id	sid	tid	token	lemma	pos
cana-2743	1	1	communications	communication	NOUN
cana-2743	1	2	on	on	ADP
cana-2743	1	3	applied	apply	VERB
cana-2743	1	4	nonlinear	nonlinear	ADJ
cana-2743	1	5	analysis	analysis	NOUN
cana-2743	1	6	issn	issn	NOUN
cana-2743	1	7	:	:	PUNCT
cana-2743	1	8	1074	1074	NUM
cana-2743	1	9	-	-	PUNCT
cana-2743	1	10	133x	133x	NUM
cana-2743	1	11	vol	vol	NOUN
cana-2743	1	12	32	32	NUM
cana-2743	1	13	no	no	NOUN
cana-2743	1	14	.	.	PUNCT
cana-2743	2	1	4s	4s	NUM
cana-2743	2	2	(	(	PUNCT
cana-2743	2	3	2025	2025	NUM
cana-2743	2	4	)	)	PUNCT
cana-2743	2	5	118	118	NUM
cana-2743	2	6	https://internationalpubls.com	https://internationalpubls.com	NOUN
cana-2743	2	7	degree	degree	NOUN
cana-2743	2	8	of	of	ADP
cana-2743	2	9	approximation	approximation	NOUN
cana-2743	2	10	of	of	ADP
cana-2743	2	11	the	the	DET
cana-2743	2	12	conjugate	conjugate	NOUN
cana-2743	2	13	of	of	ADP
cana-2743	2	14	functions	function	NOUN
cana-2743	2	15	belonging	belong	VERB
cana-2743	2	16	to	to	ADP
cana-2743	2	17	lip	lip	NOUN
cana-2743	2	18	(	(	PUNCT
cana-2743	2	19	𝜶	𝜶	NUM
cana-2743	2	20	,	,	PUNCT
cana-2743	2	21	𝒓	𝒓	NOUN
cana-2743	2	22	)	)	PUNCT
cana-2743	2	23	−class	−class	NOUN
cana-2743	2	24	by	by	ADP
cana-2743	2	25	(	(	PUNCT
cana-2743	2	26	𝑪	𝑪	PROPN
cana-2743	2	27	,	,	PUNCT
cana-2743	2	28	𝟏)(𝑬	𝟏)(𝑬	PROPN
cana-2743	2	29	,	,	PUNCT
cana-2743	2	30	𝒒)(𝑬	𝒒)(𝑬	PROPN
cana-2743	2	31	,	,	PUNCT
cana-2743	2	32	𝒒	𝒒	X
cana-2743	2	33	)	)	PUNCT
cana-2743	2	34	means	mean	NOUN
cana-2743	2	35	of	of	ADP
cana-2743	2	36	conjugate	conjugate	ADJ
cana-2743	2	37	fourier	fourier	NOUN
cana-2743	2	38	series	series	PROPN
cana-2743	2	39	rupesh	rupesh	PROPN
cana-2743	2	40	kumar	kumar	PROPN
cana-2743	2	41	mishra	mishra	PROPN
cana-2743	2	42	𝟏	𝟏	PROPN
cana-2743	2	43	and	and	CCONJ
cana-2743	2	44	shambhu	shambhu	VERB
cana-2743	2	45	kumar	kumar	PROPN
cana-2743	2	46	mishra	mishra	PROPN
cana-2743	2	47	𝟐	𝟐	PROPN
cana-2743	2	48	1research	1research	NUM
cana-2743	2	49	scholar	scholar	NOUN
cana-2743	2	50	,	,	PUNCT
cana-2743	2	51	department	department	NOUN
cana-2743	2	52	of	of	ADP
cana-2743	2	53	mathematics	mathematics	PROPN
cana-2743	2	54	,	,	PUNCT
cana-2743	2	55	patliputra	patliputra	PROPN
cana-2743	2	56	university	university	PROPN
cana-2743	2	57	,	,	PUNCT
cana-2743	2	58	patna	patna	PROPN
cana-2743	2	59	,	,	PUNCT
cana-2743	2	60	bihar	bihar	PROPN
cana-2743	2	61	,	,	PUNCT
cana-2743	2	62	india	india	PROPN
cana-2743	2	63	2professor	2professor	PROPN
cana-2743	2	64	,	,	PUNCT
cana-2743	2	65	department	department	NOUN
cana-2743	2	66	of	of	ADP
cana-2743	2	67	mathematics	mathematics	PROPN
cana-2743	2	68	,	,	PUNCT
cana-2743	2	69	patliputra	patliputra	PROPN
cana-2743	2	70	university	university	PROPN
cana-2743	2	71	,	,	PUNCT
cana-2743	2	72	patna	patna	PROPN
cana-2743	2	73	,	,	PUNCT
cana-2743	2	74	bihar	bihar	PROPN
cana-2743	2	75	,	,	PUNCT
cana-2743	2	76	india	india	PROPN
cana-2743	2	77	e	e	PROPN
cana-2743	2	78	-	-	NOUN
cana-2743	2	79	mail	mail	NOUN
cana-2743	2	80	:	:	PUNCT
cana-2743	2	81	1rupeshmishra043@gmail.com	1rupeshmishra043@gmail.com	NUM
cana-2743	2	82	,	,	PUNCT
cana-2743	2	83	2shambhumishra5@gmail.com	2shambhumishra5@gmail.com	NUM
cana-2743	2	84	article	article	NOUN
cana-2743	2	85	history	history	NOUN
cana-2743	2	86	:	:	PUNCT
cana-2743	2	87	received	receive	VERB
cana-2743	2	88	:	:	PUNCT
cana-2743	2	89	16	16	NUM
cana-2743	2	90	-	-	SYM
cana-2743	2	91	09	09	NUM
cana-2743	2	92	-	-	PUNCT
cana-2743	2	93	2024	2024	NUM
cana-2743	2	94	revised	revise	VERB
cana-2743	2	95	:	:	PUNCT
cana-2743	2	96	21	21	NUM
cana-2743	2	97	-	-	SYM
cana-2743	2	98	11	11	NUM
cana-2743	2	99	-	-	PUNCT
cana-2743	2	100	2024	2024	NUM
cana-2743	2	101	accepted	accept	VERB
cana-2743	2	102	:	:	PUNCT
cana-2743	2	103	29	29	NUM
cana-2743	2	104	-	-	SYM
cana-2743	2	105	11	11	NUM
cana-2743	2	106	-	-	PUNCT
cana-2743	2	107	2024	2024	NUM
cana-2743	2	108	abstract	abstract	NOUN
cana-2743	2	109	:	:	PUNCT
cana-2743	2	110	this	this	DET
cana-2743	2	111	research	research	NOUN
cana-2743	2	112	paper	paper	NOUN
cana-2743	2	113	is	be	AUX
cana-2743	2	114	related	relate	VERB
cana-2743	2	115	to	to	ADP
cana-2743	2	116	the	the	DET
cana-2743	2	117	degree	degree	NOUN
cana-2743	2	118	of	of	ADP
cana-2743	2	119	approximation	approximation	NOUN
cana-2743	2	120	of	of	ADP
cana-2743	2	121	the	the	DET
cana-2743	2	122	conjugate	conjugate	NOUN
cana-2743	2	123	of	of	ADP
cana-2743	2	124	2𝜋	2𝜋	NOUN
cana-2743	2	125	−periodic	−periodic	ADJ
cana-2743	2	126	function	function	NOUN
cana-2743	2	127	belonging	belong	VERB
cana-2743	2	128	to	to	ADP
cana-2743	2	129	the	the	DET
cana-2743	2	130	lip(𝛼	lip(𝛼	PROPN
cana-2743	2	131	,	,	PUNCT
cana-2743	2	132	𝑟)(0	𝑟)(0	NUM
cana-2743	2	133	<	<	X
cana-2743	2	134	𝛼	𝛼	X
cana-2743	2	135	≤	≤	NUM
cana-2743	2	136	1	1	NUM
cana-2743	2	137	,	,	PUNCT
cana-2743	2	138	𝑟	𝑟	PRON
cana-2743	2	139	≥	≥	NUM
cana-2743	2	140	1)class	1)class	NUM
cana-2743	2	141	by	by	ADP
cana-2743	2	142	using	use	VERB
cana-2743	2	143	(	(	PUNCT
cana-2743	2	144	𝐶	𝐶	PROPN
cana-2743	2	145	,	,	PUNCT
cana-2743	2	146	1)(𝐸	1)(𝐸	NUM
cana-2743	2	147	,	,	PUNCT
cana-2743	2	148	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	2	149	,	,	PUNCT
cana-2743	2	150	𝑞	𝑞	NOUN
cana-2743	2	151	)	)	PUNCT
cana-2743	2	152	means	mean	NOUN
cana-2743	2	153	of	of	ADP
cana-2743	2	154	the	the	DET
cana-2743	2	155	conjugate	conjugate	ADJ
cana-2743	2	156	fourier	fourier	NOUN
cana-2743	2	157	series	series	NOUN
cana-2743	2	158	.	.	PUNCT
cana-2743	3	1	our	our	PRON
cana-2743	3	2	result	result	NOUN
cana-2743	3	3	may	may	AUX
cana-2743	3	4	be	be	AUX
cana-2743	3	5	useful	useful	ADJ
cana-2743	3	6	for	for	ADP
cana-2743	3	7	the	the	DET
cana-2743	3	8	coming	come	VERB
cana-2743	3	9	researchers	researcher	NOUN
cana-2743	3	10	in	in	ADP
cana-2743	3	11	the	the	DET
cana-2743	3	12	future	future	NOUN
cana-2743	3	13	.	.	PUNCT
cana-2743	4	1	keywords	keyword	NOUN
cana-2743	4	2	:	:	PUNCT
cana-2743	4	3	lip(𝛼	lip(𝛼	PROPN
cana-2743	4	4	,	,	PUNCT
cana-2743	4	5	𝑟	𝑟	NOUN
cana-2743	4	6	)	)	PUNCT
cana-2743	4	7	−	−	PROPN
cana-2743	4	8	class	class	NOUN
cana-2743	4	9	,	,	PUNCT
cana-2743	4	10	conjugate	conjugate	ADJ
cana-2743	4	11	fourier	fouri	ADJ
cana-2743	4	12	series	series	NOUN
cana-2743	4	13	,	,	PUNCT
cana-2743	4	14	(	(	PUNCT
cana-2743	4	15	𝐶	𝐶	PROPN
cana-2743	4	16	,	,	PUNCT
cana-2743	4	17	1)(𝐸	1)(𝐸	NUM
cana-2743	4	18	,	,	PUNCT
cana-2743	4	19	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	4	20	,	,	PUNCT
cana-2743	4	21	𝑞	𝑞	NOUN
cana-2743	4	22	)	)	PUNCT
cana-2743	4	23	means	mean	NOUN
cana-2743	4	24	.	.	PUNCT
cana-2743	5	1	1	1	X
cana-2743	5	2	.	.	X
cana-2743	5	3	introduction	introduction	NOUN
cana-2743	5	4	let	let	VERB
cana-2743	5	5	∑∞	∑∞	PROPN
cana-2743	5	6	𝑛=0	𝑛=0	PROPN
cana-2743	5	7	𝑢𝑛	𝑢𝑛	NOUN
cana-2743	5	8	be	be	AUX
cana-2743	5	9	a	a	DET
cana-2743	5	10	given	give	VERB
cana-2743	5	11	infinite	infinite	ADJ
cana-2743	5	12	series	series	NOUN
cana-2743	5	13	and	and	CCONJ
cana-2743	5	14	the	the	DET
cana-2743	5	15	sequence	sequence	NOUN
cana-2743	5	16	{	{	PUNCT
cana-2743	5	17	𝑠𝑛	𝑠𝑛	NOUN
cana-2743	5	18	}	}	PUNCT
cana-2743	5	19	its	its	PRON
cana-2743	5	20	nth	nth	NOUN
cana-2743	5	21	partial	partial	ADJ
cana-2743	5	22	sum.the	sum.the	DET
cana-2743	5	23	sequence	sequence	NOUN
cana-2743	5	24	-to	-to	SCONJ
cana-2743	5	25	sequence	sequence	NOUN
cana-2743	5	26	transform	transform	VERB
cana-2743	5	27	𝐶𝑛	𝐶𝑛	NOUN
cana-2743	5	28	1	1	NUM
cana-2743	5	29	=	=	SYM
cana-2743	5	30	1	1	NUM
cana-2743	5	31	𝑛+1	𝑛+1	X
cana-2743	5	32	∑𝑛	∑𝑛	PROPN
cana-2743	5	33	𝑘=0	𝑘=0	ADP
cana-2743	5	34	𝑠𝑘	𝑠𝑘	NOUN
cana-2743	5	35	,	,	PUNCT
cana-2743	5	36	𝑛	𝑛	NOUN
cana-2743	5	37	=	=	NOUN
cana-2743	5	38	0,1,2	0,1,2	NOUN
cana-2743	5	39	,	,	PUNCT
cana-2743	5	40	.	.	PUNCT
cana-2743	5	41	..	..	PUNCT
cana-2743	6	1	(	(	PUNCT
cana-2743	6	2	1	1	X
cana-2743	6	3	)	)	PUNCT
cana-2743	6	4	defines	define	VERB
cana-2743	6	5	the	the	DET
cana-2743	6	6	cesàro	cesàro	PROPN
cana-2743	6	7	means	mean	VERB
cana-2743	6	8	of	of	ADP
cana-2743	6	9	order	order	NOUN
cana-2743	6	10	one	one	NUM
cana-2743	6	11	of	of	ADP
cana-2743	6	12	{	{	PUNCT
cana-2743	6	13	𝑠𝑛	𝑠𝑛	NOUN
cana-2743	6	14	}	}	PUNCT
cana-2743	6	15	.	.	PUNCT
cana-2743	7	1	if	if	SCONJ
cana-2743	7	2	𝑙𝑖𝑚𝑛→∞𝐶𝑛	𝑙𝑖𝑚𝑛→∞𝐶𝑛	NOUN
cana-2743	7	3	1	1	NUM
cana-2743	7	4	=	=	SYM
cana-2743	7	5	𝑠	𝑠	PROPN
cana-2743	7	6	,	,	PUNCT
cana-2743	7	7	the	the	DET
cana-2743	7	8	series	series	NOUN
cana-2743	7	9	∑∞	∑∞	PROPN
cana-2743	7	10	𝑛=0	𝑛=0	PROPN
cana-2743	7	11	𝑢𝑛	𝑢𝑛	NOUN
cana-2743	7	12	is	be	AUX
cana-2743	7	13	said	say	VERB
cana-2743	7	14	to	to	PART
cana-2743	7	15	be	be	AUX
cana-2743	7	16	(	(	PUNCT
cana-2743	7	17	𝐶	𝐶	PROPN
cana-2743	7	18	,	,	PUNCT
cana-2743	7	19	1	1	NUM
cana-2743	7	20	)	)	PUNCT
cana-2743	7	21	summable	summable	ADJ
cana-2743	7	22	to	to	ADP
cana-2743	7	23	s.	s.	PROPN
cana-2743	7	24	the	the	DET
cana-2743	7	25	sequence	sequence	NOUN
cana-2743	7	26	-	-	PUNCT
cana-2743	7	27	to	to	ADP
cana-2743	7	28	-	-	PUNCT
cana-2743	7	29	sequence	sequence	NOUN
cana-2743	7	30	transform	transform	NOUN
cana-2743	7	31	𝐸𝑛	𝐸𝑛	ADP
cana-2743	7	32	𝑞	𝑞	NOUN
cana-2743	7	33	=	=	SYM
cana-2743	7	34	1	1	NUM
cana-2743	7	35	(	(	PUNCT
cana-2743	7	36	1+𝑞)𝑛	1+𝑞)𝑛	NUM
cana-2743	7	37	∑𝑛	∑𝑛	ADJ
cana-2743	7	38	𝑘=0	𝑘=0	PROPN
cana-2743	7	39	(	(	PUNCT
cana-2743	7	40	𝑛	𝑛	PRON
cana-2743	7	41	𝑘	𝑘	NOUN
cana-2743	7	42	)	)	PUNCT
cana-2743	7	43	𝑞𝑛−𝑘𝑠𝑘	𝑞𝑛−𝑘𝑠𝑘	PROPN
cana-2743	7	44	,	,	PUNCT
cana-2743	7	45	𝑞	𝑞	X
cana-2743	7	46	>	>	X
cana-2743	7	47	0	0	PROPN
cana-2743	7	48	,	,	PUNCT
cana-2743	7	49	𝑛	𝑛	NOUN
cana-2743	7	50	=	=	NOUN
cana-2743	7	51	0,1,2	0,1,2	NUM
cana-2743	7	52	,	,	PUNCT
cana-2743	7	53	.	.	PUNCT
cana-2743	7	54	..	..	PUNCT
cana-2743	8	1	(	(	PUNCT
cana-2743	8	2	2	2	X
cana-2743	8	3	)	)	PUNCT
cana-2743	8	4	defines	define	VERB
cana-2743	8	5	the	the	DET
cana-2743	8	6	euler	euler	NOUN
cana-2743	8	7	mean	mean	NOUN
cana-2743	8	8	of	of	ADP
cana-2743	8	9	order	order	NOUN
cana-2743	8	10	𝑞	𝑞	X
cana-2743	8	11	>	>	X
cana-2743	8	12	0	0	NUM
cana-2743	8	13	of	of	ADP
cana-2743	8	14	{	{	PUNCT
cana-2743	8	15	𝑠𝑛	𝑠𝑛	NOUN
cana-2743	8	16	}	}	PUNCT
cana-2743	8	17	.	.	PUNCT
cana-2743	9	1	𝐶𝑛	𝐶𝑛	NOUN
cana-2743	9	2	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	9	3	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	9	4	𝑞	𝑞	X
cana-2743	9	5	=	=	SYM
cana-2743	9	6	1	1	NUM
cana-2743	9	7	𝑛+1	𝑛+1	ADP
cana-2743	9	8	∑𝑛	∑𝑛	PROPN
cana-2743	9	9	𝑘=0	𝑘=0	ADP
cana-2743	9	10	1	1	NUM
cana-2743	9	11	(	(	PUNCT
cana-2743	9	12	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	9	13	∑𝑘	∑𝑘	PROPN
cana-2743	9	14	𝑢=0	𝑢=0	PROPN
cana-2743	9	15	(	(	PUNCT
cana-2743	9	16	𝑘	𝑘	NOUN
cana-2743	9	17	𝑢	𝑢	X
cana-2743	9	18	)	)	PUNCT
cana-2743	9	19	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	9	20	(	(	PUNCT
cana-2743	9	21	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	9	22	∑𝑢	∑𝑢	NOUN
cana-2743	9	23	𝑣=0	𝑣=0	NUM
cana-2743	9	24	(	(	PUNCT
cana-2743	9	25	𝑢	𝑢	PROPN
cana-2743	9	26	𝑣	𝑣	NOUN
cana-2743	9	27	)	)	PUNCT
cana-2743	9	28	𝑞𝑢−𝑣𝑠𝑣	𝑞𝑢−𝑣𝑠𝑣	NOUN
cana-2743	9	29	(	(	PUNCT
cana-2743	9	30	3	3	X
cana-2743	9	31	)	)	PUNCT
cana-2743	9	32	the	the	DET
cana-2743	9	33	series	series	NOUN
cana-2743	9	34	∑∞	∑∞	PROPN
cana-2743	9	35	𝑛=0	𝑛=0	PROPN
cana-2743	9	36	𝑢𝑛	𝑢𝑛	NOUN
cana-2743	9	37	is	be	AUX
cana-2743	9	38	said	say	VERB
cana-2743	9	39	to	to	PART
cana-2743	9	40	be	be	AUX
cana-2743	9	41	(	(	PUNCT
cana-2743	9	42	𝐶	𝐶	PROPN
cana-2743	9	43	,	,	PUNCT
cana-2743	9	44	1)(𝐸	1)(𝐸	NUM
cana-2743	9	45	,	,	PUNCT
cana-2743	9	46	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	9	47	,	,	PUNCT
cana-2743	9	48	𝑞	𝑞	NOUN
cana-2743	9	49	)	)	PUNCT
cana-2743	9	50	summable	summable	ADJ
cana-2743	9	51	to	to	ADP
cana-2743	9	52	s	s	PRON
cana-2743	9	53	,	,	PUNCT
cana-2743	10	1	if	if	SCONJ
cana-2743	10	2	i	i	PRON
cana-2743	10	3	m	m	VERB
cana-2743	10	4	𝑛→∞𝐶𝑛	𝑛→∞𝐶𝑛	PROPN
cana-2743	10	5	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	10	6	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	10	7	𝑞	𝑞	X
cana-2743	10	8	=	=	X
cana-2743	10	9	𝑠.	𝑠.	NOUN
cana-2743	10	10	for	for	ADP
cana-2743	10	11	a	a	DET
cana-2743	10	12	2𝜋	2𝜋	NUM
cana-2743	10	13	periodic	periodic	NOUN
cana-2743	10	14	signal	signal	NOUN
cana-2743	10	15	which	which	PRON
cana-2743	10	16	is	be	AUX
cana-2743	10	17	integrable	integrable	ADJ
cana-2743	10	18	in	in	ADP
cana-2743	10	19	the	the	DET
cana-2743	10	20	sense	sense	NOUN
cana-2743	10	21	of	of	ADP
cana-2743	10	22	lebesgue	lebesgue	NOUN
cana-2743	10	23	over	over	ADP
cana-2743	10	24	(	(	PUNCT
cana-2743	10	25	−𝜋	−𝜋	ADJ
cana-2743	10	26	,	,	PUNCT
cana-2743	10	27	𝜋	𝜋	NOUN
cana-2743	10	28	)	)	PUNCT
cana-2743	10	29	.	.	PUNCT
cana-2743	11	1	the	the	DET
cana-2743	11	2	conjugate	conjugate	NOUN
cana-2743	11	3	of	of	ADP
cana-2743	11	4	fourier	fourier	ADJ
cana-2743	11	5	series	series	NOUN
cana-2743	11	6	is	be	AUX
cana-2743	11	7	defined	define	VERB
cana-2743	11	8	by	by	ADP
cana-2743	11	9	∑∞	∑∞	NOUN
cana-2743	11	10	𝑘=1	𝑘=1	X
cana-2743	11	11	(	(	PUNCT
cana-2743	11	12	𝑏𝑘𝑐𝑜𝑠𝑘𝑥	𝑏𝑘𝑐𝑜𝑠𝑘𝑥	NOUN
cana-2743	11	13	−	−	PROPN
cana-2743	11	14	𝑎𝑘𝑐𝑜𝑠𝑘𝑥	𝑎𝑘𝑐𝑜𝑠𝑘𝑥	NOUN
cana-2743	11	15	)	)	PUNCT
cana-2743	11	16	(	(	PUNCT
cana-2743	11	17	4	4	NUM
cana-2743	11	18	)	)	PUNCT
cana-2743	11	19	and	and	CCONJ
cana-2743	11	20	nth	nth	NOUN
cana-2743	11	21	partial	partial	ADJ
cana-2743	11	22	sum	sum	NOUN
cana-2743	11	23	is	be	AUX
cana-2743	11	24	defined	define	VERB
cana-2743	11	25	by	by	ADP
cana-2743	11	26	�	�	PROPN
cana-2743	11	27	̃	̃	PROPN
cana-2743	11	28	�	�	NOUN
cana-2743	11	29	𝑛(𝑓	𝑛(𝑓	ADJ
cana-2743	11	30	;	;	PUNCT
cana-2743	11	31	𝑥	𝑥	X
cana-2743	11	32	)	)	PUNCT
cana-2743	11	33	=	=	NOUN
cana-2743	11	34	∑∞	∑∞	NOUN
cana-2743	11	35	𝑘=1	𝑘=1	X
cana-2743	11	36	(	(	PUNCT
cana-2743	11	37	𝑏𝑘𝑐𝑜𝑠𝑘𝑥	𝑏𝑘𝑐𝑜𝑠𝑘𝑥	NOUN
cana-2743	11	38	−	−	PROPN
cana-2743	11	39	𝑎𝑘𝑐𝑜𝑠𝑘𝑥	𝑎𝑘𝑐𝑜𝑠𝑘𝑥	NOUN
cana-2743	11	40	)	)	PUNCT
cana-2743	11	41	(	(	PUNCT
cana-2743	11	42	5	5	X
cana-2743	11	43	)	)	PUNCT
cana-2743	11	44	communications	communication	NOUN
cana-2743	11	45	on	on	ADP
cana-2743	11	46	applied	apply	VERB
cana-2743	11	47	nonlinear	nonlinear	ADJ
cana-2743	11	48	analysis	analysis	NOUN
cana-2743	11	49	issn	issn	NOUN
cana-2743	11	50	:	:	PUNCT
cana-2743	11	51	1074	1074	NUM
cana-2743	11	52	-	-	PUNCT
cana-2743	11	53	133x	133x	NUM
cana-2743	11	54	vol	vol	NOUN
cana-2743	11	55	32	32	NUM
cana-2743	11	56	no	no	NOUN
cana-2743	11	57	.	.	PUNCT
cana-2743	12	1	4s	4s	NUM
cana-2743	12	2	(	(	PUNCT
cana-2743	12	3	2025	2025	NUM
cana-2743	12	4	)	)	PUNCT
cana-2743	12	5	119	119	NUM
cana-2743	12	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	12	7	the	the	DET
cana-2743	12	8	conjugate	conjugate	NOUN
cana-2743	12	9	of	of	ADP
cana-2743	12	10	f	f	PROPN
cana-2743	12	11	denoted	denote	VERB
cana-2743	12	12	by	by	ADP
cana-2743	12	13	f̃	f̃	PROPN
cana-2743	12	14	is	be	AUX
cana-2743	12	15	defined	define	VERB
cana-2743	12	16	by	by	ADP
cana-2743	12	17	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	12	18	)	)	PUNCT
cana-2743	13	1	=	=	SYM
cana-2743	13	2	−	−	PROPN
cana-2743	13	3	1	1	NUM
cana-2743	13	4	2𝜋	2𝜋	NOUN
cana-2743	13	5	𝑙𝑖𝑚𝜉→0	𝑙𝑖𝑚𝜉→0	PROPN
cana-2743	13	6	∫	∫	PROPN
cana-2743	14	1	𝜋	𝜋	NOUN
cana-2743	14	2	𝜉	𝜉	X
cana-2743	14	3	𝜓(𝑡)𝑐𝑜𝑠	𝜓(𝑡)𝑐𝑜𝑠	PROPN
cana-2743	14	4	(	(	PUNCT
cana-2743	14	5	𝑡	𝑡	PROPN
cana-2743	14	6	2	2	NUM
cana-2743	14	7	)	)	PUNCT
cana-2743	14	8	𝑑𝑡	𝑑𝑡	ADP
cana-2743	14	9	,	,	PUNCT
cana-2743	14	10	where	where	SCONJ
cana-2743	14	11	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2743	14	12	)	)	PUNCT
cana-2743	14	13	=	=	PUNCT
cana-2743	14	14	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	14	15	+	+	NUM
cana-2743	14	16	𝑡	𝑡	NOUN
cana-2743	14	17	)	)	PUNCT
cana-2743	14	18	−	−	PROPN
cana-2743	14	19	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	14	20	−	−	NUM
cana-2743	14	21	𝑡	𝑡	NOUN
cana-2743	14	22	)	)	PUNCT
cana-2743	14	23	a	a	DET
cana-2743	14	24	function	function	NOUN
cana-2743	14	25	𝑓	𝑓	DET
cana-2743	14	26	∈	∈	PROPN
cana-2743	14	27	lip𝛼	lip𝛼	NOUN
cana-2743	14	28	,	,	PUNCT
cana-2743	14	29	if	if	SCONJ
cana-2743	14	30	|𝑓(𝑥	|𝑓(𝑥	NOUN
cana-2743	14	31	+	+	SYM
cana-2743	14	32	𝑡	𝑡	NOUN
cana-2743	14	33	)	)	PUNCT
cana-2743	14	34	−	−	NOUN
cana-2743	14	35	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	14	36	+	+	CCONJ
cana-2743	14	37	𝑡)|	𝑡)|	NOUN
cana-2743	14	38	=	=	SYM
cana-2743	14	39	𝑂(|𝑡|𝛼	𝑂(|𝑡|𝛼	X
cana-2743	14	40	)	)	PUNCT
cana-2743	14	41	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2743	14	42	0	0	NUM
cana-2743	14	43	<	<	X
cana-2743	14	44	𝛼	𝛼	X
cana-2743	14	45	≤	≤	NUM
cana-2743	14	46	1	1	NUM
cana-2743	14	47	.	.	PUNCT
cana-2743	14	48	and	and	CCONJ
cana-2743	14	49	𝑓	𝑓	DET
cana-2743	14	50	∈	∈	PROPN
cana-2743	14	51	𝐿𝑖𝑝(𝛼	𝐿𝑖𝑝(𝛼	PROPN
cana-2743	14	52	,	,	PUNCT
cana-2743	14	53	𝑟	𝑟	NOUN
cana-2743	14	54	)	)	PUNCT
cana-2743	14	55	if	if	SCONJ
cana-2743	14	56	(	(	PUNCT
cana-2743	14	57	∫	∫	PROPN
cana-2743	14	58	2𝜋	2𝜋	PROPN
cana-2743	14	59	0	0	NUM
cana-2743	14	60	|𝑓(𝑥)|𝑟	|𝑓(𝑥)|𝑟	PROPN
cana-2743	14	61	)	)	PUNCT
cana-2743	14	62	1	1	NUM
cana-2743	14	63	𝑟	𝑟	NOUN
cana-2743	14	64	=	=	SYM
cana-2743	14	65	𝑂(𝑡𝛼	𝑂(𝑡𝛼	PROPN
cana-2743	14	66	)	)	PUNCT
cana-2743	14	67	,	,	PUNCT
cana-2743	14	68	0	0	PUNCT
cana-2743	14	69	<	<	X
cana-2743	14	70	𝛼	𝛼	X
cana-2743	14	71	≤	≤	NUM
cana-2743	14	72	1	1	NUM
cana-2743	14	73	,	,	PUNCT
cana-2743	14	74	𝑟	𝑟	PRON
cana-2743	14	75	≥	≥	NUM
cana-2743	14	76	1	1	NUM
cana-2743	14	77	.	.	PUNCT
cana-2743	15	1	𝐿𝑝norm	𝐿𝑝norm	PROPN
cana-2743	15	2	is	be	AUX
cana-2743	15	3	defined	define	VERB
cana-2743	15	4	by	by	ADP
cana-2743	15	5	𝑓𝑝	𝑓𝑝	NOUN
cana-2743	15	6	=	=	SYM
cana-2743	15	7	(	(	PUNCT
cana-2743	15	8	∫	∫	PROPN
cana-2743	15	9	2𝜋	2𝜋	PROPN
cana-2743	15	10	0	0	SYM
cana-2743	15	11	|𝑓(𝑥)|𝑝	|𝑓(𝑥)|𝑝	PROPN
cana-2743	15	12	)	)	PUNCT
cana-2743	15	13	1	1	NUM
cana-2743	15	14	𝑝	𝑝	PROPN
cana-2743	15	15	,	,	PUNCT
cana-2743	15	16	𝑝	𝑝	PROPN
cana-2743	15	17	≥	≥	NOUN
cana-2743	15	18	1	1	NUM
cana-2743	15	19	.	.	PUNCT
cana-2743	16	1	l∞-norm	l∞-norm	NOUN
cana-2743	16	2	of	of	ADP
cana-2743	16	3	a	a	DET
cana-2743	16	4	function	function	NOUN
cana-2743	16	5	𝑓	𝑓	NOUN
cana-2743	16	6	:	:	PUNCT
cana-2743	16	7	𝑅	𝑅	PROPN
cana-2743	16	8	→	→	SYM
cana-2743	16	9	𝑅	𝑅	PROPN
cana-2743	16	10	is	be	AUX
cana-2743	16	11	defined	define	VERB
cana-2743	16	12	by	by	ADP
cana-2743	16	13	𝑓∞	𝑓∞	X
cana-2743	16	14	𝑓∞	𝑓∞	X
cana-2743	16	15	=	=	SYM
cana-2743	16	16	𝑠𝑢𝑝{|𝑓(𝑥)|/𝑓	𝑠𝑢𝑝{|𝑓(𝑥)|/𝑓	PROPN
cana-2743	16	17	:	:	PUNCT
cana-2743	16	18	𝑅	𝑅	PROPN
cana-2743	16	19	→	→	SYM
cana-2743	16	20	𝑅	𝑅	PROPN
cana-2743	16	21	}	}	PUNCT
cana-2743	16	22	the	the	DET
cana-2743	16	23	degree	degree	NOUN
cana-2743	16	24	of	of	ADP
cana-2743	16	25	approximation	approximation	NOUN
cana-2743	16	26	of	of	ADP
cana-2743	16	27	function	function	NOUN
cana-2743	16	28	𝑓	𝑓	PROPN
cana-2743	16	29	:	:	PUNCT
cana-2743	16	30	𝑅	𝑅	PROPN
cana-2743	16	31	→	→	SYM
cana-2743	16	32	𝑅	𝑅	PROPN
cana-2743	16	33	by	by	ADP
cana-2743	16	34	a	a	DET
cana-2743	16	35	trigonometric	trigonometric	ADJ
cana-2743	16	36	polynomial	polynomial	NOUN
cana-2743	16	37	𝑡𝑛[1	𝑡𝑛[1	PROPN
cana-2743	16	38	]	]	PUNCT
cana-2743	16	39	is	be	AUX
cana-2743	16	40	defined	define	VERB
cana-2743	16	41	by	by	ADP
cana-2743	16	42	𝑡𝑛	𝑡𝑛	NOUN
cana-2743	16	43	−	−	PROPN
cana-2743	16	44	𝑓∞	𝑓∞	X
cana-2743	16	45	=	=	SYM
cana-2743	16	46	𝑠𝑢𝑝{|𝑡𝑛	𝑠𝑢𝑝{|𝑡𝑛	NOUN
cana-2743	16	47	−	−	PROPN
cana-2743	16	48	𝑓|	𝑓|	PROPN
cana-2743	16	49	:	:	PUNCT
cana-2743	16	50	𝑥	𝑥	PROPN
cana-2743	17	1	∈	∈	PROPN
cana-2743	17	2	𝑅}𝑜𝑟𝑡𝑛	𝑅}𝑜𝑟𝑡𝑛	NOUN
cana-2743	17	3	−	−	NOUN
cana-2743	17	4	𝑓𝑝	𝑓𝑝	NOUN
cana-2743	17	5	=	=	NOUN
cana-2743	17	6	𝑚𝑖𝑛𝑡𝑛	𝑚𝑖𝑛𝑡𝑛	PROPN
cana-2743	17	7	−	−	PROPN
cana-2743	17	8	𝑓.	𝑓.	NOUN
cana-2743	17	9	this	this	DET
cana-2743	17	10	method	method	NOUN
cana-2743	17	11	of	of	ADP
cana-2743	17	12	approximation	approximation	NOUN
cana-2743	17	13	is	be	AUX
cana-2743	17	14	called	call	VERB
cana-2743	17	15	trigonometric	trigonometric	ADJ
cana-2743	17	16	fourier	fourier	NOUN
cana-2743	17	17	approximation	approximation	NOUN
cana-2743	17	18	.	.	PUNCT
cana-2743	18	1	we	we	PRON
cana-2743	18	2	also	also	ADV
cana-2743	18	3	write	write	VERB
cana-2743	18	4	𝐶𝑛	𝐶𝑛	PROPN
cana-2743	18	5	1𝐸𝑛	1𝐸𝑛	PROPN
cana-2743	18	6	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	18	7	𝑞	𝑞	X
cana-2743	18	8	=	=	SYM
cana-2743	18	9	1	1	NUM
cana-2743	18	10	𝑛	𝑛	NOUN
cana-2743	18	11	+	+	NUM
cana-2743	18	12	1	1	NUM
cana-2743	18	13	∑	∑	ADP
cana-2743	18	14	𝑛	𝑛	PRON
cana-2743	18	15	𝑘=0	𝑘=0	PROPN
cana-2743	18	16	1	1	NUM
cana-2743	18	17	(	(	PUNCT
cana-2743	18	18	1	1	NUM
cana-2743	18	19	+	+	CCONJ
cana-2743	18	20	𝑞)𝑘	𝑞)𝑘	NOUN
cana-2743	18	21	∑	∑	PUNCT
cana-2743	18	22	𝑘	𝑘	DET
cana-2743	18	23	𝑢=0	𝑢=0	PROPN
cana-2743	18	24	(	(	PUNCT
cana-2743	18	25	𝑘	𝑘	NOUN
cana-2743	18	26	𝑢	𝑢	X
cana-2743	18	27	)	)	PUNCT
cana-2743	18	28	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	18	29	(	(	PUNCT
cana-2743	18	30	1	1	NUM
cana-2743	18	31	+	+	NUM
cana-2743	18	32	𝑞)𝑢	𝑞)𝑢	X
cana-2743	18	33	∑	∑	ADP
cana-2743	18	34	𝑢	𝑢	PROPN
cana-2743	18	35	𝑣=0	𝑣=0	NUM
cana-2743	18	36	(	(	PUNCT
cana-2743	18	37	𝑢	𝑢	PROPN
cana-2743	18	38	𝑣	𝑣	NOUN
cana-2743	18	39	)	)	PUNCT
cana-2743	18	40	𝑞𝑢−𝑣	𝑞𝑢−𝑣	NOUN
cana-2743	18	41	𝑐𝑜𝑠	𝑐𝑜𝑠	NOUN
cana-2743	18	42	(	(	PUNCT
cana-2743	18	43	𝑣	𝑣	X
cana-2743	18	44	+	+	NOUN
cana-2743	18	45	1	1	NUM
cana-2743	18	46	2	2	NUM
cana-2743	18	47	)	)	PUNCT
cana-2743	18	48	𝑡	𝑡	PROPN
cana-2743	18	49	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	18	50	(	(	PUNCT
cana-2743	18	51	𝑡	𝑡	PROPN
cana-2743	18	52	2	2	NUM
cana-2743	18	53	)	)	PUNCT
cana-2743	18	54	and	and	CCONJ
cana-2743	18	55	𝜏	𝜏	X
cana-2743	18	56	=	=	SYM
cana-2743	18	57	[	[	PUNCT
cana-2743	18	58	1	1	NUM
cana-2743	18	59	𝑡	𝑡	PROPN
cana-2743	18	60	]	]	PUNCT
cana-2743	18	61	,	,	PUNCT
cana-2743	18	62	the	the	DET
cana-2743	18	63	integral	integral	ADJ
cana-2743	18	64	part	part	NOUN
cana-2743	18	65	of	of	ADP
cana-2743	18	66	1	1	NUM
cana-2743	18	67	𝑡	𝑡	NOUN
cana-2743	18	68	.	.	PUNCT
cana-2743	19	1	2	2	X
cana-2743	19	2	.	.	X
cana-2743	19	3	known	know	VERB
cana-2743	19	4	theorem	theorem	VERB
cana-2743	19	5	various	various	ADJ
cana-2743	19	6	investigators	investigator	NOUN
cana-2743	19	7	such	such	ADJ
cana-2743	19	8	as	as	ADP
cana-2743	19	9	dhakal[2	dhakal[2	PROPN
cana-2743	19	10	]	]	PUNCT
cana-2743	19	11	,	,	PUNCT
cana-2743	19	12	lal	lal	PROPN
cana-2743	19	13	and	and	CCONJ
cana-2743	19	14	singh[8	singh[8	PROPN
cana-2743	19	15	]	]	PROPN
cana-2743	19	16	,	,	PUNCT
cana-2743	19	17	mittal	mittal	PROPN
cana-2743	19	18	et	et	PROPN
cana-2743	19	19	al	al	PROPN
cana-2743	19	20	.	.	PUNCT
cana-2743	20	1	[	[	X
cana-2743	20	2	6,7	6,7	NUM
cana-2743	20	3	]	]	PUNCT
cana-2743	20	4	,	,	PUNCT
cana-2743	20	5	qureshi[4,5	qureshi[4,5	NOUN
cana-2743	20	6	]	]	X
cana-2743	20	7	sonker	sonker	NOUN
cana-2743	20	8	and	and	CCONJ
cana-2743	20	9	singh[9	singh[9	NUM
cana-2743	20	10	]	]	PUNCT
cana-2743	20	11	have	have	AUX
cana-2743	20	12	studied	study	VERB
cana-2743	20	13	the	the	DET
cana-2743	20	14	degree	degree	NOUN
cana-2743	20	15	of	of	ADP
cana-2743	20	16	approximation	approximation	NOUN
cana-2743	20	17	in	in	ADP
cana-2743	20	18	various	various	ADJ
cana-2743	20	19	function	function	NOUN
cana-2743	20	20	spaces	space	NOUN
cana-2743	20	21	such	such	ADJ
cana-2743	20	22	as	as	ADP
cana-2743	20	23	lip	lip	NOUN
cana-2743	20	24	𝛼	𝛼	INTJ
cana-2743	20	25	,	,	PUNCT
cana-2743	20	26	lip(𝛼	lip(𝛼	PROPN
cana-2743	20	27	,	,	PUNCT
cana-2743	20	28	𝑟	𝑟	NOUN
cana-2743	20	29	)	)	PUNCT
cana-2743	20	30	,	,	PUNCT
cana-2743	20	31	lip(𝜉(𝑡	lip(𝜉(𝑡	ADV
cana-2743	20	32	)	)	PUNCT
cana-2743	20	33	,	,	PUNCT
cana-2743	20	34	𝑟	𝑟	X
cana-2743	20	35	)	)	PUNCT
cana-2743	20	36	and	and	CCONJ
cana-2743	20	37	weighted	weight	VERB
cana-2743	20	38	(	(	PUNCT
cana-2743	20	39	𝐿𝑟	𝐿𝑟	PROPN
cana-2743	20	40	,	,	PUNCT
cana-2743	20	41	𝜉(𝑡	𝜉(𝑡	PROPN
cana-2743	20	42	)	)	PUNCT
cana-2743	20	43	)	)	PUNCT
cana-2743	20	44	by	by	ADP
cana-2743	20	45	using	use	VERB
cana-2743	20	46	triangular	triangular	NOUN
cana-2743	20	47	matrix	matrix	NOUN
cana-2743	20	48	summability	summability	NOUN
cana-2743	20	49	and	and	CCONJ
cana-2743	20	50	product	product	NOUN
cana-2743	20	51	summability	summability	NOUN
cana-2743	20	52	(	(	PUNCT
cana-2743	20	53	c,1)(e,1	c,1)(e,1	NOUN
cana-2743	20	54	)	)	PUNCT
cana-2743	20	55	,	,	PUNCT
cana-2743	20	56	(	(	PUNCT
cana-2743	20	57	n,𝑝𝑛)(e,1	n,𝑝𝑛)(e,1	NOUN
cana-2743	20	58	)	)	PUNCT
cana-2743	20	59	.	.	PUNCT
cana-2743	21	1	sonker	sonker	NOUN
cana-2743	21	2	and	and	CCONJ
cana-2743	21	3	singh[9	singh[9	NUM
cana-2743	21	4	]	]	PUNCT
cana-2743	21	5	have	have	AUX
cana-2743	21	6	determined	determine	VERB
cana-2743	21	7	the	the	DET
cana-2743	21	8	degree	degree	NOUN
cana-2743	21	9	of	of	ADP
cana-2743	21	10	approximation	approximation	NOUN
cana-2743	21	11	of	of	ADP
cana-2743	21	12	the	the	DET
cana-2743	21	13	conjugate	conjugate	NOUN
cana-2743	21	14	of	of	ADP
cana-2743	21	15	signals	signal	NOUN
cana-2743	21	16	(	(	PUNCT
cana-2743	21	17	functions	function	NOUN
cana-2743	21	18	)	)	PUNCT
cana-2743	21	19	belonging	belong	VERB
cana-2743	21	20	to	to	ADP
cana-2743	21	21	lip(𝛼	lip(𝛼	PROPN
cana-2743	21	22	,	,	PUNCT
cana-2743	21	23	𝑟)-class	𝑟)-class	NOUN
cana-2743	21	24	by(𝐶	by(𝐶	PROPN
cana-2743	21	25	,	,	PUNCT
cana-2743	21	26	1)(𝐸	1)(𝐸	NUM
cana-2743	21	27	,	,	PUNCT
cana-2743	21	28	𝑞	𝑞	NOUN
cana-2743	21	29	)	)	PUNCT
cana-2743	21	30	means	mean	NOUN
cana-2743	21	31	of	of	ADP
cana-2743	21	32	conjugate	conjugate	ADJ
cana-2743	21	33	trigonometric	trigonometric	ADJ
cana-2743	21	34	fourier	fourier	NOUN
cana-2743	21	35	series	series	NOUN
cana-2743	21	36	.	.	PUNCT
cana-2743	22	1	sonker	sonker	PROPN
cana-2743	22	2	and	and	CCONJ
cana-2743	22	3	singh	singh	PROPN
cana-2743	22	4	have	have	AUX
cana-2743	22	5	proved	prove	VERB
cana-2743	22	6	the	the	DET
cana-2743	22	7	following	following	NOUN
cana-2743	22	8	:	:	PUNCT
cana-2743	22	9	theorem	theorem	VERB
cana-2743	22	10	1	1	NUM
cana-2743	23	1	[	[	X
cana-2743	23	2	9	9	NUM
cana-2743	23	3	]	]	PUNCT
cana-2743	23	4	let	let	AUX
cana-2743	23	5	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	23	6	)	)	PUNCT
cana-2743	23	7	be	be	AUX
cana-2743	23	8	a	a	DET
cana-2743	23	9	2𝜋-periodic	2𝜋-periodic	NUM
cana-2743	23	10	,	,	PUNCT
cana-2743	23	11	lebesgue	lebesgue	NOUN
cana-2743	23	12	integrable	integrable	ADJ
cana-2743	23	13	function	function	NOUN
cana-2743	23	14	and	and	CCONJ
cana-2743	23	15	belonging	belong	VERB
cana-2743	23	16	to	to	ADP
cana-2743	23	17	the	the	DET
cana-2743	23	18	lip(𝛼	lip(𝛼	PROPN
cana-2743	23	19	,	,	PUNCT
cana-2743	23	20	𝑟)class	𝑟)class	PROPN
cana-2743	23	21	with	with	ADP
cana-2743	23	22	𝑟	𝑟	PRON
cana-2743	23	23	≥	≥	NUM
cana-2743	23	24	1	1	NUM
cana-2743	23	25	and	and	CCONJ
cana-2743	23	26	𝛼𝑟	𝛼𝑟	ADV
cana-2743	23	27	≥	≥	NUM
cana-2743	23	28	1	1	NUM
cana-2743	23	29	.	.	PUNCT
cana-2743	24	1	then	then	ADV
cana-2743	24	2	the	the	DET
cana-2743	24	3	degree	degree	NOUN
cana-2743	24	4	of	of	ADP
cana-2743	24	5	approximation	approximation	NOUN
cana-2743	24	6	of	of	ADP
cana-2743	24	7	f̃(x	f̃(x	PROPN
cana-2743	24	8	)	)	PUNCT
cana-2743	24	9	,	,	PUNCT
cana-2743	24	10	the	the	DET
cana-2743	24	11	conjugate	conjugate	NOUN
cana-2743	24	12	of	of	ADP
cana-2743	24	13	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	24	14	)	)	PUNCT
cana-2743	24	15	by	by	ADP
cana-2743	24	16	(	(	PUNCT
cana-2743	24	17	𝐶	𝐶	PROPN
cana-2743	24	18	,	,	PUNCT
cana-2743	24	19	1)(𝐸	1)(𝐸	NUM
cana-2743	24	20	,	,	PUNCT
cana-2743	24	21	𝑞	𝑞	NOUN
cana-2743	24	22	)	)	PUNCT
cana-2743	24	23	means	mean	NOUN
cana-2743	24	24	of	of	ADP
cana-2743	24	25	its	its	PRON
cana-2743	24	26	conjugate	conjugate	ADJ
cana-2743	24	27	fourier	fourier	NOUN
cana-2743	24	28	series	series	NOUN
cana-2743	24	29	is	be	AUX
cana-2743	24	30	given	give	VERB
cana-2743	24	31	by	by	ADP
cana-2743	24	32	communications	communication	NOUN
cana-2743	24	33	on	on	ADP
cana-2743	24	34	applied	apply	VERB
cana-2743	24	35	nonlinear	nonlinear	ADJ
cana-2743	24	36	analysis	analysis	NOUN
cana-2743	24	37	issn	issn	NOUN
cana-2743	24	38	:	:	PUNCT
cana-2743	24	39	1074	1074	NUM
cana-2743	24	40	-	-	PUNCT
cana-2743	24	41	133x	133x	NUM
cana-2743	24	42	vol	vol	NOUN
cana-2743	24	43	32	32	NUM
cana-2743	24	44	no	no	NOUN
cana-2743	24	45	.	.	PUNCT
cana-2743	25	1	4s	4s	NUM
cana-2743	25	2	(	(	PUNCT
cana-2743	25	3	2025	2025	NUM
cana-2743	25	4	)	)	PUNCT
cana-2743	25	5	120	120	NUM
cana-2743	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	26	1	𝐶𝑛	𝐶𝑛	PROPN
cana-2743	26	2	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	26	3	𝑞	𝑞	NOUN
cana-2743	26	4	−	−	PROPN
cana-2743	26	5	𝑓𝑟	𝑓𝑟	PROPN
cana-2743	26	6	=	=	SYM
cana-2743	26	7	𝑂	𝑂	PROPN
cana-2743	26	8	(	(	PUNCT
cana-2743	26	9	𝑛	𝑛	PROPN
cana-2743	26	10	1	1	NUM
cana-2743	26	11	𝑟	𝑟	NOUN
cana-2743	26	12	−𝛼	−𝛼	ADJ
cana-2743	26	13	)	)	PUNCT
cana-2743	26	14	,	,	PUNCT
cana-2743	26	15	𝑛	𝑛	PROPN
cana-2743	26	16	=	=	PUNCT
cana-2743	26	17	0,1,2	0,1,2	NUM
cana-2743	26	18	,	,	PUNCT
cana-2743	26	19	.	.	PUNCT
cana-2743	26	20	.	.	PUNCT
cana-2743	26	21	.	.	PUNCT
cana-2743	26	22	.	.	PUNCT
cana-2743	27	1	.	.	PUNCT
cana-2743	28	1	,	,	PUNCT
cana-2743	28	2	(	(	PUNCT
cana-2743	28	3	6	6	X
cana-2743	28	4	)	)	PUNCT
cana-2743	28	5	main	main	ADJ
cana-2743	28	6	theorem	theorem	NOUN
cana-2743	28	7	the	the	DET
cana-2743	28	8	objective	objective	NOUN
cana-2743	28	9	of	of	ADP
cana-2743	28	10	this	this	DET
cana-2743	28	11	paper	paper	NOUN
cana-2743	28	12	is	be	AUX
cana-2743	28	13	to	to	PART
cana-2743	28	14	establish	establish	VERB
cana-2743	28	15	the	the	DET
cana-2743	28	16	following	follow	VERB
cana-2743	28	17	theorem	theorem	VERB
cana-2743	28	18	.	.	PUNCT
cana-2743	29	1	theorem	theorem	ADJ
cana-2743	29	2	2	2	NUM
cana-2743	29	3	let	let	VERB
cana-2743	29	4	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	29	5	)	)	PUNCT
cana-2743	29	6	be	be	AUX
cana-2743	29	7	a	a	DET
cana-2743	29	8	2𝜋-periodic	2𝜋-periodic	NUM
cana-2743	29	9	,	,	PUNCT
cana-2743	29	10	lebesgue	lebesgue	NOUN
cana-2743	29	11	integrable	integrable	ADJ
cana-2743	29	12	function	function	NOUN
cana-2743	29	13	and	and	CCONJ
cana-2743	29	14	belonging	belong	VERB
cana-2743	29	15	to	to	ADP
cana-2743	29	16	the	the	DET
cana-2743	29	17	lip(𝛼	lip(𝛼	PROPN
cana-2743	29	18	,	,	PUNCT
cana-2743	29	19	𝑟)class	𝑟)class	PROPN
cana-2743	29	20	with	with	ADP
cana-2743	29	21	𝑟	𝑟	PRON
cana-2743	29	22	≥	≥	NUM
cana-2743	29	23	1	1	NUM
cana-2743	29	24	and	and	CCONJ
cana-2743	29	25	𝛼𝑟	𝛼𝑟	ADV
cana-2743	29	26	≥	≥	NUM
cana-2743	30	1	1	1	NUM
cana-2743	30	2	.	.	PUNCT
cana-2743	31	1	then	then	ADV
cana-2743	31	2	the	the	DET
cana-2743	31	3	degree	degree	NOUN
cana-2743	31	4	of	of	ADP
cana-2743	31	5	approximation	approximation	NOUN
cana-2743	31	6	of	of	ADP
cana-2743	31	7	f̃(x	f̃(x	PROPN
cana-2743	31	8	)	)	PUNCT
cana-2743	31	9	,	,	PUNCT
cana-2743	31	10	the	the	DET
cana-2743	31	11	conjugate	conjugate	NOUN
cana-2743	31	12	of	of	ADP
cana-2743	31	13	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	31	14	)	)	PUNCT
cana-2743	31	15	by	by	ADP
cana-2743	31	16	(	(	PUNCT
cana-2743	31	17	𝐶	𝐶	PROPN
cana-2743	31	18	,	,	PUNCT
cana-2743	31	19	1)(𝐸	1)(𝐸	NUM
cana-2743	31	20	,	,	PUNCT
cana-2743	31	21	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	31	22	,	,	PUNCT
cana-2743	31	23	𝑞	𝑞	NOUN
cana-2743	31	24	)	)	PUNCT
cana-2743	31	25	means	mean	NOUN
cana-2743	31	26	of	of	ADP
cana-2743	31	27	its	its	PRON
cana-2743	31	28	conjugate	conjugate	ADJ
cana-2743	31	29	fourier	fourier	NOUN
cana-2743	31	30	series	series	NOUN
cana-2743	31	31	is	be	AUX
cana-2743	31	32	given	give	VERB
cana-2743	31	33	by	by	ADP
cana-2743	31	34	𝐶𝑛	𝐶𝑛	PROPN
cana-2743	31	35	1𝐸𝑛	1𝐸𝑛	PROPN
cana-2743	31	36	𝑞	𝑞	X
cana-2743	32	1	𝐸𝑛	𝐸𝑛	ADP
cana-2743	32	2	𝑞	𝑞	NOUN
cana-2743	32	3	−	−	PROPN
cana-2743	32	4	𝑓𝑟	𝑓𝑟	PROPN
cana-2743	32	5	=	=	SYM
cana-2743	32	6	𝑂	𝑂	PROPN
cana-2743	32	7	(	(	PUNCT
cana-2743	32	8	𝑛	𝑛	PROPN
cana-2743	32	9	1	1	NUM
cana-2743	32	10	𝑟	𝑟	NOUN
cana-2743	32	11	−𝛼	−𝛼	ADJ
cana-2743	32	12	)	)	PUNCT
cana-2743	32	13	,	,	PUNCT
cana-2743	32	14	𝑛	𝑛	PROPN
cana-2743	32	15	=	=	PUNCT
cana-2743	32	16	0,1,2	0,1,2	NUM
cana-2743	32	17	,	,	PUNCT
cana-2743	32	18	.	.	PUNCT
cana-2743	32	19	.	.	PUNCT
cana-2743	32	20	.	.	PUNCT
cana-2743	32	21	.	.	PUNCT
cana-2743	33	1	.	.	PUNCT
cana-2743	34	1	,	,	PUNCT
cana-2743	34	2	(	(	PUNCT
cana-2743	34	3	7	7	X
cana-2743	34	4	)	)	PUNCT
cana-2743	34	5	provided	provide	VERB
cana-2743	34	6	(	(	PUNCT
cana-2743	34	7	∫	∫	PROPN
cana-2743	34	8	𝜋	𝜋	PROPN
cana-2743	34	9	𝑛+1	𝑛+1	PROPN
cana-2743	34	10	0	0	NUM
cana-2743	34	11	(	(	PUNCT
cana-2743	34	12	|𝜓(𝑡)|/𝑡𝛼)𝑟𝑑𝑡	|𝜓(𝑡)|/𝑡𝛼)𝑟𝑑𝑡	NOUN
cana-2743	34	13	)	)	PUNCT
cana-2743	34	14	1	1	NUM
cana-2743	34	15	𝑟	𝑟	NOUN
cana-2743	34	16	=	=	SYM
cana-2743	34	17	𝑂	𝑂	PROPN
cana-2743	34	18	(	(	PUNCT
cana-2743	34	19	1	1	NUM
cana-2743	34	20	𝑛+1	𝑛+1	PROPN
cana-2743	34	21	)	)	PUNCT
cana-2743	34	22	,	,	PUNCT
cana-2743	34	23	(	(	PUNCT
cana-2743	34	24	8)	8)	NUM
cana-2743	34	25	(	(	PUNCT
cana-2743	34	26	∫	∫	PROPN
cana-2743	34	27	𝜋	𝜋	X
cana-2743	34	28	𝜋	𝜋	X
cana-2743	34	29	𝑛+1	𝑛+1	PROPN
cana-2743	34	30	(	(	PUNCT
cana-2743	34	31	𝑡−𝛿|𝜓(𝑡)|/𝑡𝛼	𝑡−𝛿|𝜓(𝑡)|/𝑡𝛼	NOUN
cana-2743	34	32	)	)	PUNCT
cana-2743	34	33	𝑟	𝑟	NOUN
cana-2743	34	34	𝑑𝑡	𝑑𝑡	X
cana-2743	34	35	)	)	PUNCT
cana-2743	34	36	1	1	NUM
cana-2743	34	37	𝑟	𝑟	NOUN
cana-2743	34	38	=	=	PRON
cana-2743	34	39	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	34	40	+	+	CCONJ
cana-2743	34	41	1)𝛿	1)𝛿	NUM
cana-2743	34	42	)	)	PUNCT
cana-2743	34	43	,	,	PUNCT
cana-2743	34	44	(	(	PUNCT
cana-2743	34	45	9	9	X
cana-2743	34	46	)	)	PUNCT
cana-2743	34	47	where	where	SCONJ
cana-2743	34	48	𝛿	𝛿	NOUN
cana-2743	34	49	is	be	AUX
cana-2743	34	50	an	an	DET
cana-2743	34	51	arbitrary	arbitrary	ADJ
cana-2743	34	52	number	number	NOUN
cana-2743	34	53	such	such	ADJ
cana-2743	34	54	that	that	SCONJ
cana-2743	34	55	(	(	PUNCT
cana-2743	34	56	𝛼	𝛼	NOUN
cana-2743	34	57	+	+	NUM
cana-2743	34	58	𝛿)𝑠	𝛿)𝑠	X
cana-2743	34	59	<	<	X
cana-2743	34	60	−1	−1	NOUN
cana-2743	34	61	and	and	CCONJ
cana-2743	34	62	1/𝑠	1/𝑠	NUM
cana-2743	34	63	=	=	SYM
cana-2743	34	64	1	1	NUM
cana-2743	34	65	−	−	PROPN
cana-2743	34	66	1/𝑟	1/𝑟	NUM
cana-2743	34	67	for	for	ADP
cana-2743	34	68	𝑟	𝑟	X
cana-2743	34	69	>	>	X
cana-2743	34	70	1	1	NUM
cana-2743	34	71	.	.	NOUN
cana-2743	34	72	4	4	NUM
cana-2743	34	73	.	.	PUNCT
cana-2743	35	1	lemmas	lemmas	PROPN
cana-2743	35	2	we	we	PRON
cana-2743	35	3	need	need	VERB
cana-2743	35	4	the	the	DET
cana-2743	35	5	following	follow	VERB
cana-2743	35	6	lemmas	lemma	NOUN
cana-2743	35	7	for	for	ADP
cana-2743	35	8	the	the	DET
cana-2743	35	9	proof	proof	NOUN
cana-2743	35	10	of	of	ADP
cana-2743	35	11	our	our	PRON
cana-2743	35	12	theorem	theorem	NOUN
cana-2743	35	13	.	.	PROPN
cana-2743	35	14	4.1	4.1	NUM
cana-2743	35	15	lemma	lemma	PROPN
cana-2743	35	16	|𝐾𝑛(𝑡)|	|𝐾𝑛(𝑡)|	NOUN
cana-2743	35	17	=	=	PUNCT
cana-2743	35	18	𝑂	𝑂	PROPN
cana-2743	35	19	(	(	PUNCT
cana-2743	35	20	1	1	NUM
cana-2743	35	21	𝑡	𝑡	NOUN
cana-2743	35	22	)	)	PUNCT
cana-2743	35	23	+	+	CCONJ
cana-2743	35	24	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	35	25	+	+	CCONJ
cana-2743	35	26	1)𝑡	1)𝑡	NUM
cana-2743	35	27	)	)	PUNCT
cana-2743	35	28	for	for	ADP
cana-2743	35	29	0	0	NUM
cana-2743	35	30	≤	≤	NUM
cana-2743	35	31	𝑡	𝑡	NOUN
cana-2743	35	32	≤	≤	ADJ
cana-2743	35	33	𝜋	𝜋	NOUN
cana-2743	35	34	𝑛+1	𝑛+1	ADP
cana-2743	35	35	≤	≤	NOUN
cana-2743	35	36	𝜋	𝜋	NOUN
cana-2743	35	37	𝑣+1	𝑣+1	NOUN
cana-2743	35	38	proof	proof	NOUN
cana-2743	35	39	.	.	PUNCT
cana-2743	36	1	|𝐾𝑛(𝑡)|	|𝐾𝑛(𝑡)|	NOUN
cana-2743	36	2	=	=	PUNCT
cana-2743	36	3	1	1	NUM
cana-2743	36	4	2𝜋(𝑛+1	2𝜋(𝑛+1	NUM
cana-2743	36	5	)	)	PUNCT
cana-2743	37	1	∑𝑛	∑𝑛	ADJ
cana-2743	37	2	𝑘=0	𝑘=0	ADP
cana-2743	37	3	1	1	NUM
cana-2743	37	4	(	(	PUNCT
cana-2743	37	5	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	37	6	∑𝑘	∑𝑘	PROPN
cana-2743	37	7	𝑢=0	𝑢=0	PROPN
cana-2743	37	8	(	(	PUNCT
cana-2743	37	9	𝑘	𝑘	NOUN
cana-2743	37	10	𝑢	𝑢	X
cana-2743	37	11	)	)	PUNCT
cana-2743	37	12	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	37	13	(	(	PUNCT
cana-2743	37	14	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	37	15	∑𝑢	∑𝑢	NOUN
cana-2743	37	16	𝑣=0	𝑣=0	NUM
cana-2743	37	17	(	(	PUNCT
cana-2743	37	18	𝑢	𝑢	PROPN
cana-2743	37	19	𝑣	𝑣	NOUN
cana-2743	37	20	)	)	PUNCT
cana-2743	37	21	𝑞𝑢−𝑣	𝑞𝑢−𝑣	VERB
cana-2743	37	22	𝑐𝑜𝑠(𝑣+	𝑐𝑜𝑠(𝑣+	PRON
cana-2743	37	23	1	1	NUM
cana-2743	37	24	2	2	NUM
cana-2743	37	25	)	)	PUNCT
cana-2743	37	26	𝑡	𝑡	PROPN
cana-2743	37	27	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	37	28	(	(	PUNCT
cana-2743	37	29	𝑡	𝑡	X
cana-2743	37	30	2	2	NUM
cana-2743	37	31	)	)	PUNCT
cana-2743	37	32	=	=	SYM
cana-2743	37	33	1	1	NUM
cana-2743	37	34	2𝜋(𝑛+1	2𝜋(𝑛+1	NUM
cana-2743	37	35	)	)	PUNCT
cana-2743	38	1	∑𝑛	∑𝑛	ADJ
cana-2743	38	2	𝑘=0	𝑘=0	ADP
cana-2743	38	3	1	1	NUM
cana-2743	38	4	(	(	PUNCT
cana-2743	38	5	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	38	6	∑𝑘	∑𝑘	PROPN
cana-2743	38	7	𝑢=0	𝑢=0	PROPN
cana-2743	38	8	(	(	PUNCT
cana-2743	38	9	𝑘	𝑘	NOUN
cana-2743	38	10	𝑢	𝑢	X
cana-2743	38	11	)	)	PUNCT
cana-2743	38	12	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	38	13	(	(	PUNCT
cana-2743	38	14	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	38	15	∑𝑢	∑𝑢	NOUN
cana-2743	38	16	𝑣=0	𝑣=0	NUM
cana-2743	38	17	(	(	PUNCT
cana-2743	38	18	𝑢	𝑢	PROPN
cana-2743	38	19	𝑣	𝑣	NOUN
cana-2743	38	20	)	)	PUNCT
cana-2743	38	21	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	38	22	𝑐𝑜𝑠(𝑣+1−	𝑐𝑜𝑠(𝑣+1−	PROPN
cana-2743	38	23	1	1	NUM
cana-2743	38	24	2	2	NUM
cana-2743	38	25	)	)	PUNCT
cana-2743	39	1	𝑡	𝑡	PROPN
cana-2743	39	2	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	39	3	(	(	PUNCT
cana-2743	39	4	𝑡	𝑡	PROPN
cana-2743	39	5	2	2	NUM
cana-2743	39	6	)	)	PUNCT
cana-2743	39	7	≤	≤	NOUN
cana-2743	39	8	1	1	NUM
cana-2743	39	9	(	(	PUNCT
cana-2743	39	10	𝑛+1	𝑛+1	NOUN
cana-2743	39	11	)	)	PUNCT
cana-2743	39	12	∑𝑛	∑𝑛	PROPN
cana-2743	39	13	𝑘=0	𝑘=0	ADP
cana-2743	39	14	1	1	NUM
cana-2743	39	15	(	(	PUNCT
cana-2743	39	16	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	39	17	∑𝑘	∑𝑘	PROPN
cana-2743	39	18	𝑢=0	𝑢=0	PROPN
cana-2743	39	19	(	(	PUNCT
cana-2743	39	20	𝑘	𝑘	NOUN
cana-2743	39	21	𝑢	𝑢	X
cana-2743	39	22	)	)	PUNCT
cana-2743	39	23	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	39	24	(	(	PUNCT
cana-2743	39	25	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	39	26	∑𝑢	∑𝑢	NOUN
cana-2743	39	27	𝑣=0	𝑣=0	NUM
cana-2743	39	28	(	(	PUNCT
cana-2743	39	29	𝑢	𝑢	PROPN
cana-2743	39	30	𝑣	𝑣	X
cana-2743	39	31	)	)	PUNCT
cana-2743	39	32	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	39	33	𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠	𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠	PROPN
cana-2743	39	34	(	(	PUNCT
cana-2743	39	35	𝑡	𝑡	PROPN
cana-2743	39	36	2	2	NUM
cana-2743	39	37	)	)	PUNCT
cana-2743	39	38	+	+	NOUN
cana-2743	39	39	𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛	𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛	PROPN
cana-2743	39	40	(	(	PUNCT
cana-2743	39	41	𝑡	𝑡	PROPN
cana-2743	39	42	2	2	NUM
cana-2743	39	43	)	)	PUNCT
cana-2743	39	44	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	39	45	(	(	PUNCT
cana-2743	39	46	𝑡	𝑡	PROPN
cana-2743	39	47	2	2	NUM
cana-2743	39	48	)	)	PUNCT
cana-2743	39	49	=	=	SYM
cana-2743	39	50	1	1	NUM
cana-2743	39	51	(	(	PUNCT
cana-2743	39	52	𝑛+1	𝑛+1	NOUN
cana-2743	39	53	)	)	PUNCT
cana-2743	39	54	∑𝑛	∑𝑛	PROPN
cana-2743	39	55	𝑘=0	𝑘=0	ADP
cana-2743	39	56	1	1	NUM
cana-2743	39	57	(	(	PUNCT
cana-2743	39	58	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	39	59	∑𝑘	∑𝑘	PROPN
cana-2743	39	60	𝑢=0	𝑢=0	PROPN
cana-2743	39	61	(	(	PUNCT
cana-2743	39	62	𝑘	𝑘	NOUN
cana-2743	39	63	𝑢	𝑢	X
cana-2743	39	64	)	)	PUNCT
cana-2743	39	65	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	39	66	(	(	PUNCT
cana-2743	39	67	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	39	68	∑𝑢	∑𝑢	NOUN
cana-2743	39	69	𝑣=0	𝑣=0	NUM
cana-2743	39	70	(	(	PUNCT
cana-2743	39	71	𝑢	𝑢	PROPN
cana-2743	39	72	𝑣	𝑣	NOUN
cana-2743	39	73	)	)	PUNCT
cana-2743	39	74	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	40	1	[	[	X
cana-2743	40	2	𝑂	𝑂	PROPN
cana-2743	40	3	(	(	PUNCT
cana-2743	40	4	1	1	NUM
cana-2743	40	5	𝑡	𝑡	PROPN
cana-2743	40	6	)	)	PUNCT
cana-2743	41	1	+	+	NUM
cana-2743	41	2	𝑂(𝑠𝑖𝑛(𝑣	𝑂(𝑠𝑖𝑛(𝑣	PROPN
cana-2743	41	3	+	+	CCONJ
cana-2743	41	4	1)𝑡	1)𝑡	NUM
cana-2743	41	5	)	)	PUNCT
cana-2743	41	6	]	]	PUNCT
cana-2743	42	1	=	=	PUNCT
cana-2743	42	2	[	[	PUNCT
cana-2743	42	3	1	1	NUM
cana-2743	42	4	(	(	PUNCT
cana-2743	42	5	𝑛+1)𝑡	𝑛+1)𝑡	PROPN
cana-2743	42	6	∑𝑛	∑𝑛	PROPN
cana-2743	42	7	𝑘=0	𝑘=0	ADP
cana-2743	42	8	1	1	NUM
cana-2743	42	9	(	(	PUNCT
cana-2743	42	10	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	42	11	∑𝑘	∑𝑘	PROPN
cana-2743	42	12	𝑢=0	𝑢=0	PROPN
cana-2743	42	13	(	(	PUNCT
cana-2743	42	14	𝑘	𝑘	NOUN
cana-2743	42	15	𝑢	𝑢	X
cana-2743	42	16	)	)	PUNCT
cana-2743	42	17	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	42	18	(	(	PUNCT
cana-2743	42	19	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	42	20	∑𝑢	∑𝑢	NOUN
cana-2743	42	21	𝑣=0	𝑣=0	NUM
cana-2743	42	22	(	(	PUNCT
cana-2743	42	23	𝑢	𝑢	PROPN
cana-2743	42	24	𝑣	𝑣	NOUN
cana-2743	42	25	)	)	PUNCT
cana-2743	42	26	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	42	27	]	]	X
cana-2743	43	1	+	+	CCONJ
cana-2743	43	2	[	[	PUNCT
cana-2743	43	3	1	1	NUM
cana-2743	43	4	(	(	PUNCT
cana-2743	43	5	𝑛+1	𝑛+1	NOUN
cana-2743	43	6	)	)	PUNCT
cana-2743	43	7	∑𝑛	∑𝑛	PROPN
cana-2743	43	8	𝑘=0	𝑘=0	ADP
cana-2743	43	9	1	1	NUM
cana-2743	43	10	(	(	PUNCT
cana-2743	43	11	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	43	12	∑𝑘	∑𝑘	PROPN
cana-2743	43	13	𝑢=0	𝑢=0	PROPN
cana-2743	43	14	(	(	PUNCT
cana-2743	43	15	𝑘	𝑘	NOUN
cana-2743	43	16	𝑢	𝑢	X
cana-2743	43	17	)	)	PUNCT
cana-2743	43	18	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	43	19	(	(	PUNCT
cana-2743	43	20	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	43	21	∑𝑢	∑𝑢	NOUN
cana-2743	43	22	𝑣=0	𝑣=0	NUM
cana-2743	43	23	(	(	PUNCT
cana-2743	43	24	𝑢	𝑢	NOUN
cana-2743	43	25	𝑣	𝑣	NOUN
cana-2743	43	26	)	)	PUNCT
cana-2743	43	27	𝑞𝑢−𝑣(𝑣	𝑞𝑢−𝑣(𝑣	PROPN
cana-2743	43	28	+	+	CCONJ
cana-2743	43	29	1)𝑡	1)𝑡	NUM
cana-2743	43	30	]	]	X
cana-2743	43	31	=	=	SYM
cana-2743	43	32	𝑂	𝑂	PROPN
cana-2743	43	33	[	[	PUNCT
cana-2743	43	34	1	1	NUM
cana-2743	43	35	(	(	PUNCT
cana-2743	43	36	𝑛+1)𝑡	𝑛+1)𝑡	PROPN
cana-2743	43	37	(	(	PUNCT
cana-2743	43	38	𝑛	𝑛	PROPN
cana-2743	43	39	+	+	NOUN
cana-2743	43	40	1	1	NUM
cana-2743	43	41	)	)	PUNCT
cana-2743	43	42	]	]	PUNCT
cana-2743	44	1	+	+	CCONJ
cana-2743	44	2	𝑂	𝑂	PROPN
cana-2743	44	3	[	[	PUNCT
cana-2743	44	4	1	1	NUM
cana-2743	44	5	(	(	PUNCT
cana-2743	44	6	𝑛+1	𝑛+1	NOUN
cana-2743	44	7	)	)	PUNCT
cana-2743	44	8	(	(	PUNCT
cana-2743	44	9	𝑛	𝑛	PROPN
cana-2743	44	10	+	+	NUM
cana-2743	44	11	1)(𝑛	1)(𝑛	NUM
cana-2743	44	12	+	+	SYM
cana-2743	44	13	1)𝑡	1)𝑡	NUM
cana-2743	44	14	]	]	PUNCT
cana-2743	44	15	communications	communication	NOUN
cana-2743	44	16	on	on	ADP
cana-2743	44	17	applied	apply	VERB
cana-2743	44	18	nonlinear	nonlinear	ADJ
cana-2743	44	19	analysis	analysis	NOUN
cana-2743	44	20	issn	issn	NOUN
cana-2743	44	21	:	:	PUNCT
cana-2743	44	22	1074	1074	NUM
cana-2743	44	23	-	-	PUNCT
cana-2743	44	24	133x	133x	NUM
cana-2743	44	25	vol	vol	NOUN
cana-2743	44	26	32	32	NUM
cana-2743	44	27	no	no	NOUN
cana-2743	44	28	.	.	PUNCT
cana-2743	45	1	4s	4s	NUM
cana-2743	45	2	(	(	PUNCT
cana-2743	45	3	2025	2025	NUM
cana-2743	45	4	)	)	PUNCT
cana-2743	45	5	121	121	NUM
cana-2743	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	45	7	=	=	PUNCT
cana-2743	45	8	𝑂	𝑂	PROPN
cana-2743	45	9	(	(	PUNCT
cana-2743	45	10	1	1	NUM
cana-2743	45	11	𝑡	𝑡	NOUN
cana-2743	45	12	)	)	PUNCT
cana-2743	46	1	+	+	CCONJ
cana-2743	46	2	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	46	3	+	+	CCONJ
cana-2743	46	4	1)𝑡	1)𝑡	NUM
cana-2743	46	5	)	)	PUNCT
cana-2743	46	6	,	,	PUNCT
cana-2743	46	7	in	in	ADP
cana-2743	46	8	view	view	NOUN
cana-2743	46	9	of	of	ADP
cana-2743	46	10	sin(𝑣	sin(𝑣	PROPN
cana-2743	46	11	+	+	CCONJ
cana-2743	46	12	1)𝑡	1)𝑡	NUM
cana-2743	46	13	≤	≤	NOUN
cana-2743	46	14	(	(	PUNCT
cana-2743	46	15	𝑣	𝑣	X
cana-2743	46	16	+	+	X
cana-2743	46	17	1)𝑡	1)𝑡	NUM
cana-2743	46	18	for	for	ADP
cana-2743	46	19	0	0	NUM
cana-2743	46	20	≤	≤	NUM
cana-2743	46	21	𝑡	𝑡	NOUN
cana-2743	46	22	≤	≤	NUM
cana-2743	46	23	𝜋	𝜋	NOUN
cana-2743	46	24	𝑣+1	𝑣+1	PROPN
cana-2743	46	25	and	and	CCONJ
cana-2743	46	26	(	(	PUNCT
cana-2743	46	27	𝑠𝑖𝑛	𝑠𝑖𝑛	X
cana-2743	46	28	(	(	PUNCT
cana-2743	46	29	𝑡	𝑡	PROPN
cana-2743	46	30	2	2	NUM
cana-2743	46	31	)	)	PUNCT
cana-2743	46	32	)	)	PUNCT
cana-2743	47	1	−1	−1	NOUN
cana-2743	47	2	<	<	X
cana-2743	47	3	𝜋	𝜋	X
cana-2743	47	4	𝑡	𝑡	NOUN
cana-2743	47	5	for	for	ADP
cana-2743	47	6	0	0	NUM
cana-2743	47	7	<	<	X
cana-2743	47	8	t≤	t≤	X
cana-2743	47	9	𝜋	𝜋	NOUN
cana-2743	47	10	[	[	X
cana-2743	47	11	3	3	NUM
cana-2743	47	12	,	,	PUNCT
cana-2743	47	13	p.247	p.247	NOUN
cana-2743	47	14	]	]	X
cana-2743	47	15	.	.	PUNCT
cana-2743	48	1	4.2	4.2	NUM
cana-2743	48	2	lemma	lemma	PROPN
cana-2743	48	3	|𝐾𝑛(𝑡)|	|𝐾𝑛(𝑡)|	NOUN
cana-2743	48	4	=	=	PUNCT
cana-2743	48	5	𝑂	𝑂	PROPN
cana-2743	48	6	(	(	PUNCT
cana-2743	48	7	1	1	NUM
cana-2743	48	8	𝑡	𝑡	PROPN
cana-2743	48	9	)	)	PUNCT
cana-2743	48	10	+	+	CCONJ
cana-2743	48	11	𝑂(1	𝑂(1	NOUN
cana-2743	48	12	)	)	PUNCT
cana-2743	48	13	for	for	ADP
cana-2743	48	14	𝜋	𝜋	PRON
cana-2743	48	15	𝑣+1	𝑣+1	PROPN
cana-2743	48	16	≤	≤	NUM
cana-2743	48	17	𝑡	𝑡	PROPN
cana-2743	48	18	≤	≤	ADJ
cana-2743	48	19	𝜋.	𝜋.	ADJ
cana-2743	48	20	proof	proof	NOUN
cana-2743	48	21	.	.	PUNCT
cana-2743	49	1	|𝐾𝑛(𝑡)|	|𝐾𝑛(𝑡)|	PROPN
cana-2743	49	2	≤	≤	NUM
cana-2743	49	3	1	1	NUM
cana-2743	49	4	2𝜋(𝑛+1	2𝜋(𝑛+1	NOUN
cana-2743	49	5	)	)	PUNCT
cana-2743	50	1	∑𝑛	∑𝑛	ADJ
cana-2743	50	2	𝑘=0	𝑘=0	ADP
cana-2743	50	3	1	1	NUM
cana-2743	50	4	(	(	PUNCT
cana-2743	50	5	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	50	6	∑𝑘	∑𝑘	PROPN
cana-2743	50	7	𝑢=0	𝑢=0	PROPN
cana-2743	50	8	(	(	PUNCT
cana-2743	50	9	𝑘	𝑘	NOUN
cana-2743	50	10	𝑢	𝑢	X
cana-2743	50	11	)	)	PUNCT
cana-2743	50	12	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	50	13	(	(	PUNCT
cana-2743	50	14	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	50	15	∑𝑢	∑𝑢	NOUN
cana-2743	50	16	𝑣=0	𝑣=0	NUM
cana-2743	50	17	(	(	PUNCT
cana-2743	50	18	𝑢	𝑢	PROPN
cana-2743	50	19	𝑣	𝑣	NOUN
cana-2743	50	20	)	)	PUNCT
cana-2743	50	21	𝑞𝑢−𝑣	𝑞𝑢−𝑣	VERB
cana-2743	50	22	𝑐𝑜𝑠(𝑣+	𝑐𝑜𝑠(𝑣+	PRON
cana-2743	50	23	1	1	NUM
cana-2743	50	24	2	2	NUM
cana-2743	50	25	)	)	PUNCT
cana-2743	50	26	𝑡	𝑡	PROPN
cana-2743	50	27	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	50	28	(	(	PUNCT
cana-2743	50	29	𝑡	𝑡	X
cana-2743	50	30	2	2	NUM
cana-2743	50	31	)	)	PUNCT
cana-2743	50	32	=	=	SYM
cana-2743	50	33	1	1	NUM
cana-2743	50	34	2𝜋(𝑛+1	2𝜋(𝑛+1	NUM
cana-2743	50	35	)	)	PUNCT
cana-2743	51	1	∑𝑛	∑𝑛	ADJ
cana-2743	51	2	𝑘=0	𝑘=0	ADP
cana-2743	51	3	1	1	NUM
cana-2743	51	4	(	(	PUNCT
cana-2743	51	5	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	51	6	∑𝑘	∑𝑘	PROPN
cana-2743	51	7	𝑢=0	𝑢=0	PROPN
cana-2743	51	8	(	(	PUNCT
cana-2743	51	9	𝑘	𝑘	NOUN
cana-2743	51	10	𝑢	𝑢	X
cana-2743	51	11	)	)	PUNCT
cana-2743	51	12	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	51	13	(	(	PUNCT
cana-2743	51	14	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	51	15	∑𝑢	∑𝑢	NOUN
cana-2743	51	16	𝑣=0	𝑣=0	NUM
cana-2743	51	17	(	(	PUNCT
cana-2743	51	18	𝑢	𝑢	PROPN
cana-2743	51	19	𝑣	𝑣	NOUN
cana-2743	51	20	)	)	PUNCT
cana-2743	51	21	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	51	22	𝑐𝑜𝑠(𝑣+1−	𝑐𝑜𝑠(𝑣+1−	PROPN
cana-2743	51	23	1	1	NUM
cana-2743	51	24	2	2	NUM
cana-2743	51	25	)	)	PUNCT
cana-2743	52	1	𝑡	𝑡	PROPN
cana-2743	52	2	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	52	3	(	(	PUNCT
cana-2743	52	4	𝑡	𝑡	PROPN
cana-2743	52	5	2	2	NUM
cana-2743	52	6	)	)	PUNCT
cana-2743	52	7	≤	≤	NOUN
cana-2743	52	8	1	1	NUM
cana-2743	52	9	(	(	PUNCT
cana-2743	52	10	𝑛+1	𝑛+1	NOUN
cana-2743	52	11	)	)	PUNCT
cana-2743	52	12	∑𝑛	∑𝑛	PROPN
cana-2743	52	13	𝑘=0	𝑘=0	ADP
cana-2743	52	14	1	1	NUM
cana-2743	52	15	(	(	PUNCT
cana-2743	52	16	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	52	17	∑𝑘	∑𝑘	PROPN
cana-2743	52	18	𝑢=0	𝑢=0	PROPN
cana-2743	52	19	(	(	PUNCT
cana-2743	52	20	𝑘	𝑘	NOUN
cana-2743	52	21	𝑢	𝑢	X
cana-2743	52	22	)	)	PUNCT
cana-2743	52	23	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	52	24	(	(	PUNCT
cana-2743	52	25	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	52	26	∑𝑢	∑𝑢	NOUN
cana-2743	52	27	𝑣=0	𝑣=0	NUM
cana-2743	52	28	(	(	PUNCT
cana-2743	52	29	𝑢	𝑢	PROPN
cana-2743	52	30	𝑣	𝑣	X
cana-2743	52	31	)	)	PUNCT
cana-2743	52	32	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	52	33	𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠	𝑐𝑜𝑠(𝑣+1)𝑡𝑐𝑜𝑠	PROPN
cana-2743	52	34	(	(	PUNCT
cana-2743	52	35	𝑡	𝑡	PROPN
cana-2743	52	36	2	2	NUM
cana-2743	52	37	)	)	PUNCT
cana-2743	52	38	+	+	NOUN
cana-2743	52	39	𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛	𝑠𝑖𝑛(𝑣+1)𝑡𝑠𝑖𝑛	PROPN
cana-2743	52	40	(	(	PUNCT
cana-2743	52	41	𝑡	𝑡	PROPN
cana-2743	52	42	2	2	NUM
cana-2743	52	43	)	)	PUNCT
cana-2743	52	44	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	52	45	(	(	PUNCT
cana-2743	52	46	𝑡	𝑡	PROPN
cana-2743	52	47	2	2	NUM
cana-2743	52	48	)	)	PUNCT
cana-2743	52	49	=	=	SYM
cana-2743	52	50	1	1	NUM
cana-2743	52	51	(	(	PUNCT
cana-2743	52	52	𝑛+1	𝑛+1	NOUN
cana-2743	52	53	)	)	PUNCT
cana-2743	52	54	∑𝑛	∑𝑛	PROPN
cana-2743	52	55	𝑘=0	𝑘=0	ADP
cana-2743	52	56	1	1	NUM
cana-2743	52	57	(	(	PUNCT
cana-2743	52	58	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	52	59	∑𝑘	∑𝑘	PROPN
cana-2743	52	60	𝑢=0	𝑢=0	PROPN
cana-2743	52	61	(	(	PUNCT
cana-2743	52	62	𝑘	𝑘	NOUN
cana-2743	52	63	𝑢	𝑢	X
cana-2743	52	64	)	)	PUNCT
cana-2743	52	65	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	52	66	(	(	PUNCT
cana-2743	52	67	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	52	68	∑𝑢	∑𝑢	NOUN
cana-2743	52	69	𝑣=0	𝑣=0	NUM
cana-2743	52	70	(	(	PUNCT
cana-2743	52	71	𝑢	𝑢	PROPN
cana-2743	52	72	𝑣	𝑣	NOUN
cana-2743	52	73	)	)	PUNCT
cana-2743	52	74	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	53	1	[	[	X
cana-2743	53	2	𝑂	𝑂	PROPN
cana-2743	53	3	(	(	PUNCT
cana-2743	53	4	1	1	NUM
cana-2743	53	5	𝑡	𝑡	PROPN
cana-2743	53	6	)	)	PUNCT
cana-2743	54	1	+	+	CCONJ
cana-2743	54	2	𝑂(1	𝑂(1	NOUN
cana-2743	54	3	)	)	PUNCT
cana-2743	54	4	]	]	PUNCT
cana-2743	55	1	=	=	PUNCT
cana-2743	55	2	[	[	PUNCT
cana-2743	55	3	1	1	NUM
cana-2743	55	4	(	(	PUNCT
cana-2743	55	5	𝑛+1)𝑡	𝑛+1)𝑡	PROPN
cana-2743	55	6	∑𝑛	∑𝑛	PROPN
cana-2743	55	7	𝑘=0	𝑘=0	ADP
cana-2743	55	8	1	1	NUM
cana-2743	55	9	(	(	PUNCT
cana-2743	55	10	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	55	11	∑𝑘	∑𝑘	PROPN
cana-2743	55	12	𝑢=0	𝑢=0	PROPN
cana-2743	55	13	(	(	PUNCT
cana-2743	55	14	𝑘	𝑘	NOUN
cana-2743	55	15	𝑢	𝑢	X
cana-2743	55	16	)	)	PUNCT
cana-2743	55	17	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	55	18	(	(	PUNCT
cana-2743	55	19	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	55	20	∑𝑢	∑𝑢	NOUN
cana-2743	55	21	𝑣=0	𝑣=0	NUM
cana-2743	55	22	(	(	PUNCT
cana-2743	55	23	𝑢	𝑢	PROPN
cana-2743	55	24	𝑣	𝑣	NOUN
cana-2743	55	25	)	)	PUNCT
cana-2743	55	26	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	55	27	]	]	X
cana-2743	56	1	+	+	CCONJ
cana-2743	56	2	[	[	PUNCT
cana-2743	56	3	1	1	NUM
cana-2743	56	4	(	(	PUNCT
cana-2743	56	5	𝑛+1	𝑛+1	NOUN
cana-2743	56	6	)	)	PUNCT
cana-2743	56	7	∑𝑛	∑𝑛	PROPN
cana-2743	56	8	𝑘=0	𝑘=0	ADP
cana-2743	56	9	1	1	NUM
cana-2743	56	10	(	(	PUNCT
cana-2743	56	11	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	56	12	∑𝑘	∑𝑘	PROPN
cana-2743	56	13	𝑢=0	𝑢=0	PROPN
cana-2743	56	14	(	(	PUNCT
cana-2743	56	15	𝑘	𝑘	NOUN
cana-2743	56	16	𝑢	𝑢	X
cana-2743	56	17	)	)	PUNCT
cana-2743	56	18	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	56	19	(	(	PUNCT
cana-2743	56	20	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	56	21	∑𝑢	∑𝑢	NOUN
cana-2743	56	22	𝑣=0	𝑣=0	NUM
cana-2743	56	23	(	(	PUNCT
cana-2743	56	24	𝑢	𝑢	PROPN
cana-2743	56	25	𝑣	𝑣	NOUN
cana-2743	56	26	)	)	PUNCT
cana-2743	56	27	𝑞𝑢−𝑣	𝑞𝑢−𝑣	PROPN
cana-2743	56	28	]	]	X
cana-2743	57	1	=	=	SYM
cana-2743	57	2	𝑂	𝑂	PROPN
cana-2743	57	3	[	[	PUNCT
cana-2743	57	4	1	1	NUM
cana-2743	57	5	(	(	PUNCT
cana-2743	57	6	𝑛+1)𝑡	𝑛+1)𝑡	PROPN
cana-2743	57	7	(	(	PUNCT
cana-2743	57	8	𝑛	𝑛	PROPN
cana-2743	57	9	+	+	NOUN
cana-2743	57	10	1	1	NUM
cana-2743	57	11	)	)	PUNCT
cana-2743	57	12	]	]	PUNCT
cana-2743	58	1	+	+	CCONJ
cana-2743	58	2	[	[	PUNCT
cana-2743	58	3	1	1	NUM
cana-2743	58	4	(	(	PUNCT
cana-2743	58	5	𝑛+1	𝑛+1	NOUN
cana-2743	58	6	)	)	PUNCT
cana-2743	58	7	(	(	PUNCT
cana-2743	58	8	𝑛	𝑛	PROPN
cana-2743	58	9	+	+	NOUN
cana-2743	58	10	1	1	NUM
cana-2743	58	11	)	)	PUNCT
cana-2743	58	12	]	]	PUNCT
cana-2743	59	1	=	=	PUNCT
cana-2743	59	2	𝑂	𝑂	PROPN
cana-2743	59	3	(	(	PUNCT
cana-2743	59	4	1	1	NUM
cana-2743	59	5	𝑡	𝑡	PROPN
cana-2743	59	6	)	)	PUNCT
cana-2743	59	7	+	+	CCONJ
cana-2743	59	8	𝑂(1	𝑂(1	NOUN
cana-2743	59	9	)	)	PUNCT
cana-2743	59	10	,	,	PUNCT
cana-2743	59	11	in	in	ADP
cana-2743	59	12	view	view	NOUN
cana-2743	59	13	of	of	ADP
cana-2743	59	14	|𝑠𝑖𝑛(𝑣	|𝑠𝑖𝑛(𝑣	NOUN
cana-2743	59	15	+	+	CCONJ
cana-2743	59	16	1)𝑡|	1)𝑡|	NUM
cana-2743	59	17	≤	≤	NUM
cana-2743	59	18	1	1	NUM
cana-2743	59	19	and	and	CCONJ
cana-2743	59	20	(	(	PUNCT
cana-2743	59	21	𝑠𝑖𝑛	𝑠𝑖𝑛	X
cana-2743	59	22	(	(	PUNCT
cana-2743	59	23	𝑡	𝑡	PROPN
cana-2743	59	24	2	2	NUM
cana-2743	59	25	)	)	PUNCT
cana-2743	59	26	)	)	PUNCT
cana-2743	59	27	−1	−1	NOUN
cana-2743	59	28	≤	≤	ADV
cana-2743	59	29	𝜋	𝜋	NUM
cana-2743	59	30	𝑡	𝑡	NOUN
cana-2743	59	31	for	for	ADP
cana-2743	59	32	0	0	NUM
cana-2743	59	33	<	<	X
cana-2743	59	34	𝑡	𝑡	X
cana-2743	59	35	≤	≤	NUM
cana-2743	59	36	𝜋	𝜋	NOUN
cana-2743	60	1	[	[	X
cana-2743	60	2	3	3	NUM
cana-2743	60	3	,	,	PUNCT
cana-2743	60	4	p.247	p.247	X
cana-2743	60	5	]	]	X
cana-2743	60	6	5	5	NUM
cana-2743	60	7	.	.	PUNCT
cana-2743	60	8	proof	proof	NOUN
cana-2743	60	9	of	of	ADP
cana-2743	60	10	main	main	ADJ
cana-2743	60	11	theorem	theorem	NOUN
cana-2743	60	12	the	the	DET
cana-2743	60	13	integral	integral	ADJ
cana-2743	60	14	representation	representation	NOUN
cana-2743	60	15	of	of	ADP
cana-2743	60	16	�	�	PROPN
cana-2743	60	17	̃	̃	PROPN
cana-2743	60	18	�	�	NOUN
cana-2743	60	19	𝑛(𝑓	𝑛(𝑓	ADJ
cana-2743	60	20	;	;	PUNCT
cana-2743	60	21	𝑥	𝑥	X
cana-2743	60	22	)	)	PUNCT
cana-2743	60	23	is	be	AUX
cana-2743	60	24	given	give	VERB
cana-2743	60	25	by	by	ADP
cana-2743	60	26	�	�	PROPN
cana-2743	60	27	̃	̃	PROPN
cana-2743	60	28	�	�	NOUN
cana-2743	60	29	𝑛(𝑓	𝑛(𝑓	ADJ
cana-2743	60	30	;	;	PUNCT
cana-2743	60	31	𝑥	𝑥	X
cana-2743	60	32	)	)	PUNCT
cana-2743	60	33	=	=	SYM
cana-2743	61	1	−	−	PROPN
cana-2743	61	2	1	1	NUM
cana-2743	62	1	𝜋	𝜋	NOUN
cana-2743	62	2	∫	∫	PROPN
cana-2743	62	3	𝜋	𝜋	NOUN
cana-2743	62	4	0	0	NUM
cana-2743	62	5	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2743	62	6	)	)	PUNCT
cana-2743	62	7	𝑐𝑜𝑠	𝑐𝑜𝑠	NOUN
cana-2743	62	8	(	(	PUNCT
cana-2743	62	9	𝑡	𝑡	PROPN
cana-2743	62	10	2	2	NUM
cana-2743	62	11	)	)	PUNCT
cana-2743	62	12	−𝑐𝑜𝑠(𝑛+	−𝑐𝑜𝑠(𝑛+	NOUN
cana-2743	62	13	1	1	NUM
cana-2743	62	14	2	2	NUM
cana-2743	62	15	)	)	PUNCT
cana-2743	62	16	𝑡	𝑡	PROPN
cana-2743	62	17	2𝑠𝑖𝑛	2𝑠𝑖𝑛	NUM
cana-2743	62	18	(	(	PUNCT
cana-2743	62	19	𝑡	𝑡	PROPN
cana-2743	62	20	2	2	NUM
cana-2743	62	21	)	)	PUNCT
cana-2743	62	22	𝑑𝑡.	𝑑𝑡.	NOUN
cana-2743	62	23	therefore	therefore	ADV
cana-2743	62	24	,	,	PUNCT
cana-2743	62	25	we	we	PRON
cana-2743	62	26	have	have	VERB
cana-2743	62	27	�	�	PROPN
cana-2743	62	28	̃	̃	PROPN
cana-2743	62	29	�	�	NOUN
cana-2743	62	30	𝑛(𝑓	𝑛(𝑓	ADJ
cana-2743	62	31	;	;	PUNCT
cana-2743	62	32	𝑥	𝑥	X
cana-2743	62	33	)	)	PUNCT
cana-2743	62	34	−	−	NOUN
cana-2743	62	35	𝑓(𝑥	𝑓(𝑥	NOUN
cana-2743	62	36	)	)	PUNCT
cana-2743	62	37	=	=	SYM
cana-2743	62	38	1	1	NUM
cana-2743	62	39	2𝜋	2𝜋	NUM
cana-2743	62	40	∫	∫	NOUN
cana-2743	62	41	𝜋	𝜋	X
cana-2743	62	42	0	0	NUM
cana-2743	62	43	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2743	62	44	)	)	PUNCT
cana-2743	62	45	𝑐𝑜𝑠(𝑛+	𝑐𝑜𝑠(𝑛+	NOUN
cana-2743	62	46	1	1	NUM
cana-2743	62	47	2	2	NUM
cana-2743	62	48	)	)	PUNCT
cana-2743	62	49	𝑡	𝑡	PROPN
cana-2743	62	50	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	62	51	(	(	PUNCT
cana-2743	62	52	𝑡	𝑡	PROPN
cana-2743	62	53	2	2	NUM
cana-2743	62	54	)	)	PUNCT
cana-2743	62	55	𝑑𝑡.	𝑑𝑡.	NOUN
cana-2743	62	56	now	now	ADV
cana-2743	62	57	,	,	PUNCT
cana-2743	62	58	denoting	denote	VERB
cana-2743	62	59	(	(	PUNCT
cana-2743	62	60	𝐶	𝐶	PROPN
cana-2743	62	61	,	,	PUNCT
cana-2743	62	62	1)(𝐸	1)(𝐸	NUM
cana-2743	62	63	,	,	PUNCT
cana-2743	62	64	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	62	65	,	,	PUNCT
cana-2743	62	66	𝑞	𝑞	NOUN
cana-2743	62	67	)	)	PUNCT
cana-2743	62	68	transform	transform	NOUN
cana-2743	62	69	of	of	ADP
cana-2743	62	70	�	�	PROPN
cana-2743	62	71	̃	̃	PROPN
cana-2743	62	72	�	�	NOUN
cana-2743	62	73	𝑛(𝑓	𝑛(𝑓	ADJ
cana-2743	62	74	;	;	PUNCT
cana-2743	62	75	𝑥	𝑥	X
cana-2743	62	76	)	)	PUNCT
cana-2743	62	77	by	by	ADP
cana-2743	62	78	𝐶𝑛	𝐶𝑛	PROPN
cana-2743	62	79	1𝐸𝑛	1𝐸𝑛	PROPN
cana-2743	62	80	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	62	81	𝑞	𝑞	VERB
cana-2743	62	82	,	,	PUNCT
cana-2743	62	83	we	we	PRON
cana-2743	62	84	write	write	VERB
cana-2743	62	85	communications	communication	NOUN
cana-2743	62	86	on	on	ADP
cana-2743	62	87	applied	apply	VERB
cana-2743	62	88	nonlinear	nonlinear	ADJ
cana-2743	62	89	analysis	analysis	NOUN
cana-2743	62	90	issn	issn	NOUN
cana-2743	62	91	:	:	PUNCT
cana-2743	62	92	1074	1074	NUM
cana-2743	62	93	-	-	PUNCT
cana-2743	62	94	133x	133x	NUM
cana-2743	62	95	vol	vol	NOUN
cana-2743	62	96	32	32	NUM
cana-2743	62	97	no	no	NOUN
cana-2743	62	98	.	.	PUNCT
cana-2743	63	1	4s	4s	NUM
cana-2743	63	2	(	(	PUNCT
cana-2743	63	3	2025	2025	NUM
cana-2743	63	4	)	)	PUNCT
cana-2743	63	5	122	122	NUM
cana-2743	64	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	64	2	𝐶𝑛	𝐶𝑛	NOUN
cana-2743	64	3	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	64	4	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	64	5	𝑞	𝑞	X
cana-2743	64	6	−	−	PROPN
cana-2743	64	7	𝑓	𝑓	PROPN
cana-2743	64	8	=	=	SYM
cana-2743	64	9	1	1	NUM
cana-2743	64	10	2𝜋(𝑛+1	2𝜋(𝑛+1	NOUN
cana-2743	64	11	)	)	PUNCT
cana-2743	65	1	[	[	X
cana-2743	65	2	∑𝑛	∑𝑛	ADJ
cana-2743	65	3	𝑘=0	𝑘=0	ADP
cana-2743	65	4	1	1	NUM
cana-2743	65	5	(	(	PUNCT
cana-2743	65	6	1+𝑞)𝑘	1+𝑞)𝑘	NUM
cana-2743	65	7	∫	∫	NOUN
cana-2743	65	8	𝜋	𝜋	NOUN
cana-2743	65	9	0	0	NUM
cana-2743	65	10	𝜓(𝑡	𝜓(𝑡	PROPN
cana-2743	65	11	)	)	PUNCT
cana-2743	65	12	𝑠𝑖𝑛	𝑠𝑖𝑛	NOUN
cana-2743	65	13	𝑡	𝑡	PROPN
cana-2743	65	14	2	2	NUM
cana-2743	65	15	∑𝑘	∑𝑘	PROPN
cana-2743	65	16	𝑢=0	𝑢=0	NOUN
cana-2743	65	17	(	(	PUNCT
cana-2743	65	18	𝑘	𝑘	NOUN
cana-2743	65	19	𝑢	𝑢	X
cana-2743	65	20	)	)	PUNCT
cana-2743	65	21	𝑞𝑘−𝑢	𝑞𝑘−𝑢	ADJ
cana-2743	65	22	(	(	PUNCT
cana-2743	65	23	1+𝑞)𝑢	1+𝑞)𝑢	NUM
cana-2743	65	24	∑𝑢	∑𝑢	NOUN
cana-2743	65	25	𝑣=0	𝑣=0	NUM
cana-2743	65	26	(	(	PUNCT
cana-2743	65	27	𝑢	𝑢	NOUN
cana-2743	65	28	𝑣	𝑣	NOUN
cana-2743	65	29	)	)	PUNCT
cana-2743	65	30	𝑞𝑢−𝑣𝑐𝑜𝑠	𝑞𝑢−𝑣𝑐𝑜𝑠	NOUN
cana-2743	65	31	(	(	PUNCT
cana-2743	65	32	𝑣	𝑣	X
cana-2743	65	33	+	+	NOUN
cana-2743	65	34	1	1	NUM
cana-2743	65	35	2	2	NUM
cana-2743	65	36	)	)	PUNCT
cana-2743	65	37	𝑡	𝑡	X
cana-2743	65	38	]	]	X
cana-2743	65	39	(	(	PUNCT
cana-2743	65	40	10	10	NUM
cana-2743	65	41	)	)	PUNCT
cana-2743	65	42	=	=	NOUN
cana-2743	66	1	[	[	X
cana-2743	66	2	∫	∫	X
cana-2743	66	3	𝜋	𝜋	X
cana-2743	66	4	𝑛+1	𝑛+1	PROPN
cana-2743	66	5	0	0	NUM
cana-2743	66	6	+	+	NUM
cana-2743	67	1	∫	∫	PROPN
cana-2743	67	2	𝜋	𝜋	X
cana-2743	67	3	𝜋	𝜋	X
cana-2743	67	4	𝑛+1	𝑛+1	X
cana-2743	67	5	]	]	PUNCT
cana-2743	67	6	𝜓(𝑡)𝐾𝑛(𝑡)𝑑𝑡	𝜓(𝑡)𝐾𝑛(𝑡)𝑑𝑡	PROPN
cana-2743	67	7	=	=	SYM
cana-2743	67	8	𝐼1	𝐼1	NOUN
cana-2743	67	9	+	+	CCONJ
cana-2743	67	10	𝐼2	𝐼2	NOUN
cana-2743	67	11	,	,	PUNCT
cana-2743	67	12	𝑠𝑎𝑦.	𝑠𝑎𝑦.	X
cana-2743	67	13	(	(	PUNCT
cana-2743	67	14	11	11	NUM
cana-2743	67	15	)	)	PUNCT
cana-2743	67	16	using	use	VERB
cana-2743	67	17	lemma	lemma	PROPN
cana-2743	67	18	4.1	4.1	NUM
cana-2743	67	19	,	,	PUNCT
cana-2743	67	20	hölder	hölder	PROPN
cana-2743	67	21	’s	’s	PART
cana-2743	67	22	inequality	inequality	NOUN
cana-2743	67	23	,	,	PUNCT
cana-2743	67	24	condition	condition	NOUN
cana-2743	67	25	(	(	PUNCT
cana-2743	67	26	8)	8)	NUM
cana-2743	67	27	and	and	CCONJ
cana-2743	67	28	minkwiski	minkwiski	NOUN
cana-2743	67	29	’s	’s	PART
cana-2743	67	30	inequality	inequality	NOUN
cana-2743	67	31	,	,	PUNCT
cana-2743	67	32	we	we	PRON
cana-2743	67	33	have	have	VERB
cana-2743	67	34	|𝐼1|	|𝐼1|	PROPN
cana-2743	67	35	=	=	SYM
cana-2743	67	36	∫	∫	PROPN
cana-2743	67	37	𝜋	𝜋	X
cana-2743	67	38	𝑛+1	𝑛+1	PROPN
cana-2743	67	39	0	0	NUM
cana-2743	67	40	|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡	|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡	VERB
cana-2743	67	41	≤	≤	NOUN
cana-2743	68	1	[	[	X
cana-2743	68	2	∫	∫	X
cana-2743	68	3	𝜋	𝜋	X
cana-2743	68	4	𝑛+1	𝑛+1	PROPN
cana-2743	68	5	0	0	NUM
cana-2743	68	6	(	(	PUNCT
cana-2743	68	7	|𝜓(𝑡)/𝑡𝛼)𝑟	|𝜓(𝑡)/𝑡𝛼)𝑟	PROPN
cana-2743	68	8	]	]	X
cana-2743	68	9	1	1	NUM
cana-2743	68	10	𝑟	𝑟	NOUN
cana-2743	68	11	[	[	X
cana-2743	68	12	𝑙𝑖𝑚𝜖→0	𝑙𝑖𝑚𝜖→0	PROPN
cana-2743	68	13	∫	∫	PROPN
cana-2743	69	1	𝜋	𝜋	PROPN
cana-2743	69	2	𝑛+1	𝑛+1	PROPN
cana-2743	69	3	𝜖	𝜖	X
cana-2743	69	4	(	(	PUNCT
cana-2743	69	5	𝑡𝛼|𝐾𝑛(𝑡)|)𝑠𝑑𝑡	𝑡𝛼|𝐾𝑛(𝑡)|)𝑠𝑑𝑡	PROPN
cana-2743	69	6	]	]	X
cana-2743	69	7	1	1	NUM
cana-2743	69	8	𝑠	𝑠	X
cana-2743	69	9	=	=	PRON
cana-2743	69	10	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	69	11	+	+	X
cana-2743	69	12	1)−1	1)−1	NUM
cana-2743	69	13	)	)	PUNCT
cana-2743	70	1	[	[	X
cana-2743	70	2	𝑙𝑖𝑚𝜖→0	𝑙𝑖𝑚𝜖→0	PROPN
cana-2743	70	3	∫	∫	PROPN
cana-2743	70	4	𝜋	𝜋	PROPN
cana-2743	70	5	𝑛+1	𝑛+1	PROPN
cana-2743	70	6	𝜖	𝜖	X
cana-2743	70	7	(	(	PUNCT
cana-2743	70	8	𝑡𝛼−1	𝑡𝛼−1	PROPN
cana-2743	70	9	+	+	PUNCT
cana-2743	70	10	(	(	PUNCT
cana-2743	70	11	𝑛	𝑛	PROPN
cana-2743	70	12	+	+	NOUN
cana-2743	70	13	1)𝑡𝛼+1)𝑠𝑑𝑡	1)𝑡𝛼+1)𝑠𝑑𝑡	NUM
cana-2743	70	14	]	]	SYM
cana-2743	70	15	1	1	NUM
cana-2743	70	16	𝑠	𝑠	NOUN
cana-2743	70	17	=	=	PRON
cana-2743	70	18	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	70	19	+	+	X
cana-2743	70	20	1)−1	1)−1	NUM
cana-2743	70	21	)	)	PUNCT
cana-2743	71	1	[	[	X
cana-2743	71	2	(	(	PUNCT
cana-2743	71	3	𝑙𝑖𝑚𝜖→0	𝑙𝑖𝑚𝜖→0	PROPN
cana-2743	71	4	∫	∫	PROPN
cana-2743	71	5	𝜋	𝜋	X
cana-2743	71	6	𝑛+1	𝑛+1	PROPN
cana-2743	71	7	𝜖	𝜖	X
cana-2743	71	8	𝑡(𝛼−1)𝑠𝑑𝑡	𝑡(𝛼−1)𝑠𝑑𝑡	NOUN
cana-2743	71	9	)	)	PUNCT
cana-2743	71	10	1	1	NUM
cana-2743	71	11	𝑠	𝑠	NOUN
cana-2743	71	12	+	+	CCONJ
cana-2743	71	13	(	(	PUNCT
cana-2743	71	14	𝑙𝑖𝑚𝜖→0	𝑙𝑖𝑚𝜖→0	PROPN
cana-2743	71	15	∫	∫	PROPN
cana-2743	71	16	𝜋	𝜋	PROPN
cana-2743	71	17	𝑛+1	𝑛+1	PROPN
cana-2743	71	18	𝜖	𝜖	X
cana-2743	71	19	(	(	PUNCT
cana-2743	71	20	𝑛	𝑛	PROPN
cana-2743	71	21	+	+	NOUN
cana-2743	71	22	1)𝑡(𝛼+1)𝑠𝑑𝑡	1)𝑡(𝛼+1)𝑠𝑑𝑡	NUM
cana-2743	71	23	)	)	PUNCT
cana-2743	71	24	1	1	NUM
cana-2743	71	25	𝑠	𝑠	NOUN
cana-2743	71	26	]	]	X
cana-2743	71	27	=	=	VERB
cana-2743	71	28	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	71	29	+	+	CCONJ
cana-2743	72	1	1)−1)[(𝑛	1)−1)[(𝑛	NUM
cana-2743	73	1	+	+	CCONJ
cana-2743	73	2	1)−𝛼+1−1/𝑠	1)−𝛼+1−1/𝑠	NUM
cana-2743	73	3	+	+	CCONJ
cana-2743	73	4	(	(	PUNCT
cana-2743	73	5	𝑛	𝑛	PROPN
cana-2743	73	6	+	+	NUM
cana-2743	73	7	1)(𝑛	1)(𝑛	NUM
cana-2743	74	1	+	+	SYM
cana-2743	74	2	1)−𝛼−1−1/𝑠	1)−𝛼−1−1/𝑠	NUM
cana-2743	74	3	]	]	X
cana-2743	74	4	=	=	VERB
cana-2743	74	5	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	74	6	+	+	CCONJ
cana-2743	75	1	1)−1)[(𝑛	1)−1)[(𝑛	NUM
cana-2743	75	2	+	+	NUM
cana-2743	75	3	1)−𝛼+1/𝑟	1)−𝛼+1/𝑟	NUM
cana-2743	75	4	+	+	CCONJ
cana-2743	75	5	(	(	PUNCT
cana-2743	75	6	𝑛	𝑛	PROPN
cana-2743	75	7	+	+	NUM
cana-2743	75	8	1)(𝑛	1)(𝑛	NUM
cana-2743	75	9	+	+	CCONJ
cana-2743	75	10	1)−𝛼−1−1	1)−𝛼−1−1	NUM
cana-2743	75	11	+	+	NOUN
cana-2743	75	12	1/𝑟	1/𝑟	NUM
cana-2743	75	13	]	]	X
cana-2743	75	14	=	=	SYM
cana-2743	75	15	𝑂	𝑂	PROPN
cana-2743	75	16	[	[	X
cana-2743	75	17	(	(	PUNCT
cana-2743	75	18	𝑛	𝑛	PROPN
cana-2743	75	19	+	+	NUM
cana-2743	75	20	1)−𝛼+	1)−𝛼+	NUM
cana-2743	75	21	1	1	NUM
cana-2743	75	22	𝑟	𝑟	SYM
cana-2743	75	23	−1	−1	NOUN
cana-2743	75	24	+	+	CCONJ
cana-2743	75	25	(	(	PUNCT
cana-2743	75	26	𝑛	𝑛	PROPN
cana-2743	75	27	+	+	NUM
cana-2743	75	28	1)−𝛼−2	1)−𝛼−2	NOUN
cana-2743	75	29	+	+	NOUN
cana-2743	75	30	1	1	NUM
cana-2743	75	31	𝑟	𝑟	NOUN
cana-2743	75	32	]	]	X
cana-2743	75	33	=	=	SYM
cana-2743	75	34	𝑂	𝑂	PROPN
cana-2743	75	35	(	(	PUNCT
cana-2743	75	36	(	(	PUNCT
cana-2743	75	37	𝑛	𝑛	PRON
cana-2743	75	38	+	+	NOUN
cana-2743	75	39	1)−𝛼−1	1)−𝛼−1	NOUN
cana-2743	75	40	+	+	SYM
cana-2743	75	41	1	1	NUM
cana-2743	75	42	𝑟	𝑟	NOUN
cana-2743	75	43	)	)	PUNCT
cana-2743	75	44	(	(	PUNCT
cana-2743	75	45	12	12	NUM
cana-2743	75	46	)	)	PUNCT
cana-2743	75	47	now	now	ADV
cana-2743	75	48	,	,	PUNCT
cana-2743	75	49	we	we	PRON
cana-2743	75	50	consider	consider	VERB
cana-2743	75	51	|𝐼2|	|𝐼2|	ADV
cana-2743	75	52	=	=	SYM
cana-2743	75	53	∫	∫	PROPN
cana-2743	75	54	𝜋	𝜋	X
cana-2743	75	55	𝑛+1	𝑛+1	PROPN
cana-2743	75	56	𝜋	𝜋	PRON
cana-2743	75	57	𝑛+1	𝑛+1	PROPN
cana-2743	75	58	|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡.	|𝜓(𝑡)||𝐾𝑛(𝑡)|𝑑𝑡.	PROPN
cana-2743	75	59	using	use	VERB
cana-2743	75	60	lemma	lemma	PROPN
cana-2743	75	61	4.2	4.2	NUM
cana-2743	75	62	,	,	PUNCT
cana-2743	75	63	condition	condition	NOUN
cana-2743	75	64	(	(	PUNCT
cana-2743	75	65	9	9	NUM
cana-2743	75	66	)	)	PUNCT
cana-2743	75	67	and	and	CCONJ
cana-2743	75	68	minkowiski	minkowiski	PROPN
cana-2743	75	69	’s	’s	PART
cana-2743	75	70	inequality	inequality	NOUN
cana-2743	75	71	,	,	PUNCT
cana-2743	75	72	we	we	PRON
cana-2743	75	73	have	have	VERB
cana-2743	75	74	|𝐼2|	|𝐼2|	ADV
cana-2743	75	75	≤	≤	NOUN
cana-2743	76	1	[	[	X
cana-2743	76	2	∫	∫	X
cana-2743	76	3	𝜋	𝜋	X
cana-2743	76	4	𝜋	𝜋	X
cana-2743	76	5	𝑛+1	𝑛+1	PROPN
cana-2743	76	6	(	(	PUNCT
cana-2743	76	7	𝑡−𝛿|𝜓(𝑡)|	𝑡−𝛿|𝜓(𝑡)|	X
cana-2743	76	8	𝑡𝛼	𝑡𝛼	X
cana-2743	76	9	)	)	PUNCT
cana-2743	76	10	𝑟	𝑟	X
cana-2743	76	11	𝑑𝑡	𝑑𝑡	ADP
cana-2743	76	12	]	]	PUNCT
cana-2743	76	13	1	1	NUM
cana-2743	76	14	𝑟	𝑟	NOUN
cana-2743	77	1	[	[	X
cana-2743	77	2	∫	∫	X
cana-2743	77	3	𝜋	𝜋	X
cana-2743	77	4	𝜋	𝜋	X
cana-2743	77	5	𝑛+1	𝑛+1	PROPN
cana-2743	77	6	(	(	PUNCT
cana-2743	77	7	𝑡𝛼|𝐾𝑛(𝑡)|	𝑡𝛼|𝐾𝑛(𝑡)|	NUM
cana-2743	77	8	𝑡−𝛿	𝑡−𝛿	NUM
cana-2743	77	9	)	)	PUNCT
cana-2743	77	10	𝑠	𝑠	X
cana-2743	77	11	]	]	PUNCT
cana-2743	77	12	1	1	NUM
cana-2743	77	13	𝑠	𝑠	X
cana-2743	77	14	=	=	PRON
cana-2743	77	15	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	77	16	+	+	X
cana-2743	77	17	1)𝛿	1)𝛿	NUM
cana-2743	77	18	)	)	PUNCT
cana-2743	78	1	[	[	X
cana-2743	78	2	∫	∫	X
cana-2743	78	3	𝜋	𝜋	X
cana-2743	78	4	𝜋	𝜋	X
cana-2743	78	5	𝑛+1	𝑛+1	PROPN
cana-2743	78	6	(	(	PUNCT
cana-2743	78	7	𝑡𝛼	𝑡𝛼	PROPN
cana-2743	78	8	𝑡−𝛿	𝑡−𝛿	PROPN
cana-2743	78	9	(	(	PUNCT
cana-2743	78	10	𝑂	𝑂	PROPN
cana-2743	78	11	(	(	PUNCT
cana-2743	78	12	1	1	NUM
cana-2743	78	13	𝑡	𝑡	PROPN
cana-2743	78	14	)	)	PUNCT
cana-2743	78	15	+	+	CCONJ
cana-2743	78	16	𝑂(1	𝑂(1	NOUN
cana-2743	78	17	)	)	PUNCT
cana-2743	78	18	)	)	PUNCT
cana-2743	78	19	)	)	PUNCT
cana-2743	79	1	𝑠	𝑠	X
cana-2743	79	2	𝑑𝑡	𝑑𝑡	ADP
cana-2743	79	3	]	]	X
cana-2743	79	4	1	1	NUM
cana-2743	79	5	𝑠	𝑠	NOUN
cana-2743	79	6	=	=	NOUN
cana-2743	79	7	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	79	8	+	+	X
cana-2743	79	9	1)𝛿	1)𝛿	NUM
cana-2743	79	10	)	)	PUNCT
cana-2743	80	1	[	[	X
cana-2743	80	2	∫	∫	X
cana-2743	80	3	𝜋	𝜋	X
cana-2743	80	4	𝜋	𝜋	X
cana-2743	80	5	𝑛+1	𝑛+1	PROPN
cana-2743	80	6	(	(	PUNCT
cana-2743	80	7	𝑡𝛼+𝛿−1	𝑡𝛼+𝛿−1	X
cana-2743	80	8	+	+	CCONJ
cana-2743	80	9	𝑡𝛼+𝛿	𝑡𝛼+𝛿	PROPN
cana-2743	80	10	)	)	PUNCT
cana-2743	80	11	𝑠	𝑠	PRON
cana-2743	80	12	𝑑𝑡	𝑑𝑡	ADP
cana-2743	80	13	]	]	X
cana-2743	80	14	1	1	NUM
cana-2743	80	15	𝑠	𝑠	NOUN
cana-2743	80	16	=	=	NOUN
cana-2743	80	17	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	80	18	+	+	X
cana-2743	80	19	1)𝛿	1)𝛿	NUM
cana-2743	80	20	)	)	PUNCT
cana-2743	81	1	[	[	X
cana-2743	81	2	(	(	PUNCT
cana-2743	81	3	∫	∫	PROPN
cana-2743	81	4	𝜋	𝜋	X
cana-2743	81	5	𝜋	𝜋	X
cana-2743	81	6	𝑛+1	𝑛+1	PROPN
cana-2743	81	7	𝑡(𝛼+𝛿−1)𝑠𝑑𝑡	𝑡(𝛼+𝛿−1)𝑠𝑑𝑡	NOUN
cana-2743	81	8	)	)	PUNCT
cana-2743	81	9	1	1	NUM
cana-2743	81	10	𝑠	𝑠	INTJ
cana-2743	81	11	+	+	CCONJ
cana-2743	81	12	(	(	PUNCT
cana-2743	81	13	∫	∫	PROPN
cana-2743	81	14	𝜋	𝜋	X
cana-2743	81	15	𝜋	𝜋	X
cana-2743	81	16	𝑛+1	𝑛+1	X
cana-2743	81	17	𝑡(𝛼+𝛿)𝑠𝑑𝑡	𝑡(𝛼+𝛿)𝑠𝑑𝑡	NOUN
cana-2743	81	18	)	)	PUNCT
cana-2743	81	19	1	1	NUM
cana-2743	81	20	𝑠	𝑠	NOUN
cana-2743	81	21	]	]	PUNCT
cana-2743	81	22	communications	communication	NOUN
cana-2743	81	23	on	on	ADP
cana-2743	81	24	applied	apply	VERB
cana-2743	81	25	nonlinear	nonlinear	ADJ
cana-2743	81	26	analysis	analysis	NOUN
cana-2743	81	27	issn	issn	NOUN
cana-2743	81	28	:	:	PUNCT
cana-2743	81	29	1074	1074	NUM
cana-2743	81	30	-	-	PUNCT
cana-2743	81	31	133x	133x	NUM
cana-2743	81	32	vol	vol	NOUN
cana-2743	81	33	32	32	NUM
cana-2743	81	34	no	no	NOUN
cana-2743	81	35	.	.	PUNCT
cana-2743	82	1	4s	4s	NUM
cana-2743	82	2	(	(	PUNCT
cana-2743	82	3	2025	2025	NUM
cana-2743	82	4	)	)	PUNCT
cana-2743	82	5	123	123	NUM
cana-2743	82	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	82	7	=	=	SYM
cana-2743	82	8	𝑂((𝑛	𝑂((𝑛	NOUN
cana-2743	82	9	+	+	X
cana-2743	82	10	1)𝛿	1)𝛿	NUM
cana-2743	82	11	)	)	PUNCT
cana-2743	83	1	[	[	X
cana-2743	83	2	(	(	PUNCT
cana-2743	83	3	𝑛	𝑛	PROPN
cana-2743	83	4	+	+	NUM
cana-2743	83	5	1)(−𝛼−𝛿+1)−	1)(−𝛼−𝛿+1)−	NUM
cana-2743	83	6	1	1	NUM
cana-2743	83	7	𝑠	𝑠	INTJ
cana-2743	83	8	+	+	CCONJ
cana-2743	83	9	(	(	PUNCT
cana-2743	83	10	𝑛	𝑛	ADP
cana-2743	83	11	+	+	NUM
cana-2743	83	12	1)(−𝛼−𝛿)−	1)(−𝛼−𝛿)−	NUM
cana-2743	83	13	1	1	NUM
cana-2743	83	14	𝑠	𝑠	NOUN
cana-2743	83	15	]	]	X
cana-2743	83	16	(	(	PUNCT
cana-2743	83	17	1	1	NUM
cana-2743	83	18	+	+	CCONJ
cana-2743	83	19	(	(	PUNCT
cana-2743	83	20	𝛼	𝛼	X
cana-2743	83	21	+	+	NUM
cana-2743	83	22	𝛿)𝑠	𝛿)𝑠	NOUN
cana-2743	83	23	≤	≤	NUM
cana-2743	83	24	0	0	NUM
cana-2743	83	25	)	)	PUNCT
cana-2743	84	1	=	=	SYM
cana-2743	84	2	𝑂	𝑂	PROPN
cana-2743	84	3	[	[	X
cana-2743	84	4	(	(	PUNCT
cana-2743	84	5	𝑛	𝑛	PROPN
cana-2743	84	6	+	+	SYM
cana-2743	84	7	1)−𝛼+1−	1)−𝛼+1−	NUM
cana-2743	84	8	1	1	NUM
cana-2743	84	9	𝑠	𝑠	PROPN
cana-2743	84	10	+	+	CCONJ
cana-2743	84	11	(	(	PUNCT
cana-2743	84	12	𝑛	𝑛	PROPN
cana-2743	84	13	+	+	NOUN
cana-2743	84	14	1)−𝛼−	1)−𝛼−	NUM
cana-2743	84	15	1	1	NUM
cana-2743	84	16	𝑠	𝑠	NOUN
cana-2743	84	17	]	]	X
cana-2743	84	18	=	=	SYM
cana-2743	84	19	𝑂	𝑂	PROPN
cana-2743	84	20	[	[	X
cana-2743	84	21	(	(	PUNCT
cana-2743	84	22	𝑛	𝑛	PROPN
cana-2743	84	23	+	+	NUM
cana-2743	84	24	1)−𝛼+	1)−𝛼+	NUM
cana-2743	84	25	1	1	NUM
cana-2743	84	26	𝑟	𝑟	NOUN
cana-2743	84	27	+	+	CCONJ
cana-2743	84	28	(	(	PUNCT
cana-2743	84	29	𝑛	𝑛	PRON
cana-2743	84	30	+	+	NUM
cana-2743	84	31	1)−𝛼−1	1)−𝛼−1	NOUN
cana-2743	84	32	+	+	NOUN
cana-2743	84	33	1	1	NUM
cana-2743	84	34	𝑟	𝑟	NOUN
cana-2743	84	35	]	]	X
cana-2743	84	36	=	=	SYM
cana-2743	84	37	𝑂	𝑂	PROPN
cana-2743	84	38	[	[	X
cana-2743	84	39	(	(	PUNCT
cana-2743	84	40	𝑛	𝑛	PROPN
cana-2743	84	41	+	+	CCONJ
cana-2743	84	42	1)−𝛼+	1)−𝛼+	NUM
cana-2743	84	43	1	1	NUM
cana-2743	84	44	𝑟(1	𝑟(1	PROPN
cana-2743	84	45	+	+	CCONJ
cana-2743	84	46	(	(	PUNCT
cana-2743	84	47	𝑛	𝑛	PROPN
cana-2743	84	48	+	+	NUM
cana-2743	84	49	1)−1	1)−1	NUM
cana-2743	84	50	)	)	PUNCT
cana-2743	84	51	]	]	PUNCT
cana-2743	85	1	=	=	PUNCT
cana-2743	85	2	𝑂	𝑂	PROPN
cana-2743	85	3	(	(	PUNCT
cana-2743	85	4	(	(	PUNCT
cana-2743	85	5	𝑛	𝑛	PROPN
cana-2743	85	6	+	+	NUM
cana-2743	85	7	1)−𝛼+	1)−𝛼+	NUM
cana-2743	85	8	1	1	NUM
cana-2743	85	9	𝑟	𝑟	NOUN
cana-2743	85	10	)	)	PUNCT
cana-2743	85	11	(	(	PUNCT
cana-2743	85	12	13	13	X
cana-2743	85	13	)	)	PUNCT
cana-2743	85	14	combining	combine	VERB
cana-2743	85	15	(	(	PUNCT
cana-2743	85	16	12	12	NUM
cana-2743	85	17	)	)	PUNCT
cana-2743	85	18	and	and	CCONJ
cana-2743	85	19	(	(	PUNCT
cana-2743	85	20	13	13	NUM
cana-2743	85	21	)	)	PUNCT
cana-2743	85	22	,	,	PUNCT
cana-2743	85	23	we	we	PRON
cana-2743	85	24	have	have	VERB
cana-2743	85	25	|𝐶𝑛	|𝐶𝑛	ADJ
cana-2743	85	26	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	85	27	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	85	28	𝑞	𝑞	NOUN
cana-2743	85	29	−	−	X
cana-2743	85	30	𝑓|	𝑓|	PROPN
cana-2743	85	31	=	=	SYM
cana-2743	85	32	𝑂	𝑂	PROPN
cana-2743	85	33	(	(	PUNCT
cana-2743	85	34	(	(	PUNCT
cana-2743	85	35	𝑛	𝑛	PROPN
cana-2743	85	36	+	+	NUM
cana-2743	85	37	1)−𝛼+	1)−𝛼+	NUM
cana-2743	85	38	1	1	NUM
cana-2743	85	39	𝑟	𝑟	NOUN
cana-2743	85	40	)	)	PUNCT
cana-2743	85	41	.	.	PUNCT
cana-2743	86	1	hence	hence	ADV
cana-2743	86	2	,	,	PUNCT
cana-2743	86	3	𝐶𝑛	𝐶𝑛	PROPN
cana-2743	86	4	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	86	5	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	86	6	𝑞	𝑞	X
cana-2743	86	7	−	−	PROPN
cana-2743	86	8	𝑓𝑟	𝑓𝑟	PROPN
cana-2743	86	9	=	=	PUNCT
cana-2743	86	10	(	(	PUNCT
cana-2743	86	11	∫	∫	PROPN
cana-2743	86	12	2𝜋	2𝜋	NOUN
cana-2743	86	13	0	0	PUNCT
cana-2743	87	1	|𝐶𝑛	|𝐶𝑛	ADJ
cana-2743	87	2	1𝐸𝑛	1𝐸𝑛	NUM
cana-2743	87	3	𝑞𝐸𝑛	𝑞𝐸𝑛	NOUN
cana-2743	87	4	𝑞	𝑞	NOUN
cana-2743	87	5	−	−	NOUN
cana-2743	87	6	𝑓(𝑥)|𝑟𝑑𝑥	𝑓(𝑥)|𝑟𝑑𝑥	NOUN
cana-2743	87	7	)	)	PUNCT
cana-2743	87	8	1	1	NUM
cana-2743	87	9	𝑟	𝑟	NOUN
cana-2743	87	10	=	=	SYM
cana-2743	87	11	𝑂	𝑂	PROPN
cana-2743	87	12	(	(	PUNCT
cana-2743	87	13	𝑛−𝛼+	𝑛−𝛼+	PROPN
cana-2743	87	14	1	1	NUM
cana-2743	87	15	𝑟	𝑟	NOUN
cana-2743	87	16	)	)	PUNCT
cana-2743	87	17	.	.	PUNCT
cana-2743	88	1	this	this	PRON
cana-2743	88	2	completes	complete	VERB
cana-2743	88	3	the	the	DET
cana-2743	88	4	proof	proof	NOUN
cana-2743	88	5	of	of	ADP
cana-2743	88	6	theorem	theorem	ADJ
cana-2743	88	7	2	2	NUM
cana-2743	88	8	.	.	NOUN
cana-2743	88	9	6	6	NUM
cana-2743	88	10	corollaries	corollary	NOUN
cana-2743	88	11	6.1	6.1	NUM
cana-2743	88	12	corollary	corollary	NOUN
cana-2743	88	13	if	if	SCONJ
cana-2743	88	14	one	one	NUM
cana-2743	88	15	(	(	PUNCT
cana-2743	88	16	𝐸	𝐸	PROPN
cana-2743	88	17	,	,	PUNCT
cana-2743	88	18	𝑞	𝑞	PROPN
cana-2743	88	19	)	)	PUNCT
cana-2743	88	20	=	=	SYM
cana-2743	88	21	1	1	NUM
cana-2743	88	22	,	,	PUNCT
cana-2743	88	23	then	then	ADV
cana-2743	88	24	(	(	PUNCT
cana-2743	88	25	𝐶	𝐶	PROPN
cana-2743	88	26	,	,	PUNCT
cana-2743	88	27	1)(𝐸	1)(𝐸	NUM
cana-2743	88	28	,	,	PUNCT
cana-2743	88	29	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	88	30	,	,	PUNCT
cana-2743	88	31	𝑞	𝑞	NOUN
cana-2743	88	32	)	)	PUNCT
cana-2743	88	33	means	mean	VERB
cana-2743	88	34	reduces	reduce	VERB
cana-2743	88	35	to	to	ADP
cana-2743	88	36	(	(	PUNCT
cana-2743	88	37	𝐶	𝐶	PROPN
cana-2743	88	38	,	,	PUNCT
cana-2743	88	39	1)(𝐸	1)(𝐸	PROPN
cana-2743	88	40	,	,	PUNCT
cana-2743	88	41	𝑞	𝑞	NOUN
cana-2743	88	42	)	)	PUNCT
cana-2743	88	43	means	mean	NOUN
cana-2743	88	44	.	.	PUNCT
cana-2743	89	1	hence	hence	ADV
cana-2743	89	2	,	,	PUNCT
cana-2743	89	3	theorem	theorem	ADJ
cana-2743	89	4	2	2	NUM
cana-2743	89	5	reduces	reduce	VERB
cana-2743	89	6	to	to	PART
cana-2743	89	7	theorem	theorem	VERB
cana-2743	89	8	1	1	NUM
cana-2743	89	9	.	.	NOUN
cana-2743	89	10	6.2	6.2	NUM
cana-2743	89	11	corollary	corollary	NOUN
cana-2743	89	12	when	when	SCONJ
cana-2743	89	13	𝑞	𝑞	X
cana-2743	89	14	=	=	NOUN
cana-2743	89	15	1	1	NUM
cana-2743	89	16	then	then	ADV
cana-2743	89	17	(	(	PUNCT
cana-2743	89	18	𝐶	𝐶	PROPN
cana-2743	89	19	,	,	PUNCT
cana-2743	89	20	1)(𝐸	1)(𝐸	NUM
cana-2743	89	21	,	,	PUNCT
cana-2743	89	22	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	89	23	,	,	PUNCT
cana-2743	89	24	𝑞	𝑞	NOUN
cana-2743	89	25	)	)	PUNCT
cana-2743	89	26	means	mean	VERB
cana-2743	89	27	reduces	reduce	VERB
cana-2743	89	28	to	to	ADP
cana-2743	89	29	(	(	PUNCT
cana-2743	89	30	𝐶	𝐶	PROPN
cana-2743	89	31	,	,	PUNCT
cana-2743	89	32	1)(𝐸	1)(𝐸	NUM
cana-2743	89	33	,	,	PUNCT
cana-2743	89	34	1)(𝐸	1)(𝐸	NUM
cana-2743	89	35	,	,	PUNCT
cana-2743	89	36	1	1	NUM
cana-2743	89	37	)	)	PUNCT
cana-2743	89	38	means	mean	NOUN
cana-2743	89	39	.	.	PUNCT
cana-2743	90	1	6.3	6.3	NUM
cana-2743	90	2	corollary	corollary	NOUN
cana-2743	90	3	if	if	SCONJ
cana-2743	90	4	(	(	PUNCT
cana-2743	90	5	𝐶	𝐶	PROPN
cana-2743	90	6	,	,	PUNCT
cana-2743	90	7	1	1	NUM
cana-2743	90	8	)	)	PUNCT
cana-2743	90	9	=	=	SYM
cana-2743	90	10	1	1	NUM
cana-2743	90	11	,	,	PUNCT
cana-2743	90	12	then	then	ADV
cana-2743	90	13	(	(	PUNCT
cana-2743	90	14	𝐶	𝐶	PROPN
cana-2743	90	15	,	,	PUNCT
cana-2743	90	16	1)(𝐸	1)(𝐸	NUM
cana-2743	90	17	,	,	PUNCT
cana-2743	90	18	𝑞)(𝐸	𝑞)(𝐸	PROPN
cana-2743	90	19	,	,	PUNCT
cana-2743	90	20	𝑞	𝑞	NOUN
cana-2743	90	21	)	)	PUNCT
cana-2743	90	22	means	mean	VERB
cana-2743	90	23	reduces	reduce	VERB
cana-2743	90	24	to	to	ADP
cana-2743	90	25	(	(	PUNCT
cana-2743	90	26	𝐸	𝐸	PROPN
cana-2743	90	27	,	,	PUNCT
cana-2743	90	28	𝑞)(𝐸	𝑞)(𝐸	ADJ
cana-2743	90	29	,	,	PUNCT
cana-2743	90	30	𝑞	𝑞	NOUN
cana-2743	90	31	)	)	PUNCT
cana-2743	90	32	means	mean	NOUN
cana-2743	90	33	.	.	PUNCT
cana-2743	91	1	7	7	X
cana-2743	91	2	.	.	X
cana-2743	91	3	conclusion	conclusion	NOUN
cana-2743	91	4	the	the	DET
cana-2743	91	5	result	result	NOUN
cana-2743	91	6	established	establish	VERB
cana-2743	91	7	here	here	ADV
cana-2743	91	8	is	be	AUX
cana-2743	91	9	more	more	ADV
cana-2743	91	10	general	general	ADJ
cana-2743	91	11	form	form	NOUN
cana-2743	91	12	than	than	ADP
cana-2743	91	13	some	some	DET
cana-2743	91	14	earlier	early	ADV
cana-2743	91	15	existing	exist	VERB
cana-2743	91	16	results	result	NOUN
cana-2743	91	17	in	in	ADP
cana-2743	91	18	the	the	DET
cana-2743	91	19	sense	sense	NOUN
cana-2743	91	20	that	that	SCONJ
cana-2743	91	21	,	,	PUNCT
cana-2743	91	22	one	one	NUM
cana-2743	91	23	(	(	PUNCT
cana-2743	91	24	𝐸	𝐸	PROPN
cana-2743	91	25	,	,	PUNCT
cana-2743	91	26	𝑞	𝑞	PROPN
cana-2743	91	27	)	)	PUNCT
cana-2743	91	28	=	=	SYM
cana-2743	91	29	1	1	NUM
cana-2743	91	30	our	our	PRON
cana-2743	91	31	proposed	propose	VERB
cana-2743	91	32	mean	mean	NOUN
cana-2743	91	33	reduces	reduce	VERB
cana-2743	91	34	to	to	ADP
cana-2743	91	35	(	(	PUNCT
cana-2743	91	36	𝐶	𝐶	PROPN
cana-2743	91	37	,	,	PUNCT
cana-2743	91	38	1)(𝐸	1)(𝐸	PROPN
cana-2743	91	39	,	,	PUNCT
cana-2743	91	40	𝑞	𝑞	NOUN
cana-2743	91	41	)	)	PUNCT
cana-2743	91	42	mean	mean	NOUN
cana-2743	91	43	.	.	PUNCT
cana-2743	92	1	8	8	X
cana-2743	92	2	.	.	PUNCT
cana-2743	92	3	acknowledgments	acknowledgment	NOUN
cana-2743	92	4	the	the	DET
cana-2743	92	5	first	first	ADJ
cana-2743	92	6	author	author	NOUN
cana-2743	92	7	expresses	express	VERB
cana-2743	92	8	his	his	PRON
cana-2743	92	9	gratitude	gratitude	NOUN
cana-2743	92	10	towards	towards	ADP
cana-2743	92	11	his	his	PRON
cana-2743	92	12	parents	parent	NOUN
cana-2743	92	13	for	for	ADP
cana-2743	92	14	their	their	PRON
cana-2743	92	15	blessings	blessing	NOUN
cana-2743	92	16	.	.	PUNCT
cana-2743	93	1	the	the	DET
cana-2743	93	2	second	second	ADJ
cana-2743	93	3	author	author	NOUN
cana-2743	93	4	expresses	express	VERB
cana-2743	93	5	his	his	PRON
cana-2743	93	6	gratitude	gratitude	NOUN
cana-2743	93	7	towards	towards	ADP
cana-2743	93	8	his	his	PRON
cana-2743	93	9	parents	parent	NOUN
cana-2743	93	10	for	for	ADP
cana-2743	93	11	their	their	PRON
cana-2743	93	12	blessing	blessing	NOUN
cana-2743	93	13	.	.	PUNCT
cana-2743	94	1	both	both	DET
cana-2743	94	2	authors	author	NOUN
cana-2743	94	3	are	be	AUX
cana-2743	94	4	also	also	ADV
cana-2743	94	5	grateful	grateful	ADJ
cana-2743	94	6	to	to	ADP
cana-2743	94	7	the	the	DET
cana-2743	94	8	honorable	honorable	ADJ
cana-2743	94	9	vice	vice	NOUN
cana-2743	94	10	-	-	NOUN
cana-2743	94	11	chancellor	chancellor	ADJ
cana-2743	94	12	,	,	PUNCT
cana-2743	94	13	patliputra	patliputra	PROPN
cana-2743	94	14	university	university	PROPN
cana-2743	94	15	,	,	PUNCT
cana-2743	94	16	patna	patna	PROPN
cana-2743	94	17	,	,	PUNCT
cana-2743	94	18	bihar	bihar	PROPN
cana-2743	94	19	,	,	PUNCT
cana-2743	94	20	india	india	PROPN
cana-2743	94	21	for	for	ADP
cana-2743	94	22	the	the	DET
cana-2743	94	23	inspiration	inspiration	NOUN
cana-2743	94	24	for	for	ADP
cana-2743	94	25	this	this	DET
cana-2743	94	26	work	work	NOUN
cana-2743	94	27	.	.	PUNCT
cana-2743	95	1	9	9	X
cana-2743	95	2	.	.	X
cana-2743	95	3	refrences	refrence	VERB
cana-2743	96	1	[	[	X
cana-2743	96	2	1	1	NUM
cana-2743	96	3	]	]	PUNCT
cana-2743	96	4	a.	a.	NOUN
cana-2743	96	5	zygmund	zygmund	PROPN
cana-2743	96	6	,	,	PUNCT
cana-2743	96	7	trigonometric	trigonometric	ADJ
cana-2743	96	8	series	series	NOUN
cana-2743	96	9	,	,	PUNCT
cana-2743	96	10	cambridge	cambridge	PROPN
cana-2743	96	11	univ	univ	PROPN
cana-2743	96	12	.	.	PUNCT
cana-2743	97	1	press	press	PROPN
cana-2743	97	2	,	,	PUNCT
cana-2743	97	3	cambridge	cambridge	PROPN
cana-2743	97	4	,	,	PUNCT
cana-2743	97	5	3rd	3rd	PROPN
cana-2743	97	6	rev	rev	PROPN
cana-2743	97	7	.	.	PUNCT
cana-2743	98	1	ed	ed	NOUN
cana-2743	98	2	.	.	PROPN
cana-2743	98	3	,	,	PUNCT
cana-2743	98	4	2002	2002	NUM
cana-2743	98	5	.	.	PUNCT
cana-2743	99	1	[	[	X
cana-2743	99	2	2	2	NUM
cana-2743	99	3	]	]	X
cana-2743	99	4	b.p	b.p	PROPN
cana-2743	99	5	.	.	PROPN
cana-2743	99	6	dhakal	dhakal	PROPN
cana-2743	99	7	:	:	PUNCT
cana-2743	100	1	approximation	approximation	NOUN
cana-2743	100	2	of	of	ADP
cana-2743	100	3	the	the	DET
cana-2743	100	4	conjugate	conjugate	NOUN
cana-2743	100	5	of	of	ADP
cana-2743	100	6	a	a	DET
cana-2743	100	7	function	function	NOUN
cana-2743	100	8	belonging	belong	VERB
cana-2743	100	9	to	to	ADP
cana-2743	100	10	the	the	DET
cana-2743	100	11	w𝐿𝑝	w𝐿𝑝	NOUN
cana-2743	100	12	,	,	PUNCT
cana-2743	100	13	𝜉(𝑡	𝜉(𝑡	NOUN
cana-2743	100	14	)	)	PUNCT
cana-2743	100	15	class	class	NOUN
cana-2743	100	16	by	by	ADP
cana-2743	100	17	(	(	PUNCT
cana-2743	100	18	𝑁	𝑁	PROPN
cana-2743	100	19	,	,	PUNCT
cana-2743	100	20	𝑝𝑛)(𝐸	𝑝𝑛)(𝐸	NOUN
cana-2743	100	21	,	,	PUNCT
cana-2743	100	22	1	1	NUM
cana-2743	100	23	)	)	PUNCT
cana-2743	100	24	means	mean	NOUN
cana-2743	100	25	of	of	ADP
cana-2743	100	26	the	the	DET
cana-2743	100	27	the	the	DET
cana-2743	100	28	conjugate	conjugate	ADJ
cana-2743	100	29	series	series	NOUN
cana-2743	100	30	of	of	ADP
cana-2743	100	31	the	the	DET
cana-2743	100	32	fourier	fourier	NOUN
cana-2743	100	33	series	series	NOUN
cana-2743	100	34	.	.	PUNCT
cana-2743	101	1	j.	j.	PROPN
cana-2743	101	2	sci	sci	PROPN
cana-2743	101	3	.	.	PUNCT
cana-2743	102	1	eng	eng	PROPN
cana-2743	102	2	.	.	PROPN
cana-2743	103	1	technol	technol	PROPN
cana-2743	103	2	.	.	PROPN
cana-2743	103	3	5(ii	5(ii	NUM
cana-2743	103	4	)	)	PUNCT
cana-2743	103	5	,	,	PUNCT
cana-2743	103	6	30	30	NUM
cana-2743	103	7	-	-	SYM
cana-2743	103	8	36	36	NUM
cana-2743	103	9	(	(	PUNCT
cana-2743	103	10	2009	2009	NUM
cana-2743	103	11	)	)	PUNCT
cana-2743	103	12	.	.	PUNCT
cana-2743	104	1	communications	communication	NOUN
cana-2743	104	2	on	on	ADP
cana-2743	104	3	applied	apply	VERB
cana-2743	104	4	nonlinear	nonlinear	ADJ
cana-2743	104	5	analysis	analysis	NOUN
cana-2743	104	6	issn	issn	NOUN
cana-2743	104	7	:	:	PUNCT
cana-2743	104	8	1074	1074	NUM
cana-2743	104	9	-	-	PUNCT
cana-2743	104	10	133x	133x	NUM
cana-2743	104	11	vol	vol	NOUN
cana-2743	104	12	32	32	NUM
cana-2743	104	13	no	no	NOUN
cana-2743	104	14	.	.	PUNCT
cana-2743	105	1	4s	4s	NUM
cana-2743	105	2	(	(	PUNCT
cana-2743	105	3	2025	2025	NUM
cana-2743	105	4	)	)	PUNCT
cana-2743	105	5	124	124	NUM
cana-2743	105	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2743	106	1	[	[	X
cana-2743	106	2	3	3	NUM
cana-2743	106	3	]	]	X
cana-2743	106	4	g.bachman	g.bachman	NOUN
cana-2743	106	5	,	,	PUNCT
cana-2743	106	6	l.	l.	PROPN
cana-2743	106	7	narici	narici	PROPN
cana-2743	106	8	,	,	PUNCT
cana-2743	106	9	e.	e.	PROPN
cana-2743	106	10	beckenstein	beckenstein	PROPN
cana-2743	106	11	:	:	PUNCT
cana-2743	106	12	fourier	fourier	NOUN
cana-2743	106	13	and	and	CCONJ
cana-2743	106	14	wavelet	wavelet	NOUN
cana-2743	106	15	analysis	analysis	NOUN
cana-2743	106	16	.	.	PUNCT
cana-2743	107	1	springer	springer	NOUN
cana-2743	107	2	,	,	PUNCT
cana-2743	107	3	new	new	PROPN
cana-2743	107	4	york	york	PROPN
cana-2743	107	5	(	(	PUNCT
cana-2743	107	6	2000	2000	NUM
cana-2743	107	7	)	)	PUNCT
cana-2743	107	8	.	.	PUNCT
cana-2743	108	1	[	[	X
cana-2743	108	2	4	4	X
cana-2743	108	3	]	]	PUNCT
cana-2743	108	4	k.	k.	PROPN
cana-2743	108	5	qureshi	qureshi	PROPN
cana-2743	108	6	:	:	PUNCT
cana-2743	108	7	on	on	ADP
cana-2743	108	8	the	the	DET
cana-2743	108	9	degree	degree	NOUN
cana-2743	108	10	of	of	ADP
cana-2743	108	11	approximation	approximation	NOUN
cana-2743	108	12	of	of	ADP
cana-2743	108	13	functions	function	NOUN
cana-2743	108	14	belonging	belong	VERB
cana-2743	108	15	to	to	ADP
cana-2743	108	16	the	the	DET
cana-2743	108	17	lipschitz	lipschitz	NOUN
cana-2743	108	18	class	class	NOUN
cana-2743	108	19	by	by	ADP
cana-2743	108	20	means	mean	NOUN
cana-2743	108	21	of	of	ADP
cana-2743	108	22	a	a	DET
cana-2743	108	23	conjugate	conjugate	ADJ
cana-2743	108	24	series	series	NOUN
cana-2743	108	25	,	,	PUNCT
cana-2743	108	26	indian	indian	PROPN
cana-2743	108	27	j.	j.	PROPN
cana-2743	108	28	pure	pure	PROPN
cana-2743	108	29	appl	appl	PROPN
cana-2743	108	30	.	.	PUNCT
cana-2743	108	31	math	math	PROPN
cana-2743	108	32	.	.	PUNCT
cana-2743	108	33	,	,	PUNCT
cana-2743	108	34	12(9	12(9	NUM
cana-2743	108	35	)	)	PUNCT
cana-2743	108	36	(	(	PUNCT
cana-2743	108	37	1981	1981	NUM
cana-2743	108	38	)	)	PUNCT
cana-2743	108	39	,	,	PUNCT
cana-2743	108	40	1120	1120	NUM
cana-2743	108	41	-	-	SYM
cana-2743	108	42	1123	1123	NUM
cana-2743	108	43	.	.	PUNCT
cana-2743	109	1	[	[	X
cana-2743	109	2	5	5	X
cana-2743	109	3	]	]	PUNCT
cana-2743	109	4	k.	k.	PROPN
cana-2743	109	5	qureshi	qureshi	PROPN
cana-2743	109	6	:	:	PUNCT
cana-2743	109	7	on	on	ADP
cana-2743	109	8	the	the	DET
cana-2743	109	9	degree	degree	NOUN
cana-2743	109	10	of	of	ADP
cana-2743	109	11	approximation	approximation	NOUN
cana-2743	109	12	of	of	ADP
cana-2743	109	13	function	function	NOUN
cana-2743	109	14	belonging	belong	VERB
cana-2743	109	15	to	to	ADP
cana-2743	109	16	the	the	DET
cana-2743	109	17	lip(𝛼	lip(𝛼	PROPN
cana-2743	109	18	,	,	PUNCT
cana-2743	109	19	𝑝	𝑝	NOUN
cana-2743	109	20	)	)	PUNCT
cana-2743	109	21	by	by	ADP
cana-2743	109	22	meansof	meansof	ADJ
cana-2743	109	23	conjugate	conjugate	ADJ
cana-2743	109	24	series	series	NOUN
cana-2743	109	25	.	.	PUNCT
cana-2743	110	1	indian	indian	PROPN
cana-2743	110	2	j.	j.	PROPN
cana-2743	110	3	pure	pure	PROPN
cana-2743	110	4	appl	appl	PROPN
cana-2743	110	5	.	.	PUNCT
cana-2743	110	6	math	math	NOUN
cana-2743	110	7	.	.	PUNCT
cana-2743	111	1	13(5	13(5	X
cana-2743	111	2	)	)	PUNCT
cana-2743	111	3	,	,	PUNCT
cana-2743	111	4	560	560	NUM
cana-2743	111	5	-	-	SYM
cana-2743	111	6	563	563	NUM
cana-2743	111	7	(	(	PUNCT
cana-2743	111	8	1982	1982	NUM
cana-2743	111	9	)	)	PUNCT
cana-2743	111	10	.	.	PUNCT
cana-2743	112	1	[	[	X
cana-2743	112	2	6	6	NUM
cana-2743	112	3	]	]	X
cana-2743	112	4	m.l.mittal	m.l.mittal	ADJ
cana-2743	112	5	,	,	PUNCT
cana-2743	112	6	u.	u.	PROPN
cana-2743	112	7	singh	singh	PROPN
cana-2743	112	8	,	,	PUNCT
cana-2743	112	9	v.n	v.n	PROPN
cana-2743	112	10	.	.	PROPN
cana-2743	112	11	mishra	mishra	PROPN
cana-2743	112	12	,	,	PUNCT
cana-2743	112	13	s.	s.	PROPN
cana-2743	112	14	priti	priti	PROPN
cana-2743	112	15	,	,	PUNCT
cana-2743	112	16	s.s	s.s	PROPN
cana-2743	112	17	.	.	PROPN
cana-2743	112	18	mittal	mittal	PROPN
cana-2743	112	19	:	:	PUNCT
cana-2743	112	20	approximation	approximation	NOUN
cana-2743	112	21	of	of	ADP
cana-2743	112	22	functions	function	NOUN
cana-2743	112	23	belonging	belong	VERB
cana-2743	112	24	to	to	ADP
cana-2743	112	25	lip(𝜉(𝑡	lip(𝜉(𝑡	ADV
cana-2743	112	26	)	)	PUNCT
cana-2743	112	27	,	,	PUNCT
cana-2743	112	28	𝑝)class	𝑝)clas	VERB
cana-2743	112	29	by	by	ADP
cana-2743	112	30	means	mean	NOUN
cana-2743	112	31	of	of	ADP
cana-2743	112	32	conjugate	conjugate	ADJ
cana-2743	112	33	fourier	fourier	NOUN
cana-2743	112	34	series	series	NOUN
cana-2743	112	35	using	use	VERB
cana-2743	112	36	linear	linear	PROPN
cana-2743	112	37	operators	operator	NOUN
cana-2743	112	38	.	.	PUNCT
cana-2743	113	1	indian	indian	PROPN
cana-2743	113	2	j.	j.	PROPN
cana-2743	113	3	math	math	PROPN
cana-2743	113	4	.	.	PUNCT
cana-2743	114	1	47	47	NUM
cana-2743	114	2	,	,	PUNCT
cana-2743	114	3	217	217	NUM
cana-2743	114	4	-	-	SYM
cana-2743	114	5	229	229	NUM
cana-2743	114	6	(	(	PUNCT
cana-2743	114	7	2005	2005	NUM
cana-2743	114	8	)	)	PUNCT
cana-2743	114	9	.	.	PUNCT
cana-2743	115	1	[	[	X
cana-2743	115	2	7	7	X
cana-2743	115	3	]	]	X
cana-2743	115	4	m.l	m.l	PROPN
cana-2743	115	5	.	.	PROPN
cana-2743	115	6	mittal	mittal	PROPN
cana-2743	115	7	,	,	PUNCT
cana-2743	115	8	b.e	b.e	PROPN
cana-2743	115	9	.	.	PROPN
cana-2743	115	10	rhoades	rhoades	PROPN
cana-2743	115	11	,	,	PUNCT
cana-2743	115	12	v.n	v.n	PROPN
cana-2743	115	13	.	.	PUNCT
cana-2743	115	14	mishra	mishra	PROPN
cana-2743	115	15	:	:	PUNCT
cana-2743	115	16	approximation	approximation	NOUN
cana-2743	115	17	of	of	ADP
cana-2743	115	18	signals	signal	NOUN
cana-2743	115	19	(	(	PUNCT
cana-2743	115	20	functions	function	NOUN
cana-2743	115	21	)	)	PUNCT
cana-2743	115	22	belonging	belong	VERB
cana-2743	115	23	to	to	ADP
cana-2743	115	24	the	the	DET
cana-2743	115	25	weighted	weighted	ADJ
cana-2743	115	26	w(𝐿𝑃	w(𝐿𝑃	PROPN
cana-2743	115	27	,	,	PUNCT
cana-2743	115	28	𝜉(𝑡))-class	𝜉(𝑡))-class	NOUN
cana-2743	115	29	by	by	ADP
cana-2743	115	30	linear	linear	PROPN
cana-2743	115	31	operators	operator	NOUN
cana-2743	115	32	.	.	PUNCT
cana-2743	116	1	int	int	NOUN
cana-2743	116	2	.	.	PUNCT
cana-2743	117	1	j.	j.	PROPN
cana-2743	117	2	math	math	PROPN
cana-2743	117	3	.	.	PUNCT
cana-2743	118	1	math	math	NOUN
cana-2743	118	2	.	.	PUNCT
cana-2743	119	1	sci	sci	PROPN
cana-2743	119	2	.	.	PUNCT
cana-2743	120	1	2006	2006	NUM
cana-2743	120	2	,	,	PUNCT
cana-2743	120	3	article	article	NOUN
cana-2743	120	4	i.d	i.d	PROPN
cana-2743	120	5	.	.	PROPN
cana-2743	120	6	53538	53538	NUM
cana-2743	120	7	(	(	PUNCT
cana-2743	120	8	2006	2006	NUM
cana-2743	120	9	)	)	PUNCT
cana-2743	120	10	.	.	PUNCT
cana-2743	121	1	[	[	X
cana-2743	121	2	8	8	NUM
cana-2743	121	3	]	]	SYM
cana-2743	121	4	s.lal	s.lal	PROPN
cana-2743	121	5	,	,	PUNCT
cana-2743	121	6	p.n.singh	p.n.singh	ADJ
cana-2743	121	7	:	:	PUNCT
cana-2743	121	8	degree	degree	NOUN
cana-2743	121	9	of	of	ADP
cana-2743	121	10	approximation	approximation	NOUN
cana-2743	121	11	of	of	ADP
cana-2743	121	12	conjugate	conjugate	NOUN
cana-2743	121	13	of	of	ADP
cana-2743	121	14	lip(𝛼	lip(𝛼	PROPN
cana-2743	121	15	,	,	PUNCT
cana-2743	121	16	𝑝	𝑝	NOUN
cana-2743	121	17	)	)	PUNCT
cana-2743	121	18	function	function	NOUN
cana-2743	121	19	by	by	ADP
cana-2743	121	20	(	(	PUNCT
cana-2743	121	21	𝐶	𝐶	PROPN
cana-2743	121	22	,	,	PUNCT
cana-2743	121	23	1)(𝐸	1)(𝐸	NUM
cana-2743	121	24	,	,	PUNCT
cana-2743	121	25	1	1	NUM
cana-2743	121	26	)	)	PUNCT
cana-2743	121	27	means	mean	NOUN
cana-2743	121	28	of	of	ADP
cana-2743	121	29	conjugate	conjugate	ADJ
cana-2743	121	30	series	series	NOUN
cana-2743	121	31	of	of	ADP
cana-2743	121	32	a	a	DET
cana-2743	121	33	fourier	fourier	NOUN
cana-2743	121	34	series	series	NOUN
cana-2743	121	35	.	.	PUNCT
cana-2743	122	1	tamkang	tamkang	PROPN
cana-2743	122	2	j.	j.	PROPN
cana-2743	122	3	math	math	PROPN
cana-2743	122	4	.	.	PUNCT
cana-2743	123	1	33(3	33(3	NOUN
cana-2743	123	2	)	)	PUNCT
cana-2743	123	3	,	,	PUNCT
cana-2743	123	4	269	269	NUM
cana-2743	123	5	-	-	SYM
cana-2743	123	6	274	274	NUM
cana-2743	123	7	(	(	PUNCT
cana-2743	123	8	2002	2002	NUM
cana-2743	123	9	)	)	PUNCT
cana-2743	123	10	.	.	PUNCT
cana-2743	124	1	[	[	X
cana-2743	124	2	9	9	NUM
cana-2743	124	3	]	]	SYM
cana-2743	124	4	s.sonker	s.sonker	NOUN
cana-2743	124	5	,	,	PUNCT
cana-2743	124	6	u.	u.	PROPN
cana-2743	124	7	singh	singh	PROPN
cana-2743	124	8	:	:	PUNCT
cana-2743	124	9	degree	degree	NOUN
cana-2743	124	10	of	of	ADP
cana-2743	124	11	approximation	approximation	NOUN
cana-2743	124	12	of	of	ADP
cana-2743	124	13	the	the	DET
cana-2743	124	14	conjugate	conjugate	NOUN
cana-2743	124	15	of	of	ADP
cana-2743	124	16	signals	signal	NOUN
cana-2743	124	17	(	(	PUNCT
cana-2743	124	18	functions	function	NOUN
cana-2743	124	19	)	)	PUNCT
cana-2743	124	20	belonging	belong	VERB
cana-2743	124	21	to	to	ADP
cana-2743	124	22	-class	-class	NOUN
cana-2743	124	23	by	by	ADP
cana-2743	124	24	means	mean	NOUN
cana-2743	124	25	of	of	ADP
cana-2743	124	26	conjugate	conjugate	ADJ
cana-2743	124	27	trigonometric	trigonometric	ADJ
cana-2743	124	28	fourier	fourier	NOUN
cana-2743	124	29	series	series	NOUN
cana-2743	124	30	.	.	PUNCT
cana-2743	125	1	j	j	PROPN
cana-2743	125	2	inequal	inequal	PROPN
cana-2743	125	3	appl	appl	PROPN
cana-2743	125	4	2012	2012	NUM
cana-2743	125	5	,	,	PUNCT
cana-2743	125	6	278	278	NUM
cana-2743	125	7	(	(	PUNCT
cana-2743	125	8	2012	2012	NUM
cana-2743	125	9	)	)	PUNCT
cana-2743	125	10	.	.	PUNCT
cana-2743	126	1	https://doi.org/10.1186/1029-242x-2012278	https://doi.org/10.1186/1029-242x-2012278	ADJ
cana-2743	126	2	.	.	PUNCT