id	sid	tid	token	lemma	pos
cana-708	1	1	communications	communication	NOUN
cana-708	1	2	on	on	ADP
cana-708	1	3	applied	apply	VERB
cana-708	1	4	nonlinear	nonlinear	ADJ
cana-708	1	5	analysis	analysis	NOUN
cana-708	1	6	issn	issn	NOUN
cana-708	1	7	:	:	PUNCT
cana-708	1	8	1074	1074	NUM
cana-708	1	9	-	-	PUNCT
cana-708	1	10	133x	133x	NUM
cana-708	1	11	vol	vol	NOUN
cana-708	1	12	31	31	NUM
cana-708	1	13	no	no	NOUN
cana-708	1	14	.	.	PUNCT
cana-708	2	1	2s	2s	NUM
cana-708	2	2	(	(	PUNCT
cana-708	2	3	2024	2024	NUM
cana-708	2	4	)	)	PUNCT
cana-708	2	5	696	696	NUM
cana-708	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	2	7	on	on	ADP
cana-708	2	8	mean	mean	ADJ
cana-708	2	9	convergence	convergence	NOUN
cana-708	2	10	of	of	ADP
cana-708	2	11	random	random	ADJ
cana-708	2	12	fourier	fourier	NOUN
cana-708	2	13	hermite	hermite	PROPN
cana-708	2	14	series	series	NOUN
cana-708	2	15	bharatee	bharatee	VERB
cana-708	2	16	mangaraj[1	mangaraj[1	NUM
cana-708	2	17	]	]	PUNCT
cana-708	2	18	,	,	PUNCT
cana-708	2	19	sabita	sabita	PROPN
cana-708	2	20	sahoo[2	sahoo[2	PROPN
cana-708	2	21	]	]	X
cana-708	3	1	[	[	X
cana-708	3	2	1]phd	1]phd	NOUN
cana-708	3	3	scholar	scholar	NOUN
cana-708	3	4	,	,	PUNCT
cana-708	3	5	department	department	NOUN
cana-708	3	6	of	of	ADP
cana-708	3	7	mathematics	mathematic	NOUN
cana-708	3	8	,	,	PUNCT
cana-708	3	9	sambalpur	sambalpur	NOUN
cana-708	3	10	university	university	PROPN
cana-708	3	11	,	,	PUNCT
cana-708	3	12	odisha	odisha	PROPN
cana-708	3	13	,	,	PUNCT
cana-708	3	14	india	india	PROPN
cana-708	4	1	[	[	X
cana-708	4	2	2	2	NUM
cana-708	4	3	]	]	X
cana-708	4	4	retired	retired	ADJ
cana-708	4	5	professor	professor	NOUN
cana-708	4	6	,	,	PUNCT
cana-708	4	7	department	department	NOUN
cana-708	4	8	of	of	ADP
cana-708	4	9	mathematics	mathematic	NOUN
cana-708	4	10	,	,	PUNCT
cana-708	4	11	sambalpur	sambalpur	NOUN
cana-708	4	12	university	university	PROPN
cana-708	4	13	,	,	PUNCT
cana-708	4	14	odisha	odisha	PROPN
cana-708	4	15	,	,	PUNCT
cana-708	4	16	india	india	PROPN
cana-708	5	1	[	[	X
cana-708	5	2	1	1	NUM
cana-708	5	3	]	]	PUNCT
cana-708	5	4	mangarajbharatee@suniv.ac.in	mangarajbharatee@suniv.ac.in	ADV
cana-708	5	5	,	,	PUNCT
cana-708	5	6	[	[	X
cana-708	5	7	2	2	NUM
cana-708	5	8	]	]	PUNCT
cana-708	5	9	sabitamath@suniv.ac.in	sabitamath@suniv.ac.in	NOUN
cana-708	5	10	article	article	NOUN
cana-708	5	11	history	history	NOUN
cana-708	5	12	:	:	PUNCT
cana-708	5	13	received	receive	VERB
cana-708	5	14	:	:	PUNCT
cana-708	5	15	10	10	NUM
cana-708	5	16	-	-	PUNCT
cana-708	5	17	04	04	NUM
cana-708	5	18	-	-	PUNCT
cana-708	5	19	2024	2024	NUM
cana-708	5	20	revised	revise	VERB
cana-708	5	21	:	:	PUNCT
cana-708	5	22	20	20	NUM
cana-708	5	23	-	-	SYM
cana-708	5	24	05	05	NUM
cana-708	5	25	-	-	PUNCT
cana-708	5	26	2024	2024	NUM
cana-708	5	27	accepted	accept	VERB
cana-708	5	28	:	:	PUNCT
cana-708	5	29	03	03	NUM
cana-708	5	30	-	-	PUNCT
cana-708	5	31	06	06	NUM
cana-708	5	32	-	-	PUNCT
cana-708	5	33	2024	2024	NUM
cana-708	5	34	abstract	abstract	NOUN
cana-708	5	35	:	:	PUNCT
cana-708	5	36	the	the	DET
cana-708	5	37	work	work	NOUN
cana-708	5	38	in	in	ADP
cana-708	5	39	this	this	DET
cana-708	5	40	article	article	NOUN
cana-708	5	41	is	be	AUX
cana-708	5	42	an	an	DET
cana-708	5	43	initiative	initiative	NOUN
cana-708	5	44	to	to	PART
cana-708	5	45	explore	explore	VERB
cana-708	5	46	random	random	ADJ
cana-708	5	47	fourier	fouri	ADJ
cana-708	5	48	hermite	hermite	ADJ
cana-708	5	49	series	series	NOUN
cana-708	5	50	in	in	ADP
cana-708	5	51	orthogonal	orthogonal	ADJ
cana-708	5	52	hermite	hermite	ADJ
cana-708	5	53	polynomials	polynomial	NOUN
cana-708	5	54	.	.	PUNCT
cana-708	6	1	we	we	PRON
cana-708	6	2	choose	choose	VERB
cana-708	6	3	the	the	DET
cana-708	6	4	random	random	ADJ
cana-708	6	5	coefficients	coefficient	NOUN
cana-708	6	6	in	in	ADP
cana-708	6	7	the	the	DET
cana-708	6	8	series	series	NOUN
cana-708	6	9	to	to	PART
cana-708	6	10	be	be	AUX
cana-708	6	11	the	the	DET
cana-708	6	12	fourier	fourier	ADJ
cana-708	6	13	-	-	PUNCT
cana-708	6	14	hermite	hermite	ADJ
cana-708	6	15	coefficients	coefficient	NOUN
cana-708	6	16	of	of	ADP
cana-708	6	17	a	a	DET
cana-708	6	18	symmetric	symmetric	ADJ
cana-708	6	19	stable	stable	ADJ
cana-708	6	20	process	process	NOUN
cana-708	6	21	with	with	ADP
cana-708	6	22	weight	weight	NOUN
cana-708	6	23	function	function	NOUN
cana-708	6	24	𝑈(𝑣	𝑈(𝑣	NOUN
cana-708	6	25	,	,	PUNCT
cana-708	6	26	𝑏	𝑏	NOUN
cana-708	6	27	)	)	PUNCT
cana-708	6	28	=	=	PUNCT
cana-708	6	29	𝑒	𝑒	PROPN
cana-708	6	30	−𝑣2	−𝑣2	NOUN
cana-708	6	31	2	2	NUM
cana-708	6	32	(	(	PUNCT
cana-708	6	33	1	1	NUM
cana-708	6	34	+	+	CCONJ
cana-708	6	35	|𝑣|)𝑏	|𝑣|)𝑏	ADP
cana-708	6	36	,	,	PUNCT
cana-708	6	37	where	where	SCONJ
cana-708	6	38	𝑏	𝑏	PROPN
cana-708	6	39	<	<	X
cana-708	6	40	1	1	NUM
cana-708	6	41	2	2	NUM
cana-708	6	42	.	.	PUNCT
cana-708	7	1	the	the	DET
cana-708	7	2	existence	existence	NOUN
cana-708	7	3	of	of	ADP
cana-708	7	4	these	these	DET
cana-708	7	5	random	random	ADJ
cana-708	7	6	coefficients	coefficient	NOUN
cana-708	7	7	,	,	PUNCT
cana-708	7	8	which	which	PRON
cana-708	7	9	we	we	PRON
cana-708	7	10	find	find	VERB
cana-708	7	11	to	to	PART
cana-708	7	12	be	be	AUX
cana-708	7	13	dependent	dependent	ADJ
cana-708	7	14	random	random	ADJ
cana-708	7	15	variables	variable	NOUN
cana-708	7	16	,	,	PUNCT
cana-708	7	17	is	be	AUX
cana-708	7	18	established	establish	VERB
cana-708	7	19	.	.	PUNCT
cana-708	8	1	the	the	DET
cana-708	8	2	random	random	ADJ
cana-708	8	3	fourier	fourier	ADJ
cana-708	8	4	-	-	PUNCT
cana-708	8	5	hermite	hermite	ADJ
cana-708	8	6	series	series	NOUN
cana-708	8	7	is	be	AUX
cana-708	8	8	proven	prove	VERB
cana-708	8	9	to	to	PART
cana-708	8	10	be	be	AUX
cana-708	8	11	convergent	convergent	ADJ
cana-708	8	12	in	in	ADP
cana-708	8	13	the	the	DET
cana-708	8	14	sense	sense	NOUN
cana-708	8	15	of	of	ADP
cana-708	8	16	mean	mean	VERB
cana-708	8	17	if	if	SCONJ
cana-708	8	18	the	the	DET
cana-708	8	19	scalars	scalar	NOUN
cana-708	8	20	in	in	ADP
cana-708	8	21	the	the	DET
cana-708	8	22	series	series	NOUN
cana-708	8	23	are	be	AUX
cana-708	8	24	the	the	DET
cana-708	8	25	fourier	fourier	ADJ
cana-708	8	26	-	-	PUNCT
cana-708	8	27	hermite	hermite	ADJ
cana-708	8	28	coefficients	coefficient	NOUN
cana-708	8	29	of	of	ADP
cana-708	8	30	a	a	DET
cana-708	8	31	function	function	NOUN
cana-708	8	32	𝑔	𝑔	NOUN
cana-708	8	33	in	in	ADP
cana-708	8	34	the	the	DET
cana-708	8	35	weighted	weighted	ADJ
cana-708	8	36	space	space	NOUN
cana-708	8	37	𝐿𝑊(𝑣,𝐵	𝐿𝑊(𝑣,𝐵	NOUN
cana-708	8	38	)	)	PUNCT
cana-708	8	39	2	2	NUM
cana-708	8	40	(	(	PUNCT
cana-708	8	41	ℝ	ℝ	PROPN
cana-708	8	42	)	)	PUNCT
cana-708	8	43	,	,	PUNCT
cana-708	8	44	where	where	SCONJ
cana-708	8	45	the	the	DET
cana-708	8	46	weights	weight	NOUN
cana-708	8	47	are	be	AUX
cana-708	8	48	given	give	VERB
cana-708	8	49	by	by	ADP
cana-708	8	50	𝑊(𝑣	𝑊(𝑣	PROPN
cana-708	8	51	,	,	PUNCT
cana-708	8	52	𝐵	𝐵	NOUN
cana-708	8	53	)	)	PUNCT
cana-708	8	54	=	=	PUNCT
cana-708	8	55	𝑒	𝑒	PROPN
cana-708	8	56	−𝑣2	−𝑣2	NOUN
cana-708	8	57	2	2	NUM
cana-708	8	58	(	(	PUNCT
cana-708	8	59	1	1	NUM
cana-708	8	60	+	+	CCONJ
cana-708	8	61	|𝑣|)𝐵	|𝑣|)𝐵	NOUN
cana-708	8	62	with	with	ADP
cana-708	8	63	𝐵	𝐵	PROPN
cana-708	8	64	>	>	X
cana-708	8	65	−1	−1	NOUN
cana-708	8	66	2	2	NUM
cana-708	8	67	such	such	ADJ
cana-708	8	68	that	that	SCONJ
cana-708	8	69	𝑏	𝑏	PROPN
cana-708	8	70	<	<	X
cana-708	8	71	𝐵.	𝐵.	X
cana-708	8	72	the	the	DET
cana-708	8	73	sum	sum	NOUN
cana-708	8	74	functions	function	NOUN
cana-708	8	75	of	of	ADP
cana-708	8	76	the	the	DET
cana-708	8	77	series	series	NOUN
cana-708	8	78	is	be	AUX
cana-708	8	79	obtained	obtain	VERB
cana-708	8	80	to	to	ADP
cana-708	8	81	the	the	DET
cana-708	8	82	stochastic	stochastic	ADJ
cana-708	8	83	integral	integral	ADJ
cana-708	8	84	∫	∫	NOUN
cana-708	8	85	∞	∞	PROPN
cana-708	8	86	−∞	−∞	ADP
cana-708	8	87	𝑔(𝑣)𝑈(𝑣	𝑔(𝑣)𝑈(𝑣	NOUN
cana-708	8	88	,	,	PUNCT
cana-708	8	89	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	8	90	,	,	PUNCT
cana-708	8	91	𝜔	𝜔	NOUN
cana-708	8	92	)	)	PUNCT
cana-708	8	93	.	.	PUNCT
cana-708	9	1	2020	2020	NUM
cana-708	9	2	msc	msc	PROPN
cana-708	9	3	.	.	PROPN
cana-708	9	4	primary	primary	NOUN
cana-708	9	5	:	:	PUNCT
cana-708	9	6	42a38	42a38	NUM
cana-708	9	7	;	;	PUNCT
cana-708	9	8	secondary	secondary	ADJ
cana-708	9	9	:	:	PUNCT
cana-708	9	10	40a35	40a35	NUM
cana-708	9	11	keywords	keyword	NOUN
cana-708	9	12	:	:	PUNCT
cana-708	9	13	symmetric	symmetric	ADJ
cana-708	9	14	stable	stable	ADJ
cana-708	9	15	process	process	NOUN
cana-708	9	16	,	,	PUNCT
cana-708	9	17	stochastic	stochastic	ADJ
cana-708	9	18	integral	integral	ADJ
cana-708	9	19	,	,	PUNCT
cana-708	9	20	convergence	convergence	NOUN
cana-708	9	21	in	in	ADP
cana-708	9	22	mean	mean	ADJ
cana-708	9	23	,	,	PUNCT
cana-708	9	24	convergence	convergence	NOUN
cana-708	9	25	in	in	ADP
cana-708	9	26	law	law	NOUN
cana-708	9	27	,	,	PUNCT
cana-708	9	28	continuity	continuity	NOUN
cana-708	9	29	theorem	theorem	NOUN
cana-708	9	30	.	.	PROPN
cana-708	10	1	1	1	X
cana-708	10	2	.	.	X
cana-708	10	3	introduction	introduction	NOUN
cana-708	10	4	fourier	fourier	PROPN
cana-708	10	5	series	series	PROPN
cana-708	10	6	in	in	ADP
cana-708	10	7	orthogonal	orthogonal	ADJ
cana-708	10	8	functions	function	NOUN
cana-708	10	9	𝑒𝑖𝑛𝑢	𝑒𝑖𝑛𝑢	NOUN
cana-708	10	10	and	and	CCONJ
cana-708	10	11	other	other	ADJ
cana-708	10	12	orthogonal	orthogonal	ADJ
cana-708	10	13	polynomials	polynomial	NOUN
cana-708	10	14	like	like	ADP
cana-708	10	15	hermite	hermite	ADJ
cana-708	10	16	polynomials	polynomial	NOUN
cana-708	10	17	,	,	PUNCT
cana-708	10	18	jacobi	jacobi	PROPN
cana-708	10	19	polynomials	polynomial	NOUN
cana-708	10	20	,	,	PUNCT
cana-708	10	21	etc	etc	X
cana-708	10	22	.	.	X
cana-708	10	23	,	,	PUNCT
cana-708	10	24	has	have	VERB
cana-708	10	25	a	a	DET
cana-708	10	26	widespread	widespread	ADJ
cana-708	10	27	application	application	NOUN
cana-708	10	28	in	in	ADP
cana-708	10	29	physical	physical	ADJ
cana-708	10	30	sciences	science	NOUN
cana-708	10	31	.	.	PUNCT
cana-708	11	1	random	random	ADJ
cana-708	11	2	fourier	fourier	NOUN
cana-708	11	3	series(rfs	series(rf	NOUN
cana-708	11	4	)	)	PUNCT
cana-708	11	5	in	in	ADP
cana-708	11	6	orthogonal	orthogonal	ADJ
cana-708	11	7	functions	function	NOUN
cana-708	11	8	𝑒𝑖𝑛𝑢	𝑒𝑖𝑛𝑢	NOUN
cana-708	11	9	is	be	AUX
cana-708	11	10	important	important	ADJ
cana-708	11	11	in	in	ADP
cana-708	11	12	signal	signal	ADJ
cana-708	11	13	processing	processing	NOUN
cana-708	11	14	.	.	PUNCT
cana-708	12	1	for	for	ADP
cana-708	12	2	the	the	DET
cana-708	12	3	first	first	ADJ
cana-708	12	4	time	time	NOUN
cana-708	12	5	,	,	PUNCT
cana-708	12	6	the	the	DET
cana-708	12	7	application	application	NOUN
cana-708	12	8	of	of	ADP
cana-708	12	9	rfs	rfs	PROPN
cana-708	12	10	in	in	ADP
cana-708	12	11	hermite	hermite	ADJ
cana-708	12	12	polynomial	polynomial	NOUN
cana-708	12	13	is	be	AUX
cana-708	12	14	found	find	VERB
cana-708	12	15	in	in	ADP
cana-708	12	16	image	image	NOUN
cana-708	12	17	encryption	encryption	NOUN
cana-708	12	18	and	and	CCONJ
cana-708	12	19	decryption	decryption	NOUN
cana-708	12	20	in	in	ADP
cana-708	12	21	the	the	DET
cana-708	12	22	work	work	NOUN
cana-708	12	23	of	of	ADP
cana-708	12	24	liu	liu	PROPN
cana-708	12	25	and	and	CCONJ
cana-708	12	26	liu	liu	PROPN
cana-708	13	1	[	[	X
cana-708	13	2	11	11	NUM
cana-708	13	3	]	]	PUNCT
cana-708	13	4	in	in	ADP
cana-708	13	5	2007	2007	NUM
cana-708	13	6	,	,	PUNCT
cana-708	13	7	who	who	PRON
cana-708	13	8	expected	expect	VERB
cana-708	13	9	its	its	PRON
cana-708	13	10	more	more	ADJ
cana-708	13	11	application	application	NOUN
cana-708	13	12	in	in	ADP
cana-708	13	13	general	general	ADJ
cana-708	13	14	signal	signal	NOUN
cana-708	13	15	and	and	CCONJ
cana-708	13	16	image	image	NOUN
cana-708	13	17	processing	processing	NOUN
cana-708	13	18	.	.	PUNCT
cana-708	14	1	the	the	DET
cana-708	14	2	rfs	rfs	PROPN
cana-708	14	3	they	they	PRON
cana-708	14	4	used	use	VERB
cana-708	14	5	is	be	AUX
cana-708	14	6	an	an	DET
cana-708	14	7	rft	rft	NOUN
cana-708	14	8	with	with	ADP
cana-708	14	9	random	random	ADJ
cana-708	14	10	coefficients	coefficient	NOUN
cana-708	14	11	chosen	choose	VERB
cana-708	14	12	from	from	ADP
cana-708	14	13	the	the	DET
cana-708	14	14	unit	unit	NOUN
cana-708	14	15	circle	circle	NOUN
cana-708	14	16	in	in	ADP
cana-708	14	17	ℂ	ℂ	PROPN
cana-708	14	18	randomly	randomly	ADV
cana-708	14	19	.	.	PUNCT
cana-708	15	1	this	this	PRON
cana-708	15	2	motivated	motivate	VERB
cana-708	15	3	us	we	PRON
cana-708	15	4	to	to	PART
cana-708	15	5	explore	explore	VERB
cana-708	15	6	the	the	DET
cana-708	15	7	random	random	ADJ
cana-708	15	8	fourier	fourier	NOUN
cana-708	15	9	hermite	hermite	PROPN
cana-708	15	10	series(rfhs	series(rfhs	PROPN
cana-708	15	11	)	)	PUNCT
cana-708	15	12	with	with	ADP
cana-708	15	13	different	different	ADJ
cana-708	15	14	random	random	ADJ
cana-708	15	15	coefficients	coefficient	NOUN
cana-708	15	16	.	.	PUNCT
cana-708	16	1	since	since	SCONJ
cana-708	16	2	stable	stable	ADJ
cana-708	16	3	processes	process	NOUN
cana-708	16	4	are	be	AUX
cana-708	16	5	a	a	DET
cana-708	16	6	better	well	ADJ
cana-708	16	7	model	model	NOUN
cana-708	16	8	for	for	ADP
cana-708	16	9	white	white	ADJ
cana-708	16	10	noise	noise	NOUN
cana-708	16	11	,	,	PUNCT
cana-708	16	12	the	the	DET
cana-708	16	13	random	random	ADJ
cana-708	16	14	coefficients	coefficient	NOUN
cana-708	16	15	𝐷𝑛(𝜔	𝐷𝑛(𝜔	NOUN
cana-708	16	16	)	)	PUNCT
cana-708	16	17	choosen	choosen	NOUN
cana-708	16	18	in	in	ADP
cana-708	16	19	this	this	DET
cana-708	16	20	article	article	NOUN
cana-708	16	21	are	be	AUX
cana-708	16	22	fourier	fourier	ADJ
cana-708	16	23	hermite	hermite	ADJ
cana-708	16	24	coefficients(fhc	coefficients(fhc	NOUN
cana-708	16	25	)	)	PUNCT
cana-708	16	26	of	of	ADP
cana-708	16	27	a	a	DET
cana-708	16	28	symmetric	symmetric	ADJ
cana-708	16	29	stable	stable	ADJ
cana-708	16	30	process(ssp	process(ssp	NOUN
cana-708	16	31	)	)	PUNCT
cana-708	16	32	defined	define	VERB
cana-708	16	33	as	as	ADP
cana-708	16	34	∫	∫	PROPN
cana-708	16	35	∞	∞	PROPN
cana-708	16	36	−∞	−∞	ADP
cana-708	16	37	𝐻𝑛(𝑢)𝑈(𝑢	𝐻𝑛(𝑢)𝑈(𝑢	PROPN
cana-708	16	38	,	,	PUNCT
cana-708	16	39	𝑏	𝑏	NOUN
cana-708	16	40	)	)	PUNCT
cana-708	16	41	with	with	ADP
cana-708	16	42	weights	weight	NOUN
cana-708	16	43	𝑈(𝑢	𝑈(𝑢	PROPN
cana-708	16	44	,	,	PUNCT
cana-708	16	45	𝑏	𝑏	NOUN
cana-708	16	46	)	)	PUNCT
cana-708	16	47	=	=	PUNCT
cana-708	16	48	𝑒	𝑒	PROPN
cana-708	16	49	−𝑢2	−𝑢2	NOUN
cana-708	16	50	2	2	NUM
cana-708	16	51	(	(	PUNCT
cana-708	16	52	1	1	NUM
cana-708	16	53	+	+	CCONJ
cana-708	16	54	|𝑢|)𝑏	|𝑢|)𝑏	PROPN
cana-708	16	55	,	,	PUNCT
cana-708	16	56	𝑏	𝑏	PROPN
cana-708	16	57	<	<	X
cana-708	16	58	1	1	NUM
cana-708	16	59	2	2	NUM
cana-708	16	60	.	.	PUNCT
cana-708	17	1	we	we	PRON
cana-708	17	2	establish	establish	VERB
cana-708	17	3	the	the	DET
cana-708	17	4	existence	existence	NOUN
cana-708	17	5	of	of	ADP
cana-708	17	6	these	these	DET
cana-708	17	7	random	random	ADJ
cana-708	17	8	variables	variable	NOUN
cana-708	17	9	and	and	CCONJ
cana-708	17	10	demonstrate	demonstrate	VERB
cana-708	17	11	their	their	PRON
cana-708	17	12	dependence	dependence	NOUN
cana-708	17	13	.	.	PUNCT
cana-708	18	1	it	it	PRON
cana-708	18	2	is	be	AUX
cana-708	18	3	proved	prove	VERB
cana-708	18	4	that	that	SCONJ
cana-708	18	5	the	the	DET
cana-708	18	6	random	random	ADJ
cana-708	18	7	series	series	NOUN
cana-708	18	8	∑∞	∑∞	X
cana-708	18	9	𝑘=0	𝑘=0	X
cana-708	18	10	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	18	11	)	)	PUNCT
cana-708	18	12	in	in	ADP
cana-708	18	13	hermite	hermite	PROPN
cana-708	18	14	polynomials	polynomial	VERB
cana-708	18	15	𝐻𝑘(𝑢	𝐻𝑘(𝑢	NOUN
cana-708	18	16	)	)	PUNCT
cana-708	18	17	convergence	convergence	NOUN
cana-708	18	18	in	in	ADP
cana-708	18	19	mean	mean	NOUN
cana-708	18	20	to	to	ADP
cana-708	18	21	the	the	DET
cana-708	18	22	stochastic	stochastic	ADJ
cana-708	18	23	integral	integral	ADJ
cana-708	18	24	∫	∫	PROPN
cana-708	18	25	∞	∞	PROPN
cana-708	18	26	−∞	−∞	ADP
cana-708	18	27	𝑔(𝑢)𝑈(𝑢	𝑔(𝑢)𝑈(𝑢	PROPN
cana-708	18	28	,	,	PUNCT
cana-708	18	29	𝑏)𝑑𝑋(𝑢	𝑏)𝑑𝑋(𝑢	PROPN
cana-708	18	30	,	,	PUNCT
cana-708	18	31	𝜔	𝜔	NOUN
cana-708	18	32	)	)	PUNCT
cana-708	18	33	if	if	SCONJ
cana-708	18	34	the	the	DET
cana-708	18	35	scalars	scalar	NOUN
cana-708	18	36	𝑑𝑘	𝑑𝑘	VERB
cana-708	18	37	are	be	AUX
cana-708	18	38	fhc	fhc	PROPN
cana-708	18	39	of	of	ADP
cana-708	18	40	a	a	DET
cana-708	18	41	function	function	NOUN
cana-708	18	42	𝑔	𝑔	NOUN
cana-708	18	43	in	in	ADP
cana-708	18	44	the	the	DET
cana-708	18	45	space	space	NOUN
cana-708	18	46	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	NOUN
cana-708	18	47	)	)	PUNCT
cana-708	18	48	2	2	NUM
cana-708	18	49	(	(	PUNCT
cana-708	18	50	ℝ	ℝ	PROPN
cana-708	18	51	)	)	PUNCT
cana-708	18	52	,	,	PUNCT
cana-708	18	53	defined	define	VERB
cana-708	18	54	as	as	ADP
cana-708	18	55	𝑑𝑘	𝑑𝑘	ADV
cana-708	18	56	:	:	PUNCT
cana-708	18	57	=	=	NOUN
cana-708	18	58	𝑟𝑘	𝑟𝑘	ADJ
cana-708	18	59	2	2	NUM
cana-708	18	60	∫	∫	NOUN
cana-708	18	61	∞	∞	PROPN
cana-708	18	62	−∞	−∞	X
cana-708	18	63	𝑔(𝑢)𝑒−𝑢2	𝑔(𝑢)𝑒−𝑢2	X
cana-708	18	64	𝐻𝑘(𝑢)𝑑𝑢.	𝐻𝑘(𝑢)𝑑𝑢.	PROPN
cana-708	18	65	2	2	NUM
cana-708	18	66	.	.	NOUN
cana-708	18	67	preliminaries	preliminary	NOUN
cana-708	18	68	consider	consider	VERB
cana-708	18	69	𝜙𝑛(𝑢	𝜙𝑛(𝑢	ADJ
cana-708	18	70	)	)	PUNCT
cana-708	18	71	,	,	PUNCT
cana-708	18	72	𝑛	𝑛	DET
cana-708	18	73	∈	∈	PROPN
cana-708	18	74	ℕ0	ℕ0	NOUN
cana-708	18	75	,	,	PUNCT
cana-708	18	76	ℕ0	ℕ0	PROPN
cana-708	18	77	:	:	PUNCT
cana-708	18	78	=	=	X
cana-708	18	79	{	{	PUNCT
cana-708	18	80	0,1,2	0,1,2	NOUN
cana-708	18	81	,	,	PUNCT
cana-708	18	82	.	.	PUNCT
cana-708	18	83	.	.	PUNCT
cana-708	19	1	.	.	PUNCT
cana-708	20	1	}	}	PUNCT
cana-708	20	2	to	to	PART
cana-708	20	3	be	be	AUX
cana-708	20	4	a	a	DET
cana-708	20	5	sequence	sequence	NOUN
cana-708	20	6	of	of	ADP
cana-708	20	7	functions	function	NOUN
cana-708	20	8	orthonormal	orthonormal	ADJ
cana-708	20	9	concerning	concern	VERB
cana-708	20	10	a	a	DET
cana-708	20	11	measure	measure	NOUN
cana-708	20	12	ℱ(𝑢	ℱ(𝑢	NUM
cana-708	20	13	)	)	PUNCT
cana-708	20	14	that	that	PRON
cana-708	20	15	is	be	AUX
cana-708	20	16	,	,	PUNCT
cana-708	20	17	communications	communication	NOUN
cana-708	20	18	on	on	ADP
cana-708	20	19	applied	apply	VERB
cana-708	20	20	nonlinear	nonlinear	ADJ
cana-708	20	21	analysis	analysis	NOUN
cana-708	20	22	issn	issn	NOUN
cana-708	20	23	:	:	PUNCT
cana-708	20	24	1074	1074	NUM
cana-708	20	25	-	-	PUNCT
cana-708	20	26	133x	133x	NUM
cana-708	20	27	vol	vol	NOUN
cana-708	20	28	31	31	NUM
cana-708	20	29	no	no	NOUN
cana-708	20	30	.	.	PUNCT
cana-708	21	1	2s	2s	NUM
cana-708	21	2	(	(	PUNCT
cana-708	21	3	2024	2024	NUM
cana-708	21	4	)	)	PUNCT
cana-708	21	5	697	697	NUM
cana-708	21	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	21	7	∫	∫	NOUN
cana-708	21	8	𝑏	𝑏	PRON
cana-708	21	9	𝑎	𝑎	DET
cana-708	21	10	𝜙𝑛(𝑢)𝜙𝑚(𝑢)𝑑ℱ(𝑢	𝜙𝑛(𝑢)𝜙𝑚(𝑢)𝑑ℱ(𝑢	NOUN
cana-708	21	11	)	)	PUNCT
cana-708	21	12	=	=	SYM
cana-708	21	13	𝛿𝑛𝑚	𝛿𝑛𝑚	PROPN
cana-708	21	14	,	,	PUNCT
cana-708	21	15	where	where	SCONJ
cana-708	21	16	,	,	PUNCT
cana-708	21	17	𝛿𝑚𝑛	𝛿𝑚𝑛	NOUN
cana-708	21	18	is	be	AUX
cana-708	21	19	the	the	DET
cana-708	21	20	kronecher	kronecher	PROPN
cana-708	21	21	’s	’s	PART
cana-708	21	22	delta	delta	NOUN
cana-708	21	23	function	function	NOUN
cana-708	21	24	and	and	CCONJ
cana-708	21	25	let	let	VERB
cana-708	21	26	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	21	27	)	)	PUNCT
cana-708	21	28	∼	∼	NOUN
cana-708	21	29	∑∞	∑∞	NOUN
cana-708	21	30	𝑛=0	𝑛=0	X
cana-708	21	31	𝑎𝑛𝜙𝑛(𝑢	𝑎𝑛𝜙𝑛(𝑢	PROPN
cana-708	21	32	)	)	PUNCT
cana-708	21	33	(	(	PUNCT
cana-708	21	34	2.1	2.1	NUM
cana-708	21	35	)	)	PUNCT
cana-708	21	36	be	be	VERB
cana-708	21	37	the	the	DET
cana-708	21	38	formal	formal	ADJ
cana-708	21	39	expansion	expansion	NOUN
cana-708	21	40	of	of	ADP
cana-708	21	41	an	an	DET
cana-708	21	42	arbitrary	arbitrary	ADJ
cana-708	21	43	function	function	NOUN
cana-708	21	44	in	in	ADP
cana-708	21	45	terms	term	NOUN
cana-708	21	46	of	of	ADP
cana-708	21	47	this	this	DET
cana-708	21	48	sequence	sequence	NOUN
cana-708	21	49	where	where	SCONJ
cana-708	21	50	𝑎𝑛	𝑎𝑛	ADV
cana-708	21	51	:	:	PUNCT
cana-708	21	52	=	=	PUNCT
cana-708	21	53	∫	∫	PROPN
cana-708	22	1	𝑏	𝑏	PROPN
cana-708	22	2	𝑎	𝑎	PROPN
cana-708	22	3	𝑔(𝑣)𝜙𝑛(𝑣)𝑑ℱ(𝑣	𝑔(𝑣)𝜙𝑛(𝑣)𝑑ℱ(𝑣	NOUN
cana-708	22	4	)	)	PUNCT
cana-708	22	5	.	.	PUNCT
cana-708	23	1	many	many	ADJ
cana-708	23	2	researchers	researcher	NOUN
cana-708	23	3	have	have	AUX
cana-708	23	4	exhaustively	exhaustively	ADV
cana-708	23	5	explored	explore	VERB
cana-708	23	6	the	the	DET
cana-708	23	7	convergence	convergence	NOUN
cana-708	23	8	characteristics	characteristic	NOUN
cana-708	23	9	of	of	ADP
cana-708	23	10	series	series	NOUN
cana-708	23	11	of	of	ADP
cana-708	23	12	the	the	DET
cana-708	23	13	form	form	NOUN
cana-708	23	14	(	(	PUNCT
cana-708	23	15	2.1	2.1	NUM
cana-708	23	16	)	)	PUNCT
cana-708	23	17	for	for	ADP
cana-708	23	18	a	a	DET
cana-708	23	19	specific	specific	ADJ
cana-708	23	20	set	set	NOUN
cana-708	23	21	of	of	ADP
cana-708	23	22	functions	function	NOUN
cana-708	23	23	𝜙𝑛(𝑣	𝜙𝑛(𝑣	ADV
cana-708	23	24	)	)	PUNCT
cana-708	23	25	.	.	PUNCT
cana-708	24	1	specifically	specifically	ADV
cana-708	24	2	,	,	PUNCT
cana-708	24	3	the	the	DET
cana-708	24	4	exploration	exploration	NOUN
cana-708	24	5	is	be	AUX
cana-708	24	6	on	on	ADP
cana-708	24	7	the	the	DET
cana-708	24	8	following	follow	VERB
cana-708	24	9	question	question	NOUN
cana-708	24	10	:	:	PUNCT
cana-708	24	11	for	for	ADP
cana-708	24	12	what	what	PRON
cana-708	24	13	values	value	NOUN
cana-708	24	14	of	of	ADP
cana-708	24	15	𝑝	𝑝	NOUN
cana-708	24	16	,	,	PUNCT
cana-708	24	17	1	1	NUM
cana-708	24	18	≤	≤	NOUN
cana-708	24	19	𝑝	𝑝	PROPN
cana-708	24	20	<	<	X
cana-708	24	21	∞	∞	PROPN
cana-708	24	22	,	,	PUNCT
cana-708	24	23	does	do	VERB
cana-708	24	24	the	the	DET
cana-708	24	25	existence	existence	NOUN
cana-708	24	26	of	of	ADP
cana-708	24	27	the	the	DET
cana-708	24	28	integral	integral	ADJ
cana-708	24	29	∫	∫	PROPN
cana-708	24	30	𝑏	𝑏	PROPN
cana-708	24	31	𝑎	𝑎	PROPN
cana-708	24	32	|𝑔(𝑢)|𝑝𝑑ℱ(𝑢	|𝑔(𝑢)|𝑝𝑑ℱ(𝑢	NOUN
cana-708	24	33	)	)	PUNCT
cana-708	24	34	imply	imply	NOUN
cana-708	24	35	,	,	PUNCT
cana-708	24	36	lim	lim	PROPN
cana-708	24	37	𝑛→∞	𝑛→∞	NUM
cana-708	24	38	∫	∫	PROPN
cana-708	25	1	𝑏	𝑏	PROPN
cana-708	25	2	𝑎	𝑎	PROPN
cana-708	25	3	|𝑔(𝑢	|𝑔(𝑢	NOUN
cana-708	25	4	)	)	PUNCT
cana-708	25	5	−	−	PROPN
cana-708	26	1	∑𝑛	∑𝑛	PROPN
cana-708	26	2	𝑘=0	𝑘=0	PROPN
cana-708	26	3	𝑎𝑘𝜙𝑘(𝑢)|𝑝𝑑ℱ(𝑢	𝑎𝑘𝜙𝑘(𝑢)|𝑝𝑑ℱ(𝑢	PROPN
cana-708	26	4	)	)	PUNCT
cana-708	27	1	=	=	PUNCT
cana-708	28	1	0	0	X
cana-708	28	2	?	?	PUNCT
cana-708	28	3	(	(	PUNCT
cana-708	28	4	2.2	2.2	NUM
cana-708	28	5	)	)	PUNCT
cana-708	28	6	the	the	DET
cana-708	28	7	sequence	sequence	NOUN
cana-708	28	8	𝜙𝑛(𝑢	𝜙𝑛(𝑢	PRON
cana-708	28	9	)	)	PUNCT
cana-708	28	10	forms	form	VERB
cana-708	28	11	a	a	DET
cana-708	28	12	basis	basis	NOUN
cana-708	28	13	for	for	ADP
cana-708	28	14	the	the	DET
cana-708	28	15	space	space	NOUN
cana-708	28	16	of	of	ADP
cana-708	28	17	these	these	DET
cana-708	28	18	functions	function	NOUN
cana-708	28	19	when	when	SCONJ
cana-708	28	20	this	this	DET
cana-708	28	21	equation	equation	NOUN
cana-708	28	22	holds	hold	VERB
cana-708	28	23	for	for	ADP
cana-708	28	24	every	every	DET
cana-708	28	25	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	28	26	)	)	PUNCT
cana-708	28	27	such	such	ADJ
cana-708	28	28	that	that	DET
cana-708	28	29	∫	∫	PROPN
cana-708	29	1	𝑏	𝑏	PRON
cana-708	29	2	𝑎	𝑎	PROPN
cana-708	29	3	|𝑔(𝑢)|𝑝𝑑ℱ(𝑢	|𝑔(𝑢)|𝑝𝑑ℱ(𝑢	NOUN
cana-708	29	4	)	)	PUNCT
cana-708	29	5	exists	exist	VERB
cana-708	29	6	[	[	X
cana-708	29	7	25	25	NUM
cana-708	29	8	]	]	PUNCT
cana-708	29	9	.	.	PUNCT
cana-708	30	1	if	if	SCONJ
cana-708	30	2	𝑑ℱ(𝑢	𝑑ℱ(𝑢	NUM
cana-708	30	3	)	)	PUNCT
cana-708	31	1	=	=	PUNCT
cana-708	31	2	𝑊(𝑢)𝑑𝑢	𝑊(𝑢)𝑑𝑢	NOUN
cana-708	31	3	,	,	PUNCT
cana-708	31	4	𝑊(𝑢	𝑊(𝑢	NOUN
cana-708	31	5	)	)	PUNCT
cana-708	31	6	is	be	AUX
cana-708	31	7	the	the	DET
cana-708	31	8	weight	weight	NOUN
cana-708	31	9	function	function	NOUN
cana-708	31	10	then	then	ADV
cana-708	31	11	the	the	DET
cana-708	31	12	sequence	sequence	NOUN
cana-708	31	13	𝜙𝑛(𝑢)(𝑊(𝑢	𝜙𝑛(𝑢)(𝑊(𝑢	NOUN
cana-708	31	14	)	)	PUNCT
cana-708	31	15	)	)	PUNCT
cana-708	31	16	1	1	NUM
cana-708	31	17	2	2	NUM
cana-708	31	18	is	be	AUX
cana-708	31	19	orthonormal	orthonormal	ADJ
cana-708	31	20	on	on	ADP
cana-708	31	21	the	the	DET
cana-708	31	22	classical	classical	ADJ
cana-708	31	23	sense	sense	NOUN
cana-708	31	24	and	and	CCONJ
cana-708	31	25	one	one	NOUN
cana-708	31	26	led	lead	VERB
cana-708	31	27	to	to	ADP
cana-708	31	28	the	the	DET
cana-708	31	29	formal	formal	ADJ
cana-708	31	30	expansion	expansion	NOUN
cana-708	31	31	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	31	32	)	)	PUNCT
cana-708	31	33	∼	∼	NOUN
cana-708	31	34	∑∞	∑∞	NOUN
cana-708	31	35	𝑛=0	𝑛=0	X
cana-708	31	36	𝑏𝑛𝜙𝑛(𝑢)(𝑊(𝑢	𝑏𝑛𝜙𝑛(𝑢)(𝑊(𝑢	PROPN
cana-708	31	37	)	)	PUNCT
cana-708	31	38	)	)	PUNCT
cana-708	31	39	1	1	NUM
cana-708	31	40	2	2	NUM
cana-708	31	41	,	,	PUNCT
cana-708	31	42	where	where	SCONJ
cana-708	31	43	𝑏𝑛	𝑏𝑛	VERB
cana-708	31	44	:	:	PUNCT
cana-708	31	45	=	=	SYM
cana-708	31	46	∫	∫	PROPN
cana-708	31	47	𝑏	𝑏	PROPN
cana-708	31	48	𝑎	𝑎	PROPN
cana-708	31	49	𝑔(𝑣)𝜙𝑛(𝑣)(𝑊(𝑣	𝑔(𝑣)𝜙𝑛(𝑣)(𝑊(𝑣	NOUN
cana-708	31	50	)	)	PUNCT
cana-708	31	51	)	)	PUNCT
cana-708	32	1	1	1	NUM
cana-708	32	2	2𝑑𝑣.	2𝑑𝑣.	NUM
cana-708	32	3	the	the	DET
cana-708	32	4	above	above	ADJ
cana-708	32	5	question	question	NOUN
cana-708	32	6	can	can	AUX
cana-708	32	7	be	be	AUX
cana-708	32	8	read	read	VERB
cana-708	32	9	as	as	ADP
cana-708	32	10	:	:	PUNCT
cana-708	32	11	for	for	ADP
cana-708	32	12	what	what	PRON
cana-708	32	13	values	value	NOUN
cana-708	32	14	of	of	ADP
cana-708	32	15	𝑝	𝑝	NOUN
cana-708	32	16	,	,	PUNCT
cana-708	32	17	1	1	NUM
cana-708	32	18	≤	≤	NOUN
cana-708	32	19	𝑝	𝑝	PROPN
cana-708	32	20	<	<	X
cana-708	32	21	∞	∞	PROPN
cana-708	32	22	,	,	PUNCT
cana-708	32	23	does	do	VERB
cana-708	32	24	the	the	DET
cana-708	32	25	measurability	measurability	NOUN
cana-708	32	26	of	of	ADP
cana-708	32	27	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	32	28	)	)	PUNCT
cana-708	32	29	and	and	CCONJ
cana-708	32	30	the	the	DET
cana-708	32	31	relation	relation	NOUN
cana-708	32	32	∫	∫	PROPN
cana-708	33	1	𝑏	𝑏	PRON
cana-708	33	2	𝑎	𝑎	PRON
cana-708	33	3	|𝑔(𝑢)|𝑝𝑑𝑢	|𝑔(𝑢)|𝑝𝑑𝑢	NOUN
cana-708	33	4	<	<	X
cana-708	33	5	∞	∞	PROPN
cana-708	33	6	(	(	PUNCT
cana-708	33	7	i.	i.	PROPN
cana-708	33	8	e.	e.	PROPN
cana-708	33	9	,	,	PUNCT
cana-708	33	10	𝑔	𝑔	PROPN
cana-708	33	11	∈	∈	PROPN
cana-708	33	12	𝐿𝑝(ℝ	𝐿𝑝(ℝ	NUM
cana-708	33	13	)	)	PUNCT
cana-708	33	14	)	)	PUNCT
cana-708	33	15	imply	imply	ADV
cana-708	33	16	,	,	PUNCT
cana-708	33	17	lim	lim	PROPN
cana-708	33	18	𝑛→∞	𝑛→∞	NUM
cana-708	33	19	∫	∫	PROPN
cana-708	34	1	𝑏	𝑏	PROPN
cana-708	34	2	𝑎	𝑎	PROPN
cana-708	34	3	|𝑔(𝑢	|𝑔(𝑢	NOUN
cana-708	34	4	)	)	PUNCT
cana-708	34	5	−	−	PROPN
cana-708	35	1	∑𝑛	∑𝑛	PROPN
cana-708	35	2	𝑘=0	𝑘=0	VERB
cana-708	35	3	𝑏𝑘𝜙𝑘(𝑢)(𝑊(𝑢	𝑏𝑘𝜙𝑘(𝑢)(𝑊(𝑢	PROPN
cana-708	35	4	)	)	PUNCT
cana-708	35	5	)	)	PUNCT
cana-708	35	6	1	1	NUM
cana-708	35	7	2|𝑝𝑑𝑢	2|𝑝𝑑𝑢	NUM
cana-708	35	8	=	=	SYM
cana-708	35	9	0	0	X
cana-708	35	10	?	?	PUNCT
cana-708	35	11	m.	m.	NOUN
cana-708	35	12	riesz	riesz	NOUN
cana-708	36	1	[	[	X
cana-708	36	2	19]was	19]was	NUM
cana-708	36	3	the	the	DET
cana-708	36	4	first	first	ADJ
cana-708	36	5	to	to	PART
cana-708	36	6	look	look	VERB
cana-708	36	7	into	into	ADP
cana-708	36	8	this	this	DET
cana-708	36	9	kind	kind	NOUN
cana-708	36	10	of	of	ADP
cana-708	36	11	issue	issue	NOUN
cana-708	36	12	,	,	PUNCT
cana-708	36	13	focussing	focusse	VERB
cana-708	36	14	on	on	ADP
cana-708	36	15	the	the	DET
cana-708	36	16	case	case	NOUN
cana-708	36	17	of	of	ADP
cana-708	36	18	trigonometric	trigonometric	ADJ
cana-708	36	19	functions	function	NOUN
cana-708	36	20	.	.	PUNCT
cana-708	37	1	later	later	ADV
cana-708	37	2	,	,	PUNCT
cana-708	37	3	schauder	schauder	NOUN
cana-708	38	1	[	[	X
cana-708	38	2	21	21	NUM
cana-708	38	3	]	]	PUNCT
cana-708	38	4	,	,	PUNCT
cana-708	38	5	kober	kober	PROPN
cana-708	39	1	[	[	X
cana-708	39	2	9	9	NUM
cana-708	39	3	]	]	PUNCT
cana-708	39	4	,	,	PUNCT
cana-708	39	5	caton	caton	NOUN
cana-708	39	6	and	and	CCONJ
cana-708	39	7	hille	hille	PROPN
cana-708	40	1	[	[	X
cana-708	40	2	3	3	X
cana-708	40	3	]	]	PUNCT
cana-708	40	4	looked	look	VERB
cana-708	40	5	at	at	ADP
cana-708	40	6	other	other	ADJ
cana-708	40	7	sets	set	NOUN
cana-708	40	8	of	of	ADP
cana-708	40	9	functions	function	NOUN
cana-708	40	10	.	.	PUNCT
cana-708	41	1	in	in	ADP
cana-708	41	2	this	this	DET
cana-708	41	3	article	article	NOUN
cana-708	41	4	the	the	DET
cana-708	41	5	sequence	sequence	NOUN
cana-708	41	6	of	of	ADP
cana-708	41	7	functions	function	NOUN
cana-708	41	8	𝜙𝑛(𝑢	𝜙𝑛(𝑢	PRON
cana-708	41	9	)	)	PUNCT
cana-708	41	10	are	be	AUX
cana-708	41	11	considered	consider	VERB
cana-708	41	12	to	to	PART
cana-708	41	13	be	be	AUX
cana-708	41	14	the	the	DET
cana-708	41	15	orthogonal	orthogonal	ADJ
cana-708	41	16	hermite	hermite	ADJ
cana-708	41	17	polynomials	polynomial	NOUN
cana-708	41	18	𝐻𝑛(𝑢	𝐻𝑛(𝑢	VERB
cana-708	41	19	)	)	PUNCT
cana-708	41	20	with	with	ADP
cana-708	41	21	weight	weight	NOUN
cana-708	41	22	𝑒−𝑢2	𝑒−𝑢2	NOUN
cana-708	41	23	satisfy	satisfy	PROPN
cana-708	41	24	∫	∫	PROPN
cana-708	41	25	∞	∞	PROPN
cana-708	41	26	−∞	−∞	ADP
cana-708	41	27	𝐻𝑚(𝑢)𝐻𝑛(𝑢)𝑒−𝑢2	𝐻𝑚(𝑢)𝐻𝑛(𝑢)𝑒−𝑢2	NOUN
cana-708	41	28	𝑑𝑢	𝑑𝑢	X
cana-708	41	29	=	=	PROPN
cana-708	41	30	√𝜋2	√𝜋2	PROPN
cana-708	41	31	𝑛	𝑛	PROPN
cana-708	41	32	2𝑛	2𝑛	NUM
cana-708	41	33	!	!	PUNCT
cana-708	41	34	𝛿𝑚𝑛.	𝛿𝑚𝑛.	X
cana-708	41	35	(	(	PUNCT
cana-708	41	36	2.3	2.3	NUM
cana-708	41	37	)	)	PUNCT
cana-708	41	38	the	the	DET
cana-708	41	39	𝑛𝑡ℎ	𝑛𝑡ℎ	NOUN
cana-708	41	40	degree	degree	VERB
cana-708	41	41	hermite	hermite	ADJ
cana-708	41	42	polynomials	polynomial	NOUN
cana-708	41	43	defined	define	VERB
cana-708	41	44	as	as	ADP
cana-708	41	45	𝐻𝑛(𝑢	𝐻𝑛(𝑢	NOUN
cana-708	41	46	)	)	PUNCT
cana-708	41	47	=	=	PUNCT
cana-708	42	1	(	(	PUNCT
cana-708	42	2	−1)𝑛𝑒𝑢2	−1)𝑛𝑒𝑢2	NUM
cana-708	42	3	(	(	PUNCT
cana-708	42	4	𝑑	𝑑	PROPN
cana-708	42	5	𝑑𝑢	𝑑𝑢	PROPN
cana-708	42	6	)	)	PUNCT
cana-708	42	7	𝑛{𝑒−𝑢2	𝑛{𝑒−𝑢2	PROPN
cana-708	42	8	}	}	PUNCT
cana-708	43	1	[	[	X
cana-708	43	2	23	23	NUM
cana-708	43	3	]	]	PUNCT
cana-708	43	4	.	.	PUNCT
cana-708	44	1	the	the	DET
cana-708	44	2	normalized	normalize	VERB
cana-708	44	3	hermite	hermite	ADJ
cana-708	44	4	functions	function	NOUN
cana-708	44	5	of	of	ADP
cana-708	44	6	degree	degree	NOUN
cana-708	44	7	𝑛	𝑛	DET
cana-708	44	8	∈	∈	PROPN
cana-708	44	9	ℕ0	ℕ0	NOUN
cana-708	44	10	[	[	X
cana-708	44	11	4	4	NUM
cana-708	44	12	,	,	PUNCT
cana-708	44	13	5	5	NUM
cana-708	44	14	,	,	PUNCT
cana-708	44	15	6	6	NUM
cana-708	44	16	,	,	PUNCT
cana-708	44	17	16	16	NUM
cana-708	44	18	]	]	PUNCT
cana-708	44	19	defined	define	VERB
cana-708	44	20	as	as	ADP
cana-708	44	21	,	,	PUNCT
cana-708	44	22	𝜓𝑛(𝑢	𝜓𝑛(𝑢	PRON
cana-708	44	23	):	):	PUNCT
cana-708	44	24	=	=	SYM
cana-708	44	25	𝑟𝑛𝐻𝑛(𝑢)𝑒−	𝑟𝑛𝐻𝑛(𝑢)𝑒−	NUM
cana-708	44	26	𝑢2	𝑢2	PROPN
cana-708	44	27	2	2	NUM
cana-708	44	28	,	,	PUNCT
cana-708	44	29	𝑛	𝑛	DET
cana-708	44	30	≥	≥	NOUN
cana-708	44	31	0	0	NUM
cana-708	44	32	,	,	PUNCT
cana-708	44	33	𝑢	𝑢	PROPN
cana-708	44	34	∈	∈	PROPN
cana-708	44	35	ℝ	ℝ	PROPN
cana-708	44	36	,	,	PUNCT
cana-708	44	37	(	(	PUNCT
cana-708	44	38	2.4	2.4	NUM
cana-708	44	39	)	)	PUNCT
cana-708	44	40	where	where	SCONJ
cana-708	44	41	𝑟𝑛	𝑟𝑛	NOUN
cana-708	44	42	=	=	SYM
cana-708	44	43	1	1	NUM
cana-708	44	44	√2𝑛𝑛!√𝜋	√2𝑛𝑛!√𝜋	NOUN
cana-708	44	45	meet	meet	VERB
cana-708	44	46	the	the	DET
cana-708	44	47	orthonormal	orthonormal	ADJ
cana-708	44	48	condition	condition	NOUN
cana-708	44	49	∫	∫	PROPN
cana-708	44	50	∞	∞	PROPN
cana-708	44	51	−∞	−∞	ADP
cana-708	44	52	𝜓𝑚(𝑢)𝜓𝑛(𝑢)𝑑𝑢	𝜓𝑚(𝑢)𝜓𝑛(𝑢)𝑑𝑢	PROPN
cana-708	44	53	=	=	SYM
cana-708	44	54	𝛿𝑚𝑛.	𝛿𝑚𝑛.	X
cana-708	44	55	(	(	PUNCT
cana-708	44	56	2.5	2.5	NUM
cana-708	44	57	)	)	PUNCT
cana-708	44	58	these	these	DET
cana-708	44	59	𝜓𝑛(𝑢	𝜓𝑛(𝑢	NOUN
cana-708	44	60	)	)	PUNCT
cana-708	44	61	form	form	NOUN
cana-708	44	62	a	a	DET
cana-708	44	63	basis	basis	NOUN
cana-708	44	64	in	in	ADP
cana-708	44	65	𝐿𝑝(ℝ	𝐿𝑝(ℝ	NUM
cana-708	44	66	)	)	PUNCT
cana-708	44	67	,	,	PUNCT
cana-708	44	68	𝑝	𝑝	PROPN
cana-708	44	69	≥	≥	NOUN
cana-708	44	70	2	2	NUM
cana-708	44	71	[	[	X
cana-708	44	72	24	24	NUM
cana-708	44	73	,	,	PUNCT
cana-708	44	74	13	13	NUM
cana-708	44	75	]	]	PUNCT
cana-708	44	76	.	.	PUNCT
cana-708	45	1	pollard	pollard	PROPN
cana-708	45	2	[	[	X
cana-708	45	3	18	18	NUM
cana-708	45	4	]	]	PUNCT
cana-708	45	5	in	in	ADP
cana-708	45	6	1948	1948	NUM
cana-708	45	7	showed	show	VERB
cana-708	45	8	that	that	SCONJ
cana-708	45	9	,	,	PUNCT
cana-708	45	10	if	if	SCONJ
cana-708	45	11	𝑔(𝑢)𝑒	𝑔(𝑢)𝑒	NOUN
cana-708	45	12	−𝑢2	−𝑢2	VERB
cana-708	45	13	2	2	NUM
cana-708	45	14	∈	∈	PROPN
cana-708	45	15	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-708	45	16	)	)	PUNCT
cana-708	45	17	,	,	PUNCT
cana-708	45	18	then	then	ADV
cana-708	45	19	∫	∫	PROPN
cana-708	45	20	∞	∞	PROPN
cana-708	45	21	−∞	−∞	ADP
cana-708	45	22	|𝑠𝑛(𝑢)|𝑝𝑒−𝑢2	|𝑠𝑛(𝑢)|𝑝𝑒−𝑢2	X
cana-708	45	23	𝑑𝑢	𝑑𝑢	PROPN
cana-708	45	24	≤	≤	PROPN
cana-708	45	25	𝐶	𝐶	PROPN
cana-708	45	26	∫	∫	PROPN
cana-708	45	27	∞	∞	PROPN
cana-708	45	28	−∞	−∞	ADP
cana-708	45	29	|𝑔(𝑢)|2𝑒−𝑢2	|𝑔(𝑢)|2𝑒−𝑢2	NOUN
cana-708	45	30	𝑑𝑢	𝑑𝑢	NOUN
cana-708	45	31	,	,	PUNCT
cana-708	45	32	where	where	SCONJ
cana-708	45	33	,	,	PUNCT
cana-708	45	34	𝑠𝑛	𝑠𝑛	NOUN
cana-708	45	35	is	be	AUX
cana-708	45	36	the	the	DET
cana-708	45	37	𝑛𝑡ℎ	𝑛𝑡ℎ	NUM
cana-708	45	38	partial	partial	ADJ
cana-708	45	39	sum	sum	NOUN
cana-708	45	40	of	of	ADP
cana-708	45	41	the	the	DET
cana-708	45	42	hermite	hermite	ADJ
cana-708	45	43	polynomial	polynomial	ADJ
cana-708	45	44	series	series	PROPN
cana-708	45	45	∑∞	∑∞	PROPN
cana-708	45	46	𝑘=0	𝑘=0	VERB
cana-708	45	47	𝑑𝑘𝐻𝑘(𝑢	𝑑𝑘𝐻𝑘(𝑢	NOUN
cana-708	45	48	)	)	PUNCT
cana-708	45	49	for	for	ADP
cana-708	45	50	𝑑𝑘	𝑑𝑘	ADV
cana-708	45	51	:	:	PUNCT
cana-708	45	52	=	=	NOUN
cana-708	45	53	𝑟𝑘	𝑟𝑘	ADJ
cana-708	45	54	2	2	NUM
cana-708	45	55	∫	∫	NOUN
cana-708	45	56	∞	∞	PROPN
cana-708	45	57	−∞	−∞	X
cana-708	45	58	𝑔(𝑢)𝐻𝑘(𝑢)𝑒−𝑢𝑢	𝑔(𝑢)𝐻𝑘(𝑢)𝑒−𝑢𝑢	NOUN
cana-708	45	59	,	,	PUNCT
cana-708	45	60	𝑘	𝑘	PROPN
cana-708	45	61	∈	∈	PROPN
cana-708	45	62	ℕ0	ℕ0	NOUN
cana-708	45	63	.	.	PUNCT
cana-708	46	1	this	this	PRON
cana-708	46	2	suggests	suggest	VERB
cana-708	46	3	that	that	SCONJ
cana-708	46	4	communications	communication	NOUN
cana-708	46	5	on	on	ADP
cana-708	46	6	applied	apply	VERB
cana-708	46	7	nonlinear	nonlinear	ADJ
cana-708	46	8	analysis	analysis	NOUN
cana-708	46	9	issn	issn	NOUN
cana-708	46	10	:	:	PUNCT
cana-708	46	11	1074	1074	NUM
cana-708	46	12	-	-	PUNCT
cana-708	46	13	133x	133x	NUM
cana-708	46	14	vol	vol	NOUN
cana-708	46	15	31	31	NUM
cana-708	46	16	no	no	NOUN
cana-708	46	17	.	.	PUNCT
cana-708	47	1	2s	2s	NUM
cana-708	47	2	(	(	PUNCT
cana-708	47	3	2024	2024	NUM
cana-708	47	4	)	)	PUNCT
cana-708	47	5	698	698	NUM
cana-708	47	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	47	7	∥	∥	X
cana-708	47	8	𝑠𝑛	𝑠𝑛	NOUN
cana-708	47	9	−	−	PROPN
cana-708	47	10	𝑔	𝑔	PROPN
cana-708	47	11	∥2→	∥2→	X
cana-708	47	12	0	0	NUM
cana-708	47	13	(	(	PUNCT
cana-708	47	14	2.6	2.6	NUM
cana-708	47	15	)	)	PUNCT
cana-708	47	16	as	as	ADP
cana-708	47	17	𝑛	𝑛	PROPN
cana-708	47	18	→	→	SYM
cana-708	47	19	∞	∞	PROPN
cana-708	47	20	with	with	ADP
cana-708	47	21	∥.	∥.	ADJ
cana-708	47	22	∥2	∥2	PRON
cana-708	47	23	denoting	denote	VERB
cana-708	47	24	the	the	DET
cana-708	47	25	usual	usual	ADJ
cana-708	47	26	𝐿2	𝐿2	NOUN
cana-708	47	27	norm	norm	NOUN
cana-708	47	28	on	on	ADP
cana-708	47	29	ℝ.	ℝ.	PROPN
cana-708	47	30	this	this	DET
cana-708	47	31	conclusion	conclusion	NOUN
cana-708	47	32	was	be	AUX
cana-708	47	33	extended	extend	VERB
cana-708	47	34	to	to	ADP
cana-708	47	35	a	a	DET
cana-708	47	36	larger	large	ADJ
cana-708	47	37	class	class	NOUN
cana-708	47	38	of	of	ADP
cana-708	47	39	functions	function	NOUN
cana-708	47	40	𝐿𝑝(ℝ	𝐿𝑝(ℝ	NUM
cana-708	47	41	)	)	PUNCT
cana-708	47	42	,	,	PUNCT
cana-708	47	43	4	4	NUM
cana-708	47	44	3	3	NUM
cana-708	47	45	<	<	X
cana-708	47	46	𝑝	𝑝	NOUN
cana-708	47	47	<	<	X
cana-708	47	48	4	4	NUM
cana-708	47	49	by	by	ADP
cana-708	47	50	askey	askey	NOUN
cana-708	47	51	and	and	CCONJ
cana-708	47	52	wainger	wainger	NOUN
cana-708	48	1	[	[	X
cana-708	48	2	1	1	NUM
cana-708	48	3	]	]	PUNCT
cana-708	48	4	in	in	ADP
cana-708	48	5	1965	1965	NUM
cana-708	48	6	.	.	PUNCT
cana-708	49	1	for	for	ADP
cana-708	49	2	measurable	measurable	ADJ
cana-708	49	3	function	function	NOUN
cana-708	49	4	𝑔	𝑔	ADP
cana-708	49	5	such	such	ADJ
cana-708	49	6	that	that	DET
cana-708	49	7	𝑔(𝑢)𝑒	𝑔(𝑢)𝑒	NOUN
cana-708	49	8	−𝑢2	−𝑢2	PROPN
cana-708	49	9	2	2	NUM
cana-708	49	10	∈	∈	NOUN
cana-708	49	11	𝐿𝑝(ℝ	𝐿𝑝(ℝ	NUM
cana-708	49	12	)	)	PUNCT
cana-708	49	13	,	,	PUNCT
cana-708	49	14	4	4	NUM
cana-708	49	15	3	3	NUM
cana-708	49	16	<	<	X
cana-708	49	17	𝑝	𝑝	NOUN
cana-708	49	18	<	<	X
cana-708	49	19	4	4	NUM
cana-708	49	20	,	,	PUNCT
cana-708	49	21	they	they	PRON
cana-708	49	22	proved	prove	VERB
cana-708	49	23	the	the	DET
cana-708	49	24	inequality	inequality	NOUN
cana-708	49	25	∥	∥	PUNCT
cana-708	49	26	𝑠𝑛(𝑢)𝑒−	𝑠𝑛(𝑢)𝑒−	ADJ
cana-708	49	27	𝑢2	𝑢2	PROPN
cana-708	49	28	2	2	NUM
cana-708	49	29	∥𝑝≤	∥𝑝≤	NUM
cana-708	49	30	𝐶	𝐶	PROPN
cana-708	49	31	∥	∥	PRON
cana-708	49	32	𝑔(𝑢)𝑒−	𝑔(𝑢)𝑒−	NOUN
cana-708	49	33	𝑢2	𝑢2	PROPN
cana-708	49	34	2	2	NUM
cana-708	49	35	∥𝑝	∥𝑝	PROPN
cana-708	49	36	,	,	PUNCT
cana-708	49	37	where	where	SCONJ
cana-708	49	38	𝑠𝑛	𝑠𝑛	NOUN
cana-708	49	39	=	=	SYM
cana-708	49	40	∑𝑛	∑𝑛	PROPN
cana-708	49	41	𝑘=0	𝑘=0	PROPN
cana-708	49	42	𝑎𝑘𝜓𝑘(𝑢	𝑎𝑘𝜓𝑘(𝑢	NOUN
cana-708	49	43	)	)	PUNCT
cana-708	49	44	and	and	CCONJ
cana-708	49	45	𝑎𝑘	𝑎𝑘	X
cana-708	49	46	=	=	PUNCT
cana-708	50	1	∫	∫	PROPN
cana-708	50	2	∞	∞	PROPN
cana-708	50	3	−∞	−∞	ADP
cana-708	50	4	𝑓(𝑣)𝜓𝑘(𝑣)𝑑𝑣.	𝑓(𝑣)𝜓𝑘(𝑣)𝑑𝑣.	PROPN
cana-708	50	5	this	this	PRON
cana-708	50	6	implies	imply	VERB
cana-708	50	7	the	the	DET
cana-708	50	8	mean	mean	ADJ
cana-708	50	9	convergence	convergence	NOUN
cana-708	50	10	∥	∥	PUNCT
cana-708	50	11	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	50	12	)	)	PUNCT
cana-708	51	1	−	−	ADP
cana-708	52	1	∑𝑛	∑𝑛	PROPN
cana-708	52	2	𝑘=0	𝑘=0	PROPN
cana-708	52	3	𝑎𝑘𝜓𝑘(𝑢	𝑎𝑘𝜓𝑘(𝑢	PROPN
cana-708	52	4	)	)	PUNCT
cana-708	52	5	∥𝑝→	∥𝑝→	PROPN
cana-708	52	6	0	0	PUNCT
cana-708	52	7	(	(	PUNCT
cana-708	52	8	2.7	2.7	NUM
cana-708	52	9	)	)	PUNCT
cana-708	52	10	as	as	ADP
cana-708	52	11	𝑛	𝑛	PROPN
cana-708	52	12	→	→	SYM
cana-708	52	13	∞	∞	PROPN
cana-708	53	1	where	where	SCONJ
cana-708	53	2	∥	∥	PUNCT
cana-708	53	3	𝑔	𝑔	X
cana-708	53	4	∥𝑝=	∥𝑝=	PRON
cana-708	53	5	{	{	PUNCT
cana-708	53	6	∫	∫	PROPN
cana-708	53	7	∞	∞	PROPN
cana-708	53	8	−∞	−∞	PUNCT
cana-708	53	9	|𝑔|𝑝𝑑𝑢	|𝑔|𝑝𝑑𝑢	PROPN
cana-708	53	10	}	}	SYM
cana-708	53	11	1	1	NUM
cana-708	53	12	𝑝.	𝑝.	NOUN
cana-708	53	13	in	in	ADP
cana-708	53	14	1970	1970	NUM
cana-708	53	15	,	,	PUNCT
cana-708	53	16	muckenhoupt	muckenhoupt	ADJ
cana-708	53	17	[	[	X
cana-708	53	18	14	14	NUM
cana-708	53	19	]	]	PUNCT
cana-708	53	20	generalized	generalize	VERB
cana-708	53	21	the	the	DET
cana-708	53	22	askey	askey	NOUN
cana-708	53	23	and	and	CCONJ
cana-708	53	24	wainger	wainger	NOUN
cana-708	53	25	[	[	X
cana-708	53	26	1	1	NUM
cana-708	53	27	]	]	PUNCT
cana-708	53	28	result	result	NOUN
cana-708	53	29	for	for	ADP
cana-708	53	30	𝑝	𝑝	PROPN
cana-708	53	31	∈	∈	PROPN
cana-708	53	32	[	[	X
cana-708	53	33	1	1	NUM
cana-708	53	34	,	,	PUNCT
cana-708	53	35	∞	∞	PROPN
cana-708	53	36	)	)	PUNCT
cana-708	53	37	.	.	PUNCT
cana-708	54	1	he	he	PRON
cana-708	54	2	proved	prove	VERB
cana-708	54	3	inequalities	inequality	NOUN
cana-708	54	4	of	of	ADP
cana-708	54	5	the	the	DET
cana-708	54	6	form	form	NOUN
cana-708	54	7	∥	∥	PUNCT
cana-708	54	8	𝑠𝑛(𝑢)𝑈(𝑢	𝑠𝑛(𝑢)𝑈(𝑢	NUM
cana-708	54	9	)	)	PUNCT
cana-708	54	10	∥𝑝≤	∥𝑝≤	NUM
cana-708	54	11	𝒞	𝒞	PROPN
cana-708	54	12	∥	∥	PUNCT
cana-708	54	13	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	54	14	)	)	PUNCT
cana-708	54	15	∥𝑝	∥𝑝	PROPN
cana-708	54	16	,	,	PUNCT
cana-708	54	17	where	where	SCONJ
cana-708	54	18	𝑈(𝑢	𝑈(𝑢	X
cana-708	54	19	)	)	PUNCT
cana-708	54	20	,	,	PUNCT
cana-708	54	21	𝑊(𝑢	𝑊(𝑢	NOUN
cana-708	54	22	)	)	PUNCT
cana-708	54	23	are	be	AUX
cana-708	54	24	suitable	suitable	ADJ
cana-708	54	25	weight	weight	NOUN
cana-708	54	26	functions	function	NOUN
cana-708	54	27	.	.	PUNCT
cana-708	55	1	it	it	PRON
cana-708	55	2	lead	lead	VERB
cana-708	55	3	to	to	PART
cana-708	55	4	prove	prove	VERB
cana-708	55	5	∥	∥	PROPN
cana-708	55	6	(	(	PUNCT
cana-708	55	7	𝑠𝑛(𝑢	𝑠𝑛(𝑢	ADJ
cana-708	55	8	)	)	PUNCT
cana-708	55	9	−	−	PROPN
cana-708	55	10	𝑔(𝑢))𝑈(𝑢	𝑔(𝑢))𝑈(𝑢	ADJ
cana-708	55	11	)	)	PUNCT
cana-708	55	12	∥𝑝→	∥𝑝→	NOUN
cana-708	55	13	0	0	NUM
cana-708	55	14	,	,	PUNCT
cana-708	55	15	for	for	ADP
cana-708	55	16	every	every	DET
cana-708	55	17	𝑔	𝑔	PROPN
cana-708	55	18	∈	∈	PROPN
cana-708	55	19	𝐿𝑊(𝑢	𝐿𝑊(𝑢	NOUN
cana-708	55	20	)	)	PUNCT
cana-708	55	21	𝑝	𝑝	NOUN
cana-708	55	22	i.e.	i.e.	X
cana-708	55	23	,	,	PUNCT
cana-708	55	24	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	55	25	)	)	PUNCT
cana-708	55	26	∈	∈	PROPN
cana-708	55	27	𝐿𝑝.	𝐿𝑝.	PROPN
cana-708	55	28	if	if	SCONJ
cana-708	55	29	𝑈(𝑢	𝑈(𝑢	PRON
cana-708	55	30	)	)	PUNCT
cana-708	55	31	=	=	SYM
cana-708	55	32	𝑊(𝑢	𝑊(𝑢	NOUN
cana-708	55	33	)	)	PUNCT
cana-708	55	34	=	=	PUNCT
cana-708	55	35	𝑒	𝑒	PROPN
cana-708	55	36	−𝑢2	−𝑢2	NOUN
cana-708	55	37	2	2	NUM
cana-708	55	38	,	,	PUNCT
cana-708	55	39	he	he	PRON
cana-708	55	40	obtained	obtain	VERB
cana-708	55	41	the	the	DET
cana-708	55	42	result	result	NOUN
cana-708	55	43	of	of	ADP
cana-708	55	44	askey	askey	NOUN
cana-708	55	45	and	and	CCONJ
cana-708	55	46	wainger	wainger	NOUN
cana-708	56	1	[	[	X
cana-708	56	2	1	1	NUM
cana-708	56	3	]	]	PUNCT
cana-708	56	4	for	for	ADP
cana-708	56	5	4	4	NUM
cana-708	56	6	3	3	NUM
cana-708	56	7	<	<	X
cana-708	56	8	𝑝	𝑝	NOUN
cana-708	56	9	<	<	X
cana-708	56	10	4	4	NUM
cana-708	56	11	.	.	PUNCT
cana-708	57	1	if	if	SCONJ
cana-708	57	2	𝑈(𝑢	𝑈(𝑢	PROPN
cana-708	57	3	,	,	PUNCT
cana-708	57	4	𝑏	𝑏	NOUN
cana-708	57	5	)	)	PUNCT
cana-708	57	6	=	=	PUNCT
cana-708	57	7	𝑒	𝑒	PROPN
cana-708	57	8	−𝑢2	−𝑢2	NOUN
cana-708	57	9	2	2	NUM
cana-708	57	10	(	(	PUNCT
cana-708	57	11	1	1	NUM
cana-708	57	12	+	+	CCONJ
cana-708	57	13	|𝑢|)𝑏	|𝑢|)𝑏	PROPN
cana-708	57	14	and	and	CCONJ
cana-708	57	15	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	57	16	,	,	PUNCT
cana-708	57	17	𝐵	𝐵	PROPN
cana-708	57	18	)	)	PUNCT
cana-708	57	19	=	=	PUNCT
cana-708	57	20	𝑒	𝑒	PROPN
cana-708	57	21	−𝑢2	−𝑢2	NOUN
cana-708	57	22	2	2	NUM
cana-708	57	23	(	(	PUNCT
cana-708	57	24	1	1	NUM
cana-708	58	1	+	+	NUM
cana-708	58	2	|𝑢|)𝐵	|𝑢|)𝐵	PROPN
cana-708	58	3	for	for	ADP
cana-708	58	4	different	different	ADJ
cana-708	58	5	suitable	suitable	ADJ
cana-708	58	6	numbers	number	NOUN
cana-708	58	7	𝑏	𝑏	PROPN
cana-708	58	8	and	and	CCONJ
cana-708	58	9	𝐵	𝐵	PRON
cana-708	58	10	such	such	ADJ
cana-708	58	11	that	that	SCONJ
cana-708	58	12	𝑏	𝑏	PROPN
cana-708	58	13	<	<	X
cana-708	58	14	𝐵	𝐵	PROPN
cana-708	58	15	,	,	PUNCT
cana-708	58	16	he	he	PRON
cana-708	58	17	obtained	obtain	VERB
cana-708	58	18	this	this	DET
cana-708	58	19	result	result	NOUN
cana-708	58	20	for	for	ADP
cana-708	58	21	1	1	NUM
cana-708	58	22	≤	≤	NUM
cana-708	58	23	𝑝	𝑝	NOUN
cana-708	58	24	≤	≤	NUM
cana-708	58	25	4	4	NUM
cana-708	58	26	3	3	NUM
cana-708	58	27	and	and	CCONJ
cana-708	58	28	𝑝	𝑝	NOUN
cana-708	58	29	≥	≥	NOUN
cana-708	58	30	4	4	NUM
cana-708	58	31	.	.	PUNCT
cana-708	59	1	𝑈(𝑢	𝑈(𝑢	X
cana-708	59	2	,	,	PUNCT
cana-708	59	3	𝑏	𝑏	NOUN
cana-708	59	4	)	)	PUNCT
cana-708	59	5	and	and	CCONJ
cana-708	59	6	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	59	7	,	,	PUNCT
cana-708	59	8	𝐵	𝐵	NOUN
cana-708	59	9	)	)	PUNCT
cana-708	59	10	are	be	AUX
cana-708	59	11	dense	dense	ADJ
cana-708	59	12	in	in	ADP
cana-708	59	13	𝐿𝑝(ℝ)[14	𝐿𝑝(ℝ)[14	PROPN
cana-708	59	14	]	]	PUNCT
cana-708	59	15	.	.	PUNCT
cana-708	60	1	the	the	DET
cana-708	60	2	random	random	ADJ
cana-708	60	3	series	series	NOUN
cana-708	60	4	considered	consider	VERB
cana-708	60	5	in	in	ADP
cana-708	60	6	this	this	DET
cana-708	60	7	article	article	NOUN
cana-708	60	8	is	be	AUX
cana-708	60	9	expressed	express	VERB
cana-708	60	10	as	as	ADP
cana-708	60	11	∑∞	∑∞	NOUN
cana-708	60	12	𝑘=0	𝑘=0	DET
cana-708	60	13	𝑑𝑘ℛ𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘ℛ𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	60	14	)	)	PUNCT
cana-708	60	15	(	(	PUNCT
cana-708	60	16	2.8	2.8	NUM
cana-708	60	17	)	)	PUNCT
cana-708	60	18	where	where	SCONJ
cana-708	60	19	𝑑𝑘	𝑑𝑘	ADV
cana-708	60	20	represents	represent	VERB
cana-708	60	21	scalars	scalar	NOUN
cana-708	60	22	and	and	CCONJ
cana-708	60	23	ℛ𝑘	ℛ𝑘	PROPN
cana-708	60	24	denotes	denote	VERB
cana-708	60	25	random	random	ADJ
cana-708	60	26	variables	variable	NOUN
cana-708	60	27	.	.	PUNCT
cana-708	61	1	the	the	DET
cana-708	61	2	work	work	NOUN
cana-708	61	3	of	of	ADP
cana-708	61	4	nayak	nayak	PROPN
cana-708	61	5	et	et	PROPN
cana-708	61	6	al	al	PROPN
cana-708	61	7	.	.	PUNCT
cana-708	62	1	[	[	X
cana-708	62	2	15	15	NUM
cana-708	62	3	]	]	PUNCT
cana-708	62	4	and	and	CCONJ
cana-708	62	5	pattanayak	pattanayak	NOUN
cana-708	62	6	and	and	CCONJ
cana-708	62	7	sahoo[17	sahoo[17	PROPN
cana-708	62	8	]	]	PUNCT
cana-708	62	9	are	be	AUX
cana-708	62	10	followed	follow	VERB
cana-708	62	11	to	to	PART
cana-708	62	12	choose	choose	VERB
cana-708	62	13	the	the	DET
cana-708	62	14	random	random	ADJ
cana-708	62	15	variables	variable	NOUN
cana-708	62	16	ℛ𝑘(𝜔	ℛ𝑘(𝜔	NUM
cana-708	62	17	)	)	PUNCT
cana-708	62	18	,	,	PUNCT
cana-708	62	19	𝑘	𝑘	PROPN
cana-708	62	20	∈	∈	PROPN
cana-708	62	21	ℕ0	ℕ0	NOUN
cana-708	62	22	and	and	CCONJ
cana-708	62	23	to	to	PART
cana-708	62	24	study	study	VERB
cana-708	62	25	the	the	DET
cana-708	62	26	convergence	convergence	NOUN
cana-708	62	27	of	of	ADP
cana-708	62	28	the	the	DET
cana-708	62	29	random	random	ADJ
cana-708	62	30	series	series	NOUN
cana-708	62	31	(	(	PUNCT
cana-708	62	32	2.8	2.8	NUM
cana-708	62	33	)	)	PUNCT
cana-708	62	34	.	.	PUNCT
cana-708	63	1	suitable	suitable	ADJ
cana-708	63	2	real	real	ADJ
cana-708	63	3	numbers	number	NOUN
cana-708	63	4	𝑏	𝑏	PROPN
cana-708	63	5	and	and	CCONJ
cana-708	63	6	𝐵	𝐵	PROPN
cana-708	63	7	are	be	AUX
cana-708	63	8	chosen	choose	VERB
cana-708	63	9	such	such	ADJ
cana-708	63	10	that	that	SCONJ
cana-708	63	11	𝑏	𝑏	PROPN
cana-708	63	12	<	<	X
cana-708	63	13	𝐵	𝐵	PROPN
cana-708	63	14	,	,	PUNCT
cana-708	63	15	which	which	PRON
cana-708	63	16	implies	imply	VERB
cana-708	63	17	,	,	PUNCT
cana-708	63	18	∥	∥	X
cana-708	63	19	(	(	PUNCT
cana-708	63	20	𝑠𝑛(𝑢	𝑠𝑛(𝑢	ADJ
cana-708	63	21	)	)	PUNCT
cana-708	63	22	−	−	NOUN
cana-708	63	23	𝑔(𝑢))𝑈(𝑢	𝑔(𝑢))𝑈(𝑢	ADJ
cana-708	63	24	,	,	PUNCT
cana-708	63	25	𝑏	𝑏	NOUN
cana-708	63	26	)	)	PUNCT
cana-708	63	27	∥2→	∥2→	X
cana-708	63	28	0	0	NUM
cana-708	63	29	,	,	PUNCT
cana-708	63	30	by	by	ADP
cana-708	63	31	the	the	DET
cana-708	63	32	result	result	NOUN
cana-708	63	33	of	of	ADP
cana-708	63	34	muckenhoupt	muckenhoupt	ADJ
cana-708	63	35	(	(	PUNCT
cana-708	63	36	theorem	theorem	NOUN
cana-708	63	37	6	6	NUM
cana-708	63	38	,	,	PUNCT
cana-708	63	39	[	[	X
cana-708	63	40	14	14	NUM
cana-708	63	41	]	]	PUNCT
cana-708	63	42	)	)	PUNCT
cana-708	63	43	.	.	PUNCT
cana-708	64	1	in	in	ADP
cana-708	64	2	the	the	DET
cana-708	64	3	first	first	ADJ
cana-708	64	4	step	step	NOUN
cana-708	64	5	,	,	PUNCT
cana-708	64	6	the	the	DET
cana-708	64	7	existence	existence	NOUN
cana-708	64	8	of	of	ADP
cana-708	64	9	the	the	DET
cana-708	64	10	stochastic	stochastic	ADJ
cana-708	64	11	integral	integral	ADJ
cana-708	64	12	∫	∫	PROPN
cana-708	64	13	∞	∞	PROPN
cana-708	64	14	−∞	−∞	ADP
cana-708	64	15	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	64	16	,	,	PUNCT
cana-708	64	17	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	64	18	,	,	PUNCT
cana-708	64	19	𝜔	𝜔	X
cana-708	64	20	)	)	PUNCT
cana-708	64	21	is	be	AUX
cana-708	64	22	established	establish	VERB
cana-708	64	23	for	for	ADP
cana-708	64	24	𝑔	𝑔	PROPN
cana-708	64	25	∈	∈	PROPN
cana-708	64	26	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	X
cana-708	64	27	)	)	PUNCT
cana-708	64	28	2	2	NUM
cana-708	64	29	(	(	PUNCT
cana-708	64	30	ℝ	ℝ	PROPN
cana-708	64	31	)	)	PUNCT
cana-708	64	32	.	.	PUNCT
cana-708	65	1	since	since	SCONJ
cana-708	65	2	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	65	3	,	,	PUNCT
cana-708	65	4	𝐵	𝐵	NOUN
cana-708	65	5	)	)	PUNCT
cana-708	65	6	is	be	AUX
cana-708	65	7	continuous	continuous	ADJ
cana-708	65	8	for	for	ADP
cana-708	65	9	−	−	PROPN
cana-708	65	10	1	1	NUM
cana-708	65	11	2	2	NUM
cana-708	65	12	<	<	X
cana-708	65	13	𝐵	𝐵	PROPN
cana-708	65	14	,	,	PUNCT
cana-708	65	15	𝐻𝑘(𝑢)𝑊(𝑢	𝐻𝑘(𝑢)𝑊(𝑢	PROPN
cana-708	65	16	,	,	PUNCT
cana-708	65	17	𝐵	𝐵	NOUN
cana-708	65	18	)	)	PUNCT
cana-708	65	19	∈	∈	PROPN
cana-708	65	20	𝐿2(ℝ	𝐿2(ℝ	PROPN
cana-708	65	21	)	)	PUNCT
cana-708	65	22	and	and	CCONJ
cana-708	65	23	the	the	DET
cana-708	65	24	integral	integral	ADJ
cana-708	65	25	∫	∫	PROPN
cana-708	65	26	∞	∞	PROPN
cana-708	65	27	−∞	−∞	ADP
cana-708	65	28	𝐻𝑘(𝑢)𝑊(𝑢	𝐻𝑘(𝑢)𝑊(𝑢	PROPN
cana-708	65	29	,	,	PUNCT
cana-708	65	30	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	65	31	,	,	PUNCT
cana-708	65	32	𝜔	𝜔	PRON
cana-708	65	33	)	)	PUNCT
cana-708	65	34	exists	exist	VERB
cana-708	65	35	.	.	PUNCT
cana-708	66	1	this	this	DET
cana-708	66	2	integral	integral	NOUN
cana-708	66	3	is	be	AUX
cana-708	66	4	a	a	DET
cana-708	66	5	random	random	ADJ
cana-708	66	6	variable	variable	NOUN
cana-708	66	7	.	.	PUNCT
cana-708	67	1	denote	denote	VERB
cana-708	67	2	it	it	PRON
cana-708	67	3	as	as	ADP
cana-708	67	4	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	67	5	)	)	PUNCT
cana-708	67	6	.	.	PUNCT
cana-708	68	1	choose	choose	VERB
cana-708	68	2	these	these	DET
cana-708	68	3	𝒟𝑘(𝜔	𝒟𝑘(𝜔	NOUN
cana-708	68	4	)	)	PUNCT
cana-708	68	5	as	as	ADP
cana-708	68	6	the	the	DET
cana-708	68	7	random	random	ADJ
cana-708	68	8	coefficients	coefficient	NOUN
cana-708	68	9	in	in	ADP
cana-708	68	10	the	the	DET
cana-708	68	11	series	series	NOUN
cana-708	68	12	(	(	PUNCT
cana-708	68	13	2.8	2.8	NUM
cana-708	68	14	)	)	PUNCT
cana-708	68	15	.	.	PUNCT
cana-708	69	1	the	the	DET
cana-708	69	2	convergence	convergence	NOUN
cana-708	69	3	of	of	ADP
cana-708	69	4	the	the	DET
cana-708	69	5	series	series	NOUN
cana-708	69	6	(	(	PUNCT
cana-708	69	7	2.8	2.8	NUM
cana-708	69	8	)	)	PUNCT
cana-708	69	9	in	in	ADP
cana-708	69	10	mean	mean	VERB
cana-708	69	11	if	if	SCONJ
cana-708	69	12	the	the	DET
cana-708	69	13	scalars	scalar	NOUN
cana-708	69	14	𝑑𝑘	𝑑𝑘	VERB
cana-708	69	15	:	:	PUNCT
cana-708	69	16	=	=	NOUN
cana-708	69	17	𝑟𝑘	𝑟𝑘	ADJ
cana-708	69	18	2	2	NUM
cana-708	69	19	∫	∫	NOUN
cana-708	69	20	∞	∞	PROPN
cana-708	69	21	−∞	−∞	X
cana-708	69	22	𝑔(𝑢)𝑒−𝑢2	𝑔(𝑢)𝑒−𝑢2	X
cana-708	69	23	𝐻𝑘(𝑢)𝑑𝑢.	𝐻𝑘(𝑢)𝑑𝑢.	NUM
cana-708	69	24	are	be	AUX
cana-708	69	25	the	the	DET
cana-708	69	26	fhc	fhc	NOUN
cana-708	69	27	of	of	ADP
cana-708	69	28	a	a	DET
cana-708	69	29	function	function	NOUN
cana-708	69	30	𝑔	𝑔	NOUN
cana-708	69	31	in	in	ADP
cana-708	69	32	the	the	DET
cana-708	69	33	weighted	weighted	ADJ
cana-708	69	34	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	NOUN
cana-708	69	35	)	)	PUNCT
cana-708	69	36	2	2	NUM
cana-708	69	37	(	(	PUNCT
cana-708	69	38	ℝ	ℝ	PROPN
cana-708	69	39	)	)	PUNCT
cana-708	69	40	space	space	NOUN
cana-708	69	41	with	with	ADP
cana-708	69	42	weights	weight	NOUN
cana-708	69	43	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	69	44	,	,	PUNCT
cana-708	69	45	𝐵	𝐵	NOUN
cana-708	69	46	)	)	PUNCT
cana-708	69	47	of	of	ADP
cana-708	69	48	the	the	DET
cana-708	69	49	form	form	NOUN
cana-708	69	50	𝑒	𝑒	PROPN
cana-708	69	51	−𝑢2	−𝑢2	NOUN
cana-708	69	52	2	2	NUM
cana-708	69	53	(	(	PUNCT
cana-708	69	54	1	1	NUM
cana-708	69	55	+	+	NUM
cana-708	69	56	|𝑢|)𝐵	|𝑢|)𝐵	PROPN
cana-708	69	57	for	for	ADP
cana-708	69	58	a	a	DET
cana-708	69	59	suitable	suitable	ADJ
cana-708	69	60	𝐵.	𝐵.	NOUN
cana-708	69	61	the	the	DET
cana-708	69	62	stochastic	stochastic	ADJ
cana-708	69	63	integral	integral	ADJ
cana-708	69	64	communications	communication	NOUN
cana-708	69	65	on	on	ADP
cana-708	69	66	applied	apply	VERB
cana-708	69	67	nonlinear	nonlinear	ADJ
cana-708	69	68	analysis	analysis	NOUN
cana-708	69	69	issn	issn	NOUN
cana-708	69	70	:	:	PUNCT
cana-708	69	71	1074	1074	NUM
cana-708	69	72	-	-	PUNCT
cana-708	69	73	133x	133x	NUM
cana-708	69	74	vol	vol	NOUN
cana-708	69	75	31	31	NUM
cana-708	69	76	no	no	NOUN
cana-708	69	77	.	.	PUNCT
cana-708	70	1	2s	2s	NUM
cana-708	70	2	(	(	PUNCT
cana-708	70	3	2024	2024	NUM
cana-708	70	4	)	)	PUNCT
cana-708	70	5	699	699	NUM
cana-708	70	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	70	7	∫	∫	PROPN
cana-708	70	8	∞	∞	PROPN
cana-708	70	9	−∞	−∞	ADP
cana-708	70	10	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	70	11	,	,	PUNCT
cana-708	70	12	𝑣)𝑒	𝑣)𝑒	ADV
cana-708	70	13	−𝑣2	−𝑣2	PROPN
cana-708	70	14	2	2	NUM
cana-708	70	15	(	(	PUNCT
cana-708	70	16	1	1	NUM
cana-708	70	17	+	+	CCONJ
cana-708	70	18	|𝑣|)𝐵𝑑𝑋(𝑣	|𝑣|)𝐵𝑑𝑋(𝑣	ADJ
cana-708	70	19	,	,	PUNCT
cana-708	70	20	𝜔	𝜔	NOUN
cana-708	70	21	)	)	PUNCT
cana-708	70	22	,	,	PUNCT
cana-708	70	23	(	(	PUNCT
cana-708	70	24	2.9	2.9	NUM
cana-708	70	25	)	)	PUNCT
cana-708	70	26	is	be	AUX
cana-708	70	27	seen	see	VERB
cana-708	70	28	to	to	PART
cana-708	70	29	be	be	AUX
cana-708	70	30	the	the	DET
cana-708	70	31	sum	sum	NOUN
cana-708	70	32	function	function	NOUN
cana-708	70	33	of	of	ADP
cana-708	70	34	this	this	DET
cana-708	70	35	series	series	NOUN
cana-708	70	36	.	.	PUNCT
cana-708	71	1	throughout	throughout	ADP
cana-708	71	2	the	the	DET
cana-708	71	3	sections	section	NOUN
cana-708	71	4	3	3	NUM
cana-708	71	5	and	and	CCONJ
cana-708	71	6	4	4	NUM
cana-708	71	7	below	below	ADV
cana-708	71	8	,	,	PUNCT
cana-708	71	9	𝑋(𝑢	𝑋(𝑢	NOUN
cana-708	71	10	,	,	PUNCT
cana-708	71	11	𝜔	𝜔	PRON
cana-708	71	12	)	)	PUNCT
cana-708	71	13	is	be	AUX
cana-708	71	14	considered	consider	VERB
cana-708	71	15	to	to	PART
cana-708	71	16	be	be	AUX
cana-708	71	17	the	the	DET
cana-708	71	18	ssp	ssp	NOUN
cana-708	71	19	of	of	ADP
cana-708	71	20	index	index	NOUN
cana-708	71	21	𝜇	𝜇	ADP
cana-708	71	22	=	=	PROPN
cana-708	71	23	2	2	NUM
cana-708	71	24	and	and	CCONJ
cana-708	71	25	the	the	DET
cana-708	71	26	weight	weight	NOUN
cana-708	71	27	functions	function	NOUN
cana-708	71	28	𝑈(𝑢	𝑈(𝑢	NOUN
cana-708	71	29	,	,	PUNCT
cana-708	71	30	𝑏	𝑏	NOUN
cana-708	71	31	)	)	PUNCT
cana-708	71	32	=	=	SYM
cana-708	72	1	𝑒𝑥𝑝(−	𝑒𝑥𝑝(−	NUM
cana-708	72	2	1	1	NUM
cana-708	72	3	2	2	NUM
cana-708	72	4	𝑢2)(1	𝑢2)(1	NOUN
cana-708	72	5	+	+	CCONJ
cana-708	72	6	|𝑢|)𝑏	|𝑢|)𝑏	PROPN
cana-708	72	7	,	,	PUNCT
cana-708	72	8	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	72	9	,	,	PUNCT
cana-708	72	10	𝐵	𝐵	NOUN
cana-708	72	11	)	)	PUNCT
cana-708	72	12	=	=	SYM
cana-708	73	1	𝑒𝑥𝑝(−	𝑒𝑥𝑝(−	NUM
cana-708	73	2	1	1	NUM
cana-708	73	3	2	2	NUM
cana-708	73	4	𝑢2)(1	𝑢2)(1	NOUN
cana-708	73	5	+	+	CCONJ
cana-708	73	6	|𝑢|)𝐵	|𝑢|)𝐵	PROPN
cana-708	73	7	where	where	SCONJ
cana-708	73	8	𝑏	𝑏	PROPN
cana-708	73	9	<	<	X
cana-708	73	10	1	1	NUM
cana-708	73	11	2	2	NUM
cana-708	73	12	and	and	CCONJ
cana-708	73	13	𝐵	𝐵	NOUN
cana-708	73	14	>	>	X
cana-708	73	15	−	−	PROPN
cana-708	73	16	1	1	NUM
cana-708	73	17	2	2	NUM
cana-708	73	18	such	such	ADJ
cana-708	73	19	that	that	SCONJ
cana-708	73	20	𝑏	𝑏	PROPN
cana-708	73	21	<	<	X
cana-708	73	22	𝐵.	𝐵.	PROPN
cana-708	73	23	3	3	PROPN
cana-708	73	24	.	.	PUNCT
cana-708	73	25	existence	existence	NOUN
cana-708	73	26	of	of	ADP
cana-708	73	27	the	the	DET
cana-708	73	28	stochastic	stochastic	ADJ
cana-708	73	29	integral	integral	NOUN
cana-708	73	30	the	the	DET
cana-708	73	31	following	follow	VERB
cana-708	73	32	result	result	NOUN
cana-708	73	33	is	be	AUX
cana-708	73	34	required	require	VERB
cana-708	73	35	to	to	PART
cana-708	73	36	prove	prove	VERB
cana-708	73	37	the	the	DET
cana-708	73	38	existence	existence	NOUN
cana-708	73	39	of	of	ADP
cana-708	73	40	the	the	DET
cana-708	73	41	integral	integral	ADJ
cana-708	73	42	∫	∫	PROPN
cana-708	73	43	∞	∞	PROPN
cana-708	73	44	−∞	−∞	ADP
cana-708	73	45	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	73	46	,	,	PUNCT
cana-708	73	47	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NOUN
cana-708	73	48	,	,	PUNCT
cana-708	73	49	𝜔	𝜔	NOUN
cana-708	73	50	)	)	PUNCT
cana-708	73	51	.	.	PUNCT
cana-708	74	1	lemma	lemma	PROPN
cana-708	74	2	1	1	NUM
cana-708	74	3	:	:	PUNCT
cana-708	75	1	[	[	X
cana-708	75	2	17	17	NUM
cana-708	75	3	]	]	PUNCT
cana-708	75	4	suppose	suppose	VERB
cana-708	75	5	𝑋(𝑢	𝑋(𝑢	X
cana-708	75	6	,	,	PUNCT
cana-708	75	7	𝜔	𝜔	PRON
cana-708	75	8	)	)	PUNCT
cana-708	75	9	is	be	AUX
cana-708	75	10	of	of	ADP
cana-708	75	11	index	index	NOUN
cana-708	75	12	𝜇	𝜇	ADP
cana-708	75	13	,	,	PUNCT
cana-708	75	14	𝜇	𝜇	ADP
cana-708	75	15	∈	∈	PROPN
cana-708	75	16	(	(	PUNCT
cana-708	75	17	1,2	1,2	NUM
cana-708	75	18	]	]	PUNCT
cana-708	75	19	and	and	CCONJ
cana-708	75	20	𝑔	𝑔	PROPN
cana-708	75	21	∈	∈	PROPN
cana-708	75	22	𝐿𝑝[𝑎	𝐿𝑝[𝑎	NOUN
cana-708	75	23	,	,	PUNCT
cana-708	75	24	𝑏	𝑏	NOUN
cana-708	75	25	]	]	X
cana-708	75	26	,	,	PUNCT
cana-708	75	27	𝑝	𝑝	PROPN
cana-708	75	28	≥	≥	NOUN
cana-708	75	29	𝜇	𝜇	ADP
cana-708	75	30	,	,	PUNCT
cana-708	75	31	𝑠	𝑠	PROPN
cana-708	75	32	∈	∈	PROPN
cana-708	75	33	ℝ	ℝ	PROPN
cana-708	75	34	then	then	ADV
cana-708	75	35	𝐸(|	𝐸(|	AUX
cana-708	75	36	∫	∫	PROPN
cana-708	76	1	𝑏	𝑏	PROPN
cana-708	76	2	𝑎	𝑎	PROPN
cana-708	76	3	𝑔(𝑢)𝑑𝑋(𝑢	𝑔(𝑢)𝑑𝑋(𝑢	NOUN
cana-708	76	4	,	,	PUNCT
cana-708	76	5	𝜔)|	𝜔)|	NOUN
cana-708	76	6	)	)	PUNCT
cana-708	76	7	≤	≤	NUM
cana-708	76	8	4	4	NUM
cana-708	76	9	𝜋(𝜇−1	𝜋(𝜇−1	NOUN
cana-708	76	10	)	)	PUNCT
cana-708	76	11	∫	∫	PROPN
cana-708	77	1	𝑏	𝑏	NUM
cana-708	77	2	𝑎	𝑎	PRON
cana-708	77	3	|𝑔(𝑢)|𝜇𝑑𝑢	|𝑔(𝑢)|𝜇𝑑𝑢	NOUN
cana-708	77	4	+	+	CCONJ
cana-708	77	5	2	2	NUM
cana-708	77	6	𝜋	𝜋	NOUN
cana-708	77	7	∫	∫	NOUN
cana-708	77	8	|𝑠|>1	|𝑠|>1	NOUN
cana-708	77	9	1−𝑒𝑥𝑝(−|𝑠|𝜇	1−𝑒𝑥𝑝(−|𝑠|𝜇	NUM
cana-708	77	10	∫	∫	NOUN
cana-708	78	1	𝑏	𝑏	DET
cana-708	78	2	𝑎	𝑎	ADJ
cana-708	78	3	|𝑔(𝑢)|𝜇𝑑𝑢	|𝑔(𝑢)|𝜇𝑑𝑢	NOUN
cana-708	78	4	)	)	PUNCT
cana-708	78	5	𝑠2	𝑠2	NOUN
cana-708	78	6	𝑑𝑠.	𝑑𝑠.	NOUN
cana-708	78	7	theorem	theorem	VERB
cana-708	78	8	2	2	NUM
cana-708	78	9	:	:	PUNCT
cana-708	78	10	if	if	SCONJ
cana-708	78	11	𝑋(𝑢	𝑋(𝑢	ADP
cana-708	78	12	,	,	PUNCT
cana-708	78	13	𝜔	𝜔	PRON
cana-708	78	14	)	)	PUNCT
cana-708	78	15	is	be	AUX
cana-708	78	16	of	of	ADP
cana-708	78	17	index	index	NOUN
cana-708	78	18	2	2	NUM
cana-708	78	19	,	,	PUNCT
cana-708	78	20	and	and	CCONJ
cana-708	78	21	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	78	22	)	)	PUNCT
cana-708	78	23	∈	∈	PROPN
cana-708	78	24	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	PROPN
cana-708	78	25	)	)	PUNCT
cana-708	78	26	2	2	NUM
cana-708	78	27	(	(	PUNCT
cana-708	78	28	ℝ	ℝ	PROPN
cana-708	78	29	)	)	PUNCT
cana-708	78	30	,	,	PUNCT
cana-708	78	31	then	then	ADV
cana-708	78	32	the	the	DET
cana-708	78	33	integral	integral	ADJ
cana-708	78	34	∫	∫	PROPN
cana-708	78	35	∞	∞	PROPN
cana-708	78	36	−∞	−∞	ADP
cana-708	78	37	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	78	38	,	,	PUNCT
cana-708	78	39	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	78	40	,	,	PUNCT
cana-708	78	41	𝜔	𝜔	PRON
cana-708	78	42	)	)	PUNCT
cana-708	78	43	exists	exist	VERB
cana-708	78	44	in	in	ADP
cana-708	78	45	mean	mean	NOUN
cana-708	78	46	.	.	PUNCT
cana-708	79	1	proof	proof	NOUN
cana-708	79	2	:	:	PUNCT
cana-708	79	3	we	we	PRON
cana-708	79	4	are	be	AUX
cana-708	79	5	aware	aware	ADJ
cana-708	79	6	that	that	SCONJ
cana-708	79	7	𝐶𝑐(ℝ	𝐶𝑐(ℝ	VERB
cana-708	79	8	)	)	PUNCT
cana-708	79	9	is	be	AUX
cana-708	79	10	dense	dense	ADJ
cana-708	79	11	in	in	ADP
cana-708	79	12	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-708	79	13	)	)	PUNCT
cana-708	79	14	.	.	PUNCT
cana-708	80	1	so	so	ADV
cana-708	80	2	for	for	ADP
cana-708	80	3	𝑔	𝑔	PROPN
cana-708	80	4	∈	∈	PROPN
cana-708	80	5	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	X
cana-708	80	6	)	)	PUNCT
cana-708	80	7	2	2	NUM
cana-708	80	8	there	there	PRON
cana-708	80	9	exist	exist	VERB
cana-708	80	10	a	a	DET
cana-708	80	11	sequence	sequence	NOUN
cana-708	80	12	of	of	ADP
cana-708	80	13	functions	function	NOUN
cana-708	80	14	{	{	PUNCT
cana-708	80	15	ℎ𝑘	ℎ𝑘	NOUN
cana-708	80	16	}	}	PUNCT
cana-708	80	17	in	in	ADP
cana-708	80	18	𝐶𝑐(ℝ	𝐶𝑐(ℝ	ADV
cana-708	80	19	)	)	PUNCT
cana-708	81	1	such	such	ADJ
cana-708	81	2	that	that	SCONJ
cana-708	81	3	(	(	PUNCT
cana-708	81	4	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	81	5	,	,	PUNCT
cana-708	81	6	𝐵	𝐵	NOUN
cana-708	81	7	)	)	PUNCT
cana-708	81	8	−	−	PROPN
cana-708	81	9	ℎ𝑘	ℎ𝑘	NOUN
cana-708	81	10	)	)	PUNCT
cana-708	81	11	∈	∈	PROPN
cana-708	81	12	𝐿2(ℝ	𝐿2(ℝ	PROPN
cana-708	81	13	)	)	PUNCT
cana-708	81	14	and	and	CCONJ
cana-708	81	15	∥	∥	SYM
cana-708	81	16	𝑔𝑊(𝑢	𝑔𝑊(𝑢	NOUN
cana-708	81	17	,	,	PUNCT
cana-708	81	18	𝐵	𝐵	NOUN
cana-708	81	19	)	)	PUNCT
cana-708	81	20	−	−	PROPN
cana-708	82	1	ℎ𝑘	ℎ𝑘	NOUN
cana-708	82	2	∥2	∥2	PROPN
cana-708	82	3	approaches	approach	VERB
cana-708	82	4	to	to	ADP
cana-708	82	5	0	0	NUM
cana-708	82	6	as	as	ADP
cana-708	82	7	𝑘	𝑘	PROPN
cana-708	82	8	→	→	SYM
cana-708	82	9	0	0	NUM
cana-708	82	10	.	.	PUNCT
cana-708	83	1	consider	consider	VERB
cana-708	83	2	two	two	NUM
cana-708	83	3	functions	function	NOUN
cana-708	83	4	ℎ𝑚	ℎ𝑚	NOUN
cana-708	83	5	and	and	CCONJ
cana-708	83	6	ℎ𝑛	ℎ𝑛	NOUN
cana-708	83	7	from	from	ADP
cana-708	83	8	this	this	DET
cana-708	83	9	sequence	sequence	NOUN
cana-708	83	10	{	{	PUNCT
cana-708	83	11	ℎ𝑘	ℎ𝑘	PROPN
cana-708	83	12	}	}	PUNCT
cana-708	83	13	.	.	PUNCT
cana-708	84	1	without	without	ADP
cana-708	84	2	loss	loss	NOUN
cana-708	84	3	of	of	ADP
cana-708	84	4	generality	generality	NOUN
cana-708	84	5	assume	assume	VERB
cana-708	84	6	that	that	SCONJ
cana-708	84	7	the	the	DET
cana-708	84	8	compact	compact	ADJ
cana-708	84	9	support	support	NOUN
cana-708	84	10	of	of	ADP
cana-708	84	11	ℎ𝑚	ℎ𝑚	PRON
cana-708	84	12	and	and	CCONJ
cana-708	84	13	ℎ𝑛	ℎ𝑛	PROPN
cana-708	84	14	can	can	AUX
cana-708	84	15	be	be	AUX
cana-708	84	16	in	in	ADP
cana-708	84	17	[	[	X
cana-708	84	18	𝑎	𝑎	X
cana-708	84	19	,	,	PUNCT
cana-708	84	20	𝑏	𝑏	NOUN
cana-708	84	21	]	]	PUNCT
cana-708	84	22	and	and	CCONJ
cana-708	84	23	[	[	X
cana-708	84	24	c	c	X
cana-708	84	25	,	,	PUNCT
cana-708	84	26	d	d	X
cana-708	84	27	]	]	PUNCT
cana-708	84	28	respectively	respectively	ADV
cana-708	84	29	.	.	PUNCT
cana-708	85	1	so	so	ADV
cana-708	85	2	ℎ𝑚	ℎ𝑚	NOUN
cana-708	85	3	and	and	CCONJ
cana-708	85	4	ℎ𝑛	ℎ𝑛	PROPN
cana-708	85	5	can	can	AUX
cana-708	85	6	be	be	AUX
cana-708	85	7	considered	consider	VERB
cana-708	85	8	to	to	PART
cana-708	85	9	be	be	AUX
cana-708	85	10	in	in	ADP
cana-708	85	11	𝐿2[𝑎	𝐿2[𝑎	PROPN
cana-708	85	12	,	,	PUNCT
cana-708	85	13	𝑏	𝑏	NOUN
cana-708	85	14	]	]	PUNCT
cana-708	85	15	and	and	CCONJ
cana-708	85	16	𝐿2[𝑐	𝐿2[𝑐	NOUN
cana-708	85	17	,	,	PUNCT
cana-708	85	18	𝑑	𝑑	NOUN
cana-708	85	19	]	]	X
cana-708	85	20	.	.	PUNCT
cana-708	86	1	let	let	VERB
cana-708	86	2	[	[	X
cana-708	86	3	𝑝	𝑝	NOUN
cana-708	86	4	,	,	PUNCT
cana-708	86	5	𝑞	𝑞	X
cana-708	86	6	]	]	X
cana-708	86	7	be	be	VERB
cana-708	86	8	the	the	DET
cana-708	86	9	smallest	small	ADJ
cana-708	86	10	closed	close	VERB
cana-708	86	11	sub	sub	NOUN
cana-708	86	12	-	-	NOUN
cana-708	86	13	interval	interval	NOUN
cana-708	86	14	of	of	ADP
cana-708	86	15	ℝ	ℝ	PROPN
cana-708	86	16	which	which	PRON
cana-708	86	17	contains	contain	VERB
cana-708	86	18	[	[	X
cana-708	86	19	𝑎	𝑎	X
cana-708	86	20	,	,	PUNCT
cana-708	86	21	𝑏	𝑏	NOUN
cana-708	86	22	]	]	PUNCT
cana-708	86	23	∪	∪	ADP
cana-708	86	24	[	[	X
cana-708	86	25	𝑐	𝑐	NOUN
cana-708	86	26	,	,	PUNCT
cana-708	86	27	𝑑	𝑑	NOUN
cana-708	86	28	]	]	X
cana-708	86	29	.	.	PUNCT
cana-708	87	1	now	now	ADV
cana-708	87	2	both	both	DET
cana-708	87	3	ℎ𝑚	ℎ𝑚	NOUN
cana-708	87	4	and	and	CCONJ
cana-708	87	5	ℎ𝑛	ℎ𝑛	PROPN
cana-708	87	6	can	can	AUX
cana-708	87	7	be	be	AUX
cana-708	87	8	considered	consider	VERB
cana-708	87	9	to	to	PART
cana-708	87	10	be	be	AUX
cana-708	87	11	in	in	ADP
cana-708	87	12	𝐿2[𝑝	𝐿2[𝑝	PROPN
cana-708	87	13	,	,	PUNCT
cana-708	87	14	𝑞	𝑞	X
cana-708	87	15	]	]	X
cana-708	87	16	.	.	PUNCT
cana-708	88	1	since	since	SCONJ
cana-708	88	2	ℎ𝑚	ℎ𝑚	NOUN
cana-708	88	3	and	and	CCONJ
cana-708	88	4	ℎ𝑛	ℎ𝑛	NOUN
cana-708	88	5	can	can	AUX
cana-708	88	6	be	be	AUX
cana-708	88	7	continuous	continuous	ADJ
cana-708	88	8	,	,	PUNCT
cana-708	88	9	the	the	DET
cana-708	88	10	stochastic	stochastic	ADJ
cana-708	88	11	integrals	integral	NOUN
cana-708	88	12	∫	∫	PROPN
cana-708	88	13	𝑞	𝑞	X
cana-708	88	14	𝑝	𝑝	PROPN
cana-708	88	15	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	88	16	,	,	PUNCT
cana-708	88	17	𝜔	𝜔	NOUN
cana-708	88	18	)	)	PUNCT
cana-708	88	19	=	=	SYM
cana-708	89	1	∫	∫	PROPN
cana-708	90	1	𝑏	𝑏	PROPN
cana-708	90	2	𝑎	𝑎	ADJ
cana-708	90	3	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	NOUN
cana-708	90	4	,	,	PUNCT
cana-708	90	5	𝜔	𝜔	NOUN
cana-708	90	6	)	)	PUNCT
cana-708	90	7	=	=	SYM
cana-708	91	1	∫	∫	PROPN
cana-708	91	2	∞	∞	PROPN
cana-708	91	3	−∞	−∞	ADP
cana-708	91	4	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	91	5	,	,	PUNCT
cana-708	91	6	𝜔	𝜔	PRON
cana-708	91	7	)	)	PUNCT
cana-708	91	8	and	and	CCONJ
cana-708	91	9	∫	∫	PROPN
cana-708	91	10	𝑞	𝑞	PROPN
cana-708	91	11	𝑝	𝑝	PROPN
cana-708	91	12	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	91	13	,	,	PUNCT
cana-708	91	14	𝜔	𝜔	NOUN
cana-708	91	15	)	)	PUNCT
cana-708	91	16	=	=	SYM
cana-708	92	1	∫	∫	PROPN
cana-708	93	1	𝑏	𝑏	PROPN
cana-708	93	2	𝑎	𝑎	NOUN
cana-708	93	3	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	NOUN
cana-708	93	4	,	,	PUNCT
cana-708	93	5	𝜔	𝜔	NOUN
cana-708	93	6	)	)	PUNCT
cana-708	93	7	=	=	SYM
cana-708	94	1	∫	∫	PROPN
cana-708	94	2	∞	∞	PROPN
cana-708	95	1	−∞	−∞	ADP
cana-708	95	2	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	95	3	,	,	PUNCT
cana-708	95	4	𝜔	𝜔	NOUN
cana-708	95	5	)	)	PUNCT
cana-708	95	6	exists	exist	VERB
cana-708	95	7	in	in	ADP
cana-708	95	8	the	the	DET
cana-708	95	9	sense	sense	NOUN
cana-708	95	10	of	of	ADP
cana-708	95	11	mean[17	mean[17	NOUN
cana-708	95	12	]	]	PUNCT
cana-708	95	13	.	.	PUNCT
cana-708	96	1	denote	denote	VERB
cana-708	96	2	𝑌𝑚(𝜔	𝑌𝑚(𝜔	NOUN
cana-708	96	3	):	):	PUNCT
cana-708	96	4	=	=	NOUN
cana-708	96	5	∫	∫	PROPN
cana-708	96	6	𝑞	𝑞	PROPN
cana-708	96	7	𝑝	𝑝	PROPN
cana-708	96	8	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	96	9	,	,	PUNCT
cana-708	96	10	𝜔	𝜔	NOUN
cana-708	96	11	)	)	PUNCT
cana-708	96	12	and	and	CCONJ
cana-708	96	13	𝑌𝑛(𝜔	𝑌𝑛(𝜔	NUM
cana-708	96	14	):	):	PUNCT
cana-708	97	1	=	=	VERB
cana-708	97	2	∫	∫	PROPN
cana-708	97	3	𝑞	𝑞	PROPN
cana-708	97	4	𝑝	𝑝	PROPN
cana-708	97	5	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	97	6	,	,	PUNCT
cana-708	97	7	𝜔	𝜔	NOUN
cana-708	97	8	)	)	PUNCT
cana-708	97	9	.	.	PUNCT
cana-708	98	1	now	now	ADV
cana-708	98	2	applying	apply	VERB
cana-708	98	3	lemma	lemma	PROPN
cana-708	98	4	1	1	NUM
cana-708	98	5	for	for	ADP
cana-708	98	6	𝜇	𝜇	X
cana-708	98	7	=	=	SYM
cana-708	98	8	2	2	NUM
cana-708	98	9	,	,	PUNCT
cana-708	98	10	we	we	PRON
cana-708	98	11	get	get	VERB
cana-708	98	12	𝐸|𝑌𝑛(𝜔	𝐸|𝑌𝑛(𝜔	PROPN
cana-708	98	13	)	)	PUNCT
cana-708	99	1	−	−	ADP
cana-708	100	1	𝑌𝑚(𝜔)|	𝑌𝑚(𝜔)|	ADJ
cana-708	100	2	communications	communication	NOUN
cana-708	100	3	on	on	ADP
cana-708	100	4	applied	apply	VERB
cana-708	100	5	nonlinear	nonlinear	ADJ
cana-708	100	6	analysis	analysis	NOUN
cana-708	100	7	issn	issn	NOUN
cana-708	100	8	:	:	PUNCT
cana-708	100	9	1074	1074	NUM
cana-708	100	10	-	-	PUNCT
cana-708	100	11	133x	133x	NUM
cana-708	100	12	vol	vol	NOUN
cana-708	100	13	31	31	NUM
cana-708	100	14	no	no	NOUN
cana-708	100	15	.	.	PUNCT
cana-708	101	1	2s	2s	NUM
cana-708	101	2	(	(	PUNCT
cana-708	101	3	2024	2024	NUM
cana-708	101	4	)	)	PUNCT
cana-708	101	5	700	700	NUM
cana-708	101	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	101	7	=	=	SYM
cana-708	101	8	𝐸(|	𝐸(|	NUM
cana-708	101	9	∫	∫	NOUN
cana-708	101	10	𝑞	𝑞	PROPN
cana-708	101	11	𝑝	𝑝	PROPN
cana-708	101	12	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	101	13	,	,	PUNCT
cana-708	101	14	𝜔	𝜔	NOUN
cana-708	101	15	)	)	PUNCT
cana-708	101	16	−	−	NUM
cana-708	101	17	∫	∫	PROPN
cana-708	101	18	𝑞	𝑞	PROPN
cana-708	101	19	𝑝	𝑝	PROPN
cana-708	101	20	ℎ𝑚(𝑢)𝑑𝑋(𝑢	ℎ𝑚(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	101	21	,	,	PUNCT
cana-708	101	22	𝜔)|	𝜔)|	NOUN
cana-708	101	23	)	)	PUNCT
cana-708	101	24	=	=	SYM
cana-708	102	1	𝐸(|	𝐸(|	NUM
cana-708	102	2	∫	∫	PROPN
cana-708	102	3	𝑞	𝑞	PROPN
cana-708	102	4	𝑝	𝑝	PROPN
cana-708	102	5	(	(	PUNCT
cana-708	102	6	ℎ𝑛(𝑢	ℎ𝑛(𝑢	PROPN
cana-708	102	7	)	)	PUNCT
cana-708	102	8	−	−	PROPN
cana-708	102	9	ℎ𝑚(𝑡))𝑑𝑋(𝑢	ℎ𝑚(𝑡))𝑑𝑋(𝑢	NOUN
cana-708	102	10	,	,	PUNCT
cana-708	102	11	𝜔)|	𝜔)|	NOUN
cana-708	102	12	)	)	PUNCT
cana-708	102	13	≤	≤	NOUN
cana-708	102	14	4	4	NUM
cana-708	102	15	𝜋	𝜋	NOUN
cana-708	102	16	∫	∫	PROPN
cana-708	102	17	𝑞	𝑞	PROPN
cana-708	102	18	𝑝	𝑝	PROPN
cana-708	102	19	|ℎ𝑛(𝑢	|ℎ𝑛(𝑢	PROPN
cana-708	102	20	)	)	PUNCT
cana-708	102	21	−	−	NOUN
cana-708	102	22	ℎ𝑚(𝑢)|2𝑑𝑢	ℎ𝑚(𝑢)|2𝑑𝑢	VERB
cana-708	103	1	+	+	CCONJ
cana-708	103	2	2	2	NUM
cana-708	103	3	𝜋	𝜋	NOUN
cana-708	103	4	∫	∫	NOUN
cana-708	103	5	|𝑠|>1	|𝑠|>1	VERB
cana-708	103	6	1−exp(−|𝑠|2	1−exp(−|𝑠|2	NUM
cana-708	103	7	∫	∫	NOUN
cana-708	103	8	𝑞	𝑞	PROPN
cana-708	103	9	𝑝	𝑝	PROPN
cana-708	103	10	|ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢	|ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢	PROPN
cana-708	103	11	)	)	PUNCT
cana-708	103	12	𝑠2	𝑠2	NOUN
cana-708	103	13	𝑑𝑠	𝑑𝑠	ADP
cana-708	103	14	≤	≤	NUM
cana-708	103	15	4	4	NUM
cana-708	103	16	𝜋	𝜋	NOUN
cana-708	103	17	∫	∫	PROPN
cana-708	103	18	∞	∞	PROPN
cana-708	103	19	−∞	−∞	PROPN
cana-708	103	20	|ℎ𝑛(𝑢	|ℎ𝑛(𝑢	NUM
cana-708	103	21	)	)	PUNCT
cana-708	103	22	−	−	NOUN
cana-708	103	23	ℎ𝑚(𝑢)|2𝑑𝑢	ℎ𝑚(𝑢)|2𝑑𝑢	VERB
cana-708	104	1	+	+	CCONJ
cana-708	104	2	2	2	NUM
cana-708	104	3	𝜋	𝜋	NOUN
cana-708	104	4	∫	∫	NOUN
cana-708	104	5	|𝑠|>1	|𝑠|>1	NOUN
cana-708	104	6	1−exp(−𝑐|𝑠|2	1−exp(−𝑐|𝑠|2	NUM
cana-708	104	7	∫	∫	NOUN
cana-708	104	8	∞	∞	PROPN
cana-708	104	9	−∞	−∞	ADP
cana-708	104	10	|ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢	|ℎ𝑛(𝑢)−ℎ𝑚(𝑢)|2𝑑𝑢	PROPN
cana-708	104	11	)	)	PUNCT
cana-708	104	12	𝑠2	𝑠2	NOUN
cana-708	104	13	𝑑𝑠.	𝑑𝑠.	VERB
cana-708	104	14	the	the	DET
cana-708	104	15	integrand	integrand	NOUN
cana-708	104	16	in	in	ADP
cana-708	104	17	the	the	DET
cana-708	104	18	2𝑛𝑑	2𝑛𝑑	ADJ
cana-708	104	19	integral	integral	NOUN
cana-708	104	20	is	be	AUX
cana-708	104	21	dominated	dominate	VERB
cana-708	104	22	by	by	ADP
cana-708	104	23	the	the	DET
cana-708	104	24	integrable	integrable	ADJ
cana-708	104	25	function	function	NOUN
cana-708	104	26	1	1	NUM
cana-708	104	27	𝑠2	𝑠2	NOUN
cana-708	104	28	over	over	ADP
cana-708	104	29	(	(	PUNCT
cana-708	104	30	−∞	−∞	NOUN
cana-708	104	31	,	,	PUNCT
cana-708	104	32	−1	−1	NOUN
cana-708	104	33	]	]	PUNCT
cana-708	104	34	and	and	CCONJ
cana-708	104	35	[	[	X
cana-708	104	36	1	1	NUM
cana-708	104	37	,	,	PUNCT
cana-708	104	38	∞	∞	PROPN
cana-708	104	39	)	)	PUNCT
cana-708	104	40	.	.	PUNCT
cana-708	105	1	since	since	SCONJ
cana-708	105	2	∥	∥	PRON
cana-708	105	3	ℎ𝑛(𝑢	ℎ𝑛(𝑢	X
cana-708	105	4	)	)	PUNCT
cana-708	105	5	−	−	PROPN
cana-708	105	6	ℎ𝑚(𝑢	ℎ𝑚(𝑢	NOUN
cana-708	105	7	)	)	PUNCT
cana-708	105	8	∥2=	∥2=	NOUN
cana-708	105	9	∫	∫	PROPN
cana-708	105	10	∞	∞	PROPN
cana-708	105	11	−∞	−∞	PROPN
cana-708	105	12	|ℎ𝑛(𝑢	|ℎ𝑛(𝑢	NUM
cana-708	105	13	)	)	PUNCT
cana-708	105	14	−	−	NOUN
cana-708	105	15	ℎ𝑚(𝑢)|2𝑑𝑢	ℎ𝑚(𝑢)|2𝑑𝑢	PUNCT
cana-708	105	16	approaches	approach	VERB
cana-708	105	17	0	0	NUM
cana-708	105	18	as	as	ADP
cana-708	105	19	𝑚	𝑚	PROPN
cana-708	105	20	,	,	PUNCT
cana-708	105	21	𝑛	𝑛	PROPN
cana-708	105	22	→	→	SYM
cana-708	105	23	∞	∞	PROPN
cana-708	105	24	,	,	PUNCT
cana-708	105	25	the	the	DET
cana-708	105	26	2𝑛𝑑	2𝑛𝑑	ADJ
cana-708	105	27	integral	integral	ADJ
cana-708	105	28	converges	converge	NOUN
cana-708	105	29	to	to	ADP
cana-708	105	30	0	0	NUM
cana-708	105	31	by	by	ADP
cana-708	105	32	dct	dct	PROPN
cana-708	105	33	and	and	CCONJ
cana-708	105	34	we	we	PRON
cana-708	105	35	obtained	obtain	VERB
cana-708	105	36	lim	lim	PROPN
cana-708	105	37	𝑚,𝑛→∞	𝑚,𝑛→∞	PROPN
cana-708	105	38	𝐸|𝑌𝑛(𝜔	𝐸|𝑌𝑛(𝜔	PROPN
cana-708	105	39	)	)	PUNCT
cana-708	106	1	−	−	ADP
cana-708	106	2	𝑌𝑚(𝜔)|	𝑌𝑚(𝜔)|	NOUN
cana-708	107	1	=	=	PUNCT
cana-708	107	2	0	0	X
cana-708	107	3	.	.	PUNCT
cana-708	108	1	𝑌𝑛(𝜔	𝑌𝑛(𝜔	VERB
cana-708	108	2	)	)	PUNCT
cana-708	109	1	is	be	AUX
cana-708	109	2	a	a	DET
cana-708	109	3	cauchy	cauchy	ADJ
cana-708	109	4	sequence	sequence	NOUN
cana-708	109	5	in	in	ADP
cana-708	109	6	the	the	DET
cana-708	109	7	sense	sense	NOUN
cana-708	109	8	of	of	ADP
cana-708	109	9	mean	mean	NOUN
cana-708	109	10	.	.	PUNCT
cana-708	110	1	hence	hence	ADV
cana-708	110	2	there	there	PRON
cana-708	110	3	exists	exist	VERB
cana-708	110	4	a	a	DET
cana-708	110	5	random	random	ADJ
cana-708	110	6	variable	variable	NOUN
cana-708	110	7	𝑌(𝜔	𝑌(𝜔	PRON
cana-708	110	8	)	)	PUNCT
cana-708	110	9	such	such	ADJ
cana-708	110	10	that	that	DET
cana-708	110	11	𝐸|𝑌𝑛(𝜔	𝐸|𝑌𝑛(𝜔	PROPN
cana-708	110	12	)	)	PUNCT
cana-708	111	1	−	−	PROPN
cana-708	111	2	𝑌(𝜔)|	𝑌(𝜔)|	PRON
cana-708	112	1	=	=	NOUN
cana-708	113	1	0	0	PROPN
cana-708	113	2	.	.	PUNCT
cana-708	114	1	this	this	PRON
cana-708	114	2	𝑌(𝜔	𝑌(𝜔	X
cana-708	114	3	)	)	PUNCT
cana-708	114	4	is	be	AUX
cana-708	114	5	independent	independent	ADJ
cana-708	114	6	of	of	ADP
cana-708	114	7	the	the	DET
cana-708	114	8	choice	choice	NOUN
cana-708	114	9	of	of	ADP
cana-708	114	10	the	the	DET
cana-708	114	11	sequence	sequence	NOUN
cana-708	114	12	of	of	ADP
cana-708	114	13	functions	function	NOUN
cana-708	114	14	ℎ𝑛.	ℎ𝑛.	NOUN
cana-708	114	15	in	in	ADP
cana-708	114	16	fact	fact	NOUN
cana-708	114	17	,	,	PUNCT
cana-708	114	18	if	if	SCONJ
cana-708	114	19	another	another	DET
cana-708	114	20	sequence	sequence	NOUN
cana-708	114	21	𝑓𝑛	𝑓𝑛	ADJ
cana-708	114	22	in	in	ADP
cana-708	114	23	𝐶𝑐(ℝ	𝐶𝑐(ℝ	ADV
cana-708	114	24	)	)	PUNCT
cana-708	114	25	converges	converge	VERB
cana-708	114	26	to	to	ADP
cana-708	114	27	𝑔	𝑔	PROPN
cana-708	114	28	i.e.	i.e.	X
cana-708	114	29	lim	lim	PROPN
cana-708	114	30	𝑛→∞	𝑛→∞	NUM
cana-708	114	31	∫	∫	PROPN
cana-708	114	32	∞	∞	PROPN
cana-708	114	33	−∞	−∞	ADP
cana-708	114	34	|𝑓𝑛(𝑢	|𝑓𝑛(𝑢	NUM
cana-708	114	35	)	)	PUNCT
cana-708	114	36	−	−	NOUN
cana-708	114	37	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	114	38	,	,	PUNCT
cana-708	114	39	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	114	40	=	=	SYM
cana-708	114	41	0	0	PROPN
cana-708	114	42	𝑎𝑠	𝑎𝑠	PROPN
cana-708	114	43	𝑛	𝑛	PROPN
cana-708	114	44	→	→	SYM
cana-708	114	45	∞.	∞.	PROPN
cana-708	114	46	then	then	ADV
cana-708	114	47	lim	lim	PROPN
cana-708	114	48	𝑛→∞	𝑛→∞	NUM
cana-708	114	49	∫	∫	PROPN
cana-708	114	50	∞	∞	PROPN
cana-708	114	51	−∞	−∞	PROPN
cana-708	114	52	|𝑓𝑛(𝑢	|𝑓𝑛(𝑢	NUM
cana-708	114	53	)	)	PUNCT
cana-708	114	54	−	−	NOUN
cana-708	114	55	ℎ𝑛(𝑢)|2𝑑𝑢	ℎ𝑛(𝑢)|2𝑑𝑢	NOUN
cana-708	115	1	=	=	SYM
cana-708	115	2	lim	lim	PROPN
cana-708	115	3	𝑛→∞	𝑛→∞	NUM
cana-708	115	4	∫	∫	PROPN
cana-708	115	5	∞	∞	PROPN
cana-708	115	6	−∞	−∞	ADP
cana-708	115	7	|𝑓𝑛(𝑢	|𝑓𝑛(𝑢	NUM
cana-708	115	8	)	)	PUNCT
cana-708	115	9	−	−	NOUN
cana-708	115	10	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	115	11	,	,	PUNCT
cana-708	115	12	𝐵	𝐵	NOUN
cana-708	115	13	)	)	PUNCT
cana-708	115	14	+	+	NUM
cana-708	115	15	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	115	16	,	,	PUNCT
cana-708	115	17	𝐵	𝐵	NOUN
cana-708	115	18	)	)	PUNCT
cana-708	115	19	−	−	NOUN
cana-708	115	20	ℎ𝑛(𝑢)|2𝑑𝑢	ℎ𝑛(𝑢)|2𝑑𝑢	NOUN
cana-708	115	21	=	=	SYM
cana-708	115	22	lim	lim	PROPN
cana-708	115	23	𝑛→∞	𝑛→∞	NUM
cana-708	115	24	∫	∫	PROPN
cana-708	115	25	∞	∞	PROPN
cana-708	115	26	−∞	−∞	ADP
cana-708	115	27	|𝑓𝑛(𝑢	|𝑓𝑛(𝑢	NUM
cana-708	115	28	)	)	PUNCT
cana-708	115	29	−	−	NOUN
cana-708	115	30	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	115	31	,	,	PUNCT
cana-708	115	32	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	PUNCT
cana-708	115	33	+	+	CCONJ
cana-708	115	34	lim	lim	PROPN
cana-708	115	35	𝑛→∞	𝑛→∞	NUM
cana-708	115	36	∫	∫	PROPN
cana-708	115	37	∞	∞	PROPN
cana-708	115	38	−∞	−∞	ADP
cana-708	115	39	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	115	40	,	,	PUNCT
cana-708	115	41	𝐵	𝐵	NOUN
cana-708	115	42	)	)	PUNCT
cana-708	115	43	−	−	PROPN
cana-708	115	44	ℎ𝑛(𝑢)|2𝑑𝑢	ℎ𝑛(𝑢)|2𝑑𝑢	NOUN
cana-708	115	45	which	which	PRON
cana-708	115	46	converges	converge	VERB
cana-708	115	47	to	to	ADP
cana-708	115	48	0	0	NUM
cana-708	115	49	.	.	PUNCT
cana-708	116	1	thus	thus	ADV
cana-708	116	2	we	we	PRON
cana-708	116	3	obtain	obtain	VERB
cana-708	116	4	lim	lim	PROPN
cana-708	116	5	𝑛→∞	𝑛→∞	NUM
cana-708	116	6	𝐸(|	𝐸(|	NUM
cana-708	116	7	∫	∫	PROPN
cana-708	116	8	∞	∞	PROPN
cana-708	116	9	−∞	−∞	ADP
cana-708	116	10	𝑓𝑛(𝑢)𝑑𝑋(𝑢	𝑓𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	11	,	,	PUNCT
cana-708	116	12	𝜔	𝜔	NOUN
cana-708	116	13	)	)	PUNCT
cana-708	116	14	−	−	NOUN
cana-708	116	15	𝑌(𝜔)|	𝑌(𝜔)|	NUM
cana-708	116	16	)	)	PUNCT
cana-708	116	17	=	=	SYM
cana-708	116	18	lim	lim	NOUN
cana-708	116	19	𝑛→∞	𝑛→∞	NUM
cana-708	116	20	𝐸(|	𝐸(|	NUM
cana-708	116	21	∫	∫	PROPN
cana-708	116	22	∞	∞	PROPN
cana-708	116	23	−∞	−∞	ADP
cana-708	116	24	𝑓𝑛(𝑢)𝑑𝑋(𝑢	𝑓𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	25	,	,	PUNCT
cana-708	116	26	𝜔	𝜔	PRON
cana-708	116	27	)	)	PUNCT
cana-708	116	28	−	−	NOUN
cana-708	116	29	∫	∫	PROPN
cana-708	116	30	∞	∞	PROPN
cana-708	116	31	−∞	−∞	ADP
cana-708	116	32	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	33	,	,	PUNCT
cana-708	116	34	𝜔	𝜔	PRON
cana-708	116	35	)	)	PUNCT
cana-708	116	36	+	+	NUM
cana-708	116	37	∫	∫	X
cana-708	116	38	∞	∞	PROPN
cana-708	116	39	−∞	−∞	ADP
cana-708	116	40	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	41	,	,	PUNCT
cana-708	116	42	𝜔	𝜔	NOUN
cana-708	116	43	)	)	PUNCT
cana-708	116	44	−	−	NOUN
cana-708	116	45	𝑌(𝜔)|	𝑌(𝜔)|	NUM
cana-708	116	46	)	)	PUNCT
cana-708	116	47	=	=	SYM
cana-708	116	48	lim	lim	NOUN
cana-708	116	49	𝑛→∞	𝑛→∞	NUM
cana-708	116	50	𝐸(|	𝐸(|	NUM
cana-708	116	51	∫	∫	PROPN
cana-708	116	52	∞	∞	PROPN
cana-708	116	53	−∞	−∞	ADP
cana-708	116	54	𝑓𝑛(𝑢)𝑑𝑋(𝑢	𝑓𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	55	,	,	PUNCT
cana-708	116	56	𝜔	𝜔	PRON
cana-708	116	57	)	)	PUNCT
cana-708	116	58	−	−	NOUN
cana-708	116	59	∫	∫	PROPN
cana-708	116	60	∞	∞	PROPN
cana-708	116	61	−∞	−∞	ADP
cana-708	116	62	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	63	,	,	PUNCT
cana-708	116	64	𝜔)|	𝜔)|	NOUN
cana-708	116	65	)	)	PUNCT
cana-708	116	66	+	+	CCONJ
cana-708	116	67	lim	lim	PROPN
cana-708	116	68	𝑛→∞	𝑛→∞	NUM
cana-708	116	69	𝐸(|	𝐸(|	NUM
cana-708	116	70	∫	∫	PROPN
cana-708	116	71	∞	∞	PROPN
cana-708	116	72	−∞	−∞	ADP
cana-708	116	73	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	116	74	,	,	PUNCT
cana-708	116	75	𝜔	𝜔	NOUN
cana-708	116	76	)	)	PUNCT
cana-708	116	77	−	−	NOUN
cana-708	116	78	𝑌(𝜔)|	𝑌(𝜔)|	NUM
cana-708	116	79	)	)	PUNCT
cana-708	116	80	=	=	SYM
cana-708	116	81	0	0	NUM
cana-708	116	82	𝑏𝑦	𝑏𝑦	NOUN
cana-708	116	83	𝐿𝑒𝑚𝑚𝑎	𝐿𝑒𝑚𝑚𝑎	PROPN
cana-708	116	84	1	1	NUM
cana-708	116	85	.	.	PUNCT
cana-708	117	1	hence	hence	ADV
cana-708	117	2	the	the	DET
cana-708	117	3	stochastic	stochastic	ADJ
cana-708	117	4	integral	integral	ADJ
cana-708	117	5	∫	∫	NOUN
cana-708	117	6	∞	∞	PROPN
cana-708	117	7	−∞	−∞	ADP
cana-708	117	8	ℎ𝑛(𝑢)𝑑𝑋(𝑢	ℎ𝑛(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	117	9	,	,	PUNCT
cana-708	117	10	𝜔	𝜔	NOUN
cana-708	117	11	)	)	PUNCT
cana-708	117	12	converges	converge	VERB
cana-708	117	13	uniquely	uniquely	ADV
cana-708	117	14	to	to	ADP
cana-708	117	15	𝑌(𝜔	𝑌(𝜔	NOUN
cana-708	117	16	)	)	PUNCT
cana-708	117	17	,	,	PUNCT
cana-708	117	18	in	in	ADP
cana-708	117	19	the	the	DET
cana-708	117	20	sense	sense	NOUN
cana-708	117	21	of	of	ADP
cana-708	117	22	mean	mean	VERB
cana-708	117	23	.	.	PUNCT
cana-708	118	1	define	define	VERB
cana-708	118	2	this	this	DET
cana-708	118	3	random	random	ADJ
cana-708	118	4	variable	variable	NOUN
cana-708	118	5	𝑌(𝜔	𝑌(𝜔	PRON
cana-708	118	6	)	)	PUNCT
cana-708	118	7	to	to	PART
cana-708	118	8	be	be	AUX
cana-708	118	9	the	the	DET
cana-708	118	10	stochastic	stochastic	ADJ
cana-708	118	11	integral	integral	ADJ
cana-708	118	12	,	,	PUNCT
cana-708	118	13	𝑌(𝜔	𝑌(𝜔	PRON
cana-708	118	14	)	)	PUNCT
cana-708	118	15	=	=	SYM
cana-708	119	1	∫	∫	PROPN
cana-708	119	2	∞	∞	PROPN
cana-708	119	3	−∞	−∞	ADP
cana-708	119	4	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	119	5	,	,	PUNCT
cana-708	119	6	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NOUN
cana-708	119	7	,	,	PUNCT
cana-708	119	8	𝜔	𝜔	PRON
cana-708	119	9	)	)	PUNCT
cana-708	119	10	.	.	PUNCT
cana-708	120	1	this	this	DET
cana-708	120	2	theorem	theorem	NOUN
cana-708	120	3	implies	imply	VERB
cana-708	120	4	the	the	DET
cana-708	120	5	existence	existence	NOUN
cana-708	120	6	of	of	ADP
cana-708	120	7	the	the	DET
cana-708	120	8	integral	integral	ADJ
cana-708	120	9	∫	∫	PROPN
cana-708	120	10	∞	∞	PROPN
cana-708	120	11	−∞	−∞	ADP
cana-708	120	12	𝐻𝑘(𝑢)𝑊(𝑢	𝐻𝑘(𝑢)𝑊(𝑢	PROPN
cana-708	120	13	,	,	PUNCT
cana-708	120	14	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	120	15	,	,	PUNCT
cana-708	120	16	𝜔	𝜔	PRON
cana-708	120	17	)	)	PUNCT
cana-708	120	18	for	for	ADP
cana-708	120	19	𝐵	𝐵	NOUN
cana-708	120	20	>	>	X
cana-708	120	21	−1	−1	NOUN
cana-708	120	22	2	2	NUM
cana-708	120	23	.	.	PUNCT
cana-708	121	1	the	the	DET
cana-708	121	2	random	random	ADJ
cana-708	121	3	variables	variable	NOUN
cana-708	121	4	𝒟𝑘(𝜔	𝒟𝑘(𝜔	NOUN
cana-708	121	5	)	)	PUNCT
cana-708	121	6	=	=	SYM
cana-708	122	1	∫	∫	PROPN
cana-708	122	2	∞	∞	PROPN
cana-708	122	3	−∞	−∞	ADP
cana-708	122	4	𝐻𝑘(𝑢)𝑊(𝑢	𝐻𝑘(𝑢)𝑊(𝑢	PROPN
cana-708	122	5	,	,	PUNCT
cana-708	122	6	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	122	7	,	,	PUNCT
cana-708	122	8	𝜔	𝜔	PRON
cana-708	122	9	)	)	PUNCT
cana-708	122	10	are	be	AUX
cana-708	122	11	found	find	VERB
cana-708	122	12	to	to	PART
cana-708	122	13	be	be	AUX
cana-708	122	14	dependent	dependent	ADJ
cana-708	122	15	.	.	PUNCT
cana-708	123	1	it	it	PRON
cana-708	123	2	is	be	AUX
cana-708	123	3	established	establish	VERB
cana-708	123	4	by	by	ADP
cana-708	123	5	showing	show	VERB
cana-708	123	6	the	the	DET
cana-708	123	7	fact	fact	NOUN
cana-708	123	8	that	that	SCONJ
cana-708	123	9	the	the	DET
cana-708	123	10	characteristic	characteristic	ADJ
cana-708	123	11	function(cf	function(cf	NOUN
cana-708	123	12	)	)	PUNCT
cana-708	123	13	of	of	ADP
cana-708	123	14	(	(	PUNCT
cana-708	123	15	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	123	16	)	)	PUNCT
cana-708	123	17	+	+	CCONJ
cana-708	123	18	𝒟𝑙(𝜔	𝒟𝑙(𝜔	ADJ
cana-708	123	19	)	)	PUNCT
cana-708	123	20	)	)	PUNCT
cana-708	123	21	is	be	AUX
cana-708	123	22	not	not	PART
cana-708	123	23	equal	equal	ADJ
cana-708	123	24	to	to	ADP
cana-708	123	25	the	the	DET
cana-708	123	26	product	product	NOUN
cana-708	123	27	of	of	ADP
cana-708	123	28	cf	cf	NOUN
cana-708	123	29	of	of	ADP
cana-708	123	30	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	123	31	)	)	PUNCT
cana-708	123	32	and	and	CCONJ
cana-708	123	33	the	the	DET
cana-708	123	34	cf	cf	NOUN
cana-708	123	35	𝒟𝑙(𝜔	𝒟𝑙(𝜔	NOUN
cana-708	123	36	)	)	PUNCT
cana-708	123	37	.	.	PUNCT
cana-708	124	1	the	the	DET
cana-708	124	2	cf	cf	NOUN
cana-708	124	3	of	of	ADP
cana-708	124	4	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	124	5	)	)	PUNCT
cana-708	124	6	is	be	AUX
cana-708	124	7	computed	compute	VERB
cana-708	124	8	in	in	ADP
cana-708	124	9	the	the	DET
cana-708	124	10	following	follow	VERB
cana-708	124	11	theorem	theorem	NOUN
cana-708	124	12	.	.	PUNCT
cana-708	125	1	communications	communication	NOUN
cana-708	125	2	on	on	ADP
cana-708	125	3	applied	apply	VERB
cana-708	125	4	nonlinear	nonlinear	ADJ
cana-708	125	5	analysis	analysis	NOUN
cana-708	125	6	issn	issn	NOUN
cana-708	125	7	:	:	PUNCT
cana-708	125	8	1074	1074	NUM
cana-708	125	9	-	-	PUNCT
cana-708	125	10	133x	133x	NUM
cana-708	125	11	vol	vol	NOUN
cana-708	125	12	31	31	NUM
cana-708	125	13	no	no	NOUN
cana-708	125	14	.	.	PUNCT
cana-708	126	1	2s	2s	NUM
cana-708	126	2	(	(	PUNCT
cana-708	126	3	2024	2024	NUM
cana-708	126	4	)	)	PUNCT
cana-708	126	5	701	701	NUM
cana-708	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	126	7	theorem	theorem	VERB
cana-708	126	8	3	3	NUM
cana-708	126	9	the	the	DET
cana-708	126	10	cf	cf	NOUN
cana-708	126	11	of	of	ADP
cana-708	126	12	∫	∫	PROPN
cana-708	126	13	∞	∞	PROPN
cana-708	126	14	−∞	−∞	ADP
cana-708	126	15	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	126	16	,	,	PUNCT
cana-708	126	17	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	126	18	,	,	PUNCT
cana-708	126	19	𝜔	𝜔	PRON
cana-708	126	20	)	)	PUNCT
cana-708	126	21	is	be	AUX
cana-708	126	22	𝑒𝑥𝑝(−𝑐|𝑠|2	𝑒𝑥𝑝(−𝑐|𝑠|2	X
cana-708	126	23	∫	∫	PROPN
cana-708	126	24	∞	∞	PROPN
cana-708	126	25	−∞	−∞	ADP
cana-708	126	26	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	126	27	,	,	PUNCT
cana-708	126	28	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	126	29	)	)	PUNCT
cana-708	126	30	for	for	ADP
cana-708	126	31	𝑔(𝑢	𝑔(𝑢	NOUN
cana-708	126	32	)	)	PUNCT
cana-708	126	33	∈	∈	PROPN
cana-708	126	34	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	PROPN
cana-708	126	35	)	)	PUNCT
cana-708	126	36	2	2	NUM
cana-708	126	37	(	(	PUNCT
cana-708	126	38	ℝ	ℝ	PROPN
cana-708	126	39	)	)	PUNCT
cana-708	126	40	.	.	PUNCT
cana-708	127	1	proof	proof	NOUN
cana-708	127	2	:	:	PUNCT
cana-708	127	3	as	as	SCONJ
cana-708	127	4	we	we	PRON
cana-708	127	5	know	know	VERB
cana-708	127	6	𝐶𝑐(ℝ	𝐶𝑐(ℝ	ADV
cana-708	127	7	)	)	PUNCT
cana-708	127	8	is	be	AUX
cana-708	127	9	dense	dense	ADJ
cana-708	127	10	in	in	ADP
cana-708	127	11	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-708	127	12	)	)	PUNCT
cana-708	127	13	,	,	PUNCT
cana-708	127	14	there	there	PRON
cana-708	127	15	exist	exist	VERB
cana-708	127	16	a	a	DET
cana-708	127	17	sequence	sequence	NOUN
cana-708	127	18	of	of	ADP
cana-708	127	19	functions	function	NOUN
cana-708	127	20	{	{	PUNCT
cana-708	127	21	ℎ𝑘	ℎ𝑘	NOUN
cana-708	127	22	}	}	PUNCT
cana-708	127	23	in	in	ADP
cana-708	127	24	𝐶𝑐(ℝ	𝐶𝑐(ℝ	ADV
cana-708	127	25	)	)	PUNCT
cana-708	127	26	for	for	ADP
cana-708	127	27	𝑔	𝑔	PROPN
cana-708	127	28	∈	∈	PROPN
cana-708	127	29	𝐿2(ℝ	𝐿2(ℝ	PROPN
cana-708	127	30	)	)	PUNCT
cana-708	127	31	such	such	ADJ
cana-708	127	32	that	that	SCONJ
cana-708	127	33	∥	∥	PROPN
cana-708	127	34	ℎ𝑘	ℎ𝑘	PROPN
cana-708	127	35	−	−	PROPN
cana-708	127	36	𝑔𝑊(𝑢	𝑔𝑊(𝑢	PROPN
cana-708	127	37	,	,	PUNCT
cana-708	127	38	𝐵	𝐵	NOUN
cana-708	127	39	)	)	PUNCT
cana-708	127	40	∥2→	∥2→	X
cana-708	127	41	0	0	NUM
cana-708	127	42	.	.	PUNCT
cana-708	128	1	further	far	ADV
cana-708	128	2	it	it	PRON
cana-708	128	3	is	be	AUX
cana-708	128	4	known	know	VERB
cana-708	128	5	that	that	SCONJ
cana-708	128	6	the	the	DET
cana-708	128	7	stochastic	stochastic	ADJ
cana-708	128	8	integrals	integral	NOUN
cana-708	128	9	∫	∫	X
cana-708	128	10	∞	∞	PROPN
cana-708	128	11	−∞	−∞	ADP
cana-708	128	12	ℎ𝑘𝑑𝑋(𝑢	ℎ𝑘𝑑𝑋(𝑢	NOUN
cana-708	128	13	,	,	PUNCT
cana-708	128	14	𝜔	𝜔	NOUN
cana-708	128	15	)	)	PUNCT
cana-708	128	16	and	and	CCONJ
cana-708	128	17	∫	∫	PROPN
cana-708	128	18	∞	∞	PROPN
cana-708	128	19	−∞	−∞	ADP
cana-708	128	20	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	128	21	,	,	PUNCT
cana-708	128	22	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	128	23	,	,	PUNCT
cana-708	128	24	𝜔	𝜔	PRON
cana-708	128	25	)	)	PUNCT
cana-708	128	26	exists	exist	VERB
cana-708	128	27	by	by	ADP
cana-708	128	28	theorem	theorem	NOUN
cana-708	128	29	2	2	NUM
cana-708	128	30	.	.	PUNCT
cana-708	128	31	denote	denote	VERB
cana-708	128	32	these	these	DET
cana-708	128	33	random	random	ADJ
cana-708	128	34	variables	variable	NOUN
cana-708	128	35	as	as	ADP
cana-708	128	36	𝑌𝑘	𝑌𝑘	PROPN
cana-708	128	37	:	:	PUNCT
cana-708	129	1	=	=	SYM
cana-708	129	2	∫	∫	PROPN
cana-708	129	3	∞	∞	PROPN
cana-708	129	4	−∞	−∞	ADP
cana-708	129	5	ℎ𝑘(𝑢)𝑑𝑋(𝑢	ℎ𝑘(𝑢)𝑑𝑋(𝑢	PROPN
cana-708	129	6	,	,	PUNCT
cana-708	129	7	𝜔	𝜔	PRON
cana-708	129	8	)	)	PUNCT
cana-708	129	9	and	and	CCONJ
cana-708	129	10	𝑌	𝑌	PROPN
cana-708	129	11	:	:	PUNCT
cana-708	129	12	=	=	SYM
cana-708	129	13	∫	∫	PROPN
cana-708	129	14	∞	∞	PROPN
cana-708	129	15	−∞	−∞	ADP
cana-708	129	16	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	129	17	,	,	PUNCT
cana-708	129	18	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	129	19	,	,	PUNCT
cana-708	129	20	𝜔	𝜔	PRON
cana-708	129	21	)	)	PUNCT
cana-708	129	22	in	in	ADP
cana-708	129	23	mean	mean	NOUN
cana-708	129	24	.	.	PUNCT
cana-708	130	1	𝑌𝑘	𝑌𝑘	PROPN
cana-708	130	2	converges	converge	VERB
cana-708	130	3	to	to	ADP
cana-708	130	4	𝑌	𝑌	PROPN
cana-708	130	5	in	in	ADP
cana-708	130	6	mean	mean	ADJ
cana-708	130	7	⇒	⇒	NOUN
cana-708	130	8	𝑌𝑘	𝑌𝑘	PROPN
cana-708	130	9	converges	converge	VERB
cana-708	130	10	to	to	ADP
cana-708	130	11	𝑌	𝑌	PROPN
cana-708	130	12	in	in	ADP
cana-708	130	13	law	law	NOUN
cana-708	130	14	⇒	⇒	NOUN
cana-708	130	15	distribution	distribution	NOUN
cana-708	130	16	of	of	ADP
cana-708	130	17	𝑌𝑘	𝑌𝑘	PROPN
cana-708	130	18	weakly	weakly	ADJ
cana-708	130	19	converges	converge	NOUN
cana-708	130	20	to	to	ADP
cana-708	130	21	distribution	distribution	NOUN
cana-708	130	22	of	of	ADP
cana-708	130	23	𝑌[12	𝑌[12	NOUN
cana-708	130	24	]	]	PUNCT
cana-708	130	25	.	.	PUNCT
cana-708	131	1	now	now	ADV
cana-708	131	2	the	the	DET
cana-708	131	3	cf	cf	NOUN
cana-708	131	4	of	of	ADP
cana-708	131	5	𝑌𝑘	𝑌𝑘	PROPN
cana-708	131	6	:	:	PUNCT
cana-708	131	7	=	=	SYM
cana-708	131	8	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	X
cana-708	131	9	∫	∫	X
cana-708	131	10	∞	∞	PROPN
cana-708	131	11	−∞	−∞	ADP
cana-708	131	12	|ℎ𝑘(𝑢)|2𝑑𝑢	|ℎ𝑘(𝑢)|2𝑑𝑢	NOUN
cana-708	131	13	)	)	PUNCT
cana-708	131	14	.	.	PUNCT
cana-708	132	1	for	for	ADP
cana-708	132	2	1	1	NUM
cana-708	132	3	≤	≤	NOUN
cana-708	132	4	𝑝	𝑝	PROPN
cana-708	132	5	<	<	X
cana-708	132	6	∞	∞	PROPN
cana-708	132	7	,	,	PUNCT
cana-708	132	8	it	it	PRON
cana-708	132	9	is	be	AUX
cana-708	132	10	true	true	ADJ
cana-708	132	11	that	that	SCONJ
cana-708	132	12	(	(	PUNCT
cana-708	132	13	[	[	X
cana-708	132	14	20	20	NUM
cana-708	132	15	]	]	PUNCT
cana-708	132	16	,	,	PUNCT
cana-708	132	17	page	page	NOUN
cana-708	132	18	no	no	INTJ
cana-708	132	19	.	.	NOUN
cana-708	132	20	75	75	NUM
cana-708	132	21	)	)	PUNCT
cana-708	132	22	∫	∫	PROPN
cana-708	132	23	∞	∞	PROPN
cana-708	133	1	−∞	−∞	ADP
cana-708	133	2	||ℎ𝑘(𝑢)|2	||ℎ𝑘(𝑢)|2	NOUN
cana-708	133	3	−	−	ADP
cana-708	133	4	|𝑔(𝑢)|2|𝑑𝑢	|𝑔(𝑢)|2|𝑑𝑢	VERB
cana-708	133	5	≤	≤	X
cana-708	133	6	4𝑅	4𝑅	ADJ
cana-708	133	7	∫	∫	NOUN
cana-708	133	8	∞	∞	PROPN
cana-708	133	9	−∞	−∞	PROPN
cana-708	133	10	|ℎ𝑘(𝑢	|ℎ𝑘(𝑢	NUM
cana-708	133	11	)	)	PUNCT
cana-708	133	12	−	−	PROPN
cana-708	133	13	𝑔(𝑢)|2𝑑𝑢	𝑔(𝑢)|2𝑑𝑢	NUM
cana-708	133	14	→	→	SYM
cana-708	133	15	0	0	NUM
cana-708	133	16	,	,	PUNCT
cana-708	133	17	and	and	CCONJ
cana-708	133	18	since	since	SCONJ
cana-708	133	19	𝑊(𝑢	𝑊(𝑢	PROPN
cana-708	133	20	,	,	PUNCT
cana-708	133	21	𝐵	𝐵	NOUN
cana-708	133	22	)	)	PUNCT
cana-708	133	23	are	be	AUX
cana-708	133	24	dense	dense	ADJ
cana-708	133	25	in	in	ADP
cana-708	133	26	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-708	133	27	)	)	PUNCT
cana-708	133	28	,	,	PUNCT
cana-708	133	29	this	this	PRON
cana-708	133	30	implies	imply	VERB
cana-708	133	31	∫	∫	PROPN
cana-708	133	32	∞	∞	PROPN
cana-708	133	33	−∞	−∞	PUNCT
cana-708	133	34	|ℎ𝑘(𝑢)|2𝑑𝑢	|ℎ𝑘(𝑢)|2𝑑𝑢	NUM
cana-708	133	35	𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠	𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠	PROPN
cana-708	133	36	𝑡𝑜	𝑡𝑜	PROPN
cana-708	133	37	∫	∫	PROPN
cana-708	133	38	∞	∞	PROPN
cana-708	133	39	−∞	−∞	ADP
cana-708	133	40	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	133	41	,	,	PUNCT
cana-708	133	42	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	PUNCT
cana-708	133	43	⇒	⇒	PROPN
cana-708	133	44	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	133	45	∫	∫	PROPN
cana-708	133	46	∞	∞	PROPN
cana-708	133	47	−∞	−∞	ADP
cana-708	133	48	|ℎ𝑘(𝑢)|2𝑑𝑢	|ℎ𝑘(𝑢)|2𝑑𝑢	NOUN
cana-708	133	49	)	)	PUNCT
cana-708	133	50	𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠	𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠	NOUN
cana-708	133	51	𝑡𝑜	𝑡𝑜	PROPN
cana-708	133	52	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	ADJ
cana-708	133	53	∫	∫	PROPN
cana-708	133	54	∞	∞	PROPN
cana-708	133	55	−∞	−∞	ADP
cana-708	133	56	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	133	57	,	,	PUNCT
cana-708	133	58	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	133	59	)	)	PUNCT
cana-708	133	60	.	.	PUNCT
cana-708	134	1	lhs	lhs	PROPN
cana-708	134	2	is	be	AUX
cana-708	134	3	the	the	DET
cana-708	134	4	cf	cf	NOUN
cana-708	134	5	of	of	ADP
cana-708	134	6	𝑌𝑘	𝑌𝑘	PROPN
cana-708	134	7	,	,	PUNCT
cana-708	134	8	which	which	PRON
cana-708	134	9	converges	converge	VERB
cana-708	134	10	to	to	ADP
cana-708	134	11	the	the	DET
cana-708	134	12	continuous	continuous	ADJ
cana-708	134	13	function	function	NOUN
cana-708	134	14	on	on	ADP
cana-708	134	15	the	the	DET
cana-708	134	16	rsh	rsh	PROPN
cana-708	134	17	.	.	PUNCT
cana-708	134	18	by	by	ADP
cana-708	134	19	continuity	continuity	NOUN
cana-708	134	20	theorem([12	theorem([12	NOUN
cana-708	134	21	]	]	X
cana-708	134	22	,	,	PUNCT
cana-708	134	23	theorem	theorem	VERB
cana-708	134	24	1.3.7	1.3.7	NOUN
cana-708	134	25	,	,	PUNCT
cana-708	134	26	page	page	NOUN
cana-708	134	27	no	no	NOUN
cana-708	134	28	.	.	NOUN
cana-708	134	29	15	15	NUM
cana-708	134	30	)	)	PUNCT
cana-708	134	31	,	,	PUNCT
cana-708	134	32	rhs	rhs	PROPN
cana-708	134	33	is	be	AUX
cana-708	134	34	the	the	DET
cana-708	134	35	cf	cf	NOUN
cana-708	134	36	of	of	ADP
cana-708	134	37	the	the	DET
cana-708	134	38	limiting	limit	VERB
cana-708	134	39	function	function	NOUN
cana-708	134	40	of	of	ADP
cana-708	134	41	𝑌𝑘	𝑌𝑘	PROPN
cana-708	134	42	,	,	PUNCT
cana-708	134	43	which	which	PRON
cana-708	134	44	is	be	AUX
cana-708	134	45	𝑌	𝑌	PROPN
cana-708	134	46	.	.	PUNCT
cana-708	135	1	this	this	PRON
cana-708	135	2	proves	prove	VERB
cana-708	135	3	that	that	SCONJ
cana-708	135	4	the	the	DET
cana-708	135	5	cf	cf	NOUN
cana-708	135	6	of	of	ADP
cana-708	135	7	∫	∫	PROPN
cana-708	135	8	∞	∞	PROPN
cana-708	135	9	−∞	−∞	ADP
cana-708	135	10	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	135	11	,	,	PUNCT
cana-708	135	12	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	135	13	,	,	PUNCT
cana-708	135	14	𝜔	𝜔	PRON
cana-708	135	15	)	)	PUNCT
cana-708	135	16	is	be	AUX
cana-708	135	17	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	135	18	∫	∫	PROPN
cana-708	135	19	∞	∞	PROPN
cana-708	135	20	−∞	−∞	ADP
cana-708	135	21	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	135	22	,	,	PUNCT
cana-708	135	23	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	135	24	)	)	PUNCT
cana-708	135	25	,	,	PUNCT
cana-708	135	26	implies	imply	VERB
cana-708	135	27	the	the	DET
cana-708	135	28	pointwise	pointwise	ADJ
cana-708	135	29	convergence	convergence	NOUN
cana-708	135	30	of	of	ADP
cana-708	135	31	𝐶𝑘(𝑠	𝐶𝑘(𝑠	NOUN
cana-708	135	32	)	)	PUNCT
cana-708	135	33	to	to	ADP
cana-708	135	34	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	135	35	∫	∫	PROPN
cana-708	135	36	∞	∞	PROPN
cana-708	135	37	−∞	−∞	ADP
cana-708	135	38	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	135	39	,	,	PUNCT
cana-708	135	40	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	135	41	)	)	PUNCT
cana-708	135	42	as	as	ADP
cana-708	135	43	𝑘	𝑘	PROPN
cana-708	135	44	→	→	SYM
cana-708	135	45	∞[2	∞[2	PROPN
cana-708	135	46	]	]	PUNCT
cana-708	135	47	.	.	PUNCT
cana-708	136	1	the	the	DET
cana-708	136	2	following	follow	VERB
cana-708	136	3	theorem	theorem	NOUN
cana-708	136	4	proves	prove	VERB
cana-708	136	5	that	that	SCONJ
cana-708	136	6	the	the	DET
cana-708	136	7	random	random	ADJ
cana-708	136	8	variables	variable	NOUN
cana-708	136	9	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	136	10	)	)	PUNCT
cana-708	136	11	are	be	AUX
cana-708	136	12	dependent	dependent	ADJ
cana-708	136	13	.	.	PUNCT
cana-708	137	1	theorem	theorem	VERB
cana-708	137	2	4	4	NUM
cana-708	137	3	the	the	DET
cana-708	137	4	random	random	ADJ
cana-708	137	5	variables	variable	NOUN
cana-708	137	6	𝒟𝑛(𝜔	𝒟𝑛(𝜔	VERB
cana-708	137	7	)	)	PUNCT
cana-708	138	1	=	=	SYM
cana-708	138	2	∫	∫	PROPN
cana-708	138	3	∞	∞	PROPN
cana-708	138	4	−∞	−∞	X
cana-708	138	5	𝐻𝑛(𝑢)𝑊(𝑢	𝐻𝑛(𝑢)𝑊(𝑢	PROPN
cana-708	138	6	,	,	PUNCT
cana-708	138	7	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	PROPN
cana-708	138	8	,	,	PUNCT
cana-708	138	9	𝜔	𝜔	PRON
cana-708	138	10	)	)	PUNCT
cana-708	138	11	are	be	AUX
cana-708	138	12	dependent	dependent	ADJ
cana-708	138	13	.	.	PUNCT
cana-708	139	1	proof	proof	NOUN
cana-708	139	2	:	:	PUNCT
cana-708	139	3	by	by	ADP
cana-708	139	4	theorem	theorem	NOUN
cana-708	139	5	3	3	NUM
cana-708	139	6	,	,	PUNCT
cana-708	139	7	the	the	DET
cana-708	139	8	cf	cf	NOUN
cana-708	139	9	of	of	ADP
cana-708	139	10	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	139	11	)	)	PUNCT
cana-708	139	12	is	be	AUX
cana-708	139	13	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	139	14	∫	∫	PROPN
cana-708	139	15	∞	∞	PROPN
cana-708	139	16	−∞	−∞	ADP
cana-708	139	17	|𝐻𝑛(𝑥)𝑊(𝑢	|𝐻𝑛(𝑥)𝑊(𝑢	PROPN
cana-708	139	18	,	,	PUNCT
cana-708	139	19	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	139	20	)	)	PUNCT
cana-708	139	21	.	.	PUNCT
cana-708	140	1	hence	hence	ADV
cana-708	140	2	,	,	PUNCT
cana-708	140	3	the	the	DET
cana-708	140	4	cf	cf	NOUN
cana-708	140	5	of	of	ADP
cana-708	140	6	(	(	PUNCT
cana-708	140	7	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	140	8	)	)	PUNCT
cana-708	140	9	+	+	CCONJ
cana-708	140	10	𝒟𝑚(𝜔	𝒟𝑚(𝜔	NOUN
cana-708	140	11	)	)	PUNCT
cana-708	140	12	)	)	PUNCT
cana-708	140	13	is	be	AUX
cana-708	140	14	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	140	15	∫	∫	PROPN
cana-708	140	16	∞	∞	PROPN
cana-708	140	17	−∞	−∞	X
cana-708	140	18	|𝐻𝑛(𝑢)𝑊(𝑢	|𝐻𝑛(𝑢)𝑊(𝑢	PROPN
cana-708	140	19	,	,	PUNCT
cana-708	140	20	𝐵	𝐵	PROPN
cana-708	140	21	)	)	PUNCT
cana-708	140	22	+	+	CCONJ
cana-708	140	23	𝐻𝑚(𝑢)𝑊(𝑢	𝐻𝑚(𝑢)𝑊(𝑢	PROPN
cana-708	140	24	,	,	PUNCT
cana-708	140	25	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	140	26	)	)	PUNCT
cana-708	140	27	,	,	PUNCT
cana-708	140	28	whereas	whereas	SCONJ
cana-708	140	29	the	the	DET
cana-708	140	30	product	product	NOUN
cana-708	140	31	of	of	ADP
cana-708	140	32	cf	cf	NOUN
cana-708	140	33	of	of	ADP
cana-708	140	34	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	140	35	)	)	PUNCT
cana-708	140	36	and	and	CCONJ
cana-708	140	37	the	the	DET
cana-708	140	38	cf	cf	NOUN
cana-708	140	39	of	of	ADP
cana-708	140	40	𝒟𝑚(𝜔	𝒟𝑚(𝜔	NOUN
cana-708	140	41	)	)	PUNCT
cana-708	140	42	is	be	AUX
cana-708	140	43	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	140	44	∫	∫	PROPN
cana-708	140	45	∞	∞	PROPN
cana-708	140	46	−∞	−∞	X
cana-708	140	47	|𝐻𝑛(𝑢)𝑊(𝑢	|𝐻𝑛(𝑢)𝑊(𝑢	PROPN
cana-708	140	48	,	,	PUNCT
cana-708	140	49	𝐵)|2𝑑𝑢)exp(−𝑐|𝑠|2	𝐵)|2𝑑𝑢)exp(−𝑐|𝑠|2	PROPN
cana-708	140	50	∫	∫	PROPN
cana-708	140	51	∞	∞	PROPN
cana-708	140	52	−∞	−∞	ADP
cana-708	140	53	|𝐻𝑚(𝑢)𝑊(𝑢	|𝐻𝑚(𝑢)𝑊(𝑢	PROPN
cana-708	140	54	,	,	PUNCT
cana-708	140	55	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	140	56	)	)	PUNCT
cana-708	140	57	=	=	PUNCT
cana-708	140	58	exp(−𝑐|𝑠|2	exp(−𝑐|𝑠|2	PROPN
cana-708	140	59	∫	∫	X
cana-708	140	60	∞	∞	PROPN
cana-708	140	61	−∞	−∞	X
cana-708	140	62	(	(	PUNCT
cana-708	140	63	|𝐻𝑛(𝑢)𝑊(𝑢	|𝐻𝑛(𝑢)𝑊(𝑢	PROPN
cana-708	140	64	,	,	PUNCT
cana-708	140	65	𝐵)|2	𝐵)|2	X
cana-708	140	66	+	+	CCONJ
cana-708	140	67	|𝐻𝑚(𝑥)𝑊(𝑢	|𝐻𝑚(𝑥)𝑊(𝑢	PROPN
cana-708	140	68	,	,	PUNCT
cana-708	140	69	𝐵)|2)𝑑𝑢	𝐵)|2)𝑑𝑢	NUM
cana-708	140	70	)	)	PUNCT
cana-708	140	71	.	.	PUNCT
cana-708	141	1	since	since	SCONJ
cana-708	141	2	cf	cf	NOUN
cana-708	141	3	of	of	ADP
cana-708	141	4	(	(	PUNCT
cana-708	141	5	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	141	6	)	)	PUNCT
cana-708	141	7	+	+	CCONJ
cana-708	141	8	𝒟𝑚(𝜔	𝒟𝑚(𝜔	NOUN
cana-708	141	9	)	)	PUNCT
cana-708	141	10	)	)	PUNCT
cana-708	141	11	is	be	AUX
cana-708	141	12	not	not	PART
cana-708	141	13	equal	equal	ADJ
cana-708	141	14	to	to	ADP
cana-708	141	15	the	the	DET
cana-708	141	16	product	product	NOUN
cana-708	141	17	of	of	ADP
cana-708	141	18	cf	cf	NOUN
cana-708	141	19	of	of	ADP
cana-708	141	20	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	141	21	)	)	PUNCT
cana-708	141	22	and	and	CCONJ
cana-708	141	23	cf	cf	NOUN
cana-708	141	24	of	of	ADP
cana-708	141	25	𝒟𝑚(𝜔	𝒟𝑚(𝜔	NOUN
cana-708	141	26	)	)	PUNCT
cana-708	141	27	,	,	PUNCT
cana-708	141	28	𝒟𝑛(𝜔	𝒟𝑛(𝜔	NOUN
cana-708	141	29	)	)	PUNCT
cana-708	141	30	are	be	AUX
cana-708	141	31	dependent	dependent	ADJ
cana-708	141	32	random	random	ADJ
cana-708	141	33	variables	variable	NOUN
cana-708	141	34	.	.	PUNCT
cana-708	142	1	communications	communication	NOUN
cana-708	142	2	on	on	ADP
cana-708	142	3	applied	apply	VERB
cana-708	142	4	nonlinear	nonlinear	ADJ
cana-708	142	5	analysis	analysis	NOUN
cana-708	142	6	issn	issn	NOUN
cana-708	142	7	:	:	PUNCT
cana-708	142	8	1074	1074	NUM
cana-708	142	9	-	-	PUNCT
cana-708	142	10	133x	133x	NUM
cana-708	142	11	vol	vol	NOUN
cana-708	142	12	31	31	NUM
cana-708	142	13	no	no	NOUN
cana-708	142	14	.	.	PUNCT
cana-708	143	1	2s	2s	NUM
cana-708	143	2	(	(	PUNCT
cana-708	143	3	2024	2024	NUM
cana-708	143	4	)	)	PUNCT
cana-708	143	5	702	702	NUM
cana-708	143	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	143	7	4	4	NUM
cana-708	143	8	.	.	X
cana-708	143	9	convergence	convergence	NOUN
cana-708	143	10	of	of	ADP
cana-708	143	11	random	random	ADJ
cana-708	143	12	fourier	fourier	NOUN
cana-708	143	13	hermite	hermite	PROPN
cana-708	143	14	series	series	PROPN
cana-708	143	15	∑∞	∑∞	PROPN
cana-708	143	16	𝑘=0	𝑘=0	VERB
cana-708	143	17	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	143	18	)	)	PUNCT
cana-708	143	19	to	to	PART
cana-708	143	20	prove	prove	VERB
cana-708	143	21	the	the	DET
cana-708	143	22	convergence	convergence	NOUN
cana-708	143	23	of	of	ADP
cana-708	143	24	rfhs	rfhs	ADJ
cana-708	143	25	,	,	PUNCT
cana-708	143	26	we	we	PRON
cana-708	143	27	employ	employ	VERB
cana-708	143	28	the	the	DET
cana-708	143	29	following	follow	VERB
cana-708	143	30	inequality	inequality	NOUN
cana-708	143	31	.	.	PUNCT
cana-708	144	1	lemma	lemma	PROPN
cana-708	144	2	5	5	NUM
cana-708	144	3	let	let	VERB
cana-708	144	4	𝑔	𝑔	PART
cana-708	144	5	be	be	AUX
cana-708	144	6	any	any	DET
cana-708	144	7	function	function	NOUN
cana-708	144	8	in	in	ADP
cana-708	144	9	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	ADJ
cana-708	144	10	)	)	PUNCT
cana-708	144	11	2	2	NUM
cana-708	144	12	(	(	PUNCT
cana-708	144	13	ℝ	ℝ	PROPN
cana-708	144	14	)	)	PUNCT
cana-708	144	15	then	then	ADV
cana-708	144	16	𝐸(|	𝐸(|	AUX
cana-708	144	17	∫	∫	PROPN
cana-708	144	18	∞	∞	PROPN
cana-708	144	19	−∞	−∞	ADP
cana-708	144	20	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	144	21	,	,	PUNCT
cana-708	144	22	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	144	23	,	,	PUNCT
cana-708	144	24	𝜔)|	𝜔)|	NOUN
cana-708	144	25	)	)	PUNCT
cana-708	144	26	≤	≤	NOUN
cana-708	144	27	4	4	NUM
cana-708	144	28	𝜋	𝜋	NOUN
cana-708	144	29	∫	∫	PROPN
cana-708	144	30	∞	∞	PROPN
cana-708	144	31	−∞	−∞	ADP
cana-708	144	32	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	144	33	,	,	PUNCT
cana-708	144	34	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	144	35	+	+	CCONJ
cana-708	144	36	2	2	NUM
cana-708	144	37	𝜋	𝜋	NOUN
cana-708	144	38	∫	∫	NOUN
cana-708	144	39	|𝑠|>1	|𝑠|>1	VERB
cana-708	144	40	1−exp(−|𝑠|2	1−exp(−|𝑠|2	NUM
cana-708	144	41	∫	∫	NOUN
cana-708	144	42	∞	∞	PROPN
cana-708	144	43	−∞	−∞	ADP
cana-708	144	44	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	PROPN
cana-708	144	45	)	)	PUNCT
cana-708	144	46	𝑠2	𝑠2	NOUN
cana-708	144	47	𝑑𝑠.	𝑑𝑠.	VERB
cana-708	144	48	its	its	PRON
cana-708	144	49	proof	proof	NOUN
cana-708	144	50	requires	require	VERB
cana-708	144	51	the	the	DET
cana-708	144	52	following	follow	VERB
cana-708	144	53	two	two	NUM
cana-708	144	54	results	result	NOUN
cana-708	144	55	.	.	PUNCT
cana-708	145	1	lemma	lemma	PROPN
cana-708	145	2	6	6	NUM
cana-708	146	1	[	[	X
cana-708	146	2	22	22	NUM
cana-708	146	3	]	]	PUNCT
cana-708	146	4	a	a	DET
cana-708	146	5	stable	stable	ADJ
cana-708	146	6	random	random	ADJ
cana-708	146	7	variable	variable	NOUN
cana-708	146	8	𝑋(𝑢	𝑋(𝑢	NOUN
cana-708	146	9	,	,	PUNCT
cana-708	146	10	𝜔	𝜔	PRON
cana-708	146	11	)	)	PUNCT
cana-708	146	12	always	always	ADV
cana-708	146	13	satisfies	satisfy	VERB
cana-708	146	14	the	the	DET
cana-708	146	15	inequality	inequality	NOUN
cana-708	146	16	𝐸|𝑋|𝑖	𝐸|𝑋|𝑖	ADV
cana-708	146	17	<	<	X
cana-708	146	18	∞	∞	PROPN
cana-708	146	19	for	for	ADP
cana-708	146	20	all	all	DET
cana-708	146	21	𝑖	𝑖	SYM
cana-708	146	22	∈	∈	PROPN
cana-708	146	23	(	(	PUNCT
cana-708	146	24	0	0	NUM
cana-708	146	25	,	,	PUNCT
cana-708	146	26	𝜇	𝜇	ADP
cana-708	146	27	)	)	PUNCT
cana-708	146	28	,	,	PUNCT
cana-708	146	29	0	0	NUM
cana-708	146	30	<	<	X
cana-708	146	31	𝜇	𝜇	ADP
cana-708	146	32	≤	≤	ADJ
cana-708	146	33	2	2	NUM
cana-708	146	34	.	.	PUNCT
cana-708	147	1	lemma	lemma	PROPN
cana-708	147	2	7	7	NUM
cana-708	148	1	[	[	X
cana-708	148	2	7	7	X
cana-708	148	3	]	]	X
cana-708	148	4	if	if	SCONJ
cana-708	148	5	𝛹	𝛹	PROPN
cana-708	148	6	is	be	AUX
cana-708	148	7	the	the	DET
cana-708	148	8	cf	cf	NOUN
cana-708	148	9	of	of	ADP
cana-708	148	10	a	a	DET
cana-708	148	11	random	random	ADJ
cana-708	148	12	variable	variable	NOUN
cana-708	148	13	𝑋	𝑋	NOUN
cana-708	148	14	and	and	CCONJ
cana-708	148	15	𝐹(𝑋	𝐹(𝑋	PROPN
cana-708	148	16	)	)	PUNCT
cana-708	148	17	is	be	AUX
cana-708	148	18	the	the	DET
cana-708	148	19	distribution	distribution	NOUN
cana-708	148	20	function	function	NOUN
cana-708	148	21	of	of	ADP
cana-708	148	22	𝑋	𝑋	PROPN
cana-708	148	23	then	then	ADV
cana-708	148	24	,	,	PUNCT
cana-708	148	25	𝐸|𝑋|	𝐸|𝑋|	PROPN
cana-708	148	26	=	=	SYM
cana-708	148	27	∫	∫	PROPN
cana-708	148	28	∞	∞	PROPN
cana-708	148	29	−∞	−∞	ADP
cana-708	148	30	|𝑋|𝑑𝐹(𝑋	|𝑋|𝑑𝐹(𝑋	NOUN
cana-708	148	31	)	)	PUNCT
cana-708	148	32	=	=	SYM
cana-708	149	1	2	2	NUM
cana-708	149	2	𝜋	𝜋	NOUN
cana-708	149	3	∫	∫	PROPN
cana-708	149	4	∞	∞	PROPN
cana-708	149	5	−∞	−∞	ADP
cana-708	149	6	1−𝑅𝑒𝛹(𝑠	1−𝑅𝑒𝛹(𝑠	NUM
cana-708	149	7	)	)	PUNCT
cana-708	149	8	𝑠2	𝑠2	NOUN
cana-708	149	9	𝑑𝑠.	𝑑𝑠.	NOUN
cana-708	149	10	proof	proof	NOUN
cana-708	149	11	of	of	ADP
cana-708	149	12	lemma	lemma	PROPN
cana-708	149	13	5	5	NUM
cana-708	149	14	:	:	PUNCT
cana-708	149	15	we	we	PRON
cana-708	149	16	know	know	VERB
cana-708	149	17	that	that	SCONJ
cana-708	149	18	,	,	PUNCT
cana-708	149	19	by	by	ADP
cana-708	149	20	theorem	theorem	NOUN
cana-708	149	21	2	2	NUM
cana-708	149	22	,	,	PUNCT
cana-708	149	23	∫	∫	PROPN
cana-708	149	24	∞	∞	PROPN
cana-708	149	25	−∞	−∞	ADP
cana-708	149	26	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	149	27	,	,	PUNCT
cana-708	149	28	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	149	29	,	,	PUNCT
cana-708	149	30	𝜔	𝜔	PRON
cana-708	149	31	)	)	PUNCT
cana-708	149	32	exists	exist	VERB
cana-708	149	33	in	in	ADP
cana-708	149	34	mean	mean	NOUN
cana-708	149	35	.	.	PUNCT
cana-708	150	1	now	now	ADV
cana-708	150	2	using	use	VERB
cana-708	150	3	lemma	lemma	PROPN
cana-708	150	4	6	6	NUM
cana-708	150	5	and	and	CCONJ
cana-708	150	6	7	7	NUM
cana-708	150	7	,	,	PUNCT
cana-708	150	8	we	we	PRON
cana-708	150	9	have	have	VERB
cana-708	150	10	𝐸(|	𝐸(|	VERB
cana-708	150	11	∫	∫	PROPN
cana-708	150	12	∞	∞	PROPN
cana-708	150	13	−∞	−∞	ADP
cana-708	150	14	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	150	15	,	,	PUNCT
cana-708	150	16	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	150	17	,	,	PUNCT
cana-708	150	18	𝜔)|	𝜔)|	NOUN
cana-708	150	19	)	)	PUNCT
cana-708	151	1	=	=	SYM
cana-708	151	2	2	2	NUM
cana-708	151	3	𝜋	𝜋	NOUN
cana-708	151	4	∫	∫	PROPN
cana-708	151	5	∞	∞	PROPN
cana-708	151	6	−∞	−∞	ADP
cana-708	151	7	1−𝑅𝑒ψ(𝑠	1−𝑅𝑒ψ(𝑠	X
cana-708	151	8	)	)	PUNCT
cana-708	151	9	𝑠2	𝑠2	NOUN
cana-708	151	10	𝑑𝑠	𝑑𝑠	PROPN
cana-708	151	11	=	=	SYM
cana-708	151	12	2	2	NUM
cana-708	151	13	𝜋	𝜋	NOUN
cana-708	151	14	∫	∫	PROPN
cana-708	151	15	|𝑠|≤1	|𝑠|≤1	PROPN
cana-708	151	16	1−𝑅𝑒ψ(𝑠	1−𝑅𝑒ψ(𝑠	NUM
cana-708	151	17	)	)	PUNCT
cana-708	151	18	𝑠2	𝑠2	NOUN
cana-708	151	19	𝑑𝑠	𝑑𝑠	PROPN
cana-708	151	20	+	+	NOUN
cana-708	151	21	2	2	NUM
cana-708	151	22	𝜋	𝜋	NOUN
cana-708	151	23	∫	∫	NOUN
cana-708	151	24	|𝑠|>1	|𝑠|>1	NOUN
cana-708	151	25	1−𝑅𝑒ψ(𝑠	1−𝑅𝑒ψ(𝑠	NUM
cana-708	151	26	)	)	PUNCT
cana-708	151	27	𝑠2	𝑠2	PROPN
cana-708	151	28	𝑑𝑠.	𝑑𝑠.	NOUN
cana-708	151	29	here	here	ADV
cana-708	151	30	∫	∫	PROPN
cana-708	151	31	|𝑠|≤1	|𝑠|≤1	PROPN
cana-708	151	32	1−𝑅𝑒ψ(𝑠	1−𝑅𝑒ψ(𝑠	NUM
cana-708	151	33	)	)	PUNCT
cana-708	151	34	𝑠2	𝑠2	NOUN
cana-708	151	35	𝑑𝑠	𝑑𝑠	PROPN
cana-708	151	36	=	=	SYM
cana-708	151	37	∫	∫	PROPN
cana-708	151	38	1	1	NUM
cana-708	151	39	−1	−1	NOUN
cana-708	151	40	1−exp(−|𝑠|2	1−exp(−|𝑠|2	NUM
cana-708	151	41	∫	∫	NOUN
cana-708	151	42	∞	∞	PROPN
cana-708	151	43	−∞	−∞	ADP
cana-708	151	44	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	PROPN
cana-708	151	45	)	)	PUNCT
cana-708	151	46	𝑠2	𝑠2	NOUN
cana-708	151	47	𝑑𝑠	𝑑𝑠	ADP
cana-708	151	48	≤	≤	NUM
cana-708	151	49	∫	∫	NOUN
cana-708	151	50	1	1	NUM
cana-708	151	51	−1	−1	NOUN
cana-708	151	52	|𝑠|2	|𝑠|2	PROPN
cana-708	151	53	∫	∫	PROPN
cana-708	151	54	∞	∞	PROPN
cana-708	151	55	−∞	−∞	ADP
cana-708	151	56	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	NOUN
cana-708	151	57	𝑠2	𝑠2	PROPN
cana-708	151	58	𝑑𝑠	𝑑𝑠	X
cana-708	151	59	(	(	PUNCT
cana-708	151	60	∵	∵	NOUN
cana-708	151	61	1	1	NUM
cana-708	151	62	−	−	NOUN
cana-708	151	63	𝑒−𝑢	𝑒−𝑢	NOUN
cana-708	151	64	<	<	X
cana-708	151	65	𝑢	𝑢	X
cana-708	151	66	𝑓𝑜𝑟	𝑓𝑜𝑟	X
cana-708	151	67	𝑢	𝑢	X
cana-708	151	68	>	>	X
cana-708	151	69	0	0	NUM
cana-708	151	70	)	)	PUNCT
cana-708	151	71	=	=	SYM
cana-708	151	72	2	2	NUM
cana-708	151	73	∫	∫	NOUN
cana-708	151	74	1	1	NUM
cana-708	151	75	0	0	NUM
cana-708	151	76	𝑑𝑠	𝑑𝑠	ADP
cana-708	151	77	∫	∫	PROPN
cana-708	151	78	∞	∞	PROPN
cana-708	151	79	−∞	−∞	ADP
cana-708	151	80	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	151	81	,	,	PUNCT
cana-708	151	82	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	151	83	=	=	SYM
cana-708	151	84	2	2	NUM
cana-708	151	85	∫	∫	NOUN
cana-708	151	86	∞	∞	PROPN
cana-708	151	87	−∞	−∞	X
cana-708	151	88	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	151	89	,	,	PUNCT
cana-708	151	90	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	151	91	hence	hence	ADV
cana-708	151	92	we	we	PRON
cana-708	151	93	have	have	VERB
cana-708	151	94	𝐸(|	𝐸(|	VERB
cana-708	151	95	∫	∫	PROPN
cana-708	151	96	∞	∞	PROPN
cana-708	151	97	−∞	−∞	ADP
cana-708	151	98	𝑔(𝑢)𝑊(𝑢	𝑔(𝑢)𝑊(𝑢	NOUN
cana-708	151	99	,	,	PUNCT
cana-708	151	100	𝐵)𝑑𝑋(𝑢	𝐵)𝑑𝑋(𝑢	NUM
cana-708	151	101	,	,	PUNCT
cana-708	151	102	𝜔)|	𝜔)|	NOUN
cana-708	151	103	)	)	PUNCT
cana-708	151	104	≤	≤	NOUN
cana-708	151	105	4	4	NUM
cana-708	151	106	𝜋	𝜋	NOUN
cana-708	151	107	∫	∫	PROPN
cana-708	151	108	∞	∞	PROPN
cana-708	151	109	−∞	−∞	ADP
cana-708	151	110	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	151	111	,	,	PUNCT
cana-708	151	112	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	ADV
cana-708	151	113	+	+	CCONJ
cana-708	151	114	2	2	NUM
cana-708	151	115	𝜋	𝜋	NOUN
cana-708	151	116	∫	∫	NOUN
cana-708	151	117	|𝑠|>1	|𝑠|>1	VERB
cana-708	151	118	1−exp(−|𝑠|2	1−exp(−|𝑠|2	NUM
cana-708	151	119	∫	∫	NOUN
cana-708	151	120	∞	∞	PROPN
cana-708	151	121	−∞	−∞	ADP
cana-708	151	122	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	|𝑔(𝑢)𝑊(𝑢,𝐵)|2𝑑𝑢	PROPN
cana-708	151	123	)	)	PUNCT
cana-708	151	124	𝑠2	𝑠2	NOUN
cana-708	151	125	𝑑𝑠.	𝑑𝑠.	VERB
cana-708	151	126	the	the	DET
cana-708	151	127	following	follow	VERB
cana-708	151	128	theorem	theorem	NOUN
cana-708	151	129	establishes	establish	VERB
cana-708	151	130	the	the	DET
cana-708	151	131	convergence	convergence	NOUN
cana-708	151	132	of	of	ADP
cana-708	151	133	the	the	DET
cana-708	151	134	series	series	NOUN
cana-708	151	135	∑	∑	PROPN
cana-708	151	136	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	151	137	)	)	PUNCT
cana-708	151	138	,	,	PUNCT
cana-708	151	139	to	to	ADP
cana-708	151	140	the	the	DET
cana-708	151	141	integral	integral	ADJ
cana-708	151	142	∫	∫	PROPN
cana-708	151	143	∞	∞	PROPN
cana-708	151	144	−∞	−∞	ADP
cana-708	151	145	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	151	146	,	,	PUNCT
cana-708	151	147	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	151	148	,	,	PUNCT
cana-708	151	149	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	151	150	,	,	PUNCT
cana-708	151	151	𝜔	𝜔	NOUN
cana-708	151	152	)	)	PUNCT
cana-708	151	153	,	,	PUNCT
cana-708	151	154	(	(	PUNCT
cana-708	151	155	4.1	4.1	NUM
cana-708	151	156	)	)	PUNCT
cana-708	151	157	in	in	ADP
cana-708	151	158	the	the	DET
cana-708	151	159	sense	sense	NOUN
cana-708	151	160	of	of	ADP
cana-708	151	161	mean	mean	ADJ
cana-708	151	162	,	,	PUNCT
cana-708	151	163	if	if	SCONJ
cana-708	151	164	𝑑𝑘	𝑑𝑘	ADV
cana-708	151	165	:	:	PUNCT
cana-708	151	166	=	=	NOUN
cana-708	151	167	𝑟𝑘	𝑟𝑘	ADJ
cana-708	151	168	2	2	NUM
cana-708	151	169	∫	∫	NOUN
cana-708	151	170	∞	∞	PROPN
cana-708	151	171	−∞	−∞	ADP
cana-708	151	172	𝑔(𝑣)𝐻𝑘(𝑣)𝑒−𝑣2	𝑔(𝑣)𝐻𝑘(𝑣)𝑒−𝑣2	PROPN
cana-708	151	173	𝑑𝑣	𝑑𝑣	NUM
cana-708	151	174	(	(	PUNCT
cana-708	151	175	4.2	4.2	NUM
cana-708	151	176	)	)	PUNCT
cana-708	151	177	are	be	AUX
cana-708	151	178	the	the	DET
cana-708	151	179	fhc	fhc	NOUN
cana-708	151	180	of	of	ADP
cana-708	151	181	𝑔	𝑔	PROPN
cana-708	151	182	∈	∈	PROPN
cana-708	151	183	𝐿𝑊(𝑣,𝐵	𝐿𝑊(𝑣,𝐵	PROPN
cana-708	151	184	)	)	PUNCT
cana-708	151	185	2	2	NUM
cana-708	151	186	(	(	PUNCT
cana-708	151	187	ℝ	ℝ	PROPN
cana-708	151	188	)	)	PUNCT
cana-708	151	189	.	.	PUNCT
cana-708	152	1	here	here	ADV
cana-708	152	2	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	152	3	)	)	PUNCT
cana-708	152	4	are	be	AUX
cana-708	152	5	defined	define	VERB
cana-708	152	6	as	as	ADP
cana-708	152	7	,	,	PUNCT
cana-708	152	8	𝒟𝑘(𝜔	𝒟𝑘(𝜔	PROPN
cana-708	152	9	):	):	PUNCT
cana-708	153	1	=	=	PUNCT
cana-708	153	2	∫	∫	PROPN
cana-708	153	3	∞	∞	PROPN
cana-708	153	4	−∞	−∞	ADP
cana-708	153	5	𝐻𝑘(𝑣)𝑊(𝑣	𝐻𝑘(𝑣)𝑊(𝑣	PROPN
cana-708	153	6	,	,	PUNCT
cana-708	153	7	𝐵)𝑑𝑋(𝑣	𝐵)𝑑𝑋(𝑣	PROPN
cana-708	153	8	,	,	PUNCT
cana-708	153	9	𝜔	𝜔	NOUN
cana-708	153	10	)	)	PUNCT
cana-708	153	11	.	.	PUNCT
cana-708	154	1	(	(	PUNCT
cana-708	154	2	4.3	4.3	NUM
cana-708	154	3	)	)	PUNCT
cana-708	154	4	communications	communication	NOUN
cana-708	154	5	on	on	ADP
cana-708	154	6	applied	apply	VERB
cana-708	154	7	nonlinear	nonlinear	ADJ
cana-708	154	8	analysis	analysis	NOUN
cana-708	154	9	issn	issn	NOUN
cana-708	154	10	:	:	PUNCT
cana-708	154	11	1074	1074	NUM
cana-708	154	12	-	-	PUNCT
cana-708	154	13	133x	133x	NUM
cana-708	154	14	vol	vol	NOUN
cana-708	154	15	31	31	NUM
cana-708	154	16	no	no	NOUN
cana-708	154	17	.	.	PUNCT
cana-708	155	1	2s	2s	NUM
cana-708	155	2	(	(	PUNCT
cana-708	155	3	2024	2024	NUM
cana-708	155	4	)	)	PUNCT
cana-708	155	5	703	703	NUM
cana-708	155	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	155	7	its	its	PRON
cana-708	155	8	proof	proof	NOUN
cana-708	155	9	requires	require	VERB
cana-708	155	10	the	the	DET
cana-708	155	11	following	follow	VERB
cana-708	155	12	lemma	lemma	PROPN
cana-708	155	13	,	,	PUNCT
cana-708	155	14	which	which	PRON
cana-708	155	15	is	be	AUX
cana-708	155	16	the	the	DET
cana-708	155	17	statement	statement	NOUN
cana-708	155	18	of	of	ADP
cana-708	155	19	theorem	theorem	ADJ
cana-708	155	20	1	1	NUM
cana-708	155	21	and	and	CCONJ
cana-708	155	22	theorem	theorem	VERB
cana-708	155	23	6	6	NUM
cana-708	155	24	of	of	ADP
cana-708	155	25	muckenhoupt	muckenhoupt	ADJ
cana-708	155	26	[	[	X
cana-708	155	27	14	14	NUM
cana-708	155	28	]	]	PUNCT
cana-708	155	29	for	for	ADP
cana-708	155	30	𝑝	𝑝	NOUN
cana-708	155	31	=	=	SYM
cana-708	155	32	2	2	X
cana-708	155	33	.	.	X
cana-708	156	1	lemma	lemma	PROPN
cana-708	156	2	8	8	NUM
cana-708	157	1	[	[	X
cana-708	157	2	14	14	NUM
cana-708	157	3	]	]	PUNCT
cana-708	157	4	let	let	VERB
cana-708	157	5	𝑔	𝑔	PROPN
cana-708	157	6	∈	∈	PROPN
cana-708	157	7	𝐿𝑊(𝑣	𝐿𝑊(𝑣	NOUN
cana-708	157	8	,	,	PUNCT
cana-708	157	9	𝐵	𝐵	NOUN
cana-708	157	10	)	)	PUNCT
cana-708	157	11	2	2	NUM
cana-708	157	12	(	(	PUNCT
cana-708	157	13	ℝ	ℝ	PROPN
cana-708	157	14	)	)	PUNCT
cana-708	157	15	then	then	ADV
cana-708	157	16	,	,	PUNCT
cana-708	157	17	∫	∫	PROPN
cana-708	157	18	∞	∞	PROPN
cana-708	157	19	−∞	−∞	X
cana-708	157	20	|𝑠𝑛(𝑔	|𝑠𝑛(𝑔	X
cana-708	157	21	,	,	PUNCT
cana-708	157	22	𝑢)𝑈(𝑢	𝑢)𝑈(𝑢	PROPN
cana-708	157	23	,	,	PUNCT
cana-708	157	24	𝑏)|2𝑑𝑢	𝑏)|2𝑑𝑢	X
cana-708	157	25	≤	≤	NOUN
cana-708	157	26	𝒞	𝒞	PROPN
cana-708	157	27	∫	∫	PROPN
cana-708	157	28	∞	∞	PROPN
cana-708	157	29	−∞	−∞	ADP
cana-708	157	30	|𝑔(𝑢)𝑊(𝑢	|𝑔(𝑢)𝑊(𝑢	PROPN
cana-708	157	31	,	,	PUNCT
cana-708	157	32	𝐵)|2𝑑𝑢	𝐵)|2𝑑𝑢	PUNCT
cana-708	157	33	and	and	CCONJ
cana-708	157	34	∥	∥	NUM
cana-708	157	35	(	(	PUNCT
cana-708	157	36	𝑠𝑛(𝑢	𝑠𝑛(𝑢	ADJ
cana-708	157	37	)	)	PUNCT
cana-708	157	38	−	−	NOUN
cana-708	157	39	𝑔(𝑢))𝑈(𝑢	𝑔(𝑢))𝑈(𝑢	ADJ
cana-708	157	40	,	,	PUNCT
cana-708	157	41	𝑏	𝑏	NOUN
cana-708	157	42	)	)	PUNCT
cana-708	157	43	∥2→	∥2→	X
cana-708	157	44	0	0	X
cana-708	157	45	.	.	PUNCT
cana-708	158	1	(	(	PUNCT
cana-708	158	2	4.4	4.4	NUM
cana-708	158	3	)	)	PUNCT
cana-708	158	4	theorem	theorem	VERB
cana-708	158	5	9	9	NUM
cana-708	158	6	for	for	ADP
cana-708	158	7	all	all	DET
cana-708	158	8	measurable	measurable	ADJ
cana-708	158	9	functions	function	NOUN
cana-708	158	10	𝑔	𝑔	PROPN
cana-708	158	11	∈	∈	PROPN
cana-708	158	12	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	X
cana-708	158	13	)	)	PUNCT
cana-708	158	14	2	2	NUM
cana-708	158	15	(	(	PUNCT
cana-708	158	16	ℝ	ℝ	PROPN
cana-708	158	17	)	)	PUNCT
cana-708	158	18	,	,	PUNCT
cana-708	158	19	the	the	DET
cana-708	158	20	series	series	PROPN
cana-708	158	21	∑∞	∑∞	PROPN
cana-708	158	22	𝑘=0	𝑘=0	VERB
cana-708	158	23	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	X
cana-708	158	24	)	)	PUNCT
cana-708	158	25	converges	converge	VERB
cana-708	158	26	in	in	ADP
cana-708	158	27	mean	mean	NOUN
cana-708	158	28	to	to	ADP
cana-708	158	29	the	the	DET
cana-708	158	30	integral	integral	ADJ
cana-708	158	31	(	(	PUNCT
cana-708	158	32	4.1	4.1	NUM
cana-708	158	33	)	)	PUNCT
cana-708	158	34	.	.	PUNCT
cana-708	159	1	proof	proof	NOUN
cana-708	159	2	:	:	PUNCT
cana-708	159	3	for	for	ADP
cana-708	159	4	𝑔	𝑔	PROPN
cana-708	159	5	∈	∈	PROPN
cana-708	159	6	𝐿𝑊(𝑢,𝐵	𝐿𝑊(𝑢,𝐵	X
cana-708	159	7	)	)	PUNCT
cana-708	159	8	2	2	NUM
cana-708	159	9	(	(	PUNCT
cana-708	159	10	ℝ	ℝ	PROPN
cana-708	159	11	)	)	PUNCT
cana-708	159	12	,	,	PUNCT
cana-708	159	13	let	let	VERB
cana-708	159	14	the	the	DET
cana-708	159	15	fourier	fouri	ADJ
cana-708	159	16	hermite	hermite	ADJ
cana-708	159	17	series	series	NOUN
cana-708	159	18	expansion	expansion	NOUN
cana-708	159	19	of	of	ADP
cana-708	159	20	𝑔	𝑔	PROPN
cana-708	159	21	be	be	AUX
cana-708	159	22	∑∞	∑∞	NOUN
cana-708	159	23	𝑘=−∞	𝑘=−∞	X
cana-708	160	1	𝑑𝑘𝐻𝑘(𝑢	𝑑𝑘𝐻𝑘(𝑢	NOUN
cana-708	160	2	)	)	PUNCT
cana-708	161	1	[	[	X
cana-708	161	2	1	1	NUM
cana-708	161	3	]	]	PUNCT
cana-708	161	4	.	.	PUNCT
cana-708	162	1	let	let	VERB
cana-708	162	2	its	its	PRON
cana-708	162	3	partial	partial	ADJ
cana-708	162	4	sum	sum	NOUN
cana-708	162	5	be	be	AUX
cana-708	162	6	𝑠𝑛(𝑢	𝑠𝑛(𝑢	ADJ
cana-708	162	7	):	):	PUNCT
cana-708	162	8	=	=	PUNCT
cana-708	162	9	∑𝑛	∑𝑛	ADJ
cana-708	162	10	𝑘=0	𝑘=0	VERB
cana-708	162	11	𝑑𝑘𝐻𝑘(𝑢	𝑑𝑘𝐻𝑘(𝑢	NOUN
cana-708	162	12	)	)	PUNCT
cana-708	162	13	.	.	PUNCT
cana-708	163	1	let	let	VERB
cana-708	163	2	𝒮𝑛(𝑢	𝒮𝑛(𝑢	ADJ
cana-708	163	3	,	,	PUNCT
cana-708	163	4	𝜔	𝜔	ADJ
cana-708	163	5	)	)	PUNCT
cana-708	163	6	=	=	SYM
cana-708	164	1	∑𝑛	∑𝑛	ADJ
cana-708	164	2	𝑘=0	𝑘=0	VERB
cana-708	164	3	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	164	4	)	)	PUNCT
cana-708	164	5	be	be	VERB
cana-708	164	6	the	the	DET
cana-708	164	7	𝑛𝑡ℎ	𝑛𝑡ℎ	NUM
cana-708	164	8	partial	partial	ADJ
cana-708	164	9	sum	sum	NOUN
cana-708	164	10	of	of	ADP
cana-708	164	11	the	the	DET
cana-708	164	12	rfhs	rfhs	ADJ
cana-708	164	13	∑∞	∑∞	NOUN
cana-708	164	14	𝑘=0	𝑘=0	X
cana-708	164	15	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	164	16	)	)	PUNCT
cana-708	164	17	.	.	PUNCT
cana-708	165	1	now	now	ADV
cana-708	165	2	,	,	PUNCT
cana-708	165	3	𝒮𝑛(𝑢	𝒮𝑛(𝑢	ADJ
cana-708	165	4	,	,	PUNCT
cana-708	165	5	𝜔	𝜔	NOUN
cana-708	165	6	)	)	PUNCT
cana-708	165	7	=	=	SYM
cana-708	165	8	∑𝑛	∑𝑛	ADJ
cana-708	165	9	𝑘=0	𝑘=0	VERB
cana-708	165	10	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	𝑑𝑘𝒟𝑘(𝜔)𝐻𝑘(𝑢	NOUN
cana-708	165	11	)	)	PUNCT
cana-708	166	1	=	=	PUNCT
cana-708	167	1	∑𝑛	∑𝑛	ADJ
cana-708	167	2	𝑘=0	𝑘=0	VERB
cana-708	167	3	𝑑𝑘(∫	𝑑𝑘(∫	VERB
cana-708	167	4	∞	∞	PROPN
cana-708	167	5	−∞	−∞	X
cana-708	167	6	𝐻𝑘(𝑣)𝑈(𝑣	𝐻𝑘(𝑣)𝑈(𝑣	NOUN
cana-708	167	7	,	,	PUNCT
cana-708	167	8	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	PROPN
cana-708	167	9	,	,	PUNCT
cana-708	167	10	𝜔))𝐻𝑘(𝑢	𝜔))𝐻𝑘(𝑢	NUM
cana-708	167	11	)	)	PUNCT
cana-708	167	12	=	=	SYM
cana-708	168	1	∫	∫	PROPN
cana-708	168	2	∞	∞	PROPN
cana-708	169	1	−∞	−∞	X
cana-708	169	2	(	(	PUNCT
cana-708	169	3	∑𝑛	∑𝑛	ADJ
cana-708	169	4	𝑘=0	𝑘=0	ADJ
cana-708	169	5	𝑑𝑘𝐻𝑘(𝑣)𝐻𝑘(𝑢))𝑈(𝑣	𝑑𝑘𝐻𝑘(𝑣)𝐻𝑘(𝑢))𝑈(𝑣	NOUN
cana-708	169	6	,	,	PUNCT
cana-708	169	7	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	169	8	,	,	PUNCT
cana-708	169	9	𝜔	𝜔	NOUN
cana-708	169	10	)	)	PUNCT
cana-708	169	11	since	since	SCONJ
cana-708	169	12	,	,	PUNCT
cana-708	169	13	the	the	DET
cana-708	169	14	series	series	NOUN
cana-708	169	15	∑𝑛	∑𝑛	PROPN
cana-708	169	16	𝑘=0	𝑘=0	VERB
cana-708	169	17	𝑑𝑘𝐻𝑘(𝑥)𝐻𝑘(𝑣	𝑑𝑘𝐻𝑘(𝑥)𝐻𝑘(𝑣	NOUN
cana-708	169	18	)	)	PUNCT
cana-708	169	19	exists	exist	VERB
cana-708	169	20	,	,	PUNCT
cana-708	169	21	let	let	VERB
cana-708	169	22	𝑠𝑛(𝑢	𝑠𝑛(𝑢	ADJ
cana-708	169	23	,	,	PUNCT
cana-708	169	24	𝑣	𝑣	NOUN
cana-708	169	25	)	)	PUNCT
cana-708	169	26	be	be	AUX
cana-708	169	27	the	the	DET
cana-708	169	28	𝑛𝑡ℎ	𝑛𝑡ℎ	NUM
cana-708	169	29	partial	partial	ADJ
cana-708	169	30	sum	sum	NOUN
cana-708	169	31	of	of	ADP
cana-708	169	32	the	the	DET
cana-708	169	33	series	series	NOUN
cana-708	169	34	∑∞	∑∞	PROPN
cana-708	169	35	𝑘=0	𝑘=0	AUX
cana-708	169	36	𝑑𝑘𝐻𝑘(𝑢)𝐻𝑘(𝑣	𝑑𝑘𝐻𝑘(𝑢)𝐻𝑘(𝑣	NOUN
cana-708	169	37	)	)	PUNCT
cana-708	169	38	.	.	PUNCT
cana-708	170	1	this	this	PRON
cana-708	170	2	implies	imply	VERB
cana-708	170	3	,	,	PUNCT
cana-708	170	4	𝒮𝑛(𝑢	𝒮𝑛(𝑢	ADJ
cana-708	170	5	,	,	PUNCT
cana-708	170	6	𝜔	𝜔	NOUN
cana-708	170	7	)	)	PUNCT
cana-708	170	8	=	=	SYM
cana-708	171	1	∫	∫	PROPN
cana-708	171	2	∞	∞	PROPN
cana-708	171	3	−∞	−∞	X
cana-708	171	4	𝑠𝑛(𝑢	𝑠𝑛(𝑢	PROPN
cana-708	171	5	,	,	PUNCT
cana-708	171	6	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	171	7	,	,	PUNCT
cana-708	171	8	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	171	9	,	,	PUNCT
cana-708	171	10	𝜔	𝜔	NOUN
cana-708	171	11	)	)	PUNCT
cana-708	171	12	.	.	PUNCT
cana-708	172	1	by	by	ADP
cana-708	172	2	lemma	lemma	PROPN
cana-708	172	3	2	2	NUM
cana-708	172	4	,	,	PUNCT
cana-708	172	5	we	we	PRON
cana-708	172	6	know	know	VERB
cana-708	172	7	that	that	PRON
cana-708	172	8	,	,	PUNCT
cana-708	172	9	∫	∫	PROPN
cana-708	172	10	∞	∞	PROPN
cana-708	172	11	−∞	−∞	ADP
cana-708	172	12	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	172	13	,	,	PUNCT
cana-708	172	14	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	172	15	,	,	PUNCT
cana-708	172	16	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	172	17	,	,	PUNCT
cana-708	172	18	𝜔	𝜔	NOUN
cana-708	172	19	)	)	PUNCT
cana-708	172	20	exists	exist	VERB
cana-708	172	21	in	in	ADP
cana-708	172	22	mean	mean	NOUN
cana-708	172	23	.	.	PUNCT
cana-708	173	1	now	now	ADV
cana-708	173	2	,	,	PUNCT
cana-708	173	3	𝐸(|𝑆𝑛(𝑢	𝐸(|𝑆𝑛(𝑢	PROPN
cana-708	173	4	,	,	PUNCT
cana-708	173	5	𝜔	𝜔	NOUN
cana-708	173	6	)	)	PUNCT
cana-708	173	7	−	−	NOUN
cana-708	173	8	∫	∫	PROPN
cana-708	173	9	∞	∞	PROPN
cana-708	173	10	−∞	−∞	ADP
cana-708	173	11	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	173	12	,	,	PUNCT
cana-708	173	13	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	173	14	,	,	PUNCT
cana-708	173	15	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	PROPN
cana-708	173	16	,	,	PUNCT
cana-708	173	17	𝜔)|	𝜔)|	NOUN
cana-708	173	18	)	)	PUNCT
cana-708	173	19	=	=	SYM
cana-708	174	1	𝐸(|	𝐸(|	NUM
cana-708	174	2	∫	∫	NOUN
cana-708	174	3	∞	∞	PROPN
cana-708	174	4	−∞	−∞	X
cana-708	174	5	𝑠𝑛(𝑢	𝑠𝑛(𝑢	PROPN
cana-708	174	6	,	,	PUNCT
cana-708	174	7	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	174	8	,	,	PUNCT
cana-708	174	9	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	NOUN
cana-708	174	10	,	,	PUNCT
cana-708	174	11	𝜔	𝜔	NOUN
cana-708	174	12	)	)	PUNCT
cana-708	174	13	−	−	NOUN
cana-708	174	14	∫	∫	PROPN
cana-708	174	15	∞	∞	PROPN
cana-708	174	16	−∞	−∞	ADP
cana-708	174	17	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	174	18	,	,	PUNCT
cana-708	174	19	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	ADJ
cana-708	174	20	,	,	PUNCT
cana-708	174	21	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	PROPN
cana-708	174	22	,	,	PUNCT
cana-708	174	23	𝜔)|	𝜔)|	NOUN
cana-708	174	24	)	)	PUNCT
cana-708	174	25	=	=	SYM
cana-708	175	1	𝐸(|	𝐸(|	NUM
cana-708	175	2	∫	∫	NOUN
cana-708	175	3	∞	∞	PROPN
cana-708	176	1	−∞	−∞	X
cana-708	176	2	(	(	PUNCT
cana-708	176	3	𝑠𝑛(𝑢	𝑠𝑛(𝑢	PROPN
cana-708	176	4	,	,	PUNCT
cana-708	176	5	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	PROPN
cana-708	176	6	,	,	PUNCT
cana-708	176	7	𝑏	𝑏	NOUN
cana-708	176	8	)	)	PUNCT
cana-708	176	9	−	−	PROPN
cana-708	176	10	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	176	11	,	,	PUNCT
cana-708	176	12	𝑣)𝑈(𝑣	𝑣)𝑈(𝑣	PROPN
cana-708	176	13	,	,	PUNCT
cana-708	176	14	𝑏)𝑑𝑋(𝑣	𝑏)𝑑𝑋(𝑣	PROPN
cana-708	176	15	,	,	PUNCT
cana-708	176	16	𝜔)|	𝜔)|	NOUN
cana-708	176	17	)	)	PUNCT
cana-708	176	18	≤	≤	NOUN
cana-708	176	19	4	4	NUM
cana-708	176	20	𝜋	𝜋	NOUN
cana-708	176	21	∫	∫	PROPN
cana-708	176	22	∞	∞	PROPN
cana-708	176	23	−∞	−∞	ADP
cana-708	176	24	|(𝑠𝑛(𝑢	|(𝑠𝑛(𝑢	ADJ
cana-708	176	25	,	,	PUNCT
cana-708	176	26	𝑣	𝑣	NOUN
cana-708	176	27	)	)	PUNCT
cana-708	176	28	−	−	PROPN
cana-708	176	29	𝑔(𝑢	𝑔(𝑢	PROPN
cana-708	176	30	,	,	PUNCT
cana-708	176	31	𝑣))𝑈(𝑣	𝑣))𝑈(𝑣	PROPN
cana-708	176	32	,	,	PUNCT
cana-708	176	33	𝑏)|2𝑑𝑣	𝑏)|2𝑑𝑣	X
cana-708	176	34	+	+	CCONJ
cana-708	176	35	2	2	NUM
cana-708	176	36	𝜋	𝜋	NOUN
cana-708	176	37	∫	∫	NOUN
cana-708	176	38	|𝑠|>1	|𝑠|>1	VERB
cana-708	176	39	1	1	NUM
cana-708	176	40	−	−	PROPN
cana-708	176	41	exp(−|𝑠|2	exp(−|𝑠|2	SYM
cana-708	176	42	∫	∫	NOUN
cana-708	176	43	∞	∞	PROPN
cana-708	176	44	−∞	−∞	ADP
cana-708	176	45	|(𝑠𝑛(𝑢	|(𝑠𝑛(𝑢	ADJ
cana-708	176	46	,	,	PUNCT
cana-708	176	47	𝑣)−𝑔(𝑢	𝑣)−𝑔(𝑢	NOUN
cana-708	176	48	,	,	PUNCT
cana-708	176	49	𝑣))𝑈(𝑣	𝑣))𝑈(𝑣	PROPN
cana-708	176	50	,	,	PUNCT
cana-708	176	51	𝑏)|2𝑑𝑣	𝑏)|2𝑑𝑣	NOUN
cana-708	176	52	)	)	PUNCT
cana-708	176	53	𝑠2	𝑠2	NOUN
cana-708	176	54	𝑑𝑠	𝑑𝑠	ADP
cana-708	176	55	lemma	lemma	PROPN
cana-708	176	56	8	8	NUM
cana-708	176	57	and	and	CCONJ
cana-708	176	58	the	the	DET
cana-708	176	59	dominance	dominance	NOUN
cana-708	176	60	of	of	ADP
cana-708	176	61	1	1	NUM
cana-708	176	62	𝑠2	𝑠2	NOUN
cana-708	176	63	lead	lead	VERB
cana-708	176	64	both	both	PRON
cana-708	176	65	of	of	ADP
cana-708	176	66	these	these	DET
cana-708	176	67	integrals	integral	NOUN
cana-708	176	68	to	to	PART
cana-708	176	69	tend	tend	VERB
cana-708	176	70	to	to	ADP
cana-708	176	71	zero	zero	NUM
cana-708	176	72	.	.	PUNCT
cana-708	177	1	thus	thus	ADV
cana-708	177	2	,	,	PUNCT
cana-708	177	3	the	the	DET
cana-708	177	4	theorem	theorem	NOUN
cana-708	177	5	is	be	AUX
cana-708	177	6	established	establish	VERB
cana-708	177	7	.	.	PUNCT
cana-708	178	1	acknowledgments	acknowledgment	NOUN
cana-708	178	2	this	this	DET
cana-708	178	3	research	research	NOUN
cana-708	178	4	work	work	NOUN
cana-708	178	5	was	be	AUX
cana-708	178	6	supported	support	VERB
cana-708	178	7	by	by	ADP
cana-708	178	8	ugc	ugc	PROPN
cana-708	178	9	(	(	PUNCT
cana-708	178	10	rgnf	rgnf	PROPN
cana-708	178	11	)	)	PUNCT
cana-708	178	12	with	with	ADP
cana-708	178	13	letter	letter	NOUN
cana-708	178	14	no	no	DET
cana-708	178	15	-	-	PUNCT
cana-708	178	16	f./2015	f./2015	NOUN
cana-708	178	17	-	-	PUNCT
cana-708	178	18	16	16	NUM
cana-708	178	19	/	/	SYM
cana-708	178	20	rgnf	rgnf	NOUN
cana-708	178	21	-	-	PUNCT
cana-708	178	22	sc-2015	sc-2015	NOUN
cana-708	178	23	-	-	PUNCT
cana-708	178	24	16	16	NUM
cana-708	178	25	-	-	PUNCT
cana-708	178	26	sc	sc	NOUN
cana-708	178	27	-	-	PUNCT
cana-708	178	28	ori-20053	ori-20053	NOUN
cana-708	178	29	.	.	PUNCT
cana-708	179	1	references	reference	NOUN
cana-708	179	2	[	[	X
cana-708	179	3	1	1	NUM
cana-708	179	4	]	]	X
cana-708	179	5	askey	askey	NOUN
cana-708	179	6	,	,	PUNCT
cana-708	179	7	r.	r.	PROPN
cana-708	179	8	and	and	CCONJ
cana-708	179	9	wainger	wainger	NOUN
cana-708	179	10	,	,	PUNCT
cana-708	179	11	s.	s.	PROPN
cana-708	179	12	,	,	PUNCT
cana-708	179	13	mean	mean	VERB
cana-708	179	14	convergence	convergence	NOUN
cana-708	179	15	of	of	ADP
cana-708	179	16	expansions	expansion	NOUN
cana-708	179	17	in	in	ADP
cana-708	179	18	laguerre	laguerre	NOUN
cana-708	179	19	and	and	CCONJ
cana-708	179	20	hermite	hermite	ADJ
cana-708	179	21	series	series	NOUN
cana-708	179	22	,	,	PUNCT
cana-708	179	23	amer	amer	PROPN
cana-708	179	24	.	.	PUNCT
cana-708	180	1	j.	j.	PROPN
cana-708	180	2	math	math	PROPN
cana-708	180	3	.	.	PUNCT
cana-708	181	1	,	,	PUNCT
cana-708	181	2	87	87	NUM
cana-708	181	3	(	(	PUNCT
cana-708	181	4	1965	1965	NUM
cana-708	181	5	)	)	PUNCT
cana-708	181	6	,	,	PUNCT
cana-708	181	7	695–708	695–708	NUM
cana-708	181	8	.	.	PUNCT
cana-708	182	1	communications	communication	NOUN
cana-708	182	2	on	on	ADP
cana-708	182	3	applied	apply	VERB
cana-708	182	4	nonlinear	nonlinear	ADJ
cana-708	182	5	analysis	analysis	NOUN
cana-708	182	6	issn	issn	NOUN
cana-708	182	7	:	:	PUNCT
cana-708	182	8	1074	1074	NUM
cana-708	182	9	-	-	PUNCT
cana-708	182	10	133x	133x	NUM
cana-708	182	11	vol	vol	NOUN
cana-708	182	12	31	31	NUM
cana-708	182	13	no	no	NOUN
cana-708	182	14	.	.	PUNCT
cana-708	183	1	2s	2s	NUM
cana-708	183	2	(	(	PUNCT
cana-708	183	3	2024	2024	NUM
cana-708	183	4	)	)	PUNCT
cana-708	183	5	704	704	NUM
cana-708	183	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-708	184	1	[	[	X
cana-708	184	2	2	2	NUM
cana-708	184	3	]	]	X
cana-708	184	4	bachman	bachman	NOUN
cana-708	184	5	,	,	PUNCT
cana-708	184	6	g.	g.	PROPN
cana-708	184	7	,	,	PUNCT
cana-708	184	8	narici	narici	NOUN
cana-708	184	9	,	,	PUNCT
cana-708	184	10	l.	l.	NOUN
cana-708	184	11	and	and	CCONJ
cana-708	184	12	beckenstein	beckenstein	PROPN
cana-708	184	13	,	,	PUNCT
cana-708	184	14	e.	e.	PROPN
cana-708	184	15	,	,	PUNCT
cana-708	184	16	fourier	fourier	NOUN
cana-708	184	17	and	and	CCONJ
cana-708	184	18	wavelet	wavelet	NOUN
cana-708	184	19	analysis	analysis	NOUN
cana-708	184	20	,	,	PUNCT
cana-708	184	21	springer	springer	NOUN
cana-708	184	22	verlag	verlag	PROPN
cana-708	184	23	,	,	PUNCT
cana-708	184	24	new	new	PROPN
cana-708	184	25	york	york	PROPN
cana-708	184	26	,	,	PUNCT
cana-708	184	27	inc	inc	PROPN
cana-708	184	28	.	.	PROPN
cana-708	184	29	(	(	PUNCT
cana-708	184	30	2000	2000	NUM
cana-708	184	31	)	)	PUNCT
cana-708	184	32	.	.	PUNCT
cana-708	185	1	[	[	X
cana-708	185	2	3	3	NUM
cana-708	185	3	]	]	X
cana-708	185	4	caton	caton	PROPN
cana-708	185	5	,	,	PUNCT
cana-708	185	6	w.	w.	PROPN
cana-708	185	7	b.	b.	PROPN
cana-708	185	8	,	,	PUNCT
cana-708	185	9	hille	hille	PROPN
cana-708	185	10	,	,	PUNCT
cana-708	185	11	e.	e.	PROPN
cana-708	185	12	,	,	PUNCT
cana-708	185	13	laguerre	laguerre	NOUN
cana-708	185	14	polynomials	polynomial	NOUN
cana-708	185	15	and	and	CCONJ
cana-708	185	16	laplace	laplace	NOUN
cana-708	185	17	integrals	integral	NOUN
cana-708	185	18	,	,	PUNCT
cana-708	185	19	duke	duke	PROPN
cana-708	185	20	math	math	PROPN
cana-708	185	21	.	.	PUNCT
cana-708	186	1	j.	j.	PROPN
cana-708	186	2	12(2	12(2	PROPN
cana-708	186	3	)	)	PUNCT
cana-708	186	4	,	,	PUNCT
cana-708	186	5	(	(	PUNCT
cana-708	186	6	1945	1945	NUM
cana-708	186	7	)	)	PUNCT
cana-708	186	8	,	,	PUNCT
cana-708	186	9	217–242	217–242	NUM
cana-708	186	10	.	.	PUNCT
cana-708	187	1	[	[	X
cana-708	187	2	4	4	NUM
cana-708	187	3	]	]	X
cana-708	187	4	celeghini	celeghini	PROPN
cana-708	187	5	,	,	PUNCT
cana-708	187	6	e.	e.	PROPN
cana-708	187	7	,	,	PUNCT
cana-708	187	8	theory	theory	NOUN
cana-708	187	9	of	of	ADP
cana-708	187	10	image	image	NOUN
cana-708	187	11	and	and	CCONJ
cana-708	187	12	quantum	quantum	NOUN
cana-708	187	13	mechanics	mechanic	NOUN
cana-708	187	14	,	,	PUNCT
cana-708	187	15	a	a	DET
cana-708	187	16	common	common	ADJ
cana-708	187	17	paradigm	paradigm	NOUN
cana-708	187	18	,	,	PUNCT
cana-708	187	19	j.	j.	PROPN
cana-708	187	20	phys	phys	PROPN
cana-708	187	21	.	.	PUNCT
cana-708	187	22	conf	conf	PROPN
cana-708	187	23	.	.	PUNCT
cana-708	188	1	ser	ser	PROPN
cana-708	188	2	.	.	PROPN
cana-708	188	3	,	,	PUNCT
cana-708	188	4	626	626	NUM
cana-708	188	5	,	,	PUNCT
cana-708	188	6	(	(	PUNCT
cana-708	188	7	2015	2015	NUM
cana-708	188	8	)	)	PUNCT
cana-708	188	9	,	,	PUNCT
cana-708	188	10	012–047	012–047	NUM
cana-708	188	11	.	.	PUNCT
cana-708	189	1	[	[	X
cana-708	189	2	5	5	NUM
cana-708	189	3	]	]	X
cana-708	189	4	celeghini	celeghini	PROPN
cana-708	189	5	,	,	PUNCT
cana-708	189	6	e.	e.	PROPN
cana-708	189	7	,	,	PUNCT
cana-708	189	8	del	del	PROPN
cana-708	189	9	olmo	olmo	PROPN
cana-708	189	10	,	,	PUNCT
cana-708	189	11	m.	m.	NOUN
cana-708	189	12	a.	a.	PROPN
cana-708	189	13	,	,	PUNCT
cana-708	189	14	coherent	coherent	ADJ
cana-708	189	15	orthogonal	orthogonal	ADJ
cana-708	189	16	polynomials	polynomial	NOUN
cana-708	189	17	,	,	PUNCT
cana-708	189	18	ann	ann	PROPN
cana-708	189	19	.	.	PUNCT
cana-708	189	20	phys	phys	PROPN
cana-708	189	21	.	.	PUNCT
cana-708	189	22	,	,	PUNCT
cana-708	189	23	335	335	NUM
cana-708	189	24	,	,	PUNCT
cana-708	189	25	(	(	PUNCT
cana-708	189	26	2013	2013	NUM
cana-708	189	27	)	)	PUNCT
cana-708	189	28	,	,	PUNCT
cana-708	189	29	78–85	78–85	NUM
cana-708	189	30	.	.	PUNCT
cana-708	190	1	[	[	X
cana-708	190	2	6	6	NUM
cana-708	190	3	]	]	PUNCT
cana-708	190	4	celeghini	celeghini	PROPN
cana-708	190	5	,	,	PUNCT
cana-708	190	6	e.	e.	PROPN
cana-708	190	7	,	,	PUNCT
cana-708	190	8	del	del	PROPN
cana-708	190	9	olmo	olmo	PROPN
cana-708	190	10	,	,	PUNCT
cana-708	190	11	m.	m.	NOUN
cana-708	190	12	a.	a.	PROPN
cana-708	190	13	,	,	PUNCT
cana-708	190	14	quantum	quantum	ADJ
cana-708	190	15	physics	physics	NOUN
cana-708	190	16	and	and	CCONJ
cana-708	190	17	signal	signal	NOUN
cana-708	190	18	processing	processing	NOUN
cana-708	190	19	in	in	ADP
cana-708	190	20	rigged	rig	VERB
cana-708	190	21	hilbert	hilbert	NOUN
cana-708	190	22	spaces	space	NOUN
cana-708	190	23	by	by	ADP
cana-708	190	24	means	mean	NOUN
cana-708	190	25	of	of	ADP
cana-708	190	26	special	special	ADJ
cana-708	190	27	functions	function	NOUN
cana-708	190	28	,	,	PUNCT
cana-708	190	29	lie	lie	NOUN
cana-708	190	30	algebras	algebra	NOUN
cana-708	190	31	and	and	CCONJ
cana-708	190	32	fourier	fourier	NOUN
cana-708	190	33	and	and	CCONJ
cana-708	190	34	fourier	fourier	ADV
cana-708	190	35	-	-	PUNCT
cana-708	190	36	like	like	ADJ
cana-708	190	37	transforms	transform	NOUN
cana-708	190	38	,	,	PUNCT
cana-708	190	39	j.	j.	PROPN
cana-708	190	40	phys	phys	PROPN
cana-708	190	41	.	.	PUNCT
cana-708	190	42	conf	conf	PROPN
cana-708	190	43	.	.	PUNCT
cana-708	191	1	ser	ser	PROPN
cana-708	191	2	.	.	PROPN
cana-708	191	3	,	,	PUNCT
cana-708	191	4	597	597	NUM
cana-708	191	5	,	,	PUNCT
cana-708	191	6	(	(	PUNCT
cana-708	191	7	2015	2015	NUM
cana-708	191	8	)	)	PUNCT
cana-708	191	9	,	,	PUNCT
cana-708	191	10	012–022	012–022	NUM
cana-708	191	11	.	.	PUNCT
cana-708	192	1	[	[	X
cana-708	192	2	7	7	NUM
cana-708	192	3	]	]	X
cana-708	192	4	chow	chow	NOUN
cana-708	192	5	,	,	PUNCT
cana-708	192	6	y.	y.	PROPN
cana-708	192	7	s.	s.	PROPN
cana-708	192	8	and	and	CCONJ
cana-708	192	9	teicher	teicher	PROPN
cana-708	192	10	,	,	PUNCT
cana-708	192	11	h.	h.	PROPN
cana-708	192	12	,	,	PUNCT
cana-708	192	13	probability	probability	NOUN
cana-708	192	14	theory	theory	NOUN
cana-708	192	15	;	;	PUNCT
cana-708	192	16	independence	independence	NOUN
cana-708	192	17	,	,	PUNCT
cana-708	192	18	interchangrability	interchangrability	NOUN
cana-708	192	19	,	,	PUNCT
cana-708	192	20	martingales	martingale	NOUN
cana-708	192	21	,	,	PUNCT
cana-708	192	22	springer	springer	NOUN
cana-708	192	23	verlag	verlag	NOUN
cana-708	192	24	,	,	PUNCT
cana-708	192	25	new	new	PROPN
cana-708	192	26	york	york	PROPN
cana-708	192	27	,	,	PUNCT
cana-708	192	28	(	(	PUNCT
cana-708	192	29	1978	1978	NUM
cana-708	192	30	)	)	PUNCT
cana-708	192	31	.	.	PUNCT
cana-708	193	1	[	[	X
cana-708	193	2	8	8	NUM
cana-708	193	3	]	]	X
cana-708	193	4	indritz	indritz	PROPN
cana-708	193	5	,	,	PUNCT
cana-708	193	6	j.	j.	PROPN
cana-708	193	7	,	,	PUNCT
cana-708	193	8	an	an	DET
cana-708	193	9	inequality	inequality	NOUN
cana-708	193	10	for	for	ADP
cana-708	193	11	hermite	hermite	ADJ
cana-708	193	12	polynomials	polynomial	NOUN
cana-708	193	13	,	,	PUNCT
cana-708	193	14	proc	proc	NOUN
cana-708	193	15	.	.	PUNCT
cana-708	193	16	of	of	ADP
cana-708	193	17	the	the	DET
cana-708	193	18	amer	amer	PROPN
cana-708	193	19	.	.	PUNCT
cana-708	193	20	math	math	PROPN
cana-708	193	21	.	.	PUNCT
cana-708	194	1	soc	soc	PROPN
cana-708	194	2	.	.	PUNCT
cana-708	194	3	,	,	PUNCT
cana-708	194	4	12(6	12(6	NUM
cana-708	194	5	)	)	PUNCT
cana-708	194	6	,	,	PUNCT
cana-708	194	7	(	(	PUNCT
cana-708	194	8	1961	1961	NUM
cana-708	194	9	)	)	PUNCT
cana-708	194	10	,	,	PUNCT
cana-708	194	11	981–983	981–983	NUM
cana-708	194	12	;	;	PUNCT
cana-708	194	13	[	[	X
cana-708	194	14	9	9	NUM
cana-708	194	15	]	]	SYM
cana-708	194	16	kober	kober	PROPN
cana-708	194	17	,	,	PUNCT
cana-708	194	18	h.	h.	PROPN
cana-708	194	19	,	,	PUNCT
cana-708	194	20	a	a	DET
cana-708	194	21	note	note	NOUN
cana-708	194	22	on	on	ADP
cana-708	194	23	approximation	approximation	NOUN
cana-708	194	24	by	by	ADP
cana-708	194	25	rational	rational	ADJ
cana-708	194	26	functions	function	NOUN
cana-708	194	27	,	,	PUNCT
cana-708	194	28	proceedings	proceeding	NOUN
cana-708	194	29	of	of	ADP
cana-708	194	30	the	the	DET
cana-708	194	31	edinburgh	edinburgh	PROPN
cana-708	194	32	math	math	PROPN
cana-708	194	33	.	.	PUNCT
cana-708	195	1	soc	soc	PROPN
cana-708	195	2	.	.	PUNCT
cana-708	195	3	,	,	PUNCT
cana-708	195	4	7	7	NUM
cana-708	195	5	,	,	PUNCT
cana-708	195	6	(	(	PUNCT
cana-708	195	7	1946	1946	NUM
cana-708	195	8	)	)	PUNCT
cana-708	195	9	,	,	PUNCT
cana-708	195	10	123–133	123–133	NUM
cana-708	195	11	;	;	PUNCT
cana-708	195	12	[	[	X
cana-708	195	13	10	10	NUM
cana-708	195	14	]	]	X
cana-708	195	15	kwapien	kwapien	NOUN
cana-708	195	16	,	,	PUNCT
cana-708	195	17	s.	s.	PROPN
cana-708	195	18	and	and	CCONJ
cana-708	195	19	woyczynski	woyczynski	NOUN
cana-708	195	20	,	,	PUNCT
cana-708	195	21	w.	w.	PROPN
cana-708	195	22	a.	a.	PROPN
cana-708	195	23	,	,	PUNCT
cana-708	195	24	random	random	ADJ
cana-708	195	25	series	series	NOUN
cana-708	195	26	and	and	CCONJ
cana-708	195	27	stochastic	stochastic	ADJ
cana-708	195	28	integrals	integral	NOUN
cana-708	195	29	:	:	PUNCT
cana-708	195	30	single	single	ADJ
cana-708	195	31	and	and	CCONJ
cana-708	195	32	multiple	multiple	ADJ
cana-708	195	33	,	,	PUNCT
cana-708	195	34	birkhauser	birkhaus	ADJ
cana-708	195	35	,	,	PUNCT
cana-708	195	36	(	(	PUNCT
cana-708	195	37	1992	1992	NUM
cana-708	195	38	)	)	PUNCT
cana-708	195	39	.	.	PUNCT
cana-708	196	1	[	[	X
cana-708	196	2	11	11	NUM
cana-708	196	3	]	]	X
cana-708	196	4	liu	liu	PROPN
cana-708	196	5	,	,	PUNCT
cana-708	196	6	z.	z.	PROPN
cana-708	196	7	,	,	PUNCT
cana-708	196	8	liu	liu	PROPN
cana-708	196	9	,	,	PUNCT
cana-708	196	10	s.	s.	PROPN
cana-708	196	11	,	,	PUNCT
cana-708	196	12	randomization	randomization	NOUN
cana-708	196	13	of	of	ADP
cana-708	196	14	fourier	fouri	ADJ
cana-708	196	15	transform	transform	NOUN
cana-708	196	16	,	,	PUNCT
cana-708	196	17	opt	opt	PROPN
cana-708	196	18	.	.	PUNCT
cana-708	197	1	lett	lett	PROPN
cana-708	197	2	.	.	PUNCT
cana-708	198	1	32	32	NUM
cana-708	198	2	,	,	PUNCT
cana-708	198	3	(	(	PUNCT
cana-708	198	4	2007	2007	NUM
cana-708	198	5	)	)	PUNCT
cana-708	198	6	,	,	PUNCT
cana-708	198	7	478–480	478–480	NUM
cana-708	198	8	.	.	PUNCT
cana-708	199	1	[	[	X
cana-708	199	2	12	12	NUM
cana-708	199	3	]	]	PUNCT
cana-708	199	4	lukacs	lukacs	PROPN
cana-708	199	5	,	,	PUNCT
cana-708	199	6	e.	e.	PROPN
cana-708	199	7	,	,	PUNCT
cana-708	199	8	stochastic	stochastic	ADJ
cana-708	199	9	convergence	convergence	NOUN
cana-708	199	10	,	,	PUNCT
cana-708	199	11	second	second	ADJ
cana-708	199	12	ed	ed	NOUN
cana-708	199	13	.	.	PUNCT
cana-708	199	14	,	,	PUNCT
cana-708	199	15	academic	academic	ADJ
cana-708	199	16	press	press	NOUN
cana-708	199	17	,	,	PUNCT
cana-708	199	18	(	(	PUNCT
cana-708	199	19	1975	1975	NUM
cana-708	199	20	)	)	PUNCT
cana-708	199	21	.	.	PUNCT
cana-708	200	1	[	[	X
cana-708	200	2	13	13	NUM
cana-708	200	3	]	]	X
cana-708	200	4	muckenhoupt	muckenhoupt	PROPN
cana-708	200	5	,	,	PUNCT
cana-708	200	6	b.	b.	PROPN
cana-708	200	7	,	,	PUNCT
cana-708	200	8	mean	mean	VERB
cana-708	200	9	convergence	convergence	NOUN
cana-708	200	10	of	of	ADP
cana-708	200	11	hermite	hermite	PROPN
cana-708	200	12	and	and	CCONJ
cana-708	200	13	laguerre	laguerre	PROPN
cana-708	200	14	series	series	PROPN
cana-708	200	15	i	i	PROPN
cana-708	200	16	,	,	PUNCT
cana-708	200	17	trans	trans	PROPN
cana-708	200	18	.	.	PROPN
cana-708	201	1	amer	amer	PROPN
cana-708	201	2	.	.	PUNCT
cana-708	201	3	math	math	PROPN
cana-708	201	4	.	.	PUNCT
cana-708	202	1	soc	soc	PROPN
cana-708	202	2	.	.	PUNCT
cana-708	203	1	147	147	NUM
cana-708	203	2	,	,	PUNCT
cana-708	203	3	(	(	PUNCT
cana-708	203	4	1970	1970	NUM
cana-708	203	5	)	)	PUNCT
cana-708	203	6	,	,	PUNCT
cana-708	203	7	419–431	419–431	NUM
cana-708	203	8	;	;	PUNCT
cana-708	203	9	[	[	X
cana-708	203	10	14	14	NUM
cana-708	203	11	]	]	X
cana-708	203	12	muckenhoupt	muckenhoupt	PROPN
cana-708	203	13	,	,	PUNCT
cana-708	203	14	b.	b.	PROPN
cana-708	203	15	,	,	PUNCT
cana-708	203	16	mean	mean	VERB
cana-708	203	17	convergence	convergence	NOUN
cana-708	203	18	of	of	ADP
cana-708	203	19	hermite	hermite	PROPN
cana-708	203	20	and	and	CCONJ
cana-708	203	21	laguerre	laguerre	PROPN
cana-708	203	22	series	series	PROPN
cana-708	203	23	.	.	PUNCT
cana-708	204	1	ii	ii	PROPN
cana-708	204	2	,	,	PUNCT
cana-708	204	3	trans	trans	PROPN
cana-708	204	4	.	.	PROPN
cana-708	205	1	amer	amer	PROPN
cana-708	205	2	.	.	PUNCT
cana-708	205	3	math	math	PROPN
cana-708	205	4	.	.	PUNCT
cana-708	206	1	soc	soc	PROPN
cana-708	206	2	.	.	PUNCT
cana-708	207	1	147	147	NUM
cana-708	207	2	,	,	PUNCT
cana-708	207	3	(	(	PUNCT
cana-708	207	4	1970	1970	NUM
cana-708	207	5	)	)	PUNCT
cana-708	207	6	,	,	PUNCT
cana-708	208	1	433–460	433–460	NUM
cana-708	208	2	;	;	PUNCT
cana-708	208	3	[	[	X
cana-708	208	4	15	15	NUM
cana-708	208	5	]	]	X
cana-708	208	6	nayak	nayak	PROPN
cana-708	208	7	,	,	PUNCT
cana-708	208	8	c.	c.	NOUN
cana-708	208	9	,	,	PUNCT
cana-708	208	10	pattanayak	pattanayak	NOUN
cana-708	208	11	,	,	PUNCT
cana-708	208	12	s.	s.	PROPN
cana-708	208	13	and	and	CCONJ
cana-708	208	14	mishra	mishra	PROPN
cana-708	208	15	,	,	PUNCT
cana-708	208	16	m.	m.	NOUN
cana-708	208	17	n.	n.	PROPN
cana-708	208	18	,	,	PUNCT
cana-708	208	19	random	random	ADJ
cana-708	208	20	fourier	fourier	ADJ
cana-708	208	21	-	-	PUNCT
cana-708	208	22	stieltjes	stieltjes	NOUN
cana-708	208	23	series	series	NOUN
cana-708	208	24	associated	associate	VERB
cana-708	208	25	with	with	ADP
cana-708	208	26	stable	stable	ADJ
cana-708	208	27	process	process	NOUN
cana-708	208	28	,	,	PUNCT
cana-708	208	29	tohoku	tohoku	PROPN
cana-708	208	30	math	math	PROPN
cana-708	208	31	.	.	PUNCT
cana-708	209	1	j.	j.	PROPN
cana-708	209	2	,	,	PUNCT
cana-708	209	3	39	39	NUM
cana-708	209	4	(	(	PUNCT
cana-708	209	5	1	1	NUM
cana-708	209	6	)	)	PUNCT
cana-708	209	7	,	,	PUNCT
cana-708	209	8	(	(	PUNCT
cana-708	209	9	1987	1987	NUM
cana-708	209	10	)	)	PUNCT
cana-708	209	11	,	,	PUNCT
cana-708	209	12	1–15	1–15	NUM
cana-708	209	13	;	;	PUNCT
cana-708	209	14	[	[	X
cana-708	209	15	16	16	NUM
cana-708	209	16	]	]	PUNCT
cana-708	209	17	olver	olver	ADV
cana-708	209	18	,	,	PUNCT
cana-708	209	19	f.	f.	PROPN
cana-708	209	20	w.	w.	PROPN
cana-708	209	21	,	,	PUNCT
cana-708	209	22	lozier	lozier	PROPN
cana-708	209	23	,	,	PUNCT
cana-708	209	24	d.	d.	PROPN
cana-708	209	25	w.	w.	PROPN
cana-708	209	26	,	,	PUNCT
cana-708	209	27	boisvert	boisvert	PROPN
cana-708	209	28	,	,	PUNCT
cana-708	209	29	r.	r.	PROPN
cana-708	209	30	f.	f.	PROPN
cana-708	209	31	,	,	PUNCT
cana-708	209	32	clark	clark	PROPN
cana-708	209	33	,	,	PUNCT
cana-708	209	34	c.	c.	PROPN
cana-708	209	35	w.	w.	PROPN
cana-708	209	36	,	,	PUNCT
cana-708	209	37	nist	nist	PROPN
cana-708	209	38	handbook	handbook	NOUN
cana-708	209	39	of	of	ADP
cana-708	209	40	mathematical	mathematical	ADJ
cana-708	209	41	functions	function	NOUN
cana-708	209	42	,	,	PUNCT
cana-708	209	43	cambridge	cambridge	PROPN
cana-708	209	44	university	university	PROPN
cana-708	209	45	press	press	NOUN
cana-708	209	46	,	,	PUNCT
cana-708	209	47	newyork	newyork	PROPN
cana-708	209	48	,	,	PUNCT
cana-708	209	49	ny	ny	PROPN
cana-708	209	50	,	,	PUNCT
cana-708	209	51	usa	usa	PROPN
cana-708	209	52	,	,	PUNCT
cana-708	209	53	(	(	PUNCT
cana-708	209	54	2010	2010	NUM
cana-708	209	55	)	)	PUNCT
cana-708	209	56	.	.	PUNCT
cana-708	210	1	[	[	X
cana-708	210	2	17	17	NUM
cana-708	210	3	]	]	X
cana-708	210	4	pattanayak	pattanayak	NOUN
cana-708	210	5	,	,	PUNCT
cana-708	210	6	s.	s.	PROPN
cana-708	210	7	,	,	PUNCT
cana-708	210	8	sahoo	sahoo	PROPN
cana-708	210	9	,	,	PUNCT
cana-708	210	10	s.	s.	PROPN
cana-708	210	11	,	,	PUNCT
cana-708	210	12	on	on	ADP
cana-708	210	13	summability	summability	NOUN
cana-708	210	14	of	of	ADP
cana-708	210	15	random	random	ADJ
cana-708	210	16	fourier	fourier	ADJ
cana-708	210	17	-	-	PUNCT
cana-708	210	18	stieltjes	stieltjes	NOUN
cana-708	210	19	series	series	NOUN
cana-708	210	20	,	,	PUNCT
cana-708	210	21	j.	j.	PROPN
cana-708	210	22	int	int	PROPN
cana-708	210	23	.	.	PUNCT
cana-708	211	1	acad	acad	PROPN
cana-708	211	2	.	.	PUNCT
cana-708	212	1	phys	phy	NOUN
cana-708	212	2	.	.	PUNCT
cana-708	213	1	sci	sci	PROPN
cana-708	213	2	.	.	PROPN
cana-708	213	3	,	,	PUNCT
cana-708	213	4	9	9	NUM
cana-708	213	5	,	,	PUNCT
cana-708	213	6	(	(	PUNCT
cana-708	213	7	2005	2005	NUM
cana-708	213	8	)	)	PUNCT
cana-708	213	9	,	,	PUNCT
cana-708	213	10	9–17	9–17	NOUN
cana-708	213	11	;	;	PUNCT
cana-708	213	12	[	[	X
cana-708	213	13	18	18	NUM
cana-708	213	14	]	]	X
cana-708	213	15	pollard	pollard	PROPN
cana-708	213	16	,	,	PUNCT
cana-708	213	17	h.	h.	PROPN
cana-708	213	18	,	,	PUNCT
cana-708	213	19	the	the	DET
cana-708	213	20	mean	mean	ADJ
cana-708	213	21	convergence	convergence	NOUN
cana-708	213	22	of	of	ADP
cana-708	213	23	orthogonal	orthogonal	ADJ
cana-708	213	24	series	series	PROPN
cana-708	213	25	ii	ii	PROPN
cana-708	213	26	,	,	PUNCT
cana-708	213	27	trans	trans	PROPN
cana-708	213	28	.	.	PROPN
cana-708	214	1	amer	amer	PROPN
cana-708	214	2	.	.	PUNCT
cana-708	214	3	math	math	PROPN
cana-708	214	4	.	.	PUNCT
cana-708	215	1	soc	soc	PROPN
cana-708	215	2	.	.	PUNCT
cana-708	216	1	63	63	NUM
cana-708	216	2	(	(	PUNCT
cana-708	216	3	1948	1948	NUM
cana-708	216	4	)	)	PUNCT
cana-708	216	5	,	,	PUNCT
cana-708	216	6	355–367	355–367	NUM
cana-708	216	7	.	.	PUNCT
cana-708	217	1	[	[	X
cana-708	217	2	19	19	NUM
cana-708	217	3	]	]	PUNCT
cana-708	217	4	riesz	riesz	NOUN
cana-708	217	5	,	,	PUNCT
cana-708	217	6	m.	m.	NOUN
cana-708	217	7	,	,	PUNCT
cana-708	217	8	surles	surle	NOUN
cana-708	217	9	fonctions	fonction	NOUN
cana-708	217	10	conjuge𝑒′s	conjuge𝑒′s	PROPN
cana-708	217	11	,	,	PUNCT
cana-708	217	12	mathematische	mathematische	NOUN
cana-708	217	13	zeitschrift	zeitschrift	NOUN
cana-708	217	14	,	,	PUNCT
cana-708	217	15	vol	vol	NOUN
cana-708	217	16	27(1927	27(1927	NUM
cana-708	217	17	)	)	PUNCT
cana-708	217	18	,	,	PUNCT
cana-708	217	19	218–244	218–244	NUM
cana-708	217	20	.	.	PUNCT
cana-708	218	1	[	[	X
cana-708	218	2	20	20	NUM
cana-708	218	3	]	]	X
cana-708	218	4	rudin	rudin	PROPN
cana-708	218	5	,	,	PUNCT
cana-708	218	6	w.	w.	PROPN
cana-708	218	7	,	,	PUNCT
cana-708	218	8	real	real	ADJ
cana-708	218	9	and	and	CCONJ
cana-708	218	10	complex	complex	ADJ
cana-708	218	11	analysis	analysis	NOUN
cana-708	218	12	,	,	PUNCT
cana-708	218	13	3𝑟𝑑	3𝑟𝑑	ADJ
cana-708	218	14	edition	edition	NOUN
cana-708	218	15	,	,	PUNCT
cana-708	218	16	mcgraw	mcgraw	PROPN
cana-708	218	17	-	-	PUNCT
cana-708	218	18	hill	hill	PROPN
cana-708	218	19	,	,	PUNCT
cana-708	218	20	new	new	PROPN
cana-708	218	21	york	york	PROPN
cana-708	218	22	,	,	PUNCT
cana-708	218	23	(	(	PUNCT
cana-708	218	24	1987	1987	NUM
cana-708	218	25	)	)	PUNCT
cana-708	218	26	.	.	PUNCT
cana-708	219	1	[	[	X
cana-708	219	2	21	21	NUM
cana-708	219	3	]	]	SYM
cana-708	219	4	schauder	schauder	NOUN
cana-708	219	5	,	,	PUNCT
cana-708	219	6	j.	j.	PROPN
cana-708	219	7	,	,	PUNCT
cana-708	219	8	eine	eine	PROPN
cana-708	219	9	eigenschaft	eigenschaft	ADJ
cana-708	219	10	der	der	NOUN
cana-708	219	11	haarschen	haarschen	NOUN
cana-708	219	12	orthogonalsystem	orthogonalsystem	NOUN
cana-708	219	13	,	,	PUNCT
cana-708	219	14	ibid	ibid	NOUN
cana-708	219	15	,	,	PUNCT
cana-708	219	16	28	28	NUM
cana-708	219	17	,	,	PUNCT
cana-708	219	18	(	(	PUNCT
cana-708	219	19	1928	1928	NUM
cana-708	219	20	)	)	PUNCT
cana-708	219	21	,	,	PUNCT
cana-708	219	22	317–320	317–320	NUM
cana-708	219	23	.	.	PUNCT
cana-708	220	1	[	[	X
cana-708	220	2	22	22	NUM
cana-708	220	3	]	]	X
cana-708	220	4	shiryayev	shiryayev	PROPN
cana-708	220	5	,	,	PUNCT
cana-708	220	6	a.	a.	NOUN
cana-708	220	7	n.	n.	PROPN
cana-708	220	8	,	,	PUNCT
cana-708	220	9	probability	probability	NOUN
cana-708	220	10	,	,	PUNCT
cana-708	220	11	springer	springer	NOUN
cana-708	220	12	verlag	verlag	PROPN
cana-708	220	13	,	,	PUNCT
cana-708	220	14	new	new	PROPN
cana-708	220	15	york	york	PROPN
cana-708	220	16	,	,	PUNCT
cana-708	220	17	(	(	PUNCT
cana-708	220	18	1984	1984	NUM
cana-708	220	19	)	)	PUNCT
cana-708	220	20	.	.	PUNCT
cana-708	221	1	[	[	X
cana-708	221	2	23	23	NUM
cana-708	221	3	]	]	PUNCT
cana-708	221	4	szegő	szegő	PROPN
cana-708	221	5	,	,	PUNCT
cana-708	221	6	g.	g.	PROPN
cana-708	221	7	,	,	PUNCT
cana-708	221	8	orthogonal	orthogonal	ADJ
cana-708	221	9	polynomials	polynomial	NOUN
cana-708	221	10	,	,	PUNCT
cana-708	221	11	colloquium	colloquium	NOUN
cana-708	221	12	publications	publication	NOUN
cana-708	221	13	,	,	PUNCT
cana-708	221	14	amer	amer	PROPN
cana-708	221	15	.	.	PROPN
cana-708	221	16	math	math	PROPN
cana-708	221	17	.	.	PUNCT
cana-708	222	1	soc	soc	PROPN
cana-708	222	2	.	.	PUNCT
cana-708	223	1	23	23	NUM
cana-708	223	2	,	,	PUNCT
cana-708	223	3	(	(	PUNCT
cana-708	223	4	1939	1939	NUM
cana-708	223	5	)	)	PUNCT
cana-708	223	6	.	.	PUNCT
cana-708	224	1	[	[	X
cana-708	224	2	24	24	NUM
cana-708	224	3	]	]	PUNCT
cana-708	224	4	wiener	wiener	NOUN
cana-708	224	5	,	,	PUNCT
cana-708	224	6	n.	n.	NOUN
cana-708	224	7	,	,	PUNCT
cana-708	224	8	the	the	DET
cana-708	224	9	fourier	fourier	NOUN
cana-708	224	10	integral	integral	ADJ
cana-708	224	11	and	and	CCONJ
cana-708	224	12	certain	certain	ADJ
cana-708	224	13	of	of	ADP
cana-708	224	14	its	its	PRON
cana-708	224	15	applications	application	NOUN
cana-708	224	16	,	,	PUNCT
cana-708	224	17	dover	dover	PROPN
cana-708	224	18	,	,	PUNCT
cana-708	224	19	newyork	newyork	PROPN
cana-708	224	20	,	,	PUNCT
cana-708	224	21	(	(	PUNCT
cana-708	224	22	1933	1933	NUM
cana-708	224	23	)	)	PUNCT
cana-708	224	24	.	.	PUNCT
cana-708	225	1	[	[	X
cana-708	225	2	25	25	NUM
cana-708	225	3	]	]	PUNCT
cana-708	225	4	wing	wing	NOUN
cana-708	225	5	,	,	PUNCT
cana-708	225	6	g.	g.	PROPN
cana-708	225	7	m.	m.	PROPN
cana-708	225	8	,	,	PUNCT
cana-708	225	9	the	the	DET
cana-708	225	10	mean	mean	ADJ
cana-708	225	11	convergence	convergence	NOUN
cana-708	225	12	of	of	ADP
cana-708	225	13	orthogonal	orthogonal	ADJ
cana-708	225	14	series	series	NOUN
cana-708	225	15	,	,	PUNCT
cana-708	225	16	amer	amer	PROPN
cana-708	225	17	.	.	PUNCT
cana-708	226	1	j.	j.	PROPN
cana-708	226	2	math	math	PROPN
cana-708	226	3	.	.	PUNCT
cana-708	226	4	,	,	PUNCT
cana-708	226	5	72	72	NUM
cana-708	226	6	,	,	PUNCT
cana-708	226	7	(	(	PUNCT
cana-708	226	8	1950	1950	NUM
cana-708	226	9	)	)	PUNCT
cana-708	226	10	,	,	PUNCT
cana-708	226	11	792–808	792–808	NUM
cana-708	226	12	.	.	PUNCT