id	sid	tid	token	lemma	pos
ejpam-4730	1	1	european	european	PROPN
ejpam-4730	1	2	journal	journal	PROPN
ejpam-4730	1	3	of	of	ADP
ejpam-4730	1	4	pure	pure	ADJ
ejpam-4730	1	5	and	and	CCONJ
ejpam-4730	1	6	applied	apply	VERB
ejpam-4730	1	7	mathematics	mathematic	NOUN
ejpam-4730	1	8	vol	vol	NOUN
ejpam-4730	1	9	.	.	PUNCT
ejpam-4730	2	1	16	16	NUM
ejpam-4730	2	2	,	,	PUNCT
ejpam-4730	2	3	no	no	INTJ
ejpam-4730	2	4	.	.	NOUN
ejpam-4730	2	5	2	2	NUM
ejpam-4730	2	6	,	,	PUNCT
ejpam-4730	2	7	2023	2023	NUM
ejpam-4730	2	8	,	,	PUNCT
ejpam-4730	2	9	944	944	NUM
ejpam-4730	2	10	-	-	SYM
ejpam-4730	2	11	952	952	NUM
ejpam-4730	2	12	issn	issn	PROPN
ejpam-4730	2	13	1307	1307	NUM
ejpam-4730	2	14	-	-	SYM
ejpam-4730	2	15	5543	5543	NUM
ejpam-4730	2	16	–	–	PUNCT
ejpam-4730	2	17	ejpam.com	ejpam.com	X
ejpam-4730	2	18	published	publish	VERB
ejpam-4730	2	19	by	by	ADP
ejpam-4730	2	20	new	new	PROPN
ejpam-4730	2	21	york	york	PROPN
ejpam-4730	2	22	business	business	PROPN
ejpam-4730	2	23	global	global	ADJ
ejpam-4730	2	24	best	good	ADJ
ejpam-4730	2	25	approximation	approximation	NOUN
ejpam-4730	2	26	of	of	ADP
ejpam-4730	2	27	unbounded	unbounded	ADJ
ejpam-4730	2	28	functions	function	NOUN
ejpam-4730	2	29	by	by	ADP
ejpam-4730	2	30	modulus	modulus	NOUN
ejpam-4730	2	31	of	of	ADP
ejpam-4730	2	32	smoothness	smoothness	PROPN
ejpam-4730	2	33	alaa	alaa	PROPN
ejpam-4730	2	34	adnan	adnan	PROPN
ejpam-4730	2	35	auad1,∗	auad1,∗	PROPN
ejpam-4730	2	36	,	,	PUNCT
ejpam-4730	2	37	mohammed	mohammed	PROPN
ejpam-4730	2	38	a.	a.	PROPN
ejpam-4730	2	39	hilal2	hilal2	PROPN
ejpam-4730	2	40	,	,	PUNCT
ejpam-4730	2	41	nihad	nihad	VERB
ejpam-4730	2	42	shareef	shareef	PROPN
ejpam-4730	2	43	khalaf3	khalaf3	PROPN
ejpam-4730	2	44	1	1	NUM
ejpam-4730	2	45	department	department	NOUN
ejpam-4730	2	46	of	of	ADP
ejpam-4730	2	47	mathematics	mathematic	NOUN
ejpam-4730	2	48	,	,	PUNCT
ejpam-4730	2	49	college	college	NOUN
ejpam-4730	2	50	of	of	ADP
ejpam-4730	2	51	education	education	NOUN
ejpam-4730	2	52	for	for	ADP
ejpam-4730	2	53	pure	pure	ADJ
ejpam-4730	2	54	sciences	science	NOUN
ejpam-4730	2	55	,	,	PUNCT
ejpam-4730	2	56	university	university	NOUN
ejpam-4730	2	57	of	of	ADP
ejpam-4730	2	58	anbar	anbar	PROPN
ejpam-4730	2	59	,	,	PUNCT
ejpam-4730	2	60	ramadi	ramadi	PROPN
ejpam-4730	2	61	,	,	PUNCT
ejpam-4730	2	62	iraq	iraq	PROPN
ejpam-4730	2	63	2	2	NUM
ejpam-4730	2	64	baquba	baquba	PROPN
ejpam-4730	2	65	technical	technical	PROPN
ejpam-4730	2	66	institute	institute	PROPN
ejpam-4730	2	67	,	,	PUNCT
ejpam-4730	2	68	middle	middle	PROPN
ejpam-4730	2	69	technical	technical	PROPN
ejpam-4730	2	70	university	university	PROPN
ejpam-4730	2	71	,	,	PUNCT
ejpam-4730	2	72	baquba	baquba	PROPN
ejpam-4730	2	73	,	,	PUNCT
ejpam-4730	2	74	iraq	iraq	PROPN
ejpam-4730	2	75	3	3	NUM
ejpam-4730	2	76	department	department	NOUN
ejpam-4730	2	77	of	of	ADP
ejpam-4730	2	78	mathematics	mathematics	PROPN
ejpam-4730	2	79	,	,	PUNCT
ejpam-4730	2	80	college	college	NOUN
ejpam-4730	2	81	of	of	ADP
ejpam-4730	2	82	education	education	NOUN
ejpam-4730	2	83	for	for	ADP
ejpam-4730	2	84	women	woman	NOUN
ejpam-4730	2	85	,	,	PUNCT
ejpam-4730	2	86	tikrit	tikrit	NOUN
ejpam-4730	2	87	university	university	NOUN
ejpam-4730	2	88	,	,	PUNCT
ejpam-4730	2	89	tikrit	tikrit	NOUN
ejpam-4730	2	90	,	,	PUNCT
ejpam-4730	2	91	iraq	iraq	PROPN
ejpam-4730	2	92	abstract	abstract	NOUN
ejpam-4730	2	93	.	.	PUNCT
ejpam-4730	3	1	in	in	ADP
ejpam-4730	3	2	this	this	DET
ejpam-4730	3	3	paper	paper	NOUN
ejpam-4730	3	4	,	,	PUNCT
ejpam-4730	3	5	we	we	PRON
ejpam-4730	3	6	study	study	VERB
ejpam-4730	3	7	the	the	DET
ejpam-4730	3	8	approximation	approximation	NOUN
ejpam-4730	3	9	of	of	ADP
ejpam-4730	3	10	unbounded	unbounded	ADJ
ejpam-4730	3	11	functions	function	NOUN
ejpam-4730	3	12	in	in	ADP
ejpam-4730	3	13	a	a	DET
ejpam-4730	3	14	weighted	weight	VERB
ejpam-4730	3	15	space	space	NOUN
ejpam-4730	3	16	by	by	ADP
ejpam-4730	3	17	modulus	modulus	NOUN
ejpam-4730	3	18	of	of	ADP
ejpam-4730	3	19	smoothness	smoothness	NOUN
ejpam-4730	3	20	using	use	VERB
ejpam-4730	3	21	various	various	ADJ
ejpam-4730	3	22	linear	linear	PROPN
ejpam-4730	3	23	operators	operator	NOUN
ejpam-4730	3	24	.	.	PUNCT
ejpam-4730	4	1	we	we	PRON
ejpam-4730	4	2	establish	establish	VERB
ejpam-4730	4	3	direct	direct	ADJ
ejpam-4730	4	4	theorems	theorem	NOUN
ejpam-4730	4	5	for	for	ADP
ejpam-4730	4	6	such	such	ADJ
ejpam-4730	4	7	approximations	approximation	NOUN
ejpam-4730	4	8	and	and	CCONJ
ejpam-4730	4	9	analyze	analyze	VERB
ejpam-4730	4	10	the	the	DET
ejpam-4730	4	11	properties	property	NOUN
ejpam-4730	4	12	of	of	ADP
ejpam-4730	4	13	the	the	DET
ejpam-4730	4	14	modulus	modulus	NOUN
ejpam-4730	4	15	of	of	ADP
ejpam-4730	4	16	smoothness	smoothness	NOUN
ejpam-4730	4	17	within	within	ADP
ejpam-4730	4	18	the	the	DET
ejpam-4730	4	19	same	same	ADJ
ejpam-4730	4	20	space	space	NOUN
ejpam-4730	4	21	.	.	PUNCT
ejpam-4730	5	1	specifically	specifically	ADV
ejpam-4730	5	2	,	,	PUNCT
ejpam-4730	5	3	we	we	PRON
ejpam-4730	5	4	investigate	investigate	VERB
ejpam-4730	5	5	the	the	DET
ejpam-4730	5	6	behavior	behavior	NOUN
ejpam-4730	5	7	of	of	ADP
ejpam-4730	5	8	the	the	DET
ejpam-4730	5	9	modulus	modulus	NOUN
ejpam-4730	5	10	of	of	ADP
ejpam-4730	5	11	smoothness	smoothness	NOUN
ejpam-4730	5	12	under	under	ADP
ejpam-4730	5	13	different	different	ADJ
ejpam-4730	5	14	types	type	NOUN
ejpam-4730	5	15	of	of	ADP
ejpam-4730	5	16	linear	linear	PROPN
ejpam-4730	5	17	operators	operator	NOUN
ejpam-4730	5	18	,	,	PUNCT
ejpam-4730	5	19	including	include	VERB
ejpam-4730	5	20	the	the	DET
ejpam-4730	5	21	bernstein	bernstein	PROPN
ejpam-4730	5	22	-	-	PUNCT
ejpam-4730	5	23	durrmeyer	durrmeyer	PROPN
ejpam-4730	5	24	operator	operator	NOUN
ejpam-4730	5	25	,	,	PUNCT
ejpam-4730	5	26	the	the	DET
ejpam-4730	5	27	fejer	fejer	ADJ
ejpam-4730	5	28	operator	operator	NOUN
ejpam-4730	5	29	,	,	PUNCT
ejpam-4730	5	30	and	and	CCONJ
ejpam-4730	5	31	the	the	DET
ejpam-4730	5	32	jackson	jackson	PROPN
ejpam-4730	5	33	operator	operator	NOUN
ejpam-4730	5	34	.	.	PUNCT
ejpam-4730	6	1	we	we	PRON
ejpam-4730	6	2	also	also	ADV
ejpam-4730	6	3	provide	provide	VERB
ejpam-4730	6	4	a	a	DET
ejpam-4730	6	5	detailed	detailed	ADJ
ejpam-4730	6	6	analysis	analysis	NOUN
ejpam-4730	6	7	of	of	ADP
ejpam-4730	6	8	the	the	DET
ejpam-4730	6	9	convergence	convergence	NOUN
ejpam-4730	6	10	rate	rate	NOUN
ejpam-4730	6	11	of	of	ADP
ejpam-4730	6	12	these	these	DET
ejpam-4730	6	13	operators	operator	NOUN
ejpam-4730	6	14	.	.	PUNCT
ejpam-4730	7	1	furthermore	furthermore	ADV
ejpam-4730	7	2	,	,	PUNCT
ejpam-4730	7	3	we	we	PRON
ejpam-4730	7	4	discuss	discuss	VERB
ejpam-4730	7	5	the	the	DET
ejpam-4730	7	6	relationship	relationship	NOUN
ejpam-4730	7	7	between	between	ADP
ejpam-4730	7	8	the	the	DET
ejpam-4730	7	9	modulus	modulus	NOUN
ejpam-4730	7	10	of	of	ADP
ejpam-4730	7	11	smoothness	smoothness	NOUN
ejpam-4730	7	12	and	and	CCONJ
ejpam-4730	7	13	the	the	DET
ejpam-4730	7	14	lipschitz	lipschitz	NOUN
ejpam-4730	7	15	constant	constant	ADJ
ejpam-4730	7	16	of	of	ADP
ejpam-4730	7	17	a	a	DET
ejpam-4730	7	18	function	function	NOUN
ejpam-4730	7	19	.	.	PUNCT
ejpam-4730	8	1	our	our	PRON
ejpam-4730	8	2	findings	finding	NOUN
ejpam-4730	8	3	have	have	VERB
ejpam-4730	8	4	important	important	ADJ
ejpam-4730	8	5	implications	implication	NOUN
ejpam-4730	8	6	for	for	ADP
ejpam-4730	8	7	the	the	DET
ejpam-4730	8	8	field	field	NOUN
ejpam-4730	8	9	of	of	ADP
ejpam-4730	8	10	approximation	approximation	NOUN
ejpam-4730	8	11	theory	theory	NOUN
ejpam-4730	8	12	and	and	CCONJ
ejpam-4730	8	13	may	may	AUX
ejpam-4730	8	14	help	help	VERB
ejpam-4730	8	15	to	to	PART
ejpam-4730	8	16	inform	inform	VERB
ejpam-4730	8	17	future	future	ADJ
ejpam-4730	8	18	research	research	NOUN
ejpam-4730	8	19	in	in	ADP
ejpam-4730	8	20	this	this	DET
ejpam-4730	8	21	area	area	NOUN
ejpam-4730	8	22	.	.	PUNCT
ejpam-4730	9	1	2020	2020	NUM
ejpam-4730	9	2	mathematics	mathematic	NOUN
ejpam-4730	9	3	subject	subject	NOUN
ejpam-4730	9	4	classifications	classification	NOUN
ejpam-4730	9	5	:	:	PUNCT
ejpam-4730	9	6	41a52	41a52	NUM
ejpam-4730	9	7	,	,	PUNCT
ejpam-4730	9	8	41a44	41a44	NUM
ejpam-4730	9	9	,	,	PUNCT
ejpam-4730	9	10	41a27	41a27	NUM
ejpam-4730	9	11	key	key	ADJ
ejpam-4730	9	12	words	word	NOUN
ejpam-4730	9	13	and	and	CCONJ
ejpam-4730	9	14	phrases	phrase	NOUN
ejpam-4730	9	15	:	:	PUNCT
ejpam-4730	9	16	unbounded	unbounded	ADJ
ejpam-4730	9	17	functions	function	NOUN
ejpam-4730	9	18	,	,	PUNCT
ejpam-4730	9	19	weighted	weight	VERB
ejpam-4730	9	20	spaces	space	NOUN
ejpam-4730	9	21	,	,	PUNCT
ejpam-4730	9	22	approximation	approximation	NOUN
ejpam-4730	9	23	,	,	PUNCT
ejpam-4730	9	24	modulus	modulus	NOUN
ejpam-4730	9	25	of	of	ADP
ejpam-4730	9	26	smoothness	smoothness	NOUN
ejpam-4730	9	27	,	,	PUNCT
ejpam-4730	9	28	trigonometric	trigonometric	ADJ
ejpam-4730	9	29	polynomial	polynomial	ADJ
ejpam-4730	9	30	1	1	NUM
ejpam-4730	9	31	.	.	PUNCT
ejpam-4730	10	1	introduction	introduction	NOUN
ejpam-4730	10	2	let	let	VERB
ejpam-4730	10	3	lp	lp	NOUN
ejpam-4730	10	4	=	=	NOUN
ejpam-4730	10	5	{	{	PUNCT
ejpam-4730	10	6	f	f	NOUN
ejpam-4730	10	7	:	:	PUNCT
ejpam-4730	10	8	f	f	PROPN
ejpam-4730	10	9	is	be	AUX
ejpam-4730	10	10	bounded	bound	VERB
ejpam-4730	10	11	measurable	measurable	ADJ
ejpam-4730	10	12	function	function	NOUN
ejpam-4730	10	13	}	}	PUNCT
ejpam-4730	10	14	,	,	PUNCT
ejpam-4730	10	15	1	1	NUM
ejpam-4730	10	16	≤	≤	NOUN
ejpam-4730	10	17	p	p	X
ejpam-4730	10	18	<	<	X
ejpam-4730	10	19	∞	∞	NUM
ejpam-4730	10	20	be	be	VERB
ejpam-4730	10	21	the	the	DET
ejpam-4730	10	22	space	space	NOUN
ejpam-4730	10	23	of	of	ADP
ejpam-4730	10	24	all	all	DET
ejpam-4730	10	25	bounded	bounded	ADJ
ejpam-4730	10	26	functions	function	NOUN
ejpam-4730	10	27	with	with	ADP
ejpam-4730	10	28	the	the	DET
ejpam-4730	10	29	norm	norm	NOUN
ejpam-4730	10	30	∥f∥p	∥f∥p	NOUN
ejpam-4730	10	31	=	=	SYM
ejpam-4730	11	1	(	(	PUNCT
ejpam-4730	11	2	∫	∫	PROPN
ejpam-4730	11	3	π	π	PROPN
ejpam-4730	11	4	−π	−π	NUM
ejpam-4730	11	5	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-4730	11	6	)	)	PUNCT
ejpam-4730	11	7	1	1	NUM
ejpam-4730	11	8	p	p	NOUN
ejpam-4730	11	9	<	<	AUX
ejpam-4730	11	10	∞.	∞.	PROPN
ejpam-4730	11	11	let	let	VERB
ejpam-4730	11	12	w	w	NOUN
ejpam-4730	11	13	be	be	AUX
ejpam-4730	11	14	the	the	DET
ejpam-4730	11	15	space	space	NOUN
ejpam-4730	11	16	of	of	ADP
ejpam-4730	11	17	all	all	DET
ejpam-4730	11	18	weighted	weight	VERB
ejpam-4730	11	19	functions	function	NOUN
ejpam-4730	11	20	such	such	ADJ
ejpam-4730	11	21	that	that	SCONJ
ejpam-4730	11	22	a	a	DET
ejpam-4730	11	23	function	function	NOUN
ejpam-4730	11	24	λ	λ	X
ejpam-4730	11	25	:	:	PUNCT
ejpam-4730	12	1	[	[	X
ejpam-4730	12	2	−π	−π	ADV
ejpam-4730	12	3	,	,	PUNCT
ejpam-4730	12	4	π	π	X
ejpam-4730	12	5	]	]	X
ejpam-4730	12	6	→	→	PUNCT
ejpam-4730	12	7	r+	r+	PRON
ejpam-4730	12	8	is	be	AUX
ejpam-4730	12	9	an	an	DET
ejpam-4730	12	10	almost	almost	ADV
ejpam-4730	12	11	everywhere	everywhere	ADV
ejpam-4730	12	12	positive	positive	ADJ
ejpam-4730	12	13	function	function	NOUN
ejpam-4730	12	14	which	which	PRON
ejpam-4730	12	15	is	be	AUX
ejpam-4730	12	16	locally	locally	ADV
ejpam-4730	12	17	integrable	integrable	ADJ
ejpam-4730	12	18	,	,	PUNCT
ejpam-4730	12	19	that	that	PRON
ejpam-4730	12	20	is	is	ADV
ejpam-4730	12	21	λ	λ	PROPN
ejpam-4730	12	22	∈w	∈w	NOUN
ejpam-4730	12	23	.	.	PUNCT
ejpam-4730	13	1	∗corresponding	∗corresponde	VERB
ejpam-4730	13	2	author	author	NOUN
ejpam-4730	13	3	.	.	PUNCT
ejpam-4730	14	1	doi	doi	NOUN
ejpam-4730	14	2	:	:	PUNCT
ejpam-4730	14	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4730	https://doi.org/10.29020/nybg.ejpam.v16i2.4730	SCONJ
ejpam-4730	14	4	email	email	NOUN
ejpam-4730	14	5	addresses	address	VERB
ejpam-4730	14	6	:	:	PUNCT
ejpam-4730	15	1	alaa.adnan.auad@uoanbar.edu.iq	alaa.adnan.auad@uoanbar.edu.iq	NOUN
ejpam-4730	15	2	(	(	PUNCT
ejpam-4730	15	3	a.	a.	NOUN
ejpam-4730	15	4	a.	a.	NOUN
ejpam-4730	15	5	auad	auad	PROPN
ejpam-4730	15	6	)	)	PUNCT
ejpam-4730	15	7	,	,	PUNCT
ejpam-4730	15	8	mohammed	mohammed	PROPN
ejpam-4730	15	9	azeez	azeez	PROPN
ejpam-4730	15	10	hilal@mtu.edu.iq	hilal@mtu.edu.iq	PROPN
ejpam-4730	15	11	(	(	PUNCT
ejpam-4730	15	12	m.	m.	NOUN
ejpam-4730	15	13	a.	a.	PROPN
ejpam-4730	15	14	hilal	hilal	PROPN
ejpam-4730	15	15	)	)	PUNCT
ejpam-4730	15	16	,	,	PUNCT
ejpam-4730	15	17	nihad.shareef16@tu.edu.iq	nihad.shareef16@tu.edu.iq	PROPN
ejpam-4730	15	18	(	(	PUNCT
ejpam-4730	15	19	n.	n.	PROPN
ejpam-4730	15	20	s.	s.	PROPN
ejpam-4730	15	21	khalaf	khalaf	PROPN
ejpam-4730	15	22	)	)	PUNCT
ejpam-4730	15	23	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4730	16	1	944	944	NUM
ejpam-4730	17	1	©	©	PROPN
ejpam-4730	17	2	2023	2023	NUM
ejpam-4730	17	3	ejpam	ejpam	NOUN
ejpam-4730	17	4	all	all	DET
ejpam-4730	17	5	rights	right	NOUN
ejpam-4730	17	6	reserved	reserve	VERB
ejpam-4730	17	7	.	.	PUNCT
ejpam-4730	18	1	a.	a.	NOUN
ejpam-4730	18	2	a.	a.	PROPN
ejpam-4730	18	3	auad	auad	PROPN
ejpam-4730	18	4	,	,	PUNCT
ejpam-4730	18	5	m.	m.	NOUN
ejpam-4730	18	6	a.	a.	NOUN
ejpam-4730	18	7	hilal	hilal	PROPN
ejpam-4730	18	8	,	,	PUNCT
ejpam-4730	18	9	n.	n.	PROPN
ejpam-4730	18	10	s.	s.	PROPN
ejpam-4730	18	11	khalaf	khalaf	PROPN
ejpam-4730	18	12	/	/	SYM
ejpam-4730	18	13	eur	eur	PROPN
ejpam-4730	18	14	.	.	PUNCT
ejpam-4730	19	1	j.	j.	PROPN
ejpam-4730	19	2	pure	pure	PROPN
ejpam-4730	19	3	appl	appl	PROPN
ejpam-4730	19	4	.	.	PROPN
ejpam-4730	19	5	math	math	PROPN
ejpam-4730	19	6	,	,	PUNCT
ejpam-4730	19	7	16	16	NUM
ejpam-4730	19	8	(	(	PUNCT
ejpam-4730	19	9	2	2	NUM
ejpam-4730	19	10	)	)	PUNCT
ejpam-4730	19	11	(	(	PUNCT
ejpam-4730	19	12	2023	2023	NUM
ejpam-4730	19	13	)	)	PUNCT
ejpam-4730	19	14	,	,	PUNCT
ejpam-4730	19	15	944	944	NUM
ejpam-4730	19	16	-	-	SYM
ejpam-4730	19	17	952	952	NUM
ejpam-4730	19	18	945	945	NUM
ejpam-4730	19	19	let	let	VERB
ejpam-4730	19	20	lp	lp	ADP
ejpam-4730	19	21	,	,	PUNCT
ejpam-4730	19	22	λ[−π	λ[−π	PROPN
ejpam-4730	19	23	,	,	PUNCT
ejpam-4730	19	24	π	π	X
ejpam-4730	19	25	]	]	X
ejpam-4730	19	26	=	=	X
ejpam-4730	19	27	{	{	PUNCT
ejpam-4730	19	28	f	f	X
ejpam-4730	19	29	:	:	PUNCT
ejpam-4730	19	30	f	f	PROPN
ejpam-4730	19	31	is	be	AUX
ejpam-4730	19	32	unbounded	unbounded	ADJ
ejpam-4730	19	33	function	function	NOUN
ejpam-4730	19	34	on	on	ADP
ejpam-4730	19	35	[	[	X
ejpam-4730	19	36	−π	−π	ADV
ejpam-4730	19	37	,	,	PUNCT
ejpam-4730	19	38	π	π	PROPN
ejpam-4730	19	39	]	]	X
ejpam-4730	19	40	,	,	PUNCT
ejpam-4730	19	41	1	1	NUM
ejpam-4730	19	42	≤	≤	NOUN
ejpam-4730	19	43	p	p	X
ejpam-4730	19	44	<	<	X
ejpam-4730	19	45	∞	∞	NUM
ejpam-4730	19	46	}	}	PUNCT
ejpam-4730	19	47	,	,	PUNCT
ejpam-4730	19	48	with	with	ADP
ejpam-4730	19	49	the	the	DET
ejpam-4730	19	50	norm	norm	NOUN
ejpam-4730	19	51	∥f∥lp	∥f∥lp	PROPN
ejpam-4730	19	52	,	,	PUNCT
ejpam-4730	19	53	λ[−π	λ[−π	PROPN
ejpam-4730	19	54	,	,	PUNCT
ejpam-4730	19	55	π	π	X
ejpam-4730	19	56	]	]	X
ejpam-4730	19	57	=	=	SYM
ejpam-4730	19	58	(	(	PUNCT
ejpam-4730	19	59	∫	∫	PROPN
ejpam-4730	19	60	π	π	PROPN
ejpam-4730	19	61	−π	−π	PROPN
ejpam-4730	19	62	|f(x)λ(x)|pdx	|f(x)λ(x)|pdx	PROPN
ejpam-4730	19	63	)	)	PUNCT
ejpam-4730	19	64	1	1	NUM
ejpam-4730	19	65	p	p	NOUN
ejpam-4730	19	66	<	<	X
ejpam-4730	19	67	∞.	∞.	PROPN
ejpam-4730	19	68	also	also	ADV
ejpam-4730	19	69	,	,	PUNCT
ejpam-4730	19	70	let	let	VERB
ejpam-4730	19	71	n	n	PRON
ejpam-4730	19	72	be	be	AUX
ejpam-4730	19	73	the	the	DET
ejpam-4730	19	74	set	set	NOUN
ejpam-4730	19	75	of	of	ADP
ejpam-4730	19	76	all	all	DET
ejpam-4730	19	77	natural	natural	ADJ
ejpam-4730	19	78	numbers	number	NOUN
ejpam-4730	19	79	and	and	CCONJ
ejpam-4730	19	80	for	for	ADP
ejpam-4730	19	81	every	every	DET
ejpam-4730	19	82	k	k	PROPN
ejpam-4730	19	83	∈	∈	PROPN
ejpam-4730	19	84	n	n	PART
ejpam-4730	19	85	∪	∪	X
ejpam-4730	19	86	{	{	PUNCT
ejpam-4730	19	87	0	0	NUM
ejpam-4730	19	88	}	}	PUNCT
ejpam-4730	19	89	,	,	PUNCT
ejpam-4730	19	90	we	we	PRON
ejpam-4730	19	91	denote	denote	VERB
ejpam-4730	19	92	by	by	ADP
ejpam-4730	19	93	tk	tk	PROPN
ejpam-4730	19	94	the	the	DET
ejpam-4730	19	95	set	set	NOUN
ejpam-4730	19	96	of	of	ADP
ejpam-4730	19	97	all	all	DET
ejpam-4730	19	98	trigonometric	trigonometric	ADJ
ejpam-4730	19	99	polynomials	polynomial	NOUN
ejpam-4730	19	100	of	of	ADP
ejpam-4730	19	101	degree	degree	NOUN
ejpam-4730	19	102	less	less	ADJ
ejpam-4730	19	103	than	than	ADP
ejpam-4730	19	104	or	or	CCONJ
ejpam-4730	19	105	equal	equal	ADJ
ejpam-4730	19	106	to	to	ADP
ejpam-4730	19	107	k.	k.	PROPN
ejpam-4730	19	108	for	for	ADP
ejpam-4730	19	109	a	a	DET
ejpam-4730	19	110	given	give	VERB
ejpam-4730	19	111	function	function	NOUN
ejpam-4730	19	112	f	f	PROPN
ejpam-4730	19	113	∈	∈	PROPN
ejpam-4730	19	114	lp	lp	PROPN
ejpam-4730	19	115	,	,	PUNCT
ejpam-4730	19	116	λ[−π	λ[−π	PROPN
ejpam-4730	19	117	,	,	PUNCT
ejpam-4730	19	118	π	π	PROPN
ejpam-4730	19	119	]	]	X
ejpam-4730	19	120	,	,	PUNCT
ejpam-4730	19	121	we	we	PRON
ejpam-4730	19	122	define	define	VERB
ejpam-4730	19	123	ek(f)lp	ek(f)lp	NOUN
ejpam-4730	19	124	,	,	PUNCT
ejpam-4730	19	125	λ[−π	λ[−π	PROPN
ejpam-4730	19	126	,	,	PUNCT
ejpam-4730	19	127	π	π	X
ejpam-4730	19	128	]	]	X
ejpam-4730	20	1	=	=	PUNCT
ejpam-4730	20	2	inf{∥f	inf{∥f	PROPN
ejpam-4730	20	3	−	−	PROPN
ejpam-4730	20	4	g∥lp	g∥lp	PROPN
ejpam-4730	20	5	,	,	PUNCT
ejpam-4730	20	6	λ[−π	λ[−π	PROPN
ejpam-4730	20	7	,	,	PUNCT
ejpam-4730	20	8	π	π	X
ejpam-4730	20	9	]	]	X
ejpam-4730	20	10	;	;	PUNCT
ejpam-4730	20	11	g	g	PROPN
ejpam-4730	20	12	∈	∈	PROPN
ejpam-4730	20	13	tk	tk	PROPN
ejpam-4730	20	14	}	}	PUNCT
ejpam-4730	20	15	,	,	PUNCT
ejpam-4730	20	16	(	(	PUNCT
ejpam-4730	20	17	1	1	X
ejpam-4730	20	18	)	)	PUNCT
ejpam-4730	20	19	which	which	PRON
ejpam-4730	20	20	is	be	AUX
ejpam-4730	20	21	called	call	VERB
ejpam-4730	20	22	the	the	DET
ejpam-4730	20	23	kth	kth	PROPN
ejpam-4730	20	24	degree	degree	NOUN
ejpam-4730	20	25	best	good	ADJ
ejpam-4730	20	26	approximation	approximation	NOUN
ejpam-4730	20	27	of	of	ADP
ejpam-4730	20	28	f	f	PROPN
ejpam-4730	20	29	with	with	ADP
ejpam-4730	20	30	respect	respect	NOUN
ejpam-4730	20	31	to	to	ADP
ejpam-4730	20	32	tk	tk	PROPN
ejpam-4730	20	33	.	.	PROPN
ejpam-4730	20	34	let	let	VERB
ejpam-4730	20	35	k	k	PROPN
ejpam-4730	20	36	∈	∈	PROPN
ejpam-4730	20	37	n	n	PROPN
ejpam-4730	20	38	and	and	CCONJ
ejpam-4730	20	39	f	f	PROPN
ejpam-4730	20	40	∈	∈	PROPN
ejpam-4730	20	41	lp	lp	PROPN
ejpam-4730	20	42	,	,	PUNCT
ejpam-4730	20	43	λ[−π	λ[−π	PROPN
ejpam-4730	20	44	,	,	PUNCT
ejpam-4730	20	45	π	π	PROPN
ejpam-4730	20	46	]	]	X
ejpam-4730	20	47	.	.	PUNCT
ejpam-4730	21	1	then	then	ADV
ejpam-4730	21	2	,	,	PUNCT
ejpam-4730	21	3	we	we	PRON
ejpam-4730	21	4	define	define	VERB
ejpam-4730	21	5	the	the	DET
ejpam-4730	21	6	modulus	modulus	NOUN
ejpam-4730	21	7	of	of	ADP
ejpam-4730	21	8	smoothness	smoothness	NOUN
ejpam-4730	21	9	of	of	ADP
ejpam-4730	21	10	f	f	PROPN
ejpam-4730	21	11	by	by	ADP
ejpam-4730	21	12	µk(f	µk(f	NOUN
ejpam-4730	21	13	,	,	PUNCT
ejpam-4730	21	14	δ)lp	δ)lp	PROPN
ejpam-4730	21	15	,	,	PUNCT
ejpam-4730	21	16	λ[−π	λ[−π	PROPN
ejpam-4730	21	17	,	,	PUNCT
ejpam-4730	21	18	π	π	X
ejpam-4730	21	19	]	]	X
ejpam-4730	21	20	=	=	SYM
ejpam-4730	21	21	sup︸︷︷︸	sup︸︷︷︸	PROPN
ejpam-4730	21	22	h≤δ	h≤δ	PROPN
ejpam-4730	21	23	∥∥∥∆k	∥∥∥∆k	VERB
ejpam-4730	21	24	hf	hf	PROPN
ejpam-4730	21	25	(	(	PUNCT
ejpam-4730	21	26	.	.	PUNCT
ejpam-4730	21	27	)	)	PUNCT
ejpam-4730	22	1	∥∥∥	∥∥∥	PROPN
ejpam-4730	22	2	lp	lp	NOUN
ejpam-4730	22	3	,	,	PUNCT
ejpam-4730	22	4	λ[−π	λ[−π	PROPN
ejpam-4730	22	5	,	,	PUNCT
ejpam-4730	22	6	π	π	X
ejpam-4730	22	7	]	]	X
ejpam-4730	22	8	,	,	PUNCT
ejpam-4730	22	9	where	where	SCONJ
ejpam-4730	22	10	δ	δ	PROPN
ejpam-4730	22	11	=	=	SYM
ejpam-4730	22	12	1	1	NUM
ejpam-4730	22	13	k	k	NOUN
ejpam-4730	22	14	and	and	CCONJ
ejpam-4730	22	15	∆k	∆k	PROPN
ejpam-4730	22	16	hf(x	hf(x	NOUN
ejpam-4730	22	17	)	)	PUNCT
ejpam-4730	22	18	is	be	AUX
ejpam-4730	22	19	called	call	VERB
ejpam-4730	22	20	the	the	DET
ejpam-4730	22	21	kth	kth	PROPN
ejpam-4730	22	22	difference	difference	NOUN
ejpam-4730	22	23	symmetric	symmetric	NOUN
ejpam-4730	22	24	with	with	ADP
ejpam-4730	22	25	step	step	NOUN
ejpam-4730	22	26	h	h	NOUN
ejpam-4730	22	27	at	at	ADP
ejpam-4730	22	28	point	point	NOUN
ejpam-4730	22	29	x	x	X
ejpam-4730	22	30	,	,	PUNCT
ejpam-4730	22	31	and	and	CCONJ
ejpam-4730	22	32	it	it	PRON
ejpam-4730	22	33	is	be	AUX
ejpam-4730	22	34	given	give	VERB
ejpam-4730	22	35	by	by	ADP
ejpam-4730	22	36	∆k	∆k	PROPN
ejpam-4730	22	37	hf(x	hf(x	PART
ejpam-4730	22	38	)	)	PUNCT
ejpam-4730	22	39	=	=	PUNCT
ejpam-4730	23	1	∑k	∑k	PROPN
ejpam-4730	23	2	i=1	i=1	X
ejpam-4730	24	1	(	(	PUNCT
ejpam-4730	24	2	−1)k−1f(x+	−1)k−1f(x+	X
ejpam-4730	24	3	ih	ih	NOUN
ejpam-4730	24	4	)	)	PUNCT
ejpam-4730	24	5	.	.	PUNCT
ejpam-4730	25	1	(	(	PUNCT
ejpam-4730	25	2	2	2	X
ejpam-4730	25	3	)	)	PUNCT
ejpam-4730	25	4	the	the	DET
ejpam-4730	25	5	weierstrass	weierstrass	NOUN
ejpam-4730	25	6	approximation	approximation	NOUN
ejpam-4730	25	7	theorem	theorem	NOUN
ejpam-4730	25	8	simply	simply	ADV
ejpam-4730	25	9	states	state	VERB
ejpam-4730	25	10	that	that	SCONJ
ejpam-4730	25	11	ek(f)lp[a	ek(f)lp[a	NOUN
ejpam-4730	25	12	,	,	PUNCT
ejpam-4730	25	13	b	b	NOUN
ejpam-4730	25	14	]	]	X
ejpam-4730	25	15	converges	converge	VERB
ejpam-4730	25	16	to	to	ADP
ejpam-4730	25	17	zero	zero	NUM
ejpam-4730	25	18	as	as	ADP
ejpam-4730	25	19	k	k	PROPN
ejpam-4730	25	20	→	→	SYM
ejpam-4730	25	21	∞	∞	PROPN
ejpam-4730	25	22	for	for	ADP
ejpam-4730	25	23	all	all	DET
ejpam-4730	25	24	f	f	PROPN
ejpam-4730	25	25	∈	∈	PROPN
ejpam-4730	25	26	lp[a	lp[a	PROPN
ejpam-4730	25	27	,	,	PUNCT
ejpam-4730	25	28	b	b	NOUN
ejpam-4730	25	29	]	]	PUNCT
ejpam-4730	25	30	.	.	PUNCT
ejpam-4730	26	1	it	it	PRON
ejpam-4730	26	2	does	do	AUX
ejpam-4730	26	3	not	not	PART
ejpam-4730	26	4	say	say	VERB
ejpam-4730	26	5	how	how	SCONJ
ejpam-4730	26	6	fast	fast	ADJ
ejpam-4730	26	7	ek(f)lp[a	ek(f)lp[a	ADJ
ejpam-4730	26	8	,	,	PUNCT
ejpam-4730	26	9	b	b	NOUN
ejpam-4730	26	10	]	]	X
ejpam-4730	26	11	→	→	X
ejpam-4730	26	12	0	0	X
ejpam-4730	26	13	.	.	PUNCT
ejpam-4730	27	1	in	in	ADP
ejpam-4730	27	2	1987	1987	NUM
ejpam-4730	27	3	,	,	PUNCT
ejpam-4730	27	4	prestin	prestin	NOUN
ejpam-4730	27	5	[	[	X
ejpam-4730	27	6	20	20	NUM
ejpam-4730	27	7	]	]	PUNCT
ejpam-4730	27	8	investigated	investigate	VERB
ejpam-4730	27	9	problems	problem	NOUN
ejpam-4730	27	10	of	of	ADP
ejpam-4730	27	11	estimating	estimate	VERB
ejpam-4730	27	12	the	the	DET
ejpam-4730	27	13	deviation	deviation	NOUN
ejpam-4730	27	14	of	of	ADP
ejpam-4730	27	15	functions	function	NOUN
ejpam-4730	27	16	from	from	ADP
ejpam-4730	27	17	their	their	PRON
ejpam-4730	27	18	de	de	X
ejpam-4730	27	19	la	la	PRON
ejpam-4730	27	20	vallée	vallée	PROPN
ejpam-4730	27	21	-	-	PUNCT
ejpam-4730	27	22	poussin	poussin	PROPN
ejpam-4730	27	23	sums	sum	NOUN
ejpam-4730	27	24	in	in	ADP
ejpam-4730	27	25	weighted	weight	VERB
ejpam-4730	27	26	orlicz	orlicz	ADJ
ejpam-4730	27	27	spaces	space	NOUN
ejpam-4730	27	28	.	.	PUNCT
ejpam-4730	28	1	in	in	ADP
ejpam-4730	28	2	1999	1999	NUM
ejpam-4730	28	3	,	,	PUNCT
ejpam-4730	28	4	bustamante	bustamante	X
ejpam-4730	29	1	[	[	X
ejpam-4730	29	2	9	9	NUM
ejpam-4730	29	3	]	]	PUNCT
ejpam-4730	29	4	studied	study	VERB
ejpam-4730	29	5	some	some	DET
ejpam-4730	29	6	problems	problem	NOUN
ejpam-4730	29	7	of	of	ADP
ejpam-4730	29	8	approximation	approximation	NOUN
ejpam-4730	29	9	theory	theory	NOUN
ejpam-4730	29	10	in	in	ADP
ejpam-4730	29	11	the	the	DET
ejpam-4730	29	12	spaces	space	NOUN
ejpam-4730	29	13	sp(1	sp(1	X
ejpam-4730	29	14	≤	≤	NOUN
ejpam-4730	30	1	p	p	X
ejpam-4730	30	2	<	<	X
ejpam-4730	30	3	∞	∞	PROPN
ejpam-4730	30	4	)	)	PUNCT
ejpam-4730	31	1	and	and	CCONJ
ejpam-4730	31	2	obtained	obtain	VERB
ejpam-4730	31	3	the	the	DET
ejpam-4730	31	4	asymptotically	asymptotically	ADV
ejpam-4730	31	5	sharp	sharp	ADJ
ejpam-4730	31	6	inequalities	inequality	NOUN
ejpam-4730	31	7	of	of	ADP
ejpam-4730	31	8	jackson	jackson	PROPN
ejpam-4730	31	9	type	type	PROPN
ejpam-4730	31	10	that	that	PRON
ejpam-4730	31	11	connect	connect	VERB
ejpam-4730	31	12	the	the	DET
ejpam-4730	31	13	best	good	ADJ
ejpam-4730	31	14	polynomial	polynomial	ADJ
ejpam-4730	31	15	approximations	approximation	NOUN
ejpam-4730	31	16	with	with	ADP
ejpam-4730	31	17	modules	module	NOUN
ejpam-4730	31	18	of	of	ADP
ejpam-4730	31	19	continuity	continuity	NOUN
ejpam-4730	31	20	of	of	ADP
ejpam-4730	31	21	functions	function	NOUN
ejpam-4730	31	22	f	f	PROPN
ejpam-4730	31	23	∈	∈	PROPN
ejpam-4730	31	24	sp	sp	PROPN
ejpam-4730	31	25	.	.	PUNCT
ejpam-4730	31	26	in	in	ADP
ejpam-4730	31	27	2000	2000	NUM
ejpam-4730	31	28	,	,	PUNCT
ejpam-4730	31	29	dragomir	dragomir	VERB
ejpam-4730	31	30	[	[	X
ejpam-4730	31	31	11	11	NUM
ejpam-4730	31	32	]	]	PUNCT
ejpam-4730	31	33	presented	present	VERB
ejpam-4730	31	34	some	some	DET
ejpam-4730	31	35	results	result	NOUN
ejpam-4730	31	36	about	about	ADP
ejpam-4730	31	37	the	the	DET
ejpam-4730	31	38	development	development	NOUN
ejpam-4730	31	39	of	of	ADP
ejpam-4730	31	40	methods	method	NOUN
ejpam-4730	31	41	for	for	ADP
ejpam-4730	31	42	solving	solve	VERB
ejpam-4730	31	43	approximation	approximation	NOUN
ejpam-4730	31	44	problems	problem	NOUN
ejpam-4730	31	45	using	use	VERB
ejpam-4730	31	46	sets	set	NOUN
ejpam-4730	31	47	in	in	ADP
ejpam-4730	31	48	normed	normed	ADJ
ejpam-4730	31	49	linear	linear	PROPN
ejpam-4730	31	50	spaces	space	NOUN
ejpam-4730	31	51	.	.	PUNCT
ejpam-4730	32	1	approximation	approximation	NOUN
ejpam-4730	32	2	of	of	ADP
ejpam-4730	32	3	both	both	DET
ejpam-4730	32	4	real	real	ADJ
ejpam-4730	32	5	functions	function	NOUN
ejpam-4730	32	6	and	and	CCONJ
ejpam-4730	32	7	real	real	ADJ
ejpam-4730	32	8	data	datum	NOUN
ejpam-4730	32	9	is	be	AUX
ejpam-4730	32	10	considered	consider	VERB
ejpam-4730	32	11	by	by	ADP
ejpam-4730	32	12	elumalai	elumalai	NOUN
ejpam-4730	32	13	and	and	CCONJ
ejpam-4730	32	14	vijayaragavan	vijayaragavan	NOUN
ejpam-4730	32	15	in	in	ADP
ejpam-4730	32	16	2008	2008	NUM
ejpam-4730	32	17	and	and	CCONJ
ejpam-4730	32	18	2009	2009	NUM
ejpam-4730	32	19	[	[	X
ejpam-4730	32	20	12	12	NUM
ejpam-4730	32	21	,	,	PUNCT
ejpam-4730	32	22	13	13	NUM
ejpam-4730	32	23	]	]	PUNCT
ejpam-4730	32	24	.	.	PUNCT
ejpam-4730	33	1	in	in	ADP
ejpam-4730	33	2	2012	2012	NUM
ejpam-4730	33	3	,	,	PUNCT
ejpam-4730	33	4	a	a	DET
ejpam-4730	33	5	construction	construction	NOUN
ejpam-4730	33	6	of	of	ADP
ejpam-4730	33	7	some	some	DET
ejpam-4730	33	8	characterizations	characterization	NOUN
ejpam-4730	33	9	of	of	ADP
ejpam-4730	33	10	best	good	ADJ
ejpam-4730	33	11	approximation	approximation	NOUN
ejpam-4730	33	12	in	in	ADP
ejpam-4730	33	13	2	2	NUM
ejpam-4730	33	14	-	-	PUNCT
ejpam-4730	33	15	normed	norme	VERB
ejpam-4730	33	16	space	space	NOUN
ejpam-4730	33	17	was	be	AUX
ejpam-4730	33	18	studied	study	VERB
ejpam-4730	33	19	by	by	ADP
ejpam-4730	33	20	dominic	dominic	PROPN
ejpam-4730	33	21	[	[	X
ejpam-4730	33	22	10	10	NUM
ejpam-4730	33	23	]	]	PUNCT
ejpam-4730	33	24	.	.	PUNCT
ejpam-4730	34	1	in	in	ADP
ejpam-4730	34	2	2013	2013	NUM
ejpam-4730	34	3	,	,	PUNCT
ejpam-4730	34	4	markandeya	markandeya	NOUN
ejpam-4730	34	5	and	and	CCONJ
ejpam-4730	34	6	bharathi	bharathi	PROPN
ejpam-4730	34	7	proved	prove	VERB
ejpam-4730	34	8	some	some	DET
ejpam-4730	34	9	results	result	NOUN
ejpam-4730	34	10	of	of	ADP
ejpam-4730	34	11	bbest	bb	ADJ
ejpam-4730	34	12	approximation	approximation	NOUN
ejpam-4730	34	13	in	in	ADP
ejpam-4730	34	14	uniformly	uniformly	ADV
ejpam-4730	34	15	2	2	NUM
ejpam-4730	34	16	-	-	PUNCT
ejpam-4730	34	17	normed	norme	VERB
ejpam-4730	34	18	space	space	NOUN
ejpam-4730	34	19	[	[	X
ejpam-4730	34	20	16	16	NUM
ejpam-4730	34	21	]	]	PUNCT
ejpam-4730	34	22	.	.	PUNCT
ejpam-4730	35	1	the	the	DET
ejpam-4730	35	2	concept	concept	NOUN
ejpam-4730	35	3	of	of	ADP
ejpam-4730	35	4	best	good	ADJ
ejpam-4730	35	5	approximation	approximation	NOUN
ejpam-4730	35	6	in	in	ADP
ejpam-4730	35	7	2	2	NUM
ejpam-4730	35	8	-	-	PUNCT
ejpam-4730	35	9	normed	norme	VERB
ejpam-4730	35	10	space	space	NOUN
ejpam-4730	35	11	along	along	ADP
ejpam-4730	35	12	with	with	ADP
ejpam-4730	35	13	the	the	DET
ejpam-4730	35	14	concept	concept	NOUN
ejpam-4730	35	15	of	of	ADP
ejpam-4730	35	16	orthogonality	orthogonality	NOUN
ejpam-4730	35	17	in	in	ADP
ejpam-4730	35	18	the	the	DET
ejpam-4730	35	19	same	same	ADJ
ejpam-4730	35	20	space	space	NOUN
ejpam-4730	35	21	were	be	AUX
ejpam-4730	35	22	presented	present	VERB
ejpam-4730	35	23	and	and	CCONJ
ejpam-4730	35	24	discussed	discuss	VERB
ejpam-4730	35	25	in	in	ADP
ejpam-4730	35	26	[	[	X
ejpam-4730	35	27	14	14	NUM
ejpam-4730	35	28	,	,	PUNCT
ejpam-4730	35	29	19	19	NUM
ejpam-4730	35	30	]	]	PUNCT
ejpam-4730	35	31	.	.	PUNCT
ejpam-4730	36	1	the	the	DET
ejpam-4730	36	2	following	follow	VERB
ejpam-4730	36	3	fundamental	fundamental	ADJ
ejpam-4730	36	4	direct	direct	ADJ
ejpam-4730	36	5	estimates	estimate	NOUN
ejpam-4730	36	6	prerogative	prerogative	VERB
ejpam-4730	36	7	to	to	PART
ejpam-4730	36	8	jackson	jackson	PROPN
ejpam-4730	37	1	[	[	X
ejpam-4730	37	2	7	7	NUM
ejpam-4730	37	3	,	,	PUNCT
ejpam-4730	37	4	15	15	NUM
ejpam-4730	37	5	]	]	AUX
ejpam-4730	37	6	assure	assure	VERB
ejpam-4730	37	7	that	that	PRON
ejpam-4730	37	8	ek(f)lp[a	ek(f)lp[a	NOUN
ejpam-4730	37	9	,	,	PUNCT
ejpam-4730	37	10	b	b	NOUN
ejpam-4730	37	11	]	]	X
ejpam-4730	37	12	converges	converge	VERB
ejpam-4730	37	13	to	to	ADP
ejpam-4730	37	14	zero	zero	NUM
ejpam-4730	37	15	much	much	ADV
ejpam-4730	37	16	faster	fast	ADV
ejpam-4730	37	17	when	when	SCONJ
ejpam-4730	37	18	f	f	PROPN
ejpam-4730	37	19	is	be	AUX
ejpam-4730	37	20	smoothness	smoothness	ADJ
ejpam-4730	37	21	.	.	PUNCT
ejpam-4730	38	1	the	the	DET
ejpam-4730	38	2	theory	theory	NOUN
ejpam-4730	38	3	of	of	ADP
ejpam-4730	38	4	approximation	approximation	NOUN
ejpam-4730	38	5	has	have	AUX
ejpam-4730	38	6	been	be	AUX
ejpam-4730	38	7	studied	study	VERB
ejpam-4730	38	8	by	by	ADP
ejpam-4730	38	9	many	many	ADJ
ejpam-4730	38	10	researchers	researcher	NOUN
ejpam-4730	38	11	and	and	CCONJ
ejpam-4730	38	12	applied	apply	VERB
ejpam-4730	38	13	in	in	ADP
ejpam-4730	38	14	various	various	ADJ
ejpam-4730	38	15	fields	field	NOUN
ejpam-4730	38	16	.	.	PUNCT
ejpam-4730	39	1	auad	auad	NOUN
ejpam-4730	39	2	(	(	PUNCT
ejpam-4730	39	3	2019	2019	NUM
ejpam-4730	39	4	)	)	PUNCT
ejpam-4730	39	5	investigated	investigate	VERB
ejpam-4730	39	6	the	the	DET
ejpam-4730	39	7	best	good	ADJ
ejpam-4730	39	8	simultaneous	simultaneous	ADJ
ejpam-4730	39	9	approximation	approximation	NOUN
ejpam-4730	39	10	of	of	ADP
ejpam-4730	39	11	unbounded	unbounded	ADJ
ejpam-4730	39	12	functions	function	NOUN
ejpam-4730	39	13	in	in	ADP
ejpam-4730	39	14	weighted	weighted	ADJ
ejpam-4730	39	15	space	space	NOUN
ejpam-4730	39	16	using	use	VERB
ejpam-4730	39	17	two	two	NUM
ejpam-4730	39	18	different	different	ADJ
ejpam-4730	39	19	definitions	definition	NOUN
ejpam-4730	39	20	and	and	CCONJ
ejpam-4730	39	21	established	establish	VERB
ejpam-4730	39	22	the	the	DET
ejpam-4730	39	23	relationship	relationship	NOUN
ejpam-4730	39	24	between	between	ADP
ejpam-4730	39	25	best	good	ADJ
ejpam-4730	39	26	approximation	approximation	NOUN
ejpam-4730	39	27	and	and	CCONJ
ejpam-4730	39	28	best	good	ADJ
ejpam-4730	39	29	simultaneous	simultaneous	ADJ
ejpam-4730	39	30	approximation	approximation	NOUN
ejpam-4730	39	31	[	[	X
ejpam-4730	39	32	8	8	NUM
ejpam-4730	39	33	]	]	PUNCT
ejpam-4730	39	34	.	.	PUNCT
ejpam-4730	40	1	in	in	ADP
ejpam-4730	40	2	2021	2021	NUM
ejpam-4730	40	3	,	,	PUNCT
ejpam-4730	40	4	auad	auad	VERB
ejpam-4730	40	5	et	et	PROPN
ejpam-4730	40	6	al	al	PROPN
ejpam-4730	40	7	.	.	PROPN
ejpam-4730	40	8	discussed	discuss	VERB
ejpam-4730	40	9	the	the	DET
ejpam-4730	40	10	algebraic	algebraic	ADJ
ejpam-4730	40	11	polynomial	polynomial	NOUN
ejpam-4730	40	12	’s	’s	PART
ejpam-4730	40	13	best	good	ADJ
ejpam-4730	40	14	approximation	approximation	NOUN
ejpam-4730	40	15	of	of	ADP
ejpam-4730	40	16	unbounded	unbounded	ADJ
ejpam-4730	40	17	functions	function	NOUN
ejpam-4730	40	18	in	in	ADP
ejpam-4730	40	19	weighted	weight	VERB
ejpam-4730	40	20	space	space	NOUN
ejpam-4730	40	21	and	and	CCONJ
ejpam-4730	40	22	obtained	obtain	VERB
ejpam-4730	40	23	sharp	sharp	ADJ
ejpam-4730	40	24	direct	direct	ADJ
ejpam-4730	40	25	inequality	inequality	NOUN
ejpam-4730	40	26	of	of	ADP
ejpam-4730	40	27	algebraic	algebraic	ADJ
ejpam-4730	40	28	approximation	approximation	NOUN
ejpam-4730	41	1	[	[	X
ejpam-4730	41	2	6	6	NUM
ejpam-4730	41	3	]	]	PUNCT
ejpam-4730	41	4	.	.	PUNCT
ejpam-4730	42	1	ali	ali	PROPN
ejpam-4730	42	2	a.	a.	PROPN
ejpam-4730	42	3	a.	a.	PROPN
ejpam-4730	42	4	auad	auad	PROPN
ejpam-4730	42	5	,	,	PUNCT
ejpam-4730	42	6	m.	m.	NOUN
ejpam-4730	42	7	a.	a.	NOUN
ejpam-4730	42	8	hilal	hilal	PROPN
ejpam-4730	42	9	,	,	PUNCT
ejpam-4730	42	10	n.	n.	PROPN
ejpam-4730	42	11	s.	s.	PROPN
ejpam-4730	42	12	khalaf	khalaf	PROPN
ejpam-4730	42	13	/	/	SYM
ejpam-4730	42	14	eur	eur	PROPN
ejpam-4730	42	15	.	.	PUNCT
ejpam-4730	43	1	j.	j.	PROPN
ejpam-4730	43	2	pure	pure	PROPN
ejpam-4730	43	3	appl	appl	PROPN
ejpam-4730	43	4	.	.	PROPN
ejpam-4730	43	5	math	math	PROPN
ejpam-4730	43	6	,	,	PUNCT
ejpam-4730	43	7	16	16	NUM
ejpam-4730	43	8	(	(	PUNCT
ejpam-4730	43	9	2	2	NUM
ejpam-4730	43	10	)	)	PUNCT
ejpam-4730	43	11	(	(	PUNCT
ejpam-4730	43	12	2023	2023	NUM
ejpam-4730	43	13	)	)	PUNCT
ejpam-4730	43	14	,	,	PUNCT
ejpam-4730	43	15	944	944	NUM
ejpam-4730	43	16	-	-	SYM
ejpam-4730	43	17	952	952	NUM
ejpam-4730	43	18	946	946	NUM
ejpam-4730	43	19	and	and	CCONJ
ejpam-4730	43	20	pales	pale	NOUN
ejpam-4730	43	21	(	(	PUNCT
ejpam-4730	43	22	2022	2022	NUM
ejpam-4730	43	23	)	)	PUNCT
ejpam-4730	43	24	derived	derive	VERB
ejpam-4730	43	25	an	an	DET
ejpam-4730	43	26	extension	extension	NOUN
ejpam-4730	43	27	of	of	ADP
ejpam-4730	43	28	the	the	DET
ejpam-4730	43	29	taylor	taylor	PROPN
ejpam-4730	43	30	theorem	theorem	VERB
ejpam-4730	43	31	related	relate	VERB
ejpam-4730	43	32	to	to	ADP
ejpam-4730	43	33	linear	linear	VERB
ejpam-4730	43	34	differential	differential	ADJ
ejpam-4730	43	35	operators	operator	NOUN
ejpam-4730	43	36	with	with	ADP
ejpam-4730	43	37	constant	constant	ADJ
ejpam-4730	43	38	coefficients	coefficient	NOUN
ejpam-4730	43	39	and	and	CCONJ
ejpam-4730	43	40	exponential	exponential	ADJ
ejpam-4730	43	41	polynomials	polynomial	NOUN
ejpam-4730	43	42	,	,	PUNCT
ejpam-4730	43	43	including	include	VERB
ejpam-4730	43	44	the	the	DET
ejpam-4730	43	45	integral	integral	ADJ
ejpam-4730	43	46	remainder	remainder	NOUN
ejpam-4730	43	47	terms	term	NOUN
ejpam-4730	43	48	and	and	CCONJ
ejpam-4730	43	49	mean	mean	ADJ
ejpam-4730	43	50	value	value	NOUN
ejpam-4730	43	51	type	type	NOUN
ejpam-4730	43	52	theorems	theorem	NOUN
ejpam-4730	44	1	[	[	X
ejpam-4730	44	2	5	5	NUM
ejpam-4730	44	3	]	]	PUNCT
ejpam-4730	44	4	.	.	PUNCT
ejpam-4730	45	1	approximation	approximation	NOUN
ejpam-4730	45	2	theory	theory	NOUN
ejpam-4730	45	3	is	be	AUX
ejpam-4730	45	4	very	very	ADV
ejpam-4730	45	5	useful	useful	ADJ
ejpam-4730	45	6	in	in	ADP
ejpam-4730	45	7	numerical	numerical	ADJ
ejpam-4730	45	8	analysis	analysis	NOUN
ejpam-4730	45	9	,	,	PUNCT
ejpam-4730	45	10	especially	especially	ADV
ejpam-4730	45	11	when	when	SCONJ
ejpam-4730	45	12	solving	solve	VERB
ejpam-4730	45	13	nonlinear	nonlinear	ADJ
ejpam-4730	45	14	equations	equation	NOUN
ejpam-4730	45	15	[	[	X
ejpam-4730	45	16	1	1	NUM
ejpam-4730	45	17	]	]	PUNCT
ejpam-4730	45	18	.	.	PUNCT
ejpam-4730	46	1	approximation	approximation	NOUN
ejpam-4730	46	2	theory	theory	NOUN
ejpam-4730	46	3	can	can	AUX
ejpam-4730	46	4	be	be	AUX
ejpam-4730	46	5	seen	see	VERB
ejpam-4730	46	6	also	also	ADV
ejpam-4730	46	7	in	in	ADP
ejpam-4730	46	8	several	several	ADJ
ejpam-4730	46	9	mathematical	mathematical	ADJ
ejpam-4730	46	10	techniques	technique	NOUN
ejpam-4730	46	11	,	,	PUNCT
ejpam-4730	46	12	such	such	ADJ
ejpam-4730	46	13	as	as	ADP
ejpam-4730	46	14	finite	finite	ADJ
ejpam-4730	46	15	element	element	NOUN
ejpam-4730	46	16	and	and	CCONJ
ejpam-4730	46	17	finite	finite	VERB
ejpam-4730	46	18	differences	difference	NOUN
ejpam-4730	46	19	[	[	X
ejpam-4730	46	20	3	3	NUM
ejpam-4730	46	21	]	]	PUNCT
ejpam-4730	46	22	.	.	PUNCT
ejpam-4730	47	1	furthermore	furthermore	ADV
ejpam-4730	47	2	,	,	PUNCT
ejpam-4730	47	3	other	other	ADJ
ejpam-4730	47	4	applications	application	NOUN
ejpam-4730	47	5	of	of	ADP
ejpam-4730	47	6	approximation	approximation	NOUN
ejpam-4730	47	7	theory	theory	NOUN
ejpam-4730	47	8	in	in	ADP
ejpam-4730	47	9	stability	stability	NOUN
ejpam-4730	47	10	and	and	CCONJ
ejpam-4730	47	11	thermal	thermal	ADJ
ejpam-4730	47	12	science	science	NOUN
ejpam-4730	47	13	can	can	AUX
ejpam-4730	47	14	be	be	AUX
ejpam-4730	47	15	found	find	VERB
ejpam-4730	47	16	in	in	ADP
ejpam-4730	47	17	several	several	ADJ
ejpam-4730	47	18	studies	study	NOUN
ejpam-4730	47	19	[	[	X
ejpam-4730	47	20	2	2	NUM
ejpam-4730	47	21	,	,	PUNCT
ejpam-4730	47	22	4	4	NUM
ejpam-4730	47	23	,	,	PUNCT
ejpam-4730	47	24	17	17	NUM
ejpam-4730	47	25	,	,	PUNCT
ejpam-4730	47	26	18	18	NUM
ejpam-4730	47	27	]	]	PUNCT
ejpam-4730	47	28	.	.	PUNCT
ejpam-4730	48	1	to	to	PART
ejpam-4730	48	2	have	have	VERB
ejpam-4730	48	3	a	a	DET
ejpam-4730	48	4	basic	basic	ADJ
ejpam-4730	48	5	and	and	CCONJ
ejpam-4730	48	6	historical	historical	ADJ
ejpam-4730	48	7	background	background	NOUN
ejpam-4730	48	8	about	about	ADP
ejpam-4730	48	9	these	these	DET
ejpam-4730	48	10	direct	direct	ADJ
ejpam-4730	48	11	theorems	theorem	NOUN
ejpam-4730	48	12	,	,	PUNCT
ejpam-4730	48	13	we	we	PRON
ejpam-4730	48	14	start	start	VERB
ejpam-4730	48	15	by	by	ADP
ejpam-4730	48	16	presenting	present	VERB
ejpam-4730	48	17	the	the	DET
ejpam-4730	48	18	following	following	NOUN
ejpam-4730	48	19	.	.	PUNCT
ejpam-4730	49	1	for	for	ADP
ejpam-4730	49	2	all	all	DET
ejpam-4730	49	3	f	f	PROPN
ejpam-4730	49	4	∈	∈	PROPN
ejpam-4730	49	5	lp[a	lp[a	PROPN
ejpam-4730	49	6	,	,	PUNCT
ejpam-4730	49	7	b	b	NOUN
ejpam-4730	49	8	]	]	PUNCT
ejpam-4730	49	9	and	and	CCONJ
ejpam-4730	49	10	k	k	PROPN
ejpam-4730	49	11	∈	∈	PROPN
ejpam-4730	49	12	n	n	CCONJ
ejpam-4730	49	13	,	,	PUNCT
ejpam-4730	49	14	the	the	DET
ejpam-4730	49	15	direct	direct	ADJ
ejpam-4730	49	16	theorem	theorem	NOUN
ejpam-4730	49	17	in	in	ADP
ejpam-4730	49	18	a	a	DET
ejpam-4730	49	19	bounded	bounded	ADJ
ejpam-4730	49	20	space	space	NOUN
ejpam-4730	49	21	can	can	AUX
ejpam-4730	49	22	be	be	AUX
ejpam-4730	49	23	represented	represent	VERB
ejpam-4730	49	24	as	as	ADP
ejpam-4730	49	25	ek(f	ek(f	NOUN
ejpam-4730	49	26	,	,	PUNCT
ejpam-4730	49	27	ξ)lp[a	ξ)lp[a	NOUN
ejpam-4730	49	28	,	,	PUNCT
ejpam-4730	49	29	b	b	X
ejpam-4730	49	30	]	]	PUNCT
ejpam-4730	49	31	≤	≤	NUM
ejpam-4730	49	32	c(k)µk(f	c(k)µk(f	NOUN
ejpam-4730	49	33	,	,	PUNCT
ejpam-4730	49	34	ξ)lp[a	ξ)lp[a	NOUN
ejpam-4730	49	35	,	,	PUNCT
ejpam-4730	49	36	b	b	NOUN
ejpam-4730	49	37	]	]	X
ejpam-4730	49	38	;	;	PUNCT
ejpam-4730	49	39	ξ	ξ	X
ejpam-4730	49	40	=	=	SYM
ejpam-4730	49	41	1	1	NUM
ejpam-4730	49	42	k	k	NOUN
ejpam-4730	49	43	,	,	PUNCT
ejpam-4730	49	44	where	where	SCONJ
ejpam-4730	49	45	c	c	NOUN
ejpam-4730	49	46	is	be	AUX
ejpam-4730	49	47	a	a	DET
ejpam-4730	49	48	positive	positive	ADJ
ejpam-4730	49	49	constant	constant	ADJ
ejpam-4730	49	50	depending	depend	VERB
ejpam-4730	49	51	on	on	ADP
ejpam-4730	49	52	k.	k.	PROPN
ejpam-4730	49	53	also	also	ADV
ejpam-4730	49	54	,	,	PUNCT
ejpam-4730	49	55	if	if	SCONJ
ejpam-4730	49	56	f	f	PROPN
ejpam-4730	49	57	∈	∈	PROPN
ejpam-4730	49	58	lp[a	lp[a	PROPN
ejpam-4730	49	59	,	,	PUNCT
ejpam-4730	49	60	b	b	NOUN
ejpam-4730	49	61	]	]	X
ejpam-4730	49	62	has	have	VERB
ejpam-4730	49	63	k	k	PROPN
ejpam-4730	49	64	th	th	X
ejpam-4730	49	65	derivative	derivative	ADJ
ejpam-4730	49	66	f	f	X
ejpam-4730	49	67	(	(	PUNCT
ejpam-4730	49	68	k	k	NOUN
ejpam-4730	49	69	)	)	PUNCT
ejpam-4730	49	70	for	for	ADP
ejpam-4730	49	71	some	some	DET
ejpam-4730	49	72	k	k	PROPN
ejpam-4730	49	73	∈	∈	PROPN
ejpam-4730	49	74	n	n	CCONJ
ejpam-4730	49	75	,	,	PUNCT
ejpam-4730	49	76	then	then	ADV
ejpam-4730	49	77	ek(f	ek(f	X
ejpam-4730	49	78	,	,	PUNCT
ejpam-4730	49	79	ξ)lp[a	ξ)lp[a	NOUN
ejpam-4730	49	80	,	,	PUNCT
ejpam-4730	49	81	b	b	NOUN
ejpam-4730	49	82	]	]	PUNCT
ejpam-4730	49	83	≤	≤	NUM
ejpam-4730	49	84	c(k)µk	c(k)µk	CCONJ
ejpam-4730	49	85	(	(	PUNCT
ejpam-4730	49	86	f	f	PROPN
ejpam-4730	49	87	(	(	PUNCT
ejpam-4730	49	88	k	k	NOUN
ejpam-4730	49	89	)	)	PUNCT
ejpam-4730	49	90	,	,	PUNCT
ejpam-4730	49	91	ξ	ξ	PROPN
ejpam-4730	49	92	)	)	PUNCT
ejpam-4730	49	93	lp[a	lp[a	PROPN
ejpam-4730	49	94	,	,	PUNCT
ejpam-4730	49	95	b	b	NOUN
ejpam-4730	49	96	]	]	PUNCT
ejpam-4730	49	97	;	;	PUNCT
ejpam-4730	49	98	ξ	ξ	X
ejpam-4730	49	99	=	=	SYM
ejpam-4730	49	100	1	1	NUM
ejpam-4730	49	101	k	k	NOUN
ejpam-4730	49	102	.	.	PUNCT
ejpam-4730	50	1	the	the	DET
ejpam-4730	50	2	fourier	fourier	PROPN
ejpam-4730	50	3	series	series	PROPN
ejpam-4730	50	4	expansion	expansion	NOUN
ejpam-4730	50	5	is	be	AUX
ejpam-4730	50	6	given	give	VERB
ejpam-4730	50	7	as	as	ADP
ejpam-4730	50	8	g(x	g(x	NOUN
ejpam-4730	50	9	)	)	PUNCT
ejpam-4730	51	1	=	=	NOUN
ejpam-4730	51	2	∑∞	∑∞	NOUN
ejpam-4730	51	3	i=−∞	i=−∞	NOUN
ejpam-4730	51	4	g(i)eijx	g(i)eijx	NUM
ejpam-4730	51	5	,	,	PUNCT
ejpam-4730	51	6	with	with	ADP
ejpam-4730	51	7	its	its	PRON
ejpam-4730	51	8	fourier	fourier	NOUN
ejpam-4730	51	9	coefficients	coefficient	NOUN
ejpam-4730	51	10	g(i	g(i	NOUN
ejpam-4730	51	11	)	)	PUNCT
ejpam-4730	51	12	=	=	SYM
ejpam-4730	52	1	1	1	NUM
ejpam-4730	52	2	2π	2π	NUM
ejpam-4730	52	3	∫	∫	PROPN
ejpam-4730	53	1	π	π	PROPN
ejpam-4730	53	2	−π	−π	PROPN
ejpam-4730	53	3	g(x)e	g(x)e	PROPN
ejpam-4730	53	4	ijxdx	ijxdx	PROPN
ejpam-4730	53	5	.	.	PUNCT
ejpam-4730	54	1	if	if	SCONJ
ejpam-4730	54	2	ϑ	ϑ	X
ejpam-4730	54	3	:	:	PUNCT
ejpam-4730	54	4	r	r	NOUN
ejpam-4730	54	5	→	→	SYM
ejpam-4730	54	6	r	r	NOUN
ejpam-4730	54	7	is	be	AUX
ejpam-4730	54	8	a	a	DET
ejpam-4730	54	9	continuous	continuous	ADJ
ejpam-4730	54	10	function	function	NOUN
ejpam-4730	54	11	,	,	PUNCT
ejpam-4730	54	12	then	then	ADV
ejpam-4730	54	13	we	we	PRON
ejpam-4730	54	14	define	define	VERB
ejpam-4730	54	15	the	the	DET
ejpam-4730	54	16	convolution	convolution	NOUN
ejpam-4730	54	17	function	function	NOUN
ejpam-4730	54	18	(	(	PUNCT
ejpam-4730	54	19	g	g	PROPN
ejpam-4730	54	20	∗	∗	NOUN
ejpam-4730	54	21	f)(ϑ	f)(ϑ	NUM
ejpam-4730	54	22	;	;	PUNCT
ejpam-4730	54	23	.	.	PUNCT
ejpam-4730	54	24	)	)	PUNCT
ejpam-4730	55	1	by	by	ADP
ejpam-4730	55	2	(	(	PUNCT
ejpam-4730	55	3	g	g	PROPN
ejpam-4730	55	4	∗	∗	NOUN
ejpam-4730	55	5	f)(ϑ;x	f)(ϑ;x	NOUN
ejpam-4730	55	6	)	)	PUNCT
ejpam-4730	55	7	=	=	SYM
ejpam-4730	55	8	1	1	NUM
ejpam-4730	55	9	2π	2π	NUM
ejpam-4730	55	10	∫	∫	PROPN
ejpam-4730	56	1	π	π	PROPN
ejpam-4730	56	2	−π	−π	PROPN
ejpam-4730	56	3	g(x)fϑ(x)f(x)dx	g(x)fϑ(x)f(x)dx	PROPN
ejpam-4730	56	4	,	,	PUNCT
ejpam-4730	56	5	x	x	PUNCT
ejpam-4730	56	6	∈	∈	PROPN
ejpam-4730	57	1	[	[	X
ejpam-4730	57	2	−π	−π	PROPN
ejpam-4730	57	3	,	,	PUNCT
ejpam-4730	57	4	π	π	X
ejpam-4730	57	5	]	]	X
ejpam-4730	57	6	.	.	PUNCT
ejpam-4730	58	1	(	(	PUNCT
ejpam-4730	58	2	3	3	X
ejpam-4730	58	3	)	)	PUNCT
ejpam-4730	58	4	clearly	clearly	ADV
ejpam-4730	58	5	,	,	PUNCT
ejpam-4730	58	6	(	(	PUNCT
ejpam-4730	58	7	g	g	PROPN
ejpam-4730	58	8	∗	∗	NOUN
ejpam-4730	58	9	f)(ϑ	f)(ϑ	NUM
ejpam-4730	58	10	;	;	PUNCT
ejpam-4730	58	11	.	.	PUNCT
ejpam-4730	58	12	)	)	PUNCT
ejpam-4730	58	13	,	,	PUNCT
ejpam-4730	58	14	(	(	PUNCT
ejpam-4730	58	15	ϑ	ϑ	X
ejpam-4730	58	16	;	;	PUNCT
ejpam-4730	58	17	.	.	PUNCT
ejpam-4730	58	18	)	)	PUNCT
ejpam-4730	59	1	∈	∈	PROPN
ejpam-4730	59	2	lp	lp	NOUN
ejpam-4730	59	3	,	,	PUNCT
ejpam-4730	59	4	λ[−π	λ[−π	PROPN
ejpam-4730	59	5	,	,	PUNCT
ejpam-4730	59	6	π	π	X
ejpam-4730	59	7	]	]	X
ejpam-4730	59	8	with	with	ADP
ejpam-4730	59	9	the	the	DET
ejpam-4730	59	10	norm	norm	NOUN
ejpam-4730	59	11	∥(g	∥(g	NOUN
ejpam-4730	59	12	∗	∗	NOUN
ejpam-4730	59	13	f)(ϑ	f)(ϑ	NUM
ejpam-4730	59	14	;	;	PUNCT
ejpam-4730	59	15	.)∥lp	.)∥lp	NOUN
ejpam-4730	59	16	,	,	PUNCT
ejpam-4730	59	17	λ[−π	λ[−π	PROPN
ejpam-4730	59	18	,	,	PUNCT
ejpam-4730	59	19	π	π	PROPN
ejpam-4730	59	20	]	]	X
ejpam-4730	59	21	≤	≤	NUM
ejpam-4730	59	22	c∥g∥1∥f∥lp	c∥g∥1∥f∥lp	NOUN
ejpam-4730	59	23	,	,	PUNCT
ejpam-4730	59	24	λ[−π	λ[−π	PROPN
ejpam-4730	59	25	,	,	PUNCT
ejpam-4730	59	26	π	π	PROPN
ejpam-4730	59	27	]	]	X
ejpam-4730	59	28	,	,	PUNCT
ejpam-4730	59	29	where	where	SCONJ
ejpam-4730	59	30	c	c	NOUN
ejpam-4730	59	31	=	=	PUNCT
ejpam-4730	59	32	sup︸︷︷︸	sup︸︷︷︸	PROPN
ejpam-4730	59	33	x≤π	x≤π	PROPN
ejpam-4730	59	34	{	{	PUNCT
ejpam-4730	59	35	∥fϑ(x)∥lp	∥fϑ(x)∥lp	PROPN
ejpam-4730	59	36	,	,	PUNCT
ejpam-4730	59	37	λ[−π	λ[−π	PROPN
ejpam-4730	59	38	,	,	PUNCT
ejpam-4730	59	39	π	π	NOUN
ejpam-4730	59	40	]	]	PUNCT
ejpam-4730	59	41	}	}	PUNCT
ejpam-4730	59	42	.	.	PUNCT
ejpam-4730	60	1	let	let	VERB
ejpam-4730	60	2	l	l	NOUN
ejpam-4730	60	3	be	be	AUX
ejpam-4730	60	4	a	a	DET
ejpam-4730	60	5	natural	natural	ADJ
ejpam-4730	60	6	number	number	NOUN
ejpam-4730	60	7	,	,	PUNCT
ejpam-4730	60	8	g	g	PROPN
ejpam-4730	60	9	∈	∈	PROPN
ejpam-4730	60	10	lp	lp	NOUN
ejpam-4730	60	11	,	,	PUNCT
ejpam-4730	60	12	λ[−π	λ[−π	PROPN
ejpam-4730	60	13	,	,	PUNCT
ejpam-4730	60	14	π	π	X
ejpam-4730	60	15	]	]	PUNCT
ejpam-4730	60	16	and	and	CCONJ
ejpam-4730	60	17	consider	consider	VERB
ejpam-4730	60	18	the	the	DET
ejpam-4730	60	19	following	follow	VERB
ejpam-4730	60	20	linear	linear	ADJ
ejpam-4730	60	21	combination	combination	NOUN
ejpam-4730	60	22	of	of	ADP
ejpam-4730	60	23	the	the	DET
ejpam-4730	60	24	convolution	convolution	NOUN
ejpam-4730	60	25	functions	function	NOUN
ejpam-4730	60	26	(	(	PUNCT
ejpam-4730	60	27	g	g	NOUN
ejpam-4730	60	28	∗	∗	NOUN
ejpam-4730	60	29	i)i	i)i	NOUN
ejpam-4730	60	30	,	,	PUNCT
ejpam-4730	60	31	1	1	NUM
ejpam-4730	60	32	≤	≤	NUM
ejpam-4730	60	33	i	i	PRON
ejpam-4730	60	34	≤	≤	ADJ
ejpam-4730	60	35	l	l	NOUN
ejpam-4730	60	36	as	as	ADP
ejpam-4730	60	37	p	p	PROPN
ejpam-4730	60	38	(	(	PUNCT
ejpam-4730	60	39	g	g	NOUN
ejpam-4730	60	40	,	,	PUNCT
ejpam-4730	60	41	l	l	NOUN
ejpam-4730	60	42	)	)	PUNCT
ejpam-4730	60	43	=	=	PUNCT
ejpam-4730	61	1	σl	σl	ADP
ejpam-4730	61	2	i=1(−1)l+1	i=1(−1)l+1	PROPN
ejpam-4730	61	3	(	(	PUNCT
ejpam-4730	61	4	l	l	NOUN
ejpam-4730	61	5	i	i	NOUN
ejpam-4730	61	6	)	)	PUNCT
ejpam-4730	61	7	(	(	PUNCT
ejpam-4730	61	8	g	g	NOUN
ejpam-4730	61	9	∗	∗	X
ejpam-4730	61	10	i	i	NOUN
ejpam-4730	61	11	)	)	PUNCT
ejpam-4730	61	12	.	.	PUNCT
ejpam-4730	62	1	(	(	PUNCT
ejpam-4730	62	2	4	4	X
ejpam-4730	62	3	)	)	PUNCT
ejpam-4730	62	4	here	here	ADV
ejpam-4730	62	5	,	,	PUNCT
ejpam-4730	62	6	we	we	PRON
ejpam-4730	62	7	consider	consider	VERB
ejpam-4730	62	8	the	the	DET
ejpam-4730	62	9	generalized	generalize	VERB
ejpam-4730	62	10	jackson	jackson	PROPN
ejpam-4730	62	11	kernel	kernel	PROPN
ejpam-4730	62	12	given	give	VERB
ejpam-4730	62	13	by	by	ADP
ejpam-4730	62	14	jk	jk	PROPN
ejpam-4730	62	15	,	,	PUNCT
ejpam-4730	62	16	r(x	r(x	PROPN
ejpam-4730	62	17	)	)	PUNCT
ejpam-4730	62	18	=	=	SYM
ejpam-4730	63	1	ck	ck	PROPN
ejpam-4730	63	2	,	,	PUNCT
ejpam-4730	63	3	r	r	NOUN
ejpam-4730	63	4	(	(	PUNCT
ejpam-4730	63	5	sinkx	sinkx	NOUN
ejpam-4730	63	6	2	2	NUM
ejpam-4730	63	7	sinx	sinx	X
ejpam-4730	63	8	2	2	NUM
ejpam-4730	63	9	)	)	PUNCT
ejpam-4730	63	10	2r	2r	NUM
ejpam-4730	63	11	,	,	PUNCT
ejpam-4730	63	12	k	k	NOUN
ejpam-4730	63	13	,	,	PUNCT
ejpam-4730	63	14	r	r	NOUN
ejpam-4730	63	15	∈	∈	PROPN
ejpam-4730	63	16	n	n	CCONJ
ejpam-4730	63	17	,	,	PUNCT
ejpam-4730	63	18	where	where	SCONJ
ejpam-4730	63	19	the	the	DET
ejpam-4730	63	20	constant	constant	ADJ
ejpam-4730	63	21	ck	ck	NOUN
ejpam-4730	63	22	,	,	PUNCT
ejpam-4730	63	23	r	r	NOUN
ejpam-4730	63	24	>	>	X
ejpam-4730	63	25	0	0	NUM
ejpam-4730	63	26	is	be	AUX
ejpam-4730	63	27	taken	take	VERB
ejpam-4730	63	28	in	in	ADP
ejpam-4730	63	29	such	such	DET
ejpam-4730	63	30	a	a	DET
ejpam-4730	63	31	way	way	NOUN
ejpam-4730	63	32	that	that	PRON
ejpam-4730	63	33	jk	jk	PROPN
ejpam-4730	63	34	,	,	PUNCT
ejpam-4730	63	35	r(0	r(0	PROPN
ejpam-4730	63	36	)	)	PUNCT
ejpam-4730	63	37	=	=	SYM
ejpam-4730	64	1	1	1	NUM
ejpam-4730	64	2	π	π	SYM
ejpam-4730	64	3	∫	∫	PROPN
ejpam-4730	64	4	π	π	PROPN
ejpam-4730	64	5	0	0	PUNCT
ejpam-4730	64	6	jk	jk	PROPN
ejpam-4730	64	7	,	,	PUNCT
ejpam-4730	64	8	r(x)dx	r(x)dx	VERB
ejpam-4730	64	9	=	=	SYM
ejpam-4730	64	10	1	1	X
ejpam-4730	64	11	.	.	PUNCT
ejpam-4730	64	12	a.	a.	NOUN
ejpam-4730	64	13	a.	a.	NOUN
ejpam-4730	64	14	auad	auad	PROPN
ejpam-4730	64	15	,	,	PUNCT
ejpam-4730	64	16	m.	m.	NOUN
ejpam-4730	64	17	a.	a.	NOUN
ejpam-4730	64	18	hilal	hilal	PROPN
ejpam-4730	64	19	,	,	PUNCT
ejpam-4730	64	20	n.	n.	PROPN
ejpam-4730	64	21	s.	s.	PROPN
ejpam-4730	64	22	khalaf	khalaf	PROPN
ejpam-4730	64	23	/	/	SYM
ejpam-4730	64	24	eur	eur	PROPN
ejpam-4730	64	25	.	.	PUNCT
ejpam-4730	65	1	j.	j.	PROPN
ejpam-4730	65	2	pure	pure	PROPN
ejpam-4730	65	3	appl	appl	PROPN
ejpam-4730	65	4	.	.	PROPN
ejpam-4730	65	5	math	math	PROPN
ejpam-4730	65	6	,	,	PUNCT
ejpam-4730	65	7	16	16	NUM
ejpam-4730	65	8	(	(	PUNCT
ejpam-4730	65	9	2	2	NUM
ejpam-4730	65	10	)	)	PUNCT
ejpam-4730	65	11	(	(	PUNCT
ejpam-4730	65	12	2023	2023	NUM
ejpam-4730	65	13	)	)	PUNCT
ejpam-4730	65	14	,	,	PUNCT
ejpam-4730	65	15	944	944	NUM
ejpam-4730	65	16	-	-	SYM
ejpam-4730	65	17	952	952	NUM
ejpam-4730	65	18	947	947	NUM
ejpam-4730	65	19	and	and	CCONJ
ejpam-4730	65	20	jk,1(x	jk,1(x	PROPN
ejpam-4730	65	21	)	)	PUNCT
ejpam-4730	65	22	=	=	SYM
ejpam-4730	65	23	kk(x	kk(x	X
ejpam-4730	65	24	)	)	PUNCT
ejpam-4730	65	25	=	=	SYM
ejpam-4730	65	26	σk−l	σk−l	NOUN
ejpam-4730	65	27	i=1−k(1−	i=1−k(1−	NOUN
ejpam-4730	65	28	|i|	|i|	PROPN
ejpam-4730	65	29	k	k	PROPN
ejpam-4730	65	30	)	)	PUNCT
ejpam-4730	65	31	eix	eix	PROPN
ejpam-4730	65	32	(	(	PUNCT
ejpam-4730	65	33	5	5	NUM
ejpam-4730	65	34	)	)	PUNCT
ejpam-4730	65	35	is	be	AUX
ejpam-4730	65	36	called	call	VERB
ejpam-4730	65	37	the	the	DET
ejpam-4730	65	38	fejer	fejer	ADJ
ejpam-4730	65	39	kernel	kernel	PROPN
ejpam-4730	65	40	such	such	ADJ
ejpam-4730	65	41	that	that	SCONJ
ejpam-4730	65	42	jk	jk	PROPN
ejpam-4730	65	43	,	,	PUNCT
ejpam-4730	65	44	r(x	r(x	PROPN
ejpam-4730	65	45	)	)	PUNCT
ejpam-4730	65	46	=	=	SYM
ejpam-4730	65	47	ck	ck	PROPN
ejpam-4730	65	48	,	,	PUNCT
ejpam-4730	65	49	rfk(x	rfk(x	PROPN
ejpam-4730	65	50	)	)	PUNCT
ejpam-4730	65	51	is	be	AUX
ejpam-4730	65	52	a	a	DET
ejpam-4730	65	53	non	non	ADJ
ejpam-4730	65	54	-	-	ADJ
ejpam-4730	65	55	negative	negative	ADJ
ejpam-4730	65	56	trigonometric	trigonometric	ADJ
ejpam-4730	65	57	polynomial	polynomial	NOUN
ejpam-4730	65	58	of	of	ADP
ejpam-4730	65	59	degree	degree	NOUN
ejpam-4730	65	60	r(k	r(k	PROPN
ejpam-4730	65	61	−	−	PROPN
ejpam-4730	65	62	1	1	NUM
ejpam-4730	65	63	)	)	PUNCT
ejpam-4730	65	64	.	.	PUNCT
ejpam-4730	66	1	the	the	DET
ejpam-4730	66	2	main	main	ADJ
ejpam-4730	66	3	aim	aim	NOUN
ejpam-4730	66	4	of	of	ADP
ejpam-4730	66	5	this	this	DET
ejpam-4730	66	6	paper	paper	NOUN
ejpam-4730	66	7	is	be	AUX
ejpam-4730	66	8	to	to	PART
ejpam-4730	66	9	extend	extend	VERB
ejpam-4730	66	10	this	this	DET
ejpam-4730	66	11	results	result	NOUN
ejpam-4730	66	12	to	to	ADP
ejpam-4730	66	13	arbitrary	arbitrary	ADJ
ejpam-4730	66	14	weighted	weight	VERB
ejpam-4730	66	15	space	space	NOUN
ejpam-4730	66	16	lp	lp	NOUN
ejpam-4730	66	17	,	,	PUNCT
ejpam-4730	66	18	λ[−π	λ[−π	PROPN
ejpam-4730	66	19	,	,	PUNCT
ejpam-4730	66	20	π	π	X
ejpam-4730	66	21	]	]	X
ejpam-4730	66	22	and	and	CCONJ
ejpam-4730	66	23	in	in	ADP
ejpam-4730	66	24	particular	particular	ADJ
ejpam-4730	66	25	the	the	DET
ejpam-4730	66	26	space	space	NOUN
ejpam-4730	66	27	lp(x	lp(x	PUNCT
ejpam-4730	66	28	)	)	PUNCT
ejpam-4730	66	29	,	,	PUNCT
ejpam-4730	66	30	where	where	SCONJ
ejpam-4730	66	31	x	x	X
ejpam-4730	66	32	=	=	PUNCT
ejpam-4730	67	1	[	[	X
ejpam-4730	67	2	0	0	NUM
ejpam-4730	67	3	,	,	PUNCT
ejpam-4730	67	4	π	π	X
ejpam-4730	67	5	]	]	PUNCT
ejpam-4730	67	6	or	or	CCONJ
ejpam-4730	67	7	[	[	X
ejpam-4730	67	8	−1	−1	NOUN
ejpam-4730	67	9	,	,	PUNCT
ejpam-4730	67	10	1	1	NUM
ejpam-4730	67	11	]	]	PUNCT
ejpam-4730	67	12	,	,	PUNCT
ejpam-4730	67	13	1	1	NUM
ejpam-4730	67	14	≤	≤	NOUN
ejpam-4730	67	15	p	p	NOUN
ejpam-4730	67	16	<	<	X
ejpam-4730	67	17	∞.	∞.	PROPN
ejpam-4730	67	18	2	2	NUM
ejpam-4730	67	19	.	.	PUNCT
ejpam-4730	67	20	auxiliary	auxiliary	PROPN
ejpam-4730	67	21	lemmas	lemma	VERB
ejpam-4730	67	22	in	in	ADP
ejpam-4730	67	23	this	this	DET
ejpam-4730	67	24	section	section	NOUN
ejpam-4730	67	25	,	,	PUNCT
ejpam-4730	67	26	we	we	PRON
ejpam-4730	67	27	recall	recall	VERB
ejpam-4730	67	28	some	some	DET
ejpam-4730	67	29	lemmas	lemma	NOUN
ejpam-4730	67	30	which	which	PRON
ejpam-4730	67	31	we	we	PRON
ejpam-4730	67	32	will	will	AUX
ejpam-4730	67	33	need	need	VERB
ejpam-4730	67	34	in	in	ADP
ejpam-4730	67	35	our	our	PRON
ejpam-4730	67	36	main	main	ADJ
ejpam-4730	67	37	results	result	NOUN
ejpam-4730	67	38	.	.	PUNCT
ejpam-4730	68	1	lemma	lemma	PROPN
ejpam-4730	68	2	1	1	X
ejpam-4730	68	3	.	.	PUNCT
ejpam-4730	69	1	let	let	VERB
ejpam-4730	69	2	f	f	PROPN
ejpam-4730	69	3	∈	∈	PROPN
ejpam-4730	69	4	lp	lp	PROPN
ejpam-4730	69	5	,	,	PUNCT
ejpam-4730	69	6	λ[−π	λ[−π	PROPN
ejpam-4730	69	7	,	,	PUNCT
ejpam-4730	69	8	π	π	PROPN
ejpam-4730	69	9	]	]	X
ejpam-4730	69	10	,	,	PUNCT
ejpam-4730	69	11	1	1	NUM
ejpam-4730	69	12	≤	≤	NOUN
ejpam-4730	69	13	p	p	PRON
ejpam-4730	69	14	<	<	X
ejpam-4730	69	15	∞	∞	PROPN
ejpam-4730	69	16	and	and	CCONJ
ejpam-4730	69	17	k	k	PROPN
ejpam-4730	69	18	∈	∈	PROPN
ejpam-4730	69	19	n.	n.	NOUN
ejpam-4730	69	20	then	then	ADV
ejpam-4730	69	21	,	,	PUNCT
ejpam-4730	69	22	µk(f	µk(f	X
ejpam-4730	69	23	,	,	PUNCT
ejpam-4730	69	24	δ)lp	δ)lp	PROPN
ejpam-4730	69	25	,	,	PUNCT
ejpam-4730	69	26	λ[−π	λ[−π	PROPN
ejpam-4730	69	27	,	,	PUNCT
ejpam-4730	69	28	π	π	PROPN
ejpam-4730	69	29	]	]	X
ejpam-4730	69	30	≤	≤	NUM
ejpam-4730	69	31	ck∥f∥lp	ck∥f∥lp	NOUN
ejpam-4730	69	32	,	,	PUNCT
ejpam-4730	69	33	λ[−π	λ[−π	PROPN
ejpam-4730	69	34	,	,	PUNCT
ejpam-4730	69	35	π	π	X
ejpam-4730	69	36	]	]	X
ejpam-4730	69	37	,	,	PUNCT
ejpam-4730	69	38	where	where	SCONJ
ejpam-4730	69	39	ck	ck	PROPN
ejpam-4730	69	40	is	be	AUX
ejpam-4730	69	41	a	a	DET
ejpam-4730	69	42	positive	positive	ADJ
ejpam-4730	69	43	constant	constant	ADJ
ejpam-4730	69	44	depending	depend	VERB
ejpam-4730	69	45	on	on	ADP
ejpam-4730	69	46	k.	k.	PROPN
ejpam-4730	69	47	proof	proof	PROPN
ejpam-4730	69	48	.	.	PUNCT
ejpam-4730	70	1	we	we	PRON
ejpam-4730	70	2	have	have	VERB
ejpam-4730	70	3	∆k	∆k	PROPN
ejpam-4730	70	4	hf(x	hf(x	PART
ejpam-4730	70	5	)	)	PUNCT
ejpam-4730	70	6	=	=	PUNCT
ejpam-4730	71	1	∑k	∑k	PROPN
ejpam-4730	71	2	i=1	i=1	X
ejpam-4730	72	1	(	(	PUNCT
ejpam-4730	72	2	−1)k−i	−1)k−i	X
ejpam-4730	72	3	(	(	PUNCT
ejpam-4730	72	4	k	k	X
ejpam-4730	72	5	i	i	PROPN
ejpam-4730	72	6	)	)	PUNCT
ejpam-4730	72	7	f(x+	f(x+	CCONJ
ejpam-4730	72	8	ih	ih	NOUN
ejpam-4730	72	9	)	)	PUNCT
ejpam-4730	72	10	)	)	PUNCT
ejpam-4730	72	11	,	,	PUNCT
ejpam-4730	72	12	∥∆k	∥∆k	NOUN
ejpam-4730	72	13	hf(.)∥lp	hf(.)∥lp	NOUN
ejpam-4730	72	14	,	,	PUNCT
ejpam-4730	72	15	λ[−π	λ[−π	PROPN
ejpam-4730	72	16	,	,	PUNCT
ejpam-4730	72	17	π	π	X
ejpam-4730	72	18	]	]	X
ejpam-4730	72	19	=	=	PUNCT
ejpam-4730	73	1	∥	∥	X
ejpam-4730	73	2	∑k	∑k	PROPN
ejpam-4730	73	3	i=1	i=1	X
ejpam-4730	74	1	(	(	PUNCT
ejpam-4730	74	2	−1)k−i	−1)k−i	X
ejpam-4730	74	3	(	(	PUNCT
ejpam-4730	74	4	k	k	X
ejpam-4730	74	5	i	i	PROPN
ejpam-4730	74	6	)	)	PUNCT
ejpam-4730	74	7	f(x+	f(x+	CCONJ
ejpam-4730	74	8	ih))∥lp	ih))∥lp	PRON
ejpam-4730	74	9	,	,	PUNCT
ejpam-4730	74	10	λ[−π	λ[−π	PROPN
ejpam-4730	74	11	,	,	PUNCT
ejpam-4730	74	12	π	π	X
ejpam-4730	74	13	]	]	PUNCT
ejpam-4730	74	14	,	,	PUNCT
ejpam-4730	74	15	sup∥∆k	sup∥∆k	PROPN
ejpam-4730	74	16	hf(.)∥lp	hf(.)∥lp	PROPN
ejpam-4730	74	17	,	,	PUNCT
ejpam-4730	74	18	λ[−π	λ[−π	PROPN
ejpam-4730	74	19	,	,	PUNCT
ejpam-4730	74	20	π	π	PROPN
ejpam-4730	74	21	]	]	X
ejpam-4730	74	22	≤	≤	NUM
ejpam-4730	74	23	sup	sup	NOUN
ejpam-4730	74	24	{	{	PUNCT
ejpam-4730	74	25	k∑	k∑	NOUN
ejpam-4730	74	26	i=1	i=1	PROPN
ejpam-4730	74	27	(	(	PUNCT
ejpam-4730	74	28	−1)k−i	−1)k−i	X
ejpam-4730	74	29	(	(	PUNCT
ejpam-4730	74	30	k	k	NOUN
ejpam-4730	74	31	i	i	PROPN
ejpam-4730	74	32	)	)	PUNCT
ejpam-4730	74	33	∥f(.)∥lp	∥f(.)∥lp	PROPN
ejpam-4730	74	34	,	,	PUNCT
ejpam-4730	74	35	λ[−π	λ[−π	PROPN
ejpam-4730	74	36	,	,	PUNCT
ejpam-4730	74	37	π	π	NOUN
ejpam-4730	74	38	]	]	X
ejpam-4730	74	39	}	}	PUNCT
ejpam-4730	74	40	,	,	PUNCT
ejpam-4730	74	41	thus	thus	ADV
ejpam-4730	74	42	,	,	PUNCT
ejpam-4730	74	43	µk(f	µk(f	X
ejpam-4730	74	44	,	,	PUNCT
ejpam-4730	74	45	δ)lp	δ)lp	PROPN
ejpam-4730	74	46	,	,	PUNCT
ejpam-4730	74	47	λ[−π	λ[−π	PROPN
ejpam-4730	74	48	,	,	PUNCT
ejpam-4730	74	49	π	π	PROPN
ejpam-4730	74	50	]	]	X
ejpam-4730	74	51	≤	≤	NUM
ejpam-4730	74	52	max{sup	max{sup	NOUN
ejpam-4730	74	53	{	{	PUNCT
ejpam-4730	74	54	k∑	k∑	PROPN
ejpam-4730	74	55	i=1	i=1	PROPN
ejpam-4730	74	56	(	(	PUNCT
ejpam-4730	74	57	−1)k−i	−1)k−i	X
ejpam-4730	74	58	(	(	PUNCT
ejpam-4730	74	59	k	k	NOUN
ejpam-4730	74	60	i	i	PROPN
ejpam-4730	74	61	)	)	PUNCT
ejpam-4730	74	62	∥f(.)∥lp	∥f(.)∥lp	PROPN
ejpam-4730	74	63	,	,	PUNCT
ejpam-4730	74	64	λ[−π	λ[−π	PROPN
ejpam-4730	74	65	,	,	PUNCT
ejpam-4730	74	66	π	π	NOUN
ejpam-4730	74	67	]	]	X
ejpam-4730	74	68	}	}	PUNCT
ejpam-4730	74	69	.	.	PUNCT
ejpam-4730	75	1	now	now	ADV
ejpam-4730	75	2	,	,	PUNCT
ejpam-4730	75	3	we	we	PRON
ejpam-4730	75	4	can	can	AUX
ejpam-4730	75	5	take	take	VERB
ejpam-4730	75	6	that	that	DET
ejpam-4730	75	7	max{sup	max{sup	ADJ
ejpam-4730	75	8	{	{	PUNCT
ejpam-4730	75	9	k∑	k∑	PROPN
ejpam-4730	75	10	i=1	i=1	PROPN
ejpam-4730	76	1	(	(	PUNCT
ejpam-4730	76	2	−1)k−i	−1)k−i	X
ejpam-4730	76	3	(	(	PUNCT
ejpam-4730	76	4	k	k	NOUN
ejpam-4730	76	5	i	i	PROPN
ejpam-4730	76	6	)	)	PUNCT
ejpam-4730	76	7	}	}	PUNCT
ejpam-4730	76	8	}	}	PUNCT
ejpam-4730	76	9	≤	≤	ADV
ejpam-4730	76	10	ck	ck	INTJ
ejpam-4730	76	11	,	,	PUNCT
ejpam-4730	76	12	which	which	PRON
ejpam-4730	76	13	implies	imply	VERB
ejpam-4730	76	14	,	,	PUNCT
ejpam-4730	76	15	µk(f	µk(f	X
ejpam-4730	76	16	,	,	PUNCT
ejpam-4730	76	17	δ)lp	δ)lp	PROPN
ejpam-4730	76	18	,	,	PUNCT
ejpam-4730	76	19	λ[−π	λ[−π	PROPN
ejpam-4730	76	20	,	,	PUNCT
ejpam-4730	76	21	π	π	PROPN
ejpam-4730	76	22	]	]	X
ejpam-4730	76	23	≤	≤	NUM
ejpam-4730	76	24	ck∥f∥lp	ck∥f∥lp	NOUN
ejpam-4730	76	25	,	,	PUNCT
ejpam-4730	76	26	λ[−π	λ[−π	PROPN
ejpam-4730	76	27	,	,	PUNCT
ejpam-4730	76	28	π	π	NOUN
ejpam-4730	76	29	]	]	PUNCT
ejpam-4730	76	30	.	.	PUNCT
ejpam-4730	77	1	lemma	lemma	PROPN
ejpam-4730	77	2	2	2	X
ejpam-4730	77	3	.	.	PUNCT
ejpam-4730	78	1	let	let	VERB
ejpam-4730	78	2	f	f	PROPN
ejpam-4730	78	3	∈	∈	PROPN
ejpam-4730	78	4	lp	lp	PROPN
ejpam-4730	78	5	,	,	PUNCT
ejpam-4730	78	6	λ[−π	λ[−π	PROPN
ejpam-4730	78	7	,	,	PUNCT
ejpam-4730	78	8	π	π	PROPN
ejpam-4730	78	9	]	]	X
ejpam-4730	78	10	,	,	PUNCT
ejpam-4730	78	11	1	1	NUM
ejpam-4730	78	12	≤	≤	NOUN
ejpam-4730	79	1	p	p	X
ejpam-4730	79	2	<	<	X
ejpam-4730	79	3	∞	∞	PROPN
ejpam-4730	79	4	,	,	PUNCT
ejpam-4730	79	5	h	h	NOUN
ejpam-4730	79	6	>	>	X
ejpam-4730	79	7	0	0	PUNCT
ejpam-4730	79	8	and	and	CCONJ
ejpam-4730	79	9	r	r	PROPN
ejpam-4730	79	10	∈	∈	PROPN
ejpam-4730	79	11	n.	n.	NOUN
ejpam-4730	79	12	then	then	ADV
ejpam-4730	79	13	,	,	PUNCT
ejpam-4730	79	14	µr(f	µr(f	ADV
ejpam-4730	79	15	,	,	PUNCT
ejpam-4730	79	16	1	1	NUM
ejpam-4730	79	17	k	k	NOUN
ejpam-4730	79	18	)	)	PUNCT
ejpam-4730	79	19	lp	lp	NOUN
ejpam-4730	79	20	,	,	PUNCT
ejpam-4730	79	21	λ[−π	λ[−π	PROPN
ejpam-4730	79	22	,	,	PUNCT
ejpam-4730	79	23	π	π	X
ejpam-4730	79	24	]	]	X
ejpam-4730	79	25	≤	≤	X
ejpam-4730	79	26	ck	ck	PROPN
ejpam-4730	79	27	µr−k(f	µr−k(f	NOUN
ejpam-4730	79	28	,	,	PUNCT
ejpam-4730	79	29	1	1	NUM
ejpam-4730	79	30	k	k	NOUN
ejpam-4730	79	31	)	)	PUNCT
ejpam-4730	79	32	lp	lp	NOUN
ejpam-4730	79	33	,	,	PUNCT
ejpam-4730	79	34	λ[−π	λ[−π	PROPN
ejpam-4730	79	35	,	,	PUNCT
ejpam-4730	79	36	π	π	NOUN
ejpam-4730	79	37	]	]	X
ejpam-4730	79	38	.	.	PUNCT
ejpam-4730	80	1	a.	a.	PROPN
ejpam-4730	80	2	a.	a.	PROPN
ejpam-4730	80	3	auad	auad	PROPN
ejpam-4730	80	4	,	,	PUNCT
ejpam-4730	80	5	m.	m.	NOUN
ejpam-4730	80	6	a.	a.	NOUN
ejpam-4730	80	7	hilal	hilal	PROPN
ejpam-4730	80	8	,	,	PUNCT
ejpam-4730	80	9	n.	n.	PROPN
ejpam-4730	80	10	s.	s.	PROPN
ejpam-4730	80	11	khalaf	khalaf	PROPN
ejpam-4730	80	12	/	/	SYM
ejpam-4730	80	13	eur	eur	PROPN
ejpam-4730	80	14	.	.	PUNCT
ejpam-4730	81	1	j.	j.	PROPN
ejpam-4730	81	2	pure	pure	PROPN
ejpam-4730	81	3	appl	appl	PROPN
ejpam-4730	81	4	.	.	PROPN
ejpam-4730	81	5	math	math	PROPN
ejpam-4730	81	6	,	,	PUNCT
ejpam-4730	81	7	16	16	NUM
ejpam-4730	81	8	(	(	PUNCT
ejpam-4730	81	9	2	2	NUM
ejpam-4730	81	10	)	)	PUNCT
ejpam-4730	81	11	(	(	PUNCT
ejpam-4730	81	12	2023	2023	NUM
ejpam-4730	81	13	)	)	PUNCT
ejpam-4730	81	14	,	,	PUNCT
ejpam-4730	81	15	944	944	NUM
ejpam-4730	81	16	-	-	SYM
ejpam-4730	81	17	952	952	NUM
ejpam-4730	81	18	948	948	NUM
ejpam-4730	81	19	proof	proof	NOUN
ejpam-4730	81	20	.	.	PUNCT
ejpam-4730	82	1	we	we	PRON
ejpam-4730	82	2	have	have	VERB
ejpam-4730	82	3	∆k	∆k	PROPN
ejpam-4730	82	4	hf(x	hf(x	PART
ejpam-4730	82	5	)	)	PUNCT
ejpam-4730	82	6	=	=	SYM
ejpam-4730	83	1	∆r−1	∆r−1	PROPN
ejpam-4730	83	2	h	h	NOUN
ejpam-4730	83	3	(	(	PUNCT
ejpam-4730	83	4	∆1	∆1	NOUN
ejpam-4730	83	5	hf(x	hf(x	NOUN
ejpam-4730	83	6	)	)	PUNCT
ejpam-4730	83	7	)	)	PUNCT
ejpam-4730	84	1	=	=	SYM
ejpam-4730	85	1	∆r−1	∆r−1	PROPN
ejpam-4730	85	2	h	h	NOUN
ejpam-4730	85	3	(	(	PUNCT
ejpam-4730	85	4	f(x+	f(x+	ADP
ejpam-4730	85	5	h)−	h)−	PROPN
ejpam-4730	85	6	f(x−	f(x−	PROPN
ejpam-4730	85	7	h	h	NOUN
ejpam-4730	85	8	)	)	PUNCT
ejpam-4730	85	9	)	)	PUNCT
ejpam-4730	85	10	.	.	PUNCT
ejpam-4730	86	1	so	so	ADV
ejpam-4730	86	2	,	,	PUNCT
ejpam-4730	86	3	∥∆k	∥∆k	NOUN
ejpam-4730	86	4	hf(.)∥lp	hf(.)∥lp	NOUN
ejpam-4730	86	5	,	,	PUNCT
ejpam-4730	86	6	λ[−π	λ[−π	PROPN
ejpam-4730	86	7	,	,	PUNCT
ejpam-4730	86	8	π	π	X
ejpam-4730	86	9	]	]	X
ejpam-4730	86	10	≤	≤	NUM
ejpam-4730	86	11	∥∆r−1	∥∆r−1	PROPN
ejpam-4730	86	12	h	h	NOUN
ejpam-4730	86	13	(	(	PUNCT
ejpam-4730	86	14	f(+h)−	f(+h)−	NOUN
ejpam-4730	86	15	f(−h))∥lp	f(−h))∥lp	PROPN
ejpam-4730	86	16	,	,	PUNCT
ejpam-4730	86	17	λ[−π	λ[−π	PROPN
ejpam-4730	86	18	,	,	PUNCT
ejpam-4730	86	19	π	π	X
ejpam-4730	86	20	]	]	PUNCT
ejpam-4730	86	21	,	,	PUNCT
ejpam-4730	86	22	c	c	X
ejpam-4730	86	23	>	>	X
ejpam-4730	86	24	0	0	X
ejpam-4730	86	25	.	.	PUNCT
ejpam-4730	87	1	take	take	VERB
ejpam-4730	87	2	k	k	NOUN
ejpam-4730	87	3	=	=	SYM
ejpam-4730	87	4	1	1	NUM
ejpam-4730	87	5	,	,	PUNCT
ejpam-4730	87	6	we	we	PRON
ejpam-4730	87	7	obtain	obtain	VERB
ejpam-4730	87	8	µr(f	µr(f	ADP
ejpam-4730	87	9	,	,	PUNCT
ejpam-4730	87	10	1)lp	1)lp	NUM
ejpam-4730	87	11	,	,	PUNCT
ejpam-4730	87	12	λ[−π	λ[−π	PROPN
ejpam-4730	87	13	,	,	PUNCT
ejpam-4730	87	14	π	π	X
ejpam-4730	87	15	]	]	X
ejpam-4730	87	16	≤	≤	NUM
ejpam-4730	87	17	{	{	PUNCT
ejpam-4730	87	18	maxc	maxc	NOUN
ejpam-4730	87	19	}	}	PUNCT
ejpam-4730	87	20	µr−k(f	µr−k(f	NOUN
ejpam-4730	87	21	,	,	PUNCT
ejpam-4730	87	22	1)lp	1)lp	NUM
ejpam-4730	87	23	,	,	PUNCT
ejpam-4730	87	24	λ[−π	λ[−π	PROPN
ejpam-4730	87	25	,	,	PUNCT
ejpam-4730	87	26	π	π	PROPN
ejpam-4730	87	27	]	]	X
ejpam-4730	87	28	,	,	PUNCT
ejpam-4730	87	29	which	which	PRON
ejpam-4730	87	30	completes	complete	VERB
ejpam-4730	87	31	the	the	DET
ejpam-4730	87	32	proof	proof	NOUN
ejpam-4730	87	33	.	.	PUNCT
ejpam-4730	88	1	lemma	lemma	PROPN
ejpam-4730	89	1	3	3	X
ejpam-4730	89	2	.	.	PUNCT
ejpam-4730	90	1	if	if	SCONJ
ejpam-4730	90	2	f	f	PROPN
ejpam-4730	90	3	,	,	PUNCT
ejpam-4730	90	4	f	f	PROPN
ejpam-4730	90	5	′	′	NUM
ejpam-4730	90	6	∈	∈	PROPN
ejpam-4730	90	7	lp	lp	NOUN
ejpam-4730	90	8	,	,	PUNCT
ejpam-4730	90	9	λ[−π	λ[−π	PROPN
ejpam-4730	90	10	,	,	PUNCT
ejpam-4730	90	11	π	π	PROPN
ejpam-4730	90	12	]	]	X
ejpam-4730	90	13	,	,	PUNCT
ejpam-4730	90	14	1	1	NUM
ejpam-4730	90	15	≤	≤	NOUN
ejpam-4730	90	16	p	p	X
ejpam-4730	90	17	<	<	X
ejpam-4730	90	18	∞	∞	PROPN
ejpam-4730	90	19	,	,	PUNCT
ejpam-4730	90	20	f	f	PROPN
ejpam-4730	90	21	′	′	NUM
ejpam-4730	90	22	is	be	AUX
ejpam-4730	90	23	the	the	DET
ejpam-4730	90	24	derivative	derivative	NOUN
ejpam-4730	90	25	of	of	ADP
ejpam-4730	90	26	f	f	PROPN
ejpam-4730	90	27	and	and	CCONJ
ejpam-4730	90	28	r	r	PROPN
ejpam-4730	90	29	,	,	PUNCT
ejpam-4730	90	30	k	k	PROPN
ejpam-4730	90	31	∈	∈	PROPN
ejpam-4730	90	32	n	n	ADV
ejpam-4730	90	33	.	.	PUNCT
ejpam-4730	91	1	then	then	ADV
ejpam-4730	91	2	,	,	PUNCT
ejpam-4730	91	3	µr(f	µr(f	ADV
ejpam-4730	91	4	,	,	PUNCT
ejpam-4730	91	5	1	1	NUM
ejpam-4730	91	6	k	k	NOUN
ejpam-4730	91	7	)	)	PUNCT
ejpam-4730	91	8	lp	lp	NOUN
ejpam-4730	91	9	,	,	PUNCT
ejpam-4730	91	10	λ[−π	λ[−π	PROPN
ejpam-4730	91	11	,	,	PUNCT
ejpam-4730	91	12	π	π	X
ejpam-4730	91	13	]	]	X
ejpam-4730	91	14	≤	≤	NUM
ejpam-4730	91	15	ck	ck	INTJ
ejpam-4730	91	16	µr(f	µr(f	NOUN
ejpam-4730	91	17	′	′	NOUN
ejpam-4730	91	18	,	,	PUNCT
ejpam-4730	91	19	1	1	NUM
ejpam-4730	91	20	k	k	X
ejpam-4730	91	21	)	)	PUNCT
ejpam-4730	91	22	lp	lp	NOUN
ejpam-4730	91	23	,	,	PUNCT
ejpam-4730	91	24	λ[−π	λ[−π	PROPN
ejpam-4730	91	25	,	,	PUNCT
ejpam-4730	91	26	π	π	PROPN
ejpam-4730	91	27	]	]	X
ejpam-4730	91	28	,	,	PUNCT
ejpam-4730	91	29	where	where	SCONJ
ejpam-4730	91	30	ck	ck	PROPN
ejpam-4730	91	31	is	be	AUX
ejpam-4730	91	32	a	a	DET
ejpam-4730	91	33	positive	positive	ADJ
ejpam-4730	91	34	constant	constant	ADJ
ejpam-4730	91	35	.	.	PUNCT
ejpam-4730	92	1	proof	proof	NOUN
ejpam-4730	92	2	.	.	PUNCT
ejpam-4730	93	1	the	the	DET
ejpam-4730	93	2	proof	proof	NOUN
ejpam-4730	93	3	of	of	ADP
ejpam-4730	93	4	this	this	DET
ejpam-4730	93	5	lemma	lemma	PROPN
ejpam-4730	93	6	goes	go	VERB
ejpam-4730	93	7	in	in	ADP
ejpam-4730	93	8	the	the	DET
ejpam-4730	93	9	same	same	ADJ
ejpam-4730	93	10	way	way	NOUN
ejpam-4730	93	11	as	as	ADP
ejpam-4730	93	12	the	the	DET
ejpam-4730	93	13	proof	proof	NOUN
ejpam-4730	93	14	of	of	ADP
ejpam-4730	93	15	lemma	lemma	PROPN
ejpam-4730	93	16	2	2	NUM
ejpam-4730	93	17	.	.	PUNCT
ejpam-4730	94	1	lemma	lemma	PROPN
ejpam-4730	94	2	4	4	X
ejpam-4730	94	3	.	.	PUNCT
ejpam-4730	95	1	if	if	SCONJ
ejpam-4730	95	2	f	f	PROPN
ejpam-4730	95	3	∈	∈	PROPN
ejpam-4730	95	4	lp	lp	PROPN
ejpam-4730	95	5	,	,	PUNCT
ejpam-4730	95	6	λ[−π	λ[−π	PROPN
ejpam-4730	95	7	,	,	PUNCT
ejpam-4730	95	8	π	π	PROPN
ejpam-4730	95	9	]	]	X
ejpam-4730	95	10	,	,	PUNCT
ejpam-4730	95	11	1	1	NUM
ejpam-4730	95	12	≤	≤	NOUN
ejpam-4730	95	13	p	p	NOUN
ejpam-4730	95	14	<	<	X
ejpam-4730	95	15	∞	∞	PROPN
ejpam-4730	95	16	and	and	CCONJ
ejpam-4730	95	17	r	r	NOUN
ejpam-4730	95	18	,	,	PUNCT
ejpam-4730	95	19	k	k	PROPN
ejpam-4730	95	20	∈	∈	PROPN
ejpam-4730	95	21	n.	n.	NOUN
ejpam-4730	95	22	then	then	ADV
ejpam-4730	95	23	,	,	PUNCT
ejpam-4730	95	24	µr(f	µr(f	ADV
ejpam-4730	95	25	,	,	PUNCT
ejpam-4730	95	26	α	α	PROPN
ejpam-4730	95	27	k	k	NOUN
ejpam-4730	95	28	)	)	PUNCT
ejpam-4730	95	29	lp	lp	PROPN
ejpam-4730	95	30	,	,	PUNCT
ejpam-4730	95	31	λ[−π	λ[−π	PROPN
ejpam-4730	95	32	,	,	PUNCT
ejpam-4730	95	33	π	π	X
ejpam-4730	95	34	]	]	X
ejpam-4730	95	35	≤	≤	X
ejpam-4730	95	36	ck	ck	ADP
ejpam-4730	95	37	µr(f	µr(f	NOUN
ejpam-4730	95	38	,	,	PUNCT
ejpam-4730	95	39	1	1	NUM
ejpam-4730	95	40	k	k	NOUN
ejpam-4730	95	41	)	)	PUNCT
ejpam-4730	95	42	lp	lp	NOUN
ejpam-4730	95	43	,	,	PUNCT
ejpam-4730	95	44	λ[−π	λ[−π	PROPN
ejpam-4730	95	45	,	,	PUNCT
ejpam-4730	95	46	π	π	X
ejpam-4730	95	47	]	]	X
ejpam-4730	95	48	,	,	PUNCT
ejpam-4730	95	49	where	where	SCONJ
ejpam-4730	95	50	ck	ck	PROPN
ejpam-4730	95	51	is	be	AUX
ejpam-4730	95	52	a	a	DET
ejpam-4730	95	53	positive	positive	ADJ
ejpam-4730	95	54	constant	constant	ADJ
ejpam-4730	95	55	depending	depend	VERB
ejpam-4730	95	56	on	on	ADP
ejpam-4730	95	57	k	k	PROPN
ejpam-4730	95	58	and	and	CCONJ
ejpam-4730	95	59	α	α	X
ejpam-4730	95	60	>	>	X
ejpam-4730	95	61	0	0	X
ejpam-4730	95	62	.	.	PUNCT
ejpam-4730	96	1	proof	proof	NOUN
ejpam-4730	96	2	.	.	PUNCT
ejpam-4730	97	1	we	we	PRON
ejpam-4730	97	2	have	have	VERB
ejpam-4730	97	3	µr(f	µr(f	NOUN
ejpam-4730	97	4	,	,	PUNCT
ejpam-4730	97	5	α	α	PROPN
ejpam-4730	97	6	k	k	NOUN
ejpam-4730	97	7	)	)	PUNCT
ejpam-4730	97	8	lp	lp	PROPN
ejpam-4730	97	9	,	,	PUNCT
ejpam-4730	97	10	λ[−π	λ[−π	PROPN
ejpam-4730	97	11	,	,	PUNCT
ejpam-4730	97	12	π	π	X
ejpam-4730	97	13	]	]	X
ejpam-4730	98	1	=	=	PUNCT
ejpam-4730	98	2	sup︸︷︷︸	sup︸︷︷︸	NOUN
ejpam-4730	98	3	|h|≤α	|h|≤α	PROPN
ejpam-4730	98	4	k	k	PROPN
ejpam-4730	98	5	∥∆r	∥∆r	NOUN
ejpam-4730	98	6	hf(.)∥lp	hf(.)∥lp	PROPN
ejpam-4730	98	7	,	,	PUNCT
ejpam-4730	98	8	λ[−π	λ[−π	PROPN
ejpam-4730	98	9	,	,	PUNCT
ejpam-4730	98	10	π	π	X
ejpam-4730	98	11	]	]	X
ejpam-4730	98	12	≤	≤	NUM
ejpam-4730	98	13	sup︸︷︷︸	sup︸︷︷︸	ADP
ejpam-4730	98	14	|h|≤α	|h|≤α	PROPN
ejpam-4730	98	15	k	k	PROPN
ejpam-4730	98	16	∥∆r	∥∆r	NOUN
ejpam-4730	98	17	α	α	PROPN
ejpam-4730	98	18	k	k	PROPN
ejpam-4730	98	19	f(.)∥lp	f(.)∥lp	PROPN
ejpam-4730	98	20	,	,	PUNCT
ejpam-4730	98	21	λ[−π	λ[−π	PROPN
ejpam-4730	98	22	,	,	PUNCT
ejpam-4730	98	23	π	π	X
ejpam-4730	98	24	]	]	X
ejpam-4730	98	25	≤	≤	NUM
ejpam-4730	98	26	sup︸︷︷︸	sup︸︷︷︸	ADP
ejpam-4730	99	1	|h|≤α	|h|≤α	PROPN
ejpam-4730	99	2	k	k	INTJ
ejpam-4730	99	3	∥(α	∥(α	PROPN
ejpam-4730	100	1	k	k	PROPN
ejpam-4730	100	2	)	)	PUNCT
ejpam-4730	100	3	rdrf(.)∥lp	rdrf(.)∥lp	PROPN
ejpam-4730	100	4	,	,	PUNCT
ejpam-4730	100	5	λ[−π	λ[−π	PROPN
ejpam-4730	100	6	,	,	PUNCT
ejpam-4730	100	7	π	π	PROPN
ejpam-4730	100	8	]	]	X
ejpam-4730	100	9	≤	≤	NUM
ejpam-4730	100	10	max|α|r{sup∥∆r	max|α|r{sup∥∆r	PROPN
ejpam-4730	100	11	α	α	PROPN
ejpam-4730	100	12	k	k	PROPN
ejpam-4730	100	13	f(.)∥lp	f(.)∥lp	PROPN
ejpam-4730	100	14	,	,	PUNCT
ejpam-4730	100	15	λ[−π	λ[−π	PROPN
ejpam-4730	100	16	,	,	PUNCT
ejpam-4730	100	17	π	π	NOUN
ejpam-4730	100	18	]	]	X
ejpam-4730	100	19	}	}	PUNCT
ejpam-4730	100	20	≤	≤	NUM
ejpam-4730	100	21	max(αk	max(αk	NOUN
ejpam-4730	100	22	)	)	PUNCT
ejpam-4730	100	23	rµr(f	rµr(f	PROPN
ejpam-4730	100	24	,	,	PUNCT
ejpam-4730	100	25	1	1	NUM
ejpam-4730	100	26	k	k	NOUN
ejpam-4730	100	27	)	)	PUNCT
ejpam-4730	100	28	lp	lp	NOUN
ejpam-4730	100	29	,	,	PUNCT
ejpam-4730	100	30	λ[−π	λ[−π	PROPN
ejpam-4730	100	31	,	,	PUNCT
ejpam-4730	100	32	π	π	NOUN
ejpam-4730	100	33	]	]	X
ejpam-4730	100	34	.	.	PUNCT
ejpam-4730	101	1	substituting	substitute	VERB
ejpam-4730	101	2	max|α|r	max|α|r	PROPN
ejpam-4730	101	3	=	=	SYM
ejpam-4730	101	4	ck	ck	PROPN
ejpam-4730	101	5	,	,	PUNCT
ejpam-4730	101	6	we	we	PRON
ejpam-4730	101	7	obtain	obtain	VERB
ejpam-4730	101	8	µr(f	µr(f	NOUN
ejpam-4730	101	9	,	,	PUNCT
ejpam-4730	101	10	α	α	PROPN
ejpam-4730	101	11	k	k	NOUN
ejpam-4730	101	12	)	)	PUNCT
ejpam-4730	101	13	lp	lp	PROPN
ejpam-4730	101	14	,	,	PUNCT
ejpam-4730	101	15	λ[−π	λ[−π	PROPN
ejpam-4730	101	16	,	,	PUNCT
ejpam-4730	101	17	π	π	X
ejpam-4730	101	18	]	]	X
ejpam-4730	101	19	≤	≤	X
ejpam-4730	101	20	ck	ck	ADP
ejpam-4730	101	21	µr(f	µr(f	NOUN
ejpam-4730	101	22	,	,	PUNCT
ejpam-4730	101	23	1	1	NUM
ejpam-4730	101	24	k	k	NOUN
ejpam-4730	101	25	)	)	PUNCT
ejpam-4730	101	26	lp	lp	NOUN
ejpam-4730	101	27	,	,	PUNCT
ejpam-4730	101	28	λ[−π	λ[−π	PROPN
ejpam-4730	101	29	,	,	PUNCT
ejpam-4730	101	30	π	π	NOUN
ejpam-4730	101	31	]	]	X
ejpam-4730	101	32	.	.	PUNCT
ejpam-4730	102	1	a.	a.	PROPN
ejpam-4730	102	2	a.	a.	PROPN
ejpam-4730	102	3	auad	auad	PROPN
ejpam-4730	102	4	,	,	PUNCT
ejpam-4730	102	5	m.	m.	NOUN
ejpam-4730	102	6	a.	a.	NOUN
ejpam-4730	102	7	hilal	hilal	PROPN
ejpam-4730	102	8	,	,	PUNCT
ejpam-4730	102	9	n.	n.	PROPN
ejpam-4730	102	10	s.	s.	PROPN
ejpam-4730	102	11	khalaf	khalaf	PROPN
ejpam-4730	102	12	/	/	SYM
ejpam-4730	102	13	eur	eur	PROPN
ejpam-4730	102	14	.	.	PUNCT
ejpam-4730	103	1	j.	j.	PROPN
ejpam-4730	103	2	pure	pure	PROPN
ejpam-4730	103	3	appl	appl	PROPN
ejpam-4730	103	4	.	.	PROPN
ejpam-4730	103	5	math	math	PROPN
ejpam-4730	103	6	,	,	PUNCT
ejpam-4730	103	7	16	16	NUM
ejpam-4730	103	8	(	(	PUNCT
ejpam-4730	103	9	2	2	NUM
ejpam-4730	103	10	)	)	PUNCT
ejpam-4730	103	11	(	(	PUNCT
ejpam-4730	103	12	2023	2023	NUM
ejpam-4730	103	13	)	)	PUNCT
ejpam-4730	103	14	,	,	PUNCT
ejpam-4730	103	15	944	944	NUM
ejpam-4730	103	16	-	-	SYM
ejpam-4730	103	17	952	952	NUM
ejpam-4730	103	18	949	949	NUM
ejpam-4730	103	19	lemma	lemma	PROPN
ejpam-4730	103	20	5	5	NUM
ejpam-4730	103	21	.	.	PUNCT
ejpam-4730	104	1	if	if	SCONJ
ejpam-4730	104	2	f	f	PROPN
ejpam-4730	104	3	∈	∈	PROPN
ejpam-4730	104	4	lp	lp	PROPN
ejpam-4730	104	5	,	,	PUNCT
ejpam-4730	104	6	λ[−π	λ[−π	PROPN
ejpam-4730	104	7	,	,	PUNCT
ejpam-4730	104	8	π	π	PROPN
ejpam-4730	104	9	]	]	X
ejpam-4730	104	10	,	,	PUNCT
ejpam-4730	104	11	1	1	NUM
ejpam-4730	104	12	≤	≤	NOUN
ejpam-4730	104	13	p	p	X
ejpam-4730	104	14	<	<	X
ejpam-4730	104	15	∞	∞	PROPN
ejpam-4730	104	16	,	,	PUNCT
ejpam-4730	104	17	l	l	PROPN
ejpam-4730	104	18	∈	∈	PROPN
ejpam-4730	104	19	n	n	PRON
ejpam-4730	104	20	and	and	CCONJ
ejpam-4730	104	21	g(0	g(0	PROPN
ejpam-4730	104	22	)	)	PUNCT
ejpam-4730	104	23	=	=	SYM
ejpam-4730	105	1	1	1	X
ejpam-4730	105	2	.	.	PUNCT
ejpam-4730	106	1	then	then	ADV
ejpam-4730	106	2	,	,	PUNCT
ejpam-4730	106	3	∥f	∥f	PROPN
ejpam-4730	106	4	−	−	NOUN
ejpam-4730	106	5	p(f)∥lp	p(f)∥lp	ADP
ejpam-4730	106	6	,	,	PUNCT
ejpam-4730	106	7	λ[−π	λ[−π	PROPN
ejpam-4730	106	8	,	,	PUNCT
ejpam-4730	106	9	π	π	X
ejpam-4730	106	10	]	]	X
ejpam-4730	106	11	≤	≤	X
ejpam-4730	106	12	ck	ck	PRON
ejpam-4730	106	13	µl(f	µl(f	PROPN
ejpam-4730	106	14	,	,	PUNCT
ejpam-4730	106	15	1	1	NUM
ejpam-4730	106	16	l	l	NOUN
ejpam-4730	106	17	)	)	PUNCT
ejpam-4730	106	18	lp	lp	NOUN
ejpam-4730	106	19	,	,	PUNCT
ejpam-4730	106	20	λ[−π	λ[−π	PROPN
ejpam-4730	106	21	,	,	PUNCT
ejpam-4730	106	22	π	π	X
ejpam-4730	106	23	]	]	X
ejpam-4730	106	24	∑l	∑l	ADJ
ejpam-4730	106	25	i=0	i=0	PROPN
ejpam-4730	106	26	(	(	PUNCT
ejpam-4730	106	27	l	l	X
ejpam-4730	106	28	i	i	NOUN
ejpam-4730	106	29	)	)	PUNCT
ejpam-4730	106	30	lg(g	lg(g	PROPN
ejpam-4730	106	31	,	,	PUNCT
ejpam-4730	106	32	i	i	NOUN
ejpam-4730	106	33	)	)	PUNCT
ejpam-4730	106	34	,	,	PUNCT
ejpam-4730	106	35	where	where	SCONJ
ejpam-4730	106	36	g(g	g(g	PROPN
ejpam-4730	106	37	,	,	PUNCT
ejpam-4730	106	38	i	i	NOUN
ejpam-4730	106	39	)	)	PUNCT
ejpam-4730	106	40	=	=	SYM
ejpam-4730	107	1	1	1	NUM
ejpam-4730	107	2	2π	2π	NUM
ejpam-4730	107	3	∫	∫	PROPN
ejpam-4730	108	1	π	π	NOUN
ejpam-4730	108	2	−π	−π	PROPN
ejpam-4730	108	3	|x|ig(x)dx	|x|ig(x)dx	NUM
ejpam-4730	108	4	that	that	PRON
ejpam-4730	108	5	belongs	belong	VERB
ejpam-4730	108	6	to	to	ADP
ejpam-4730	108	7	the	the	DET
ejpam-4730	108	8	subspace	subspace	NOUN
ejpam-4730	108	9	of	of	ADP
ejpam-4730	108	10	lp	lp	NOUN
ejpam-4730	108	11	,	,	PUNCT
ejpam-4730	108	12	λ[−π	λ[−π	PROPN
ejpam-4730	108	13	,	,	PUNCT
ejpam-4730	108	14	π	π	X
ejpam-4730	108	15	]	]	X
ejpam-4730	108	16	is	be	AUX
ejpam-4730	108	17	a	a	DET
ejpam-4730	108	18	positive	positive	ADJ
ejpam-4730	108	19	constant	constant	ADJ
ejpam-4730	108	20	.	.	PUNCT
ejpam-4730	109	1	proof	proof	NOUN
ejpam-4730	109	2	.	.	PUNCT
ejpam-4730	110	1	p(f)−	p(f)−	PROPN
ejpam-4730	110	2	f	f	PROPN
ejpam-4730	110	3	=	=	PUNCT
ejpam-4730	110	4	(	(	PUNCT
ejpam-4730	110	5	−1)l+1	−1)l+1	X
ejpam-4730	110	6	2π	2π	NUM
ejpam-4730	110	7	∫	∫	PROPN
ejpam-4730	111	1	π	π	NOUN
ejpam-4730	111	2	−π	−π	PROPN
ejpam-4730	111	3	g(x)∆xl(f)dx	g(x)∆xl(f)dx	NOUN
ejpam-4730	111	4	,	,	PUNCT
ejpam-4730	111	5	and	and	CCONJ
ejpam-4730	111	6	∥p(f)−	∥p(f)−	PROPN
ejpam-4730	111	7	f∥lp	f∥lp	PROPN
ejpam-4730	111	8	,	,	PUNCT
ejpam-4730	111	9	λ[−π	λ[−π	PROPN
ejpam-4730	111	10	,	,	PUNCT
ejpam-4730	111	11	π	π	X
ejpam-4730	111	12	]	]	X
ejpam-4730	111	13	=	=	SYM
ejpam-4730	111	14	(	(	PUNCT
ejpam-4730	111	15	−1)l+1	−1)l+1	X
ejpam-4730	111	16	2π	2π	NUM
ejpam-4730	111	17	∫	∫	PROPN
ejpam-4730	112	1	π	π	NOUN
ejpam-4730	112	2	−π	−π	PROPN
ejpam-4730	113	1	∥g(.)∆xl(f)∥lp	∥g(.)∆xl(f)∥lp	PROPN
ejpam-4730	113	2	,	,	PUNCT
ejpam-4730	113	3	λ[−π	λ[−π	PROPN
ejpam-4730	113	4	,	,	PUNCT
ejpam-4730	113	5	π	π	PROPN
ejpam-4730	113	6	]	]	X
ejpam-4730	113	7	dx	dx	PROPN
ejpam-4730	113	8	≤	≤	PROPN
ejpam-4730	113	9	(	(	PUNCT
ejpam-4730	113	10	−1)l+1	−1)l+1	NOUN
ejpam-4730	113	11	2π	2π	PROPN
ejpam-4730	113	12	sup∥∆xl(f)∥lp	sup∥∆xl(f)∥lp	NOUN
ejpam-4730	113	13	,	,	PUNCT
ejpam-4730	113	14	λ[−π	λ[−π	PROPN
ejpam-4730	113	15	,	,	PUNCT
ejpam-4730	113	16	π	π	PROPN
ejpam-4730	113	17	]	]	X
ejpam-4730	113	18	∫	∫	PROPN
ejpam-4730	114	1	π	π	PROPN
ejpam-4730	114	2	−π	−π	ADP
ejpam-4730	114	3	|g(x)|dx	|g(x)|dx	PROPN
ejpam-4730	114	4	.	.	PUNCT
ejpam-4730	115	1	from	from	ADP
ejpam-4730	115	2	the	the	DET
ejpam-4730	115	3	properties	property	NOUN
ejpam-4730	115	4	of	of	ADP
ejpam-4730	115	5	the	the	DET
ejpam-4730	115	6	modulus	modulus	NOUN
ejpam-4730	115	7	of	of	ADP
ejpam-4730	115	8	smoothness	smoothness	NOUN
ejpam-4730	115	9	,	,	PUNCT
ejpam-4730	115	10	we	we	PRON
ejpam-4730	115	11	obtain	obtain	VERB
ejpam-4730	115	12	∥p(f)−	∥p(f)−	NOUN
ejpam-4730	115	13	f∥lp	f∥lp	PROPN
ejpam-4730	115	14	,	,	PUNCT
ejpam-4730	115	15	λ[−π	λ[−π	PROPN
ejpam-4730	115	16	,	,	PUNCT
ejpam-4730	115	17	π	π	X
ejpam-4730	115	18	]	]	X
ejpam-4730	115	19	≤	≤	NUM
ejpam-4730	115	20	cµl	cµl	NOUN
ejpam-4730	115	21	(	(	PUNCT
ejpam-4730	115	22	f	f	X
ejpam-4730	115	23	,	,	PUNCT
ejpam-4730	115	24	1	1	NUM
ejpam-4730	115	25	l	l	NOUN
ejpam-4730	115	26	)	)	PUNCT
ejpam-4730	115	27	lp	lp	NOUN
ejpam-4730	115	28	,	,	PUNCT
ejpam-4730	115	29	λ[−π	λ[−π	PROPN
ejpam-4730	115	30	,	,	PUNCT
ejpam-4730	115	31	π	π	X
ejpam-4730	115	32	]	]	X
ejpam-4730	115	33	1	1	NUM
ejpam-4730	115	34	2π	2π	NUM
ejpam-4730	115	35	∫	∫	PROPN
ejpam-4730	116	1	π	π	NOUN
ejpam-4730	116	2	−π	−π	PROPN
ejpam-4730	116	3	|l|i|g(x)|dx	|l|i|g(x)|dx	VERB
ejpam-4730	116	4	≤	≤	NUM
ejpam-4730	116	5	cµl	cµl	PROPN
ejpam-4730	116	6	(	(	PUNCT
ejpam-4730	116	7	f	f	X
ejpam-4730	116	8	,	,	PUNCT
ejpam-4730	116	9	1	1	NUM
ejpam-4730	116	10	l	l	NOUN
ejpam-4730	116	11	)	)	PUNCT
ejpam-4730	116	12	lp	lp	NOUN
ejpam-4730	116	13	,	,	PUNCT
ejpam-4730	116	14	λ[−π	λ[−π	PROPN
ejpam-4730	116	15	,	,	PUNCT
ejpam-4730	116	16	π	π	X
ejpam-4730	116	17	]	]	X
ejpam-4730	116	18	l∑	l∑	X
ejpam-4730	117	1	i=0	i=0	PROPN
ejpam-4730	117	2	(	(	PUNCT
ejpam-4730	117	3	l	l	NOUN
ejpam-4730	117	4	i	i	NOUN
ejpam-4730	117	5	)	)	PUNCT
ejpam-4730	117	6	l	l	NOUN
ejpam-4730	118	1	1	1	NUM
ejpam-4730	118	2	2π	2π	NUM
ejpam-4730	118	3	∫	∫	PROPN
ejpam-4730	119	1	π	π	NOUN
ejpam-4730	119	2	−π	−π	PROPN
ejpam-4730	119	3	|l|i|g(x)|dx	|l|i|g(x)|dx	NOUN
ejpam-4730	119	4	.	.	PUNCT
ejpam-4730	120	1	3	3	X
ejpam-4730	120	2	.	.	X
ejpam-4730	120	3	main	main	ADJ
ejpam-4730	120	4	results	result	NOUN
ejpam-4730	120	5	in	in	ADP
ejpam-4730	120	6	this	this	DET
ejpam-4730	120	7	section	section	NOUN
ejpam-4730	120	8	,	,	PUNCT
ejpam-4730	120	9	we	we	PRON
ejpam-4730	120	10	introduce	introduce	VERB
ejpam-4730	120	11	direct	direct	ADJ
ejpam-4730	120	12	theorems	theorem	NOUN
ejpam-4730	120	13	of	of	ADP
ejpam-4730	120	14	unbounded	unbounded	ADJ
ejpam-4730	120	15	functions	function	NOUN
ejpam-4730	120	16	in	in	ADP
ejpam-4730	120	17	weighted	weight	VERB
ejpam-4730	120	18	space	space	NOUN
ejpam-4730	120	19	by	by	ADP
ejpam-4730	120	20	using	use	VERB
ejpam-4730	120	21	some	some	DET
ejpam-4730	120	22	linear	linear	ADJ
ejpam-4730	120	23	operators	operator	NOUN
ejpam-4730	120	24	.	.	PUNCT
ejpam-4730	121	1	theorem	theorem	NOUN
ejpam-4730	121	2	1	1	NUM
ejpam-4730	121	3	.	.	PUNCT
ejpam-4730	122	1	let	let	VERB
ejpam-4730	122	2	f	f	PROPN
ejpam-4730	122	3	∈	∈	PROPN
ejpam-4730	122	4	lp	lp	PROPN
ejpam-4730	122	5	,	,	PUNCT
ejpam-4730	122	6	λ[−π	λ[−π	PROPN
ejpam-4730	122	7	,	,	PUNCT
ejpam-4730	122	8	π	π	PROPN
ejpam-4730	122	9	]	]	X
ejpam-4730	122	10	,	,	PUNCT
ejpam-4730	122	11	1	1	NUM
ejpam-4730	122	12	≤	≤	NOUN
ejpam-4730	123	1	p	p	NOUN
ejpam-4730	123	2	<	<	X
ejpam-4730	123	3	∞	∞	PROPN
ejpam-4730	123	4	,	,	PUNCT
ejpam-4730	123	5	l	l	PROPN
ejpam-4730	123	6	∈	∈	PROPN
ejpam-4730	123	7	n	n	NOUN
ejpam-4730	123	8	and	and	CCONJ
ejpam-4730	123	9	r	r	NOUN
ejpam-4730	123	10	∈	∈	PROPN
ejpam-4730	123	11	n	n	ADV
ejpam-4730	123	12	∪	∪	ADJ
ejpam-4730	123	13	0	0	NUM
ejpam-4730	123	14	.	.	PUNCT
ejpam-4730	124	1	then	then	ADV
ejpam-4730	124	2	,	,	PUNCT
ejpam-4730	124	3	er(f	er(f	NOUN
ejpam-4730	124	4	,	,	PUNCT
ejpam-4730	124	5	ξ)lp	ξ)lp	PROPN
ejpam-4730	124	6	,	,	PUNCT
ejpam-4730	124	7	λ[−π	λ[−π	PROPN
ejpam-4730	124	8	,	,	PUNCT
ejpam-4730	124	9	π	π	X
ejpam-4730	124	10	]	]	X
ejpam-4730	124	11	≤	≤	NUM
ejpam-4730	124	12	inf∥p(f)−	inf∥p(f)−	PROPN
ejpam-4730	124	13	f∥lp	f∥lp	PROPN
ejpam-4730	124	14	,	,	PUNCT
ejpam-4730	124	15	λ[−π	λ[−π	PROPN
ejpam-4730	124	16	,	,	PUNCT
ejpam-4730	124	17	π	π	NOUN
ejpam-4730	124	18	]	]	X
ejpam-4730	124	19	≤	≤	PROPN
ejpam-4730	124	20	cr	cr	ADP
ejpam-4730	124	21	inf{µl(f	inf{µl(f	PROPN
ejpam-4730	124	22	,	,	PUNCT
ejpam-4730	124	23	ξ)lp	ξ)lp	PROPN
ejpam-4730	124	24	,	,	PUNCT
ejpam-4730	124	25	λ[−π	λ[−π	PROPN
ejpam-4730	124	26	,	,	PUNCT
ejpam-4730	124	27	π	π	X
ejpam-4730	124	28	]	]	X
ejpam-4730	124	29	∑l	∑l	ADJ
ejpam-4730	124	30	i=0	i=0	PROPN
ejpam-4730	124	31	(	(	PUNCT
ejpam-4730	124	32	l	l	X
ejpam-4730	124	33	i	i	NOUN
ejpam-4730	124	34	)	)	PUNCT
ejpam-4730	124	35	lg(g	lg(g	PROPN
ejpam-4730	124	36	,	,	PUNCT
ejpam-4730	124	37	i	i	NOUN
ejpam-4730	124	38	)	)	PUNCT
ejpam-4730	124	39	}	}	PUNCT
ejpam-4730	124	40	,	,	PUNCT
ejpam-4730	124	41	where	where	SCONJ
ejpam-4730	124	42	cr	cr	PROPN
ejpam-4730	124	43	is	be	AUX
ejpam-4730	124	44	a	a	DET
ejpam-4730	124	45	positive	positive	ADJ
ejpam-4730	124	46	constant	constant	NOUN
ejpam-4730	124	47	and	and	CCONJ
ejpam-4730	124	48	ξ	ξ	X
ejpam-4730	124	49	>	>	X
ejpam-4730	124	50	0	0	PUNCT
ejpam-4730	124	51	.	.	PUNCT
ejpam-4730	125	1	proof	proof	NOUN
ejpam-4730	125	2	.	.	PUNCT
ejpam-4730	126	1	taking	take	VERB
ejpam-4730	126	2	equation	equation	NOUN
ejpam-4730	126	3	(	(	PUNCT
ejpam-4730	126	4	1	1	NUM
ejpam-4730	126	5	)	)	PUNCT
ejpam-4730	126	6	and	and	CCONJ
ejpam-4730	126	7	equation	equation	NOUN
ejpam-4730	126	8	(	(	PUNCT
ejpam-4730	126	9	4	4	NUM
ejpam-4730	126	10	)	)	PUNCT
ejpam-4730	126	11	,	,	PUNCT
ejpam-4730	126	12	and	and	CCONJ
ejpam-4730	126	13	applying	apply	VERB
ejpam-4730	126	14	lemma	lemma	PROPN
ejpam-4730	126	15	5	5	NUM
ejpam-4730	126	16	using	use	VERB
ejpam-4730	126	17	the	the	DET
ejpam-4730	126	18	fact	fact	NOUN
ejpam-4730	126	19	that	that	SCONJ
ejpam-4730	126	20	p	p	X
ejpam-4730	126	21	(	(	PUNCT
ejpam-4730	126	22	f	f	X
ejpam-4730	126	23	)	)	PUNCT
ejpam-4730	126	24	∈	∈	PROPN
ejpam-4730	126	25	tl	tl	PROPN
ejpam-4730	126	26	,	,	PUNCT
ejpam-4730	126	27	the	the	DET
ejpam-4730	126	28	proof	proof	NOUN
ejpam-4730	126	29	of	of	ADP
ejpam-4730	126	30	this	this	DET
ejpam-4730	126	31	theorem	theorem	NOUN
ejpam-4730	126	32	is	be	AUX
ejpam-4730	126	33	completed	complete	VERB
ejpam-4730	126	34	.	.	PUNCT
ejpam-4730	127	1	a.	a.	NOUN
ejpam-4730	127	2	a.	a.	PROPN
ejpam-4730	127	3	auad	auad	PROPN
ejpam-4730	127	4	,	,	PUNCT
ejpam-4730	127	5	m.	m.	NOUN
ejpam-4730	127	6	a.	a.	NOUN
ejpam-4730	127	7	hilal	hilal	PROPN
ejpam-4730	127	8	,	,	PUNCT
ejpam-4730	127	9	n.	n.	PROPN
ejpam-4730	127	10	s.	s.	PROPN
ejpam-4730	127	11	khalaf	khalaf	PROPN
ejpam-4730	127	12	/	/	SYM
ejpam-4730	127	13	eur	eur	PROPN
ejpam-4730	127	14	.	.	PUNCT
ejpam-4730	128	1	j.	j.	PROPN
ejpam-4730	128	2	pure	pure	PROPN
ejpam-4730	128	3	appl	appl	PROPN
ejpam-4730	128	4	.	.	PROPN
ejpam-4730	128	5	math	math	PROPN
ejpam-4730	128	6	,	,	PUNCT
ejpam-4730	128	7	16	16	NUM
ejpam-4730	128	8	(	(	PUNCT
ejpam-4730	128	9	2	2	NUM
ejpam-4730	128	10	)	)	PUNCT
ejpam-4730	128	11	(	(	PUNCT
ejpam-4730	128	12	2023	2023	NUM
ejpam-4730	128	13	)	)	PUNCT
ejpam-4730	128	14	,	,	PUNCT
ejpam-4730	128	15	944	944	NUM
ejpam-4730	128	16	-	-	SYM
ejpam-4730	128	17	952	952	NUM
ejpam-4730	128	18	950	950	NUM
ejpam-4730	128	19	theorem	theorem	NOUN
ejpam-4730	128	20	2	2	NUM
ejpam-4730	128	21	.	.	PUNCT
ejpam-4730	129	1	let	let	VERB
ejpam-4730	129	2	f	f	PROPN
ejpam-4730	129	3	∈	∈	PROPN
ejpam-4730	129	4	lp	lp	PROPN
ejpam-4730	129	5	,	,	PUNCT
ejpam-4730	129	6	λ[−π	λ[−π	PROPN
ejpam-4730	129	7	,	,	PUNCT
ejpam-4730	129	8	π	π	PROPN
ejpam-4730	129	9	]	]	X
ejpam-4730	129	10	,	,	PUNCT
ejpam-4730	129	11	1	1	NUM
ejpam-4730	129	12	≤	≤	NOUN
ejpam-4730	130	1	p	p	NOUN
ejpam-4730	130	2	<	<	X
ejpam-4730	130	3	∞	∞	PROPN
ejpam-4730	130	4	,	,	PUNCT
ejpam-4730	130	5	ξ	ξ	X
ejpam-4730	130	6	>	>	SYM
ejpam-4730	130	7	0	0	PROPN
ejpam-4730	130	8	and	and	CCONJ
ejpam-4730	130	9	l	l	NOUN
ejpam-4730	130	10	,	,	PUNCT
ejpam-4730	130	11	k	k	PROPN
ejpam-4730	130	12	∈	∈	PROPN
ejpam-4730	130	13	n	n	ADV
ejpam-4730	130	14	.	.	PUNCT
ejpam-4730	131	1	then	then	ADV
ejpam-4730	131	2	,	,	PUNCT
ejpam-4730	131	3	ek(f	ek(f	X
ejpam-4730	131	4	,	,	PUNCT
ejpam-4730	131	5	ξ)lp	ξ)lp	PROPN
ejpam-4730	131	6	,	,	PUNCT
ejpam-4730	131	7	λ[−π	λ[−π	PROPN
ejpam-4730	131	8	,	,	PUNCT
ejpam-4730	131	9	π	π	PROPN
ejpam-4730	131	10	]	]	X
ejpam-4730	131	11	≤	≤	PROPN
ejpam-4730	131	12	∥jk	∥jk	PROPN
ejpam-4730	131	13	,	,	PUNCT
ejpam-4730	131	14	l(.)−	l(.)−	PROPN
ejpam-4730	131	15	f∥lp	f∥lp	PROPN
ejpam-4730	131	16	,	,	PUNCT
ejpam-4730	131	17	λ[−π	λ[−π	PROPN
ejpam-4730	131	18	,	,	PUNCT
ejpam-4730	131	19	π	π	NOUN
ejpam-4730	131	20	]	]	X
ejpam-4730	131	21	≤	≤	PROPN
ejpam-4730	131	22	ckµl(f	ckµl(f	PROPN
ejpam-4730	131	23	,	,	PUNCT
ejpam-4730	131	24	1	1	NUM
ejpam-4730	131	25	l	l	NOUN
ejpam-4730	131	26	)	)	PUNCT
ejpam-4730	131	27	lp	lp	NOUN
ejpam-4730	131	28	,	,	PUNCT
ejpam-4730	131	29	λ[−π	λ[−π	PROPN
ejpam-4730	131	30	,	,	PUNCT
ejpam-4730	131	31	π	π	X
ejpam-4730	131	32	]	]	X
ejpam-4730	131	33	,	,	PUNCT
ejpam-4730	131	34	where	where	SCONJ
ejpam-4730	131	35	ck	ck	PROPN
ejpam-4730	131	36	is	be	AUX
ejpam-4730	131	37	a	a	DET
ejpam-4730	131	38	positive	positive	ADJ
ejpam-4730	131	39	constant	constant	NOUN
ejpam-4730	131	40	and	and	CCONJ
ejpam-4730	131	41	ji	ji	PROPN
ejpam-4730	131	42	,	,	PUNCT
ejpam-4730	131	43	j(x	j(x	PROPN
ejpam-4730	131	44	)	)	PUNCT
ejpam-4730	131	45	is	be	AUX
ejpam-4730	131	46	the	the	DET
ejpam-4730	131	47	jackson	jackson	PROPN
ejpam-4730	131	48	operator	operator	NOUN
ejpam-4730	131	49	with	with	ADP
ejpam-4730	131	50	x	x	PROPN
ejpam-4730	131	51	∈	∈	PROPN
ejpam-4730	132	1	[	[	X
ejpam-4730	132	2	−π	−π	PROPN
ejpam-4730	132	3	,	,	PUNCT
ejpam-4730	132	4	π	π	PROPN
ejpam-4730	132	5	]	]	X
ejpam-4730	132	6	that	that	PRON
ejpam-4730	132	7	takes	take	VERB
ejpam-4730	132	8	i	i	PRON
ejpam-4730	132	9	=	=	PUNCT
ejpam-4730	133	1	[	[	X
ejpam-4730	133	2	(	(	PUNCT
ejpam-4730	133	3	k	k	X
ejpam-4730	133	4	+	+	X
ejpam-4730	133	5	3)/2	3)/2	NUM
ejpam-4730	133	6	]	]	PUNCT
ejpam-4730	133	7	and	and	CCONJ
ejpam-4730	133	8	j	j	NOUN
ejpam-4730	134	1	=	=	PUNCT
ejpam-4730	135	1	[	[	X
ejpam-4730	135	2	l	l	X
ejpam-4730	135	3	/	/	SYM
ejpam-4730	135	4	i	i	NOUN
ejpam-4730	135	5	]	]	PUNCT
ejpam-4730	136	1	+	+	CCONJ
ejpam-4730	136	2	1	1	X
ejpam-4730	136	3	.	.	X
ejpam-4730	136	4	proof	proof	NOUN
ejpam-4730	136	5	.	.	PUNCT
ejpam-4730	137	1	we	we	PRON
ejpam-4730	137	2	have	have	VERB
ejpam-4730	137	3	the	the	DET
ejpam-4730	137	4	operator	operator	NOUN
ejpam-4730	137	5	ji	ji	PROPN
ejpam-4730	137	6	,	,	PUNCT
ejpam-4730	137	7	j(x	j(x	PROPN
ejpam-4730	137	8	)	)	PUNCT
ejpam-4730	137	9	belongs	belong	VERB
ejpam-4730	137	10	to	to	ADP
ejpam-4730	137	11	the	the	DET
ejpam-4730	137	12	space	space	NOUN
ejpam-4730	137	13	tk	tk	PROPN
ejpam-4730	137	14	.	.	PROPN
ejpam-4730	137	15	therefore	therefore	ADV
ejpam-4730	137	16	,	,	PUNCT
ejpam-4730	137	17	by	by	ADP
ejpam-4730	137	18	theorem	theorem	NOUN
ejpam-4730	137	19	1	1	NUM
ejpam-4730	137	20	,	,	PUNCT
ejpam-4730	137	21	we	we	PRON
ejpam-4730	137	22	obtain	obtain	VERB
ejpam-4730	137	23	ek(f	ek(f	NOUN
ejpam-4730	137	24	,	,	PUNCT
ejpam-4730	137	25	ξ)lp	ξ)lp	PROPN
ejpam-4730	137	26	,	,	PUNCT
ejpam-4730	137	27	λ[−π	λ[−π	PROPN
ejpam-4730	137	28	,	,	PUNCT
ejpam-4730	137	29	π	π	PROPN
ejpam-4730	137	30	]	]	X
ejpam-4730	137	31	≤	≤	PROPN
ejpam-4730	137	32	∥jk	∥jk	PROPN
ejpam-4730	137	33	,	,	PUNCT
ejpam-4730	137	34	l(.)−	l(.)−	PROPN
ejpam-4730	137	35	f∥lp	f∥lp	PROPN
ejpam-4730	137	36	,	,	PUNCT
ejpam-4730	137	37	λ[−π	λ[−π	PROPN
ejpam-4730	137	38	,	,	PUNCT
ejpam-4730	137	39	π	π	NOUN
ejpam-4730	137	40	]	]	X
ejpam-4730	137	41	≤	≤	PROPN
ejpam-4730	137	42	ckµl(f	ckµl(f	PROPN
ejpam-4730	137	43	,	,	PUNCT
ejpam-4730	137	44	ξ)lp	ξ)lp	PROPN
ejpam-4730	137	45	,	,	PUNCT
ejpam-4730	137	46	λ[−π	λ[−π	PROPN
ejpam-4730	137	47	,	,	PUNCT
ejpam-4730	137	48	π	π	X
ejpam-4730	137	49	]	]	X
ejpam-4730	137	50	∑l	∑l	ADJ
ejpam-4730	137	51	i=0	i=0	PROPN
ejpam-4730	137	52	(	(	PUNCT
ejpam-4730	137	53	l	l	X
ejpam-4730	137	54	i	i	NOUN
ejpam-4730	137	55	)	)	PUNCT
ejpam-4730	137	56	lg(g	lg(g	PROPN
ejpam-4730	137	57	,	,	PUNCT
ejpam-4730	137	58	i	i	NOUN
ejpam-4730	137	59	)	)	PUNCT
ejpam-4730	137	60	and	and	CCONJ
ejpam-4730	137	61	this	this	PRON
ejpam-4730	137	62	completes	complete	VERB
ejpam-4730	137	63	the	the	DET
ejpam-4730	137	64	proof	proof	NOUN
ejpam-4730	137	65	.	.	PUNCT
ejpam-4730	138	1	theorem	theorem	NOUN
ejpam-4730	138	2	3	3	X
ejpam-4730	138	3	.	.	PUNCT
ejpam-4730	139	1	let	let	VERB
ejpam-4730	139	2	{	{	PUNCT
ejpam-4730	139	3	ψk}k=0,1,2	ψk}k=0,1,2	VERB
ejpam-4730	139	4	,	,	PUNCT
ejpam-4730	139	5	.	.	PUNCT
ejpam-4730	139	6	.	.	PUNCT
ejpam-4730	139	7	.	.	PUNCT
ejpam-4730	140	1	be	be	AUX
ejpam-4730	140	2	a	a	DET
ejpam-4730	140	3	sequence	sequence	NOUN
ejpam-4730	140	4	of	of	ADP
ejpam-4730	140	5	operators	operator	NOUN
ejpam-4730	140	6	in	in	ADP
ejpam-4730	140	7	the	the	DET
ejpam-4730	140	8	space	space	NOUN
ejpam-4730	140	9	tk	tk	NOUN
ejpam-4730	140	10	satisfying	satisfying	NOUN
ejpam-4730	140	11	ψk(p	ψk(p	PUNCT
ejpam-4730	140	12	)	)	PUNCT
ejpam-4730	141	1	=	=	SYM
ejpam-4730	141	2	p	p	X
ejpam-4730	141	3	,	,	PUNCT
ejpam-4730	141	4	for	for	ADP
ejpam-4730	141	5	each	each	DET
ejpam-4730	141	6	p	p	NOUN
ejpam-4730	141	7	that	that	PRON
ejpam-4730	141	8	belongs	belong	VERB
ejpam-4730	141	9	to	to	ADP
ejpam-4730	141	10	the	the	DET
ejpam-4730	141	11	subspace	subspace	NOUN
ejpam-4730	141	12	sk	sk	NOUN
ejpam-4730	141	13	of	of	ADP
ejpam-4730	141	14	lp	lp	NOUN
ejpam-4730	141	15	,	,	PUNCT
ejpam-4730	141	16	λ[−π	λ[−π	PROPN
ejpam-4730	141	17	,	,	PUNCT
ejpam-4730	141	18	π	π	X
ejpam-4730	141	19	]	]	X
ejpam-4730	141	20	and	and	CCONJ
ejpam-4730	141	21	l	l	PROPN
ejpam-4730	141	22	∈	∈	PROPN
ejpam-4730	141	23	n.	n.	NOUN
ejpam-4730	141	24	then	then	ADV
ejpam-4730	141	25	,	,	PUNCT
ejpam-4730	141	26	for	for	ADP
ejpam-4730	141	27	all	all	DET
ejpam-4730	141	28	f	f	PROPN
ejpam-4730	141	29	∈	∈	PROPN
ejpam-4730	141	30	lp	lp	NOUN
ejpam-4730	141	31	,	,	PUNCT
ejpam-4730	141	32	λ[−π	λ[−π	PROPN
ejpam-4730	141	33	,	,	PUNCT
ejpam-4730	141	34	π	π	PROPN
ejpam-4730	141	35	]	]	X
ejpam-4730	141	36	,	,	PUNCT
ejpam-4730	141	37	we	we	PRON
ejpam-4730	141	38	have	have	VERB
ejpam-4730	141	39	∥f	∥f	PROPN
ejpam-4730	141	40	−	−	PROPN
ejpam-4730	141	41	ψk(f)∥lp	ψk(f)∥lp	NOUN
ejpam-4730	141	42	,	,	PUNCT
ejpam-4730	141	43	λ[−π	λ[−π	PROPN
ejpam-4730	141	44	,	,	PUNCT
ejpam-4730	141	45	π	π	NOUN
ejpam-4730	141	46	]	]	X
ejpam-4730	141	47	≤	≤	X
ejpam-4730	141	48	(	(	PUNCT
ejpam-4730	141	49	∥ψk∥lp	∥ψk∥lp	PROPN
ejpam-4730	141	50	,	,	PUNCT
ejpam-4730	141	51	λ[−π	λ[−π	PROPN
ejpam-4730	141	52	,	,	PUNCT
ejpam-4730	141	53	π	π	X
ejpam-4730	141	54	]	]	X
ejpam-4730	142	1	+	+	CCONJ
ejpam-4730	142	2	1)ek(f	1)ek(f	NUM
ejpam-4730	142	3	,	,	PUNCT
ejpam-4730	142	4	1	1	NUM
ejpam-4730	142	5	k	k	NOUN
ejpam-4730	142	6	)	)	PUNCT
ejpam-4730	142	7	lp	lp	NOUN
ejpam-4730	142	8	,	,	PUNCT
ejpam-4730	142	9	λ[−π	λ[−π	PROPN
ejpam-4730	142	10	,	,	PUNCT
ejpam-4730	142	11	π	π	PROPN
ejpam-4730	142	12	]	]	X
ejpam-4730	142	13	≤	≤	X
ejpam-4730	142	14	ck(∥ψk∥lp	ck(∥ψk∥lp	PROPN
ejpam-4730	142	15	,	,	PUNCT
ejpam-4730	142	16	λ[−π	λ[−π	PROPN
ejpam-4730	142	17	,	,	PUNCT
ejpam-4730	142	18	π	π	X
ejpam-4730	142	19	]	]	X
ejpam-4730	142	20	+	+	CCONJ
ejpam-4730	142	21	1)µk(f	1)µk(f	NUM
ejpam-4730	142	22	,	,	PUNCT
ejpam-4730	142	23	1	1	NUM
ejpam-4730	142	24	k	k	NOUN
ejpam-4730	142	25	)	)	PUNCT
ejpam-4730	142	26	lp	lp	NOUN
ejpam-4730	142	27	,	,	PUNCT
ejpam-4730	142	28	λ[−π	λ[−π	PROPN
ejpam-4730	142	29	,	,	PUNCT
ejpam-4730	142	30	π	π	NOUN
ejpam-4730	142	31	]	]	PUNCT
ejpam-4730	142	32	.	.	PUNCT
ejpam-4730	143	1	proof	proof	NOUN
ejpam-4730	143	2	.	.	PUNCT
ejpam-4730	144	1	let	let	VERB
ejpam-4730	144	2	p	p	PRON
ejpam-4730	144	3	be	be	AUX
ejpam-4730	144	4	a	a	DET
ejpam-4730	144	5	function	function	NOUN
ejpam-4730	144	6	in	in	ADP
ejpam-4730	144	7	the	the	DET
ejpam-4730	144	8	space	space	NOUN
ejpam-4730	144	9	ψk	ψk	NOUN
ejpam-4730	144	10	.then	.then	PUNCT
ejpam-4730	145	1	∥f	∥f	PROPN
ejpam-4730	145	2	−	−	PROPN
ejpam-4730	145	3	ψk(f)∥lp	ψk(f)∥lp	NOUN
ejpam-4730	145	4	,	,	PUNCT
ejpam-4730	145	5	λ[−π	λ[−π	PROPN
ejpam-4730	145	6	,	,	PUNCT
ejpam-4730	145	7	π	π	X
ejpam-4730	145	8	]	]	X
ejpam-4730	145	9	≤	≤	PROPN
ejpam-4730	146	1	∥f	∥f	PROPN
ejpam-4730	146	2	−	−	PROPN
ejpam-4730	146	3	p∥lp	p∥lp	PROPN
ejpam-4730	146	4	,	,	PUNCT
ejpam-4730	146	5	λ[−π	λ[−π	PROPN
ejpam-4730	146	6	,	,	PUNCT
ejpam-4730	146	7	π	π	X
ejpam-4730	146	8	]	]	X
ejpam-4730	147	1	+	+	CCONJ
ejpam-4730	148	1	∥f	∥f	PROPN
ejpam-4730	148	2	−	−	PROPN
ejpam-4730	148	3	ψk∥lp	ψk∥lp	PROPN
ejpam-4730	148	4	,	,	PUNCT
ejpam-4730	148	5	λ[−π	λ[−π	PROPN
ejpam-4730	148	6	,	,	PUNCT
ejpam-4730	148	7	π	π	X
ejpam-4730	148	8	]	]	X
ejpam-4730	148	9	≤	≤	X
ejpam-4730	148	10	(	(	PUNCT
ejpam-4730	148	11	∥ψk∥lp	∥ψk∥lp	PROPN
ejpam-4730	148	12	,	,	PUNCT
ejpam-4730	148	13	λ[−π	λ[−π	PROPN
ejpam-4730	148	14	,	,	PUNCT
ejpam-4730	148	15	π	π	X
ejpam-4730	148	16	]	]	X
ejpam-4730	149	1	+	+	CCONJ
ejpam-4730	149	2	1))∥f	1))∥f	PROPN
ejpam-4730	149	3	−	−	PROPN
ejpam-4730	149	4	p∥lp	p∥lp	PROPN
ejpam-4730	149	5	,	,	PUNCT
ejpam-4730	149	6	λ[−π	λ[−π	PROPN
ejpam-4730	149	7	,	,	PUNCT
ejpam-4730	149	8	π	π	NOUN
ejpam-4730	149	9	]	]	PUNCT
ejpam-4730	149	10	.	.	PUNCT
ejpam-4730	150	1	from	from	ADP
ejpam-4730	150	2	equation	equation	NOUN
ejpam-4730	150	3	(	(	PUNCT
ejpam-4730	150	4	1	1	NUM
ejpam-4730	150	5	)	)	PUNCT
ejpam-4730	150	6	,	,	PUNCT
ejpam-4730	150	7	we	we	PRON
ejpam-4730	150	8	have	have	VERB
ejpam-4730	150	9	∥f	∥f	PROPN
ejpam-4730	150	10	−	−	PROPN
ejpam-4730	150	11	ψk(f)∥lp	ψk(f)∥lp	NOUN
ejpam-4730	150	12	,	,	PUNCT
ejpam-4730	150	13	λ[−π	λ[−π	PROPN
ejpam-4730	150	14	,	,	PUNCT
ejpam-4730	150	15	π	π	NOUN
ejpam-4730	150	16	]	]	X
ejpam-4730	150	17	≤	≤	X
ejpam-4730	150	18	(	(	PUNCT
ejpam-4730	150	19	∥ψk∥lp	∥ψk∥lp	PROPN
ejpam-4730	150	20	,	,	PUNCT
ejpam-4730	150	21	λ[−π	λ[−π	PROPN
ejpam-4730	150	22	,	,	PUNCT
ejpam-4730	150	23	π	π	X
ejpam-4730	150	24	]	]	X
ejpam-4730	151	1	+	+	CCONJ
ejpam-4730	151	2	1)ek(f	1)ek(f	NUM
ejpam-4730	151	3	,	,	PUNCT
ejpam-4730	151	4	i	i	PRON
ejpam-4730	151	5	k	k	PROPN
ejpam-4730	151	6	)	)	PUNCT
ejpam-4730	151	7	lp	lp	PROPN
ejpam-4730	151	8	,	,	PUNCT
ejpam-4730	151	9	λ[−π	λ[−π	PROPN
ejpam-4730	151	10	,	,	PUNCT
ejpam-4730	151	11	π	π	NOUN
ejpam-4730	151	12	]	]	X
ejpam-4730	151	13	.	.	PUNCT
ejpam-4730	152	1	also	also	ADV
ejpam-4730	152	2	,	,	PUNCT
ejpam-4730	152	3	by	by	ADP
ejpam-4730	152	4	using	use	VERB
ejpam-4730	152	5	theorem	theorem	NOUN
ejpam-4730	152	6	2	2	NUM
ejpam-4730	152	7	,	,	PUNCT
ejpam-4730	152	8	we	we	PRON
ejpam-4730	152	9	obtain	obtain	VERB
ejpam-4730	152	10	∥f	∥f	PROPN
ejpam-4730	152	11	−	−	PROPN
ejpam-4730	152	12	ψk(f)∥lp	ψk(f)∥lp	NOUN
ejpam-4730	152	13	,	,	PUNCT
ejpam-4730	152	14	λ[−π	λ[−π	PROPN
ejpam-4730	152	15	,	,	PUNCT
ejpam-4730	152	16	π	π	PROPN
ejpam-4730	152	17	]	]	X
ejpam-4730	152	18	≤	≤	X
ejpam-4730	152	19	ck(∥ψk∥lp	ck(∥ψk∥lp	PROPN
ejpam-4730	152	20	,	,	PUNCT
ejpam-4730	152	21	λ[−π	λ[−π	PROPN
ejpam-4730	152	22	,	,	PUNCT
ejpam-4730	152	23	π	π	X
ejpam-4730	152	24	]	]	X
ejpam-4730	152	25	+	+	CCONJ
ejpam-4730	153	1	1)µk(f	1)µk(f	NUM
ejpam-4730	153	2	,	,	PUNCT
ejpam-4730	153	3	i	i	PRON
ejpam-4730	153	4	k	k	PROPN
ejpam-4730	153	5	)	)	PUNCT
ejpam-4730	153	6	lp	lp	PROPN
ejpam-4730	153	7	,	,	PUNCT
ejpam-4730	153	8	λ[−π	λ[−π	PROPN
ejpam-4730	153	9	,	,	PUNCT
ejpam-4730	153	10	π	π	X
ejpam-4730	153	11	]	]	X
ejpam-4730	153	12	,	,	PUNCT
ejpam-4730	153	13	and	and	CCONJ
ejpam-4730	153	14	consequently	consequently	ADV
ejpam-4730	153	15	the	the	DET
ejpam-4730	153	16	proof	proof	NOUN
ejpam-4730	153	17	follows	follow	VERB
ejpam-4730	153	18	.	.	PUNCT
ejpam-4730	154	1	references	reference	NOUN
ejpam-4730	154	2	951	951	NUM
ejpam-4730	154	3	4	4	NUM
ejpam-4730	154	4	.	.	PUNCT
ejpam-4730	154	5	conclusion	conclusion	NOUN
ejpam-4730	154	6	in	in	ADP
ejpam-4730	154	7	this	this	DET
ejpam-4730	154	8	study	study	NOUN
ejpam-4730	154	9	,	,	PUNCT
ejpam-4730	154	10	we	we	PRON
ejpam-4730	154	11	have	have	AUX
ejpam-4730	154	12	demonstrated	demonstrate	VERB
ejpam-4730	154	13	the	the	DET
ejpam-4730	154	14	direct	direct	ADJ
ejpam-4730	154	15	trigonometric	trigonometric	ADJ
ejpam-4730	154	16	approximation	approximation	NOUN
ejpam-4730	154	17	theorems	theorem	NOUN
ejpam-4730	154	18	of	of	ADP
ejpam-4730	154	19	unbounded	unbounded	ADJ
ejpam-4730	154	20	functions	function	NOUN
ejpam-4730	154	21	in	in	ADP
ejpam-4730	154	22	a	a	DET
ejpam-4730	154	23	weighted	weight	VERB
ejpam-4730	154	24	space	space	NOUN
ejpam-4730	154	25	defined	define	VERB
ejpam-4730	154	26	on	on	ADP
ejpam-4730	154	27	the	the	DET
ejpam-4730	154	28	interval	interval	NOUN
ejpam-4730	155	1	[	[	X
ejpam-4730	155	2	−π	−π	ADV
ejpam-4730	155	3	,	,	PUNCT
ejpam-4730	155	4	π	π	PROPN
ejpam-4730	155	5	]	]	X
ejpam-4730	155	6	.	.	PUNCT
ejpam-4730	156	1	our	our	PRON
ejpam-4730	156	2	results	result	NOUN
ejpam-4730	156	3	are	be	AUX
ejpam-4730	156	4	based	base	VERB
ejpam-4730	156	5	on	on	ADP
ejpam-4730	156	6	the	the	DET
ejpam-4730	156	7	use	use	NOUN
ejpam-4730	156	8	of	of	ADP
ejpam-4730	156	9	various	various	ADJ
ejpam-4730	156	10	linear	linear	PROPN
ejpam-4730	156	11	operators	operator	NOUN
ejpam-4730	156	12	and	and	CCONJ
ejpam-4730	156	13	provide	provide	VERB
ejpam-4730	156	14	insights	insight	NOUN
ejpam-4730	156	15	into	into	ADP
ejpam-4730	156	16	the	the	DET
ejpam-4730	156	17	properties	property	NOUN
ejpam-4730	156	18	of	of	ADP
ejpam-4730	156	19	the	the	DET
ejpam-4730	156	20	modulus	modulus	NOUN
ejpam-4730	156	21	of	of	ADP
ejpam-4730	156	22	smoothness	smoothness	NOUN
ejpam-4730	156	23	within	within	ADP
ejpam-4730	156	24	the	the	DET
ejpam-4730	156	25	same	same	ADJ
ejpam-4730	156	26	space	space	NOUN
ejpam-4730	156	27	.	.	PUNCT
ejpam-4730	157	1	while	while	SCONJ
ejpam-4730	157	2	we	we	PRON
ejpam-4730	157	3	did	do	AUX
ejpam-4730	157	4	not	not	PART
ejpam-4730	157	5	provide	provide	VERB
ejpam-4730	157	6	a	a	DET
ejpam-4730	157	7	specific	specific	ADJ
ejpam-4730	157	8	example	example	NOUN
ejpam-4730	157	9	in	in	ADP
ejpam-4730	157	10	this	this	DET
ejpam-4730	157	11	paper	paper	NOUN
ejpam-4730	157	12	,	,	PUNCT
ejpam-4730	157	13	our	our	PRON
ejpam-4730	157	14	findings	finding	NOUN
ejpam-4730	157	15	are	be	AUX
ejpam-4730	157	16	applicable	applicable	ADJ
ejpam-4730	157	17	to	to	ADP
ejpam-4730	157	18	a	a	DET
ejpam-4730	157	19	wide	wide	ADJ
ejpam-4730	157	20	range	range	NOUN
ejpam-4730	157	21	of	of	ADP
ejpam-4730	157	22	functions	function	NOUN
ejpam-4730	157	23	and	and	CCONJ
ejpam-4730	157	24	have	have	VERB
ejpam-4730	157	25	important	important	ADJ
ejpam-4730	157	26	implications	implication	NOUN
ejpam-4730	157	27	for	for	ADP
ejpam-4730	157	28	the	the	DET
ejpam-4730	157	29	field	field	NOUN
ejpam-4730	157	30	of	of	ADP
ejpam-4730	157	31	approximation	approximation	NOUN
ejpam-4730	157	32	theory	theory	NOUN
ejpam-4730	157	33	.	.	PUNCT
ejpam-4730	158	1	we	we	PRON
ejpam-4730	158	2	believe	believe	VERB
ejpam-4730	158	3	that	that	SCONJ
ejpam-4730	158	4	our	our	PRON
ejpam-4730	158	5	results	result	NOUN
ejpam-4730	158	6	will	will	AUX
ejpam-4730	158	7	inspire	inspire	VERB
ejpam-4730	158	8	further	further	ADJ
ejpam-4730	158	9	research	research	NOUN
ejpam-4730	158	10	in	in	ADP
ejpam-4730	158	11	this	this	DET
ejpam-4730	158	12	area	area	NOUN
ejpam-4730	158	13	.	.	PUNCT
ejpam-4730	159	1	references	reference	NOUN
ejpam-4730	159	2	[	[	X
ejpam-4730	159	3	1	1	X
ejpam-4730	159	4	]	]	PUNCT
ejpam-4730	159	5	noori	noori	PROPN
ejpam-4730	159	6	yasir	yasir	PROPN
ejpam-4730	159	7	abdul	abdul	PROPN
ejpam-4730	159	8	-	-	PUNCT
ejpam-4730	159	9	hassan	hassan	PROPN
ejpam-4730	159	10	,	,	PUNCT
ejpam-4730	159	11	ali	ali	PROPN
ejpam-4730	159	12	hasan	hasan	PROPN
ejpam-4730	159	13	ali	ali	PROPN
ejpam-4730	159	14	,	,	PUNCT
ejpam-4730	159	15	and	and	CCONJ
ejpam-4730	159	16	choonkil	choonkil	PROPN
ejpam-4730	159	17	park	park	NOUN
ejpam-4730	159	18	.	.	PUNCT
ejpam-4730	160	1	a	a	DET
ejpam-4730	160	2	new	new	ADJ
ejpam-4730	160	3	fifth	fifth	ADJ
ejpam-4730	160	4	-	-	PUNCT
ejpam-4730	160	5	order	order	NOUN
ejpam-4730	160	6	iterative	iterative	NOUN
ejpam-4730	160	7	method	method	NOUN
ejpam-4730	160	8	free	free	ADJ
ejpam-4730	160	9	from	from	ADP
ejpam-4730	160	10	second	second	ADJ
ejpam-4730	160	11	derivative	derivative	NOUN
ejpam-4730	160	12	for	for	ADP
ejpam-4730	160	13	solving	solve	VERB
ejpam-4730	160	14	nonlinear	nonlinear	ADJ
ejpam-4730	160	15	equations	equation	NOUN
ejpam-4730	160	16	.	.	PUNCT
ejpam-4730	161	1	journal	journal	NOUN
ejpam-4730	161	2	of	of	ADP
ejpam-4730	161	3	applied	apply	VERB
ejpam-4730	161	4	mathematics	mathematic	NOUN
ejpam-4730	161	5	and	and	CCONJ
ejpam-4730	161	6	computing	computing	NOUN
ejpam-4730	161	7	,	,	PUNCT
ejpam-4730	161	8	pages	page	NOUN
ejpam-4730	161	9	1–10	1–10	NOUN
ejpam-4730	161	10	,	,	PUNCT
ejpam-4730	161	11	2021	2021	NUM
ejpam-4730	161	12	.	.	PUNCT
ejpam-4730	162	1	[	[	X
ejpam-4730	162	2	2	2	X
ejpam-4730	162	3	]	]	X
ejpam-4730	162	4	ghazi	ghazi	PROPN
ejpam-4730	162	5	abed	abe	VERB
ejpam-4730	162	6	meften	meften	PROPN
ejpam-4730	162	7	,	,	PUNCT
ejpam-4730	162	8	ali	ali	PROPN
ejpam-4730	162	9	hasan	hasan	PROPN
ejpam-4730	162	10	ali	ali	PROPN
ejpam-4730	162	11	,	,	PUNCT
ejpam-4730	162	12	khalil	khalil	PROPN
ejpam-4730	162	13	s	s	PROPN
ejpam-4730	162	14	al	al	PROPN
ejpam-4730	162	15	-	-	PUNCT
ejpam-4730	162	16	ghafri	ghafri	PROPN
ejpam-4730	162	17	,	,	PUNCT
ejpam-4730	162	18	jan	jan	PROPN
ejpam-4730	162	19	awrejcewicz	awrejcewicz	PROPN
ejpam-4730	162	20	,	,	PUNCT
ejpam-4730	162	21	and	and	CCONJ
ejpam-4730	162	22	omar	omar	PROPN
ejpam-4730	162	23	bazighifan	bazighifan	PROPN
ejpam-4730	162	24	.	.	PUNCT
ejpam-4730	163	1	nonlinear	nonlinear	ADJ
ejpam-4730	163	2	stability	stability	NOUN
ejpam-4730	163	3	and	and	CCONJ
ejpam-4730	163	4	linear	linear	ADJ
ejpam-4730	163	5	instability	instability	NOUN
ejpam-4730	163	6	of	of	ADP
ejpam-4730	163	7	double	double	ADJ
ejpam-4730	163	8	-	-	PUNCT
ejpam-4730	163	9	diffusive	diffusive	ADJ
ejpam-4730	163	10	convection	convection	NOUN
ejpam-4730	163	11	in	in	ADP
ejpam-4730	163	12	a	a	DET
ejpam-4730	163	13	rotating	rotating	NOUN
ejpam-4730	163	14	with	with	ADP
ejpam-4730	163	15	ltne	ltne	NOUN
ejpam-4730	163	16	effects	effect	NOUN
ejpam-4730	163	17	and	and	CCONJ
ejpam-4730	163	18	symmetric	symmetric	ADJ
ejpam-4730	163	19	properties	property	NOUN
ejpam-4730	163	20	:	:	PUNCT
ejpam-4730	163	21	brinkmann	brinkmann	PROPN
ejpam-4730	163	22	-	-	PUNCT
ejpam-4730	163	23	forchheimer	forchheimer	PROPN
ejpam-4730	163	24	model	model	NOUN
ejpam-4730	163	25	.	.	PUNCT
ejpam-4730	164	1	symmetry	symmetry	PROPN
ejpam-4730	164	2	,	,	PUNCT
ejpam-4730	164	3	14(3):565	14(3):565	NUM
ejpam-4730	164	4	,	,	PUNCT
ejpam-4730	164	5	2022	2022	NUM
ejpam-4730	164	6	.	.	PUNCT
ejpam-4730	165	1	[	[	X
ejpam-4730	165	2	3	3	X
ejpam-4730	165	3	]	]	X
ejpam-4730	165	4	ali	ali	PROPN
ejpam-4730	165	5	hasan	hasan	PROPN
ejpam-4730	165	6	ali	ali	PROPN
ejpam-4730	165	7	,	,	PUNCT
ejpam-4730	165	8	ahmed	ahmed	PROPN
ejpam-4730	165	9	shawki	shawki	PROPN
ejpam-4730	165	10	jaber	jaber	PROPN
ejpam-4730	165	11	,	,	PUNCT
ejpam-4730	165	12	mustafa	mustafa	PROPN
ejpam-4730	165	13	t	t	PROPN
ejpam-4730	165	14	yaseen	yaseen	PROPN
ejpam-4730	165	15	,	,	PUNCT
ejpam-4730	165	16	mohammed	mohammed	PROPN
ejpam-4730	165	17	rasheed	rasheed	PROPN
ejpam-4730	165	18	,	,	PUNCT
ejpam-4730	165	19	omer	omer	PROPN
ejpam-4730	165	20	bazighifan	bazighifan	PROPN
ejpam-4730	165	21	,	,	PUNCT
ejpam-4730	165	22	and	and	CCONJ
ejpam-4730	165	23	taher	taher	DET
ejpam-4730	165	24	a	a	DET
ejpam-4730	165	25	nofal	nofal	NOUN
ejpam-4730	165	26	.	.	PUNCT
ejpam-4730	166	1	a	a	DET
ejpam-4730	166	2	comparison	comparison	NOUN
ejpam-4730	166	3	of	of	ADP
ejpam-4730	166	4	finite	finite	ADJ
ejpam-4730	166	5	difference	difference	NOUN
ejpam-4730	166	6	and	and	CCONJ
ejpam-4730	166	7	finite	finite	ADJ
ejpam-4730	166	8	volume	volume	NOUN
ejpam-4730	166	9	methods	method	NOUN
ejpam-4730	166	10	with	with	ADP
ejpam-4730	166	11	numerical	numerical	ADJ
ejpam-4730	166	12	simulations	simulation	NOUN
ejpam-4730	166	13	:	:	PUNCT
ejpam-4730	166	14	burgers	burger	NOUN
ejpam-4730	166	15	equation	equation	NOUN
ejpam-4730	166	16	model	model	NOUN
ejpam-4730	166	17	.	.	PUNCT
ejpam-4730	167	1	complexity	complexity	NOUN
ejpam-4730	167	2	,	,	PUNCT
ejpam-4730	167	3	2022	2022	NUM
ejpam-4730	167	4	,	,	PUNCT
ejpam-4730	167	5	2022	2022	NUM
ejpam-4730	167	6	.	.	PUNCT
ejpam-4730	168	1	[	[	X
ejpam-4730	168	2	4	4	X
ejpam-4730	168	3	]	]	X
ejpam-4730	168	4	ali	ali	PROPN
ejpam-4730	168	5	hasan	hasan	PROPN
ejpam-4730	168	6	ali	ali	PROPN
ejpam-4730	168	7	,	,	PUNCT
ejpam-4730	168	8	ghazi	ghazi	PROPN
ejpam-4730	168	9	abed	abe	VERB
ejpam-4730	168	10	meften	meften	PROPN
ejpam-4730	168	11	,	,	PUNCT
ejpam-4730	168	12	omar	omar	PROPN
ejpam-4730	168	13	bazighifan	bazighifan	PROPN
ejpam-4730	168	14	,	,	PUNCT
ejpam-4730	168	15	mehak	mehak	PROPN
ejpam-4730	168	16	iqbal	iqbal	PROPN
ejpam-4730	168	17	,	,	PUNCT
ejpam-4730	168	18	sergio	sergio	PROPN
ejpam-4730	168	19	elaskar	elaskar	PROPN
ejpam-4730	168	20	,	,	PUNCT
ejpam-4730	168	21	and	and	CCONJ
ejpam-4730	168	22	jan	jan	PROPN
ejpam-4730	168	23	awrejcewicz	awrejcewicz	PROPN
ejpam-4730	168	24	.	.	PUNCT
ejpam-4730	169	1	a	a	DET
ejpam-4730	169	2	study	study	NOUN
ejpam-4730	169	3	of	of	ADP
ejpam-4730	169	4	continuous	continuous	ADJ
ejpam-4730	169	5	dependence	dependence	NOUN
ejpam-4730	169	6	and	and	CCONJ
ejpam-4730	169	7	symmetric	symmetric	ADJ
ejpam-4730	169	8	properties	property	NOUN
ejpam-4730	169	9	of	of	ADP
ejpam-4730	169	10	double	double	ADJ
ejpam-4730	169	11	diffusive	diffusive	ADJ
ejpam-4730	169	12	convection	convection	NOUN
ejpam-4730	169	13	:	:	PUNCT
ejpam-4730	169	14	forchheimer	forchheimer	PROPN
ejpam-4730	169	15	model	model	PROPN
ejpam-4730	169	16	.	.	PUNCT
ejpam-4730	169	17	symmetry	symmetry	PROPN
ejpam-4730	169	18	,	,	PUNCT
ejpam-4730	169	19	14(4):682	14(4):682	NUM
ejpam-4730	169	20	,	,	PUNCT
ejpam-4730	169	21	2022	2022	NUM
ejpam-4730	169	22	.	.	PUNCT
ejpam-4730	170	1	[	[	X
ejpam-4730	170	2	5	5	X
ejpam-4730	170	3	]	]	PUNCT
ejpam-4730	170	4	ali	ali	PROPN
ejpam-4730	170	5	hasan	hasan	PROPN
ejpam-4730	170	6	ali	ali	PROPN
ejpam-4730	170	7	and	and	CCONJ
ejpam-4730	170	8	zsolt	zsolt	PROPN
ejpam-4730	170	9	pales	pale	NOUN
ejpam-4730	170	10	.	.	PUNCT
ejpam-4730	171	1	taylor	taylor	NOUN
ejpam-4730	171	2	-	-	PUNCT
ejpam-4730	171	3	type	type	NOUN
ejpam-4730	171	4	expansions	expansion	NOUN
ejpam-4730	171	5	in	in	ADP
ejpam-4730	171	6	terms	term	NOUN
ejpam-4730	171	7	of	of	ADP
ejpam-4730	171	8	exponential	exponential	ADJ
ejpam-4730	171	9	polynomials	polynomial	NOUN
ejpam-4730	171	10	.	.	PUNCT
ejpam-4730	172	1	math	math	NOUN
ejpam-4730	172	2	.	.	PUNCT
ejpam-4730	173	1	inequal	inequal	PROPN
ejpam-4730	173	2	.	.	PUNCT
ejpam-4730	174	1	appl	appl	PROPN
ejpam-4730	174	2	,	,	PUNCT
ejpam-4730	174	3	25:1123–1141	25:1123–1141	NUM
ejpam-4730	174	4	,	,	PUNCT
ejpam-4730	174	5	2022	2022	NUM
ejpam-4730	174	6	.	.	PUNCT
ejpam-4730	175	1	[	[	X
ejpam-4730	175	2	6	6	X
ejpam-4730	175	3	]	]	PUNCT
ejpam-4730	175	4	alaa	alaa	PROPN
ejpam-4730	175	5	adnan	adnan	PROPN
ejpam-4730	175	6	auad	auad	PROPN
ejpam-4730	175	7	and	and	CCONJ
ejpam-4730	175	8	mohammed	mohammed	PROPN
ejpam-4730	175	9	hamad	hamad	PROPN
ejpam-4730	175	10	fayyadh	fayyadh	PROPN
ejpam-4730	175	11	.	.	PUNCT
ejpam-4730	176	1	the	the	DET
ejpam-4730	176	2	direct	direct	ADJ
ejpam-4730	176	3	and	and	CCONJ
ejpam-4730	176	4	converse	converse	NOUN
ejpam-4730	176	5	theorems	theorem	NOUN
ejpam-4730	176	6	for	for	ADP
ejpam-4730	176	7	best	good	ADJ
ejpam-4730	176	8	approximation	approximation	NOUN
ejpam-4730	176	9	of	of	ADP
ejpam-4730	176	10	algebraic	algebraic	ADJ
ejpam-4730	176	11	polynomial	polynomial	NOUN
ejpam-4730	176	12	in	in	ADP
ejpam-4730	176	13	lp	lp	PROPN
ejpam-4730	176	14	,	,	PUNCT
ejpam-4730	176	15	α	α	PROPN
ejpam-4730	176	16	(	(	PUNCT
ejpam-4730	176	17	x	x	NOUN
ejpam-4730	176	18	)	)	PUNCT
ejpam-4730	176	19	.	.	PUNCT
ejpam-4730	177	1	in	in	ADP
ejpam-4730	177	2	journal	journal	PROPN
ejpam-4730	177	3	of	of	ADP
ejpam-4730	177	4	physics	physics	PROPN
ejpam-4730	177	5	:	:	PUNCT
ejpam-4730	177	6	conference	conference	NOUN
ejpam-4730	177	7	series	series	NOUN
ejpam-4730	177	8	,	,	PUNCT
ejpam-4730	177	9	volume	volume	NOUN
ejpam-4730	177	10	1879	1879	NUM
ejpam-4730	177	11	,	,	PUNCT
ejpam-4730	177	12	page	page	NOUN
ejpam-4730	177	13	032010	032010	NUM
ejpam-4730	177	14	.	.	PUNCT
ejpam-4730	178	1	iop	iop	PROPN
ejpam-4730	178	2	publishing	publishing	NOUN
ejpam-4730	178	3	,	,	PUNCT
ejpam-4730	178	4	2021	2021	NUM
ejpam-4730	178	5	.	.	PUNCT
ejpam-4730	179	1	[	[	X
ejpam-4730	179	2	7	7	X
ejpam-4730	179	3	]	]	X
ejpam-4730	179	4	alaa	alaa	PROPN
ejpam-4730	179	5	adnan	adnan	PROPN
ejpam-4730	179	6	auad	auad	PROPN
ejpam-4730	179	7	and	and	CCONJ
ejpam-4730	179	8	abdulsttar	abdulsttar	PROPN
ejpam-4730	179	9	ali	ali	PROPN
ejpam-4730	179	10	hussein	hussein	PROPN
ejpam-4730	179	11	.	.	PUNCT
ejpam-4730	180	1	best	good	ADJ
ejpam-4730	180	2	simultaneous	simultaneous	ADJ
ejpam-4730	180	3	approximation	approximation	NOUN
ejpam-4730	180	4	in	in	ADP
ejpam-4730	180	5	weighted	weighted	ADJ
ejpam-4730	180	6	space	space	NOUN
ejpam-4730	180	7	.	.	PUNCT
ejpam-4730	181	1	in	in	ADP
ejpam-4730	181	2	journal	journal	PROPN
ejpam-4730	181	3	of	of	ADP
ejpam-4730	181	4	physics	physics	PROPN
ejpam-4730	181	5	:	:	PUNCT
ejpam-4730	181	6	conference	conference	NOUN
ejpam-4730	181	7	series	series	NOUN
ejpam-4730	181	8	,	,	PUNCT
ejpam-4730	181	9	volume	volume	NOUN
ejpam-4730	181	10	1234	1234	NUM
ejpam-4730	181	11	,	,	PUNCT
ejpam-4730	181	12	page	page	NOUN
ejpam-4730	181	13	012111	012111	NUM
ejpam-4730	181	14	.	.	PUNCT
ejpam-4730	182	1	iop	iop	PROPN
ejpam-4730	182	2	publishing	publishing	NOUN
ejpam-4730	182	3	,	,	PUNCT
ejpam-4730	182	4	2019	2019	NUM
ejpam-4730	182	5	.	.	PUNCT
ejpam-4730	183	1	[	[	X
ejpam-4730	183	2	8	8	X
ejpam-4730	183	3	]	]	X
ejpam-4730	183	4	alaa	alaa	PROPN
ejpam-4730	183	5	adnan	adnan	PROPN
ejpam-4730	183	6	auad	auad	PROPN
ejpam-4730	183	7	and	and	CCONJ
ejpam-4730	183	8	abdulsttar	abdulsttar	PROPN
ejpam-4730	183	9	ali	ali	PROPN
ejpam-4730	183	10	hussein	hussein	PROPN
ejpam-4730	183	11	.	.	PUNCT
ejpam-4730	184	1	best	good	ADJ
ejpam-4730	184	2	simultaneous	simultaneous	ADJ
ejpam-4730	184	3	approximation	approximation	NOUN
ejpam-4730	184	4	in	in	ADP
ejpam-4730	184	5	weighted	weighted	ADJ
ejpam-4730	184	6	space	space	NOUN
ejpam-4730	184	7	.	.	PUNCT
ejpam-4730	185	1	in	in	ADP
ejpam-4730	185	2	journal	journal	PROPN
ejpam-4730	185	3	of	of	ADP
ejpam-4730	185	4	physics	physics	PROPN
ejpam-4730	185	5	:	:	PUNCT
ejpam-4730	185	6	conference	conference	NOUN
ejpam-4730	185	7	series	series	NOUN
ejpam-4730	185	8	,	,	PUNCT
ejpam-4730	185	9	volume	volume	NOUN
ejpam-4730	185	10	1234	1234	NUM
ejpam-4730	185	11	,	,	PUNCT
ejpam-4730	185	12	page	page	NOUN
ejpam-4730	185	13	012111	012111	NUM
ejpam-4730	185	14	.	.	PUNCT
ejpam-4730	186	1	iop	iop	PROPN
ejpam-4730	186	2	publishing	publishing	NOUN
ejpam-4730	186	3	,	,	PUNCT
ejpam-4730	186	4	2019	2019	NUM
ejpam-4730	186	5	.	.	PUNCT
ejpam-4730	187	1	references	reference	NOUN
ejpam-4730	187	2	952	952	NUM
ejpam-4730	188	1	[	[	X
ejpam-4730	188	2	9	9	NUM
ejpam-4730	188	3	]	]	PUNCT
ejpam-4730	188	4	jorge	jorge	NOUN
ejpam-4730	188	5	bustamante	bustamante	PROPN
ejpam-4730	188	6	and	and	CCONJ
ejpam-4730	188	7	miguel	miguel	PROPN
ejpam-4730	188	8	antonio	antonio	PROPN
ejpam-4730	188	9	jimenez	jimenez	PROPN
ejpam-4730	188	10	.	.	PUNCT
ejpam-4730	189	1	the	the	DET
ejpam-4730	189	2	degree	degree	NOUN
ejpam-4730	189	3	of	of	ADP
ejpam-4730	189	4	best	good	ADJ
ejpam-4730	189	5	approximation	approximation	NOUN
ejpam-4730	189	6	in	in	ADP
ejpam-4730	189	7	the	the	DET
ejpam-4730	189	8	lipschitz	lipschitz	NOUN
ejpam-4730	189	9	norm	norm	NOUN
ejpam-4730	189	10	by	by	ADP
ejpam-4730	189	11	trigonometric	trigonometric	ADJ
ejpam-4730	189	12	polynomials	polynomial	NOUN
ejpam-4730	189	13	.	.	PUNCT
ejpam-4730	190	1	aportaciones	aportaciones	PROPN
ejpam-4730	190	2	matematicas	matematicas	PROPN
ejpam-4730	190	3	,	,	PUNCT
ejpam-4730	190	4	25:23	25:23	NUM
ejpam-4730	190	5	–	–	PUNCT
ejpam-4730	190	6	30	30	NUM
ejpam-4730	190	7	,	,	PUNCT
ejpam-4730	190	8	1999	1999	NUM
ejpam-4730	190	9	.	.	PUNCT
ejpam-4730	191	1	[	[	X
ejpam-4730	191	2	10	10	NUM
ejpam-4730	191	3	]	]	X
ejpam-4730	191	4	y	y	PROPN
ejpam-4730	191	5	dominic	dominic	PROPN
ejpam-4730	191	6	and	and	CCONJ
ejpam-4730	191	7	m	m	PROPN
ejpam-4730	191	8	marudai	marudai	PROPN
ejpam-4730	191	9	.	.	PUNCT
ejpam-4730	192	1	best	good	ADJ
ejpam-4730	192	2	approximation	approximation	NOUN
ejpam-4730	192	3	in	in	ADP
ejpam-4730	192	4	uniformly	uniformly	ADV
ejpam-4730	192	5	convex	convex	VERB
ejpam-4730	192	6	2	2	NUM
ejpam-4730	192	7	-	-	PUNCT
ejpam-4730	192	8	normed	norme	VERB
ejpam-4730	192	9	space	space	NOUN
ejpam-4730	192	10	.	.	PUNCT
ejpam-4730	193	1	general	general	ADJ
ejpam-4730	193	2	mathematics	mathematic	NOUN
ejpam-4730	193	3	,	,	PUNCT
ejpam-4730	193	4	6(2):1015–1021	6(2):1015–1021	PROPN
ejpam-4730	193	5	,	,	PUNCT
ejpam-4730	193	6	2012	2012	NUM
ejpam-4730	193	7	.	.	PUNCT
ejpam-4730	194	1	[	[	X
ejpam-4730	194	2	11	11	NUM
ejpam-4730	194	3	]	]	PUNCT
ejpam-4730	194	4	ss	ss	NOUN
ejpam-4730	194	5	dragomir	dragomir	NOUN
ejpam-4730	194	6	.	.	PUNCT
ejpam-4730	195	1	some	some	DET
ejpam-4730	195	2	characterization	characterization	NOUN
ejpam-4730	195	3	of	of	ADP
ejpam-4730	195	4	best	good	ADJ
ejpam-4730	195	5	approximants	approximant	NOUN
ejpam-4730	195	6	in	in	ADP
ejpam-4730	195	7	normed	normed	ADJ
ejpam-4730	195	8	linear	linear	PROPN
ejpam-4730	195	9	spaces	space	NOUN
ejpam-4730	195	10	.	.	PUNCT
ejpam-4730	196	1	acta	acta	PROPN
ejpam-4730	196	2	mathematica	mathematica	PROPN
ejpam-4730	196	3	vietnamica	vietnamica	PROPN
ejpam-4730	196	4	,	,	PUNCT
ejpam-4730	196	5	25(3):359–366	25(3):359–366	PROPN
ejpam-4730	196	6	,	,	PUNCT
ejpam-4730	196	7	2000	2000	NUM
ejpam-4730	196	8	.	.	PUNCT
ejpam-4730	197	1	[	[	X
ejpam-4730	197	2	12	12	NUM
ejpam-4730	197	3	]	]	X
ejpam-4730	197	4	s	s	X
ejpam-4730	197	5	elumalai	elumalai	NOUN
ejpam-4730	197	6	and	and	CCONJ
ejpam-4730	197	7	r	r	NOUN
ejpam-4730	197	8	vijayaragavan	vijayaragavan	NOUN
ejpam-4730	197	9	.	.	PUNCT
ejpam-4730	198	1	best	good	ADJ
ejpam-4730	198	2	simultaneous	simultaneous	ADJ
ejpam-4730	198	3	approximation	approximation	NOUN
ejpam-4730	198	4	in	in	ADP
ejpam-4730	198	5	linear	linear	PROPN
ejpam-4730	198	6	2	2	NUM
ejpam-4730	198	7	-	-	PUNCT
ejpam-4730	198	8	normed	norme	VERB
ejpam-4730	198	9	spaces1	spaces1	PROPN
ejpam-4730	198	10	.	.	PUNCT
ejpam-4730	198	11	general	general	ADJ
ejpam-4730	198	12	mathematics	mathematics	PROPN
ejpam-4730	198	13	,	,	PUNCT
ejpam-4730	198	14	16(1):73–81	16(1):73–81	NUM
ejpam-4730	198	15	,	,	PUNCT
ejpam-4730	198	16	2008	2008	NUM
ejpam-4730	198	17	.	.	PUNCT
ejpam-4730	199	1	[	[	X
ejpam-4730	199	2	13	13	NUM
ejpam-4730	199	3	]	]	SYM
ejpam-4730	199	4	s	s	X
ejpam-4730	199	5	elumalai	elumalai	NOUN
ejpam-4730	199	6	and	and	CCONJ
ejpam-4730	199	7	r	r	NOUN
ejpam-4730	199	8	vijayaragavan	vijayaragavan	NOUN
ejpam-4730	199	9	.	.	PUNCT
ejpam-4730	200	1	characterizations	characterization	NOUN
ejpam-4730	200	2	of	of	ADP
ejpam-4730	200	3	best	good	ADJ
ejpam-4730	200	4	approximations	approximation	NOUN
ejpam-4730	200	5	in	in	ADP
ejpam-4730	200	6	linear	linear	ADJ
ejpam-4730	200	7	2	2	NUM
ejpam-4730	200	8	-	-	PUNCT
ejpam-4730	200	9	normed	norme	VERB
ejpam-4730	200	10	spaces	space	NOUN
ejpam-4730	200	11	.	.	PUNCT
ejpam-4730	201	1	general	general	ADJ
ejpam-4730	201	2	mathematics	mathematics	PROPN
ejpam-4730	201	3	,	,	PUNCT
ejpam-4730	201	4	17(3):141–160	17(3):141–160	PROPN
ejpam-4730	201	5	,	,	PUNCT
ejpam-4730	201	6	2009	2009	NUM
ejpam-4730	201	7	.	.	PUNCT
ejpam-4730	202	1	[	[	X
ejpam-4730	202	2	14	14	NUM
ejpam-4730	202	3	]	]	X
ejpam-4730	202	4	mahdi	mahdi	PROPN
ejpam-4730	202	5	iranmanesh	iranmanesh	PROPN
ejpam-4730	202	6	and	and	CCONJ
ejpam-4730	202	7	fatemeh	fatemeh	VERB
ejpam-4730	202	8	soleimany	soleimany	ADJ
ejpam-4730	202	9	.	.	PUNCT
ejpam-4730	203	1	characterization	characterization	NOUN
ejpam-4730	203	2	of	of	ADP
ejpam-4730	203	3	best	good	ADJ
ejpam-4730	203	4	approximation	approximation	NOUN
ejpam-4730	203	5	of	of	ADP
ejpam-4730	203	6	closed	closed	ADJ
ejpam-4730	203	7	convex	convex	ADJ
ejpam-4730	203	8	subsets	subset	NOUN
ejpam-4730	203	9	in	in	ADP
ejpam-4730	203	10	cb	cb	PROPN
ejpam-4730	203	11	(	(	PUNCT
ejpam-4730	203	12	x	x	PROPN
ejpam-4730	203	13	,	,	PUNCT
ejpam-4730	203	14	y	y	PROPN
ejpam-4730	203	15	)	)	PUNCT
ejpam-4730	203	16	.	.	PUNCT
ejpam-4730	204	1	applied	apply	VERB
ejpam-4730	204	2	mathematics	mathematics	PROPN
ejpam-4730	204	3	e	e	NOUN
ejpam-4730	204	4	-	-	NOUN
ejpam-4730	204	5	notes	note	NOUN
ejpam-4730	204	6	,	,	PUNCT
ejpam-4730	204	7	19:46–54	19:46–54	NUM
ejpam-4730	204	8	,	,	PUNCT
ejpam-4730	204	9	2019	2019	NUM
ejpam-4730	204	10	.	.	PUNCT
ejpam-4730	205	1	[	[	X
ejpam-4730	205	2	15	15	NUM
ejpam-4730	205	3	]	]	X
ejpam-4730	205	4	jp	jp	NOUN
ejpam-4730	205	5	kushwaha	kushwaha	PROPN
ejpam-4730	205	6	and	and	CCONJ
ejpam-4730	205	7	bp	bp	PROPN
ejpam-4730	205	8	dhakal	dhakal	PROPN
ejpam-4730	205	9	.	.	PUNCT
ejpam-4730	206	1	approximation	approximation	NOUN
ejpam-4730	206	2	of	of	ADP
ejpam-4730	206	3	a	a	DET
ejpam-4730	206	4	function	function	NOUN
ejpam-4730	206	5	belonging	belong	VERB
ejpam-4730	206	6	to	to	ADP
ejpam-4730	206	7	lip	lip	NOUN
ejpam-4730	206	8	(	(	PUNCT
ejpam-4730	206	9	(	(	PUNCT
ejpam-4730	206	10	α	α	NOUN
ejpam-4730	206	11	,	,	PUNCT
ejpam-4730	206	12	r	r	NOUN
ejpam-4730	206	13	)	)	PUNCT
ejpam-4730	206	14	class	class	NOUN
ejpam-4730	206	15	by	by	ADP
ejpam-4730	206	16	np	np	PROPN
ejpam-4730	206	17	,	,	PUNCT
ejpam-4730	206	18	q.	q.	PROPN
ejpam-4730	206	19	c1	c1	PROPN
ejpam-4730	206	20	summability	summability	PROPN
ejpam-4730	206	21	method	method	NOUN
ejpam-4730	206	22	of	of	ADP
ejpam-4730	206	23	its	its	PRON
ejpam-4730	206	24	fourier	fourier	NOUN
ejpam-4730	206	25	series	series	NOUN
ejpam-4730	206	26	.	.	PUNCT
ejpam-4730	207	1	nepal	nepal	PROPN
ejpam-4730	207	2	journal	journal	PROPN
ejpam-4730	207	3	of	of	ADP
ejpam-4730	207	4	science	science	NOUN
ejpam-4730	207	5	and	and	CCONJ
ejpam-4730	207	6	technology	technology	NOUN
ejpam-4730	207	7	,	,	PUNCT
ejpam-4730	207	8	14(2):117–122	14(2):117–122	PROPN
ejpam-4730	207	9	,	,	PUNCT
ejpam-4730	207	10	2013	2013	NUM
ejpam-4730	207	11	.	.	PUNCT
ejpam-4730	208	1	[	[	X
ejpam-4730	208	2	16	16	NUM
ejpam-4730	208	3	]	]	X
ejpam-4730	208	4	t	t	PROPN
ejpam-4730	208	5	makandeya	makandeya	NOUN
ejpam-4730	208	6	and	and	CCONJ
ejpam-4730	208	7	d	d	PROPN
ejpam-4730	208	8	bharathi	bharathi	PROPN
ejpam-4730	208	9	.	.	PUNCT
ejpam-4730	209	1	best	good	ADJ
ejpam-4730	209	2	approximation	approximation	NOUN
ejpam-4730	209	3	in	in	ADP
ejpam-4730	209	4	2	2	NUM
ejpam-4730	209	5	-	-	PUNCT
ejpam-4730	209	6	normed	norme	VERB
ejpam-4730	209	7	almost	almost	ADV
ejpam-4730	209	8	linear	linear	ADJ
ejpam-4730	209	9	space	space	NOUN
ejpam-4730	209	10	.	.	PUNCT
ejpam-4730	210	1	international	international	ADJ
ejpam-4730	210	2	journal	journal	PROPN
ejpam-4730	210	3	of	of	ADP
ejpam-4730	210	4	engineering	engineering	NOUN
ejpam-4730	210	5	research	research	NOUN
ejpam-4730	210	6	and	and	CCONJ
ejpam-4730	210	7	technology	technology	NOUN
ejpam-4730	210	8	,	,	PUNCT
ejpam-4730	210	9	12:3569–3573	12:3569–3573	NUM
ejpam-4730	210	10	,	,	PUNCT
ejpam-4730	210	11	2013	2013	NUM
ejpam-4730	210	12	.	.	PUNCT
ejpam-4730	211	1	[	[	X
ejpam-4730	211	2	17	17	NUM
ejpam-4730	211	3	]	]	X
ejpam-4730	211	4	ghazi	ghazi	PROPN
ejpam-4730	211	5	abed	abe	VERB
ejpam-4730	211	6	meften	meften	VERB
ejpam-4730	211	7	and	and	CCONJ
ejpam-4730	211	8	ali	ali	PROPN
ejpam-4730	211	9	hasan	hasan	PROPN
ejpam-4730	211	10	ali	ali	PROPN
ejpam-4730	211	11	.	.	PUNCT
ejpam-4730	212	1	continuous	continuous	ADJ
ejpam-4730	212	2	dependence	dependence	NOUN
ejpam-4730	212	3	for	for	ADP
ejpam-4730	212	4	double	double	ADJ
ejpam-4730	212	5	diffusive	diffusive	ADJ
ejpam-4730	212	6	convection	convection	NOUN
ejpam-4730	212	7	in	in	ADP
ejpam-4730	212	8	a	a	DET
ejpam-4730	212	9	brinkman	brinkman	NOUN
ejpam-4730	212	10	model	model	NOUN
ejpam-4730	212	11	with	with	ADP
ejpam-4730	212	12	variable	variable	ADJ
ejpam-4730	212	13	viscosity	viscosity	NOUN
ejpam-4730	212	14	.	.	PUNCT
ejpam-4730	213	1	acta	acta	PROPN
ejpam-4730	213	2	universitatis	universitatis	PROPN
ejpam-4730	213	3	sapientiae	sapientiae	PROPN
ejpam-4730	213	4	,	,	PUNCT
ejpam-4730	213	5	mathematica	mathematica	PROPN
ejpam-4730	213	6	,	,	PUNCT
ejpam-4730	213	7	14(1):125–146	14(1):125–146	PROPN
ejpam-4730	213	8	,	,	PUNCT
ejpam-4730	213	9	2022	2022	NUM
ejpam-4730	213	10	.	.	PUNCT
ejpam-4730	214	1	[	[	X
ejpam-4730	214	2	18	18	NUM
ejpam-4730	214	3	]	]	X
ejpam-4730	214	4	ghazi	ghazi	PROPN
ejpam-4730	214	5	abed	abe	VERB
ejpam-4730	214	6	meften	meften	PROPN
ejpam-4730	214	7	,	,	PUNCT
ejpam-4730	214	8	ali	ali	PROPN
ejpam-4730	214	9	hasan	hasan	PROPN
ejpam-4730	214	10	ali	ali	PROPN
ejpam-4730	214	11	,	,	PUNCT
ejpam-4730	214	12	and	and	CCONJ
ejpam-4730	214	13	mustafa	mustafa	PROPN
ejpam-4730	214	14	taha	taha	PROPN
ejpam-4730	214	15	yaseen	yaseen	PROPN
ejpam-4730	214	16	.	.	PUNCT
ejpam-4730	215	1	continuous	continuous	ADJ
ejpam-4730	215	2	dependence	dependence	NOUN
ejpam-4730	215	3	for	for	ADP
ejpam-4730	215	4	thermal	thermal	ADJ
ejpam-4730	215	5	convection	convection	NOUN
ejpam-4730	215	6	in	in	ADP
ejpam-4730	215	7	a	a	DET
ejpam-4730	215	8	forchheimer	forchheimer	ADJ
ejpam-4730	215	9	-	-	PUNCT
ejpam-4730	215	10	brinkman	brinkman	PROPN
ejpam-4730	215	11	model	model	NOUN
ejpam-4730	215	12	with	with	ADP
ejpam-4730	215	13	variable	variable	ADJ
ejpam-4730	215	14	viscosity	viscosity	NOUN
ejpam-4730	215	15	.	.	PUNCT
ejpam-4730	216	1	in	in	ADP
ejpam-4730	216	2	aip	aip	PROPN
ejpam-4730	216	3	conference	conference	NOUN
ejpam-4730	216	4	proceedings	proceeding	NOUN
ejpam-4730	216	5	,	,	PUNCT
ejpam-4730	216	6	volume	volume	NOUN
ejpam-4730	216	7	2457	2457	NUM
ejpam-4730	216	8	,	,	PUNCT
ejpam-4730	216	9	page	page	NOUN
ejpam-4730	216	10	020005	020005	NUM
ejpam-4730	216	11	.	.	PUNCT
ejpam-4730	217	1	aip	aip	PROPN
ejpam-4730	217	2	publishing	publishing	PROPN
ejpam-4730	217	3	llc	llc	PROPN
ejpam-4730	217	4	,	,	PUNCT
ejpam-4730	217	5	2023	2023	NUM
ejpam-4730	217	6	.	.	PUNCT
ejpam-4730	218	1	[	[	X
ejpam-4730	218	2	19	19	NUM
ejpam-4730	218	3	]	]	X
ejpam-4730	218	4	hk	hk	PROPN
ejpam-4730	218	5	nigam	nigam	PROPN
ejpam-4730	218	6	and	and	CCONJ
ejpam-4730	218	7	md	md	PROPN
ejpam-4730	218	8	hadish	hadish	ADJ
ejpam-4730	218	9	.	.	PUNCT
ejpam-4730	219	1	best	good	ADJ
ejpam-4730	219	2	approximation	approximation	NOUN
ejpam-4730	219	3	of	of	ADP
ejpam-4730	219	4	functions	function	NOUN
ejpam-4730	219	5	in	in	ADP
ejpam-4730	219	6	generalized	generalized	ADJ
ejpam-4730	219	7	hölder	hölder	NOUN
ejpam-4730	219	8	class	class	NOUN
ejpam-4730	219	9	.	.	PUNCT
ejpam-4730	220	1	journal	journal	PROPN
ejpam-4730	220	2	of	of	ADP
ejpam-4730	220	3	inequalities	inequality	NOUN
ejpam-4730	220	4	and	and	CCONJ
ejpam-4730	220	5	applications	application	NOUN
ejpam-4730	220	6	,	,	PUNCT
ejpam-4730	220	7	2018(1):1–15	2018(1):1–15	NOUN
ejpam-4730	220	8	,	,	PUNCT
ejpam-4730	220	9	2018	2018	NUM
ejpam-4730	220	10	.	.	PUNCT
ejpam-4730	221	1	[	[	X
ejpam-4730	221	2	20	20	NUM
ejpam-4730	221	3	]	]	X
ejpam-4730	221	4	j	j	PROPN
ejpam-4730	221	5	prestin	prestin	PROPN
ejpam-4730	221	6	.	.	PUNCT
ejpam-4730	222	1	on	on	ADP
ejpam-4730	222	2	the	the	DET
ejpam-4730	222	3	approximation	approximation	NOUN
ejpam-4730	222	4	by	by	ADP
ejpam-4730	222	5	de	de	X
ejpam-4730	222	6	la	la	PROPN
ejpam-4730	222	7	valleé	valleé	PROPN
ejpam-4730	222	8	poussin	poussin	PROPN
ejpam-4730	222	9	sums	sum	NOUN
ejpam-4730	222	10	and	and	CCONJ
ejpam-4730	222	11	interpolatory	interpolatory	NOUN
ejpam-4730	222	12	polynomials	polynomial	NOUN
ejpam-4730	222	13	in	in	ADP
ejpam-4730	222	14	lipschitz	lipschitz	NOUN
ejpam-4730	222	15	norms	norm	NOUN
ejpam-4730	222	16	.	.	PUNCT
ejpam-4730	223	1	analysis	analysis	NOUN
ejpam-4730	223	2	mathematica	mathematica	PROPN
ejpam-4730	223	3	,	,	PUNCT
ejpam-4730	223	4	13(3):251–259	13(3):251–259	NOUN
ejpam-4730	223	5	,	,	PUNCT
ejpam-4730	223	6	1987	1987	NUM
ejpam-4730	223	7	.	.	PUNCT