	NOUN
cana-5853	265	15	with	with	ADP
cana-5853	265	16	the	the	DET
cana-5853	265	17	specific	specific	ADJ
cana-5853	265	18	structure	structure	NOUN
cana-5853	265	19	of	of	ADP
cana-5853	265	20	modulated	modulate	VERB
cana-5853	265	21	orlicz	orlicz	ADJ
cana-5853	265	22	-	-	PUNCT
cana-5853	265	23	type	type	NOUN
cana-5853	265	24	sequence	sequence	NOUN
cana-5853	265	25	spaces	space	NOUN
cana-5853	265	26	.	.	PUNCT
cana-5853	266	1	compact	compact	ADJ
cana-5853	266	2	operators	operator	NOUN
cana-5853	266	3	exhibit	exhibit	VERB
cana-5853	266	4	a	a	DET
cana-5853	266	5	spectral	spectral	ADJ
cana-5853	266	6	structure	structure	NOUN
cana-5853	266	7	dominated	dominate	VERB
cana-5853	266	8	by	by	ADP
cana-5853	266	9	eigenvalues	eigenvalue	NOUN
cana-5853	266	10	accumulating	accumulate	VERB
cana-5853	266	11	only	only	ADV
cana-5853	266	12	at	at	ADP
cana-5853	266	13	0	0	NUM
cana-5853	266	14	,	,	PUNCT
cana-5853	266	15	while	while	SCONJ
cana-5853	266	16	diagonal	diagonal	ADJ
cana-5853	266	17	operators	operator	NOUN
cana-5853	266	18	offer	offer	VERB
cana-5853	266	19	explicit	explicit	ADJ
cana-5853	266	20	eigenvalue	eigenvalue	ADJ
cana-5853	266	21	representations	representation	NOUN
cana-5853	266	22	.	.	PUNCT
cana-5853	267	1	these	these	DET
cana-5853	267	2	properties	property	NOUN
cana-5853	267	3	are	be	AUX
cana-5853	267	4	essential	essential	ADJ
cana-5853	267	5	for	for	ADP
cana-5853	267	6	deeper	deep	ADJ
cana-5853	267	7	analyses	analysis	NOUN
cana-5853	267	8	in	in	ADP
cana-5853	267	9	approximation	approximation	NOUN
cana-5853	267	10	theory	theory	NOUN
cana-5853	267	11	,	,	PUNCT
cana-5853	267	12	iterative	iterative	NOUN
cana-5853	267	13	methods	method	NOUN
cana-5853	267	14	,	,	PUNCT
cana-5853	267	15	and	and	CCONJ
cana-5853	267	16	the	the	DET
cana-5853	267	17	spectral	spectral	ADJ
cana-5853	267	18	decomposition	decomposition	NOUN
cana-5853	267	19	of	of	ADP
cana-5853	267	20	operators	operator	NOUN
cana-5853	267	21	in	in	ADP
cana-5853	267	22	functional	functional	ADJ
cana-5853	267	23	analysis	analysis	NOUN
cana-5853	267	24	.	.	PUNCT
cana-5853	268	1	7	7	NUM
cana-5853	268	2	applications	application	NOUN
cana-5853	268	3	to	to	PART
cana-5853	268	4	discrete	discrete	VERB
cana-5853	268	5	operator	operator	NOUN
cana-5853	268	6	theory	theory	NOUN
cana-5853	268	7	beyond	beyond	ADP
cana-5853	268	8	their	their	PRON
cana-5853	268	9	intrinsic	intrinsic	ADJ
cana-5853	268	10	theoretical	theoretical	ADJ
cana-5853	268	11	interest	interest	NOUN
cana-5853	268	12	,	,	PUNCT
cana-5853	268	13	matrix	matrix	NOUN
cana-5853	268	14	transformations	transformation	NOUN
cana-5853	268	15	on	on	ADP
cana-5853	268	16	modulated	modulate	VERB
cana-5853	268	17	orlicz	orlicz	ADJ
cana-5853	268	18	-	-	PUNCT
cana-5853	268	19	type	type	NOUN
cana-5853	268	20	sequence	sequence	NOUN
cana-5853	268	21	spaces	space	VERB
cana-5853	268	22	xm	xm	PROPN
cana-5853	268	23	have	have	VERB
cana-5853	268	24	meaningful	meaningful	ADJ
cana-5853	268	25	implications	implication	NOUN
cana-5853	268	26	for	for	ADP
cana-5853	268	27	discrete	discrete	ADJ
cana-5853	268	28	operator	operator	NOUN
cana-5853	268	29	theory	theory	NOUN
cana-5853	268	30	.	.	PUNCT
cana-5853	269	1	the	the	DET
cana-5853	269	2	flexible	flexible	ADJ
cana-5853	269	3	,	,	PUNCT
cana-5853	269	4	index	index	NOUN
cana-5853	269	5	-	-	PUNCT
cana-5853	269	6	dependent	dependent	ADJ
cana-5853	269	7	modular	modular	ADJ
cana-5853	269	8	framework	framework	NOUN
cana-5853	269	9	of	of	ADP
cana-5853	269	10	xm	xm	PROPN
cana-5853	269	11	naturally	naturally	ADV
cana-5853	269	12	models	model	VERB
cana-5853	269	13	situations	situation	NOUN
cana-5853	269	14	where	where	SCONJ
cana-5853	269	15	local	local	ADJ
cana-5853	269	16	properties	property	NOUN
cana-5853	269	17	vary	vary	VERB
cana-5853	269	18	across	across	ADP
cana-5853	269	19	a	a	DET
cana-5853	269	20	sequence	sequence	NOUN
cana-5853	269	21	—	—	PUNCT
cana-5853	269	22	a	a	DET
cana-5853	269	23	scenario	scenario	NOUN
cana-5853	269	24	common	common	ADJ
cana-5853	269	25	in	in	ADP
cana-5853	269	26	applied	applied	ADJ
cana-5853	269	27	mathematics	mathematic	NOUN
cana-5853	269	28	,	,	PUNCT
cana-5853	269	29	numerical	numerical	ADJ
cana-5853	269	30	analysis	analysis	NOUN
cana-5853	269	31	,	,	PUNCT
cana-5853	269	32	and	and	CCONJ
cana-5853	269	33	engineering	engineering	NOUN
cana-5853	269	34	.	.	PUNCT
cana-5853	270	1	in	in	ADP
cana-5853	270	2	this	this	DET
cana-5853	270	3	section	section	NOUN
cana-5853	270	4	,	,	PUNCT
cana-5853	270	5	we	we	PRON
cana-5853	270	6	highlight	highlight	VERB
cana-5853	270	7	three	three	NUM
cana-5853	270	8	key	key	ADJ
cana-5853	270	9	areas	area	NOUN
cana-5853	270	10	of	of	ADP
cana-5853	270	11	application	application	NOUN
cana-5853	270	12	:	:	PUNCT
cana-5853	270	13	summability	summability	NOUN
cana-5853	270	14	methods	method	NOUN
cana-5853	270	15	,	,	PUNCT
cana-5853	270	16	approximation	approximation	NOUN
cana-5853	270	17	theory	theory	NOUN
cana-5853	270	18	in	in	ADP
cana-5853	270	19	xm	xm	PROPN
cana-5853	270	20	,	,	PUNCT
cana-5853	270	21	and	and	CCONJ
cana-5853	270	22	potential	potential	ADJ
cana-5853	270	23	uses	use	NOUN
cana-5853	270	24	in	in	ADP
cana-5853	270	25	signal	signal	ADJ
cana-5853	270	26	processing	processing	NOUN
cana-5853	270	27	.	.	PUNCT
cana-5853	271	1	communications	communication	NOUN
cana-5853	271	2	on	on	ADP
cana-5853	271	3	applied	apply	VERB
cana-5853	271	4	nonlinear	nonlinear	ADJ
cana-5853	271	5	analysis	analysis	NOUN
cana-5853	271	6	issn	issn	NOUN
cana-5853	271	7	:	:	PUNCT
cana-5853	271	8	1074	1074	NUM
cana-5853	271	9	-	-	PUNCT
cana-5853	271	10	133x	133x	NUM
cana-5853	271	11	vol	vol	NOUN
cana-5853	271	12	31	31	NUM
cana-5853	271	13	no	no	NOUN
cana-5853	271	14	.	.	NOUN
cana-5853	271	15	2	2	NUM
cana-5853	271	16	(	(	PUNCT
cana-5853	271	17	2024	2024	NUM
cana-5853	271	18	)	)	PUNCT
cana-5853	271	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	271	20	476	476	NUM
cana-5853	271	21	7.1	7.1	NUM
cana-5853	271	22	summability	summability	NOUN
cana-5853	271	23	methods	method	NOUN
cana-5853	271	24	summability	summability	NOUN
cana-5853	271	25	theory	theory	NOUN
cana-5853	271	26	traditionally	traditionally	ADV
cana-5853	271	27	studies	study	VERB
cana-5853	271	28	the	the	DET
cana-5853	271	29	transformation	transformation	NOUN
cana-5853	271	30	of	of	ADP
cana-5853	271	31	divergent	divergent	ADJ
cana-5853	271	32	or	or	CCONJ
cana-5853	271	33	slowly	slowly	ADV
cana-5853	271	34	convergent	convergent	ADJ
cana-5853	271	35	series	series	NOUN
cana-5853	271	36	into	into	ADP
cana-5853	271	37	convergent	convergent	ADJ
cana-5853	271	38	ones	one	NOUN
cana-5853	271	39	via	via	ADP
cana-5853	271	40	matrix	matrix	NOUN
cana-5853	271	41	methods	method	NOUN
cana-5853	271	42	.	.	PUNCT
cana-5853	272	1	classical	classical	ADJ
cana-5853	272	2	summability	summability	NOUN
cana-5853	272	3	matrices	matrix	NOUN
cana-5853	272	4	such	such	ADJ
cana-5853	272	5	as	as	ADP
cana-5853	272	6	cesàro	cesàro	PROPN
cana-5853	272	7	,	,	PUNCT
cana-5853	272	8	hölder	hölder	NOUN
cana-5853	272	9	,	,	PUNCT
cana-5853	272	10	and	and	CCONJ
cana-5853	272	11	riesz	riesz	NOUN
cana-5853	272	12	matrices	matrix	NOUN
cana-5853	272	13	have	have	AUX
cana-5853	272	14	been	be	AUX
cana-5853	272	15	extensively	extensively	ADV
cana-5853	272	16	analyzed	analyze	VERB
cana-5853	272	17	on	on	ADP
cana-5853	272	18	ℓp	ℓp	ADJ
cana-5853	272	19	spaces	space	NOUN
cana-5853	272	20	,	,	PUNCT
cana-5853	272	21	establishing	establish	VERB
cana-5853	272	22	criteria	criterion	NOUN
cana-5853	272	23	for	for	ADP
cana-5853	272	24	regularity	regularity	NOUN
cana-5853	272	25	,	,	PUNCT
cana-5853	272	26	boundedness	boundedness	NOUN
cana-5853	272	27	,	,	PUNCT
cana-5853	272	28	and	and	CCONJ
cana-5853	272	29	equivalence	equivalence	NOUN
cana-5853	272	30	of	of	ADP
cana-5853	272	31	summability	summability	NOUN
cana-5853	272	32	methods	method	NOUN
cana-5853	272	33	.	.	PUNCT
cana-5853	273	1	in	in	ADP
cana-5853	273	2	the	the	DET
cana-5853	273	3	context	context	NOUN
cana-5853	273	4	of	of	ADP
cana-5853	273	5	xm	xm	PROPN
cana-5853	273	6	spaces	space	NOUN
cana-5853	273	7	,	,	PUNCT
cana-5853	273	8	matrix	matrix	NOUN
cana-5853	273	9	transformations	transformation	NOUN
cana-5853	273	10	generalize	generalize	VERB
cana-5853	273	11	these	these	DET
cana-5853	273	12	summability	summability	NOUN
cana-5853	273	13	methods	method	NOUN
cana-5853	273	14	to	to	PART
cana-5853	273	15	accommodate	accommodate	VERB
cana-5853	273	16	variable	variable	ADJ
cana-5853	273	17	growth	growth	NOUN
cana-5853	273	18	or	or	CCONJ
cana-5853	273	19	weighting	weight	VERB
cana-5853	273	20	across	across	ADP
cana-5853	273	21	terms	term	NOUN
cana-5853	273	22	.	.	PUNCT
cana-5853	274	1	for	for	ADP
cana-5853	274	2	example	example	NOUN
cana-5853	274	3	:	:	PUNCT
cana-5853	274	4	•	•	NUM
cana-5853	274	5	cesàro	cesàro	ADJ
cana-5853	274	6	-	-	PUNCT
cana-5853	274	7	type	type	NOUN
cana-5853	274	8	matrices	matrix	NOUN
cana-5853	274	9	on	on	ADP
cana-5853	274	10	xm	xm	PROPN
cana-5853	274	11	allow	allow	VERB
cana-5853	274	12	inhomogeneous	inhomogeneous	ADJ
cana-5853	274	13	averaging	averaging	NOUN
cana-5853	274	14	where	where	SCONJ
cana-5853	274	15	the	the	DET
cana-5853	274	16	modular	modular	ADJ
cana-5853	274	17	penalizes	penalize	NOUN
cana-5853	274	18	higher	high	ADJ
cana-5853	274	19	-	-	PUNCT
cana-5853	274	20	index	index	NOUN
cana-5853	274	21	terms	term	NOUN
cana-5853	274	22	differently	differently	ADV
cana-5853	274	23	,	,	PUNCT
cana-5853	274	24	enabling	enable	VERB
cana-5853	274	25	adaptive	adaptive	ADJ
cana-5853	274	26	smoothing	smoothing	NOUN
cana-5853	274	27	of	of	ADP
cana-5853	274	28	sequences	sequence	NOUN
cana-5853	274	29	.	.	PUNCT
cana-5853	275	1	•	•	NUM
cana-5853	275	2	weighted	weight	VERB
cana-5853	275	3	summability	summability	NOUN
cana-5853	275	4	matrices	matrix	NOUN
cana-5853	275	5	naturally	naturally	ADV
cana-5853	275	6	fit	fit	ADJ
cana-5853	275	7	into	into	ADP
cana-5853	275	8	the	the	DET
cana-5853	275	9	modular	modular	ADJ
cana-5853	275	10	setting	setting	NOUN
cana-5853	275	11	by	by	ADP
cana-5853	275	12	adjusting	adjust	VERB
cana-5853	275	13	m(n	m(n	PROPN
cana-5853	275	14	,	,	PUNCT
cana-5853	275	15	t	t	PROPN
cana-5853	275	16	)	)	PUNCT
cana-5853	275	17	to	to	PART
cana-5853	275	18	reflect	reflect	VERB
cana-5853	275	19	position	position	NOUN
cana-5853	275	20	-	-	PUNCT
cana-5853	275	21	dependent	dependent	ADJ
cana-5853	275	22	weights	weight	NOUN
cana-5853	275	23	.	.	PUNCT
cana-5853	276	1	such	such	ADJ
cana-5853	276	2	generalizations	generalization	NOUN
cana-5853	276	3	are	be	AUX
cana-5853	276	4	particularly	particularly	ADV
cana-5853	276	5	important	important	ADJ
cana-5853	276	6	in	in	ADP
cana-5853	276	7	contexts	context	NOUN
cana-5853	276	8	where	where	SCONJ
cana-5853	276	9	uniform	uniform	ADJ
cana-5853	276	10	convergence	convergence	NOUN
cana-5853	276	11	control	control	NOUN
cana-5853	276	12	is	be	AUX
cana-5853	276	13	insufficient	insufficient	ADJ
cana-5853	276	14	or	or	CCONJ
cana-5853	276	15	too	too	ADV
cana-5853	276	16	restrictive	restrictive	ADJ
cana-5853	276	17	.	.	PUNCT
cana-5853	277	1	the	the	DET
cana-5853	277	2	operator	operator	NOUN
cana-5853	277	3	theory	theory	NOUN
cana-5853	277	4	developed	develop	VERB
cana-5853	277	5	in	in	ADP
cana-5853	277	6	this	this	DET
cana-5853	277	7	paper	paper	NOUN
cana-5853	277	8	—	—	PUNCT
cana-5853	277	9	especially	especially	ADV
cana-5853	277	10	modular	modular	ADJ
cana-5853	277	11	domination	domination	NOUN
cana-5853	277	12	and	and	CCONJ
cana-5853	277	13	tail	tail	NOUN
cana-5853	277	14	conditions	condition	NOUN
cana-5853	277	15	for	for	ADP
cana-5853	277	16	compactness	compactness	NOUN
cana-5853	277	17	—	—	PUNCT
cana-5853	277	18	provides	provide	VERB
cana-5853	277	19	systematic	systematic	ADJ
cana-5853	277	20	tools	tool	NOUN
cana-5853	277	21	for	for	ADP
cana-5853	277	22	verifying	verifying	NOUN
cana-5853	277	23	when	when	SCONJ
cana-5853	277	24	these	these	DET
cana-5853	277	25	summability	summability	NOUN
cana-5853	277	26	methods	method	NOUN
cana-5853	277	27	yield	yield	VERB
cana-5853	277	28	convergent	convergent	NOUN
cana-5853	277	29	or	or	CCONJ
cana-5853	277	30	improved	improve	VERB
cana-5853	277	31	representations	representation	NOUN
cana-5853	277	32	in	in	ADP
cana-5853	277	33	xm	xm	PROPN
cana-5853	277	34	.	.	PUNCT
cana-5853	278	1	7.2	7.2	NUM
cana-5853	278	2	approximation	approximation	NOUN
cana-5853	278	3	theory	theory	NOUN
cana-5853	278	4	in	in	ADP
cana-5853	278	5	xm	xm	PROPN
cana-5853	278	6	approximation	approximation	NOUN
cana-5853	278	7	theory	theory	NOUN
cana-5853	278	8	often	often	ADV
cana-5853	278	9	deals	deal	VERB
cana-5853	278	10	with	with	ADP
cana-5853	278	11	finding	find	VERB
cana-5853	278	12	best	good	ADJ
cana-5853	278	13	approximations	approximation	NOUN
cana-5853	278	14	of	of	ADP
cana-5853	278	15	functions	function	NOUN
cana-5853	278	16	or	or	CCONJ
cana-5853	278	17	sequences	sequence	NOUN
cana-5853	278	18	using	use	VERB
cana-5853	278	19	simpler	simple	ADJ
cana-5853	278	20	or	or	CCONJ
cana-5853	278	21	structured	structured	ADJ
cana-5853	278	22	elements	element	NOUN
cana-5853	278	23	.	.	PUNCT
cana-5853	279	1	in	in	ADP
cana-5853	279	2	classical	classical	ADJ
cana-5853	279	3	sequence	sequence	NOUN
cana-5853	279	4	spaces	space	NOUN
cana-5853	279	5	,	,	PUNCT
cana-5853	279	6	this	this	PRON
cana-5853	279	7	might	might	AUX
cana-5853	279	8	involve	involve	VERB
cana-5853	279	9	projections	projection	NOUN
cana-5853	279	10	onto	onto	ADP
cana-5853	279	11	finite	finite	ADJ
cana-5853	279	12	-	-	ADJ
cana-5853	279	13	dimensional	dimensional	ADJ
cana-5853	279	14	subspaces	subspace	NOUN
cana-5853	279	15	or	or	CCONJ
cana-5853	279	16	representations	representation	NOUN
cana-5853	279	17	via	via	ADP
cana-5853	279	18	bases	basis	NOUN
cana-5853	279	19	.	.	PUNCT
cana-5853	280	1	in	in	ADP
cana-5853	280	2	xm	xm	PROPN
cana-5853	280	3	spaces	space	NOUN
cana-5853	280	4	,	,	PUNCT
cana-5853	280	5	approximation	approximation	NOUN
cana-5853	280	6	theory	theory	NOUN
cana-5853	280	7	gains	gain	VERB
cana-5853	280	8	new	new	ADJ
cana-5853	280	9	flexibility	flexibility	NOUN
cana-5853	280	10	:	:	PUNCT
cana-5853	280	11	•	•	NUM
cana-5853	280	12	modular	modular	ADJ
cana-5853	280	13	control	control	NOUN
cana-5853	280	14	allows	allow	VERB
cana-5853	280	15	penalizing	penalize	VERB
cana-5853	280	16	errors	error	NOUN
cana-5853	280	17	differently	differently	ADV
cana-5853	280	18	at	at	ADP
cana-5853	280	19	different	different	ADJ
cana-5853	280	20	indices	index	NOUN
cana-5853	280	21	,	,	PUNCT
cana-5853	280	22	accommodating	accommodate	VERB
cana-5853	280	23	non	non	ADJ
cana-5853	280	24	-	-	ADJ
cana-5853	280	25	uniform	uniform	ADJ
cana-5853	280	26	smoothness	smoothness	NOUN
cana-5853	280	27	or	or	CCONJ
cana-5853	280	28	importance	importance	NOUN
cana-5853	280	29	across	across	ADP
cana-5853	280	30	sequence	sequence	NOUN
cana-5853	280	31	entries	entry	NOUN
cana-5853	280	32	.	.	PUNCT
cana-5853	281	1	•	•	NUM
cana-5853	281	2	operator	operator	NOUN
cana-5853	281	3	-	-	PUNCT
cana-5853	281	4	theoretic	theoretic	NOUN
cana-5853	281	5	results	result	NOUN
cana-5853	281	6	on	on	ADP
cana-5853	281	7	boundedness	boundedness	NOUN
cana-5853	281	8	and	and	CCONJ
cana-5853	281	9	compactness	compactness	NOUN
cana-5853	281	10	ensure	ensure	VERB
cana-5853	281	11	the	the	DET
cana-5853	281	12	existence	existence	NOUN
cana-5853	281	13	of	of	ADP
cana-5853	281	14	best	good	ADJ
cana-5853	281	15	approximations	approximation	NOUN
cana-5853	281	16	under	under	ADP
cana-5853	281	17	modular	modular	ADJ
cana-5853	281	18	norms	norm	NOUN
cana-5853	281	19	.	.	PUNCT
cana-5853	282	1	•	•	NUM
cana-5853	282	2	diagonal	diagonal	ADJ
cana-5853	282	3	and	and	CCONJ
cana-5853	282	4	triangular	triangular	NOUN
cana-5853	282	5	operators	operator	NOUN
cana-5853	282	6	model	model	VERB
cana-5853	282	7	natural	natural	ADJ
cana-5853	282	8	approximation	approximation	NOUN
cana-5853	282	9	schemes	scheme	NOUN
cana-5853	282	10	—	—	PUNCT
cana-5853	282	11	such	such	ADJ
cana-5853	282	12	as	as	ADP
cana-5853	282	13	truncations	truncation	NOUN
cana-5853	282	14	,	,	PUNCT
cana-5853	282	15	weighted	weight	VERB
cana-5853	282	16	interpolations	interpolation	NOUN
cana-5853	282	17	,	,	PUNCT
cana-5853	282	18	or	or	CCONJ
cana-5853	282	19	adaptive	adaptive	ADJ
cana-5853	282	20	filters	filter	NOUN
cana-5853	282	21	—	—	PUNCT
cana-5853	282	22	while	while	SCONJ
cana-5853	282	23	the	the	DET
cana-5853	282	24	modular	modular	ADJ
cana-5853	282	25	structure	structure	NOUN
cana-5853	282	26	ensures	ensure	VERB
cana-5853	282	27	convergence	convergence	NOUN
cana-5853	282	28	analysis	analysis	NOUN
cana-5853	282	29	respects	respect	VERB
cana-5853	282	30	inhomogeneous	inhomogeneous	ADJ
cana-5853	282	31	conditions	condition	NOUN
cana-5853	282	32	.	.	PUNCT
cana-5853	283	1	for	for	ADP
cana-5853	283	2	example	example	NOUN
cana-5853	283	3	,	,	PUNCT
cana-5853	283	4	consider	consider	VERB
cana-5853	283	5	approximating	approximate	VERB
cana-5853	283	6	x	x	X
cana-5853	283	7	∈	∈	PROPN
cana-5853	283	8	xm	xm	PROPN
cana-5853	283	9	by	by	ADP
cana-5853	283	10	sequences	sequence	NOUN
cana-5853	283	11	with	with	ADP
cana-5853	283	12	only	only	ADV
cana-5853	283	13	finitely	finitely	ADV
cana-5853	283	14	many	many	ADJ
cana-5853	283	15	nonzero	nonzero	ADJ
cana-5853	283	16	terms	term	NOUN
cana-5853	283	17	.	.	PUNCT
cana-5853	284	1	compactness	compactness	NOUN
cana-5853	284	2	of	of	ADP
cana-5853	284	3	certain	certain	ADJ
cana-5853	284	4	matrix	matrix	NOUN
cana-5853	284	5	operators	operator	NOUN
cana-5853	284	6	guarantees	guarantee	VERB
cana-5853	284	7	that	that	SCONJ
cana-5853	284	8	such	such	ADJ
cana-5853	284	9	approximations	approximation	NOUN
cana-5853	284	10	converge	converge	VERB
cana-5853	284	11	in	in	ADP
cana-5853	284	12	the	the	DET
cana-5853	284	13	modular	modular	ADJ
cana-5853	284	14	sense	sense	NOUN
cana-5853	284	15	,	,	PUNCT
cana-5853	284	16	while	while	SCONJ
cana-5853	284	17	modular	modular	ADJ
cana-5853	284	18	inequalities	inequality	NOUN
cana-5853	284	19	allow	allow	VERB
cana-5853	284	20	precise	precise	ADJ
cana-5853	284	21	error	error	NOUN
cana-5853	284	22	bounds	bound	NOUN
cana-5853	284	23	that	that	PRON
cana-5853	284	24	reflect	reflect	VERB
cana-5853	284	25	local	local	ADJ
cana-5853	284	26	properties	property	NOUN
cana-5853	284	27	of	of	ADP
cana-5853	284	28	x.	x.	NOUN
cana-5853	284	29	7.3	7.3	NUM
cana-5853	284	30	potential	potential	ADJ
cana-5853	284	31	applications	application	NOUN
cana-5853	284	32	to	to	PART
cana-5853	284	33	signal	signal	VERB
cana-5853	284	34	processing	processing	NOUN
cana-5853	284	35	signal	signal	NOUN
cana-5853	284	36	processing	processing	NOUN
cana-5853	284	37	frequently	frequently	ADV
cana-5853	284	38	involves	involve	VERB
cana-5853	284	39	manipulating	manipulate	VERB
cana-5853	284	40	discrete	discrete	ADJ
cana-5853	284	41	signals	signal	NOUN
cana-5853	284	42	(	(	PUNCT
cana-5853	284	43	sequences	sequence	NOUN
cana-5853	284	44	)	)	PUNCT
cana-5853	284	45	via	via	ADP
cana-5853	284	46	linear	linear	ADJ
cana-5853	284	47	or	or	CCONJ
cana-5853	284	48	nonlinear	nonlinear	ADJ
cana-5853	284	49	operators	operator	NOUN
cana-5853	284	50	to	to	PART
cana-5853	284	51	achieve	achieve	VERB
cana-5853	284	52	filtering	filtering	NOUN
cana-5853	284	53	,	,	PUNCT
cana-5853	284	54	compression	compression	NOUN
cana-5853	284	55	,	,	PUNCT
cana-5853	284	56	or	or	CCONJ
cana-5853	284	57	reconstruction	reconstruction	NOUN
cana-5853	284	58	.	.	PUNCT
cana-5853	285	1	the	the	DET
cana-5853	285	2	modulated	modulate	VERB
cana-5853	285	3	orlicz	orlicz	ADJ
cana-5853	285	4	-	-	PUNCT
cana-5853	285	5	type	type	NOUN
cana-5853	285	6	sequence	sequence	NOUN
cana-5853	285	7	spaces	space	VERB
cana-5853	285	8	xm	xm	PRON
cana-5853	285	9	provide	provide	VERB
cana-5853	285	10	a	a	DET
cana-5853	285	11	natural	natural	ADJ
cana-5853	285	12	mathematical	mathematical	ADJ
cana-5853	285	13	setting	setting	NOUN
cana-5853	285	14	for	for	ADP
cana-5853	285	15	such	such	ADJ
cana-5853	285	16	tasks	task	NOUN
cana-5853	285	17	when	when	SCONJ
cana-5853	285	18	the	the	DET
cana-5853	285	19	signal	signal	NOUN
cana-5853	285	20	exhibits	exhibit	VERB
cana-5853	285	21	non	non	ADJ
cana-5853	285	22	-	-	ADJ
cana-5853	285	23	uniform	uniform	ADJ
cana-5853	285	24	characteristics	characteristic	NOUN
cana-5853	285	25	:	:	PUNCT
cana-5853	285	26	communications	communication	NOUN
cana-5853	285	27	on	on	ADP
cana-5853	285	28	applied	apply	VERB
cana-5853	285	29	nonlinear	nonlinear	ADJ
cana-5853	285	30	analysis	analysis	NOUN
cana-5853	285	31	issn	issn	NOUN
cana-5853	285	32	:	:	PUNCT
cana-5853	285	33	1074	1074	NUM
cana-5853	285	34	-	-	PUNCT
cana-5853	285	35	133x	133x	NUM
cana-5853	285	36	vol	vol	NOUN
cana-5853	285	37	31	31	NUM
cana-5853	285	38	no	no	NOUN
cana-5853	285	39	.	.	NOUN
cana-5853	285	40	2	2	NUM
cana-5853	285	41	(	(	PUNCT
cana-5853	285	42	2024	2024	NUM
cana-5853	285	43	)	)	PUNCT
cana-5853	285	44	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	285	45	477	477	NUM
cana-5853	285	46	•	•	NUM
cana-5853	285	47	adaptive	adaptive	ADJ
cana-5853	285	48	weighting	weighting	NOUN
cana-5853	285	49	:	:	PUNCT
cana-5853	285	50	by	by	ADP
cana-5853	285	51	choosing	choose	VERB
cana-5853	285	52	m(n	m(n	PROPN
cana-5853	285	53	,	,	PUNCT
cana-5853	285	54	t	t	PROPN
cana-5853	285	55	)	)	PUNCT
cana-5853	285	56	to	to	PART
cana-5853	285	57	vary	vary	VERB
cana-5853	285	58	with	with	ADP
cana-5853	285	59	n	n	CCONJ
cana-5853	285	60	,	,	PUNCT
cana-5853	285	61	xm	xm	PROPN
cana-5853	285	62	models	model	NOUN
cana-5853	285	63	situations	situation	NOUN
cana-5853	285	64	where	where	SCONJ
cana-5853	285	65	higher	high	ADJ
cana-5853	285	66	-	-	PUNCT
cana-5853	285	67	frequency	frequency	NOUN
cana-5853	285	68	components	component	NOUN
cana-5853	285	69	are	be	AUX
cana-5853	285	70	penalized	penalize	VERB
cana-5853	285	71	more	more	ADV
cana-5853	285	72	heavily	heavily	ADV
cana-5853	285	73	to	to	PART
cana-5853	285	74	enforce	enforce	VERB
cana-5853	285	75	smoothness	smoothness	NOUN
cana-5853	285	76	or	or	CCONJ
cana-5853	285	77	denoising	denoising	NOUN
cana-5853	285	78	.	.	PUNCT
cana-5853	286	1	•	•	NUM
cana-5853	286	2	non	non	ADJ
cana-5853	286	3	-	-	ADJ
cana-5853	286	4	uniform	uniform	ADJ
cana-5853	286	5	resolution	resolution	NOUN
cana-5853	286	6	:	:	PUNCT
cana-5853	286	7	sequences	sequence	NOUN
cana-5853	286	8	sampled	sample	VERB
cana-5853	286	9	on	on	ADP
cana-5853	286	10	non	non	ADJ
cana-5853	286	11	-	-	ADJ
cana-5853	286	12	uniform	uniform	ADJ
cana-5853	286	13	grids	grid	NOUN
cana-5853	286	14	can	can	AUX
cana-5853	286	15	be	be	AUX
cana-5853	286	16	effectively	effectively	ADV
cana-5853	286	17	handled	handle	VERB
cana-5853	286	18	by	by	ADP
cana-5853	286	19	modulating	modulate	VERB
cana-5853	286	20	the	the	DET
cana-5853	286	21	growth	growth	NOUN
cana-5853	286	22	conditions	condition	NOUN
cana-5853	286	23	across	across	ADP
cana-5853	286	24	indices	index	NOUN
cana-5853	286	25	.	.	PUNCT
cana-5853	287	1	•	•	NUM
cana-5853	287	2	compression	compression	NOUN
cana-5853	287	3	schemes	scheme	NOUN
cana-5853	287	4	:	:	PUNCT
cana-5853	287	5	diagonal	diagonal	ADJ
cana-5853	287	6	operators	operator	NOUN
cana-5853	287	7	with	with	ADP
cana-5853	287	8	decaying	decay	VERB
cana-5853	287	9	eigenvalues	eigenvalue	VERB
cana-5853	287	10	model	model	NOUN
cana-5853	287	11	thresholding	thresholding	NOUN
cana-5853	287	12	and	and	CCONJ
cana-5853	287	13	compression	compression	NOUN
cana-5853	287	14	,	,	PUNCT
cana-5853	287	15	with	with	ADP
cana-5853	287	16	spectral	spectral	ADJ
cana-5853	287	17	analysis	analysis	NOUN
cana-5853	287	18	ensuring	ensure	VERB
cana-5853	287	19	controlled	control	VERB
cana-5853	287	20	loss	loss	NOUN
cana-5853	287	21	of	of	ADP
cana-5853	287	22	information	information	NOUN
cana-5853	287	23	.	.	PUNCT
cana-5853	288	1	additionally	additionally	ADV
cana-5853	288	2	,	,	PUNCT
cana-5853	288	3	matrix	matrix	NOUN
cana-5853	288	4	transformations	transformation	NOUN
cana-5853	288	5	in	in	ADP
cana-5853	288	6	xm	xm	PROPN
cana-5853	288	7	can	can	AUX
cana-5853	288	8	formalize	formalize	VERB
cana-5853	288	9	common	common	ADJ
cana-5853	288	10	filtering	filtering	NOUN
cana-5853	288	11	operations	operation	NOUN
cana-5853	288	12	.	.	PUNCT
cana-5853	289	1	for	for	ADP
cana-5853	289	2	example	example	NOUN
cana-5853	289	3	,	,	PUNCT
cana-5853	289	4	cesàro	cesàro	ADJ
cana-5853	289	5	-	-	PUNCT
cana-5853	289	6	type	type	NOUN
cana-5853	289	7	matrices	matrix	NOUN
cana-5853	289	8	represent	represent	VERB
cana-5853	289	9	averaging	average	VERB
cana-5853	289	10	filters	filter	NOUN
cana-5853	289	11	whose	whose	DET
cana-5853	289	12	weights	weight	NOUN
cana-5853	289	13	can	can	AUX
cana-5853	289	14	be	be	AUX
cana-5853	289	15	adapted	adapt	VERB
cana-5853	289	16	to	to	ADP
cana-5853	289	17	local	local	ADJ
cana-5853	289	18	signal	signal	ADJ
cana-5853	289	19	behavior	behavior	NOUN
cana-5853	289	20	via	via	ADP
cana-5853	289	21	index	index	NOUN
cana-5853	289	22	-	-	PUNCT
cana-5853	289	23	dependent	dependent	ADJ
cana-5853	289	24	modulars	modular	NOUN
cana-5853	289	25	.	.	PUNCT
cana-5853	290	1	compactness	compactness	NOUN
cana-5853	290	2	criteria	criterion	NOUN
cana-5853	290	3	guarantee	guarantee	VERB
cana-5853	290	4	that	that	SCONJ
cana-5853	290	5	such	such	ADJ
cana-5853	290	6	filters	filter	NOUN
cana-5853	290	7	suppress	suppress	VERB
cana-5853	290	8	noise	noise	NOUN
cana-5853	290	9	while	while	SCONJ
cana-5853	290	10	preserving	preserve	VERB
cana-5853	290	11	essential	essential	ADJ
cana-5853	290	12	structure	structure	NOUN
cana-5853	290	13	,	,	PUNCT
cana-5853	290	14	making	make	VERB
cana-5853	290	15	them	they	PRON
cana-5853	290	16	powerful	powerful	ADJ
cana-5853	290	17	tools	tool	NOUN
cana-5853	290	18	in	in	ADP
cana-5853	290	19	denoising	denoising	NOUN
cana-5853	290	20	and	and	CCONJ
cana-5853	290	21	reconstruction	reconstruction	NOUN
cana-5853	290	22	.	.	PUNCT
cana-5853	291	1	summary	summary	VERB
cana-5853	291	2	these	these	DET
cana-5853	291	3	applications	application	NOUN
cana-5853	291	4	demonstrate	demonstrate	VERB
cana-5853	291	5	that	that	SCONJ
cana-5853	291	6	the	the	DET
cana-5853	291	7	theory	theory	NOUN
cana-5853	291	8	of	of	ADP
cana-5853	291	9	matrix	matrix	NOUN
cana-5853	291	10	transformations	transformation	NOUN
cana-5853	291	11	on	on	ADP
cana-5853	291	12	xm	xm	PROPN
cana-5853	291	13	is	be	AUX
cana-5853	291	14	not	not	PART
cana-5853	291	15	merely	merely	ADV
cana-5853	291	16	abstract	abstract	ADJ
cana-5853	291	17	but	but	CCONJ
cana-5853	291	18	connects	connect	VERB
cana-5853	291	19	directly	directly	ADV
cana-5853	291	20	to	to	ADP
cana-5853	291	21	concrete	concrete	ADJ
cana-5853	291	22	problems	problem	NOUN
cana-5853	291	23	in	in	ADP
cana-5853	291	24	analysis	analysis	NOUN
cana-5853	291	25	and	and	CCONJ
cana-5853	291	26	engineering	engineering	NOUN
cana-5853	291	27	.	.	PUNCT
cana-5853	292	1	summability	summability	NOUN
cana-5853	292	2	methods	method	NOUN
cana-5853	292	3	extend	extend	VERB
cana-5853	292	4	naturally	naturally	ADV
cana-5853	292	5	to	to	ADP
cana-5853	292	6	variable	variable	ADJ
cana-5853	292	7	-	-	PUNCT
cana-5853	292	8	weight	weight	NOUN
cana-5853	292	9	settings	setting	NOUN
cana-5853	292	10	,	,	PUNCT
cana-5853	292	11	approximation	approximation	NOUN
cana-5853	292	12	theory	theory	NOUN
cana-5853	292	13	gains	gain	VERB
cana-5853	292	14	fine	fine	ADV
cana-5853	292	15	-	-	PUNCT
cana-5853	292	16	grained	grain	VERB
cana-5853	292	17	control	control	NOUN
cana-5853	292	18	through	through	ADP
cana-5853	292	19	modular	modular	ADJ
cana-5853	292	20	norms	norm	NOUN
cana-5853	292	21	,	,	PUNCT
cana-5853	292	22	and	and	CCONJ
cana-5853	292	23	signal	signal	NOUN
cana-5853	292	24	processing	processing	NOUN
cana-5853	292	25	applications	application	NOUN
cana-5853	292	26	benefit	benefit	VERB
cana-5853	292	27	from	from	ADP
cana-5853	292	28	adaptive	adaptive	ADJ
cana-5853	292	29	modeling	modeling	NOUN
cana-5853	292	30	of	of	ADP
cana-5853	292	31	inhomogeneous	inhomogeneous	ADJ
cana-5853	292	32	data	datum	NOUN
cana-5853	292	33	.	.	PUNCT
cana-5853	293	1	together	together	ADV
cana-5853	293	2	,	,	PUNCT
cana-5853	293	3	these	these	DET
cana-5853	293	4	areas	area	NOUN
cana-5853	293	5	showcase	showcase	VERB
cana-5853	293	6	the	the	DET
cana-5853	293	7	practical	practical	ADJ
cana-5853	293	8	relevance	relevance	NOUN
cana-5853	293	9	of	of	ADP
cana-5853	293	10	the	the	DET
cana-5853	293	11	theoretical	theoretical	ADJ
cana-5853	293	12	results	result	NOUN
cana-5853	293	13	developed	develop	VERB
cana-5853	293	14	in	in	ADP
cana-5853	293	15	this	this	DET
cana-5853	293	16	paper	paper	NOUN
cana-5853	293	17	,	,	PUNCT
cana-5853	293	18	suggesting	suggest	VERB
cana-5853	293	19	a	a	DET
cana-5853	293	20	rich	rich	ADJ
cana-5853	293	21	field	field	NOUN
cana-5853	293	22	of	of	ADP
cana-5853	293	23	future	future	ADJ
cana-5853	293	24	interdisciplinary	interdisciplinary	ADJ
cana-5853	293	25	research	research	NOUN
cana-5853	293	26	.	.	PUNCT
cana-5853	294	1	references	reference	NOUN
cana-5853	294	2	[	[	X
cana-5853	294	3	1	1	X
cana-5853	294	4	]	]	PUNCT
cana-5853	294	5	e.	e.	PROPN
cana-5853	294	6	malkowsky	malkowsky	PROPN
cana-5853	294	7	and	and	CCONJ
cana-5853	294	8	v.	v.	PROPN
cana-5853	294	9	rakočević	rakočević	PROPN
cana-5853	294	10	,	,	PUNCT
cana-5853	294	11	an	an	DET
cana-5853	294	12	introduction	introduction	NOUN
cana-5853	294	13	to	to	ADP
cana-5853	294	14	sequence	sequence	NOUN
cana-5853	294	15	spaces	space	NOUN
cana-5853	294	16	and	and	CCONJ
cana-5853	294	17	measures	measure	NOUN
cana-5853	294	18	of	of	ADP
cana-5853	294	19	noncompactness	noncompactness	ADJ
cana-5853	294	20	,	,	PUNCT
cana-5853	294	21	springer	springer	NOUN
cana-5853	294	22	,	,	PUNCT
cana-5853	294	23	2017	2017	NUM
cana-5853	294	24	.	.	PUNCT
cana-5853	295	1	[	[	X
cana-5853	295	2	2	2	X
cana-5853	295	3	]	]	PUNCT
cana-5853	295	4	w.	w.	PROPN
cana-5853	295	5	rudin	rudin	PROPN
cana-5853	295	6	,	,	PUNCT
cana-5853	295	7	functional	functional	ADJ
cana-5853	295	8	analysis	analysis	NOUN
cana-5853	295	9	,	,	PUNCT
cana-5853	295	10	2nd	2nd	ADJ
cana-5853	295	11	ed	ed	NOUN
cana-5853	295	12	.	.	PROPN
cana-5853	295	13	,	,	PUNCT
cana-5853	295	14	mcgraw	mcgraw	PROPN
cana-5853	295	15	-	-	PUNCT
cana-5853	295	16	hill	hill	NOUN
cana-5853	295	17	,	,	PUNCT
cana-5853	295	18	1991	1991	NUM
cana-5853	295	19	.	.	PUNCT
cana-5853	296	1	[	[	X
cana-5853	296	2	3	3	X
cana-5853	296	3	]	]	PUNCT
cana-5853	296	4	m.	m.	NOUN
cana-5853	296	5	demiriz	demiriz	NOUN
cana-5853	296	6	,	,	PUNCT
cana-5853	296	7	generalized	generalized	ADJ
cana-5853	296	8	sequence	sequence	NOUN
cana-5853	296	9	spaces	space	NOUN
cana-5853	296	10	and	and	CCONJ
cana-5853	296	11	applications	application	NOUN
cana-5853	296	12	,	,	PUNCT
cana-5853	296	13	forthcoming	forthcoming	ADJ
cana-5853	296	14	monograph	monograph	NOUN
cana-5853	296	15	,	,	PUNCT
cana-5853	296	16	2025	2025	NUM
cana-5853	296	17	.	.	PUNCT
cana-5853	297	1	[	[	X
cana-5853	297	2	4	4	X
cana-5853	297	3	]	]	PUNCT
cana-5853	297	4	j.	j.	PROPN
cana-5853	297	5	musielak	musielak	PROPN
cana-5853	297	6	,	,	PUNCT
cana-5853	297	7	orlicz	orlicz	NOUN
cana-5853	297	8	spaces	space	NOUN
cana-5853	297	9	and	and	CCONJ
cana-5853	297	10	modular	modular	ADJ
cana-5853	297	11	spaces	space	NOUN
cana-5853	297	12	,	,	PUNCT
cana-5853	297	13	lecture	lecture	NOUN
cana-5853	297	14	notes	note	NOUN
cana-5853	297	15	in	in	ADP
cana-5853	297	16	mathematics	mathematic	NOUN
cana-5853	297	17	,	,	PUNCT
cana-5853	297	18	vol	vol	NOUN
cana-5853	297	19	.	.	NOUN
cana-5853	297	20	1034	1034	NUM
cana-5853	297	21	,	,	PUNCT
cana-5853	297	22	springer	springer	NOUN
cana-5853	297	23	,	,	PUNCT
cana-5853	297	24	1983	1983	NUM
cana-5853	297	25	.	.	PUNCT
cana-5853	298	1	[	[	X
cana-5853	298	2	5	5	X
cana-5853	298	3	]	]	PUNCT
cana-5853	298	4	m.	m.	NOUN
cana-5853	298	5	a.	a.	NOUN
cana-5853	298	6	krasnosel’skii	krasnosel’skii	PROPN
cana-5853	298	7	and	and	CCONJ
cana-5853	298	8	ya	ya	PROPN
cana-5853	298	9	.	.	PUNCT
cana-5853	298	10	b.	b.	PROPN
cana-5853	298	11	rutickii	rutickii	PROPN
cana-5853	298	12	,	,	PUNCT
cana-5853	298	13	convex	convex	NOUN
cana-5853	298	14	functions	function	NOUN
cana-5853	298	15	and	and	CCONJ
cana-5853	298	16	orlicz	orlicz	ADJ
cana-5853	298	17	spaces	space	NOUN
cana-5853	298	18	,	,	PUNCT
cana-5853	298	19	p.	p.	PROPN
cana-5853	298	20	noordhoff	noordhoff	PROPN
cana-5853	298	21	ltd	ltd	PROPN
cana-5853	298	22	.	.	PROPN
cana-5853	298	23	,	,	PUNCT
cana-5853	298	24	groningen	groningen	PROPN
cana-5853	298	25	,	,	PUNCT
cana-5853	298	26	1961	1961	NUM
cana-5853	298	27	.	.	PUNCT
cana-5853	299	1	[	[	X
cana-5853	299	2	6	6	NUM
cana-5853	299	3	]	]	PUNCT
cana-5853	299	4	r.	r.	PROPN
cana-5853	299	5	e.	e.	PROPN
cana-5853	299	6	edwards	edwards	PROPN
cana-5853	299	7	,	,	PUNCT
cana-5853	299	8	functional	functional	ADJ
cana-5853	299	9	analysis	analysis	NOUN
cana-5853	299	10	:	:	PUNCT
cana-5853	299	11	theory	theory	NOUN
cana-5853	299	12	and	and	CCONJ
cana-5853	299	13	applications	application	NOUN
cana-5853	299	14	,	,	PUNCT
cana-5853	299	15	holt	holt	PROPN
cana-5853	299	16	,	,	PUNCT
cana-5853	299	17	rinehart	rinehart	PROPN
cana-5853	299	18	and	and	CCONJ
cana-5853	299	19	winston	winston	PROPN
cana-5853	299	20	,	,	PUNCT
cana-5853	299	21	1965	1965	NUM
cana-5853	299	22	.	.	PUNCT
cana-5853	300	1	[	[	X
cana-5853	300	2	7	7	X
cana-5853	300	3	]	]	X
cana-5853	300	4	i.	i.	PROPN
cana-5853	300	5	j.	j.	PROPN
cana-5853	300	6	maddox	maddox	PROPN
cana-5853	300	7	,	,	PUNCT
cana-5853	300	8	elements	element	NOUN
cana-5853	300	9	of	of	ADP
cana-5853	300	10	functional	functional	ADJ
cana-5853	300	11	analysis	analysis	NOUN
cana-5853	300	12	,	,	PUNCT
cana-5853	300	13	cambridge	cambridge	PROPN
cana-5853	300	14	university	university	PROPN
cana-5853	300	15	press	press	NOUN
cana-5853	300	16	,	,	PUNCT
cana-5853	300	17	1969	1969	NUM
cana-5853	300	18	.	.	PUNCT
cana-5853	301	1	communications	communication	NOUN
cana-5853	301	2	on	on	ADP
cana-5853	301	3	applied	apply	VERB
cana-5853	301	4	nonlinear	nonlinear	ADJ
cana-5853	301	5	analysis	analysis	NOUN
cana-5853	301	6	issn	issn	NOUN
cana-5853	301	7	:	:	PUNCT
cana-5853	301	8	1074	1074	NUM
cana-5853	301	9	-	-	PUNCT
cana-5853	301	10	133x	133x	NUM
cana-5853	301	11	vol	vol	NOUN
cana-5853	301	12	31	31	NUM
cana-5853	301	13	no	no	NOUN
cana-5853	301	14	.	.	NOUN
cana-5853	301	15	2	2	NUM
cana-5853	301	16	(	(	PUNCT
cana-5853	301	17	2024	2024	NUM
cana-5853	301	18	)	)	PUNCT
cana-5853	301	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	301	20	478	478	NUM
cana-5853	302	1	[	[	X
cana-5853	302	2	8	8	NUM
cana-5853	302	3	]	]	PUNCT
cana-5853	302	4	m.	m.	NOUN
cana-5853	302	5	altun	altun	NOUN
cana-5853	302	6	and	and	CCONJ
cana-5853	302	7	f.	f.	PROPN
cana-5853	302	8	başar	başar	PROPN
cana-5853	302	9	,	,	PUNCT
cana-5853	302	10	matrix	matrix	NOUN
cana-5853	302	11	domains	domain	NOUN
cana-5853	302	12	of	of	ADP
cana-5853	302	13	orlicz	orlicz	ADJ
cana-5853	302	14	sequence	sequence	NOUN
cana-5853	302	15	spaces	space	NOUN
cana-5853	302	16	,	,	PUNCT
cana-5853	302	17	journal	journal	NOUN
cana-5853	302	18	of	of	ADP
cana-5853	302	19	mathematical	mathematical	ADJ
cana-5853	302	20	analysis	analysis	NOUN
cana-5853	302	21	and	and	CCONJ
cana-5853	302	22	applications	application	NOUN
cana-5853	302	23	,	,	PUNCT
cana-5853	302	24	2007	2007	NUM
cana-5853	302	25	.	.	PUNCT
cana-5853	303	1	[	[	X
cana-5853	303	2	9	9	NUM
cana-5853	303	3	]	]	X
cana-5853	303	4	h.	h.	PROPN
cana-5853	303	5	kizmaz	kizmaz	PROPN
cana-5853	303	6	,	,	PUNCT
cana-5853	303	7	on	on	ADP
cana-5853	303	8	certain	certain	ADJ
cana-5853	303	9	sequence	sequence	NOUN
cana-5853	303	10	spaces	space	NOUN
cana-5853	303	11	,	,	PUNCT
cana-5853	303	12	canad	canad	PROPN
cana-5853	303	13	.	.	PUNCT
cana-5853	304	1	math	math	NOUN
cana-5853	304	2	.	.	PUNCT
cana-5853	305	1	bull	bull	PROPN
cana-5853	305	2	.	.	PUNCT
cana-5853	305	3	,	,	PUNCT
cana-5853	305	4	25(1982	25(1982	NUM
cana-5853	305	5	)	)	PUNCT
cana-5853	305	6	,	,	PUNCT
cana-5853	305	7	151–158	151–158	NUM
cana-5853	305	8	.	.	PUNCT
cana-5853	306	1	[	[	X
cana-5853	306	2	10	10	NUM
cana-5853	306	3	]	]	PUNCT
cana-5853	306	4	m.	m.	NOUN
cana-5853	306	5	demiriz	demiriz	NOUN
cana-5853	306	6	,	,	PUNCT
cana-5853	306	7	generalized	generalized	ADJ
cana-5853	306	8	sequence	sequence	NOUN
cana-5853	306	9	spaces	space	NOUN
cana-5853	306	10	and	and	CCONJ
cana-5853	306	11	applications	application	NOUN
cana-5853	306	12	,	,	PUNCT
cana-5853	306	13	forthcoming	forthcoming	ADJ
cana-5853	306	14	monograph	monograph	NOUN
cana-5853	306	15	,	,	PUNCT
cana-5853	306	16	.	.	PUNCT
cana-5853	307	1	[	[	X
cana-5853	307	2	11	11	NUM
cana-5853	307	3	]	]	X
cana-5853	307	4	c.	c.	PROPN
cana-5853	307	5	w.	w.	PROPN
cana-5853	307	6	groetsch	groetsch	PROPN
cana-5853	307	7	,	,	PUNCT
cana-5853	307	8	spectral	spectral	ADJ
cana-5853	307	9	theory	theory	NOUN
cana-5853	307	10	:	:	PUNCT
cana-5853	307	11	a	a	DET
cana-5853	307	12	first	first	ADJ
cana-5853	307	13	course	course	NOUN
cana-5853	307	14	,	,	PUNCT
cana-5853	307	15	marcel	marcel	PROPN
cana-5853	307	16	dekker	dekker	PROPN
cana-5853	307	17	,	,	PUNCT
cana-5853	307	18	1984	1984	NUM
cana-5853	307	19	.	.	PUNCT
cana-5853	308	1	[	[	X
cana-5853	308	2	12	12	NUM
cana-5853	308	3	]	]	X
cana-5853	308	4	c.	c.	PROPN
cana-5853	308	5	bennett	bennett	PROPN
cana-5853	308	6	and	and	CCONJ
cana-5853	308	7	r.	r.	PROPN
cana-5853	308	8	sharpley	sharpley	PROPN
cana-5853	308	9	,	,	PUNCT
cana-5853	308	10	interpolation	interpolation	NOUN
cana-5853	308	11	of	of	ADP
cana-5853	308	12	operators	operator	NOUN
cana-5853	308	13	,	,	PUNCT
cana-5853	308	14	academic	academic	ADJ
cana-5853	308	15	press	press	NOUN
cana-5853	308	16	,	,	PUNCT
cana-5853	308	17	1988	1988	NUM
cana-5853	308	18	.	.	PUNCT
cana-5853	309	1	[	[	X
cana-5853	309	2	13	13	NUM
cana-5853	309	3	]	]	X
cana-5853	309	4	g.	g.	PROPN
cana-5853	309	5	h.	h.	PROPN
cana-5853	309	6	hardy	hardy	PROPN
cana-5853	309	7	,	,	PUNCT
cana-5853	309	8	j.	j.	PROPN
cana-5853	309	9	e.	e.	PROPN
cana-5853	309	10	littlewood	littlewood	PROPN
cana-5853	309	11	,	,	PUNCT
cana-5853	309	12	and	and	CCONJ
cana-5853	309	13	g.	g.	PROPN
cana-5853	309	14	pólya	pólya	PROPN
cana-5853	309	15	,	,	PUNCT
cana-5853	309	16	inequalities	inequality	NOUN
cana-5853	309	17	,	,	PUNCT
cana-5853	309	18	cambridge	cambridge	PROPN
cana-5853	309	19	university	university	PROPN
cana-5853	309	20	press	press	NOUN
cana-5853	309	21	,	,	PUNCT
cana-5853	309	22	1934	1934	NUM
cana-5853	309	23	.	.	PUNCT
cana-5853	310	1	[	[	X
cana-5853	310	2	14	14	NUM
cana-5853	310	3	]	]	PUNCT
cana-5853	310	4	a.	a.	PROPN
cana-5853	310	5	c.	c.	PROPN
cana-5853	310	6	zaanen	zaanen	PROPN
cana-5853	310	7	,	,	PUNCT
cana-5853	310	8	linear	linear	VERB
cana-5853	310	9	analysis	analysis	NOUN
cana-5853	310	10	,	,	PUNCT
cana-5853	310	11	north	north	NOUN
cana-5853	310	12	-	-	PUNCT
cana-5853	310	13	holland	holland	NOUN
cana-5853	310	14	,	,	PUNCT
cana-5853	310	15	1957	1957	NUM
cana-5853	310	16	.	.	PUNCT
cana-5853	311	1	[	[	X
cana-5853	311	2	15	15	NUM
cana-5853	311	3	]	]	X
cana-5853	311	4	l.	l.	PROPN
cana-5853	311	5	v.	v.	CCONJ
cana-5853	311	6	kantorovich	kantorovich	PROPN
cana-5853	311	7	and	and	CCONJ
cana-5853	311	8	g.	g.	PROPN
cana-5853	311	9	p.	p.	PROPN
cana-5853	311	10	akilov	akilov	PROPN
cana-5853	311	11	,	,	PUNCT
cana-5853	311	12	functional	functional	ADJ
cana-5853	311	13	analysis	analysis	NOUN
cana-5853	311	14	,	,	PUNCT
cana-5853	311	15	2nd	2nd	ADJ
cana-5853	311	16	ed	ed	NOUN
cana-5853	311	17	.	.	PROPN
cana-5853	311	18	,	,	PUNCT
cana-5853	311	19	pergamon	pergamon	PROPN
cana-5853	311	20	press	press	PROPN
cana-5853	311	21	,	,	PUNCT
cana-5853	311	22	1984	1984	NUM
cana-5853	311	23	.	.	PUNCT
cana-5853	312	1	[	[	X
cana-5853	312	2	16	16	NUM
cana-5853	312	3	]	]	PUNCT
cana-5853	312	4	l.	l.	PROPN
cana-5853	312	5	schwartz	schwartz	PROPN
cana-5853	312	6	,	,	PUNCT
cana-5853	312	7	mathematics	mathematic	NOUN
cana-5853	312	8	for	for	ADP
cana-5853	312	9	the	the	DET
cana-5853	312	10	physical	physical	ADJ
cana-5853	312	11	sciences	sciences	PROPN
cana-5853	312	12	,	,	PUNCT
cana-5853	312	13	hermann	hermann	PROPN
cana-5853	312	14	,	,	PUNCT
cana-5853	312	15	paris	paris	PROPN
cana-5853	312	16	,	,	PUNCT
cana-5853	312	17	1964	1964	NUM
cana-5853	312	18	.	.	PUNCT
cana-5853	313	1	[	[	X
cana-5853	313	2	17	17	NUM
cana-5853	313	3	]	]	X
cana-5853	313	4	w.	w.	PROPN
cana-5853	313	5	h.	h.	PROPN
cana-5853	313	6	young	young	PROPN
cana-5853	313	7	,	,	PUNCT
cana-5853	313	8	on	on	ADP
cana-5853	313	9	classes	class	NOUN
cana-5853	313	10	of	of	ADP
cana-5853	313	11	summable	summable	ADJ
cana-5853	313	12	series	series	NOUN
cana-5853	313	13	and	and	CCONJ
cana-5853	313	14	their	their	PRON
cana-5853	313	15	properties	property	NOUN
cana-5853	313	16	,	,	PUNCT
cana-5853	313	17	proc	proc	NOUN
cana-5853	313	18	.	.	PUNCT
cana-5853	314	1	lond	lond	PROPN
cana-5853	314	2	.	.	PUNCT
cana-5853	315	1	math	math	NOUN
cana-5853	315	2	.	.	PUNCT
cana-5853	316	1	soc	soc	PROPN
cana-5853	316	2	.	.	PUNCT
cana-5853	316	3	,	,	PUNCT
cana-5853	316	4	1912	1912	NUM
cana-5853	316	5	.	.	PUNCT
cana-5853	317	1	[	[	X
cana-5853	317	2	18	18	NUM
cana-5853	317	3	]	]	PUNCT
cana-5853	317	4	m.	m.	NOUN
cana-5853	317	5	z.	z.	PROPN
cana-5853	317	6	nashed	nashe	VERB
cana-5853	317	7	(	(	PUNCT
cana-5853	317	8	ed	ed	NOUN
cana-5853	317	9	.	.	PUNCT
cana-5853	317	10	)	)	PUNCT
cana-5853	318	1	,	,	PUNCT
cana-5853	318	2	generalized	generalize	VERB
cana-5853	318	3	inverses	inverse	NOUN
cana-5853	318	4	and	and	CCONJ
cana-5853	318	5	applications	application	NOUN
cana-5853	318	6	,	,	PUNCT
cana-5853	318	7	academic	academic	ADJ
cana-5853	318	8	press	press	NOUN
cana-5853	318	9	,	,	PUNCT
cana-5853	318	10	1976	1976	NUM
cana-5853	318	11	.	.	PUNCT
cana-5853	319	1	communications	communication	NOUN
cana-5853	319	2	on	on	ADP
cana-5853	319	3	applied	apply	VERB
cana-5853	319	4	nonlinear	nonlinear	ADJ
cana-5853	319	5	analysis	analysis	NOUN
cana-5853	319	6	issn	issn	NOUN
cana-5853	319	7	:	:	PUNCT
cana-5853	319	8	1074	1074	NUM
cana-5853	319	9	-	-	PUNCT
cana-5853	319	10	133x	133x	NUM
cana-5853	319	11	vol	vol	NOUN
cana-5853	319	12	31	31	NUM
cana-5853	319	13	no	no	NOUN
cana-5853	319	14	.	.	NOUN
cana-5853	319	15	2	2	NUM
cana-5853	319	16	(	(	PUNCT
cana-5853	319	17	2024	2024	NUM
cana-5853	319	18	)	)	PUNCT
cana-5853	319	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-5853	319	20	479	479	NUM
cana-5853	319	21	introduction	introduction	NOUN
cana-5853	319	22	preliminaries	preliminary	NOUN
cana-5853	319	23	modular	modular	ADJ
cana-5853	319	24	functions	function	NOUN
cana-5853	319	25	m(n	m(n	PROPN
cana-5853	319	26	,	,	PUNCT
cana-5853	319	27	t	t	PROPN
cana-5853	319	28	)	)	PUNCT
cana-5853	319	29	the	the	DET
cana-5853	319	30	space	space	NOUN
cana-5853	319	31	xm	xm	PROPN
cana-5853	319	32	and	and	CCONJ
cana-5853	319	33	its	its	PRON
cana-5853	319	34	norm	norm	NOUN
cana-5853	319	35	/	/	SYM
cana-5853	319	36	modular	modular	ADJ
cana-5853	319	37	complementary	complementary	ADJ
cana-5853	319	38	modular	modular	ADJ
cana-5853	319	39	functions	function	NOUN
cana-5853	319	40	m*(n	m*(n	ADJ
cana-5853	319	41	,	,	PUNCT
cana-5853	319	42	y	y	PROPN
cana-5853	319	43	)	)	PUNCT
cana-5853	319	44	examples	example	NOUN
cana-5853	319	45	bounded	bound	VERB
cana-5853	319	46	linear	linear	PROPN
cana-5853	319	47	operators	operator	NOUN
cana-5853	319	48	on	on	ADP
cana-5853	319	49	xm	xm	PROPN
cana-5853	319	50	definition	definition	NOUN
cana-5853	319	51	and	and	CCONJ
cana-5853	319	52	general	general	ADJ
cana-5853	319	53	criteria	criterion	NOUN
cana-5853	319	54	matrix	matrix	NOUN
cana-5853	319	55	transformations	transformation	NOUN
cana-5853	319	56	as	as	ADP
cana-5853	319	57	operators	operator	NOUN
cana-5853	319	58	conditions	condition	NOUN
cana-5853	319	59	for	for	ADP
cana-5853	319	60	boundedness	boundedness	NOUN
cana-5853	319	61	young	young	ADJ
cana-5853	319	62	-	-	PUNCT
cana-5853	319	63	type	type	NOUN
cana-5853	319	64	inequalities	inequality	NOUN
cana-5853	319	65	in	in	ADP
cana-5853	319	66	the	the	DET
cana-5853	319	67	modular	modular	ADJ
cana-5853	319	68	setting	set	VERB
cana-5853	319	69	diagonal	diagonal	ADJ
cana-5853	319	70	and	and	CCONJ
cana-5853	319	71	triangular	triangular	NOUN
cana-5853	319	72	matrices	matrix	NOUN
cana-5853	319	73	boundedness	boundedness	NOUN
cana-5853	319	74	criteria	criterion	NOUN
cana-5853	319	75	compactness	compactness	NOUN
cana-5853	319	76	characterizations	characterization	NOUN
cana-5853	319	77	cesàro	cesàro	NOUN
cana-5853	319	78	-	-	PUNCT
cana-5853	319	79	type	type	NOUN
cana-5853	319	80	and	and	CCONJ
cana-5853	319	81	summability	summability	NOUN
cana-5853	319	82	matrices	matrix	NOUN
cana-5853	319	83	generalizations	generalization	NOUN
cana-5853	319	84	of	of	ADP
cana-5853	319	85	classical	classical	ADJ
cana-5853	319	86	cesàro	cesàro	PROPN
cana-5853	319	87	matrices	matrix	NOUN
cana-5853	319	88	boundedness	boundedness	NOUN
cana-5853	319	89	in	in	ADP
cana-5853	319	90	xm	xm	PROPN
cana-5853	319	91	compactness	compactness	NOUN
cana-5853	319	92	analysis	analysis	NOUN
cana-5853	319	93	spectral	spectral	ADJ
cana-5853	319	94	properties	property	NOUN
cana-5853	319	95	of	of	ADP
cana-5853	319	96	matrix	matrix	NOUN
cana-5853	319	97	operators	operator	NOUN
cana-5853	319	98	spectral	spectral	ADJ
cana-5853	319	99	theory	theory	NOUN
cana-5853	319	100	in	in	ADP
cana-5853	319	101	sequence	sequence	NOUN
cana-5853	319	102	spaces	space	NOUN
cana-5853	319	103	compact	compact	ADJ
cana-5853	319	104	operator	operator	NOUN
cana-5853	319	105	spectra	spectra	NOUN
cana-5853	319	106	in	in	ADP
cana-5853	319	107	xm	xm	PROPN
cana-5853	319	108	diagonal	diagonal	ADJ
cana-5853	319	109	operators	operator	NOUN
cana-5853	319	110	and	and	CCONJ
cana-5853	319	111	eigenvalue	eigenvalue	VERB
cana-5853	319	112	analysis	analysis	NOUN
cana-5853	319	113	applications	application	NOUN
cana-5853	319	114	to	to	PART
cana-5853	319	115	discrete	discrete	VERB
cana-5853	319	116	operator	operator	NOUN
cana-5853	319	117	theory	theory	NOUN
cana-5853	319	118	summability	summability	NOUN
cana-5853	319	119	methods	method	NOUN
cana-5853	319	120	approximation	approximation	NOUN
cana-5853	319	121	theory	theory	NOUN
cana-5853	319	122	in	in	ADP
cana-5853	319	123	xm	xm	PROPN
cana-5853	319	124	potential	potential	ADJ
cana-5853	319	125	applications	application	NOUN
cana-5853	319	126	to	to	PART
cana-5853	319	127	signal	signal	VERB
cana-5853	319	128	processing	processing	NOUN