id	sid	tid	token	lemma	pos
cana-2208	1	1	communications	communication	NOUN
cana-2208	1	2	on	on	ADP
cana-2208	1	3	applied	apply	VERB
cana-2208	1	4	nonlinear	nonlinear	ADJ
cana-2208	1	5	analysis	analysis	NOUN
cana-2208	1	6	issn	issn	NOUN
cana-2208	1	7	:	:	PUNCT
cana-2208	1	8	1074	1074	NUM
cana-2208	1	9	-	-	PUNCT
cana-2208	1	10	133x	133x	NUM
cana-2208	1	11	vol	vol	NOUN
cana-2208	1	12	32	32	NUM
cana-2208	1	13	no	no	NOUN
cana-2208	1	14	.	.	PUNCT
cana-2208	2	1	1s	1s	NUM
cana-2208	2	2	(	(	PUNCT
cana-2208	2	3	2025	2025	NUM
cana-2208	2	4	)	)	PUNCT
cana-2208	2	5	445	445	NUM
cana-2208	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	3	2	approximation	approximation	NOUN
cana-2208	3	3	of	of	ADP
cana-2208	3	4	functions	function	NOUN
cana-2208	3	5	by	by	ADP
cana-2208	3	6	𝒏-euler	𝒏-euler	NOUN
cana-2208	3	7	product	product	NOUN
cana-2208	3	8	means	mean	NOUN
cana-2208	3	9	of	of	ADP
cana-2208	3	10	fourier	fourier	ADJ
cana-2208	3	11	series	series	NOUN
cana-2208	3	12	and	and	CCONJ
cana-2208	3	13	its	its	PRON
cana-2208	3	14	conjugate	conjugate	ADJ
cana-2208	3	15	md	md	PROPN
cana-2208	3	16	hadish	hadish	VERB
cana-2208	3	17	𝟏	𝟏	PROPN
cana-2208	3	18	,	,	PUNCT
cana-2208	3	19	rupesh	rupesh	PROPN
cana-2208	3	20	kumar	kumar	PROPN
cana-2208	3	21	mishra	mishra	PROPN
cana-2208	3	22	𝟐	𝟐	PROPN
cana-2208	3	23	and	and	CCONJ
cana-2208	3	24	shambhu	shambhu	VERB
cana-2208	3	25	kumar	kumar	PROPN
cana-2208	3	26	mishra	mishra	PROPN
cana-2208	3	27	𝟑	𝟑	PROPN
cana-2208	3	28	1assistant	1assistant	PROPN
cana-2208	3	29	professor	professor	NOUN
cana-2208	3	30	(	(	PUNCT
cana-2208	3	31	mathematics	mathematics	PROPN
cana-2208	3	32	)	)	PUNCT
cana-2208	3	33	,	,	PUNCT
cana-2208	3	34	department	department	NOUN
cana-2208	3	35	of	of	ADP
cana-2208	3	36	applied	apply	VERB
cana-2208	3	37	science	science	NOUN
cana-2208	3	38	and	and	CCONJ
cana-2208	3	39	humanities	humanity	NOUN
cana-2208	3	40	government	government	NOUN
cana-2208	3	41	engineering	engineering	PROPN
cana-2208	3	42	college	college	PROPN
cana-2208	3	43	west	west	PROPN
cana-2208	3	44	champaran	champaran	PROPN
cana-2208	3	45	,	,	PUNCT
cana-2208	3	46	bihar	bihar	PROPN
cana-2208	3	47	,	,	PUNCT
cana-2208	3	48	india	india	PROPN
cana-2208	3	49	2research	2research	NUM
cana-2208	3	50	scholar	scholar	NOUN
cana-2208	3	51	,	,	PUNCT
cana-2208	3	52	department	department	NOUN
cana-2208	3	53	of	of	ADP
cana-2208	3	54	mathematics	mathematics	PROPN
cana-2208	3	55	,	,	PUNCT
cana-2208	3	56	patliputra	patliputra	PROPN
cana-2208	3	57	university	university	NOUN
cana-2208	3	58	patna	patna	PROPN
cana-2208	3	59	bihar	bihar	PROPN
cana-2208	3	60	,	,	PUNCT
cana-2208	3	61	india	india	PROPN
cana-2208	3	62	.	.	PUNCT
cana-2208	4	1	3professor	3professor	NUM
cana-2208	4	2	,	,	PUNCT
cana-2208	4	3	department	department	NOUN
cana-2208	4	4	of	of	ADP
cana-2208	4	5	mathematics	mathematics	PROPN
cana-2208	4	6	,	,	PUNCT
cana-2208	4	7	patliputra	patliputra	PROPN
cana-2208	4	8	university	university	NOUN
cana-2208	4	9	patna	patna	PROPN
cana-2208	4	10	bihar	bihar	PROPN
cana-2208	4	11	,	,	PUNCT
cana-2208	4	12	india	india	PROPN
cana-2208	4	13	e	e	PROPN
cana-2208	4	14	-	-	NOUN
cana-2208	4	15	mail	mail	NOUN
cana-2208	4	16	:	:	PUNCT
cana-2208	4	17	1h.ahmad043@gmail.com	1h.ahmad043@gmail.com	NUM
cana-2208	4	18	,	,	PUNCT
cana-2208	4	19	2rupeshmishra043@gmail.com	2rupeshmishra043@gmail.com	NUM
cana-2208	4	20	,	,	PUNCT
cana-2208	4	21	3shambhumishra5@gmail.com	3shambhumishra5@gmail.com	NUM
cana-2208	4	22	article	article	NOUN
cana-2208	4	23	history	history	NOUN
cana-2208	4	24	:	:	PUNCT
cana-2208	4	25	received	receive	VERB
cana-2208	4	26	:	:	PUNCT
cana-2208	4	27	22	22	NUM
cana-2208	4	28	-	-	SYM
cana-2208	4	29	08	08	NUM
cana-2208	4	30	-	-	PUNCT
cana-2208	4	31	2024	2024	NUM
cana-2208	4	32	revised	revise	VERB
cana-2208	4	33	:	:	PUNCT
cana-2208	4	34	07	07	NUM
cana-2208	4	35	-	-	SYM
cana-2208	4	36	10	10	NUM
cana-2208	4	37	-	-	PUNCT
cana-2208	4	38	2024	2024	NUM
cana-2208	4	39	accepted	accept	VERB
cana-2208	4	40	:	:	PUNCT
cana-2208	4	41	23	23	NUM
cana-2208	4	42	-	-	SYM
cana-2208	4	43	10	10	NUM
cana-2208	4	44	-	-	PUNCT
cana-2208	4	45	2024	2024	NUM
cana-2208	4	46	abstract	abstract	NOUN
cana-2208	4	47	:	:	PUNCT
cana-2208	4	48	our	our	PRON
cana-2208	4	49	research	research	NOUN
cana-2208	4	50	in	in	ADP
cana-2208	4	51	this	this	DET
cana-2208	4	52	work	work	NOUN
cana-2208	4	53	produced	produce	VERB
cana-2208	4	54	new	new	ADJ
cana-2208	4	55	theorems	theorem	NOUN
cana-2208	4	56	on	on	ADP
cana-2208	4	57	approximation	approximation	NOUN
cana-2208	4	58	of	of	ADP
cana-2208	4	59	functions	function	NOUN
cana-2208	4	60	by	by	ADP
cana-2208	4	61	neulers	neuler	NOUN
cana-2208	4	62	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	4	63	product	product	NOUN
cana-2208	4	64	means	mean	VERB
cana-2208	4	65	of	of	ADP
cana-2208	4	66	fourier	fourier	ADJ
cana-2208	4	67	series	series	NOUN
cana-2208	4	68	and	and	CCONJ
cana-2208	4	69	its	its	PRON
cana-2208	4	70	conjugate	conjugate	ADJ
cana-2208	4	71	fourier	fourier	NOUN
cana-2208	4	72	series	series	NOUN
cana-2208	4	73	.	.	PUNCT
cana-2208	5	1	this	this	DET
cana-2208	5	2	new	new	ADJ
cana-2208	5	3	results	result	NOUN
cana-2208	5	4	generalize	generalize	VERB
cana-2208	5	5	the	the	DET
cana-2208	5	6	results	result	NOUN
cana-2208	5	7	of	of	ADP
cana-2208	5	8	[	[	X
cana-2208	5	9	5	5	NUM
cana-2208	5	10	]	]	PUNCT
cana-2208	5	11	,	,	PUNCT
cana-2208	5	12	[	[	X
cana-2208	5	13	7	7	X
cana-2208	5	14	]	]	PUNCT
cana-2208	5	15	and	and	CCONJ
cana-2208	5	16	[	[	X
cana-2208	5	17	9	9	NUM
cana-2208	5	18	]	]	PUNCT
cana-2208	5	19	.	.	PUNCT
cana-2208	6	1	key	key	ADJ
cana-2208	6	2	words	word	NOUN
cana-2208	6	3	:	:	PUNCT
cana-2208	6	4	approximation	approximation	NOUN
cana-2208	6	5	of	of	ADP
cana-2208	6	6	function	function	NOUN
cana-2208	6	7	,	,	PUNCT
cana-2208	6	8	fourier	fourier	NOUN
cana-2208	6	9	series	series	NOUN
cana-2208	6	10	,	,	PUNCT
cana-2208	6	11	conjugate	conjugate	ADJ
cana-2208	6	12	fourier	fouri	ADJ
cana-2208	6	13	series	series	NOUN
cana-2208	6	14	,	,	PUNCT
cana-2208	6	15	𝐿𝑝	𝐿𝑝	ADP
cana-2208	6	16	space	space	NOUN
cana-2208	6	17	,	,	PUNCT
cana-2208	6	18	lebesgue	lebesgue	NOUN
cana-2208	6	19	integral	integral	ADJ
cana-2208	6	20	.	.	PUNCT
cana-2208	7	1	1	1	NUM
cana-2208	7	2	introduction	introduction	NOUN
cana-2208	7	3	estimating	estimate	VERB
cana-2208	7	4	functions	function	NOUN
cana-2208	7	5	using	use	VERB
cana-2208	7	6	generalized	generalized	ADJ
cana-2208	7	7	fourier	fourier	NOUN
cana-2208	7	8	series	series	NOUN
cana-2208	7	9	based	base	VERB
cana-2208	7	10	on	on	ADP
cana-2208	7	11	trigonometric	trigonometric	ADJ
cana-2208	7	12	polynomials	polynomial	NOUN
cana-2208	7	13	has	have	AUX
cana-2208	7	14	become	become	VERB
cana-2208	7	15	increasingly	increasingly	ADV
cana-2208	7	16	important	important	ADJ
cana-2208	7	17	in	in	ADP
cana-2208	7	18	the	the	DET
cana-2208	7	19	development	development	NOUN
cana-2208	7	20	of	of	ADP
cana-2208	7	21	mathematics	mathematic	NOUN
cana-2208	7	22	and	and	CCONJ
cana-2208	7	23	engineering	engineering	NOUN
cana-2208	7	24	fields	field	NOUN
cana-2208	7	25	in	in	ADP
cana-2208	7	26	recent	recent	ADJ
cana-2208	7	27	years	year	NOUN
cana-2208	7	28	.	.	PUNCT
cana-2208	8	1	for	for	ADP
cana-2208	8	2	instance	instance	NOUN
cana-2208	8	3	,	,	PUNCT
cana-2208	8	4	a	a	DET
cana-2208	8	5	new	new	ADJ
cana-2208	8	6	𝐿2	𝐿2	NOUN
cana-2208	8	7	-	-	PUNCT
cana-2208	8	8	bassed	basse	VERB
cana-2208	8	9	technique	technique	NOUN
cana-2208	8	10	for	for	ADP
cana-2208	8	11	creating	create	VERB
cana-2208	8	12	finit	finit	ADJ
cana-2208	8	13	impulse	impulse	ADJ
cana-2208	8	14	response	response	NOUN
cana-2208	8	15	digital	digital	ADJ
cana-2208	8	16	filters	filter	NOUN
cana-2208	8	17	to	to	PART
cana-2208	8	18	obtain	obtain	VERB
cana-2208	8	19	an	an	DET
cana-2208	8	20	optimal	optimal	ADJ
cana-2208	8	21	approximation	approximation	NOUN
cana-2208	8	22	was	be	AUX
cana-2208	8	23	devised	devise	VERB
cana-2208	8	24	by	by	ADP
cana-2208	8	25	psarakis	psaraki	NOUN
cana-2208	8	26	and	and	CCONJ
cana-2208	8	27	moustakieds	moustakied	VERB
cana-2208	8	28	[	[	X
cana-2208	8	29	2	2	NUM
cana-2208	8	30	]	]	PUNCT
cana-2208	8	31	,	,	PUNCT
cana-2208	8	32	utilising	utilise	VERB
cana-2208	8	33	the	the	DET
cana-2208	8	34	features	feature	NOUN
cana-2208	8	35	of	of	ADP
cana-2208	8	36	approximation	approximation	NOUN
cana-2208	8	37	of	of	ADP
cana-2208	8	38	functions	function	NOUN
cana-2208	8	39	.	.	PUNCT
cana-2208	9	1	in	in	ADP
cana-2208	9	2	the	the	DET
cana-2208	9	3	development	development	NOUN
cana-2208	9	4	of	of	ADP
cana-2208	9	5	digital	digital	ADJ
cana-2208	9	6	filters	filter	NOUN
cana-2208	9	7	,	,	PUNCT
cana-2208	9	8	𝐿𝑝-space	𝐿𝑝-space	SYM
cana-2208	9	9	,	,	PUNCT
cana-2208	9	10	𝐿2	𝐿2	ADJ
cana-2208	9	11	-	-	PUNCT
cana-2208	9	12	space	space	NOUN
cana-2208	9	13	,	,	PUNCT
cana-2208	9	14	and	and	CCONJ
cana-2208	9	15	𝐿∞-space	𝐿∞-space	PROPN
cana-2208	9	16	are	be	AUX
cana-2208	9	17	also	also	ADV
cana-2208	9	18	very	very	ADV
cana-2208	9	19	important	important	ADJ
cana-2208	9	20	.	.	PUNCT
cana-2208	10	1	in	in	ADP
cana-2208	10	2	the	the	DET
cana-2208	10	3	past	past	ADJ
cana-2208	10	4	few	few	ADJ
cana-2208	10	5	years	year	NOUN
cana-2208	10	6	,	,	PUNCT
cana-2208	10	7	numerous	numerous	ADJ
cana-2208	10	8	researchers	researcher	NOUN
cana-2208	10	9	have	have	AUX
cana-2208	10	10	grown	grow	VERB
cana-2208	10	11	interested	interested	ADJ
cana-2208	10	12	in	in	ADP
cana-2208	10	13	the	the	DET
cana-2208	10	14	inaccuracy	inaccuracy	NOUN
cana-2208	10	15	of	of	ADP
cana-2208	10	16	approximation	approximation	NOUN
cana-2208	10	17	of	of	ADP
cana-2208	10	18	periodic	periodic	ADJ
cana-2208	10	19	functions	function	NOUN
cana-2208	10	20	belonging	belong	VERB
cana-2208	10	21	to	to	ADP
cana-2208	10	22	distinct	distinct	ADJ
cana-2208	10	23	classes	class	NOUN
cana-2208	10	24	using	use	VERB
cana-2208	10	25	different	different	ADJ
cana-2208	10	26	summability	summability	NOUN
cana-2208	10	27	methods	method	NOUN
cana-2208	10	28	.	.	PUNCT
cana-2208	11	1	several	several	ADJ
cana-2208	11	2	scholars	scholar	NOUN
cana-2208	11	3	,	,	PUNCT
cana-2208	11	4	including	include	VERB
cana-2208	11	5	hadish	hadish	ADJ
cana-2208	11	6	[	[	X
cana-2208	11	7	6	6	NUM
cana-2208	11	8	]	]	PUNCT
cana-2208	11	9	,	,	PUNCT
cana-2208	11	10	sonkar	sonkar	NOUN
cana-2208	11	11	and	and	CCONJ
cana-2208	11	12	singh[8	singh[8	PROPN
cana-2208	11	13	]	]	PROPN
cana-2208	11	14	,	,	PUNCT
cana-2208	11	15	saxena	saxena	PROPN
cana-2208	11	16	and	and	CCONJ
cana-2208	11	17	prabhakar	prabhakar	PROPN
cana-2208	11	18	[	[	X
cana-2208	11	19	5	5	NUM
cana-2208	11	20	]	]	PUNCT
cana-2208	11	21	,	,	PUNCT
cana-2208	11	22	sonkar	sonkar	NOUN
cana-2208	11	23	and	and	CCONJ
cana-2208	11	24	sagwan[7	sagwan[7	PROPN
cana-2208	11	25	]	]	X
cana-2208	11	26	,	,	PUNCT
cana-2208	11	27	and	and	CCONJ
cana-2208	11	28	sachin	sachin	ADJ
cana-2208	12	1	[	[	X
cana-2208	12	2	9	9	NUM
cana-2208	12	3	]	]	PUNCT
cana-2208	12	4	,	,	PUNCT
cana-2208	12	5	have	have	AUX
cana-2208	12	6	studied	study	VERB
cana-2208	12	7	the	the	DET
cana-2208	12	8	following	follow	VERB
cana-2208	12	9	topics	topic	NOUN
cana-2208	12	10	:	:	PUNCT
cana-2208	12	11	(	(	PUNCT
cana-2208	12	12	𝐶	𝐶	PROPN
cana-2208	12	13	,	,	PUNCT
cana-2208	12	14	1)(𝐸	1)(𝐸	PROPN
cana-2208	12	15	,	,	PUNCT
cana-2208	12	16	𝑞	𝑞	NOUN
cana-2208	12	17	)	)	PUNCT
cana-2208	12	18	,	,	PUNCT
cana-2208	12	19	double	double	PROPN
cana-2208	12	20	euler	euler	NOUN
cana-2208	12	21	summability	summability	NOUN
cana-2208	12	22	,	,	PUNCT
cana-2208	12	23	triple	triple	ADJ
cana-2208	12	24	𝐸1	𝐸1	PROPN
cana-2208	12	25	euler	euler	NOUN
cana-2208	12	26	summability	summability	NOUN
cana-2208	12	27	,	,	PUNCT
cana-2208	12	28	and	and	CCONJ
cana-2208	12	29	triple	triple	ADJ
cana-2208	12	30	𝐸𝑞-euler	𝐸𝑞-euler	ADJ
cana-2208	12	31	summability	summability	NOUN
cana-2208	12	32	,	,	PUNCT
cana-2208	12	33	respectively	respectively	ADV
cana-2208	12	34	.	.	PUNCT
cana-2208	13	1	we	we	PRON
cana-2208	13	2	investigated	investigate	VERB
cana-2208	13	3	the	the	DET
cana-2208	13	4	approximation	approximation	NOUN
cana-2208	13	5	of	of	ADP
cana-2208	13	6	functions	function	NOUN
cana-2208	13	7	in	in	ADP
cana-2208	13	8	this	this	DET
cana-2208	13	9	direction	direction	NOUN
cana-2208	13	10	using	use	VERB
cana-2208	13	11	the	the	DET
cana-2208	13	12	n	n	NOUN
cana-2208	13	13	-	-	PUNCT
cana-2208	13	14	euler	euler	NOUN
cana-2208	13	15	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	13	16	product	product	NOUN
cana-2208	13	17	summability	summability	NOUN
cana-2208	13	18	approach	approach	NOUN
cana-2208	13	19	.	.	PUNCT
cana-2208	14	1	the	the	DET
cana-2208	14	2	outcomes	outcome	NOUN
cana-2208	14	3	of	of	ADP
cana-2208	14	4	[	[	X
cana-2208	14	5	7	7	NUM
cana-2208	14	6	]	]	PUNCT
cana-2208	14	7	,	,	PUNCT
cana-2208	14	8	[	[	X
cana-2208	14	9	8	8	NUM
cana-2208	14	10	]	]	PUNCT
cana-2208	14	11	,	,	PUNCT
cana-2208	14	12	and	and	CCONJ
cana-2208	14	13	[	[	X
cana-2208	14	14	9	9	NUM
cana-2208	14	15	]	]	PUNCT
cana-2208	14	16	were	be	AUX
cana-2208	14	17	generalized	generalize	VERB
cana-2208	14	18	by	by	ADP
cana-2208	14	19	our	our	PRON
cana-2208	14	20	result	result	NOUN
cana-2208	14	21	.	.	PUNCT
cana-2208	15	1	2	2	NUM
cana-2208	15	2	definitions	definition	NOUN
cana-2208	15	3	and	and	CCONJ
cana-2208	15	4	notations	notation	NOUN
cana-2208	15	5	let	let	VERB
cana-2208	15	6	𝑔	𝑔	PART
cana-2208	15	7	be	be	AUX
cana-2208	15	8	a	a	DET
cana-2208	15	9	lebesgue	lebesgue	NOUN
cana-2208	15	10	integrable	integrable	ADJ
cana-2208	15	11	function	function	NOUN
cana-2208	15	12	with	with	ADP
cana-2208	15	13	period	period	NOUN
cana-2208	15	14	2𝜋	2𝜋	NOUN
cana-2208	15	15	on	on	ADP
cana-2208	15	16	the	the	DET
cana-2208	15	17	interval	interval	NOUN
cana-2208	15	18	[	[	X
cana-2208	15	19	0,2𝜋	0,2𝜋	X
cana-2208	15	20	]	]	X
cana-2208	15	21	.	.	PUNCT
cana-2208	16	1	the	the	DET
cana-2208	16	2	fourier	fourier	PROPN
cana-2208	16	3	series	series	NOUN
cana-2208	16	4	of	of	ADP
cana-2208	16	5	a	a	DET
cana-2208	16	6	function	function	NOUN
cana-2208	16	7	𝑔	𝑔	NOUN
cana-2208	16	8	is	be	AUX
cana-2208	16	9	given	give	VERB
cana-2208	16	10	by	by	ADP
cana-2208	16	11	𝑔(𝑥	𝑔(𝑥	PROPN
cana-2208	16	12	)	)	PUNCT
cana-2208	17	1	≈	≈	PROPN
cana-2208	17	2	𝑎0	𝑎0	PROPN
cana-2208	17	3	2	2	NUM
cana-2208	17	4	+	+	NUM
cana-2208	17	5	∑∞	∑∞	NOUN
cana-2208	17	6	𝑛=1	𝑛=1	NOUN
cana-2208	17	7	(	(	PUNCT
cana-2208	17	8	𝑎𝑛𝑐𝑜𝑠𝑛𝑥	𝑎𝑛𝑐𝑜𝑠𝑛𝑥	NOUN
cana-2208	17	9	+	+	CCONJ
cana-2208	17	10	𝑏𝑛𝑠𝑖𝑛𝑛𝑥	𝑏𝑛𝑠𝑖𝑛𝑛𝑥	ADJ
cana-2208	17	11	)	)	PUNCT
cana-2208	17	12	(	(	PUNCT
cana-2208	17	13	2.1	2.1	NUM
cana-2208	17	14	)	)	PUNCT
cana-2208	17	15	�	�	PROPN
cana-2208	17	16	̃	̃	PROPN
cana-2208	17	17	�	�	NOUN
cana-2208	17	18	(𝑥	(𝑥	NOUN
cana-2208	17	19	)	)	PUNCT
cana-2208	17	20	≈	≈	PROPN
cana-2208	17	21	∑∞	∑∞	NOUN
cana-2208	17	22	𝑚=1	𝑚=1	X
cana-2208	17	23	(	(	PUNCT
cana-2208	17	24	𝑏𝑚𝑐𝑜𝑠𝑚𝑥	𝑏𝑚𝑐𝑜𝑠𝑚𝑥	NOUN
cana-2208	17	25	−	−	PROPN
cana-2208	17	26	𝑎𝑚𝑠𝑖𝑛𝑚𝑥	𝑎𝑚𝑠𝑖𝑛𝑚𝑥	PROPN
cana-2208	17	27	)	)	PUNCT
cana-2208	17	28	(	(	PUNCT
cana-2208	17	29	2.2	2.2	NUM
cana-2208	17	30	)	)	PUNCT
cana-2208	17	31	with	with	ADP
cana-2208	17	32	𝑛𝑡ℎ	𝑛𝑡ℎ	PRON
cana-2208	17	33	partial	partial	ADJ
cana-2208	17	34	sum	sum	PROPN
cana-2208	17	35	𝑔𝑛(𝑥	𝑔𝑛(𝑥	NOUN
cana-2208	17	36	)	)	PUNCT
cana-2208	17	37	.	.	PUNCT
cana-2208	18	1	the	the	DET
cana-2208	18	2	𝐿𝑝[0,2𝜋	𝐿𝑝[0,2𝜋	NOUN
cana-2208	18	3	]	]	X
cana-2208	18	4	−	−	NOUN
cana-2208	18	5	space	space	NOUN
cana-2208	18	6	can	can	AUX
cana-2208	18	7	be	be	AUX
cana-2208	18	8	defined	define	VERB
cana-2208	18	9	as	as	ADP
cana-2208	18	10	communications	communication	NOUN
cana-2208	18	11	on	on	ADP
cana-2208	18	12	applied	apply	VERB
cana-2208	18	13	nonlinear	nonlinear	ADJ
cana-2208	18	14	analysis	analysis	NOUN
cana-2208	18	15	issn	issn	NOUN
cana-2208	18	16	:	:	PUNCT
cana-2208	18	17	1074	1074	NUM
cana-2208	18	18	-	-	PUNCT
cana-2208	18	19	133x	133x	NUM
cana-2208	18	20	vol	vol	NOUN
cana-2208	18	21	32	32	NUM
cana-2208	18	22	no	no	NOUN
cana-2208	18	23	.	.	PUNCT
cana-2208	19	1	1s	1s	NUM
cana-2208	19	2	(	(	PUNCT
cana-2208	19	3	2025	2025	NUM
cana-2208	19	4	)	)	PUNCT
cana-2208	19	5	446	446	NUM
cana-2208	19	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	19	7	𝐿𝑝[0,2𝜋	𝐿𝑝[0,2𝜋	ADP
cana-2208	19	8	]	]	X
cana-2208	19	9	:	:	PUNCT
cana-2208	19	10	=	=	SYM
cana-2208	19	11	{	{	PUNCT
cana-2208	19	12	𝑔	𝑔	NOUN
cana-2208	19	13	:	:	PUNCT
cana-2208	20	1	[	[	X
cana-2208	20	2	0,2𝜋	0,2𝜋	X
cana-2208	20	3	]	]	X
cana-2208	20	4	→	→	SYM
cana-2208	20	5	𝑅	𝑅	NOUN
cana-2208	20	6	:	:	PUNCT
cana-2208	20	7	∫	∫	PROPN
cana-2208	20	8	2𝜋	2𝜋	PROPN
cana-2208	20	9	0	0	NUM
cana-2208	20	10	|𝑔(𝑥)|𝑝𝑑𝑥	|𝑔(𝑥)|𝑝𝑑𝑥	NOUN
cana-2208	20	11	<	<	X
cana-2208	20	12	∞	∞	NUM
cana-2208	20	13	}	}	PUNCT
cana-2208	20	14	,	,	PUNCT
cana-2208	20	15	𝑝	𝑝	PROPN
cana-2208	20	16	≥	≥	NOUN
cana-2208	20	17	1	1	NUM
cana-2208	20	18	.	.	PUNCT
cana-2208	21	1	let	let	VERB
cana-2208	21	2	∑∞	∑∞	NOUN
cana-2208	21	3	𝑟=0	𝑟=0	PUNCT
cana-2208	21	4	𝑢𝑟	𝑢𝑟	ADV
cana-2208	21	5	be	be	AUX
cana-2208	21	6	an	an	DET
cana-2208	21	7	infinite	infinite	ADJ
cana-2208	21	8	series	series	NOUN
cana-2208	21	9	and	and	CCONJ
cana-2208	21	10	sequence	sequence	NOUN
cana-2208	21	11	{	{	PUNCT
cana-2208	21	12	𝑠𝑟	𝑠𝑟	NOUN
cana-2208	21	13	}	}	PUNCT
cana-2208	21	14	is	be	AUX
cana-2208	21	15	(	(	PUNCT
cana-2208	21	16	𝑟	𝑟	X
cana-2208	21	17	+	+	CCONJ
cana-2208	21	18	1)𝑡ℎ	1)𝑡ℎ	NUM
cana-2208	21	19	partial	partial	ADJ
cana-2208	21	20	sum	sum	NOUN
cana-2208	21	21	of	of	ADP
cana-2208	21	22	given	give	VERB
cana-2208	21	23	series	series	NOUN
cana-2208	21	24	,	,	PUNCT
cana-2208	21	25	then	then	ADV
cana-2208	21	26	the	the	DET
cana-2208	21	27	series	series	NOUN
cana-2208	21	28	∑∞	∑∞	PROPN
cana-2208	21	29	𝑟=0	𝑟=0	PUNCT
cana-2208	21	30	𝑢𝑟	𝑢𝑟	ADV
cana-2208	21	31	is	be	AUX
cana-2208	21	32	said	say	VERB
cana-2208	21	33	to	to	PART
cana-2208	21	34	be	be	AUX
cana-2208	21	35	(	(	PUNCT
cana-2208	21	36	𝐸	𝐸	PROPN
cana-2208	21	37	,	,	PUNCT
cana-2208	21	38	𝑞	𝑞	NOUN
cana-2208	21	39	)	)	PUNCT
cana-2208	21	40	summable[1	summable[1	NOUN
cana-2208	21	41	]	]	PUNCT
cana-2208	21	42	to	to	ADP
cana-2208	21	43	s	s	PRON
cana-2208	21	44	if	if	SCONJ
cana-2208	21	45	𝑡𝑟	𝑡𝑟	VERB
cana-2208	21	46	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	21	47	=	=	SYM
cana-2208	21	48	1	1	NUM
cana-2208	21	49	(	(	PUNCT
cana-2208	21	50	1+𝑞)𝑟	1+𝑞)𝑟	NUM
cana-2208	22	1	∑𝑟	∑𝑟	NOUN
cana-2208	22	2	𝑗=0	𝑗=0	X
cana-2208	23	1	(	(	PUNCT
cana-2208	23	2	𝑟	𝑟	X
cana-2208	23	3	𝑗	𝑗	X
cana-2208	23	4	)	)	PUNCT
cana-2208	23	5	𝑞𝑟−𝑗𝑠𝑗	𝑞𝑟−𝑗𝑠𝑗	NOUN
cana-2208	23	6	→	→	SYM
cana-2208	23	7	𝑠	𝑠	PROPN
cana-2208	23	8	𝑎𝑠	𝑎𝑠	NOUN
cana-2208	23	9	𝑟	𝑟	NOUN
cana-2208	23	10	→	→	SYM
cana-2208	23	11	∞.	∞.	PROPN
cana-2208	23	12	the	the	DET
cana-2208	23	13	series	series	PROPN
cana-2208	23	14	∑∞	∑∞	PROPN
cana-2208	23	15	𝑟=0	𝑟=0	PUNCT
cana-2208	23	16	𝑢𝑟	𝑢𝑟	ADV
cana-2208	23	17	is	be	AUX
cana-2208	23	18	said	say	VERB
cana-2208	23	19	to	to	PART
cana-2208	23	20	be	be	AUX
cana-2208	23	21	(	(	PUNCT
cana-2208	23	22	𝐸	𝐸	PROPN
cana-2208	23	23	,	,	PUNCT
cana-2208	23	24	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	23	25	,	,	PUNCT
cana-2208	23	26	𝑞	𝑞	NOUN
cana-2208	23	27	)	)	PUNCT
cana-2208	23	28	summable	summable	ADJ
cana-2208	23	29	to	to	ADP
cana-2208	23	30	s	s	PRON
cana-2208	23	31	,	,	PUNCT
cana-2208	23	32	if	if	SCONJ
cana-2208	23	33	𝑡𝑟	𝑡𝑟	VERB
cana-2208	23	34	𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞	PROPN
cana-2208	23	35	=	=	SYM
cana-2208	23	36	1	1	NUM
cana-2208	23	37	(	(	PUNCT
cana-2208	23	38	1	1	NUM
cana-2208	23	39	+	+	CCONJ
cana-2208	23	40	𝑞)𝑟	𝑞)𝑟	NOUN
cana-2208	23	41	∑	∑	PUNCT
cana-2208	23	42	𝑟	𝑟	X
cana-2208	23	43	𝑗=0	𝑗=0	PUNCT
cana-2208	23	44	(	(	PUNCT
cana-2208	23	45	𝑟	𝑟	NOUN
cana-2208	23	46	𝑗	𝑗	X
cana-2208	23	47	)	)	PUNCT
cana-2208	23	48	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	23	49	(	(	PUNCT
cana-2208	23	50	1	1	NUM
cana-2208	23	51	+	+	CCONJ
cana-2208	23	52	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	23	53	∑	∑	PUNCT
cana-2208	23	54	𝑗	𝑗	INTJ
cana-2208	23	55	ℎ=0	ℎ=0	X
cana-2208	23	56	(	(	PUNCT
cana-2208	23	57	𝑗	𝑗	INTJ
cana-2208	23	58	ℎ	ℎ	PROPN
cana-2208	23	59	)	)	PUNCT
cana-2208	23	60	𝑞𝑗−ℎ𝑠ℎ	𝑞𝑗−ℎ𝑠ℎ	PROPN
cana-2208	23	61	→	→	SYM
cana-2208	23	62	𝑠	𝑠	PROPN
cana-2208	23	63	𝑎𝑠	𝑎𝑠	NOUN
cana-2208	23	64	𝑟	𝑟	NOUN
cana-2208	23	65	→	→	SYM
cana-2208	23	66	∞.	∞.	PROPN
cana-2208	23	67	the	the	DET
cana-2208	23	68	series	series	PROPN
cana-2208	23	69	∑∞	∑∞	PROPN
cana-2208	23	70	𝑟=0	𝑟=0	PUNCT
cana-2208	23	71	𝑢𝑟	𝑢𝑟	ADV
cana-2208	23	72	is	be	AUX
cana-2208	23	73	said	say	VERB
cana-2208	23	74	to	to	PART
cana-2208	23	75	be	be	AUX
cana-2208	23	76	(	(	PUNCT
cana-2208	23	77	𝐸	𝐸	PROPN
cana-2208	23	78	,	,	PUNCT
cana-2208	23	79	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	23	80	,	,	PUNCT
cana-2208	23	81	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	23	82	,	,	PUNCT
cana-2208	23	83	𝑞	𝑞	NOUN
cana-2208	23	84	)	)	PUNCT
cana-2208	23	85	summable	summable	ADJ
cana-2208	23	86	to	to	ADP
cana-2208	23	87	s	s	PRON
cana-2208	23	88	,	,	PUNCT
cana-2208	23	89	if	if	SCONJ
cana-2208	23	90	𝑡𝑟	𝑡𝑟	VERB
cana-2208	23	91	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	23	92	=	=	SYM
cana-2208	23	93	1	1	NUM
cana-2208	23	94	(	(	PUNCT
cana-2208	23	95	1+𝑞)𝑟	1+𝑞)𝑟	NUM
cana-2208	24	1	∑𝑟	∑𝑟	NOUN
cana-2208	24	2	𝑗=0	𝑗=0	X
cana-2208	25	1	(	(	PUNCT
cana-2208	25	2	𝑟	𝑟	NOUN
cana-2208	25	3	𝑗	𝑗	X
cana-2208	25	4	)	)	PUNCT
cana-2208	25	5	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	25	6	(	(	PUNCT
cana-2208	25	7	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	25	8	∑𝑗	∑𝑗	PROPN
cana-2208	25	9	ℎ=0	ℎ=0	PROPN
cana-2208	25	10	(	(	PUNCT
cana-2208	25	11	𝑗	𝑗	INTJ
cana-2208	25	12	ℎ	ℎ	PROPN
cana-2208	25	13	)	)	PUNCT
cana-2208	25	14	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	25	15	(	(	PUNCT
cana-2208	25	16	1+𝑞)ℎ	1+𝑞)ℎ	NOUN
cana-2208	25	17	∑ℎ	∑ℎ	VERB
cana-2208	25	18	𝑙=0	𝑙=0	PUNCT
cana-2208	25	19	(	(	PUNCT
cana-2208	25	20	ℎ	ℎ	X
cana-2208	25	21	𝑙	𝑙	NOUN
cana-2208	25	22	)	)	PUNCT
cana-2208	25	23	𝑞ℎ−𝑙𝑠𝑙	𝑞ℎ−𝑙𝑠𝑙	PROPN
cana-2208	25	24	→	→	SYM
cana-2208	25	25	𝑠	𝑠	PROPN
cana-2208	25	26	𝑎𝑠	𝑎𝑠	ADP
cana-2208	25	27	𝑟	𝑟	NOUN
cana-2208	25	28	→	→	SYM
cana-2208	25	29	∞.	∞.	PROPN
cana-2208	25	30	similarly	similarly	ADV
cana-2208	25	31	,	,	PUNCT
cana-2208	25	32	the	the	DET
cana-2208	25	33	series	series	NOUN
cana-2208	25	34	∑∞	∑∞	PROPN
cana-2208	25	35	𝑟=0	𝑟=0	PUNCT
cana-2208	25	36	𝑢𝑟	𝑢𝑟	ADV
cana-2208	25	37	is	be	AUX
cana-2208	25	38	said	say	VERB
cana-2208	25	39	to	to	PART
cana-2208	25	40	be	be	AUX
cana-2208	25	41	(	(	PUNCT
cana-2208	25	42	𝐸	𝐸	PROPN
cana-2208	25	43	,	,	PUNCT
cana-2208	25	44	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	25	45	,	,	PUNCT
cana-2208	25	46	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	25	47	,	,	PUNCT
cana-2208	25	48	𝑞	𝑞	NOUN
cana-2208	25	49	)	)	PUNCT
cana-2208	25	50	.	.	PUNCT
cana-2208	25	51	.	.	PUNCT
cana-2208	25	52	.	.	PUNCT
cana-2208	26	1	(	(	PUNCT
cana-2208	26	2	𝐸	𝐸	PROPN
cana-2208	26	3	,	,	PUNCT
cana-2208	26	4	𝑞	𝑞	NOUN
cana-2208	26	5	)	)	PUNCT
cana-2208	26	6	summable	summable	ADJ
cana-2208	26	7	to	to	ADP
cana-2208	26	8	s	s	PROPN
cana-2208	26	9	i.e.	i.e.	X
cana-2208	26	10	n	n	CCONJ
cana-2208	26	11	-	-	PUNCT
cana-2208	26	12	euler	euler	NOUN
cana-2208	26	13	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	26	14	summable	summable	ADJ
cana-2208	26	15	to	to	ADP
cana-2208	26	16	s.	s.	PROPN
cana-2208	26	17	if	if	SCONJ
cana-2208	26	18	𝑡𝑟	𝑡𝑟	VERB
cana-2208	26	19	𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞	PROPN
cana-2208	26	20	...	...	PUNCT
cana-2208	27	1	𝐸𝑞	𝐸𝑞	NOUN
cana-2208	27	2	=	=	SYM
cana-2208	27	3	1	1	NUM
cana-2208	27	4	(	(	PUNCT
cana-2208	27	5	1	1	NUM
cana-2208	27	6	+	+	CCONJ
cana-2208	27	7	𝑞)𝑟	𝑞)𝑟	NOUN
cana-2208	27	8	∑	∑	PUNCT
cana-2208	27	9	𝑟	𝑟	X
cana-2208	27	10	𝑗=0	𝑗=0	PUNCT
cana-2208	27	11	(	(	PUNCT
cana-2208	27	12	𝑟	𝑟	NOUN
cana-2208	27	13	𝑗	𝑗	X
cana-2208	27	14	)	)	PUNCT
cana-2208	27	15	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	27	16	(	(	PUNCT
cana-2208	27	17	1	1	NUM
cana-2208	27	18	+	+	CCONJ
cana-2208	27	19	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	27	20	∑	∑	PUNCT
cana-2208	27	21	𝑗	𝑗	INTJ
cana-2208	27	22	ℎ=0	ℎ=0	X
cana-2208	27	23	(	(	PUNCT
cana-2208	27	24	𝑗	𝑗	INTJ
cana-2208	27	25	ℎ	ℎ	PROPN
cana-2208	27	26	)	)	PUNCT
cana-2208	27	27	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	27	28	(	(	PUNCT
cana-2208	27	29	1	1	NUM
cana-2208	27	30	+	+	CCONJ
cana-2208	27	31	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	27	32	∑	∑	ADP
cana-2208	27	33	ℎ	ℎ	X
cana-2208	27	34	𝑙=0	𝑙=0	PROPN
cana-2208	27	35	(	(	PUNCT
cana-2208	27	36	ℎ	ℎ	X
cana-2208	27	37	𝑙	𝑙	PROPN
cana-2208	27	38	)	)	PUNCT
cana-2208	27	39	𝑞ℎ−𝑙.	𝑞ℎ−𝑙.	X
cana-2208	27	40	.	.	PUNCT
cana-2208	27	41	.	.	PUNCT
cana-2208	28	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	28	2	(	(	PUNCT
cana-2208	28	3	1	1	NUM
cana-2208	28	4	+	+	NUM
cana-2208	28	5	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	28	6	∑	∑	PUNCT
cana-2208	28	7	𝑖	𝑖	X
cana-2208	28	8	𝑚=0	𝑚=0	PUNCT
cana-2208	28	9	(	(	PUNCT
cana-2208	28	10	𝑖	𝑖	NOUN
cana-2208	28	11	𝑚	𝑚	NOUN
cana-2208	28	12	)	)	PUNCT
cana-2208	28	13	𝑞𝑖−𝑚𝑠𝑚	𝑞𝑖−𝑚𝑠𝑚	PROPN
cana-2208	28	14	→	→	SYM
cana-2208	28	15	𝑠	𝑠	PROPN
cana-2208	28	16	𝑎𝑠	𝑎𝑠	NOUN
cana-2208	28	17	𝑟	𝑟	NOUN
cana-2208	28	18	→	→	SYM
cana-2208	28	19	∞	∞	PROPN
cana-2208	28	20	we	we	PRON
cana-2208	28	21	also	also	ADV
cana-2208	28	22	write	write	VERB
cana-2208	28	23	,	,	PUNCT
cana-2208	28	24	𝐽𝑟(𝑡	𝐽𝑟(𝑡	PROPN
cana-2208	28	25	)	)	PUNCT
cana-2208	28	26	=	=	SYM
cana-2208	29	1	1	1	NUM
cana-2208	29	2	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	29	3	∑𝑟	∑𝑟	NOUN
cana-2208	29	4	𝑗=0	𝑗=0	PUNCT
cana-2208	30	1	[	[	X
cana-2208	30	2	(	(	PUNCT
cana-2208	30	3	𝑟	𝑟	X
cana-2208	30	4	𝑗	𝑗	X
cana-2208	30	5	)	)	PUNCT
cana-2208	30	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	30	7	(	(	PUNCT
cana-2208	30	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	30	9	∑𝑗	∑𝑗	PROPN
cana-2208	30	10	ℎ=0	ℎ=0	PROPN
cana-2208	30	11	(	(	PUNCT
cana-2208	30	12	𝑗	𝑗	INTJ
cana-2208	30	13	ℎ	ℎ	PROPN
cana-2208	30	14	)	)	PUNCT
cana-2208	30	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	30	16	(	(	PUNCT
cana-2208	30	17	1+𝑞)ℎ	1+𝑞)ℎ	NOUN
cana-2208	30	18	∑ℎ	∑ℎ	VERB
cana-2208	30	19	𝑙=0	𝑙=0	PUNCT
cana-2208	30	20	(	(	PUNCT
cana-2208	30	21	ℎ	ℎ	X
cana-2208	30	22	𝑙	𝑙	PROPN
cana-2208	30	23	)	)	PUNCT
cana-2208	30	24	𝑞ℎ−𝑙.	𝑞ℎ−𝑙.	X
cana-2208	30	25	.	.	PUNCT
cana-2208	30	26	.	.	PUNCT
cana-2208	31	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	31	2	(	(	PUNCT
cana-2208	31	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	31	4	{	{	PUNCT
cana-2208	31	5	∑𝑖	∑𝑖	PROPN
cana-2208	31	6	𝑚=0	𝑚=0	PUNCT
cana-2208	31	7	(	(	PUNCT
cana-2208	31	8	𝑖	𝑖	NOUN
cana-2208	31	9	𝑚	𝑚	NOUN
cana-2208	31	10	)	)	PUNCT
cana-2208	31	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	31	12	𝑠𝑖𝑛(𝑚+	𝑠𝑖𝑛(𝑚+	ADJ
cana-2208	31	13	1	1	NUM
cana-2208	31	14	2	2	NUM
cana-2208	31	15	)	)	PUNCT
cana-2208	31	16	𝑡	𝑡	PROPN
cana-2208	31	17	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	31	18	𝑡	𝑡	ADP
cana-2208	31	19	2	2	NUM
cana-2208	31	20	}	}	PUNCT
cana-2208	31	21	]	]	PUNCT
cana-2208	31	22	𝐽𝑟(𝑡	𝐽𝑟(𝑡	PROPN
cana-2208	31	23	)	)	PUNCT
cana-2208	31	24	=	=	SYM
cana-2208	32	1	1	1	NUM
cana-2208	32	2	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	32	3	∑𝑟	∑𝑟	NOUN
cana-2208	32	4	𝑗=0	𝑗=0	PUNCT
cana-2208	33	1	[	[	X
cana-2208	33	2	(	(	PUNCT
cana-2208	33	3	𝑟	𝑟	X
cana-2208	33	4	𝑗	𝑗	X
cana-2208	33	5	)	)	PUNCT
cana-2208	33	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	33	7	(	(	PUNCT
cana-2208	33	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	33	9	∑𝑗	∑𝑗	PROPN
cana-2208	33	10	ℎ=0	ℎ=0	PROPN
cana-2208	33	11	(	(	PUNCT
cana-2208	33	12	𝑗	𝑗	INTJ
cana-2208	33	13	ℎ	ℎ	PROPN
cana-2208	33	14	)	)	PUNCT
cana-2208	33	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	33	16	(	(	PUNCT
cana-2208	33	17	1+𝑞)ℎ	1+𝑞)ℎ	NOUN
cana-2208	33	18	∑ℎ	∑ℎ	VERB
cana-2208	33	19	𝑙=0	𝑙=0	PUNCT
cana-2208	33	20	(	(	PUNCT
cana-2208	33	21	ℎ	ℎ	X
cana-2208	33	22	𝑙	𝑙	PROPN
cana-2208	33	23	)	)	PUNCT
cana-2208	33	24	𝑞ℎ−𝑙.	𝑞ℎ−𝑙.	X
cana-2208	33	25	.	.	PUNCT
cana-2208	33	26	.	.	PUNCT
cana-2208	34	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	34	2	(	(	PUNCT
cana-2208	34	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	34	4	{	{	PUNCT
cana-2208	34	5	∑𝑖	∑𝑖	PROPN
cana-2208	34	6	𝑚=0	𝑚=0	PUNCT
cana-2208	34	7	(	(	PUNCT
cana-2208	34	8	𝑖	𝑖	NOUN
cana-2208	34	9	𝑚	𝑚	NOUN
cana-2208	34	10	)	)	PUNCT
cana-2208	34	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	34	12	𝑐𝑜𝑠(𝑚+	𝑐𝑜𝑠(𝑚+	ADP
cana-2208	34	13	1	1	NUM
cana-2208	34	14	2	2	NUM
cana-2208	34	15	𝑡	𝑡	NOUN
cana-2208	34	16	)	)	PUNCT
cana-2208	34	17	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2208	34	18	𝑡	𝑡	ADP
cana-2208	34	19	2	2	NUM
cana-2208	34	20	}	}	PUNCT
cana-2208	34	21	]	]	PUNCT
cana-2208	34	22	𝜙(𝑡	𝜙(𝑡	NOUN
cana-2208	34	23	)	)	PUNCT
cana-2208	35	1	=	=	SYM
cana-2208	36	1	𝑔(𝑥	𝑔(𝑥	PROPN
cana-2208	36	2	+	+	NUM
cana-2208	36	3	𝑡	𝑡	X
cana-2208	36	4	)	)	PUNCT
cana-2208	36	5	−	−	NOUN
cana-2208	36	6	2𝑔(𝑡	2𝑔(𝑡	NUM
cana-2208	36	7	)	)	PUNCT
cana-2208	37	1	+	+	CCONJ
cana-2208	37	2	𝑔(𝑥	𝑔(𝑥	ADP
cana-2208	37	3	−	−	NOUN
cana-2208	37	4	𝑡	𝑡	NOUN
cana-2208	37	5	)	)	PUNCT
cana-2208	37	6	and	and	CCONJ
cana-2208	37	7	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2208	37	8	)	)	PUNCT
cana-2208	37	9	=	=	SYM
cana-2208	38	1	𝑔(𝑥	𝑔(𝑥	PROPN
cana-2208	38	2	+	+	CCONJ
cana-2208	38	3	𝑡	𝑡	X
cana-2208	38	4	)	)	PUNCT
cana-2208	38	5	−	−	PROPN
cana-2208	39	1	𝑔(𝑥	𝑔(𝑥	NOUN
cana-2208	39	2	−	−	NUM
cana-2208	39	3	𝑡	𝑡	NOUN
cana-2208	39	4	)	)	PUNCT
cana-2208	39	5	2	2	NUM
cana-2208	39	6	.	.	PUNCT
cana-2208	40	1	communications	communication	NOUN
cana-2208	40	2	on	on	ADP
cana-2208	40	3	applied	apply	VERB
cana-2208	40	4	nonlinear	nonlinear	ADJ
cana-2208	40	5	analysis	analysis	NOUN
cana-2208	40	6	issn	issn	NOUN
cana-2208	40	7	:	:	PUNCT
cana-2208	40	8	1074	1074	NUM
cana-2208	40	9	-	-	PUNCT
cana-2208	40	10	133x	133x	NUM
cana-2208	40	11	vol	vol	NOUN
cana-2208	40	12	32	32	NUM
cana-2208	40	13	no	no	NOUN
cana-2208	40	14	.	.	PUNCT
cana-2208	41	1	1s	1s	NUM
cana-2208	41	2	(	(	PUNCT
cana-2208	41	3	2025	2025	NUM
cana-2208	41	4	)	)	PUNCT
cana-2208	41	5	447	447	NUM
cana-2208	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	41	7	3	3	NUM
cana-2208	41	8	main	main	ADJ
cana-2208	41	9	results	result	NOUN
cana-2208	41	10	saxena	saxena	PROPN
cana-2208	41	11	and	and	CCONJ
cana-2208	41	12	prabhakar[8	prabhakar[8	PROPN
cana-2208	41	13	]	]	PUNCT
cana-2208	41	14	obtained	obtain	VERB
cana-2208	41	15	novel	novel	ADJ
cana-2208	41	16	findings	finding	NOUN
cana-2208	41	17	in	in	ADP
cana-2208	41	18	the	the	DET
cana-2208	41	19	domain	domain	NOUN
cana-2208	41	20	of	of	ADP
cana-2208	41	21	function	function	NOUN
cana-2208	41	22	approximation	approximation	NOUN
cana-2208	41	23	.	.	PUNCT
cana-2208	42	1	the	the	DET
cana-2208	42	2	triple	triple	ADJ
cana-2208	42	3	𝐸1	𝐸1	NOUN
cana-2208	42	4	summability	summability	NOUN
cana-2208	42	5	approach	approach	NOUN
cana-2208	42	6	is	be	AUX
cana-2208	42	7	introduced	introduce	VERB
cana-2208	42	8	by	by	ADP
cana-2208	42	9	sonkar	sonkar	NOUN
cana-2208	42	10	and	and	CCONJ
cana-2208	42	11	sangwan[7	sangwan[7	NUM
cana-2208	42	12	]	]	PUNCT
cana-2208	42	13	,	,	PUNCT
cana-2208	42	14	who	who	PRON
cana-2208	42	15	used	use	VERB
cana-2208	42	16	it	it	PRON
cana-2208	42	17	to	to	PART
cana-2208	42	18	generalise	generalise	VERB
cana-2208	42	19	the	the	DET
cana-2208	42	20	findings	finding	NOUN
cana-2208	42	21	of	of	ADP
cana-2208	42	22	saxena	saxena	PROPN
cana-2208	42	23	and	and	CCONJ
cana-2208	42	24	prabhakar.the	prabhakar.the	DET
cana-2208	42	25	triple	triple	ADJ
cana-2208	42	26	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	42	27	summability	summability	NOUN
cana-2208	42	28	approach	approach	NOUN
cana-2208	42	29	was	be	AUX
cana-2208	42	30	recently	recently	ADV
cana-2208	42	31	introduced	introduce	VERB
cana-2208	42	32	by	by	ADP
cana-2208	42	33	devaiya	devaiya	PROPN
cana-2208	42	34	and	and	CCONJ
cana-2208	42	35	sriwastva[9	sriwastva[9	NOUN
cana-2208	42	36	]	]	X
cana-2208	42	37	,	,	PUNCT
cana-2208	42	38	and	and	CCONJ
cana-2208	42	39	they	they	PRON
cana-2208	42	40	used	use	VERB
cana-2208	42	41	it	it	PRON
cana-2208	42	42	to	to	PART
cana-2208	42	43	generalise	generalise	VERB
cana-2208	42	44	the	the	DET
cana-2208	42	45	findings	finding	NOUN
cana-2208	42	46	of	of	ADP
cana-2208	42	47	sonkar	sonkar	NOUN
cana-2208	42	48	and	and	CCONJ
cana-2208	42	49	sagwan[7	sagwan[7	PROPN
cana-2208	42	50	]	]	PUNCT
cana-2208	42	51	.	.	PUNCT
cana-2208	43	1	in	in	ADP
cana-2208	43	2	this	this	DET
cana-2208	43	3	work	work	NOUN
cana-2208	43	4	,	,	PUNCT
cana-2208	43	5	we	we	PRON
cana-2208	43	6	establish	establish	VERB
cana-2208	43	7	the	the	DET
cana-2208	43	8	more	more	ADV
cana-2208	43	9	broad	broad	ADJ
cana-2208	43	10	setting	setting	NOUN
cana-2208	43	11	and	and	CCONJ
cana-2208	43	12	generalise	generalise	VERB
cana-2208	43	13	the	the	DET
cana-2208	43	14	findings	finding	NOUN
cana-2208	43	15	of	of	ADP
cana-2208	43	16	sachin	sachin	PROPN
cana-2208	43	17	devaiya	devaiya	PROPN
cana-2208	43	18	and	and	CCONJ
cana-2208	43	19	shailesh	shailesh	VERB
cana-2208	43	20	kumar	kumar	PROPN
cana-2208	43	21	srivastava[9	srivastava[9	PROPN
cana-2208	43	22	]	]	X
cana-2208	43	23	.	.	PUNCT
cana-2208	44	1	3.1	3.1	NUM
cana-2208	44	2	theorem	theorem	NOUN
cana-2208	44	3	let	let	VERB
cana-2208	44	4	𝑝𝑟	𝑝𝑟	PRON
cana-2208	44	5	be	be	AUX
cana-2208	44	6	a	a	DET
cana-2208	44	7	none	none	NOUN
cana-2208	44	8	increasing	increase	VERB
cana-2208	44	9	positive	positive	ADJ
cana-2208	44	10	sequence	sequence	NOUN
cana-2208	44	11	of	of	ADP
cana-2208	44	12	real	real	ADJ
cana-2208	44	13	constant	constant	ADJ
cana-2208	44	14	such	such	ADJ
cana-2208	44	15	that	that	SCONJ
cana-2208	44	16	𝑝𝑟	𝑝𝑟	PROPN
cana-2208	44	17	=	=	PUNCT
cana-2208	44	18	∑𝑟	∑𝑟	PROPN
cana-2208	44	19	𝑧=0	𝑧=0	PROPN
cana-2208	44	20	𝑝𝑧	𝑝𝑧	PROPN
cana-2208	45	1	→	→	SYM
cana-2208	45	2	𝑠	𝑠	X
cana-2208	45	3	𝑎𝑠	𝑎𝑠	ADP
cana-2208	45	4	𝑟	𝑟	NOUN
cana-2208	45	5	→	→	SYM
cana-2208	45	6	∞	∞	PROPN
cana-2208	45	7	,	,	PUNCT
cana-2208	45	8	(	(	PUNCT
cana-2208	45	9	3.1	3.1	NUM
cana-2208	45	10	)	)	PUNCT
cana-2208	45	11	and	and	CCONJ
cana-2208	45	12	given	give	VERB
cana-2208	45	13	function	function	NOUN
cana-2208	45	14	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	45	15	)	)	PUNCT
cana-2208	45	16	satisfies	satisfie	NOUN
cana-2208	45	17	,	,	PUNCT
cana-2208	45	18	φ(𝑡	φ(𝑡	ADJ
cana-2208	45	19	)	)	PUNCT
cana-2208	45	20	=	=	SYM
cana-2208	45	21	∫	∫	PROPN
cana-2208	45	22	𝑡	𝑡	PROPN
cana-2208	45	23	0	0	NUM
cana-2208	45	24	|𝜙(𝑢)|𝑑𝑢	|𝜙(𝑢)|𝑑𝑢	NOUN
cana-2208	45	25	=	=	NOUN
cana-2208	45	26	𝑜	𝑜	X
cana-2208	45	27	[	[	PUNCT
cana-2208	45	28	𝑡	𝑡	X
cana-2208	45	29	𝛼	𝛼	PROPN
cana-2208	45	30	(	(	PUNCT
cana-2208	45	31	1	1	NUM
cana-2208	45	32	𝑡	𝑡	NOUN
cana-2208	45	33	)	)	PUNCT
cana-2208	45	34	𝑝𝑡	𝑝𝑡	NOUN
cana-2208	45	35	]	]	PUNCT
cana-2208	45	36	,	,	PUNCT
cana-2208	45	37	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	45	38	𝑡	𝑡	PROPN
cana-2208	45	39	→	→	SYM
cana-2208	45	40	0	0	NUM
cana-2208	46	1	+	+	CCONJ
cana-2208	46	2	,	,	PUNCT
cana-2208	46	3	(	(	PUNCT
cana-2208	46	4	3.2	3.2	NUM
cana-2208	46	5	)	)	PUNCT
cana-2208	46	6	where	where	SCONJ
cana-2208	46	7	𝛼(𝑡	𝛼(𝑡	NOUN
cana-2208	46	8	)	)	PUNCT
cana-2208	46	9	is	be	AUX
cana-2208	46	10	positrive	positrive	ADJ
cana-2208	46	11	,	,	PUNCT
cana-2208	46	12	monotonic	monotonic	ADJ
cana-2208	46	13	and	and	CCONJ
cana-2208	46	14	none	none	NOUN
cana-2208	46	15	increasing	increase	VERB
cana-2208	46	16	function	function	NOUN
cana-2208	46	17	of	of	ADP
cana-2208	46	18	t.	t.	PROPN
cana-2208	46	19	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	46	20	=	=	SYM
cana-2208	46	21	𝑂[𝛼(𝑟	𝑂[𝛼(𝑟	PROPN
cana-2208	46	22	)	)	PUNCT
cana-2208	46	23	.	.	PUNCT
cana-2208	47	1	𝑝𝑟	𝑝𝑟	X
cana-2208	47	2	]	]	X
cana-2208	47	3	,	,	PUNCT
cana-2208	47	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	47	5	𝑟	𝑟	X
cana-2208	47	6	→	→	SYM
cana-2208	47	7	∞	∞	PROPN
cana-2208	47	8	(	(	PUNCT
cana-2208	47	9	3.3	3.3	NUM
cana-2208	47	10	)	)	PUNCT
cana-2208	47	11	the	the	DET
cana-2208	47	12	approximation	approximation	NOUN
cana-2208	47	13	of	of	ADP
cana-2208	47	14	function	function	NOUN
cana-2208	47	15	g	g	NOUN
cana-2208	47	16	at	at	ADP
cana-2208	47	17	𝑥	𝑥	NOUN
cana-2208	47	18	=	=	PUNCT
cana-2208	47	19	𝑡	𝑡	X
cana-2208	47	20	by	by	ADP
cana-2208	47	21	neular	neular	ADJ
cana-2208	47	22	product	product	NOUN
cana-2208	47	23	means	mean	NOUN
cana-2208	47	24	of	of	ADP
cana-2208	47	25	its	its	PRON
cana-2208	47	26	fourier	fourier	NOUN
cana-2208	47	27	series	series	NOUN
cana-2208	47	28	is	be	AUX
cana-2208	47	29	given	give	VERB
cana-2208	47	30	by	by	ADP
cana-2208	47	31	|𝑡𝑟	|𝑡𝑟	DET
cana-2208	47	32	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	47	33	...	...	PUNCT
cana-2208	48	1	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	48	2	−	−	PROPN
cana-2208	48	3	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	48	4	=	=	PUNCT
cana-2208	48	5	𝑂(1	𝑂(1	ADP
cana-2208	48	6	)	)	PUNCT
cana-2208	48	7	,	,	PUNCT
cana-2208	48	8	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	48	9	𝑟	𝑟	X
cana-2208	48	10	→	→	SYM
cana-2208	48	11	∞.	∞.	PROPN
cana-2208	48	12	3.2	3.2	NUM
cana-2208	48	13	theorem	theorem	NOUN
cana-2208	48	14	let	let	VERB
cana-2208	48	15	𝑝𝑟	𝑝𝑟	PRON
cana-2208	48	16	be	be	AUX
cana-2208	48	17	a	a	DET
cana-2208	48	18	none	none	NOUN
cana-2208	48	19	-	-	PUNCT
cana-2208	48	20	increasing	increase	VERB
cana-2208	48	21	positive	positive	ADJ
cana-2208	48	22	sequence	sequence	NOUN
cana-2208	48	23	of	of	ADP
cana-2208	48	24	real	real	ADJ
cana-2208	48	25	constant	constant	ADJ
cana-2208	48	26	such	such	ADJ
cana-2208	48	27	that	that	SCONJ
cana-2208	48	28	𝑝𝑟	𝑝𝑟	PROPN
cana-2208	48	29	=	=	PUNCT
cana-2208	48	30	∑𝑟	∑𝑟	PROPN
cana-2208	48	31	𝑧=0	𝑧=0	PROPN
cana-2208	48	32	𝑝𝑧	𝑝𝑧	PROPN
cana-2208	49	1	→	→	SYM
cana-2208	49	2	𝑠	𝑠	X
cana-2208	49	3	𝑎𝑠	𝑎𝑠	ADP
cana-2208	49	4	𝑟	𝑟	NOUN
cana-2208	49	5	→	→	SYM
cana-2208	49	6	∞	∞	PROPN
cana-2208	49	7	and	and	CCONJ
cana-2208	49	8	given	give	VERB
cana-2208	49	9	function	function	NOUN
cana-2208	49	10	𝜓(𝑡	𝜓(𝑡	NOUN
cana-2208	49	11	)	)	PUNCT
cana-2208	49	12	satisfies	satisfie	NOUN
cana-2208	49	13	,	,	PUNCT
cana-2208	49	14	ψ(𝑡	ψ(𝑡	NOUN
cana-2208	49	15	)	)	PUNCT
cana-2208	49	16	=	=	SYM
cana-2208	50	1	∫	∫	PROPN
cana-2208	50	2	𝑡	𝑡	PROPN
cana-2208	50	3	0	0	NUM
cana-2208	50	4	|𝜓(𝑢)|𝑑𝑢	|𝜓(𝑢)|𝑑𝑢	NOUN
cana-2208	51	1	=	=	PUNCT
cana-2208	51	2	𝑜	𝑜	X
cana-2208	51	3	[	[	PUNCT
cana-2208	51	4	𝑡	𝑡	X
cana-2208	51	5	𝛼	𝛼	PROPN
cana-2208	51	6	(	(	PUNCT
cana-2208	51	7	1	1	NUM
cana-2208	51	8	𝑡	𝑡	NOUN
cana-2208	51	9	)	)	PUNCT
cana-2208	51	10	𝑝𝑡	𝑝𝑡	NOUN
cana-2208	51	11	]	]	PUNCT
cana-2208	51	12	,	,	PUNCT
cana-2208	51	13	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	51	14	𝑡	𝑡	PROPN
cana-2208	51	15	→	→	SYM
cana-2208	51	16	0	0	NUM
cana-2208	52	1	+	+	ADJ
cana-2208	52	2	,	,	PUNCT
cana-2208	52	3	(	(	PUNCT
cana-2208	52	4	3.4	3.4	NUM
cana-2208	52	5	)	)	PUNCT
cana-2208	52	6	where	where	SCONJ
cana-2208	52	7	𝛼(𝑡	𝛼(𝑡	NOUN
cana-2208	52	8	)	)	PUNCT
cana-2208	52	9	is	be	AUX
cana-2208	52	10	positive	positive	ADJ
cana-2208	52	11	,	,	PUNCT
cana-2208	52	12	monotonic	monotonic	ADJ
cana-2208	52	13	and	and	CCONJ
cana-2208	52	14	none	none	NOUN
cana-2208	52	15	increasing	increase	VERB
cana-2208	52	16	function	function	NOUN
cana-2208	52	17	of	of	ADP
cana-2208	52	18	t.	t.	PROPN
cana-2208	52	19	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	52	20	=	=	SYM
cana-2208	52	21	𝑂[𝛼(𝑟	𝑂[𝛼(𝑟	PROPN
cana-2208	52	22	)	)	PUNCT
cana-2208	52	23	.	.	PUNCT
cana-2208	53	1	𝑝𝑟	𝑝𝑟	X
cana-2208	53	2	]	]	X
cana-2208	53	3	,	,	PUNCT
cana-2208	53	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	53	5	𝑟	𝑟	X
cana-2208	53	6	→	→	SYM
cana-2208	53	7	∞	∞	PROPN
cana-2208	53	8	,	,	PUNCT
cana-2208	53	9	(	(	PUNCT
cana-2208	53	10	3.5	3.5	NUM
cana-2208	53	11	)	)	PUNCT
cana-2208	53	12	then	then	ADV
cana-2208	53	13	approximation	approximation	NOUN
cana-2208	53	14	of	of	ADP
cana-2208	53	15	function	function	PROPN
cana-2208	53	16	�	�	PROPN
cana-2208	53	17	̃	̃	PROPN
cana-2208	53	18	�	�	PROPN
cana-2208	53	19	at	at	ADP
cana-2208	53	20	x	x	NOUN
cana-2208	53	21	=	=	NOUN
cana-2208	53	22	t	t	X
cana-2208	53	23	by	by	ADP
cana-2208	53	24	neular	neular	ADJ
cana-2208	53	25	product	product	NOUN
cana-2208	53	26	means	mean	NOUN
cana-2208	53	27	of	of	ADP
cana-2208	53	28	its	its	PRON
cana-2208	53	29	conjugate	conjugate	ADJ
cana-2208	53	30	fourier	fourier	NOUN
cana-2208	53	31	series	series	NOUN
cana-2208	53	32	is	be	AUX
cana-2208	53	33	given	give	VERB
cana-2208	53	34	by	by	ADP
cana-2208	53	35	|	|	PRON
cana-2208	53	36	�	�	PROPN
cana-2208	53	37	̃	̃	PROPN
cana-2208	53	38	�	�	NOUN
cana-2208	53	39	𝑟	𝑟	NOUN
cana-2208	53	40	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	53	41	...	...	PUNCT
cana-2208	54	1	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	54	2	−	−	PROPN
cana-2208	54	3	�	�	PROPN
cana-2208	54	4	̃	̃	NOUN
cana-2208	54	5	�	�	PROPN
cana-2208	54	6	(𝑥)|	(𝑥)|	NOUN
cana-2208	54	7	=	=	SYM
cana-2208	54	8	𝑂(1	𝑂(1	ADP
cana-2208	54	9	)	)	PUNCT
cana-2208	54	10	,	,	PUNCT
cana-2208	54	11	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	54	12	𝑟	𝑟	X
cana-2208	54	13	→	→	SYM
cana-2208	54	14	∞.	∞.	PROPN
cana-2208	54	15	4	4	NUM
cana-2208	54	16	lemmas	lemma	VERB
cana-2208	54	17	here	here	ADV
cana-2208	54	18	few	few	ADJ
cana-2208	54	19	lemmas	lemma	NOUN
cana-2208	54	20	are	be	AUX
cana-2208	54	21	given	give	VERB
cana-2208	54	22	,	,	PUNCT
cana-2208	54	23	which	which	PRON
cana-2208	54	24	are	be	AUX
cana-2208	54	25	useful	useful	ADJ
cana-2208	54	26	to	to	PART
cana-2208	54	27	prove	prove	VERB
cana-2208	54	28	our	our	PRON
cana-2208	54	29	theorem	theorem	ADJ
cana-2208	54	30	lemma	lemma	PROPN
cana-2208	54	31	4.1	4.1	NUM
cana-2208	54	32	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	54	33	=	=	PUNCT
cana-2208	54	34	𝑂(𝑟	𝑂(𝑟	X
cana-2208	54	35	)	)	PUNCT
cana-2208	54	36	,	,	PUNCT
cana-2208	54	37	for	for	ADP
cana-2208	54	38	0	0	NUM
cana-2208	54	39	≤t≤	≤t≤	NOUN
cana-2208	54	40	1	1	NUM
cana-2208	54	41	𝑟	𝑟	NOUN
cana-2208	54	42	.	.	PUNCT
cana-2208	55	1	proof	proof	NOUN
cana-2208	55	2	.	.	PUNCT
cana-2208	56	1	using	use	VERB
cana-2208	56	2	𝑠𝑖𝑛𝑟𝑡	𝑠𝑖𝑛𝑟𝑡	PROPN
cana-2208	56	3	≤	≤	PROPN
cana-2208	56	4	𝑟𝑡	𝑟𝑡	PROPN
cana-2208	56	5	and	and	CCONJ
cana-2208	56	6	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	56	7	𝑡	𝑡	PROPN
cana-2208	56	8	2	2	NUM
cana-2208	56	9	≥	≥	NOUN
cana-2208	56	10	𝑡	𝑡	NOUN
cana-2208	56	11	𝜋	𝜋	NOUN
cana-2208	56	12	,	,	PUNCT
cana-2208	56	13	we	we	PRON
cana-2208	56	14	have	have	VERB
cana-2208	56	15	communications	communication	NOUN
cana-2208	56	16	on	on	ADP
cana-2208	56	17	applied	apply	VERB
cana-2208	56	18	nonlinear	nonlinear	ADJ
cana-2208	56	19	analysis	analysis	NOUN
cana-2208	56	20	issn	issn	NOUN
cana-2208	56	21	:	:	PUNCT
cana-2208	56	22	1074	1074	NUM
cana-2208	56	23	-	-	PUNCT
cana-2208	56	24	133x	133x	NUM
cana-2208	56	25	vol	vol	NOUN
cana-2208	56	26	32	32	NUM
cana-2208	57	1	no	no	NOUN
cana-2208	57	2	.	.	PUNCT
cana-2208	58	1	1s	1s	NUM
cana-2208	58	2	(	(	PUNCT
cana-2208	58	3	2025	2025	NUM
cana-2208	58	4	)	)	PUNCT
cana-2208	58	5	448	448	NUM
cana-2208	58	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	58	7	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	58	8	=	=	SYM
cana-2208	58	9	1	1	NUM
cana-2208	58	10	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	58	11	|∑𝑟	|∑𝑟	NOUN
cana-2208	58	12	𝑗=0	𝑗=0	PUNCT
cana-2208	59	1	[	[	X
cana-2208	59	2	(	(	PUNCT
cana-2208	59	3	𝑟	𝑟	X
cana-2208	59	4	𝑗	𝑗	X
cana-2208	59	5	)	)	PUNCT
cana-2208	59	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	59	7	(	(	PUNCT
cana-2208	59	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	59	9	∑𝑗	∑𝑗	PROPN
cana-2208	59	10	ℎ=0	ℎ=0	PROPN
cana-2208	59	11	(	(	PUNCT
cana-2208	59	12	𝑗	𝑗	INTJ
cana-2208	59	13	ℎ	ℎ	PROPN
cana-2208	59	14	)	)	PUNCT
cana-2208	59	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	59	16	(	(	PUNCT
cana-2208	59	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	59	18	.	.	PUNCT
cana-2208	59	19	.	.	PUNCT
cana-2208	59	20	.	.	PUNCT
cana-2208	60	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	60	2	(	(	PUNCT
cana-2208	60	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	60	4	{	{	PUNCT
cana-2208	60	5	∑𝑖	∑𝑖	PROPN
cana-2208	60	6	𝑚=0	𝑚=0	PUNCT
cana-2208	60	7	(	(	PUNCT
cana-2208	60	8	𝑖	𝑖	NOUN
cana-2208	60	9	𝑚	𝑚	NOUN
cana-2208	60	10	)	)	PUNCT
cana-2208	60	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	60	12	𝑠𝑖𝑛(𝑚+	𝑠𝑖𝑛(𝑚+	ADJ
cana-2208	60	13	1	1	NUM
cana-2208	60	14	2	2	NUM
cana-2208	60	15	)	)	PUNCT
cana-2208	60	16	𝑡	𝑡	PROPN
cana-2208	60	17	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	60	18	𝑡	𝑡	ADP
cana-2208	60	19	2	2	NUM
cana-2208	60	20	}	}	PUNCT
cana-2208	60	21	]	]	PUNCT
cana-2208	60	22	|	|	ADV
cana-2208	60	23	≤	≤	NUM
cana-2208	60	24	1	1	NUM
cana-2208	60	25	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	60	26	|∑𝑟	|∑𝑟	NOUN
cana-2208	60	27	𝑗=0	𝑗=0	PUNCT
cana-2208	61	1	[	[	X
cana-2208	61	2	(	(	PUNCT
cana-2208	61	3	𝑟	𝑟	X
cana-2208	61	4	𝑗	𝑗	X
cana-2208	61	5	)	)	PUNCT
cana-2208	61	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	61	7	(	(	PUNCT
cana-2208	61	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	61	9	∑𝑗	∑𝑗	PROPN
cana-2208	61	10	ℎ=0	ℎ=0	PROPN
cana-2208	61	11	(	(	PUNCT
cana-2208	61	12	𝑗	𝑗	INTJ
cana-2208	61	13	ℎ	ℎ	PROPN
cana-2208	61	14	)	)	PUNCT
cana-2208	61	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	61	16	(	(	PUNCT
cana-2208	61	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	61	18	.	.	PUNCT
cana-2208	61	19	.	.	PUNCT
cana-2208	61	20	.	.	PUNCT
cana-2208	62	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	62	2	(	(	PUNCT
cana-2208	62	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	62	4	{	{	PUNCT
cana-2208	62	5	∑𝑖	∑𝑖	PROPN
cana-2208	62	6	𝑚=0	𝑚=0	PUNCT
cana-2208	62	7	(	(	PUNCT
cana-2208	62	8	𝑖	𝑖	NOUN
cana-2208	62	9	𝑚	𝑚	NOUN
cana-2208	62	10	)	)	PUNCT
cana-2208	62	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	62	12	(	(	PUNCT
cana-2208	62	13	𝑚+	𝑚+	X
cana-2208	62	14	1	1	NUM
cana-2208	62	15	2	2	NUM
cana-2208	62	16	)	)	PUNCT
cana-2208	62	17	𝑡	𝑡	PROPN
cana-2208	62	18	𝑡	𝑡	PROPN
cana-2208	62	19	𝜋	𝜋	NOUN
cana-2208	62	20	}	}	PUNCT
cana-2208	62	21	]	]	PUNCT
cana-2208	62	22	|	|	X
cana-2208	62	23	≤	≤	NUM
cana-2208	62	24	1	1	NUM
cana-2208	62	25	4𝜋(1+𝑞)𝑟	4𝜋(1+𝑞)𝑟	NOUN
cana-2208	62	26	∑𝑟	∑𝑟	NOUN
cana-2208	62	27	𝑗=0	𝑗=0	PUNCT
cana-2208	63	1	[	[	X
cana-2208	63	2	(	(	PUNCT
cana-2208	63	3	𝑟	𝑟	X
cana-2208	63	4	𝑗	𝑗	X
cana-2208	63	5	)	)	PUNCT
cana-2208	63	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	63	7	(	(	PUNCT
cana-2208	63	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	63	9	∑𝑗	∑𝑗	PROPN
cana-2208	63	10	ℎ=0	ℎ=0	PROPN
cana-2208	63	11	(	(	PUNCT
cana-2208	63	12	𝑗	𝑗	INTJ
cana-2208	63	13	ℎ	ℎ	PROPN
cana-2208	63	14	)	)	PUNCT
cana-2208	63	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	63	16	(	(	PUNCT
cana-2208	63	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	63	18	.	.	PUNCT
cana-2208	63	19	.	.	PUNCT
cana-2208	63	20	.	.	PUNCT
cana-2208	64	1	(	(	PUNCT
cana-2208	64	2	2𝑖	2𝑖	NOUN
cana-2208	64	3	+	+	CCONJ
cana-2208	64	4	1	1	X
cana-2208	64	5	)	)	PUNCT
cana-2208	64	6	𝑞𝑣−𝑖	𝑞𝑣−𝑖	NOUN
cana-2208	64	7	(	(	PUNCT
cana-2208	64	8	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	64	9	{	{	PUNCT
cana-2208	64	10	∑𝑖	∑𝑖	PROPN
cana-2208	64	11	𝑚=0	𝑚=0	PUNCT
cana-2208	64	12	(	(	PUNCT
cana-2208	64	13	𝑖	𝑖	NOUN
cana-2208	64	14	𝑚	𝑚	NOUN
cana-2208	64	15	)	)	PUNCT
cana-2208	64	16	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	64	17	}	}	PUNCT
cana-2208	64	18	]	]	PUNCT
cana-2208	64	19	≤	≤	NUM
cana-2208	64	20	1	1	NUM
cana-2208	64	21	4(1+𝑞)𝑟	4(1+𝑞)𝑟	NUM
cana-2208	64	22	∑𝑟	∑𝑟	NOUN
cana-2208	64	23	𝑗=0	𝑗=0	PUNCT
cana-2208	65	1	[	[	X
cana-2208	65	2	(	(	PUNCT
cana-2208	65	3	𝑟	𝑟	X
cana-2208	65	4	𝑗	𝑗	X
cana-2208	65	5	)	)	PUNCT
cana-2208	65	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	65	7	(	(	PUNCT
cana-2208	65	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	65	9	∑𝑗	∑𝑗	PROPN
cana-2208	65	10	ℎ=0	ℎ=0	PROPN
cana-2208	65	11	(	(	PUNCT
cana-2208	65	12	𝑗	𝑗	INTJ
cana-2208	65	13	ℎ	ℎ	PROPN
cana-2208	65	14	)	)	PUNCT
cana-2208	65	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	65	16	(	(	PUNCT
cana-2208	65	17	1+𝑞)ℎ	1+𝑞)ℎ	NOUN
cana-2208	65	18	∑ℎ	∑ℎ	VERB
cana-2208	65	19	𝑙=0	𝑙=0	PUNCT
cana-2208	65	20	(	(	PUNCT
cana-2208	65	21	ℎ	ℎ	X
cana-2208	65	22	𝑙	𝑙	NUM
cana-2208	65	23	)	)	PUNCT
cana-2208	65	24	𝑞ℎ−𝑙(2𝑙	𝑞ℎ−𝑙(2𝑙	ADJ
cana-2208	66	1	+	+	CCONJ
cana-2208	67	1	1	1	NUM
cana-2208	67	2	)	)	PUNCT
cana-2208	67	3	]	]	PUNCT
cana-2208	68	1	≤	≤	NUM
cana-2208	68	2	1	1	NUM
cana-2208	68	3	4(1+𝑞)𝑟	4(1+𝑞)𝑟	NUM
cana-2208	68	4	∑𝑟	∑𝑟	NOUN
cana-2208	68	5	𝑗=0	𝑗=0	PUNCT
cana-2208	69	1	[	[	X
cana-2208	69	2	(	(	PUNCT
cana-2208	69	3	𝑟	𝑟	X
cana-2208	69	4	𝑗	𝑗	X
cana-2208	69	5	)	)	PUNCT
cana-2208	69	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	69	7	(	(	PUNCT
cana-2208	69	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	69	9	∑𝑗	∑𝑗	PROPN
cana-2208	69	10	ℎ=0	ℎ=0	PROPN
cana-2208	69	11	(	(	PUNCT
cana-2208	69	12	𝑗	𝑗	INTJ
cana-2208	69	13	ℎ	ℎ	PROPN
cana-2208	69	14	)	)	PUNCT
cana-2208	69	15	𝑞𝑗−ℎ(2ℎ	𝑞𝑗−ℎ(2ℎ	PROPN
cana-2208	69	16	+	+	CCONJ
cana-2208	69	17	1	1	NUM
cana-2208	69	18	)	)	PUNCT
cana-2208	69	19	]	]	PUNCT
cana-2208	69	20	≤	≤	NUM
cana-2208	69	21	1	1	NUM
cana-2208	69	22	4(1+𝑞)𝑟	4(1+𝑞)𝑟	NUM
cana-2208	69	23	∑𝑟	∑𝑟	NOUN
cana-2208	69	24	𝑗=0	𝑗=0	PUNCT
cana-2208	70	1	[	[	X
cana-2208	70	2	(	(	PUNCT
cana-2208	70	3	𝑟	𝑟	X
cana-2208	70	4	𝑗	𝑗	X
cana-2208	70	5	)	)	PUNCT
cana-2208	70	6	𝑞𝑟−𝑗(2𝑗	𝑞𝑟−𝑗(2𝑗	NOUN
cana-2208	71	1	+	+	CCONJ
cana-2208	71	2	1	1	NUM
cana-2208	71	3	)	)	PUNCT
cana-2208	71	4	]	]	PUNCT
cana-2208	72	1	≤	≤	NUM
cana-2208	72	2	1	1	NUM
cana-2208	72	3	4	4	NUM
cana-2208	72	4	(	(	PUNCT
cana-2208	72	5	2𝑟	2𝑟	NOUN
cana-2208	72	6	+	+	CCONJ
cana-2208	72	7	1	1	X
cana-2208	72	8	)	)	PUNCT
cana-2208	72	9	=	=	PUNCT
cana-2208	72	10	𝑂(𝑟	𝑂(𝑟	X
cana-2208	72	11	)	)	PUNCT
cana-2208	72	12	lemma	lemma	PROPN
cana-2208	72	13	4.2	4.2	NUM
cana-2208	72	14	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	72	15	=	=	SYM
cana-2208	72	16	𝑂	𝑂	NOUN
cana-2208	72	17	(	(	PUNCT
cana-2208	72	18	1	1	NUM
cana-2208	72	19	𝑡	𝑡	PROPN
cana-2208	72	20	)	)	PUNCT
cana-2208	72	21	for	for	ADP
cana-2208	72	22	1	1	NUM
cana-2208	72	23	𝑟	𝑟	NOUN
cana-2208	72	24	≤	≤	NUM
cana-2208	72	25	𝑡	𝑡	VERB
cana-2208	72	26	≤	≤	ADJ
cana-2208	72	27	𝜋	𝜋	NOUN
cana-2208	72	28	proof	proof	NOUN
cana-2208	72	29	.	.	PUNCT
cana-2208	73	1	using	use	VERB
cana-2208	73	2	𝑠𝑖𝑛(𝑟𝑡	𝑠𝑖𝑛(𝑟𝑡	ADV
cana-2208	73	3	)	)	PUNCT
cana-2208	73	4	≤	≤	NUM
cana-2208	73	5	1	1	NUM
cana-2208	73	6	and	and	CCONJ
cana-2208	73	7	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	73	8	(	(	PUNCT
cana-2208	73	9	𝑡	𝑡	PROPN
cana-2208	73	10	2	2	NUM
cana-2208	73	11	)	)	PUNCT
cana-2208	73	12	≥	≥	NOUN
cana-2208	73	13	𝑡	𝑡	NOUN
cana-2208	73	14	𝜋	𝜋	NOUN
cana-2208	73	15	,	,	PUNCT
cana-2208	73	16	we	we	PRON
cana-2208	73	17	have	have	VERB
cana-2208	73	18	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	73	19	=	=	SYM
cana-2208	73	20	1	1	NUM
cana-2208	73	21	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	73	22	|∑𝑟	|∑𝑟	NOUN
cana-2208	73	23	𝑗=0	𝑗=0	PUNCT
cana-2208	74	1	[	[	X
cana-2208	74	2	(	(	PUNCT
cana-2208	74	3	𝑟	𝑟	X
cana-2208	74	4	𝑗	𝑗	X
cana-2208	74	5	)	)	PUNCT
cana-2208	74	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	74	7	(	(	PUNCT
cana-2208	74	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	74	9	∑𝑗	∑𝑗	PROPN
cana-2208	74	10	ℎ=0	ℎ=0	PROPN
cana-2208	74	11	(	(	PUNCT
cana-2208	74	12	𝑗	𝑗	INTJ
cana-2208	74	13	ℎ	ℎ	PROPN
cana-2208	74	14	)	)	PUNCT
cana-2208	74	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	74	16	(	(	PUNCT
cana-2208	74	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	74	18	.	.	PUNCT
cana-2208	74	19	.	.	PUNCT
cana-2208	74	20	.	.	PUNCT
cana-2208	75	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	75	2	(	(	PUNCT
cana-2208	75	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	75	4	{	{	PUNCT
cana-2208	75	5	∑𝑖	∑𝑖	PROPN
cana-2208	75	6	𝑚=0	𝑚=0	PUNCT
cana-2208	75	7	(	(	PUNCT
cana-2208	75	8	𝑖	𝑖	NOUN
cana-2208	75	9	𝑚	𝑚	NOUN
cana-2208	75	10	)	)	PUNCT
cana-2208	75	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	75	12	𝑠𝑖𝑛(𝑚+	𝑠𝑖𝑛(𝑚+	ADJ
cana-2208	75	13	1	1	NUM
cana-2208	75	14	2	2	NUM
cana-2208	75	15	)	)	PUNCT
cana-2208	75	16	𝑡	𝑡	PROPN
cana-2208	75	17	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	75	18	𝑡	𝑡	ADP
cana-2208	75	19	2	2	NUM
cana-2208	75	20	}	}	PUNCT
cana-2208	75	21	]	]	PUNCT
cana-2208	75	22	|	|	ADV
cana-2208	75	23	≤	≤	NUM
cana-2208	75	24	1	1	NUM
cana-2208	75	25	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	76	1	∑𝑟	∑𝑟	NOUN
cana-2208	76	2	𝑗=0	𝑗=0	PUNCT
cana-2208	77	1	[	[	X
cana-2208	77	2	(	(	PUNCT
cana-2208	77	3	𝑟	𝑟	X
cana-2208	77	4	𝑗	𝑗	X
cana-2208	77	5	)	)	PUNCT
cana-2208	77	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	77	7	(	(	PUNCT
cana-2208	77	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	77	9	∑𝑗	∑𝑗	PROPN
cana-2208	77	10	ℎ=0	ℎ=0	PROPN
cana-2208	77	11	(	(	PUNCT
cana-2208	77	12	𝑗	𝑗	INTJ
cana-2208	77	13	ℎ	ℎ	PROPN
cana-2208	77	14	)	)	PUNCT
cana-2208	77	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	77	16	(	(	PUNCT
cana-2208	77	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	77	18	.	.	PUNCT
cana-2208	77	19	.	.	PUNCT
cana-2208	77	20	.	.	PUNCT
cana-2208	78	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	78	2	(	(	PUNCT
cana-2208	78	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	78	4	{	{	PUNCT
cana-2208	78	5	∑𝑖	∑𝑖	PROPN
cana-2208	78	6	𝑚=0	𝑚=0	PUNCT
cana-2208	78	7	(	(	PUNCT
cana-2208	78	8	𝑖	𝑖	NOUN
cana-2208	78	9	𝑚	𝑚	NOUN
cana-2208	78	10	)	)	PUNCT
cana-2208	78	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	78	12	𝜋	𝜋	NOUN
cana-2208	78	13	𝑡	𝑡	PROPN
cana-2208	78	14	}	}	PUNCT
cana-2208	78	15	]	]	PUNCT
cana-2208	78	16	≤	≤	NUM
cana-2208	78	17	1	1	NUM
cana-2208	78	18	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	79	1	∑𝑟	∑𝑟	NOUN
cana-2208	79	2	𝑗=0	𝑗=0	PUNCT
cana-2208	80	1	[	[	X
cana-2208	80	2	(	(	PUNCT
cana-2208	80	3	𝑟	𝑟	X
cana-2208	80	4	𝑗	𝑗	X
cana-2208	80	5	)	)	PUNCT
cana-2208	80	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	80	7	(	(	PUNCT
cana-2208	80	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	80	9	∑𝑗	∑𝑗	PROPN
cana-2208	80	10	ℎ=0	ℎ=0	PROPN
cana-2208	80	11	(	(	PUNCT
cana-2208	80	12	𝑗	𝑗	INTJ
cana-2208	80	13	ℎ	ℎ	PROPN
cana-2208	80	14	)	)	PUNCT
cana-2208	80	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	80	16	(	(	PUNCT
cana-2208	80	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	80	18	.	.	PUNCT
cana-2208	80	19	.	.	PUNCT
cana-2208	80	20	.	.	PUNCT
cana-2208	81	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	81	2	(	(	PUNCT
cana-2208	81	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	81	4	{	{	PUNCT
cana-2208	81	5	∑𝑖	∑𝑖	PROPN
cana-2208	81	6	𝑚=0	𝑚=0	PUNCT
cana-2208	81	7	(	(	PUNCT
cana-2208	81	8	𝑖	𝑖	NOUN
cana-2208	81	9	𝑚	𝑚	NOUN
cana-2208	81	10	)	)	PUNCT
cana-2208	81	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	81	12	}	}	PUNCT
cana-2208	81	13	]	]	PUNCT
cana-2208	81	14	≤	≤	NUM
cana-2208	81	15	1	1	NUM
cana-2208	81	16	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	82	1	∑𝑟	∑𝑟	NOUN
cana-2208	82	2	𝑗=0	𝑗=0	PUNCT
cana-2208	83	1	[	[	X
cana-2208	83	2	(	(	PUNCT
cana-2208	83	3	𝑟	𝑟	X
cana-2208	83	4	𝑗	𝑗	X
cana-2208	83	5	)	)	PUNCT
cana-2208	83	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	83	7	(	(	PUNCT
cana-2208	83	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	83	9	∑𝑗	∑𝑗	PROPN
cana-2208	83	10	ℎ=0	ℎ=0	PROPN
cana-2208	83	11	(	(	PUNCT
cana-2208	83	12	𝑗	𝑗	INTJ
cana-2208	83	13	ℎ	ℎ	PROPN
cana-2208	83	14	)	)	PUNCT
cana-2208	83	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	83	16	]	]	PUNCT
cana-2208	83	17	≤	≤	NUM
cana-2208	83	18	1	1	NUM
cana-2208	83	19	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	83	20	∑𝑟	∑𝑟	NOUN
cana-2208	83	21	𝑗=0	𝑗=0	PUNCT
cana-2208	84	1	[	[	X
cana-2208	84	2	(	(	PUNCT
cana-2208	84	3	𝑟	𝑟	X
cana-2208	84	4	𝑗	𝑗	X
cana-2208	84	5	)	)	PUNCT
cana-2208	84	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	84	7	]	]	PUNCT
cana-2208	84	8	≤	≤	NUM
cana-2208	84	9	1	1	NUM
cana-2208	84	10	2𝑡	2𝑡	NOUN
cana-2208	84	11	=	=	SYM
cana-2208	84	12	𝑂	𝑂	NOUN
cana-2208	84	13	(	(	PUNCT
cana-2208	84	14	1	1	NUM
cana-2208	84	15	𝑡	𝑡	NOUN
cana-2208	84	16	)	)	PUNCT
cana-2208	84	17	.	.	PUNCT
cana-2208	85	1	lemma	lemma	PROPN
cana-2208	85	2	4.3	4.3	NUM
cana-2208	85	3	|𝐽(𝑡)|	|𝐽(𝑡)|	X
cana-2208	85	4	=	=	X
cana-2208	85	5	o	o	X
cana-2208	85	6	(	(	PUNCT
cana-2208	85	7	1	1	NUM
cana-2208	85	8	𝑡	𝑡	NOUN
cana-2208	85	9	)	)	PUNCT
cana-2208	85	10	,	,	PUNCT
cana-2208	85	11	for	for	ADP
cana-2208	85	12	0	0	NUM
cana-2208	85	13	≤	≤	NUM
cana-2208	85	14	𝑡	𝑡	NOUN
cana-2208	85	15	≤	≤	NUM
cana-2208	85	16	1	1	NUM
cana-2208	85	17	𝑟	𝑟	NOUN
cana-2208	85	18	.	.	PUNCT
cana-2208	86	1	proof	proof	NOUN
cana-2208	86	2	.	.	PUNCT
cana-2208	87	1	using	use	VERB
cana-2208	87	2	|cos(rt)|	|cos(rt)|	PROPN
cana-2208	87	3	≤	≤	NUM
cana-2208	87	4	1	1	NUM
cana-2208	87	5	,	,	PUNCT
cana-2208	87	6	we	we	PRON
cana-2208	87	7	have	have	VERB
cana-2208	87	8	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	87	9	=	=	SYM
cana-2208	87	10	1	1	NUM
cana-2208	87	11	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	87	12	|∑𝑟	|∑𝑟	NOUN
cana-2208	87	13	𝑗=0	𝑗=0	PUNCT
cana-2208	88	1	[	[	X
cana-2208	88	2	(	(	PUNCT
cana-2208	88	3	𝑟	𝑟	X
cana-2208	88	4	𝑗	𝑗	X
cana-2208	88	5	)	)	PUNCT
cana-2208	88	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	88	7	(	(	PUNCT
cana-2208	88	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	88	9	∑𝑗	∑𝑗	PROPN
cana-2208	88	10	ℎ=0	ℎ=0	PROPN
cana-2208	88	11	(	(	PUNCT
cana-2208	88	12	𝑗	𝑗	INTJ
cana-2208	88	13	ℎ	ℎ	PROPN
cana-2208	88	14	)	)	PUNCT
cana-2208	88	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	88	16	(	(	PUNCT
cana-2208	88	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	88	18	.	.	PUNCT
cana-2208	88	19	.	.	PUNCT
cana-2208	88	20	.	.	PUNCT
cana-2208	89	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	89	2	(	(	PUNCT
cana-2208	89	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	89	4	{	{	PUNCT
cana-2208	89	5	∑𝑖	∑𝑖	PROPN
cana-2208	89	6	𝑚=0	𝑚=0	PUNCT
cana-2208	89	7	(	(	PUNCT
cana-2208	89	8	𝑖	𝑖	NOUN
cana-2208	89	9	𝑚	𝑚	NOUN
cana-2208	89	10	)	)	PUNCT
cana-2208	89	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	89	12	𝑐𝑜𝑠(𝑚+	𝑐𝑜𝑠(𝑚+	ADP
cana-2208	89	13	1	1	NUM
cana-2208	89	14	2	2	NUM
cana-2208	89	15	)	)	PUNCT
cana-2208	89	16	𝑡	𝑡	PROPN
cana-2208	89	17	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	89	18	𝑡	𝑡	ADP
cana-2208	89	19	2	2	NUM
cana-2208	89	20	}	}	PUNCT
cana-2208	89	21	]	]	PUNCT
cana-2208	89	22	|	|	NOUN
cana-2208	89	23	communications	communication	NOUN
cana-2208	89	24	on	on	ADP
cana-2208	89	25	applied	apply	VERB
cana-2208	89	26	nonlinear	nonlinear	ADJ
cana-2208	89	27	analysis	analysis	NOUN
cana-2208	89	28	issn	issn	NOUN
cana-2208	89	29	:	:	PUNCT
cana-2208	89	30	1074	1074	NUM
cana-2208	89	31	-	-	PUNCT
cana-2208	89	32	133x	133x	NUM
cana-2208	89	33	vol	vol	NOUN
cana-2208	89	34	32	32	NUM
cana-2208	89	35	no	no	NOUN
cana-2208	89	36	.	.	PUNCT
cana-2208	90	1	1s	1s	NUM
cana-2208	90	2	(	(	PUNCT
cana-2208	90	3	2025	2025	NUM
cana-2208	90	4	)	)	PUNCT
cana-2208	90	5	449	449	NUM
cana-2208	90	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	90	7	≤	≤	NUM
cana-2208	90	8	1	1	NUM
cana-2208	90	9	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	90	10	∑𝑟	∑𝑟	NOUN
cana-2208	91	1	𝑗=0	𝑗=0	PUNCT
cana-2208	92	1	[	[	X
cana-2208	92	2	(	(	PUNCT
cana-2208	92	3	𝑟	𝑟	X
cana-2208	92	4	𝑗	𝑗	X
cana-2208	92	5	)	)	PUNCT
cana-2208	92	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	92	7	(	(	PUNCT
cana-2208	92	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	92	9	∑𝑗	∑𝑗	PROPN
cana-2208	92	10	ℎ=0	ℎ=0	PROPN
cana-2208	92	11	(	(	PUNCT
cana-2208	92	12	𝑗	𝑗	INTJ
cana-2208	92	13	ℎ	ℎ	PROPN
cana-2208	92	14	)	)	PUNCT
cana-2208	92	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	92	16	(	(	PUNCT
cana-2208	92	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	92	18	.	.	PUNCT
cana-2208	92	19	.	.	PUNCT
cana-2208	92	20	.	.	PUNCT
cana-2208	93	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	93	2	(	(	PUNCT
cana-2208	93	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	93	4	{	{	PUNCT
cana-2208	93	5	∑𝑖	∑𝑖	PROPN
cana-2208	93	6	𝑚=0	𝑚=0	PUNCT
cana-2208	93	7	(	(	PUNCT
cana-2208	93	8	𝑖	𝑖	NOUN
cana-2208	93	9	𝑚	𝑚	NOUN
cana-2208	93	10	)	)	PUNCT
cana-2208	93	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	93	12	𝜋	𝜋	NOUN
cana-2208	93	13	𝑡	𝑡	PROPN
cana-2208	93	14	}	}	PUNCT
cana-2208	93	15	]	]	PUNCT
cana-2208	93	16	≤	≤	NUM
cana-2208	93	17	1	1	NUM
cana-2208	93	18	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	94	1	∑𝑟	∑𝑟	NOUN
cana-2208	94	2	𝑗=0	𝑗=0	PUNCT
cana-2208	95	1	[	[	X
cana-2208	95	2	(	(	PUNCT
cana-2208	95	3	𝑟	𝑟	X
cana-2208	95	4	𝑗	𝑗	X
cana-2208	95	5	)	)	PUNCT
cana-2208	95	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	95	7	(	(	PUNCT
cana-2208	95	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	95	9	∑𝑗	∑𝑗	PROPN
cana-2208	95	10	ℎ=0	ℎ=0	PROPN
cana-2208	95	11	(	(	PUNCT
cana-2208	95	12	𝑗	𝑗	INTJ
cana-2208	95	13	ℎ	ℎ	PROPN
cana-2208	95	14	)	)	PUNCT
cana-2208	95	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	95	16	(	(	PUNCT
cana-2208	95	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	95	18	.	.	PUNCT
cana-2208	95	19	.	.	PUNCT
cana-2208	95	20	.	.	PUNCT
cana-2208	96	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	96	2	(	(	PUNCT
cana-2208	96	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	96	4	{	{	PUNCT
cana-2208	96	5	∑𝑖	∑𝑖	PROPN
cana-2208	96	6	𝑚=0	𝑚=0	PUNCT
cana-2208	96	7	(	(	PUNCT
cana-2208	96	8	𝑖	𝑖	NOUN
cana-2208	96	9	𝑚	𝑚	NOUN
cana-2208	96	10	)	)	PUNCT
cana-2208	96	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	96	12	}	}	PUNCT
cana-2208	96	13	]	]	PUNCT
cana-2208	96	14	≤	≤	NUM
cana-2208	96	15	1	1	NUM
cana-2208	96	16	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	97	1	∑𝑟	∑𝑟	NOUN
cana-2208	97	2	𝑗=0	𝑗=0	PUNCT
cana-2208	98	1	[	[	X
cana-2208	98	2	(	(	PUNCT
cana-2208	98	3	𝑟	𝑟	X
cana-2208	98	4	𝑗	𝑗	X
cana-2208	98	5	)	)	PUNCT
cana-2208	98	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	98	7	(	(	PUNCT
cana-2208	98	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	98	9	∑𝑗	∑𝑗	PROPN
cana-2208	98	10	ℎ=0	ℎ=0	PROPN
cana-2208	98	11	(	(	PUNCT
cana-2208	98	12	𝑗	𝑗	INTJ
cana-2208	98	13	ℎ	ℎ	PROPN
cana-2208	98	14	)	)	PUNCT
cana-2208	98	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	98	16	]	]	PUNCT
cana-2208	98	17	≤	≤	NUM
cana-2208	98	18	1	1	NUM
cana-2208	98	19	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	98	20	∑𝑟	∑𝑟	NOUN
cana-2208	98	21	𝑗=0	𝑗=0	PUNCT
cana-2208	99	1	[	[	X
cana-2208	99	2	(	(	PUNCT
cana-2208	99	3	𝑟	𝑟	X
cana-2208	99	4	𝑗	𝑗	X
cana-2208	99	5	)	)	PUNCT
cana-2208	99	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	99	7	]	]	PUNCT
cana-2208	99	8	≤	≤	NUM
cana-2208	99	9	1	1	NUM
cana-2208	99	10	2𝑡	2𝑡	NOUN
cana-2208	99	11	=	=	SYM
cana-2208	99	12	𝑂	𝑂	NOUN
cana-2208	99	13	(	(	PUNCT
cana-2208	99	14	1	1	NUM
cana-2208	99	15	𝑡	𝑡	NOUN
cana-2208	99	16	)	)	PUNCT
cana-2208	99	17	.	.	PUNCT
cana-2208	100	1	lemma	lemma	PROPN
cana-2208	100	2	4.4	4.4	NUM
cana-2208	101	1	|𝐽(𝑡)|	|𝐽(𝑡)|	X
cana-2208	101	2	=	=	SYM
cana-2208	101	3	o	o	X
cana-2208	101	4	(	(	PUNCT
cana-2208	101	5	1	1	NUM
cana-2208	101	6	𝑡	𝑡	NOUN
cana-2208	101	7	)	)	PUNCT
cana-2208	101	8	,	,	PUNCT
cana-2208	101	9	for	for	ADP
cana-2208	101	10	1	1	NUM
cana-2208	101	11	𝑟	𝑟	NOUN
cana-2208	101	12	≤t≤	≤t≤	ADV
cana-2208	101	13	𝜋.	𝜋.	ADJ
cana-2208	101	14	proof	proof	NOUN
cana-2208	101	15	.	.	PUNCT
cana-2208	102	1	appling	apple	VERB
cana-2208	102	2	sin	sin	PROPN
cana-2208	102	3	(	(	PUNCT
cana-2208	102	4	𝑡	𝑡	PROPN
cana-2208	102	5	2	2	NUM
cana-2208	102	6	)	)	PUNCT
cana-2208	102	7	≥	≥	NOUN
cana-2208	102	8	𝑡	𝑡	NOUN
cana-2208	102	9	𝜋	𝜋	NOUN
cana-2208	102	10	,	,	PUNCT
cana-2208	102	11	we	we	PRON
cana-2208	102	12	have	have	AUX
cana-2208	102	13	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	102	14	=	=	SYM
cana-2208	102	15	1	1	NUM
cana-2208	102	16	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	102	17	|∑𝑟	|∑𝑟	NOUN
cana-2208	102	18	𝑗=0	𝑗=0	PUNCT
cana-2208	103	1	[	[	X
cana-2208	103	2	(	(	PUNCT
cana-2208	103	3	𝑟	𝑟	X
cana-2208	103	4	𝑗	𝑗	X
cana-2208	103	5	)	)	PUNCT
cana-2208	103	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	103	7	(	(	PUNCT
cana-2208	103	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	103	9	∑𝑗	∑𝑗	PROPN
cana-2208	103	10	ℎ=0	ℎ=0	PROPN
cana-2208	103	11	(	(	PUNCT
cana-2208	103	12	𝑗	𝑗	INTJ
cana-2208	103	13	ℎ	ℎ	PROPN
cana-2208	103	14	)	)	PUNCT
cana-2208	103	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	103	16	(	(	PUNCT
cana-2208	103	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	103	18	.	.	PUNCT
cana-2208	103	19	.	.	PUNCT
cana-2208	103	20	.	.	PUNCT
cana-2208	104	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	104	2	(	(	PUNCT
cana-2208	104	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	104	4	{	{	PUNCT
cana-2208	104	5	∑𝑖	∑𝑖	PROPN
cana-2208	104	6	𝑚=0	𝑚=0	PUNCT
cana-2208	104	7	(	(	PUNCT
cana-2208	104	8	𝑖	𝑖	NOUN
cana-2208	104	9	𝑚	𝑚	NOUN
cana-2208	104	10	)	)	PUNCT
cana-2208	104	11	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	104	12	𝑐𝑜𝑠(𝑚+	𝑐𝑜𝑠(𝑚+	ADP
cana-2208	104	13	1	1	NUM
cana-2208	104	14	2	2	NUM
cana-2208	104	15	)	)	PUNCT
cana-2208	104	16	𝑡	𝑡	PROPN
cana-2208	104	17	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	104	18	𝑡	𝑡	ADP
cana-2208	104	19	2	2	NUM
cana-2208	104	20	}	}	PUNCT
cana-2208	104	21	]	]	PUNCT
cana-2208	104	22	|	|	X
cana-2208	104	23	≤	≤	ADV
cana-2208	104	24	1	1	NUM
cana-2208	104	25	2𝜋(1	2𝜋(1	NUM
cana-2208	104	26	+	+	CCONJ
cana-2208	104	27	𝑞)𝑟	𝑞)𝑟	X
cana-2208	104	28	|∑	|∑	VERB
cana-2208	104	29	𝑟	𝑟	X
cana-2208	104	30	𝑗=0	𝑗=0	PUNCT
cana-2208	105	1	[	[	X
cana-2208	105	2	(	(	PUNCT
cana-2208	105	3	𝑟	𝑟	X
cana-2208	105	4	𝑗	𝑗	X
cana-2208	105	5	)	)	PUNCT
cana-2208	105	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	105	7	(	(	PUNCT
cana-2208	105	8	1	1	NUM
cana-2208	105	9	+	+	CCONJ
cana-2208	105	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	105	11	∑	∑	PUNCT
cana-2208	105	12	𝑗	𝑗	INTJ
cana-2208	105	13	ℎ=0	ℎ=0	X
cana-2208	105	14	(	(	PUNCT
cana-2208	105	15	𝑗	𝑗	INTJ
cana-2208	105	16	ℎ	ℎ	PROPN
cana-2208	105	17	)	)	PUNCT
cana-2208	105	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	105	19	(	(	PUNCT
cana-2208	105	20	1	1	NUM
cana-2208	105	21	+	+	CCONJ
cana-2208	105	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	105	23	…	…	PUNCT
cana-2208	105	24	𝑞𝑣−𝑖	𝑞𝑣−𝑖	NOUN
cana-2208	105	25	(	(	PUNCT
cana-2208	105	26	1	1	NUM
cana-2208	105	27	+	+	NOUN
cana-2208	105	28	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	105	29	{	{	PUNCT
cana-2208	105	30	∑	∑	PART
cana-2208	105	31	𝑖	𝑖	PRON
cana-2208	105	32	𝑚=0	𝑚=0	PUNCT
cana-2208	105	33	(	(	PUNCT
cana-2208	105	34	𝑖	𝑖	NOUN
cana-2208	105	35	𝑚	𝑚	NOUN
cana-2208	105	36	)	)	PUNCT
cana-2208	105	37	𝑞𝑖−𝑚	𝑞𝑖−𝑚	ADJ
cana-2208	105	38	𝑐𝑜𝑠	𝑐𝑜𝑠	NOUN
cana-2208	105	39	(	(	PUNCT
cana-2208	105	40	𝑚	𝑚	PROPN
cana-2208	105	41	+	+	NOUN
cana-2208	105	42	1	1	NUM
cana-2208	105	43	2	2	NUM
cana-2208	105	44	)	)	PUNCT
cana-2208	105	45	𝑡	𝑡	PROPN
cana-2208	105	46	𝑡	𝑡	PROPN
cana-2208	105	47	𝜋	𝜋	NOUN
cana-2208	105	48	}	}	PUNCT
cana-2208	105	49	]	]	PUNCT
cana-2208	105	50	|	|	X
cana-2208	105	51	≤	≤	NUM
cana-2208	105	52	1	1	NUM
cana-2208	105	53	2𝑡(1	2𝑡(1	NUM
cana-2208	105	54	+	+	CCONJ
cana-2208	105	55	𝑞)𝑟	𝑞)𝑟	X
cana-2208	105	56	|∑	|∑	VERB
cana-2208	105	57	𝑟	𝑟	X
cana-2208	105	58	𝑗=0	𝑗=0	PUNCT
cana-2208	106	1	[	[	X
cana-2208	106	2	(	(	PUNCT
cana-2208	106	3	𝑟	𝑟	X
cana-2208	106	4	𝑗	𝑗	X
cana-2208	106	5	)	)	PUNCT
cana-2208	106	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	106	7	(	(	PUNCT
cana-2208	106	8	1	1	NUM
cana-2208	106	9	+	+	CCONJ
cana-2208	106	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	106	11	∑	∑	PUNCT
cana-2208	106	12	𝑗	𝑗	INTJ
cana-2208	106	13	ℎ=0	ℎ=0	X
cana-2208	106	14	(	(	PUNCT
cana-2208	106	15	𝑗	𝑗	INTJ
cana-2208	106	16	ℎ	ℎ	PROPN
cana-2208	106	17	)	)	PUNCT
cana-2208	106	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	106	19	(	(	PUNCT
cana-2208	106	20	1	1	NUM
cana-2208	106	21	+	+	CCONJ
cana-2208	106	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	106	23	…	…	PUNCT
cana-2208	106	24	𝑞𝑣−𝑖	𝑞𝑣−𝑖	NOUN
cana-2208	106	25	(	(	PUNCT
cana-2208	106	26	1	1	NUM
cana-2208	106	27	+	+	NOUN
cana-2208	106	28	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	107	1	𝑅𝑒	𝑅𝑒	PROPN
cana-2208	107	2	{	{	PUNCT
cana-2208	107	3	∑	∑	PROPN
cana-2208	107	4	𝑖	𝑖	PRON
cana-2208	107	5	𝑚=0	𝑚=0	PUNCT
cana-2208	107	6	(	(	PUNCT
cana-2208	107	7	𝑖	𝑖	NOUN
cana-2208	107	8	𝑚	𝑚	NOUN
cana-2208	107	9	)	)	PUNCT
cana-2208	107	10	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	NOUN
cana-2208	107	11	≤	≤	ADJ
cana-2208	107	12	1	1	NUM
cana-2208	107	13	2𝑡(1	2𝑡(1	NUM
cana-2208	107	14	+	+	CCONJ
cana-2208	107	15	𝑞)𝑟	𝑞)𝑟	X
cana-2208	107	16	|∑	|∑	VERB
cana-2208	108	1	𝜏−1	𝜏−1	PROPN
cana-2208	108	2	𝑗=0	𝑗=0	X
cana-2208	109	1	[	[	X
cana-2208	109	2	(	(	PUNCT
cana-2208	109	3	𝑟	𝑟	X
cana-2208	109	4	𝑗	𝑗	X
cana-2208	109	5	)	)	PUNCT
cana-2208	109	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	109	7	(	(	PUNCT
cana-2208	109	8	1	1	NUM
cana-2208	109	9	+	+	CCONJ
cana-2208	109	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	109	11	∑	∑	PUNCT
cana-2208	109	12	𝑗	𝑗	INTJ
cana-2208	109	13	ℎ=0	ℎ=0	X
cana-2208	109	14	(	(	PUNCT
cana-2208	109	15	𝑗	𝑗	INTJ
cana-2208	109	16	ℎ	ℎ	PROPN
cana-2208	109	17	)	)	PUNCT
cana-2208	109	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	109	19	(	(	PUNCT
cana-2208	109	20	1	1	NUM
cana-2208	109	21	+	+	CCONJ
cana-2208	109	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	109	23	.	.	PUNCT
cana-2208	109	24	.	.	PUNCT
cana-2208	109	25	.	.	PUNCT
cana-2208	110	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	110	2	(	(	PUNCT
cana-2208	110	3	1	1	NUM
cana-2208	110	4	+	+	NOUN
cana-2208	110	5	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	111	1	𝑅𝑒	𝑅𝑒	PROPN
cana-2208	111	2	{	{	PUNCT
cana-2208	111	3	∑	∑	PROPN
cana-2208	111	4	𝑖	𝑖	PRON
cana-2208	111	5	𝑚=0	𝑚=0	PUNCT
cana-2208	111	6	(	(	PUNCT
cana-2208	111	7	𝑖	𝑖	NOUN
cana-2208	111	8	𝑚	𝑚	NOUN
cana-2208	111	9	)	)	PUNCT
cana-2208	111	10	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	NOUN
cana-2208	111	11	+	+	X
cana-2208	111	12	1	1	NUM
cana-2208	111	13	2𝑡(1	2𝑡(1	NUM
cana-2208	111	14	+	+	CCONJ
cana-2208	111	15	𝑞)𝑟	𝑞)𝑟	X
cana-2208	111	16	|∑	|∑	X
cana-2208	111	17	𝑟	𝑟	X
cana-2208	112	1	𝑗=𝜏	𝑗=𝜏	X
cana-2208	113	1	[	[	X
cana-2208	113	2	(	(	PUNCT
cana-2208	113	3	𝑟	𝑟	X
cana-2208	113	4	𝑗	𝑗	X
cana-2208	113	5	)	)	PUNCT
cana-2208	113	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	113	7	(	(	PUNCT
cana-2208	113	8	1	1	NUM
cana-2208	113	9	+	+	CCONJ
cana-2208	113	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	113	11	∑	∑	PUNCT
cana-2208	113	12	𝑗	𝑗	INTJ
cana-2208	113	13	ℎ=0	ℎ=0	X
cana-2208	113	14	(	(	PUNCT
cana-2208	113	15	𝑗	𝑗	INTJ
cana-2208	113	16	ℎ	ℎ	PROPN
cana-2208	113	17	)	)	PUNCT
cana-2208	113	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	113	19	(	(	PUNCT
cana-2208	113	20	1	1	NUM
cana-2208	113	21	+	+	CCONJ
cana-2208	113	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	113	23	.	.	PUNCT
cana-2208	113	24	.	.	PUNCT
cana-2208	113	25	.	.	PUNCT
cana-2208	114	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	114	2	(	(	PUNCT
cana-2208	114	3	1	1	NUM
cana-2208	114	4	+	+	NOUN
cana-2208	114	5	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	115	1	𝑅𝑒	𝑅𝑒	PROPN
cana-2208	115	2	{	{	PUNCT
cana-2208	115	3	∑	∑	PROPN
cana-2208	115	4	𝑖	𝑖	PRON
cana-2208	115	5	𝑚=0	𝑚=0	PUNCT
cana-2208	115	6	(	(	PUNCT
cana-2208	115	7	𝑖	𝑖	NOUN
cana-2208	115	8	𝑚	𝑚	NOUN
cana-2208	115	9	)	)	PUNCT
cana-2208	115	10	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡}]|	NOUN
cana-2208	115	11	≤	≤	ADJ
cana-2208	115	12	1	1	NUM
cana-2208	115	13	2𝑡(1	2𝑡(1	NUM
cana-2208	115	14	+	+	X
cana-2208	115	15	𝑞)𝑟	𝑞)𝑟	X
cana-2208	115	16	∑	∑	PUNCT
cana-2208	115	17	𝜏−1	𝜏−1	PROPN
cana-2208	115	18	𝑗=0	𝑗=0	PUNCT
cana-2208	116	1	[	[	X
cana-2208	116	2	(	(	PUNCT
cana-2208	116	3	𝑟	𝑟	X
cana-2208	116	4	𝑗	𝑗	X
cana-2208	116	5	)	)	PUNCT
cana-2208	116	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	116	7	(	(	PUNCT
cana-2208	116	8	1	1	NUM
cana-2208	116	9	+	+	CCONJ
cana-2208	116	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	116	11	∑	∑	PUNCT
cana-2208	116	12	𝑗	𝑗	INTJ
cana-2208	116	13	ℎ=0	ℎ=0	X
cana-2208	116	14	(	(	PUNCT
cana-2208	116	15	𝑗	𝑗	INTJ
cana-2208	116	16	ℎ	ℎ	PROPN
cana-2208	116	17	)	)	PUNCT
cana-2208	116	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	116	19	(	(	PUNCT
cana-2208	116	20	1	1	NUM
cana-2208	116	21	+	+	CCONJ
cana-2208	116	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	116	23	.	.	PUNCT
cana-2208	116	24	.	.	PUNCT
cana-2208	116	25	.	.	PUNCT
cana-2208	117	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	117	2	(	(	PUNCT
cana-2208	117	3	1	1	NUM
cana-2208	117	4	+	+	NOUN
cana-2208	117	5	𝑞)𝑖	𝑞)𝑖	NOUN
cana-2208	118	1	𝑅𝑒	𝑅𝑒	PROPN
cana-2208	118	2	{	{	PUNCT
cana-2208	118	3	∑	∑	PROPN
cana-2208	118	4	𝑖	𝑖	PRON
cana-2208	118	5	𝑚=0	𝑚=0	PUNCT
cana-2208	118	6	(	(	PUNCT
cana-2208	118	7	𝑖	𝑖	NOUN
cana-2208	118	8	𝑚	𝑚	NOUN
cana-2208	118	9	)	)	PUNCT
cana-2208	118	10	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	118	11	}	}	PUNCT
cana-2208	118	12	]	]	PUNCT
cana-2208	119	1	|𝑒𝑖𝑚𝑡|	|𝑒𝑖𝑚𝑡|	CCONJ
cana-2208	119	2	+	+	SYM
cana-2208	119	3	1	1	NUM
cana-2208	119	4	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	120	1	∑𝑟	∑𝑟	NOUN
cana-2208	120	2	𝑗=𝜏	𝑗=𝜏	PROPN
cana-2208	121	1	[	[	X
cana-2208	121	2	(	(	PUNCT
cana-2208	121	3	𝑟	𝑟	X
cana-2208	121	4	𝑗	𝑗	X
cana-2208	121	5	)	)	PUNCT
cana-2208	121	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	121	7	(	(	PUNCT
cana-2208	121	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	121	9	∑𝑗	∑𝑗	PROPN
cana-2208	121	10	ℎ=0	ℎ=0	PROPN
cana-2208	121	11	(	(	PUNCT
cana-2208	121	12	𝑗	𝑗	INTJ
cana-2208	121	13	ℎ	ℎ	PROPN
cana-2208	121	14	)	)	PUNCT
cana-2208	121	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	121	16	(	(	PUNCT
cana-2208	121	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	121	18	.	.	PUNCT
cana-2208	121	19	.	.	PUNCT
cana-2208	121	20	.	.	PUNCT
cana-2208	122	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	122	2	(	(	PUNCT
cana-2208	122	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	122	4	𝑚𝑎𝑥0≤𝑚≤𝑖	𝑚𝑎𝑥0≤𝑚≤𝑖	ADV
cana-2208	122	5	∑𝑖	∑𝑖	ADJ
cana-2208	122	6	𝑚=0	𝑚=0	PUNCT
cana-2208	122	7	(	(	PUNCT
cana-2208	122	8	𝑖	𝑖	X
cana-2208	122	9	𝑚	𝑚	NOUN
cana-2208	122	10	)	)	PUNCT
cana-2208	122	11	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡	𝑞𝑖−𝑚𝑒𝑖𝑚𝑡	NOUN
cana-2208	122	12	]	]	X
cana-2208	122	13	communications	communication	NOUN
cana-2208	122	14	on	on	ADP
cana-2208	122	15	applied	apply	VERB
cana-2208	122	16	nonlinear	nonlinear	ADJ
cana-2208	122	17	analysis	analysis	NOUN
cana-2208	122	18	issn	issn	NOUN
cana-2208	122	19	:	:	PUNCT
cana-2208	122	20	1074	1074	NUM
cana-2208	122	21	-	-	PUNCT
cana-2208	122	22	133x	133x	NUM
cana-2208	122	23	vol	vol	NOUN
cana-2208	122	24	32	32	NUM
cana-2208	122	25	no	no	NOUN
cana-2208	122	26	.	.	PUNCT
cana-2208	123	1	1s	1s	NUM
cana-2208	123	2	(	(	PUNCT
cana-2208	123	3	2025	2025	NUM
cana-2208	123	4	)	)	PUNCT
cana-2208	123	5	450	450	NUM
cana-2208	123	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	123	7	≤	≤	NUM
cana-2208	123	8	1	1	NUM
cana-2208	123	9	2𝑡(1	2𝑡(1	NUM
cana-2208	123	10	+	+	X
cana-2208	123	11	𝑞)𝑟	𝑞)𝑟	X
cana-2208	123	12	∑	∑	PUNCT
cana-2208	123	13	𝜏−1	𝜏−1	PROPN
cana-2208	123	14	𝑗=0	𝑗=0	PUNCT
cana-2208	124	1	[	[	X
cana-2208	124	2	(	(	PUNCT
cana-2208	124	3	𝑟	𝑟	X
cana-2208	124	4	𝑗	𝑗	X
cana-2208	124	5	)	)	PUNCT
cana-2208	124	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	124	7	(	(	PUNCT
cana-2208	124	8	1	1	NUM
cana-2208	124	9	+	+	CCONJ
cana-2208	124	10	𝑞)𝑗	𝑞)𝑗	VERB
cana-2208	124	11	∑	∑	PUNCT
cana-2208	124	12	𝑗	𝑗	INTJ
cana-2208	124	13	ℎ=0	ℎ=0	X
cana-2208	124	14	(	(	PUNCT
cana-2208	124	15	𝑗	𝑗	INTJ
cana-2208	124	16	ℎ	ℎ	PROPN
cana-2208	124	17	)	)	PUNCT
cana-2208	124	18	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	124	19	(	(	PUNCT
cana-2208	124	20	1	1	NUM
cana-2208	124	21	+	+	CCONJ
cana-2208	124	22	𝑞)ℎ	𝑞)ℎ	ADJ
cana-2208	124	23	.	.	PUNCT
cana-2208	124	24	.	.	PUNCT
cana-2208	124	25	.	.	PUNCT
cana-2208	125	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	125	2	]	]	PUNCT
cana-2208	126	1	+	+	CCONJ
cana-2208	127	1	1	1	NUM
cana-2208	127	2	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	127	3	∑𝑟	∑𝑟	NOUN
cana-2208	127	4	𝑗=𝜏	𝑗=𝜏	PROPN
cana-2208	128	1	[	[	X
cana-2208	128	2	(	(	PUNCT
cana-2208	128	3	𝑟	𝑟	X
cana-2208	128	4	𝑗	𝑗	X
cana-2208	128	5	)	)	PUNCT
cana-2208	128	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	128	7	(	(	PUNCT
cana-2208	128	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	128	9	∑𝑗	∑𝑗	PROPN
cana-2208	128	10	ℎ=0	ℎ=0	PROPN
cana-2208	128	11	(	(	PUNCT
cana-2208	128	12	𝑗	𝑗	INTJ
cana-2208	128	13	ℎ	ℎ	PROPN
cana-2208	128	14	)	)	PUNCT
cana-2208	128	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	128	16	(	(	PUNCT
cana-2208	128	17	1+𝑞)ℎ	1+𝑞)ℎ	ADJ
cana-2208	128	18	.	.	PUNCT
cana-2208	128	19	.	.	PUNCT
cana-2208	128	20	.	.	PUNCT
cana-2208	129	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	129	2	]	]	PUNCT
cana-2208	129	3	≤	≤	NUM
cana-2208	129	4	1	1	NUM
cana-2208	129	5	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	129	6	∑𝜏−1	∑𝜏−1	NOUN
cana-2208	129	7	𝑗=0	𝑗=0	PUNCT
cana-2208	130	1	[	[	X
cana-2208	130	2	(	(	PUNCT
cana-2208	130	3	𝑟	𝑟	X
cana-2208	130	4	𝑗	𝑗	X
cana-2208	130	5	)	)	PUNCT
cana-2208	130	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	130	7	(	(	PUNCT
cana-2208	130	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	130	9	∑𝑗	∑𝑗	PROPN
cana-2208	130	10	ℎ=0	ℎ=0	PROPN
cana-2208	130	11	(	(	PUNCT
cana-2208	130	12	𝑗	𝑗	INTJ
cana-2208	130	13	ℎ	ℎ	PROPN
cana-2208	130	14	)	)	PUNCT
cana-2208	130	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	130	16	]	]	PUNCT
cana-2208	131	1	+	+	CCONJ
cana-2208	132	1	1	1	NUM
cana-2208	132	2	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	132	3	∑𝑟	∑𝑟	NOUN
cana-2208	132	4	𝑗=𝜏	𝑗=𝜏	PROPN
cana-2208	133	1	[	[	X
cana-2208	133	2	(	(	PUNCT
cana-2208	133	3	𝑟	𝑟	X
cana-2208	133	4	𝑗	𝑗	X
cana-2208	133	5	)	)	PUNCT
cana-2208	133	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	133	7	(	(	PUNCT
cana-2208	133	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	133	9	∑𝑗	∑𝑗	PROPN
cana-2208	133	10	ℎ=0	ℎ=0	PROPN
cana-2208	133	11	(	(	PUNCT
cana-2208	133	12	𝑗	𝑗	INTJ
cana-2208	133	13	ℎ	ℎ	PROPN
cana-2208	133	14	)	)	PUNCT
cana-2208	133	15	𝑞𝑗−ℎ	𝑞𝑗−ℎ	PROPN
cana-2208	133	16	]	]	PUNCT
cana-2208	133	17	=	=	SYM
cana-2208	133	18	1	1	NUM
cana-2208	133	19	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	133	20	∑𝜏−1	∑𝜏−1	ADP
cana-2208	134	1	𝑗=0	𝑗=0	PROPN
cana-2208	134	2	(	(	PUNCT
cana-2208	134	3	𝑟	𝑟	X
cana-2208	134	4	𝑗	𝑗	X
cana-2208	134	5	)	)	PUNCT
cana-2208	134	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	134	7	+	+	CCONJ
cana-2208	134	8	1	1	NUM
cana-2208	134	9	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	134	10	∑𝑟	∑𝑟	PROPN
cana-2208	134	11	𝑗=𝜏	𝑗=𝜏	PROPN
cana-2208	134	12	(	(	PUNCT
cana-2208	134	13	𝑟	𝑟	NOUN
cana-2208	134	14	𝑗	𝑗	X
cana-2208	134	15	)	)	PUNCT
cana-2208	134	16	𝑞𝑟−𝑗	𝑞𝑟−𝑗	NOUN
cana-2208	134	17	=	=	SYM
cana-2208	134	18	1	1	NUM
cana-2208	134	19	2𝑡(1+𝑞)𝑟	2𝑡(1+𝑞)𝑟	NOUN
cana-2208	135	1	∑𝑟	∑𝑟	NOUN
cana-2208	135	2	𝑗=0	𝑗=0	X
cana-2208	136	1	(	(	PUNCT
cana-2208	136	2	𝑟	𝑟	NOUN
cana-2208	136	3	𝑗	𝑗	X
cana-2208	136	4	)	)	PUNCT
cana-2208	136	5	𝑞𝑟−𝑗	𝑞𝑟−𝑗	NOUN
cana-2208	136	6	hence	hence	ADV
cana-2208	136	7	|𝐽𝑟(𝑡)|	|𝐽𝑟(𝑡)|	NOUN
cana-2208	136	8	=	=	SYM
cana-2208	136	9	𝑂(1/𝑡	𝑂(1/𝑡	PROPN
cana-2208	136	10	)	)	PUNCT
cana-2208	136	11	.	.	PUNCT
cana-2208	137	1	5	5	NUM
cana-2208	137	2	proof	proof	NOUN
cana-2208	137	3	of	of	ADP
cana-2208	137	4	theorem	theorem	ADJ
cana-2208	137	5	3.1	3.1	NUM
cana-2208	137	6	proof	proof	NOUN
cana-2208	137	7	.	.	PUNCT
cana-2208	138	1	we	we	PRON
cana-2208	138	2	have	have	VERB
cana-2208	138	3	𝑠𝑟(𝑔	𝑠𝑟(𝑔	NOUN
cana-2208	138	4	;	;	PUNCT
cana-2208	138	5	𝑥	𝑥	X
cana-2208	138	6	)	)	PUNCT
cana-2208	138	7	−	−	PROPN
cana-2208	138	8	𝑔(𝑥	𝑔(𝑥	NOUN
cana-2208	138	9	)	)	PUNCT
cana-2208	138	10	=	=	SYM
cana-2208	139	1	1	1	NUM
cana-2208	139	2	2𝜋	2𝜋	NUM
cana-2208	139	3	∫	∫	NOUN
cana-2208	139	4	𝜋	𝜋	NOUN
cana-2208	139	5	0	0	NUM
cana-2208	139	6	𝜙(𝑡)𝑠𝑖𝑛(𝑟+	𝜙(𝑡)𝑠𝑖𝑛(𝑟+	NOUN
cana-2208	139	7	1	1	NUM
cana-2208	139	8	2	2	NUM
cana-2208	139	9	)	)	PUNCT
cana-2208	139	10	𝑡	𝑡	PROPN
cana-2208	139	11	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2208	139	12	(	(	PUNCT
cana-2208	139	13	𝑡	𝑡	PROPN
cana-2208	139	14	2	2	NUM
cana-2208	139	15	)	)	PUNCT
cana-2208	139	16	𝑑𝑡	𝑑𝑡	ADP
cana-2208	139	17	,	,	PUNCT
cana-2208	139	18	and	and	CCONJ
cana-2208	139	19	𝑡𝑟	𝑡𝑟	VERB
cana-2208	139	20	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	139	21	...	...	PUNCT
cana-2208	140	1	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	140	2	(	(	PUNCT
cana-2208	140	3	𝑔	𝑔	NOUN
cana-2208	140	4	;	;	PUNCT
cana-2208	140	5	𝑥	𝑥	X
cana-2208	140	6	)	)	PUNCT
cana-2208	140	7	−	−	PROPN
cana-2208	140	8	𝑔(𝑥	𝑔(𝑥	NOUN
cana-2208	140	9	)	)	PUNCT
cana-2208	140	10	=	=	SYM
cana-2208	141	1	1	1	NUM
cana-2208	141	2	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	142	1	|	|	ADV
cana-2208	143	1	∑𝑟	∑𝑟	PROPN
cana-2208	144	1	𝑗=0	𝑗=0	PUNCT
cana-2208	145	1	[	[	X
cana-2208	145	2	(	(	PUNCT
cana-2208	145	3	𝑟	𝑟	X
cana-2208	145	4	𝑗	𝑗	X
cana-2208	145	5	)	)	PUNCT
cana-2208	145	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	145	7	(	(	PUNCT
cana-2208	145	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	145	9	∑𝑗	∑𝑗	PROPN
cana-2208	145	10	ℎ=0	ℎ=0	PROPN
cana-2208	145	11	(	(	PUNCT
cana-2208	145	12	𝑗	𝑗	INTJ
cana-2208	145	13	ℎ	ℎ	PROPN
cana-2208	145	14	)	)	PUNCT
cana-2208	145	15	.	.	PUNCT
cana-2208	145	16	.	.	PUNCT
cana-2208	145	17	.	.	PUNCT
cana-2208	146	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	146	2	(	(	PUNCT
cana-2208	146	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	146	4	∫	∫	PROPN
cana-2208	146	5	𝜋	𝜋	NOUN
cana-2208	146	6	0	0	NUM
cana-2208	146	7	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	146	8	)	)	PUNCT
cana-2208	146	9	{	{	PUNCT
cana-2208	146	10	∑𝑖	∑𝑖	NOUN
cana-2208	146	11	𝑚=0	𝑚=0	PUNCT
cana-2208	146	12	(	(	PUNCT
cana-2208	146	13	𝑖	𝑖	NOUN
cana-2208	146	14	𝑚	𝑚	NOUN
cana-2208	146	15	)	)	PUNCT
cana-2208	146	16	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	146	17	𝑠𝑖𝑛(𝑚+	𝑠𝑖𝑛(𝑚+	ADJ
cana-2208	146	18	1	1	NUM
cana-2208	146	19	2	2	NUM
cana-2208	146	20	)	)	PUNCT
cana-2208	146	21	𝑡	𝑡	PROPN
cana-2208	146	22	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	146	23	𝑡	𝑡	ADP
cana-2208	146	24	2	2	NUM
cana-2208	146	25	}	}	PUNCT
cana-2208	146	26	]	]	PUNCT
cana-2208	147	1	|	|	NOUN
cana-2208	147	2	=	=	SYM
cana-2208	147	3	∫	∫	PROPN
cana-2208	147	4	𝜋	𝜋	NOUN
cana-2208	147	5	0	0	NOUN
cana-2208	147	6	𝜙(𝑡)𝐽𝑟(𝑡)𝑑𝑡	𝜙(𝑡)𝐽𝑟(𝑡)𝑑𝑡	NOUN
cana-2208	147	7	=	=	PUNCT
cana-2208	148	1	[	[	X
cana-2208	148	2	∫	∫	X
cana-2208	148	3	1	1	NUM
cana-2208	148	4	𝑟	𝑟	NOUN
cana-2208	148	5	0	0	PUNCT
cana-2208	148	6	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	148	7	)	)	PUNCT
cana-2208	149	1	+	+	CCONJ
cana-2208	149	2	∫	∫	PROPN
cana-2208	149	3	𝛾	𝛾	PROPN
cana-2208	149	4	1	1	NUM
cana-2208	149	5	𝑟	𝑟	NOUN
cana-2208	149	6	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	149	7	)	)	PUNCT
cana-2208	150	1	+	+	CCONJ
cana-2208	150	2	∫	∫	PROPN
cana-2208	150	3	𝜋	𝜋	NOUN
cana-2208	150	4	𝛾	𝛾	PROPN
cana-2208	150	5	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	150	6	)	)	PUNCT
cana-2208	150	7	]	]	PUNCT
cana-2208	150	8	𝐽𝑟(𝑡)𝑑𝑡	𝐽𝑟(𝑡)𝑑𝑡	NOUN
cana-2208	150	9	=	=	PUNCT
cana-2208	150	10	𝑅1	𝑅1	PROPN
cana-2208	150	11	+	+	NUM
cana-2208	150	12	𝑅2	𝑅2	NOUN
cana-2208	150	13	+	+	ADJ
cana-2208	150	14	𝑅3	𝑅3	NOUN
cana-2208	150	15	(	(	PUNCT
cana-2208	150	16	5.1	5.1	NUM
cana-2208	150	17	)	)	PUNCT
cana-2208	150	18	applying	apply	VERB
cana-2208	150	19	lemma	lemma	PROPN
cana-2208	150	20	4.1	4.1	NUM
cana-2208	150	21	,	,	PUNCT
cana-2208	150	22	condion	condion	NOUN
cana-2208	150	23	3.2	3.2	NUM
cana-2208	150	24	and	and	CCONJ
cana-2208	150	25	3.3	3.3	NUM
cana-2208	150	26	and	and	CCONJ
cana-2208	150	27	second	second	ADJ
cana-2208	150	28	mean	mean	NOUN
cana-2208	150	29	value	value	NOUN
cana-2208	150	30	theorem	theorem	NOUN
cana-2208	150	31	is	be	AUX
cana-2208	150	32	applying	apply	VERB
cana-2208	150	33	for	for	ADP
cana-2208	150	34	second	second	ADJ
cana-2208	150	35	term	term	NOUN
cana-2208	150	36	integral	integral	ADJ
cana-2208	150	37	,	,	PUNCT
cana-2208	150	38	we	we	PRON
cana-2208	150	39	have	have	VERB
cana-2208	150	40	|𝑅1|	|𝑅1|	NOUN
cana-2208	150	41	≤	≤	NUM
cana-2208	150	42	∫	∫	PROPN
cana-2208	150	43	1	1	NUM
cana-2208	150	44	𝑟	𝑟	NOUN
cana-2208	150	45	0	0	NUM
cana-2208	150	46	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	NUM
cana-2208	150	47	=	=	SYM
cana-2208	150	48	𝑂(𝑟	𝑂(𝑟	NUM
cana-2208	150	49	)	)	PUNCT
cana-2208	151	1	[	[	X
cana-2208	151	2	∫	∫	X
cana-2208	151	3	1	1	NUM
cana-2208	151	4	𝑟	𝑟	PRON
cana-2208	151	5	𝑜	𝑜	X
cana-2208	151	6	|𝜙(𝑡)|𝑑𝑡	|𝜙(𝑡)|𝑑𝑡	X
cana-2208	151	7	]	]	X
cana-2208	151	8	=	=	PUNCT
cana-2208	151	9	𝑂(𝑟	𝑂(𝑟	X
cana-2208	151	10	)	)	PUNCT
cana-2208	152	1	[	[	X
cana-2208	152	2	𝑂	𝑂	PROPN
cana-2208	152	3	(	(	PUNCT
cana-2208	152	4	1	1	NUM
cana-2208	152	5	𝑟	𝑟	NOUN
cana-2208	152	6	𝛼(𝑟).𝑃𝑟	𝛼(𝑟).𝑃𝑟	ADJ
cana-2208	152	7	)	)	PUNCT
cana-2208	152	8	]	]	PUNCT
cana-2208	152	9	=	=	PUNCT
cana-2208	152	10	𝑂	𝑂	PROPN
cana-2208	152	11	(	(	PUNCT
cana-2208	152	12	1	1	NUM
cana-2208	152	13	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	152	14	)	)	PUNCT
cana-2208	152	15	=	=	PUNCT
cana-2208	153	1	𝑂(1	𝑂(1	ADP
cana-2208	153	2	)	)	PUNCT
cana-2208	153	3	𝑎𝑠	𝑎𝑠	SCONJ
cana-2208	153	4	𝑟	𝑟	NOUN
cana-2208	153	5	→	→	SYM
cana-2208	153	6	∞	∞	PROPN
cana-2208	153	7	(	(	PUNCT
cana-2208	153	8	5.2	5.2	NUM
cana-2208	153	9	)	)	PUNCT
cana-2208	153	10	communications	communication	NOUN
cana-2208	153	11	on	on	ADP
cana-2208	153	12	applied	apply	VERB
cana-2208	153	13	nonlinear	nonlinear	ADJ
cana-2208	153	14	analysis	analysis	NOUN
cana-2208	153	15	issn	issn	NOUN
cana-2208	153	16	:	:	PUNCT
cana-2208	153	17	1074	1074	NUM
cana-2208	153	18	-	-	PUNCT
cana-2208	153	19	133x	133x	NUM
cana-2208	153	20	vol	vol	NOUN
cana-2208	153	21	32	32	NUM
cana-2208	153	22	no	no	NOUN
cana-2208	153	23	.	.	PUNCT
cana-2208	154	1	1s	1s	NUM
cana-2208	154	2	(	(	PUNCT
cana-2208	154	3	2025	2025	NUM
cana-2208	154	4	)	)	PUNCT
cana-2208	154	5	451	451	NUM
cana-2208	154	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	154	7	apply	apply	VERB
cana-2208	154	8	lemma	lemma	PROPN
cana-2208	154	9	4.2	4.2	NUM
cana-2208	154	10	,	,	PUNCT
cana-2208	154	11	condition	condition	NOUN
cana-2208	154	12	3.2	3.2	NUM
cana-2208	154	13	and	and	CCONJ
cana-2208	154	14	3.3	3.3	NUM
cana-2208	154	15	and	and	CCONJ
cana-2208	154	16	second	second	ADJ
cana-2208	154	17	mean	mean	NOUN
cana-2208	154	18	value	value	NOUN
cana-2208	154	19	theorem	theorem	NOUN
cana-2208	154	20	is	be	AUX
cana-2208	154	21	appling	apple	VERB
cana-2208	154	22	for	for	ADP
cana-2208	154	23	second	second	ADJ
cana-2208	154	24	term	term	NOUN
cana-2208	154	25	integral	integral	ADJ
cana-2208	154	26	,	,	PUNCT
cana-2208	154	27	we	we	PRON
cana-2208	154	28	have	have	VERB
cana-2208	154	29	|𝑅2|	|𝑅2|	ADJ
cana-2208	154	30	≤	≤	NUM
cana-2208	154	31	∫	∫	PROPN
cana-2208	154	32	𝛾	𝛾	PROPN
cana-2208	154	33	1	1	NUM
cana-2208	154	34	𝑟	𝑟	NOUN
cana-2208	154	35	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	NOUN
cana-2208	154	36	=	=	SYM
cana-2208	154	37	𝑂	𝑂	PROPN
cana-2208	155	1	[	[	X
cana-2208	155	2	∫	∫	PROPN
cana-2208	155	3	𝛾	𝛾	ADP
cana-2208	155	4	1	1	NUM
cana-2208	155	5	𝑟	𝑟	NOUN
cana-2208	155	6	|𝜙(𝑡)|	|𝜙(𝑡)|	SYM
cana-2208	155	7	(	(	PUNCT
cana-2208	155	8	1	1	NUM
cana-2208	155	9	𝑡	𝑡	NOUN
cana-2208	155	10	)	)	PUNCT
cana-2208	155	11	]	]	PUNCT
cana-2208	156	1	=	=	PUNCT
cana-2208	156	2	𝑂	𝑂	PROPN
cana-2208	156	3	[	[	PUNCT
cana-2208	156	4	{	{	PUNCT
cana-2208	156	5	1	1	NUM
cana-2208	156	6	𝑡	𝑡	PROPN
cana-2208	156	7	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	156	8	)	)	PUNCT
cana-2208	156	9	}	}	PUNCT
cana-2208	157	1	1	1	NUM
cana-2208	157	2	𝑟	𝑟	NOUN
cana-2208	157	3	𝛾	𝛾	NOUN
cana-2208	157	4	+	+	X
cana-2208	157	5	∫	∫	PROPN
cana-2208	157	6	𝛾	𝛾	PROPN
cana-2208	157	7	1	1	NUM
cana-2208	157	8	𝑟	𝑟	NOUN
cana-2208	157	9	𝑜	𝑜	X
cana-2208	157	10	{	{	PUNCT
cana-2208	157	11	𝜙(𝑡	𝜙(𝑡	PROPN
cana-2208	157	12	)	)	PUNCT
cana-2208	157	13	𝑡2	𝑡2	NOUN
cana-2208	157	14	}	}	PUNCT
cana-2208	157	15	𝑑𝑡	𝑑𝑡	PROPN
cana-2208	157	16	]	]	PUNCT
cana-2208	157	17	=	=	SYM
cana-2208	157	18	𝑂	𝑂	PROPN
cana-2208	158	1	[	[	NOUN
cana-2208	158	2	𝑜	𝑜	X
cana-2208	158	3	{	{	PUNCT
cana-2208	158	4	1	1	NUM
cana-2208	158	5	𝛼	𝛼	PROPN
cana-2208	158	6	(	(	PUNCT
cana-2208	158	7	1	1	NUM
cana-2208	158	8	𝑡	𝑡	NOUN
cana-2208	158	9	)	)	PUNCT
cana-2208	158	10	.𝑃𝑡	.𝑃𝑡	PUNCT
cana-2208	158	11	}	}	PUNCT
cana-2208	159	1	1	1	NUM
cana-2208	159	2	𝑟	𝑟	NOUN
cana-2208	159	3	𝛾	𝛾	NOUN
cana-2208	159	4	+	+	ADJ
cana-2208	159	5	∫	∫	PROPN
cana-2208	159	6	𝑟	𝑟	NOUN
cana-2208	159	7	1	1	NUM
cana-2208	159	8	𝑟	𝑟	X
cana-2208	159	9	𝑜	𝑜	X
cana-2208	159	10	(	(	PUNCT
cana-2208	159	11	1	1	NUM
cana-2208	159	12	𝑡	𝑡	NOUN
cana-2208	159	13	𝛼	𝛼	PROPN
cana-2208	159	14	(	(	PUNCT
cana-2208	159	15	1	1	NUM
cana-2208	159	16	𝑡	𝑡	PROPN
cana-2208	159	17	)	)	PUNCT
cana-2208	159	18	.𝑃𝑡	.𝑃𝑡	PUNCT
cana-2208	159	19	)	)	PUNCT
cana-2208	159	20	𝑑𝑡	𝑑𝑡	ADP
cana-2208	159	21	]	]	PUNCT
cana-2208	159	22	=	=	SYM
cana-2208	159	23	𝑂	𝑂	PROPN
cana-2208	160	1	[	[	NOUN
cana-2208	160	2	𝑜	𝑜	X
cana-2208	160	3	{	{	PUNCT
cana-2208	160	4	1	1	NUM
cana-2208	160	5	𝛼(𝑟)𝑃𝑟	𝛼(𝑟)𝑃𝑟	NOUN
cana-2208	160	6	}	}	PUNCT
cana-2208	160	7	+	+	NUM
cana-2208	160	8	∫	∫	PROPN
cana-2208	160	9	𝑟	𝑟	NOUN
cana-2208	160	10	1	1	NUM
cana-2208	160	11	𝛾	𝛾	NOUN
cana-2208	160	12	𝑜	𝑜	PRON
cana-2208	160	13	(	(	PUNCT
cana-2208	160	14	1	1	NUM
cana-2208	160	15	𝑢	𝑢	NOUN
cana-2208	160	16	𝛼(𝑢)𝑝𝑢	𝛼(𝑢)𝑝𝑢	NUM
cana-2208	160	17	)	)	PUNCT
cana-2208	160	18	𝑑𝑢	𝑑𝑢	X
cana-2208	160	19	]	]	X
cana-2208	160	20	=	=	SYM
cana-2208	160	21	𝑂	𝑂	PROPN
cana-2208	160	22	(	(	PUNCT
cana-2208	160	23	1	1	NUM
cana-2208	160	24	𝛼(𝑟)𝑃𝑟	𝛼(𝑟)𝑃𝑟	NOUN
cana-2208	160	25	)	)	PUNCT
cana-2208	161	1	+	+	CCONJ
cana-2208	161	2	𝑂	𝑂	PROPN
cana-2208	161	3	(	(	PUNCT
cana-2208	161	4	1	1	NUM
cana-2208	161	5	𝑟	𝑟	NOUN
cana-2208	161	6	𝛼(𝑟)𝑝𝑟	𝛼(𝑟)𝑝𝑟	NUM
cana-2208	161	7	)	)	PUNCT
cana-2208	161	8	∫	∫	PROPN
cana-2208	161	9	𝑟	𝑟	NOUN
cana-2208	161	10	1	1	NUM
cana-2208	161	11	𝑟	𝑟	NUM
cana-2208	161	12	1𝑑𝑢	1𝑑𝑢	ADJ
cana-2208	161	13	=	=	SYM
cana-2208	161	14	𝑂	𝑂	NOUN
cana-2208	161	15	(	(	PUNCT
cana-2208	161	16	1	1	NUM
cana-2208	161	17	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	161	18	)	)	PUNCT
cana-2208	162	1	+	+	CCONJ
cana-2208	162	2	𝑂	𝑂	PROPN
cana-2208	162	3	(	(	PUNCT
cana-2208	162	4	1	1	NUM
cana-2208	162	5	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	162	6	)	)	PUNCT
cana-2208	162	7	=	=	PUNCT
cana-2208	163	1	𝑂(1	𝑂(1	ADP
cana-2208	163	2	)	)	PUNCT
cana-2208	163	3	,	,	PUNCT
cana-2208	163	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	163	5	𝑟	𝑟	X
cana-2208	163	6	→	→	SYM
cana-2208	163	7	∞	∞	PROPN
cana-2208	163	8	(	(	PUNCT
cana-2208	163	9	5.3	5.3	NUM
cana-2208	163	10	)	)	PUNCT
cana-2208	163	11	applying	apply	VERB
cana-2208	163	12	riemann	riemann	PROPN
cana-2208	163	13	-	-	PUNCT
cana-2208	163	14	lebesgue	lebesgue	NOUN
cana-2208	163	15	theorem	theorem	NOUN
cana-2208	163	16	and	and	CCONJ
cana-2208	163	17	regularity	regularity	NOUN
cana-2208	163	18	condition	condition	NOUN
cana-2208	163	19	of	of	ADP
cana-2208	163	20	summability	summability	NOUN
cana-2208	163	21	,	,	PUNCT
cana-2208	163	22	we	we	PRON
cana-2208	163	23	have	have	VERB
cana-2208	163	24	|𝑅3|	|𝑅3|	VERB
cana-2208	163	25	≤	≤	NUM
cana-2208	163	26	∫	∫	PROPN
cana-2208	163	27	𝛾	𝛾	PROPN
cana-2208	163	28	1	1	NUM
cana-2208	163	29	𝑟	𝑟	NOUN
cana-2208	163	30	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜙(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	PUNCT
cana-2208	163	31	=	=	SYM
cana-2208	164	1	𝑂(1	𝑂(1	ADP
cana-2208	164	2	)	)	PUNCT
cana-2208	164	3	,	,	PUNCT
cana-2208	164	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	164	5	𝑟	𝑟	X
cana-2208	164	6	→	→	SYM
cana-2208	164	7	∞.	∞.	PROPN
cana-2208	164	8	(	(	PUNCT
cana-2208	164	9	5.4	5.4	NUM
cana-2208	164	10	)	)	PUNCT
cana-2208	164	11	collecting	collecting	NOUN
cana-2208	164	12	(	(	PUNCT
cana-2208	164	13	5.1)-(5.4	5.1)-(5.4	NUM
cana-2208	164	14	)	)	PUNCT
cana-2208	164	15	,	,	PUNCT
cana-2208	164	16	we	we	PRON
cana-2208	164	17	get	get	VERB
cana-2208	164	18	|𝑡𝑟	|𝑡𝑟	ADV
cana-2208	164	19	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	164	20	...	...	PUNCT
cana-2208	165	1	𝐸𝑞(𝑔	𝐸𝑞(𝑔	ADJ
cana-2208	165	2	;	;	PUNCT
cana-2208	165	3	𝑥	𝑥	X
cana-2208	165	4	)	)	PUNCT
cana-2208	165	5	−	−	PROPN
cana-2208	165	6	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	165	7	=	=	PUNCT
cana-2208	165	8	𝑂(1	𝑂(1	ADP
cana-2208	165	9	)	)	PUNCT
cana-2208	165	10	,	,	PUNCT
cana-2208	165	11	𝑎𝑠	𝑎𝑠	ADP
cana-2208	165	12	𝑟	𝑟	X
cana-2208	165	13	→	→	SYM
cana-2208	165	14	∞.	∞.	PROPN
cana-2208	165	15	hence	hence	ADV
cana-2208	165	16	complete	complete	VERB
cana-2208	165	17	the	the	DET
cana-2208	165	18	proof	proof	NOUN
cana-2208	165	19	.	.	PUNCT
cana-2208	166	1	6	6	NUM
cana-2208	166	2	proof	proof	NOUN
cana-2208	166	3	of	of	ADP
cana-2208	166	4	theorem	theorem	ADJ
cana-2208	166	5	3.2	3.2	NUM
cana-2208	166	6	proof	proof	NOUN
cana-2208	166	7	.	.	PUNCT
cana-2208	167	1	we	we	PRON
cana-2208	167	2	have	have	VERB
cana-2208	167	3	�	�	PROPN
cana-2208	167	4	̃	̃	NOUN
cana-2208	167	5	�	�	NOUN
cana-2208	167	6	𝑟(𝑔	𝑟(𝑔	NOUN
cana-2208	167	7	;	;	PUNCT
cana-2208	167	8	𝑥	𝑥	X
cana-2208	167	9	)	)	PUNCT
cana-2208	167	10	−	−	PROPN
cana-2208	167	11	�	�	PROPN
cana-2208	167	12	̃	̃	PROPN
cana-2208	167	13	�	�	NOUN
cana-2208	167	14	(𝑥	(𝑥	NOUN
cana-2208	167	15	)	)	PUNCT
cana-2208	167	16	=	=	SYM
cana-2208	167	17	1	1	NUM
cana-2208	167	18	2𝜋	2𝜋	NUM
cana-2208	167	19	∫	∫	NOUN
cana-2208	167	20	𝜋	𝜋	NOUN
cana-2208	167	21	0	0	NUM
cana-2208	167	22	𝜓(𝑡)𝑐𝑜𝑠(𝑟+	𝜓(𝑡)𝑐𝑜𝑠(𝑟+	NOUN
cana-2208	167	23	1	1	NUM
cana-2208	167	24	2	2	NUM
cana-2208	167	25	)	)	PUNCT
cana-2208	167	26	𝑡	𝑡	PROPN
cana-2208	167	27	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2208	167	28	(	(	PUNCT
cana-2208	167	29	𝑡	𝑡	PROPN
cana-2208	167	30	2	2	NUM
cana-2208	167	31	)	)	PUNCT
cana-2208	167	32	𝑑𝑡	𝑑𝑡	ADP
cana-2208	167	33	and	and	CCONJ
cana-2208	167	34	�	�	PROPN
cana-2208	167	35	̃	̃	PROPN
cana-2208	167	36	�	�	NOUN
cana-2208	167	37	𝑟	𝑟	PART
cana-2208	167	38	𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞	PROPN
cana-2208	167	39	...	...	PUNCT
cana-2208	168	1	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	168	2	(	(	PUNCT
cana-2208	168	3	𝑔	𝑔	NOUN
cana-2208	168	4	;	;	PUNCT
cana-2208	168	5	𝑥	𝑥	X
cana-2208	168	6	)	)	PUNCT
cana-2208	168	7	−	−	PROPN
cana-2208	168	8	�	�	PROPN
cana-2208	168	9	̃	̃	PROPN
cana-2208	168	10	�	�	NOUN
cana-2208	168	11	(𝑥	(𝑥	NOUN
cana-2208	168	12	)	)	PUNCT
cana-2208	168	13	=	=	SYM
cana-2208	168	14	1	1	NUM
cana-2208	168	15	2𝜋(1+𝑞)𝑟	2𝜋(1+𝑞)𝑟	NUM
cana-2208	168	16	∑𝑟	∑𝑟	NOUN
cana-2208	168	17	𝑗=0	𝑗=0	PUNCT
cana-2208	169	1	[	[	X
cana-2208	169	2	(	(	PUNCT
cana-2208	169	3	𝑟	𝑟	X
cana-2208	169	4	𝑗	𝑗	X
cana-2208	169	5	)	)	PUNCT
cana-2208	169	6	𝑞𝑟−𝑗	𝑞𝑟−𝑗	PROPN
cana-2208	169	7	(	(	PUNCT
cana-2208	169	8	1+𝑞)𝑗	1+𝑞)𝑗	VERB
cana-2208	169	9	∑𝑗	∑𝑗	PROPN
cana-2208	169	10	ℎ=0	ℎ=0	PROPN
cana-2208	169	11	(	(	PUNCT
cana-2208	169	12	𝑗	𝑗	INTJ
cana-2208	169	13	ℎ	ℎ	PROPN
cana-2208	169	14	)	)	PUNCT
cana-2208	169	15	.	.	PUNCT
cana-2208	169	16	.	.	PUNCT
cana-2208	169	17	.	.	PUNCT
cana-2208	170	1	𝑞𝑣−𝑖	𝑞𝑣−𝑖	PROPN
cana-2208	170	2	(	(	PUNCT
cana-2208	170	3	1+𝑞)𝑖	1+𝑞)𝑖	NUM
cana-2208	170	4	∫	∫	PROPN
cana-2208	170	5	𝜋	𝜋	X
cana-2208	170	6	0	0	NUM
cana-2208	170	7	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2208	170	8	)	)	PUNCT
cana-2208	170	9	{	{	PUNCT
cana-2208	170	10	∑𝑖	∑𝑖	NOUN
cana-2208	170	11	𝑚=0	𝑚=0	PUNCT
cana-2208	170	12	(	(	PUNCT
cana-2208	170	13	𝑖	𝑖	NOUN
cana-2208	170	14	𝑚	𝑚	NOUN
cana-2208	170	15	)	)	PUNCT
cana-2208	170	16	𝑞𝑖−𝑚	𝑞𝑖−𝑚	NOUN
cana-2208	170	17	𝑐𝑜𝑠(𝑚+	𝑐𝑜𝑠(𝑚+	ADP
cana-2208	170	18	1	1	NUM
cana-2208	170	19	2	2	NUM
cana-2208	170	20	)	)	PUNCT
cana-2208	170	21	𝑡	𝑡	PROPN
cana-2208	170	22	𝑠𝑖𝑛	𝑠𝑖𝑛	ADJ
cana-2208	170	23	𝑡	𝑡	ADP
cana-2208	170	24	2	2	NUM
cana-2208	170	25	}	}	PUNCT
cana-2208	170	26	𝑑𝑡	𝑑𝑡	ADP
cana-2208	170	27	]	]	PUNCT
cana-2208	170	28	=	=	SYM
cana-2208	170	29	∫	∫	PROPN
cana-2208	170	30	𝜋	𝜋	NOUN
cana-2208	170	31	0	0	PROPN
cana-2208	170	32	𝜓(𝑡)𝐽𝑟(𝑡)𝑑𝑡	𝜓(𝑡)𝐽𝑟(𝑡)𝑑𝑡	ADJ
cana-2208	170	33	communications	communication	NOUN
cana-2208	170	34	on	on	ADP
cana-2208	170	35	applied	apply	VERB
cana-2208	170	36	nonlinear	nonlinear	ADJ
cana-2208	170	37	analysis	analysis	NOUN
cana-2208	170	38	issn	issn	NOUN
cana-2208	170	39	:	:	PUNCT
cana-2208	170	40	1074	1074	NUM
cana-2208	170	41	-	-	PUNCT
cana-2208	170	42	133x	133x	NUM
cana-2208	170	43	vol	vol	NOUN
cana-2208	170	44	32	32	NUM
cana-2208	170	45	no	no	NOUN
cana-2208	170	46	.	.	PUNCT
cana-2208	171	1	1s	1s	NUM
cana-2208	171	2	(	(	PUNCT
cana-2208	171	3	2025	2025	NUM
cana-2208	171	4	)	)	PUNCT
cana-2208	171	5	452	452	NUM
cana-2208	171	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	171	7	=	=	PUNCT
cana-2208	172	1	[	[	X
cana-2208	172	2	∫	∫	X
cana-2208	172	3	1	1	NUM
cana-2208	172	4	𝑟	𝑟	NOUN
cana-2208	172	5	0	0	PUNCT
cana-2208	172	6	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2208	172	7	)	)	PUNCT
cana-2208	173	1	+	+	CCONJ
cana-2208	173	2	∫	∫	PROPN
cana-2208	173	3	𝛾	𝛾	PROPN
cana-2208	173	4	1	1	NUM
cana-2208	173	5	𝑟	𝑟	PRON
cana-2208	173	6	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2208	173	7	)	)	PUNCT
cana-2208	173	8	+	+	NUM
cana-2208	173	9	∫	∫	PROPN
cana-2208	173	10	𝜋	𝜋	NOUN
cana-2208	173	11	𝛾	𝛾	ADP
cana-2208	173	12	𝜓(𝑡	𝜓(𝑡	NOUN
cana-2208	173	13	)	)	PUNCT
cana-2208	173	14	]	]	PUNCT
cana-2208	174	1	𝐽𝑟(𝑡)𝑑𝑡	𝐽𝑟(𝑡)𝑑𝑡	X
cana-2208	174	2	=	=	PUNCT
cana-2208	174	3	�	�	PROPN
cana-2208	174	4	̃	̃	PROPN
cana-2208	174	5	�	�	PROPN
cana-2208	174	6	1	1	NUM
cana-2208	174	7	+	+	NUM
cana-2208	174	8	�	�	PROPN
cana-2208	174	9	̃	̃	PROPN
cana-2208	174	10	�	�	NOUN
cana-2208	174	11	2	2	NUM
cana-2208	174	12	+	+	SYM
cana-2208	174	13	�	�	PROPN
cana-2208	174	14	̃	̃	PROPN
cana-2208	174	15	�	�	NOUN
cana-2208	174	16	3	3	NUM
cana-2208	174	17	,	,	PUNCT
cana-2208	174	18	say	say	INTJ
cana-2208	174	19	.	.	PUNCT
cana-2208	175	1	(	(	PUNCT
cana-2208	175	2	6.1	6.1	NUM
cana-2208	175	3	)	)	PUNCT
cana-2208	175	4	applying	apply	VERB
cana-2208	175	5	4.3	4.3	NUM
cana-2208	175	6	,	,	PUNCT
cana-2208	175	7	condition	condition	NOUN
cana-2208	175	8	3.4	3.4	NUM
cana-2208	175	9	and	and	CCONJ
cana-2208	175	10	3.5	3.5	NUM
cana-2208	175	11	and	and	CCONJ
cana-2208	175	12	second	second	ADJ
cana-2208	175	13	mean	mean	NOUN
cana-2208	175	14	value	value	NOUN
cana-2208	175	15	theorem	theorem	NOUN
cana-2208	175	16	is	be	AUX
cana-2208	175	17	applying	apply	VERB
cana-2208	175	18	for	for	ADP
cana-2208	175	19	second	second	ADJ
cana-2208	175	20	term	term	NOUN
cana-2208	175	21	integral	integral	ADJ
cana-2208	175	22	,	,	PUNCT
cana-2208	175	23	we	we	PRON
cana-2208	175	24	have	have	VERB
cana-2208	175	25	|	|	ADV
cana-2208	175	26	�	�	PROPN
cana-2208	175	27	̃	̃	PROPN
cana-2208	175	28	�	�	NOUN
cana-2208	175	29	1|	1|	NUM
cana-2208	175	30	≤	≤	NUM
cana-2208	175	31	∫	∫	PROPN
cana-2208	175	32	1	1	NUM
cana-2208	175	33	𝑟	𝑟	NOUN
cana-2208	175	34	0	0	NUM
cana-2208	176	1	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	PROPN
cana-2208	176	2	=	=	NOUN
cana-2208	177	1	[	[	X
cana-2208	177	2	∫	∫	X
cana-2208	177	3	1	1	NUM
cana-2208	177	4	𝑟	𝑟	NOUN
cana-2208	177	5	𝑜	𝑜	PROPN
cana-2208	177	6	|𝜓(𝑡)|	|𝜓(𝑡)|	PROPN
cana-2208	177	7	1	1	NUM
cana-2208	177	8	𝑡	𝑡	NOUN
cana-2208	177	9	𝑑𝑡	𝑑𝑡	ADP
cana-2208	177	10	]	]	X
cana-2208	177	11	=	=	PUNCT
cana-2208	177	12	𝑂(𝑟	𝑂(𝑟	X
cana-2208	177	13	)	)	PUNCT
cana-2208	178	1	[	[	X
cana-2208	178	2	∫	∫	X
cana-2208	178	3	1	1	NUM
cana-2208	178	4	𝑟	𝑟	PRON
cana-2208	178	5	𝑜	𝑜	NOUN
cana-2208	178	6	|𝜓(𝑡)|𝑑𝑡	|𝜓(𝑡)|𝑑𝑡	X
cana-2208	178	7	]	]	X
cana-2208	178	8	=	=	PUNCT
cana-2208	178	9	𝑂(𝑟	𝑂(𝑟	X
cana-2208	178	10	)	)	PUNCT
cana-2208	179	1	[	[	X
cana-2208	179	2	𝑜	𝑜	X
cana-2208	179	3	(	(	PUNCT
cana-2208	179	4	1	1	NUM
cana-2208	179	5	𝑟	𝑟	NOUN
cana-2208	179	6	𝛼(𝑟).𝑃𝑟	𝛼(𝑟).𝑃𝑟	ADJ
cana-2208	179	7	)	)	PUNCT
cana-2208	179	8	]	]	PUNCT
cana-2208	180	1	=	=	PUNCT
cana-2208	180	2	𝑂	𝑂	PROPN
cana-2208	180	3	(	(	PUNCT
cana-2208	180	4	1	1	NUM
cana-2208	180	5	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	180	6	)	)	PUNCT
cana-2208	180	7	=	=	PUNCT
cana-2208	181	1	𝑂(1	𝑂(1	ADP
cana-2208	181	2	)	)	PUNCT
cana-2208	181	3	,	,	PUNCT
cana-2208	181	4	as	as	SCONJ
cana-2208	181	5	𝑟	𝑟	X
cana-2208	181	6	→	→	SYM
cana-2208	181	7	∞.	∞.	PROPN
cana-2208	181	8	(	(	PUNCT
cana-2208	181	9	6.2	6.2	NUM
cana-2208	181	10	)	)	PUNCT
cana-2208	181	11	apply	apply	VERB
cana-2208	181	12	lemma	lemma	PROPN
cana-2208	181	13	4.4	4.4	NUM
cana-2208	181	14	,	,	PUNCT
cana-2208	181	15	condition	condition	NOUN
cana-2208	181	16	3.4	3.4	NUM
cana-2208	181	17	and	and	CCONJ
cana-2208	181	18	3.5	3.5	NUM
cana-2208	181	19	and	and	CCONJ
cana-2208	181	20	second	second	ADJ
cana-2208	181	21	mean	mean	NOUN
cana-2208	181	22	value	value	NOUN
cana-2208	181	23	theorem	theorem	NOUN
cana-2208	181	24	is	be	AUX
cana-2208	181	25	applying	apply	VERB
cana-2208	181	26	for	for	ADP
cana-2208	181	27	second	second	ADJ
cana-2208	181	28	term	term	NOUN
cana-2208	181	29	integral	integral	ADJ
cana-2208	181	30	,	,	PUNCT
cana-2208	181	31	we	we	PRON
cana-2208	181	32	have	have	VERB
cana-2208	181	33	|	|	ADV
cana-2208	181	34	�	�	PROPN
cana-2208	181	35	̃	̃	PROPN
cana-2208	181	36	�	�	NOUN
cana-2208	181	37	2|	2|	NOUN
cana-2208	181	38	≤	≤	NUM
cana-2208	181	39	∫	∫	PROPN
cana-2208	181	40	𝛾	𝛾	PROPN
cana-2208	181	41	1	1	NUM
cana-2208	181	42	𝑟	𝑟	NOUN
cana-2208	181	43	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	X
cana-2208	181	44	=	=	SYM
cana-2208	181	45	𝑂	𝑂	PROPN
cana-2208	182	1	[	[	X
cana-2208	182	2	∫	∫	PROPN
cana-2208	182	3	𝛾	𝛾	ADP
cana-2208	182	4	1	1	NUM
cana-2208	182	5	𝑟	𝑟	NOUN
cana-2208	182	6	|𝜓(𝑡)|	|𝜓(𝑡)|	PROPN
cana-2208	182	7	(	(	PUNCT
cana-2208	182	8	1	1	NUM
cana-2208	182	9	𝑡	𝑡	PROPN
cana-2208	182	10	)	)	PUNCT
cana-2208	182	11	𝑑𝑡	𝑑𝑡	ADP
cana-2208	182	12	]	]	PUNCT
cana-2208	182	13	=	=	SYM
cana-2208	182	14	𝑂	𝑂	PROPN
cana-2208	182	15	[	[	PUNCT
cana-2208	182	16	{	{	PUNCT
cana-2208	182	17	1	1	NUM
cana-2208	182	18	𝑡	𝑡	PROPN
cana-2208	182	19	ψ(𝑡	ψ(𝑡	NOUN
cana-2208	182	20	)	)	PUNCT
cana-2208	182	21	}	}	PUNCT
cana-2208	182	22	1	1	NUM
cana-2208	182	23	𝑟	𝑟	NOUN
cana-2208	182	24	𝛾	𝛾	NOUN
cana-2208	183	1	+	+	X
cana-2208	183	2	∫	∫	PROPN
cana-2208	183	3	𝛾	𝛾	PROPN
cana-2208	183	4	1	1	NUM
cana-2208	183	5	𝑟	𝑟	NOUN
cana-2208	183	6	𝑜	𝑜	X
cana-2208	183	7	{	{	PUNCT
cana-2208	183	8	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2208	183	9	)	)	PUNCT
cana-2208	183	10	𝑡2	𝑡2	NOUN
cana-2208	183	11	}	}	PUNCT
cana-2208	183	12	𝑑𝑡	𝑑𝑡	PROPN
cana-2208	183	13	]	]	PUNCT
cana-2208	183	14	=	=	SYM
cana-2208	183	15	𝑂	𝑂	PROPN
cana-2208	183	16	[	[	X
cana-2208	183	17	𝑜	𝑜	X
cana-2208	183	18	{	{	PUNCT
cana-2208	183	19	1	1	NUM
cana-2208	183	20	𝛼(𝑟	𝛼(𝑟	NUM
cana-2208	183	21	)	)	PUNCT
cana-2208	183	22	.	.	PUNCT
cana-2208	184	1	𝑝𝑟	𝑝𝑟	X
cana-2208	184	2	}	}	PUNCT
cana-2208	184	3	+	+	NUM
cana-2208	184	4	∫	∫	PROPN
cana-2208	184	5	𝛾	𝛾	PROPN
cana-2208	184	6	1	1	NUM
cana-2208	184	7	𝑟	𝑟	NOUN
cana-2208	184	8	𝑜	𝑜	X
cana-2208	184	9	(	(	PUNCT
cana-2208	184	10	1	1	NUM
cana-2208	184	11	𝑡	𝑡	NOUN
cana-2208	184	12	𝛼	𝛼	PROPN
cana-2208	184	13	(	(	PUNCT
cana-2208	184	14	1	1	NUM
cana-2208	184	15	𝑡	𝑡	NOUN
cana-2208	184	16	)	)	PUNCT
cana-2208	184	17	.𝑝𝑡	.𝑝𝑡	PUNCT
cana-2208	184	18	)	)	PUNCT
cana-2208	184	19	𝑑𝑡	𝑑𝑡	ADP
cana-2208	184	20	]	]	PUNCT
cana-2208	184	21	=	=	SYM
cana-2208	184	22	𝑂	𝑂	PROPN
cana-2208	185	1	[	[	NOUN
cana-2208	185	2	𝑜	𝑜	X
cana-2208	185	3	{	{	PUNCT
cana-2208	185	4	1	1	NUM
cana-2208	185	5	𝛼(𝑟)𝑃𝑟	𝛼(𝑟)𝑃𝑟	NOUN
cana-2208	185	6	}	}	PUNCT
cana-2208	185	7	+	+	NUM
cana-2208	185	8	∫	∫	PROPN
cana-2208	185	9	𝑟	𝑟	NOUN
cana-2208	185	10	1	1	NUM
cana-2208	185	11	𝛾	𝛾	NOUN
cana-2208	185	12	𝑜	𝑜	PRON
cana-2208	185	13	(	(	PUNCT
cana-2208	185	14	1	1	NUM
cana-2208	185	15	𝑢	𝑢	NOUN
cana-2208	185	16	𝛼(𝑢)𝑝𝑢	𝛼(𝑢)𝑝𝑢	NUM
cana-2208	185	17	)	)	PUNCT
cana-2208	185	18	𝑑𝑢	𝑑𝑢	X
cana-2208	185	19	]	]	X
cana-2208	185	20	=	=	SYM
cana-2208	185	21	𝑂	𝑂	PROPN
cana-2208	185	22	(	(	PUNCT
cana-2208	185	23	1	1	NUM
cana-2208	185	24	𝛼(𝑟)𝑃𝑟	𝛼(𝑟)𝑃𝑟	NOUN
cana-2208	185	25	)	)	PUNCT
cana-2208	186	1	+	+	CCONJ
cana-2208	186	2	𝑂	𝑂	PROPN
cana-2208	186	3	(	(	PUNCT
cana-2208	186	4	1	1	NUM
cana-2208	186	5	𝑟	𝑟	NOUN
cana-2208	186	6	𝛼(𝑟)𝑝𝑟	𝛼(𝑟)𝑝𝑟	NUM
cana-2208	186	7	)	)	PUNCT
cana-2208	186	8	∫	∫	PROPN
cana-2208	186	9	𝑟	𝑟	NOUN
cana-2208	186	10	1	1	NUM
cana-2208	186	11	𝑟	𝑟	NUM
cana-2208	186	12	1𝑑𝑢	1𝑑𝑢	ADJ
cana-2208	186	13	=	=	SYM
cana-2208	186	14	𝑂	𝑂	NOUN
cana-2208	186	15	(	(	PUNCT
cana-2208	186	16	1	1	NUM
cana-2208	186	17	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	186	18	)	)	PUNCT
cana-2208	187	1	+	+	CCONJ
cana-2208	187	2	𝑂	𝑂	PROPN
cana-2208	187	3	(	(	PUNCT
cana-2208	187	4	1	1	NUM
cana-2208	187	5	𝑙𝑜𝑔𝑟	𝑙𝑜𝑔𝑟	NOUN
cana-2208	187	6	)	)	PUNCT
cana-2208	187	7	=	=	PUNCT
cana-2208	188	1	𝑂(1	𝑂(1	ADP
cana-2208	188	2	)	)	PUNCT
cana-2208	188	3	,	,	PUNCT
cana-2208	188	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	188	5	𝑟	𝑟	X
cana-2208	188	6	→	→	SYM
cana-2208	188	7	∞.	∞.	PROPN
cana-2208	188	8	(	(	PUNCT
cana-2208	188	9	6.3	6.3	NUM
cana-2208	188	10	)	)	PUNCT
cana-2208	188	11	applying	apply	VERB
cana-2208	188	12	riemann	riemann	PROPN
cana-2208	188	13	-	-	PUNCT
cana-2208	188	14	lebesgue	lebesgue	NOUN
cana-2208	188	15	theorem	theorem	NOUN
cana-2208	188	16	and	and	CCONJ
cana-2208	188	17	regularity	regularity	NOUN
cana-2208	188	18	condition	condition	NOUN
cana-2208	188	19	of	of	ADP
cana-2208	188	20	summability	summability	NOUN
cana-2208	188	21	,	,	PUNCT
cana-2208	188	22	we	we	PRON
cana-2208	188	23	have	have	VERB
cana-2208	188	24	|	|	ADV
cana-2208	188	25	�	�	PROPN
cana-2208	188	26	̃	̃	PROPN
cana-2208	188	27	�	�	PROPN
cana-2208	188	28	3|	3|	NUM
cana-2208	188	29	≤	≤	NOUN
cana-2208	188	30	∫	∫	PROPN
cana-2208	188	31	𝛾	𝛾	PROPN
cana-2208	188	32	1	1	NUM
cana-2208	188	33	𝑟	𝑟	NOUN
cana-2208	188	34	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	|𝜓(𝑡)𝐽𝑟(𝑡)|𝑑𝑡	PROPN
cana-2208	188	35	=	=	SYM
cana-2208	188	36	𝑂(1	𝑂(1	PROPN
cana-2208	188	37	)	)	PUNCT
cana-2208	188	38	,	,	PUNCT
cana-2208	188	39	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	188	40	𝑟	𝑟	X
cana-2208	188	41	→	→	SYM
cana-2208	188	42	∞.	∞.	PROPN
cana-2208	188	43	(	(	PUNCT
cana-2208	188	44	6.4	6.4	NUM
cana-2208	188	45	)	)	PUNCT
cana-2208	188	46	collecting	collecting	NOUN
cana-2208	188	47	(	(	PUNCT
cana-2208	188	48	6.1)-(6.4	6.1)-(6.4	NUM
cana-2208	188	49	)	)	PUNCT
cana-2208	188	50	,	,	PUNCT
cana-2208	188	51	we	we	PRON
cana-2208	188	52	get	get	VERB
cana-2208	188	53	communications	communication	NOUN
cana-2208	188	54	on	on	ADP
cana-2208	188	55	applied	apply	VERB
cana-2208	188	56	nonlinear	nonlinear	ADJ
cana-2208	188	57	analysis	analysis	NOUN
cana-2208	188	58	issn	issn	NOUN
cana-2208	188	59	:	:	PUNCT
cana-2208	188	60	1074	1074	NUM
cana-2208	188	61	-	-	PUNCT
cana-2208	188	62	133x	133x	NUM
cana-2208	188	63	vol	vol	NOUN
cana-2208	188	64	32	32	NUM
cana-2208	188	65	no	no	NOUN
cana-2208	188	66	.	.	PUNCT
cana-2208	189	1	1s	1s	NUM
cana-2208	189	2	(	(	PUNCT
cana-2208	189	3	2025	2025	NUM
cana-2208	189	4	)	)	PUNCT
cana-2208	189	5	453	453	NUM
cana-2208	189	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	189	7	|	|	NOUN
cana-2208	189	8	�	�	PROPN
cana-2208	189	9	̃	̃	PROPN
cana-2208	189	10	�	�	NOUN
cana-2208	189	11	𝑟	𝑟	NOUN
cana-2208	189	12	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	189	13	...	...	PUNCT
cana-2208	189	14	𝐸𝑞(𝑔	𝐸𝑞(𝑔	ADJ
cana-2208	189	15	;	;	PUNCT
cana-2208	189	16	𝑥	𝑥	X
cana-2208	189	17	)	)	PUNCT
cana-2208	189	18	−	−	PROPN
cana-2208	189	19	�	�	PROPN
cana-2208	189	20	̃	̃	NOUN
cana-2208	189	21	�	�	PROPN
cana-2208	189	22	(𝑥)|	(𝑥)|	NOUN
cana-2208	189	23	=	=	SYM
cana-2208	189	24	𝑂(1	𝑂(1	ADP
cana-2208	189	25	)	)	PUNCT
cana-2208	189	26	,	,	PUNCT
cana-2208	189	27	𝑎𝑠	𝑎𝑠	ADP
cana-2208	189	28	𝑟	𝑟	X
cana-2208	189	29	→	→	SYM
cana-2208	189	30	∞.	∞.	PROPN
cana-2208	189	31	hence	hence	ADV
cana-2208	189	32	complete	complete	VERB
cana-2208	189	33	the	the	DET
cana-2208	189	34	proof	proof	NOUN
cana-2208	189	35	.	.	PUNCT
cana-2208	190	1	7	7	NUM
cana-2208	190	2	corollaries	corollary	NOUN
cana-2208	190	3	below	below	ADP
cana-2208	190	4	some	some	DET
cana-2208	190	5	corollaries	corollary	NOUN
cana-2208	190	6	are	be	AUX
cana-2208	190	7	given	give	VERB
cana-2208	190	8	,	,	PUNCT
cana-2208	190	9	which	which	PRON
cana-2208	190	10	are	be	AUX
cana-2208	190	11	derived	derive	VERB
cana-2208	190	12	from	from	ADP
cana-2208	190	13	our	our	PRON
cana-2208	190	14	theorems	theorem	NOUN
cana-2208	190	15	3.1	3.1	NUM
cana-2208	190	16	and	and	CCONJ
cana-2208	190	17	3.2	3.2	NUM
cana-2208	190	18	7.1	7.1	NUM
cana-2208	190	19	corollary	corollary	NOUN
cana-2208	190	20	if	if	SCONJ
cana-2208	190	21	we	we	PRON
cana-2208	190	22	take	take	VERB
cana-2208	190	23	q=1	q=1	PROPN
cana-2208	190	24	in	in	ADP
cana-2208	190	25	theorem	theorem	ADJ
cana-2208	190	26	3.1	3.1	NUM
cana-2208	190	27	then	then	ADV
cana-2208	190	28	neular	neular	ADJ
cana-2208	190	29	product	product	NOUN
cana-2208	190	30	summability	summability	NOUN
cana-2208	190	31	(	(	PUNCT
cana-2208	190	32	𝐸	𝐸	PROPN
cana-2208	190	33	,	,	PUNCT
cana-2208	190	34	𝑞	𝑞	PROPN
cana-2208	190	35	)	)	PUNCT
cana-2208	190	36	(	(	PUNCT
cana-2208	190	37	𝐸	𝐸	PROPN
cana-2208	190	38	,	,	PUNCT
cana-2208	190	39	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	190	40	,	,	PUNCT
cana-2208	190	41	𝑞)	𝑞)	NUM
cana-2208	190	42	...	...	X
cana-2208	190	43	(𝐸	(𝐸	NUM
cana-2208	190	44	,	,	PUNCT
cana-2208	190	45	𝑞	𝑞	X
cana-2208	190	46	)	)	PUNCT
cana-2208	190	47	reduce	reduce	VERB
cana-2208	190	48	to	to	ADP
cana-2208	190	49	(	(	PUNCT
cana-2208	190	50	𝐸	𝐸	PROPN
cana-2208	190	51	,	,	PUNCT
cana-2208	190	52	1	1	NUM
cana-2208	190	53	)	)	PUNCT
cana-2208	190	54	(	(	PUNCT
cana-2208	190	55	𝐸	𝐸	PROPN
cana-2208	190	56	,	,	PUNCT
cana-2208	190	57	1	1	NUM
cana-2208	190	58	)	)	PUNCT
cana-2208	190	59	(	(	PUNCT
cana-2208	190	60	𝐸	𝐸	PROPN
cana-2208	190	61	,	,	PUNCT
cana-2208	190	62	1)	1)	NUM
cana-2208	190	63	...	...	PUNCT
cana-2208	190	64	(𝐸	(𝐸	NUM
cana-2208	190	65	,	,	PUNCT
cana-2208	190	66	1	1	NUM
cana-2208	190	67	)	)	PUNCT
cana-2208	190	68	,	,	PUNCT
cana-2208	190	69	then	then	ADV
cana-2208	190	70	|𝑡𝑟	|𝑡𝑟	NUM
cana-2208	190	71	𝐸1𝐸1𝐸1	𝐸1𝐸1𝐸1	NOUN
cana-2208	190	72	...	...	PUNCT
cana-2208	190	73	𝐸1(𝑔	𝐸1(𝑔	PROPN
cana-2208	190	74	;	;	PUNCT
cana-2208	190	75	𝑥	𝑥	X
cana-2208	190	76	)	)	PUNCT
cana-2208	190	77	−	−	PROPN
cana-2208	190	78	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	190	79	=	=	PUNCT
cana-2208	191	1	𝑂(1	𝑂(1	ADP
cana-2208	191	2	)	)	PUNCT
cana-2208	191	3	,	,	PUNCT
cana-2208	191	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	191	5	𝑟	𝑟	X
cana-2208	191	6	→	→	SYM
cana-2208	191	7	∞.	∞.	PROPN
cana-2208	191	8	7.2	7.2	NUM
cana-2208	191	9	corollary	corollary	NOUN
cana-2208	191	10	if	if	SCONJ
cana-2208	191	11	we	we	PRON
cana-2208	191	12	take	take	VERB
cana-2208	191	13	q=1	q=1	PROPN
cana-2208	191	14	in	in	ADP
cana-2208	191	15	theorem	theorem	NOUN
cana-2208	191	16	3.2	3.2	NUM
cana-2208	191	17	,	,	PUNCT
cana-2208	191	18	then	then	ADV
cana-2208	191	19	n	n	NUM
cana-2208	191	20	eular	eular	ADJ
cana-2208	191	21	product	product	NOUN
cana-2208	191	22	summaility	summaility	NOUN
cana-2208	191	23	(	(	PUNCT
cana-2208	191	24	𝐸	𝐸	PROPN
cana-2208	191	25	,	,	PUNCT
cana-2208	191	26	𝑞	𝑞	PROPN
cana-2208	191	27	)	)	PUNCT
cana-2208	191	28	(	(	PUNCT
cana-2208	191	29	𝐸	𝐸	PROPN
cana-2208	191	30	,	,	PUNCT
cana-2208	191	31	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	191	32	,	,	PUNCT
cana-2208	191	33	𝑞)	𝑞)	NUM
cana-2208	191	34	...	...	X
cana-2208	191	35	(𝐸	(𝐸	NUM
cana-2208	191	36	,	,	PUNCT
cana-2208	191	37	𝑞	𝑞	X
cana-2208	191	38	)	)	PUNCT
cana-2208	191	39	reduce	reduce	VERB
cana-2208	191	40	to	to	ADP
cana-2208	191	41	(	(	PUNCT
cana-2208	191	42	𝐸	𝐸	PROPN
cana-2208	191	43	,	,	PUNCT
cana-2208	191	44	1	1	NUM
cana-2208	191	45	)	)	PUNCT
cana-2208	191	46	(	(	PUNCT
cana-2208	191	47	𝐸	𝐸	PROPN
cana-2208	191	48	,	,	PUNCT
cana-2208	191	49	1	1	NUM
cana-2208	191	50	)	)	PUNCT
cana-2208	191	51	(	(	PUNCT
cana-2208	191	52	𝐸	𝐸	PROPN
cana-2208	191	53	,	,	PUNCT
cana-2208	191	54	1)	1)	NUM
cana-2208	191	55	...	...	PUNCT
cana-2208	191	56	(𝐸	(𝐸	NUM
cana-2208	191	57	,	,	PUNCT
cana-2208	191	58	1	1	NUM
cana-2208	191	59	)	)	PUNCT
cana-2208	191	60	,	,	PUNCT
cana-2208	191	61	then	then	ADV
cana-2208	191	62	|	|	ADV
cana-2208	191	63	�	�	PROPN
cana-2208	191	64	̃	̃	PROPN
cana-2208	191	65	�	�	NOUN
cana-2208	191	66	𝑟	𝑟	PART
cana-2208	191	67	𝐸1𝐸1𝐸1	𝐸1𝐸1𝐸1	NOUN
cana-2208	191	68	...	...	PUNCT
cana-2208	191	69	𝐸1(𝑔	𝐸1(𝑔	PROPN
cana-2208	191	70	;	;	PUNCT
cana-2208	191	71	𝑥	𝑥	X
cana-2208	191	72	)	)	PUNCT
cana-2208	191	73	−	−	PROPN
cana-2208	191	74	�	�	PROPN
cana-2208	191	75	̃	̃	NOUN
cana-2208	191	76	�	�	PROPN
cana-2208	191	77	(𝑥)|	(𝑥)|	NOUN
cana-2208	191	78	=	=	SYM
cana-2208	192	1	𝑂(1	𝑂(1	ADP
cana-2208	192	2	)	)	PUNCT
cana-2208	192	3	,	,	PUNCT
cana-2208	192	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	192	5	𝑟	𝑟	X
cana-2208	192	6	→	→	SYM
cana-2208	192	7	∞.	∞.	PROPN
cana-2208	192	8	7.3	7.3	NUM
cana-2208	192	9	corollary	corollary	NOUN
cana-2208	192	10	if	if	SCONJ
cana-2208	192	11	we	we	PRON
cana-2208	192	12	take	take	VERB
cana-2208	192	13	n=2	n=2	ADV
cana-2208	192	14	in	in	ADP
cana-2208	192	15	our	our	PRON
cana-2208	192	16	results	result	NOUN
cana-2208	192	17	3.1	3.1	NUM
cana-2208	192	18	then	then	ADV
cana-2208	192	19	𝑛	𝑛	DET
cana-2208	192	20	−	−	PROPN
cana-2208	192	21	euler	euler	NOUN
cana-2208	192	22	product	product	NOUN
cana-2208	192	23	summability	summability	NOUN
cana-2208	192	24	(	(	PUNCT
cana-2208	192	25	𝐸	𝐸	PROPN
cana-2208	192	26	,	,	PUNCT
cana-2208	192	27	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	192	28	,	,	PUNCT
cana-2208	192	29	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	192	30	,	,	PUNCT
cana-2208	192	31	𝑞	𝑞	NOUN
cana-2208	192	32	)	)	PUNCT
cana-2208	192	33	.	.	PUNCT
cana-2208	192	34	.	.	PUNCT
cana-2208	193	1	.	.	PUNCT
cana-2208	194	1	(	(	PUNCT
cana-2208	194	2	𝐸	𝐸	PROPN
cana-2208	194	3	,	,	PUNCT
cana-2208	194	4	𝑞	𝑞	NOUN
cana-2208	194	5	)	)	PUNCT
cana-2208	194	6	reduces	reduce	VERB
cana-2208	194	7	to	to	PART
cana-2208	194	8	(	(	PUNCT
cana-2208	194	9	𝐸	𝐸	PROPN
cana-2208	194	10	,	,	PUNCT
cana-2208	194	11	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	194	12	,	,	PUNCT
cana-2208	194	13	𝑞	𝑞	NOUN
cana-2208	194	14	)	)	PUNCT
cana-2208	194	15	double	double	ADJ
cana-2208	194	16	euler	euler	NOUN
cana-2208	194	17	summability	summability	NOUN
cana-2208	194	18	,	,	PUNCT
cana-2208	194	19	then	then	ADV
cana-2208	194	20	|𝑡𝑟	|𝑡𝑟	AUX
cana-2208	194	21	𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞	PROPN
cana-2208	194	22	(	(	PUNCT
cana-2208	194	23	𝑔	𝑔	NOUN
cana-2208	194	24	;	;	PUNCT
cana-2208	194	25	𝑥	𝑥	X
cana-2208	194	26	)	)	PUNCT
cana-2208	194	27	−	−	PROPN
cana-2208	194	28	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	194	29	=	=	PUNCT
cana-2208	195	1	𝑂(1	𝑂(1	ADP
cana-2208	195	2	)	)	PUNCT
cana-2208	195	3	,	,	PUNCT
cana-2208	195	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	195	5	𝑟	𝑟	X
cana-2208	195	6	→	→	SYM
cana-2208	195	7	∞.	∞.	PROPN
cana-2208	195	8	7.4	7.4	NUM
cana-2208	195	9	corollary	corollary	NOUN
cana-2208	195	10	if	if	SCONJ
cana-2208	195	11	we	we	PRON
cana-2208	195	12	take	take	VERB
cana-2208	195	13	n=2	n=2	ADV
cana-2208	195	14	in	in	ADP
cana-2208	195	15	our	our	PRON
cana-2208	195	16	results	result	NOUN
cana-2208	195	17	3.2	3.2	NUM
cana-2208	195	18	then	then	ADV
cana-2208	195	19	𝑛	𝑛	DET
cana-2208	195	20	−	−	PROPN
cana-2208	195	21	euler	euler	NOUN
cana-2208	195	22	product	product	NOUN
cana-2208	195	23	summability	summability	NOUN
cana-2208	195	24	(	(	PUNCT
cana-2208	195	25	𝐸	𝐸	PROPN
cana-2208	195	26	,	,	PUNCT
cana-2208	195	27	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	195	28	,	,	PUNCT
cana-2208	195	29	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	195	30	,	,	PUNCT
cana-2208	195	31	𝑞	𝑞	NOUN
cana-2208	195	32	)	)	PUNCT
cana-2208	195	33	.	.	PUNCT
cana-2208	195	34	.	.	PUNCT
cana-2208	196	1	.	.	PUNCT
cana-2208	197	1	(	(	PUNCT
cana-2208	197	2	𝐸	𝐸	PROPN
cana-2208	197	3	,	,	PUNCT
cana-2208	197	4	𝑞	𝑞	NOUN
cana-2208	197	5	)	)	PUNCT
cana-2208	197	6	reduces	reduce	VERB
cana-2208	197	7	to	to	PART
cana-2208	197	8	(	(	PUNCT
cana-2208	197	9	𝐸	𝐸	PROPN
cana-2208	197	10	,	,	PUNCT
cana-2208	197	11	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	197	12	,	,	PUNCT
cana-2208	197	13	𝑞	𝑞	NOUN
cana-2208	197	14	)	)	PUNCT
cana-2208	197	15	double	double	ADJ
cana-2208	197	16	euler	euler	NOUN
cana-2208	197	17	summability	summability	NOUN
cana-2208	197	18	,	,	PUNCT
cana-2208	197	19	then	then	ADV
cana-2208	197	20	|	|	ADV
cana-2208	197	21	�	�	PROPN
cana-2208	197	22	̃	̃	PROPN
cana-2208	197	23	�	�	NOUN
cana-2208	197	24	𝑟	𝑟	PART
cana-2208	197	25	𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞	PROPN
cana-2208	197	26	(	(	PUNCT
cana-2208	197	27	𝑔	𝑔	NOUN
cana-2208	197	28	;	;	PUNCT
cana-2208	197	29	𝑥	𝑥	X
cana-2208	197	30	)	)	PUNCT
cana-2208	197	31	−	−	PROPN
cana-2208	197	32	�	�	PROPN
cana-2208	197	33	̃	̃	NOUN
cana-2208	197	34	�	�	PROPN
cana-2208	197	35	(𝑥)|	(𝑥)|	NOUN
cana-2208	197	36	=	=	SYM
cana-2208	198	1	𝑂(1	𝑂(1	ADP
cana-2208	198	2	)	)	PUNCT
cana-2208	198	3	,	,	PUNCT
cana-2208	198	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	198	5	𝑟	𝑟	X
cana-2208	198	6	→	→	SYM
cana-2208	198	7	∞	∞	NUM
cana-2208	198	8	7.5	7.5	NUM
cana-2208	198	9	corollary	corollary	NOUN
cana-2208	198	10	if	if	SCONJ
cana-2208	198	11	we	we	PRON
cana-2208	198	12	take	take	VERB
cana-2208	198	13	n=2	n=2	ADV
cana-2208	198	14	and	and	CCONJ
cana-2208	198	15	q=1	q=1	X
cana-2208	198	16	in	in	ADP
cana-2208	198	17	our	our	PRON
cana-2208	198	18	result	result	NOUN
cana-2208	198	19	3.1	3.1	NUM
cana-2208	198	20	,	,	PUNCT
cana-2208	198	21	𝑛	𝑛	DET
cana-2208	198	22	−	−	PROPN
cana-2208	198	23	euler	euler	NOUN
cana-2208	198	24	product	product	NOUN
cana-2208	198	25	summability	summability	NOUN
cana-2208	198	26	(	(	PUNCT
cana-2208	198	27	𝐸	𝐸	PROPN
cana-2208	198	28	,	,	PUNCT
cana-2208	198	29	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	198	30	,	,	PUNCT
cana-2208	198	31	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	198	32	,	,	PUNCT
cana-2208	198	33	𝑞	𝑞	NOUN
cana-2208	198	34	)	)	PUNCT
cana-2208	198	35	.	.	PUNCT
cana-2208	198	36	.	.	PUNCT
cana-2208	199	1	.	.	PUNCT
cana-2208	200	1	(	(	PUNCT
cana-2208	200	2	𝐸	𝐸	PROPN
cana-2208	200	3	,	,	PUNCT
cana-2208	200	4	𝑞	𝑞	NOUN
cana-2208	200	5	)	)	PUNCT
cana-2208	200	6	reduces	reduce	VERB
cana-2208	200	7	to	to	PART
cana-2208	200	8	(	(	PUNCT
cana-2208	200	9	𝐸	𝐸	PROPN
cana-2208	200	10	,	,	PUNCT
cana-2208	200	11	1)(𝐸	1)(𝐸	NUM
cana-2208	200	12	,	,	PUNCT
cana-2208	200	13	1	1	NUM
cana-2208	200	14	)	)	PUNCT
cana-2208	200	15	then	then	ADV
cana-2208	200	16	|𝑡𝑟	|𝑡𝑟	PUNCT
cana-2208	200	17	𝐸1𝐸1	𝐸1𝐸1	NOUN
cana-2208	200	18	(	(	PUNCT
cana-2208	200	19	𝑔	𝑔	NOUN
cana-2208	200	20	;	;	PUNCT
cana-2208	200	21	𝑥	𝑥	X
cana-2208	200	22	)	)	PUNCT
cana-2208	200	23	−	−	PROPN
cana-2208	200	24	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	200	25	=	=	PUNCT
cana-2208	200	26	𝑂(1	𝑂(1	ADP
cana-2208	200	27	)	)	PUNCT
cana-2208	200	28	,	,	PUNCT
cana-2208	200	29	𝑎𝑠	𝑎𝑠	ADP
cana-2208	200	30	𝑟	𝑟	X
cana-2208	200	31	→	→	SYM
cana-2208	200	32	∞	∞	PROPN
cana-2208	200	33	7.6	7.6	NUM
cana-2208	200	34	corollary	corollary	NOUN
cana-2208	200	35	if	if	SCONJ
cana-2208	200	36	we	we	PRON
cana-2208	200	37	take	take	VERB
cana-2208	200	38	n=2	n=2	ADV
cana-2208	200	39	and	and	CCONJ
cana-2208	200	40	q=1	q=1	X
cana-2208	200	41	in	in	ADP
cana-2208	200	42	our	our	PRON
cana-2208	200	43	result	result	NOUN
cana-2208	200	44	3.2	3.2	NUM
cana-2208	200	45	,	,	PUNCT
cana-2208	200	46	𝑛	𝑛	DET
cana-2208	200	47	−	−	PROPN
cana-2208	200	48	euler	euler	NOUN
cana-2208	200	49	product	product	NOUN
cana-2208	200	50	summability	summability	NOUN
cana-2208	200	51	(	(	PUNCT
cana-2208	200	52	𝐸	𝐸	PROPN
cana-2208	200	53	,	,	PUNCT
cana-2208	200	54	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	200	55	,	,	PUNCT
cana-2208	200	56	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	200	57	,	,	PUNCT
cana-2208	200	58	𝑞	𝑞	NOUN
cana-2208	200	59	)	)	PUNCT
cana-2208	200	60	.	.	PUNCT
cana-2208	200	61	.	.	PUNCT
cana-2208	201	1	.	.	PUNCT
cana-2208	202	1	(	(	PUNCT
cana-2208	202	2	𝐸	𝐸	PROPN
cana-2208	202	3	,	,	PUNCT
cana-2208	202	4	𝑞	𝑞	NOUN
cana-2208	202	5	)	)	PUNCT
cana-2208	202	6	reduces	reduce	VERB
cana-2208	202	7	to	to	PART
cana-2208	202	8	(	(	PUNCT
cana-2208	202	9	𝐸	𝐸	PROPN
cana-2208	202	10	,	,	PUNCT
cana-2208	202	11	1)(𝐸	1)(𝐸	NUM
cana-2208	202	12	,	,	PUNCT
cana-2208	202	13	1	1	NUM
cana-2208	202	14	)	)	PUNCT
cana-2208	202	15	then	then	ADV
cana-2208	202	16	|	|	ADV
cana-2208	202	17	�	�	PROPN
cana-2208	202	18	̃	̃	PROPN
cana-2208	202	19	�	�	NOUN
cana-2208	202	20	𝑟	𝑟	PART
cana-2208	202	21	𝐸1𝐸1	𝐸1𝐸1	NOUN
cana-2208	202	22	(	(	PUNCT
cana-2208	202	23	𝑔	𝑔	NOUN
cana-2208	202	24	;	;	PUNCT
cana-2208	202	25	𝑥	𝑥	X
cana-2208	202	26	)	)	PUNCT
cana-2208	202	27	−	−	PROPN
cana-2208	202	28	�	�	PROPN
cana-2208	202	29	̃	̃	NOUN
cana-2208	202	30	�	�	PROPN
cana-2208	202	31	(𝑥)|	(𝑥)|	NOUN
cana-2208	202	32	=	=	SYM
cana-2208	203	1	𝑂(1	𝑂(1	ADP
cana-2208	203	2	)	)	PUNCT
cana-2208	203	3	,	,	PUNCT
cana-2208	203	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	203	5	𝑟	𝑟	X
cana-2208	203	6	→	→	SYM
cana-2208	203	7	∞.	∞.	PROPN
cana-2208	203	8	communications	communication	NOUN
cana-2208	203	9	on	on	ADP
cana-2208	203	10	applied	apply	VERB
cana-2208	203	11	nonlinear	nonlinear	ADJ
cana-2208	203	12	analysis	analysis	NOUN
cana-2208	203	13	issn	issn	NOUN
cana-2208	203	14	:	:	PUNCT
cana-2208	203	15	1074	1074	NUM
cana-2208	203	16	-	-	PUNCT
cana-2208	203	17	133x	133x	NUM
cana-2208	203	18	vol	vol	NOUN
cana-2208	203	19	32	32	NUM
cana-2208	203	20	no	no	NOUN
cana-2208	203	21	.	.	PUNCT
cana-2208	204	1	1s	1s	NUM
cana-2208	204	2	(	(	PUNCT
cana-2208	204	3	2025	2025	NUM
cana-2208	204	4	)	)	PUNCT
cana-2208	204	5	454	454	NUM
cana-2208	204	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2208	204	7	7.7	7.7	NUM
cana-2208	204	8	corollary	corollary	NOUN
cana-2208	204	9	if	if	SCONJ
cana-2208	204	10	we	we	PRON
cana-2208	204	11	consider	consider	VERB
cana-2208	204	12	n=3	n=3	PUNCT
cana-2208	204	13	in	in	ADP
cana-2208	204	14	our	our	PRON
cana-2208	204	15	theorem	theorem	NOUN
cana-2208	204	16	3.1	3.1	NUM
cana-2208	204	17	,	,	PUNCT
cana-2208	204	18	𝑛	𝑛	DET
cana-2208	204	19	−	−	PROPN
cana-2208	204	20	euler	euler	NOUN
cana-2208	204	21	product	product	NOUN
cana-2208	204	22	summability	summability	NOUN
cana-2208	204	23	(	(	PUNCT
cana-2208	204	24	𝐸	𝐸	PROPN
cana-2208	204	25	,	,	PUNCT
cana-2208	204	26	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	204	27	,	,	PUNCT
cana-2208	204	28	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	204	29	,	,	PUNCT
cana-2208	204	30	𝑞	𝑞	NOUN
cana-2208	204	31	)	)	PUNCT
cana-2208	204	32	.	.	PUNCT
cana-2208	204	33	.	.	PUNCT
cana-2208	204	34	.	.	PUNCT
cana-2208	205	1	(	(	PUNCT
cana-2208	205	2	𝐸	𝐸	PROPN
cana-2208	205	3	,	,	PUNCT
cana-2208	205	4	𝑞	𝑞	NOUN
cana-2208	205	5	)	)	PUNCT
cana-2208	205	6	reduces	reduce	VERB
cana-2208	205	7	to	to	PART
cana-2208	205	8	(	(	PUNCT
cana-2208	205	9	𝐸	𝐸	PROPN
cana-2208	205	10	,	,	PUNCT
cana-2208	205	11	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	205	12	,	,	PUNCT
cana-2208	205	13	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	205	14	,	,	PUNCT
cana-2208	205	15	𝑞	𝑞	NOUN
cana-2208	205	16	)	)	PUNCT
cana-2208	205	17	i.e.	i.e.	X
cana-2208	205	18	triple	triple	ADJ
cana-2208	205	19	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	205	20	summability	summability	NOUN
cana-2208	205	21	then	then	ADV
cana-2208	205	22	|𝑡𝑟	|𝑡𝑟	NUM
cana-2208	205	23	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	205	24	(	(	PUNCT
cana-2208	205	25	𝑔	𝑔	NUM
cana-2208	205	26	;	;	PUNCT
cana-2208	205	27	𝑥	𝑥	X
cana-2208	205	28	)	)	PUNCT
cana-2208	205	29	−	−	PROPN
cana-2208	205	30	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	205	31	=	=	PUNCT
cana-2208	206	1	𝑂(1	𝑂(1	ADP
cana-2208	206	2	)	)	PUNCT
cana-2208	206	3	,	,	PUNCT
cana-2208	206	4	𝑎𝑠	𝑎𝑠	PROPN
cana-2208	206	5	𝑟	𝑟	X
cana-2208	206	6	→	→	SYM
cana-2208	206	7	∞.	∞.	PROPN
cana-2208	206	8	7.8	7.8	NUM
cana-2208	206	9	corollary	corollary	NOUN
cana-2208	206	10	if	if	SCONJ
cana-2208	206	11	we	we	PRON
cana-2208	206	12	consider	consider	VERB
cana-2208	206	13	n=	n=	ADJ
cana-2208	206	14	3	3	NUM
cana-2208	206	15	in	in	ADP
cana-2208	206	16	our	our	PRON
cana-2208	206	17	theorem	theorem	NOUN
cana-2208	206	18	3.2	3.2	NUM
cana-2208	206	19	,	,	PUNCT
cana-2208	206	20	𝑛	𝑛	DET
cana-2208	206	21	−	−	PROPN
cana-2208	206	22	euler	euler	NOUN
cana-2208	206	23	product	product	NOUN
cana-2208	206	24	summability	summability	NOUN
cana-2208	206	25	(	(	PUNCT
cana-2208	206	26	𝐸	𝐸	PROPN
cana-2208	206	27	,	,	PUNCT
cana-2208	206	28	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	206	29	,	,	PUNCT
cana-2208	206	30	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	206	31	,	,	PUNCT
cana-2208	206	32	𝑞	𝑞	NOUN
cana-2208	206	33	)	)	PUNCT
cana-2208	206	34	.	.	PUNCT
cana-2208	206	35	.	.	PUNCT
cana-2208	207	1	.	.	PUNCT
cana-2208	208	1	(	(	PUNCT
cana-2208	208	2	𝐸	𝐸	PROPN
cana-2208	208	3	,	,	PUNCT
cana-2208	208	4	𝑞	𝑞	NOUN
cana-2208	208	5	)	)	PUNCT
cana-2208	208	6	reduces	reduce	VERB
cana-2208	208	7	to	to	PART
cana-2208	208	8	(	(	PUNCT
cana-2208	208	9	𝐸	𝐸	PROPN
cana-2208	208	10	,	,	PUNCT
cana-2208	208	11	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	208	12	,	,	PUNCT
cana-2208	208	13	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	208	14	,	,	PUNCT
cana-2208	208	15	𝑞	𝑞	NOUN
cana-2208	208	16	)	)	PUNCT
cana-2208	208	17	i.e.	i.e.	X
cana-2208	208	18	triple	triple	ADJ
cana-2208	208	19	𝐸𝑞	𝐸𝑞	PROPN
cana-2208	208	20	summability	summability	NOUN
cana-2208	208	21	then	then	ADV
cana-2208	208	22	|	|	ADV
cana-2208	208	23	�	�	PROPN
cana-2208	208	24	̃	̃	PROPN
cana-2208	208	25	�	�	NOUN
cana-2208	208	26	𝑟	𝑟	NOUN
cana-2208	208	27	𝐸𝑞𝐸𝑞𝐸𝑞	𝐸𝑞𝐸𝑞𝐸𝑞	NOUN
cana-2208	208	28	(	(	PUNCT
cana-2208	208	29	𝑔	𝑔	NUM
cana-2208	208	30	;	;	PUNCT
cana-2208	208	31	𝑥	𝑥	X
cana-2208	208	32	)	)	PUNCT
cana-2208	208	33	−	−	PROPN
cana-2208	208	34	�	�	PROPN
cana-2208	208	35	̃	̃	NOUN
cana-2208	208	36	�	�	PROPN
cana-2208	208	37	(𝑥)|	(𝑥)|	NOUN
cana-2208	208	38	=	=	SYM
cana-2208	209	1	𝑂(1	𝑂(1	ADP
cana-2208	209	2	)	)	PUNCT
cana-2208	209	3	,	,	PUNCT
cana-2208	209	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	209	5	𝑟	𝑟	X
cana-2208	209	6	→	→	SYM
cana-2208	209	7	∞.	∞.	PROPN
cana-2208	209	8	7.9	7.9	NUM
cana-2208	209	9	corollary	corollary	NOUN
cana-2208	209	10	if	if	SCONJ
cana-2208	209	11	we	we	PRON
cana-2208	209	12	take	take	VERB
cana-2208	209	13	n=3	n=3	PUNCT
cana-2208	209	14	and	and	CCONJ
cana-2208	209	15	q=1	q=1	VERB
cana-2208	209	16	in	in	ADP
cana-2208	209	17	our	our	PRON
cana-2208	209	18	result	result	NOUN
cana-2208	209	19	3.1	3.1	NUM
cana-2208	209	20	,	,	PUNCT
cana-2208	209	21	𝑛	𝑛	DET
cana-2208	209	22	−	−	PROPN
cana-2208	209	23	euler	euler	NOUN
cana-2208	209	24	product	product	NOUN
cana-2208	209	25	summability	summability	NOUN
cana-2208	209	26	(	(	PUNCT
cana-2208	209	27	𝐸	𝐸	PROPN
cana-2208	209	28	,	,	PUNCT
cana-2208	209	29	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	209	30	,	,	PUNCT
cana-2208	209	31	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	209	32	,	,	PUNCT
cana-2208	209	33	𝑞	𝑞	NOUN
cana-2208	209	34	)	)	PUNCT
cana-2208	209	35	.	.	PUNCT
cana-2208	209	36	.	.	PUNCT
cana-2208	210	1	.	.	PUNCT
cana-2208	211	1	(	(	PUNCT
cana-2208	211	2	𝐸	𝐸	PROPN
cana-2208	211	3	,	,	PUNCT
cana-2208	211	4	𝑞	𝑞	NOUN
cana-2208	211	5	)	)	PUNCT
cana-2208	211	6	reduces	reduce	VERB
cana-2208	211	7	to	to	PART
cana-2208	211	8	(	(	PUNCT
cana-2208	211	9	𝐸	𝐸	PROPN
cana-2208	211	10	,	,	PUNCT
cana-2208	211	11	1)(𝐸	1)(𝐸	NUM
cana-2208	211	12	,	,	PUNCT
cana-2208	211	13	1)(𝐸	1)(𝐸	NUM
cana-2208	211	14	,	,	PUNCT
cana-2208	211	15	1	1	NUM
cana-2208	211	16	)	)	PUNCT
cana-2208	211	17	i.e.	i.e.	X
cana-2208	211	18	triple	triple	ADJ
cana-2208	211	19	𝐸1	𝐸1	NOUN
cana-2208	211	20	summability	summability	NOUN
cana-2208	211	21	then	then	ADV
cana-2208	211	22	|𝑡𝑟	|𝑡𝑟	NUM
cana-2208	211	23	𝐸1𝐸1𝐸1	𝐸1𝐸1𝐸1	PROPN
cana-2208	211	24	(	(	PUNCT
cana-2208	211	25	𝑔	𝑔	NOUN
cana-2208	211	26	;	;	PUNCT
cana-2208	211	27	𝑥	𝑥	X
cana-2208	211	28	)	)	PUNCT
cana-2208	211	29	−	−	PROPN
cana-2208	211	30	𝑔(𝑥)|	𝑔(𝑥)|	ADV
cana-2208	211	31	=	=	PUNCT
cana-2208	212	1	𝑂(1	𝑂(1	ADP
cana-2208	212	2	)	)	PUNCT
cana-2208	212	3	,	,	PUNCT
cana-2208	212	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	212	5	𝑟	𝑟	X
cana-2208	212	6	→	→	SYM
cana-2208	212	7	∞.	∞.	PROPN
cana-2208	212	8	7.10	7.10	NUM
cana-2208	212	9	corollary	corollary	NOUN
cana-2208	212	10	if	if	SCONJ
cana-2208	212	11	we	we	PRON
cana-2208	212	12	take	take	VERB
cana-2208	212	13	n=3	n=3	PUNCT
cana-2208	212	14	and	and	CCONJ
cana-2208	212	15	q=1	q=1	VERB
cana-2208	212	16	in	in	ADP
cana-2208	212	17	our	our	PRON
cana-2208	212	18	result	result	NOUN
cana-2208	212	19	3.2	3.2	NUM
cana-2208	212	20	,	,	PUNCT
cana-2208	212	21	𝑛	𝑛	DET
cana-2208	212	22	−	−	PROPN
cana-2208	212	23	euler	euler	NOUN
cana-2208	212	24	product	product	NOUN
cana-2208	212	25	summability	summability	NOUN
cana-2208	212	26	(	(	PUNCT
cana-2208	212	27	𝐸	𝐸	PROPN
cana-2208	212	28	,	,	PUNCT
cana-2208	212	29	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	212	30	,	,	PUNCT
cana-2208	212	31	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	212	32	,	,	PUNCT
cana-2208	212	33	𝑞	𝑞	NOUN
cana-2208	212	34	)	)	PUNCT
cana-2208	212	35	.	.	PUNCT
cana-2208	212	36	.	.	PUNCT
cana-2208	213	1	.	.	PUNCT
cana-2208	214	1	(	(	PUNCT
cana-2208	214	2	𝐸	𝐸	PROPN
cana-2208	214	3	,	,	PUNCT
cana-2208	214	4	𝑞	𝑞	NOUN
cana-2208	214	5	)	)	PUNCT
cana-2208	214	6	reduces	reduce	VERB
cana-2208	214	7	to	to	PART
cana-2208	214	8	(	(	PUNCT
cana-2208	214	9	𝐸	𝐸	PROPN
cana-2208	214	10	,	,	PUNCT
cana-2208	214	11	1)(𝐸	1)(𝐸	NUM
cana-2208	214	12	,	,	PUNCT
cana-2208	214	13	1)(𝐸	1)(𝐸	NUM
cana-2208	214	14	,	,	PUNCT
cana-2208	214	15	1	1	NUM
cana-2208	214	16	)	)	PUNCT
cana-2208	214	17	i.e.	i.e.	X
cana-2208	214	18	triple	triple	ADJ
cana-2208	214	19	𝐸1	𝐸1	NOUN
cana-2208	214	20	summability	summability	NOUN
cana-2208	214	21	then	then	ADV
cana-2208	214	22	|	|	ADV
cana-2208	214	23	�	�	PROPN
cana-2208	214	24	̃	̃	PROPN
cana-2208	214	25	�	�	NOUN
cana-2208	214	26	𝑟	𝑟	NOUN
cana-2208	214	27	𝐸1𝐸1𝐸1	𝐸1𝐸1𝐸1	NOUN
cana-2208	214	28	(	(	PUNCT
cana-2208	214	29	𝑔	𝑔	NOUN
cana-2208	214	30	;	;	PUNCT
cana-2208	214	31	𝑥	𝑥	X
cana-2208	214	32	)	)	PUNCT
cana-2208	214	33	−	−	PROPN
cana-2208	214	34	�	�	PROPN
cana-2208	214	35	̃	̃	NOUN
cana-2208	214	36	�	�	PROPN
cana-2208	214	37	(𝑥)|	(𝑥)|	NOUN
cana-2208	214	38	=	=	SYM
cana-2208	215	1	𝑂(1	𝑂(1	ADP
cana-2208	215	2	)	)	PUNCT
cana-2208	215	3	,	,	PUNCT
cana-2208	215	4	𝑎𝑠	𝑎𝑠	ADP
cana-2208	215	5	𝑟	𝑟	X
cana-2208	215	6	→	→	SYM
cana-2208	215	7	∞	∞	NUM
cana-2208	215	8	8	8	NUM
cana-2208	215	9	particular	particular	ADJ
cana-2208	215	10	case	case	NOUN
cana-2208	215	11	8.1	8.1	NUM
cana-2208	215	12	in	in	ADP
cana-2208	215	13	view	view	NOUN
cana-2208	215	14	of	of	ADP
cana-2208	215	15	corollary	corollary	ADJ
cana-2208	215	16	7.3	7.3	NUM
cana-2208	215	17	and	and	CCONJ
cana-2208	215	18	7.4	7.4	NUM
cana-2208	215	19	,	,	PUNCT
cana-2208	215	20	theorem	theorem	VERB
cana-2208	215	21	1	1	NUM
cana-2208	215	22	and	and	CCONJ
cana-2208	215	23	2	2	NUM
cana-2208	216	1	[	[	SYM
cana-2208	216	2	5	5	NUM
cana-2208	216	3	]	]	PUNCT
cana-2208	216	4	are	be	AUX
cana-2208	216	5	particular	particular	ADJ
cana-2208	216	6	cases	case	NOUN
cana-2208	216	7	of	of	ADP
cana-2208	216	8	our	our	PRON
cana-2208	216	9	theorem	theorem	ADJ
cana-2208	216	10	3.1	3.1	NUM
cana-2208	216	11	and	and	CCONJ
cana-2208	216	12	3.2	3.2	NUM
cana-2208	216	13	respectively	respectively	ADV
cana-2208	216	14	.	.	PUNCT
cana-2208	217	1	8.2	8.2	NUM
cana-2208	217	2	in	in	ADP
cana-2208	217	3	view	view	NOUN
cana-2208	217	4	of	of	ADP
cana-2208	217	5	corollary	corollary	ADJ
cana-2208	217	6	7.5	7.5	NUM
cana-2208	217	7	and	and	CCONJ
cana-2208	217	8	7.6	7.6	NUM
cana-2208	217	9	,	,	PUNCT
cana-2208	217	10	theorem	theorem	VERB
cana-2208	217	11	1	1	NUM
cana-2208	217	12	and	and	CCONJ
cana-2208	217	13	2	2	NUM
cana-2208	218	1	[	[	SYM
cana-2208	218	2	4	4	NUM
cana-2208	218	3	]	]	PUNCT
cana-2208	218	4	are	be	AUX
cana-2208	218	5	particular	particular	ADJ
cana-2208	218	6	cases	case	NOUN
cana-2208	218	7	of	of	ADP
cana-2208	218	8	our	our	PRON
cana-2208	218	9	theorem	theorem	ADJ
cana-2208	218	10	3.1	3.1	NUM
cana-2208	218	11	and	and	CCONJ
cana-2208	218	12	3.2	3.2	NUM
cana-2208	218	13	respectively	respectively	ADV
cana-2208	218	14	.	.	PUNCT
cana-2208	219	1	8.3	8.3	NUM
cana-2208	219	2	in	in	ADP
cana-2208	219	3	view	view	NOUN
cana-2208	219	4	of	of	ADP
cana-2208	219	5	corollary	corollary	ADJ
cana-2208	219	6	7.7	7.7	NUM
cana-2208	219	7	and	and	CCONJ
cana-2208	219	8	7.8	7.8	NUM
cana-2208	219	9	,	,	PUNCT
cana-2208	219	10	theorem	theorem	VERB
cana-2208	219	11	3.1	3.1	NUM
cana-2208	219	12	and	and	CCONJ
cana-2208	219	13	3.2	3.2	NUM
cana-2208	220	1	[	[	SYM
cana-2208	220	2	3	3	NUM
cana-2208	220	3	]	]	PUNCT
cana-2208	220	4	are	be	AUX
cana-2208	220	5	particular	particular	ADJ
cana-2208	220	6	cases	case	NOUN
cana-2208	220	7	of	of	ADP
cana-2208	220	8	our	our	PRON
cana-2208	220	9	theorem	theorem	ADJ
cana-2208	220	10	3.1	3.1	NUM
cana-2208	220	11	and	and	CCONJ
cana-2208	220	12	3.2	3.2	NUM
cana-2208	220	13	respectively	respectively	ADV
cana-2208	220	14	.	.	PUNCT
cana-2208	221	1	8.4	8.4	NUM
cana-2208	221	2	in	in	ADP
cana-2208	221	3	view	view	NOUN
cana-2208	221	4	of	of	ADP
cana-2208	221	5	corollary	corollary	ADJ
cana-2208	221	6	7.9	7.9	NUM
cana-2208	221	7	and	and	CCONJ
cana-2208	221	8	7.10	7.10	NUM
cana-2208	221	9	,	,	PUNCT
cana-2208	221	10	theorem	theorem	VERB
cana-2208	221	11	3.1	3.1	NUM
cana-2208	221	12	and	and	CCONJ
cana-2208	221	13	3.2	3.2	NUM
cana-2208	222	1	[	[	X
cana-2208	222	2	7	7	NUM
cana-2208	222	3	]	]	PUNCT
cana-2208	222	4	are	be	AUX
cana-2208	222	5	particular	particular	ADJ
cana-2208	222	6	cases	case	NOUN
cana-2208	222	7	of	of	ADP
cana-2208	222	8	our	our	PRON
cana-2208	222	9	theorem	theorem	ADJ
cana-2208	222	10	3.1	3.1	NUM
cana-2208	222	11	and	and	CCONJ
cana-2208	222	12	3.2	3.2	NUM
cana-2208	222	13	respectively	respectively	ADV
cana-2208	222	14	.	.	PUNCT
cana-2208	223	1	communications	communication	NOUN
cana-2208	223	2	on	on	ADP
cana-2208	223	3	applied	apply	VERB
cana-2208	223	4	nonlinear	nonlinear	ADJ
cana-2208	223	5	analysis	analysis	NOUN
cana-2208	223	6	issn	issn	NOUN
cana-2208	223	7	:	:	PUNCT
cana-2208	223	8	1074	1074	NUM
cana-2208	223	9	-	-	PUNCT
cana-2208	223	10	133x	133x	NUM
cana-2208	223	11	vol	vol	NOUN
cana-2208	223	12	32	32	NUM
cana-2208	223	13	no	no	NOUN
cana-2208	223	14	.	.	PUNCT
cana-2208	224	1	1s	1s	NUM
cana-2208	224	2	(	(	PUNCT
cana-2208	224	3	2025	2025	NUM
cana-2208	224	4	)	)	PUNCT
cana-2208	224	5	455	455	NUM
cana-2208	224	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2208	224	7	9	9	NUM
cana-2208	224	8	conclusion	conclusion	NOUN
cana-2208	224	9	the	the	DET
cana-2208	224	10	results	result	NOUN
cana-2208	224	11	of	of	ADP
cana-2208	224	12	the	the	DET
cana-2208	224	13	paper	paper	NOUN
cana-2208	224	14	are	be	AUX
cana-2208	224	15	aimed	aim	VERB
cana-2208	224	16	to	to	PART
cana-2208	224	17	formulate	formulate	VERB
cana-2208	224	18	the	the	DET
cana-2208	224	19	problem	problem	NOUN
cana-2208	224	20	of	of	ADP
cana-2208	224	21	approximation	approximation	NOUN
cana-2208	224	22	of	of	ADP
cana-2208	224	23	function	function	NOUN
cana-2208	224	24	g	g	PROPN
cana-2208	224	25	and	and	CCONJ
cana-2208	224	26	their	their	PRON
cana-2208	224	27	conjugates	conjugate	NOUN
cana-2208	224	28	�	�	NOUN
cana-2208	224	29	̃	̃	NOUN
cana-2208	224	30	�	�	NOUN
cana-2208	224	31	by	by	ADP
cana-2208	224	32	iterates	iterate	NOUN
cana-2208	224	33	of	of	ADP
cana-2208	224	34	euler	euler	NOUN
cana-2208	224	35	sum	sum	NOUN
cana-2208	224	36	of	of	ADP
cana-2208	224	37	their	their	PRON
cana-2208	224	38	fourier	fourier	NOUN
cana-2208	224	39	series	series	NOUN
cana-2208	224	40	and	and	CCONJ
cana-2208	224	41	conjugate	conjugate	ADJ
cana-2208	224	42	fourier	fourier	NOUN
cana-2208	224	43	series	series	NOUN
cana-2208	224	44	respectively	respectively	ADV
cana-2208	224	45	.	.	PUNCT
cana-2208	225	1	acknowledgments	acknowledgment	NOUN
cana-2208	225	2	the	the	DET
cana-2208	225	3	first	first	ADJ
cana-2208	225	4	author	author	NOUN
cana-2208	225	5	expresses	express	VERB
cana-2208	225	6	his	his	PRON
cana-2208	225	7	gratitude	gratitude	NOUN
cana-2208	225	8	towards	towards	ADP
cana-2208	225	9	his	his	PRON
cana-2208	225	10	parents	parent	NOUN
cana-2208	225	11	for	for	ADP
cana-2208	225	12	blessings	blessing	NOUN
cana-2208	225	13	.	.	PUNCT
cana-2208	226	1	second	second	ADJ
cana-2208	226	2	author	author	NOUN
cana-2208	226	3	exress	exress	VERB
cana-2208	226	4	his	his	PRON
cana-2208	226	5	gratitude	gratitude	NOUN
cana-2208	226	6	towards	towards	ADP
cana-2208	226	7	his	his	PRON
cana-2208	226	8	parents	parent	NOUN
cana-2208	226	9	for	for	ADP
cana-2208	226	10	blessing	blessing	NOUN
cana-2208	226	11	.	.	PUNCT
cana-2208	227	1	all	all	DET
cana-2208	227	2	the	the	DET
cana-2208	227	3	authors	author	NOUN
cana-2208	227	4	are	be	AUX
cana-2208	227	5	also	also	ADV
cana-2208	227	6	grateful	grateful	ADJ
cana-2208	227	7	to	to	ADP
cana-2208	227	8	the	the	DET
cana-2208	227	9	hon’ble	hon’ble	ADJ
cana-2208	227	10	vicechancellor	vicechancellor	NOUN
cana-2208	227	11	,	,	PUNCT
cana-2208	227	12	patliputra	patliputra	PROPN
cana-2208	227	13	university	university	PROPN
cana-2208	227	14	,	,	PUNCT
cana-2208	227	15	patna	patna	PROPN
cana-2208	227	16	,	,	PUNCT
cana-2208	227	17	bihar	bihar	PROPN
cana-2208	227	18	,	,	PUNCT
cana-2208	227	19	india	india	PROPN
cana-2208	227	20	,	,	PUNCT
cana-2208	227	21	for	for	ADP
cana-2208	227	22	motivation	motivation	NOUN
cana-2208	227	23	to	to	ADP
cana-2208	227	24	this	this	DET
cana-2208	227	25	work	work	NOUN
cana-2208	227	26	.	.	PUNCT
cana-2208	228	1	refrences	refrence	VERB
cana-2208	229	1	[	[	X
cana-2208	229	2	1	1	NUM
cana-2208	229	3	]	]	PUNCT
cana-2208	229	4	a.	a.	NOUN
cana-2208	229	5	zygmund	zygmund	PROPN
cana-2208	229	6	,	,	PUNCT
cana-2208	229	7	trigonometric	trigonometric	ADJ
cana-2208	229	8	series	series	NOUN
cana-2208	229	9	,	,	PUNCT
cana-2208	229	10	cambridge	cambridge	PROPN
cana-2208	229	11	univ	univ	PROPN
cana-2208	229	12	.	.	PUNCT
cana-2208	230	1	press	press	PROPN
cana-2208	230	2	,	,	PUNCT
cana-2208	230	3	cambridge	cambridge	PROPN
cana-2208	230	4	,	,	PUNCT
cana-2208	230	5	3rd	3rd	PROPN
cana-2208	230	6	rev	rev	PROPN
cana-2208	230	7	.	.	PUNCT
cana-2208	231	1	ed	ed	NOUN
cana-2208	231	2	.	.	PROPN
cana-2208	231	3	,	,	PUNCT
cana-2208	231	4	2002	2002	NUM
cana-2208	231	5	.	.	PUNCT
cana-2208	232	1	[	[	X
cana-2208	232	2	2	2	NUM
cana-2208	232	3	]	]	X
cana-2208	232	4	e.z	e.z	PROPN
cana-2208	232	5	.	.	PROPN
cana-2208	232	6	psarakis	psarakis	PROPN
cana-2208	232	7	and	and	CCONJ
cana-2208	232	8	g.v	g.v	PROPN
cana-2208	232	9	.	.	PROPN
cana-2208	232	10	moustakides	moustakides	PROPN
cana-2208	232	11	,	,	PUNCT
cana-2208	232	12	an	an	DET
cana-2208	232	13	𝐿2	𝐿2	NOUN
cana-2208	232	14	-	-	PUNCT
cana-2208	232	15	based	base	VERB
cana-2208	232	16	method	method	NOUN
cana-2208	232	17	for	for	ADP
cana-2208	232	18	the	the	DET
cana-2208	232	19	design	design	NOUN
cana-2208	232	20	of	of	ADP
cana-2208	232	21	1	1	NUM
cana-2208	232	22	-	-	SYM
cana-2208	232	23	d	d	ADJ
cana-2208	232	24	zero	zero	NUM
cana-2208	232	25	phase	phase	NOUN
cana-2208	232	26	fir	fir	NOUN
cana-2208	232	27	digital	digital	ADJ
cana-2208	232	28	filters	filter	NOUN
cana-2208	232	29	,	,	PUNCT
cana-2208	232	30	ieee	ieee	NOUN
cana-2208	232	31	trans	tran	NOUN
cana-2208	232	32	.	.	PUNCT
cana-2208	233	1	circuits	circuit	NOUN
cana-2208	233	2	syst	syst	PROPN
cana-2208	233	3	.	.	PUNCT
cana-2208	234	1	i	i	PRON
cana-2208	234	2	,	,	PUNCT
cana-2208	234	3	fundam	fundam	PROPN
cana-2208	234	4	.	.	PUNCT
cana-2208	234	5	theory	theory	NOUN
cana-2208	234	6	appl	appl	PROPN
cana-2208	234	7	.	.	PUNCT
cana-2208	235	1	44(7	44(7	NUM
cana-2208	235	2	)	)	PUNCT
cana-2208	235	3	,	,	PUNCT
cana-2208	235	4	551	551	NUM
cana-2208	235	5	-	-	NUM
cana-2208	235	6	601(1997	601(1997	NUM
cana-2208	235	7	)	)	PUNCT
cana-2208	235	8	.	.	PUNCT
cana-2208	236	1	[	[	X
cana-2208	236	2	3	3	NUM
cana-2208	236	3	]	]	X
cana-2208	236	4	hare	hare	NOUN
cana-2208	236	5	krishna	krishna	PROPN
cana-2208	236	6	nigam	nigam	PROPN
cana-2208	236	7	and	and	CCONJ
cana-2208	236	8	md	md	PROPN
cana-2208	236	9	.	.	PROPN
cana-2208	237	1	hadish	hadish	PROPN
cana-2208	237	2	,	,	PUNCT
cana-2208	237	3	trigonometric	trigonometric	ADJ
cana-2208	237	4	approximation	approximation	NOUN
cana-2208	237	5	of	of	ADP
cana-2208	237	6	functions	function	NOUN
cana-2208	237	7	by	by	ADP
cana-2208	237	8	hausdorff	hausdorff	NOUN
cana-2208	237	9	-	-	PUNCT
cana-2208	237	10	matrix	matrix	NOUN
cana-2208	237	11	product	product	NOUN
cana-2208	237	12	operators	operator	NOUN
cana-2208	237	13	,	,	PUNCT
cana-2208	237	14	vol	vol	NOUN
cana-2208	237	15	.	.	PROPN
cana-2208	238	1	24	24	NUM
cana-2208	238	2	,	,	PUNCT
cana-2208	238	3	no	no	INTJ
cana-2208	238	4	.	.	NOUN
cana-2208	238	5	4	4	NUM
cana-2208	238	6	(	(	PUNCT
cana-2208	238	7	2019	2019	NUM
cana-2208	238	8	)	)	PUNCT
cana-2208	238	9	,	,	PUNCT
cana-2208	238	10	pp	pp	ADJ
cana-2208	238	11	.	.	PUNCT
cana-2208	239	1	675	675	NUM
cana-2208	239	2	-	-	SYM
cana-2208	239	3	689issn	689issn	NUM
cana-2208	239	4	:	:	PUNCT
cana-2208	239	5	1229	1229	NUM
cana-2208	239	6	-	-	SYM
cana-2208	239	7	1595(print	1595(print	NUM
cana-2208	239	8	)	)	PUNCT
cana-2208	239	9	,	,	PUNCT
cana-2208	239	10	2466	2466	NUM
cana-2208	239	11	-	-	SYM
cana-2208	239	12	0973(online	0973(online	NUM
cana-2208	239	13	)	)	PUNCT
cana-2208	239	14	.	.	PUNCT
cana-2208	240	1	[	[	X
cana-2208	240	2	4	4	X
cana-2208	240	3	]	]	X
cana-2208	240	4	kalpana	kalpana	PROPN
cana-2208	240	5	saxena	saxena	PROPN
cana-2208	240	6	,	,	PUNCT
cana-2208	240	7	manju	manju	PROPN
cana-2208	240	8	prabhakar	prabhakar	PROPN
cana-2208	240	9	,	,	PUNCT
cana-2208	240	10	a	a	DET
cana-2208	240	11	study	study	NOUN
cana-2208	240	12	on	on	ADP
cana-2208	240	13	(	(	PUNCT
cana-2208	240	14	𝐸	𝐸	PROPN
cana-2208	240	15	,	,	PUNCT
cana-2208	240	16	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2208	240	17	,	,	PUNCT
cana-2208	240	18	𝑞	𝑞	NOUN
cana-2208	240	19	)	)	PUNCT
cana-2208	240	20	product	product	NOUN
cana-2208	240	21	summability	summability	NOUN
cana-2208	240	22	,	,	PUNCT
cana-2208	240	23	international	international	ADJ
cana-2208	240	24	journal	journal	NOUN
cana-2208	240	25	of	of	ADP
cana-2208	240	26	advanced	advanced	ADJ
cana-2208	240	27	technology	technology	NOUN
cana-2208	240	28	and	and	CCONJ
cana-2208	240	29	engineering	engineering	NOUN
cana-2208	240	30	research(ijater	research(ijater	NOUN
cana-2208	240	31	)	)	PUNCT
cana-2208	240	32	.	.	PUNCT
cana-2208	241	1	[	[	X
cana-2208	241	2	5	5	X
cana-2208	241	3	]	]	PUNCT
cana-2208	241	4	k.	k.	PROPN
cana-2208	241	5	saxena	saxena	PROPN
cana-2208	241	6	and	and	CCONJ
cana-2208	241	7	m.	m.	PROPN
cana-2208	241	8	prabhakar	prabhakar	PROPN
cana-2208	241	9	,	,	PUNCT
cana-2208	241	10	a	a	DET
cana-2208	241	11	study	study	NOUN
cana-2208	241	12	of	of	ADP
cana-2208	241	13	double	double	ADJ
cana-2208	241	14	euler	euler	NOUN
cana-2208	241	15	summability	summability	NOUN
cana-2208	241	16	method	method	NOUN
cana-2208	241	17	of	of	ADP
cana-2208	241	18	fourier	fourier	ADJ
cana-2208	241	19	series	series	NOUN
cana-2208	241	20	and	and	CCONJ
cana-2208	241	21	its	its	PRON
cana-2208	241	22	conjugate	conjugate	ADJ
cana-2208	241	23	series	series	NOUN
cana-2208	241	24	,	,	PUNCT
cana-2208	241	25	int	int	NOUN
cana-2208	241	26	.	.	PUNCT
cana-2208	242	1	j.sci	j.sci	PROPN
cana-2208	242	2	.	.	PUNCT
cana-2208	243	1	innov.math.res	innov.math.res	PROPN
cana-2208	243	2	.	.	PUNCT
cana-2208	244	1	4(1),46	4(1),46	NUM
cana-2208	244	2	-	-	PUNCT
cana-2208	244	3	52(2016	52(2016	NUM
cana-2208	244	4	)	)	PUNCT
cana-2208	244	5	.	.	PUNCT
cana-2208	245	1	[	[	X
cana-2208	245	2	6	6	NUM
cana-2208	245	3	]	]	PUNCT
cana-2208	245	4	mohammed	mohammed	PROPN
cana-2208	245	5	hadish	hadish	PROPN
cana-2208	245	6	,	,	PUNCT
cana-2208	245	7	a	a	DET
cana-2208	245	8	study	study	NOUN
cana-2208	245	9	on	on	ADP
cana-2208	245	10	approximation	approximation	NOUN
cana-2208	245	11	of	of	ADP
cana-2208	245	12	a	a	DET
cana-2208	245	13	conjugate	conjugate	ADJ
cana-2208	245	14	function	function	NOUN
cana-2208	245	15	using	use	VERB
cana-2208	245	16	cesáro	cesáro	NOUN
cana-2208	245	17	-	-	PUNCT
cana-2208	245	18	matrix	matrix	NOUN
cana-2208	245	19	product	product	NOUN
cana-2208	245	20	operator	operator	NOUN
cana-2208	245	21	,	,	PUNCT
cana-2208	245	22	doi	doi	NOUN
cana-2208	245	23	:	:	PUNCT
cana-2208	245	24	http://dx.doi.org/10.5772	http://dx.doi.org/10.5772	PROPN
cana-2208	245	25	/	/	SYM
cana-2208	245	26	intechopen.103015	intechopen.103015	PROPN
cana-2208	245	27	.	.	PUNCT
cana-2208	246	1	[	[	X
cana-2208	246	2	7	7	NUM
cana-2208	246	3	]	]	SYM
cana-2208	246	4	s.sonkar	s.sonkar	NOUN
cana-2208	246	5	and	and	CCONJ
cana-2208	246	6	p.sagwan	p.sagwan	NOUN
cana-2208	246	7	,	,	PUNCT
cana-2208	246	8	approximation	approximation	NOUN
cana-2208	246	9	of	of	ADP
cana-2208	246	10	fourier	fourier	NOUN
cana-2208	246	11	and	and	CCONJ
cana-2208	246	12	its	its	PRON
cana-2208	246	13	conjugate	conjugate	ADJ
cana-2208	246	14	series	series	NOUN
cana-2208	246	15	by	by	ADP
cana-2208	246	16	triple	triple	PROPN
cana-2208	246	17	euler	euler	NOUN
cana-2208	246	18	product	product	NOUN
cana-2208	246	19	summability	summability	NOUN
cana-2208	246	20	,	,	PUNCT
cana-2208	246	21	j.	j.	PROPN
cana-2208	246	22	phys	phys	PROPN
cana-2208	246	23	.	.	PUNCT
cana-2208	246	24	conf	conf	PROPN
cana-2208	246	25	.	.	PUNCT
cana-2208	247	1	ser	ser	PROPN
cana-2208	247	2	.	.	PROPN
cana-2208	248	1	1770(1	1770(1	NUM
cana-2208	248	2	)	)	PUNCT
cana-2208	248	3	,	,	PUNCT
cana-2208	249	1	012003(2021	012003(2021	NUM
cana-2208	249	2	)	)	PUNCT
cana-2208	249	3	.	.	PUNCT
cana-2208	250	1	[	[	X
cana-2208	250	2	8	8	X
cana-2208	250	3	]	]	X
cana-2208	250	4	s.	s.	PROPN
cana-2208	250	5	sonkar	sonkar	PROPN
cana-2208	250	6	and	and	CCONJ
cana-2208	250	7	u.	u.	PROPN
cana-2208	250	8	singh	singh	PROPN
cana-2208	250	9	,	,	PUNCT
cana-2208	250	10	degree	degree	NOUN
cana-2208	250	11	of	of	ADP
cana-2208	250	12	approximation	approximation	NOUN
cana-2208	250	13	of	of	ADP
cana-2208	250	14	the	the	DET
cana-2208	250	15	conjugate	conjugate	NOUN
cana-2208	250	16	of	of	ADP
cana-2208	250	17	signals(functions	signals(function	NOUN
cana-2208	250	18	)	)	PUNCT
cana-2208	250	19	belonging	belong	VERB
cana-2208	250	20	to	to	ADP
cana-2208	250	21	lip(𝛼	lip(𝛼	PROPN
cana-2208	250	22	,	,	PUNCT
cana-2208	250	23	𝑟)-class	𝑟)-clas	VERB
cana-2208	250	24	by	by	ADP
cana-2208	250	25	(	(	PUNCT
cana-2208	250	26	𝐶	𝐶	PROPN
cana-2208	250	27	,	,	PUNCT
cana-2208	250	28	1)(𝐸	1)(𝐸	NUM
cana-2208	250	29	,	,	PUNCT
cana-2208	250	30	𝑞	𝑞	NOUN
cana-2208	250	31	)	)	PUNCT
cana-2208	250	32	means	mean	NOUN
cana-2208	250	33	of	of	ADP
cana-2208	250	34	conjugate	conjugate	ADJ
cana-2208	250	35	trigonometric	trigonometric	ADJ
cana-2208	250	36	fourier	fourier	NOUN
cana-2208	250	37	series	series	NOUN
cana-2208	250	38	,	,	PUNCT
cana-2208	250	39	journal	journal	NOUN
cana-2208	250	40	of	of	ADP
cana-2208	250	41	inequalities	inequality	NOUN
cana-2208	250	42	and	and	CCONJ
cana-2208	250	43	applications	application	NOUN
cana-2208	250	44	2012	2012	NUM
cana-2208	250	45	,	,	PUNCT
cana-2208	250	46	2012:278	2012:278	NOUN
cana-2208	250	47	.	.	PUNCT
cana-2208	251	1	[	[	X
cana-2208	251	2	9	9	NUM
cana-2208	251	3	]	]	X
cana-2208	251	4	sachin	sachin	PROPN
cana-2208	251	5	devaiya	devaiya	PROPN
cana-2208	251	6	and	and	CCONJ
cana-2208	251	7	shailesh	shailesh	PROPN
cana-2208	251	8	kumar	kumar	PROPN
cana-2208	251	9	srivastava	srivastava	PROPN
cana-2208	251	10	,	,	PUNCT
cana-2208	251	11	approximation	approximation	NOUN
cana-2208	251	12	of	of	ADP
cana-2208	251	13	functions	function	NOUN
cana-2208	251	14	and	and	CCONJ
cana-2208	251	15	conjugate	conjugate	NOUN
cana-2208	251	16	of	of	ADP
cana-2208	251	17	functions	function	NOUN
cana-2208	251	18	using	use	VERB
cana-2208	251	19	product	product	NOUN
cana-2208	251	20	mean	mean	NOUN
cana-2208	251	21	(	(	PUNCT
cana-2208	251	22	e	e	NOUN
cana-2208	251	23	,	,	PUNCT
cana-2208	251	24	q)(e	q)(e	PROPN
cana-2208	251	25	,	,	PUNCT
cana-2208	251	26	q)(e	q)(e	PROPN
cana-2208	251	27	,	,	PUNCT
cana-2208	251	28	q),vol	q),vol	NOUN
cana-2208	251	29	.	.	PUNCT
cana-2208	252	1	11(special	11(special	ADJ
cana-2208	252	2	issue	issue	NOUN
cana-2208	252	3	i)(2022	i)(2022	NOUN
cana-2208	252	4	)	)	PUNCT
cana-2208	252	5	,	,	PUNCT
cana-2208	252	6	29–37	29–37	NOUN
cana-2208	252	7	.	.	PUNCT