id	sid	tid	token	lemma	pos
ma-301	1	1	2025	2025	NUM
ma-301	1	2	ada	ada	PROPN
ma-301	1	3	academica	academica	PROPN
ma-301	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-301	1	5	.	.	PUNCT
ma-301	2	1	j.	j.	PROPN
ma-301	2	2	math	math	PROPN
ma-301	2	3	.	.	PUNCT
ma-301	3	1	anal	anal	ADJ
ma-301	3	2	.	.	PUNCT
ma-301	4	1	5	5	NUM
ma-301	4	2	(	(	PUNCT
ma-301	4	3	2025	2025	NUM
ma-301	4	4	)	)	PUNCT
ma-301	5	1	8doi	8doi	NUM
ma-301	5	2	:	:	PUNCT
ma-301	5	3	10.28924	10.28924	NUM
ma-301	5	4	/	/	SYM
ma-301	5	5	ada	ada	PROPN
ma-301	5	6	/	/	SYM
ma-301	5	7	ma.5.8	ma.5.8	NOUN
ma-301	5	8	uncertainty	uncertainty	NOUN
ma-301	5	9	principles	principle	NOUN
ma-301	5	10	and	and	CCONJ
ma-301	5	11	extremal	extremal	ADJ
ma-301	5	12	functions	function	NOUN
ma-301	5	13	for	for	ADP
ma-301	5	14	bessel	bessel	ADJ
ma-301	5	15	multiplier	multipli	ADJ
ma-301	5	16	operators	operator	NOUN
ma-301	5	17	in	in	ADP
ma-301	5	18	quantum	quantum	ADJ
ma-301	5	19	calculus	calculus	NOUN
ma-301	5	20	ahmed	ahmed	PROPN
ma-301	5	21	chana∗	chana∗	PROPN
ma-301	5	22	,	,	PUNCT
ma-301	5	23	abdellatif	abdellatif	NOUN
ma-301	5	24	akhlidj	akhlidj	VERB
ma-301	5	25	laboratory	laboratory	NOUN
ma-301	5	26	of	of	ADP
ma-301	5	27	fundamental	fundamental	ADJ
ma-301	5	28	and	and	CCONJ
ma-301	5	29	applied	applied	ADJ
ma-301	5	30	mathematics	mathematic	NOUN
ma-301	5	31	,	,	PUNCT
ma-301	5	32	department	department	NOUN
ma-301	5	33	of	of	ADP
ma-301	5	34	mathematics	mathematics	PROPN
ma-301	5	35	and	and	CCONJ
ma-301	5	36	informatics	informatic	NOUN
ma-301	5	37	,	,	PUNCT
ma-301	5	38	faculty	faculty	NOUN
ma-301	5	39	of	of	ADP
ma-301	5	40	sciences	science	NOUN
ma-301	5	41	ain	ain	PROPN
ma-301	5	42	chock	chock	NOUN
ma-301	5	43	,	,	PUNCT
ma-301	5	44	university	university	NOUN
ma-301	5	45	of	of	ADP
ma-301	5	46	hassan	hassan	PROPN
ma-301	5	47	ii	ii	PROPN
ma-301	5	48	,	,	PUNCT
ma-301	5	49	b.p	b.p	PROPN
ma-301	5	50	5366	5366	NUM
ma-301	5	51	maarif	maarif	PROPN
ma-301	5	52	,	,	PUNCT
ma-301	5	53	casablanca	casablanca	PROPN
ma-301	5	54	,	,	PUNCT
ma-301	5	55	morocco	morocco	PROPN
ma-301	5	56	maths.chana@gmail.com	maths.chana@gmail.com	PROPN
ma-301	5	57	,	,	PUNCT
ma-301	5	58	akhlidj@hotmail.fr	akhlidj@hotmail.fr	NOUN
ma-301	5	59	∗correspondence	∗correspondence	NOUN
ma-301	5	60	:	:	PUNCT
ma-301	5	61	maths.chana@gmail.com	maths.chana@gmail.com	X
ma-301	5	62	abstract	abstract	NOUN
ma-301	5	63	.	.	PUNCT
ma-301	6	1	using	use	VERB
ma-301	6	2	the	the	DET
ma-301	6	3	q	q	PROPN
ma-301	6	4	-	-	PUNCT
ma-301	6	5	jackson	jackson	NOUN
ma-301	6	6	integral	integral	ADJ
ma-301	6	7	and	and	CCONJ
ma-301	6	8	some	some	DET
ma-301	6	9	elements	element	NOUN
ma-301	6	10	of	of	ADP
ma-301	6	11	the	the	DET
ma-301	6	12	q	q	ADJ
ma-301	6	13	-	-	ADJ
ma-301	6	14	harmonic	harmonic	ADJ
ma-301	6	15	analysis	analysis	NOUN
ma-301	6	16	associatedwith	associatedwith	ADP
ma-301	6	17	the	the	DET
ma-301	6	18	q	q	ADJ
ma-301	6	19	-	-	PUNCT
ma-301	6	20	bessel	bessel	ADJ
ma-301	6	21	operator	operator	NOUN
ma-301	6	22	for	for	ADP
ma-301	6	23	fixed	fix	VERB
ma-301	6	24	0	0	PUNCT
ma-301	6	25	<	<	X
ma-301	6	26	q	q	X
ma-301	6	27	<	<	X
ma-301	6	28	1	1	NUM
ma-301	6	29	,	,	PUNCT
ma-301	6	30	we	we	PRON
ma-301	6	31	introduce	introduce	VERB
ma-301	6	32	the	the	DET
ma-301	6	33	q	q	ADJ
ma-301	6	34	-	-	PUNCT
ma-301	6	35	bessel	bessel	ADJ
ma-301	6	36	multiplier	multipli	ADJ
ma-301	6	37	operators	operator	NOUN
ma-301	6	38	and	and	CCONJ
ma-301	6	39	wegive	wegive	VERB
ma-301	6	40	some	some	DET
ma-301	6	41	new	new	ADJ
ma-301	6	42	results	result	NOUN
ma-301	6	43	related	relate	VERB
ma-301	6	44	to	to	ADP
ma-301	6	45	these	these	DET
ma-301	6	46	operators	operator	NOUN
ma-301	6	47	as	as	ADP
ma-301	6	48	plancherel	plancherel	PROPN
ma-301	6	49	’s	’s	PART
ma-301	6	50	,	,	PUNCT
ma-301	6	51	calderón	calderón	PROPN
ma-301	6	52	’s	’s	PART
ma-301	6	53	reproducing	reproduce	VERB
ma-301	6	54	formulas	formula	NOUN
ma-301	6	55	andheisenberg	andheisenberg	PROPN
ma-301	6	56	’s	’s	PART
ma-301	6	57	,	,	PUNCT
ma-301	6	58	donoho	donoho	NOUN
ma-301	6	59	-	-	PUNCT
ma-301	6	60	stark	stark	NOUN
ma-301	6	61	’s	’s	PART
ma-301	6	62	uncertainty	uncertainty	NOUN
ma-301	6	63	principles	principle	NOUN
ma-301	6	64	.	.	PUNCT
ma-301	7	1	next	next	ADV
ma-301	7	2	,	,	PUNCT
ma-301	7	3	using	use	VERB
ma-301	7	4	the	the	DET
ma-301	7	5	theory	theory	NOUN
ma-301	7	6	of	of	ADP
ma-301	7	7	reproducing	reproduce	VERB
ma-301	7	8	kernelswe	kernelswe	NOUN
ma-301	7	9	give	give	VERB
ma-301	7	10	best	good	ADJ
ma-301	7	11	estimates	estimate	NOUN
ma-301	7	12	and	and	CCONJ
ma-301	7	13	an	an	DET
ma-301	7	14	integral	integral	ADJ
ma-301	7	15	representation	representation	NOUN
ma-301	7	16	of	of	ADP
ma-301	7	17	the	the	DET
ma-301	7	18	extremal	extremal	ADJ
ma-301	7	19	functions	function	NOUN
ma-301	7	20	related	relate	VERB
ma-301	7	21	to	to	ADP
ma-301	7	22	theseoperators	theseoperator	NOUN
ma-301	7	23	on	on	ADP
ma-301	7	24	weighted	weight	VERB
ma-301	7	25	sobolev	sobolev	NOUN
ma-301	7	26	spaces	space	NOUN
ma-301	7	27	.	.	PUNCT
ma-301	8	1	1	1	X
ma-301	8	2	.	.	X
ma-301	8	3	introduction	introduction	NOUN
ma-301	8	4	the	the	DET
ma-301	8	5	q	q	NOUN
ma-301	8	6	-	-	NOUN
ma-301	8	7	theory	theory	NOUN
ma-301	8	8	,	,	PUNCT
ma-301	8	9	called	call	VERB
ma-301	8	10	also	also	ADV
ma-301	8	11	in	in	ADP
ma-301	8	12	some	some	DET
ma-301	8	13	literature	literature	NOUN
ma-301	8	14	quantum	quantum	NOUN
ma-301	8	15	calculus	calculus	NOUN
ma-301	8	16	began	begin	VERB
ma-301	8	17	to	to	PART
ma-301	8	18	arise	arise	VERB
ma-301	8	19	.	.	PUNCT
ma-301	9	1	interest	interest	NOUN
ma-301	9	2	in	in	ADP
ma-301	9	3	thistheory	thistheory	NOUN
ma-301	9	4	is	be	AUX
ma-301	9	5	grown	grow	VERB
ma-301	9	6	at	at	ADP
ma-301	9	7	an	an	DET
ma-301	9	8	explosive	explosive	ADJ
ma-301	9	9	note	note	NOUN
ma-301	9	10	by	by	ADP
ma-301	9	11	both	both	DET
ma-301	9	12	physicists	physicist	NOUN
ma-301	9	13	and	and	CCONJ
ma-301	9	14	mathematicians	mathematician	NOUN
ma-301	9	15	due	due	ADJ
ma-301	9	16	to	to	ADP
ma-301	9	17	a	a	DET
ma-301	9	18	large	large	ADJ
ma-301	9	19	numberof	numberof	NOUN
ma-301	9	20	its	its	PRON
ma-301	9	21	application	application	NOUN
ma-301	9	22	domains	domain	NOUN
ma-301	9	23	,	,	PUNCT
ma-301	9	24	for	for	SCONJ
ma-301	9	25	more	more	ADJ
ma-301	9	26	information	information	NOUN
ma-301	9	27	about	about	ADP
ma-301	9	28	quantum	quantum	NOUN
ma-301	9	29	calculus	calculus	NOUN
ma-301	9	30	one	one	PRON
ma-301	9	31	can	can	AUX
ma-301	9	32	see	see	VERB
ma-301	9	33	[	[	X
ma-301	9	34	20].recently	20].recently	ADV
ma-301	9	35	,	,	PUNCT
ma-301	9	36	many	many	ADJ
ma-301	9	37	reasercher	reasercher	ADV
ma-301	9	38	have	have	AUX
ma-301	9	39	been	be	AUX
ma-301	9	40	investigated	investigate	VERB
ma-301	9	41	the	the	DET
ma-301	9	42	behavior	behavior	NOUN
ma-301	9	43	of	of	ADP
ma-301	9	44	the	the	DET
ma-301	9	45	q	q	NOUN
ma-301	9	46	-	-	NOUN
ma-301	9	47	theory	theory	NOUN
ma-301	9	48	to	to	ADP
ma-301	9	49	several	several	ADJ
ma-301	9	50	alreadystudied	alreadystudie	VERB
ma-301	9	51	for	for	ADP
ma-301	9	52	the	the	DET
ma-301	9	53	fourier	fourier	ADJ
ma-301	9	54	analysis	analysis	NOUN
ma-301	9	55	,	,	PUNCT
ma-301	9	56	for	for	ADP
ma-301	9	57	example	example	NOUN
ma-301	9	58	sampling	sample	VERB
ma-301	9	59	theorem	theorem	NOUN
ma-301	9	60	[	[	X
ma-301	9	61	2	2	NUM
ma-301	9	62	]	]	PUNCT
ma-301	9	63	,	,	PUNCT
ma-301	9	64	paley	paley	ADJ
ma-301	9	65	-	-	PUNCT
ma-301	9	66	wiener	wiener	NOUN
ma-301	9	67	theorem	theorem	NOUN
ma-301	9	68	[	[	PUNCT
ma-301	9	69	1],uncertainty	1],uncertainty	NUM
ma-301	9	70	principles	principle	NOUN
ma-301	9	71	[	[	X
ma-301	9	72	31	31	NUM
ma-301	9	73	]	]	PUNCT
ma-301	9	74	,	,	PUNCT
ma-301	9	75	wavelet	wavelet	NOUN
ma-301	9	76	transform	transform	NOUN
ma-301	9	77	[	[	X
ma-301	9	78	15	15	NUM
ma-301	9	79	]	]	PUNCT
ma-301	9	80	,	,	PUNCT
ma-301	9	81	wavelet	wavelet	NOUN
ma-301	9	82	packet	packet	NOUN
ma-301	10	1	[	[	X
ma-301	10	2	6	6	NUM
ma-301	10	3	]	]	PUNCT
ma-301	10	4	,	,	PUNCT
ma-301	10	5	ramanujan	ramanujan	PROPN
ma-301	10	6	master	master	PROPN
ma-301	10	7	theorem[16	theorem[16	PROPN
ma-301	10	8	]	]	X
ma-301	10	9	,	,	PUNCT
ma-301	10	10	sobolev	sobolev	ADJ
ma-301	10	11	type	type	NOUN
ma-301	10	12	spaces	space	NOUN
ma-301	11	1	[	[	X
ma-301	11	2	27	27	NUM
ma-301	11	3	]	]	PUNCT
ma-301	11	4	and	and	CCONJ
ma-301	11	5	wave	wave	NOUN
ma-301	11	6	equation	equation	NOUN
ma-301	11	7	[	[	X
ma-301	11	8	29	29	NUM
ma-301	11	9	]	]	PUNCT
ma-301	11	10	.	.	PUNCT
ma-301	12	1	in	in	ADP
ma-301	12	2	their	their	PRON
ma-301	12	3	seminal	seminal	ADJ
ma-301	12	4	papers	paper	NOUN
ma-301	12	5	,	,	PUNCT
ma-301	12	6	hörmander	hörmander	NOUN
ma-301	12	7	’s	’s	PART
ma-301	12	8	andmikhlin	andmikhlin	PROPN
ma-301	12	9	’s	’s	PART
ma-301	13	1	[	[	X
ma-301	13	2	18,25	18,25	NUM
ma-301	13	3	]	]	PUNCT
ma-301	13	4	initiated	initiate	VERB
ma-301	13	5	the	the	DET
ma-301	13	6	study	study	NOUN
ma-301	13	7	of	of	ADP
ma-301	13	8	boundedness	boundedness	NOUN
ma-301	13	9	of	of	ADP
ma-301	13	10	the	the	DET
ma-301	13	11	translation	translation	NOUN
ma-301	13	12	invariant	invariant	PROPN
ma-301	13	13	operators	operator	NOUN
ma-301	13	14	on	on	ADP
ma-301	13	15	rd	rd	PROPN
ma-301	13	16	.	.	PUNCT
ma-301	14	1	thetranslation	thetranslation	PROPN
ma-301	14	2	invariant	invariant	ADJ
ma-301	14	3	operators	operator	NOUN
ma-301	14	4	on	on	ADP
ma-301	14	5	rd	rd	NOUN
ma-301	14	6	characterized	characterize	VERB
ma-301	14	7	using	use	VERB
ma-301	14	8	the	the	DET
ma-301	14	9	classical	classical	ADJ
ma-301	14	10	euclidean	euclidean	ADJ
ma-301	14	11	fourier	fourier	NOUN
ma-301	14	12	transform	transform	VERB
ma-301	14	13	f(f	f(f	PROPN
ma-301	14	14	)	)	PUNCT
ma-301	14	15	therefore	therefore	ADV
ma-301	14	16	they	they	PRON
ma-301	14	17	also	also	ADV
ma-301	14	18	known	know	VERB
ma-301	14	19	as	as	ADP
ma-301	14	20	fourier	fourier	NOUN
ma-301	14	21	multipliers	multiplier	NOUN
ma-301	14	22	.	.	PUNCT
ma-301	15	1	given	give	VERB
ma-301	15	2	a	a	DET
ma-301	15	3	measurable	measurable	ADJ
ma-301	15	4	function	function	NOUN
ma-301	15	5	m	m	VERB
ma-301	15	6	:	:	PUNCT
ma-301	15	7	rd	rd	AUX
ma-301	15	8	−→	−→	NOUN
ma-301	15	9	c	c	VERB
ma-301	15	10	its	its	PRON
ma-301	15	11	fourier	fourier	NOUN
ma-301	15	12	multiplier	multiplier	ADV
ma-301	15	13	is	be	AUX
ma-301	15	14	the	the	DET
ma-301	15	15	linear	linear	ADJ
ma-301	15	16	map	map	NOUN
ma-301	15	17	tm	tm	NOUN
ma-301	15	18	given	give	VERB
ma-301	15	19	for	for	ADP
ma-301	15	20	all	all	DET
ma-301	15	21	λ	λ	PROPN
ma-301	15	22	∈	∈	PROPN
ma-301	15	23	rd	rd	NOUN
ma-301	15	24	by	by	ADP
ma-301	15	25	the	the	DET
ma-301	15	26	relation	relation	NOUN
ma-301	15	27	f(tm(f	f(tm(f	PROPN
ma-301	15	28	)	)	PUNCT
ma-301	15	29	)	)	PUNCT
ma-301	15	30	(	(	PUNCT
ma-301	15	31	λ	λ	NOUN
ma-301	15	32	)	)	PUNCT
ma-301	15	33	=	=	SYM
ma-301	16	1	m(λ)f(f	m(λ)f(f	NOUN
ma-301	16	2	)	)	PUNCT
ma-301	16	3	(	(	PUNCT
ma-301	16	4	λ	λ	X
ma-301	16	5	)	)	PUNCT
ma-301	16	6	(	(	PUNCT
ma-301	16	7	1.1	1.1	NUM
ma-301	16	8	)	)	PUNCT
ma-301	16	9	received	receive	VERB
ma-301	16	10	:	:	PUNCT
ma-301	16	11	22	22	NUM
ma-301	16	12	nov	nov	PROPN
ma-301	16	13	2024	2024	NUM
ma-301	16	14	.	.	PUNCT
ma-301	17	1	key	key	ADJ
ma-301	17	2	words	word	NOUN
ma-301	17	3	and	and	CCONJ
ma-301	17	4	phrases	phrase	NOUN
ma-301	17	5	.	.	PUNCT
ma-301	18	1	quantum	quantum	NOUN
ma-301	18	2	calculus	calculus	NOUN
ma-301	18	3	;	;	PUNCT
ma-301	18	4	q	q	ADJ
ma-301	18	5	-	-	PUNCT
ma-301	18	6	bessel	bessel	ADJ
ma-301	18	7	transform	transform	NOUN
ma-301	18	8	;	;	PUNCT
ma-301	18	9	calderón	calderón	PROPN
ma-301	18	10	’s	’s	PART
ma-301	18	11	reproducing	reproduce	VERB
ma-301	18	12	formulas	formula	NOUN
ma-301	18	13	;	;	PUNCT
ma-301	18	14	extremal	extremal	PROPN
ma-301	18	15	functions;heisenberg	functions;heisenberg	PROPN
ma-301	18	16	’s	’s	PART
ma-301	18	17	uncertainty	uncertainty	NOUN
ma-301	18	18	principle	principle	NOUN
ma-301	18	19	;	;	PUNCT
ma-301	18	20	approximation	approximation	NOUN
ma-301	18	21	theory	theory	NOUN
ma-301	18	22	;	;	PUNCT
ma-301	18	23	sobolev	sobolev	NOUN
ma-301	18	24	spaces	space	VERB
ma-301	18	25	.	.	PUNCT
ma-301	19	1	1	1	NUM
ma-301	19	2	https://adac.ee	https://adac.ee	PROPN
ma-301	19	3	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	19	4	eur	eur	PROPN
ma-301	19	5	.	.	PUNCT
ma-301	20	1	j.	j.	PROPN
ma-301	20	2	math	math	PROPN
ma-301	20	3	.	.	PUNCT
ma-301	21	1	anal	anal	PROPN
ma-301	21	2	.	.	PUNCT
ma-301	22	1	10.28924	10.28924	NUM
ma-301	22	2	/	/	SYM
ma-301	22	3	ada	ada	PROPN
ma-301	22	4	/	/	PROPN
ma-301	22	5	ma.5.8	ma.5.8	VERB
ma-301	22	6	2the	2the	PROPN
ma-301	22	7	hörmander	hörmander	NOUN
ma-301	22	8	-	-	PUNCT
ma-301	22	9	mikhlin	mikhlin	NOUN
ma-301	22	10	fundamental	fundamental	ADJ
ma-301	22	11	condition	condition	NOUN
ma-301	22	12	gives	give	VERB
ma-301	22	13	a	a	DET
ma-301	22	14	criterion	criterion	NOUN
ma-301	22	15	for	for	ADP
ma-301	22	16	lp	lp	NOUN
ma-301	22	17	-	-	PUNCT
ma-301	22	18	boundedness	boundedness	NOUN
ma-301	22	19	for	for	ADP
ma-301	22	20	all	all	DET
ma-301	22	21	1	1	NUM
ma-301	22	22	<	<	X
ma-301	22	23	p	p	X
ma-301	22	24	<	<	X
ma-301	22	25	∞	∞	NOUN
ma-301	22	26	of	of	ADP
ma-301	22	27	fourier	fourier	NOUN
ma-301	22	28	multiplier	multipli	ADJ
ma-301	22	29	tm	tm	NOUN
ma-301	22	30	in	in	ADP
ma-301	22	31	terms	term	NOUN
ma-301	22	32	of	of	ADP
ma-301	22	33	derivatives	derivative	NOUN
ma-301	22	34	of	of	ADP
ma-301	22	35	the	the	DET
ma-301	22	36	symbol	symbol	NOUN
ma-301	22	37	m	m	NOUN
ma-301	22	38	,	,	PUNCT
ma-301	22	39	more	more	ADV
ma-301	22	40	precisely	precisely	ADV
ma-301	22	41	if∣∣∂γλm(λ	if∣∣∂γλm(λ	NOUN
ma-301	22	42	)	)	PUNCT
ma-301	22	43	∣∣	∣∣	NUM
ma-301	22	44	.	.	PUNCT
ma-301	23	1	|λ|−|γ|	|λ|−|γ|	VERB
ma-301	23	2	f	f	NOUN
ma-301	23	3	or	or	CCONJ
ma-301	23	4	0	0	NUM
ma-301	23	5	≤	≤	NOUN
ma-301	24	1	|γ|	|γ|	VERB
ma-301	24	2	≤	≤	NOUN
ma-301	24	3	[	[	PUNCT
ma-301	24	4	d	d	NOUN
ma-301	24	5	2	2	NUM
ma-301	24	6	]	]	PUNCT
ma-301	24	7	+	+	CCONJ
ma-301	24	8	1	1	X
ma-301	24	9	.	.	PUNCT
ma-301	24	10	(	(	PUNCT
ma-301	24	11	1.2	1.2	NUM
ma-301	24	12	)	)	PUNCT
ma-301	24	13	then	then	ADV
ma-301	24	14	,	,	PUNCT
ma-301	24	15	tm	tm	PROPN
ma-301	24	16	can	can	AUX
ma-301	24	17	be	be	AUX
ma-301	24	18	extended	extend	VERB
ma-301	24	19	to	to	ADP
ma-301	24	20	a	a	DET
ma-301	24	21	bounded	bounded	ADJ
ma-301	24	22	linear	linear	ADJ
ma-301	24	23	operator	operator	NOUN
ma-301	24	24	from	from	ADP
ma-301	24	25	lp(rd	lp(rd	ADJ
ma-301	24	26	)	)	PUNCT
ma-301	24	27	into	into	ADP
ma-301	24	28	itself	itself	PRON
ma-301	25	1	.the	.the	DET
ma-301	25	2	condition	condition	NOUN
ma-301	25	3	(	(	PUNCT
ma-301	25	4	1.2	1.2	NUM
ma-301	25	5	)	)	PUNCT
ma-301	25	6	imposes	impose	VERB
ma-301	25	7	m	m	NOUN
ma-301	25	8	to	to	PART
ma-301	25	9	be	be	AUX
ma-301	25	10	a	a	DET
ma-301	25	11	bounded	bounded	ADJ
ma-301	25	12	function	function	NOUN
ma-301	25	13	,	,	PUNCT
ma-301	25	14	smooth	smooth	VERB
ma-301	25	15	over	over	ADP
ma-301	25	16	rd\{0	rd\{0	NOUN
ma-301	25	17	}	}	PUNCT
ma-301	25	18	satisfying	satisfy	VERB
ma-301	25	19	certainlocal	certainlocal	ADJ
ma-301	25	20	and	and	CCONJ
ma-301	25	21	asymptotic	asymptotic	ADJ
ma-301	25	22	behavior	behavior	NOUN
ma-301	25	23	.	.	PUNCT
ma-301	26	1	locally	locally	ADV
ma-301	26	2	,	,	PUNCT
ma-301	26	3	m	m	PROPN
ma-301	26	4	admits	admit	VERB
ma-301	26	5	a	a	DET
ma-301	26	6	singularity	singularity	NOUN
ma-301	26	7	at	at	ADP
ma-301	26	8	0	0	NUM
ma-301	26	9	with	with	ADP
ma-301	26	10	a	a	DET
ma-301	26	11	mild	mild	ADJ
ma-301	26	12	control	control	NOUN
ma-301	26	13	of	of	ADP
ma-301	26	14	deriva	deriva	NOUN
ma-301	26	15	-	-	PUNCT
ma-301	26	16	tives	tive	NOUN
ma-301	26	17	around	around	ADP
ma-301	26	18	it	it	PRON
ma-301	26	19	up	up	ADP
ma-301	26	20	to	to	PART
ma-301	26	21	order	order	VERB
ma-301	26	22	[	[	X
ma-301	26	23	d2	d2	X
ma-301	26	24	]	]	PUNCT
ma-301	27	1	+	+	CCONJ
ma-301	28	1	1	1	X
ma-301	28	2	.	.	X
ma-301	29	1	this	this	DET
ma-301	29	2	singularity	singularity	NOUN
ma-301	29	3	links	link	NOUN
ma-301	29	4	to	to	ADP
ma-301	29	5	deep	deep	ADJ
ma-301	29	6	concepts	concept	NOUN
ma-301	29	7	in	in	ADP
ma-301	29	8	harmonic	harmonic	ADJ
ma-301	29	9	analysisand	analysisand	PROPN
ma-301	29	10	justifies	justify	VERB
ma-301	29	11	the	the	DET
ma-301	29	12	key	key	ADJ
ma-301	29	13	role	role	NOUN
ma-301	29	14	of	of	ADP
ma-301	29	15	hörmander	hörmander	NOUN
ma-301	29	16	-	-	PUNCT
ma-301	29	17	mikhlin	mikhlin	NOUN
ma-301	29	18	theorem	theorem	NOUN
ma-301	29	19	in	in	ADP
ma-301	29	20	fourier	fourier	NOUN
ma-301	29	21	multiplier	multipli	ADJ
ma-301	29	22	lp	lp	NOUN
ma-301	29	23	-	-	NOUN
ma-301	29	24	theory	theory	NOUN
ma-301	29	25	,	,	PUNCT
ma-301	29	26	this	this	DET
ma-301	29	27	con	con	NOUN
ma-301	29	28	-	-	PUNCT
ma-301	29	29	dition	dition	NOUN
ma-301	29	30	defines	define	VERB
ma-301	29	31	a	a	DET
ma-301	29	32	large	large	ADJ
ma-301	29	33	class	class	NOUN
ma-301	29	34	of	of	ADP
ma-301	29	35	fourier	fourier	NOUN
ma-301	29	36	multipliers	multiplier	NOUN
ma-301	29	37	including	include	VERB
ma-301	29	38	riesz	riesz	PROPN
ma-301	29	39	transforms	transform	NOUN
ma-301	29	40	and	and	CCONJ
ma-301	29	41	littelwood	littelwood	NOUN
ma-301	29	42	-	-	PUNCT
ma-301	29	43	paleypartitions	paleypartition	NOUN
ma-301	29	44	of	of	ADP
ma-301	29	45	unity	unity	NOUN
ma-301	29	46	which	which	PRON
ma-301	29	47	are	be	AUX
ma-301	29	48	crucial	crucial	ADJ
ma-301	29	49	in	in	ADP
ma-301	29	50	fourier	fourier	NOUN
ma-301	29	51	summability	summability	NOUN
ma-301	29	52	or	or	CCONJ
ma-301	29	53	pseudo	pseudo	NOUN
ma-301	29	54	-	-	NOUN
ma-301	29	55	differential	differential	NOUN
ma-301	29	56	operator.theboundedness	operator.theboundedness	NOUN
ma-301	29	57	of	of	ADP
ma-301	29	58	fourier	fourier	NOUN
ma-301	29	59	multipliers	multiplier	NOUN
ma-301	29	60	is	be	AUX
ma-301	29	61	useful	useful	ADJ
ma-301	29	62	to	to	PART
ma-301	29	63	solve	solve	VERB
ma-301	29	64	problems	problem	NOUN
ma-301	29	65	in	in	ADP
ma-301	29	66	the	the	DET
ma-301	29	67	area	area	NOUN
ma-301	29	68	of	of	ADP
ma-301	29	69	mathematical	mathematical	ADJ
ma-301	29	70	analysisas	analysisas	PROPN
ma-301	29	71	probability	probability	PROPN
ma-301	29	72	theory	theory	NOUN
ma-301	29	73	see	see	VERB
ma-301	29	74	[	[	X
ma-301	29	75	24	24	NUM
ma-301	29	76	]	]	PUNCT
ma-301	29	77	,	,	PUNCT
ma-301	29	78	stochastic	stochastic	ADJ
ma-301	29	79	processus	processus	NOUN
ma-301	29	80	see	see	VERB
ma-301	29	81	[	[	X
ma-301	29	82	5	5	NUM
ma-301	29	83	]	]	PUNCT
ma-301	29	84	,	,	PUNCT
ma-301	29	85	and	and	CCONJ
ma-301	29	86	the	the	DET
ma-301	29	87	study	study	NOUN
ma-301	29	88	of	of	ADP
ma-301	29	89	nonlinear	nonlinear	ADJ
ma-301	29	90	partialdifferential	partialdifferential	ADJ
ma-301	29	91	equations	equation	NOUN
ma-301	29	92	see	see	VERB
ma-301	29	93	[	[	X
ma-301	29	94	22	22	NUM
ma-301	29	95	]	]	PUNCT
ma-301	29	96	.	.	PUNCT
ma-301	30	1	for	for	ADP
ma-301	30	2	its	its	PRON
ma-301	30	3	importance	importance	NOUN
ma-301	30	4	many	many	ADJ
ma-301	30	5	researcher	researcher	NOUN
ma-301	30	6	extend	extend	VERB
ma-301	30	7	the	the	DET
ma-301	30	8	theory	theory	NOUN
ma-301	30	9	of	of	ADP
ma-301	30	10	fouriermultiplier	fouriermultiplier	NOUN
ma-301	30	11	to	to	ADP
ma-301	30	12	different	different	ADJ
ma-301	30	13	setting	setting	NOUN
ma-301	30	14	for	for	ADP
ma-301	30	15	example	example	NOUN
ma-301	30	16	in	in	ADP
ma-301	30	17	the	the	DET
ma-301	30	18	dunkl	dunkl	PROPN
ma-301	30	19	-	-	PUNCT
ma-301	30	20	weinstein	weinstein	PROPN
ma-301	30	21	setting	set	VERB
ma-301	30	22	[	[	X
ma-301	30	23	33	33	NUM
ma-301	30	24	]	]	PUNCT
ma-301	30	25	,	,	PUNCT
ma-301	30	26	in	in	ADP
ma-301	30	27	the	the	DET
ma-301	30	28	laguerre	laguerre	NOUN
ma-301	30	29	-	-	PUNCT
ma-301	30	30	bessel	bessel	NOUN
ma-301	30	31	setting	set	VERB
ma-301	30	32	[	[	X
ma-301	30	33	8	8	NUM
ma-301	30	34	]	]	PUNCT
ma-301	30	35	,	,	PUNCT
ma-301	30	36	in	in	ADP
ma-301	30	37	the	the	DET
ma-301	30	38	q	q	ADJ
ma-301	30	39	-	-	PUNCT
ma-301	30	40	fourier	fourier	NOUN
ma-301	30	41	setting	setting	NOUN
ma-301	30	42	[	[	X
ma-301	30	43	26	26	NUM
ma-301	30	44	,	,	PUNCT
ma-301	30	45	31	31	NUM
ma-301	30	46	,	,	PUNCT
ma-301	30	47	32	32	NUM
ma-301	30	48	]	]	PUNCT
ma-301	30	49	and	and	CCONJ
ma-301	30	50	the	the	DET
ma-301	30	51	q	q	ADJ
ma-301	30	52	-	-	ADJ
ma-301	30	53	cosine	cosine	ADJ
ma-301	30	54	fourier	fourier	NOUN
ma-301	30	55	setting	set	VERB
ma-301	30	56	[	[	X
ma-301	30	57	3	3	NUM
ma-301	30	58	]	]	PUNCT
ma-301	30	59	.	.	PUNCT
ma-301	31	1	thegeneral	thegeneral	ADJ
ma-301	31	2	theory	theory	NOUN
ma-301	31	3	of	of	ADP
ma-301	31	4	reproducing	reproduce	VERB
ma-301	31	5	kernels	kernel	NOUN
ma-301	31	6	is	be	AUX
ma-301	31	7	stared	stare	VERB
ma-301	31	8	with	with	ADP
ma-301	31	9	aronszajn	aronszajn	PROPN
ma-301	31	10	’s	’s	PART
ma-301	31	11	in	in	ADP
ma-301	31	12	[	[	X
ma-301	31	13	4	4	X
ma-301	31	14	]	]	PUNCT
ma-301	31	15	in	in	ADP
ma-301	31	16	1950	1950	NUM
ma-301	31	17	,	,	PUNCT
ma-301	31	18	next	next	ADP
ma-301	31	19	the	the	DET
ma-301	31	20	authorsin	authorsin	NOUN
ma-301	31	21	[	[	X
ma-301	31	22	23	23	NUM
ma-301	31	23	,	,	PUNCT
ma-301	31	24	30	30	NUM
ma-301	31	25	]	]	PUNCT
ma-301	31	26	applied	apply	VERB
ma-301	31	27	this	this	DET
ma-301	31	28	theory	theory	NOUN
ma-301	31	29	to	to	PART
ma-301	31	30	study	study	VERB
ma-301	31	31	tikhonov	tikhonov	NOUN
ma-301	31	32	regularization	regularization	NOUN
ma-301	31	33	problem	problem	NOUN
ma-301	31	34	and	and	CCONJ
ma-301	31	35	they	they	PRON
ma-301	31	36	obtained	obtain	VERB
ma-301	31	37	ap	ap	PROPN
ma-301	31	38	-	-	PUNCT
ma-301	31	39	proximate	proximate	NOUN
ma-301	31	40	solutions	solution	NOUN
ma-301	31	41	for	for	ADP
ma-301	31	42	bounded	bounded	ADJ
ma-301	31	43	linear	linear	ADJ
ma-301	31	44	operator	operator	NOUN
ma-301	31	45	equations	equation	NOUN
ma-301	31	46	on	on	ADP
ma-301	31	47	hilbert	hilbert	PROPN
ma-301	31	48	spaces	space	NOUN
ma-301	31	49	with	with	ADP
ma-301	31	50	the	the	DET
ma-301	31	51	viewpoint	viewpoint	NOUN
ma-301	31	52	ofnumerical	ofnumerical	ADJ
ma-301	31	53	solutions	solution	NOUN
ma-301	31	54	by	by	ADP
ma-301	31	55	computers	computer	NOUN
ma-301	31	56	.	.	PUNCT
ma-301	32	1	this	this	DET
ma-301	32	2	theory	theory	NOUN
ma-301	32	3	has	have	AUX
ma-301	32	4	gained	gain	VERB
ma-301	32	5	considerable	considerable	ADJ
ma-301	32	6	interest	interest	NOUN
ma-301	32	7	in	in	ADP
ma-301	32	8	various	various	ADJ
ma-301	32	9	field	field	NOUN
ma-301	32	10	ofmathematical	ofmathematical	ADJ
ma-301	32	11	sciences	science	NOUN
ma-301	32	12	especially	especially	ADV
ma-301	32	13	in	in	ADP
ma-301	32	14	engineering	engineering	NOUN
ma-301	32	15	and	and	CCONJ
ma-301	32	16	numerical	numerical	ADJ
ma-301	32	17	experiments	experiment	NOUN
ma-301	32	18	by	by	ADP
ma-301	32	19	using	use	VERB
ma-301	32	20	computerssee	computerssee	NOUN
ma-301	32	21	[	[	X
ma-301	32	22	30].this	30].this	NUM
ma-301	32	23	paper	paper	NOUN
ma-301	32	24	focuses	focus	VERB
ma-301	32	25	on	on	ADP
ma-301	32	26	the	the	DET
ma-301	32	27	generalized	generalized	ADJ
ma-301	32	28	fourier	fourier	NOUN
ma-301	32	29	transform	transform	NOUN
ma-301	32	30	associated	associate	VERB
ma-301	32	31	with	with	ADP
ma-301	32	32	the	the	DET
ma-301	32	33	q	q	ADJ
ma-301	32	34	-	-	PUNCT
ma-301	32	35	bessel	bessel	NOUN
ma-301	32	36	operatorcalled	operatorcalle	VERB
ma-301	32	37	the	the	DET
ma-301	32	38	q	q	ADJ
ma-301	32	39	-	-	PUNCT
ma-301	32	40	bessel	bessel	ADJ
ma-301	32	41	transform	transform	NOUN
ma-301	32	42	introduced	introduce	VERB
ma-301	32	43	in	in	ADP
ma-301	32	44	[	[	X
ma-301	32	45	11	11	NUM
ma-301	32	46	]	]	PUNCT
ma-301	32	47	,	,	PUNCT
ma-301	32	48	more	more	ADV
ma-301	32	49	precisely	precisely	ADV
ma-301	32	50	we	we	PRON
ma-301	32	51	define	define	VERB
ma-301	32	52	the	the	DET
ma-301	32	53	following	follow	VERB
ma-301	32	54	q	q	NOUN
ma-301	32	55	-	-	PUNCT
ma-301	32	56	differentialoperator	differentialoperator	NOUN
ma-301	32	57	for	for	ADP
ma-301	32	58	0	0	NUM
ma-301	32	59	<	<	X
ma-301	32	60	q	q	X
ma-301	32	61	<	<	X
ma-301	32	62	1	1	NUM
ma-301	32	63	by	by	ADP
ma-301	32	64	∆q	∆q	PROPN
ma-301	32	65	,	,	PUNCT
ma-301	32	66	αf	αf	ADP
ma-301	32	67	(	(	PUNCT
ma-301	32	68	x	x	X
ma-301	32	69	)	)	PUNCT
ma-301	33	1	=	=	SYM
ma-301	33	2	f	f	PROPN
ma-301	33	3	(	(	PUNCT
ma-301	33	4	q−1x	q−1x	NOUN
ma-301	33	5	)	)	PUNCT
ma-301	33	6	−	−	PROPN
ma-301	34	1	(	(	PUNCT
ma-301	34	2	1	1	NUM
ma-301	34	3	+	+	CCONJ
ma-301	34	4	q2α	q2α	PROPN
ma-301	34	5	)	)	PUNCT
ma-301	35	1	f	f	X
ma-301	35	2	(	(	PUNCT
ma-301	35	3	x	x	X
ma-301	35	4	)	)	PUNCT
ma-301	35	5	+	+	CCONJ
ma-301	35	6	q2αf	q2αf	ADJ
ma-301	35	7	(	(	PUNCT
ma-301	35	8	qx	qx	PROPN
ma-301	35	9	)	)	PUNCT
ma-301	35	10	x2	x2	NOUN
ma-301	35	11	,	,	PUNCT
ma-301	35	12	∀x	∀x	X
ma-301	35	13	6=	6=	ADP
ma-301	35	14	0	0	NUM
ma-301	35	15	.	.	PUNCT
ma-301	36	1	(	(	PUNCT
ma-301	36	2	1.3	1.3	NUM
ma-301	36	3	)	)	PUNCT
ma-301	36	4	the	the	DET
ma-301	36	5	eigenfunctions	eigenfunction	NOUN
ma-301	36	6	of	of	ADP
ma-301	36	7	the	the	DET
ma-301	36	8	operator	operator	NOUN
ma-301	36	9	(	(	PUNCT
ma-301	36	10	1.3	1.3	NUM
ma-301	36	11	)	)	PUNCT
ma-301	36	12	are	be	AUX
ma-301	36	13	related	relate	VERB
ma-301	36	14	to	to	ADP
ma-301	36	15	the	the	DET
ma-301	36	16	hahn	hahn	NOUN
ma-301	36	17	-	-	PUNCT
ma-301	36	18	exton	exton	NOUN
ma-301	36	19	q	q	ADJ
ma-301	36	20	-	-	PUNCT
ma-301	36	21	bessel	bessel	ADJ
ma-301	36	22	function	function	NOUN
ma-301	36	23	jα(x	jα(x	NOUN
ma-301	36	24	;	;	PUNCT
ma-301	36	25	q2)defined	q2)define	VERB
ma-301	36	26	in	in	ADP
ma-301	36	27	[	[	X
ma-301	36	28	15	15	NUM
ma-301	36	29	]	]	PUNCT
ma-301	36	30	.	.	PUNCT
ma-301	37	1	the	the	DET
ma-301	37	2	q	q	ADJ
ma-301	37	3	-	-	PUNCT
ma-301	37	4	bessel	bessel	NOUN
ma-301	37	5	tranform	tranform	NOUN
ma-301	37	6	hq	hq	PROPN
ma-301	37	7	,	,	PUNCT
ma-301	37	8	α	α	PROPN
ma-301	37	9	is	be	AUX
ma-301	37	10	defined	define	VERB
ma-301	37	11	on	on	ADP
ma-301	37	12	l1α(r+q	l1α(r+q	PROPN
ma-301	37	13	)	)	PUNCT
ma-301	37	14	by	by	ADP
ma-301	37	15	hq	hq	PROPN
ma-301	37	16	,	,	PUNCT
ma-301	37	17	α(f	α(f	PROPN
ma-301	37	18	)	)	PUNCT
ma-301	37	19	(	(	PUNCT
ma-301	37	20	λ	λ	X
ma-301	37	21	)	)	PUNCT
ma-301	37	22	=	=	SYM
ma-301	38	1	∫	∫	PROPN
ma-301	39	1	∞	∞	NOUN
ma-301	39	2	0	0	NUM
ma-301	39	3	jα(λx	jα(λx	NOUN
ma-301	39	4	;	;	PUNCT
ma-301	39	5	q2)f	q2)f	NOUN
ma-301	39	6	(	(	PUNCT
ma-301	39	7	x)dµq	x)dµq	PROPN
ma-301	39	8	,	,	PUNCT
ma-301	39	9	α(x	α(x	NOUN
ma-301	39	10	)	)	PUNCT
ma-301	39	11	,	,	PUNCT
ma-301	39	12	for	for	ADP
ma-301	39	13	λ	λ	PROPN
ma-301	39	14	∈	∈	PROPN
ma-301	39	15	r+q	r+q	PROPN
ma-301	39	16	where	where	SCONJ
ma-301	39	17	dµq	dµq	NOUN
ma-301	39	18	,	,	PUNCT
ma-301	39	19	α	α	PROPN
ma-301	39	20	is	be	AUX
ma-301	39	21	the	the	DET
ma-301	39	22	measure	measure	NOUN
ma-301	39	23	on	on	ADP
ma-301	39	24	r+q	r+q	PROPN
ma-301	39	25	given	give	VERB
ma-301	39	26	later	later	ADV
ma-301	39	27	.	.	PUNCT
ma-301	40	1	let	let	VERB
ma-301	40	2	σ	σ	NOUN
ma-301	40	3	be	be	AUX
ma-301	40	4	a	a	DET
ma-301	40	5	function	function	NOUN
ma-301	40	6	in	in	ADP
ma-301	40	7	l2α(r+q	l2α(r+q	PROPN
ma-301	40	8	)	)	PUNCT
ma-301	40	9	and	and	CCONJ
ma-301	40	10	β	β	X
ma-301	40	11	∈	∈	PROPN
ma-301	40	12	r+q	r+q	PROPN
ma-301	40	13	,	,	PUNCT
ma-301	40	14	theq	theq	NOUN
ma-301	40	15	-	-	PUNCT
ma-301	40	16	bessel	bessel	ADJ
ma-301	40	17	l2α	l2α	ADJ
ma-301	40	18	-	-	ADJ
ma-301	40	19	multiplier	multipli	ADJ
ma-301	40	20	operators	operator	NOUN
ma-301	40	21	are	be	AUX
ma-301	40	22	defined	define	VERB
ma-301	40	23	for	for	ADP
ma-301	40	24	smooth	smooth	ADJ
ma-301	40	25	function	function	NOUN
ma-301	40	26	f	f	PROPN
ma-301	40	27	on	on	ADP
ma-301	40	28	r+q	r+q	PROPN
ma-301	40	29	as	as	ADP
ma-301	40	30	mq	mq	PROPN
ma-301	40	31	,	,	PUNCT
ma-301	40	32	σ	σ	PROPN
ma-301	40	33	,	,	PUNCT
ma-301	40	34	β(f	β(f	PROPN
ma-301	40	35	)	)	PUNCT
ma-301	40	36	(	(	PUNCT
ma-301	40	37	x	x	X
ma-301	40	38	)	)	PUNCT
ma-301	40	39	:	:	PUNCT
ma-301	41	1	=	=	SYM
ma-301	41	2	h−1q	h−1q	PROPN
ma-301	41	3	,	,	PUNCT
ma-301	41	4	α	α	PROPN
ma-301	41	5	(	(	PUNCT
ma-301	41	6	σβhq	σβhq	PROPN
ma-301	41	7	,	,	PUNCT
ma-301	41	8	α(f	α(f	PROPN
ma-301	41	9	)	)	PUNCT
ma-301	41	10	)	)	PUNCT
ma-301	42	1	(	(	PUNCT
ma-301	42	2	x	x	X
ma-301	42	3	)	)	PUNCT
ma-301	42	4	(	(	PUNCT
ma-301	42	5	1.4	1.4	NUM
ma-301	42	6	)	)	PUNCT
ma-301	42	7	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	42	8	eur	eur	PROPN
ma-301	42	9	.	.	PUNCT
ma-301	43	1	j.	j.	PROPN
ma-301	43	2	math	math	PROPN
ma-301	43	3	.	.	PUNCT
ma-301	44	1	anal	anal	PROPN
ma-301	44	2	.	.	PUNCT
ma-301	45	1	10.28924	10.28924	NUM
ma-301	45	2	/	/	SYM
ma-301	45	3	ada	ada	PROPN
ma-301	45	4	/	/	SYM
ma-301	45	5	ma.5.8	ma.5.8	PROPN
ma-301	45	6	3where	3where	NUM
ma-301	45	7	the	the	DET
ma-301	45	8	function	function	NOUN
ma-301	45	9	σβ	σβ	NOUN
ma-301	45	10	is	be	AUX
ma-301	45	11	given	give	VERB
ma-301	45	12	by	by	ADP
ma-301	45	13	σβ(λ	σβ(λ	NOUN
ma-301	45	14	)	)	PUNCT
ma-301	45	15	:	:	PUNCT
ma-301	45	16	=	=	PUNCT
ma-301	45	17	σ(λβ	σ(λβ	PROPN
ma-301	45	18	)	)	PUNCT
ma-301	45	19	.	.	PUNCT
ma-301	46	1	(	(	PUNCT
ma-301	46	2	1.5)these	1.5)these	NUM
ma-301	46	3	operators	operator	NOUN
ma-301	46	4	are	be	AUX
ma-301	46	5	a	a	DET
ma-301	46	6	generalization	generalization	NOUN
ma-301	46	7	of	of	ADP
ma-301	46	8	all	all	DET
ma-301	46	9	classical	classical	ADJ
ma-301	46	10	multiplier	multipli	ADJ
ma-301	46	11	operators	operator	NOUN
ma-301	46	12	introduced	introduce	VERB
ma-301	46	13	in	in	ADP
ma-301	46	14	[	[	X
ma-301	46	15	3	3	NUM
ma-301	46	16	,	,	PUNCT
ma-301	46	17	10,26	10,26	NUM
ma-301	46	18	,	,	PUNCT
ma-301	46	19	31	31	NUM
ma-301	46	20	,	,	PUNCT
ma-301	46	21	32	32	NUM
ma-301	46	22	]	]	PUNCT
ma-301	46	23	.	.	PUNCT
ma-301	47	1	the	the	DET
ma-301	47	2	remainder	remainder	NOUN
ma-301	47	3	of	of	ADP
ma-301	47	4	this	this	DET
ma-301	47	5	paper	paper	NOUN
ma-301	47	6	is	be	AUX
ma-301	47	7	arranged	arrange	VERB
ma-301	47	8	as	as	SCONJ
ma-301	47	9	follows	follow	VERB
ma-301	47	10	,	,	PUNCT
ma-301	47	11	in	in	ADP
ma-301	47	12	section	section	NOUN
ma-301	47	13	2	2	NUM
ma-301	47	14	we	we	PRON
ma-301	47	15	recall	recall	VERB
ma-301	47	16	the	the	DET
ma-301	47	17	mainresults	mainresult	NOUN
ma-301	47	18	concerning	concern	VERB
ma-301	47	19	the	the	DET
ma-301	47	20	harmonic	harmonic	ADJ
ma-301	47	21	analysis	analysis	NOUN
ma-301	47	22	associated	associate	VERB
ma-301	47	23	with	with	ADP
ma-301	47	24	the	the	DET
ma-301	47	25	q	q	ADJ
ma-301	47	26	-	-	PUNCT
ma-301	47	27	bessel	bessel	ADJ
ma-301	47	28	transform	transform	NOUN
ma-301	47	29	,	,	PUNCT
ma-301	47	30	in	in	ADP
ma-301	47	31	section	section	NOUN
ma-301	47	32	3,we	3,we	NUM
ma-301	47	33	introduce	introduce	VERB
ma-301	47	34	the	the	DET
ma-301	47	35	q	q	ADJ
ma-301	47	36	-	-	PUNCT
ma-301	47	37	bessel	bessel	ADJ
ma-301	47	38	l2α	l2α	NOUN
ma-301	47	39	-	-	ADJ
ma-301	47	40	multiplier	multipli	ADJ
ma-301	47	41	operators	operator	NOUN
ma-301	47	42	mq	mq	PROPN
ma-301	47	43	,	,	PUNCT
ma-301	47	44	σ	σ	PROPN
ma-301	47	45	,	,	PUNCT
ma-301	47	46	β	β	X
ma-301	47	47	and	and	CCONJ
ma-301	47	48	we	we	PRON
ma-301	47	49	give	give	VERB
ma-301	47	50	for	for	ADP
ma-301	47	51	them	they	PRON
ma-301	47	52	a	a	DET
ma-301	47	53	plancherel’s	plancherel’s	NOUN
ma-301	47	54	,	,	PUNCT
ma-301	47	55	pointwise	pointwise	ADJ
ma-301	47	56	reproducing	reproduce	VERB
ma-301	47	57	formulas	formula	NOUN
ma-301	47	58	and	and	CCONJ
ma-301	47	59	heisenberg	heisenberg	PROPN
ma-301	47	60	’s	’s	PROPN
ma-301	47	61	,	,	PUNCT
ma-301	47	62	donoho	donoho	NOUN
ma-301	47	63	-	-	PUNCT
ma-301	47	64	stark	stark	NOUN
ma-301	47	65	’s	’s	PART
ma-301	47	66	uncertainty	uncertainty	NOUN
ma-301	47	67	principles	principle	NOUN
ma-301	47	68	.	.	PUNCT
ma-301	48	1	thelast	thelast	ADJ
ma-301	48	2	section	section	NOUN
ma-301	48	3	of	of	ADP
ma-301	48	4	this	this	DET
ma-301	48	5	paper	paper	NOUN
ma-301	48	6	is	be	AUX
ma-301	48	7	devoted	devote	VERB
ma-301	48	8	to	to	PART
ma-301	48	9	give	give	VERB
ma-301	48	10	an	an	DET
ma-301	48	11	application	application	NOUN
ma-301	48	12	of	of	ADP
ma-301	48	13	the	the	DET
ma-301	48	14	general	general	ADJ
ma-301	48	15	theory	theory	NOUN
ma-301	48	16	of	of	ADP
ma-301	48	17	reproducingkernels	reproducingkernel	NOUN
ma-301	48	18	to	to	ADP
ma-301	48	19	q	q	ADJ
ma-301	48	20	-	-	PUNCT
ma-301	48	21	bessel	bessel	ADJ
ma-301	48	22	multiplier	multipli	ADJ
ma-301	48	23	theory	theory	NOUN
ma-301	48	24	and	and	CCONJ
ma-301	48	25	to	to	PART
ma-301	48	26	give	give	VERB
ma-301	48	27	best	good	ADJ
ma-301	48	28	estimates	estimate	NOUN
ma-301	48	29	and	and	CCONJ
ma-301	48	30	an	an	DET
ma-301	48	31	integral	integral	ADJ
ma-301	48	32	representation	representation	NOUN
ma-301	48	33	ofthe	ofthe	NOUN
ma-301	48	34	extremal	extremal	ADJ
ma-301	48	35	functions	function	NOUN
ma-301	48	36	related	relate	VERB
ma-301	48	37	to	to	ADP
ma-301	48	38	the	the	DET
ma-301	48	39	q	q	ADJ
ma-301	48	40	-	-	PUNCT
ma-301	48	41	bessel	bessel	ADJ
ma-301	48	42	l2α	l2α	NOUN
ma-301	48	43	-	-	ADJ
ma-301	48	44	multiplier	multipli	ADJ
ma-301	48	45	operatorsmq	operatorsmq	NOUN
ma-301	48	46	,	,	PUNCT
ma-301	48	47	σ	σ	PROPN
ma-301	48	48	,	,	PUNCT
ma-301	48	49	β	β	X
ma-301	48	50	on	on	ADP
ma-301	48	51	weighted	weight	VERB
ma-301	48	52	sobolevspaces	sobolevspace	NOUN
ma-301	48	53	.	.	PUNCT
ma-301	49	1	2	2	X
ma-301	49	2	.	.	X
ma-301	49	3	harmonic	harmonic	ADJ
ma-301	49	4	analysis	analysis	NOUN
ma-301	49	5	associated	associate	VERB
ma-301	49	6	with	with	ADP
ma-301	49	7	the	the	DET
ma-301	49	8	q	q	ADJ
ma-301	49	9	-	-	PUNCT
ma-301	49	10	bessel	bessel	ADJ
ma-301	49	11	transform	transform	NOUN
ma-301	49	12	in	in	ADP
ma-301	49	13	this	this	DET
ma-301	49	14	section	section	NOUN
ma-301	49	15	we	we	PRON
ma-301	49	16	set	set	VERB
ma-301	49	17	some	some	DET
ma-301	49	18	notations	notation	NOUN
ma-301	49	19	and	and	CCONJ
ma-301	49	20	we	we	PRON
ma-301	49	21	recall	recall	VERB
ma-301	49	22	some	some	DET
ma-301	49	23	results	result	NOUN
ma-301	49	24	in	in	ADP
ma-301	49	25	harmonic	harmonic	ADJ
ma-301	49	26	analysis	analysis	NOUN
ma-301	49	27	related	relate	VERB
ma-301	49	28	tothe	tothe	DET
ma-301	49	29	q	q	ADJ
ma-301	49	30	-	-	PUNCT
ma-301	49	31	bessel	bessel	ADJ
ma-301	49	32	operator	operator	NOUN
ma-301	49	33	(	(	PUNCT
ma-301	49	34	1.3	1.3	NUM
ma-301	49	35	)	)	PUNCT
ma-301	49	36	,	,	PUNCT
ma-301	49	37	all	all	DET
ma-301	49	38	these	these	DET
ma-301	49	39	results	result	NOUN
ma-301	49	40	can	can	AUX
ma-301	49	41	be	be	AUX
ma-301	49	42	founded	found	VERB
ma-301	49	43	in	in	ADP
ma-301	49	44	[	[	X
ma-301	49	45	11,17,19–21,28	11,17,19–21,28	NUM
ma-301	49	46	]	]	X
ma-301	49	47	.	.	PUNCT
ma-301	50	1	2.1	2.1	NUM
ma-301	50	2	.	.	PUNCT
ma-301	50	3	notations	notation	NOUN
ma-301	50	4	and	and	CCONJ
ma-301	50	5	preliminaries	preliminary	NOUN
ma-301	50	6	.	.	PUNCT
ma-301	51	1	in	in	ADP
ma-301	51	2	this	this	DET
ma-301	51	3	subsection	subsection	NOUN
ma-301	51	4	,	,	PUNCT
ma-301	51	5	we	we	PRON
ma-301	51	6	give	give	VERB
ma-301	51	7	some	some	DET
ma-301	51	8	notations	notation	NOUN
ma-301	51	9	,	,	PUNCT
ma-301	51	10	definitions	definition	NOUN
ma-301	51	11	and	and	CCONJ
ma-301	51	12	prop	prop	NOUN
ma-301	51	13	-	-	PUNCT
ma-301	51	14	erties	ertie	NOUN
ma-301	51	15	of	of	ADP
ma-301	51	16	the	the	DET
ma-301	51	17	q	q	ADJ
ma-301	51	18	-	-	PUNCT
ma-301	51	19	shifted	shift	VERB
ma-301	51	20	factorial	factorial	NOUN
ma-301	51	21	,	,	PUNCT
ma-301	51	22	the	the	DET
ma-301	51	23	jackson	jackson	PROPN
ma-301	51	24	’s	’s	PART
ma-301	51	25	q	q	NOUN
ma-301	51	26	-	-	PUNCT
ma-301	51	27	derivatives	derivative	NOUN
ma-301	51	28	and	and	CCONJ
ma-301	51	29	the	the	DET
ma-301	51	30	jackson	jackson	PROPN
ma-301	51	31	’s	’s	PART
ma-301	51	32	q	q	NOUN
ma-301	51	33	-	-	PUNCT
ma-301	51	34	integrals	integral	NOUN
ma-301	51	35	introducedin	introducedin	VERB
ma-301	51	36	[	[	X
ma-301	51	37	19].let	19].let	NUM
ma-301	51	38	a	a	DET
ma-301	51	39	∈	∈	PROPN
ma-301	51	40	c	c	NOUN
ma-301	51	41	,	,	PUNCT
ma-301	51	42	the	the	DET
ma-301	51	43	q	q	ADJ
ma-301	51	44	-	-	PUNCT
ma-301	51	45	shifted	shift	VERB
ma-301	51	46	factorial	factorial	NOUN
ma-301	51	47	are	be	AUX
ma-301	51	48	defined	define	VERB
ma-301	51	49	by	by	ADP
ma-301	51	50	:	:	PUNCT
ma-301	51	51	(	(	PUNCT
ma-301	51	52	a	a	X
ma-301	51	53	;	;	PUNCT
ma-301	51	54	q)0	q)0	PROPN
ma-301	51	55	=	=	SYM
ma-301	51	56	1	1	NUM
ma-301	51	57	,	,	PUNCT
ma-301	51	58	(	(	PUNCT
ma-301	51	59	a	a	X
ma-301	51	60	;	;	PUNCT
ma-301	51	61	q)n	q)n	SYM
ma-301	51	62	=	=	SYM
ma-301	51	63	n−1∏	n−1∏	PROPN
ma-301	51	64	k=0	k=0	PROPN
ma-301	51	65	(	(	PUNCT
ma-301	51	66	1−	1−	NUM
ma-301	51	67	aqk	aqk	NOUN
ma-301	51	68	)	)	PUNCT
ma-301	51	69	,	,	PUNCT
ma-301	51	70	(	(	PUNCT
ma-301	51	71	a	a	X
ma-301	51	72	;	;	PUNCT
ma-301	51	73	q)∞	q)∞	ADJ
ma-301	51	74	=	=	SYM
ma-301	51	75	∞∏	∞∏	X
ma-301	51	76	k=0	k=0	PROPN
ma-301	51	77	(	(	PUNCT
ma-301	51	78	1−	1−	NUM
ma-301	51	79	aqk	aqk	NOUN
ma-301	51	80	)	)	PUNCT
ma-301	51	81	.	.	PUNCT
ma-301	52	1	the	the	DET
ma-301	52	2	jackson	jackson	PROPN
ma-301	52	3	’s	’s	PART
ma-301	52	4	q	q	NOUN
ma-301	52	5	-	-	NOUN
ma-301	52	6	derivative	derivative	NOUN
ma-301	52	7	of	of	ADP
ma-301	52	8	a	a	DET
ma-301	52	9	function	function	NOUN
ma-301	52	10	f	f	NOUN
ma-301	52	11	is	be	AUX
ma-301	52	12	given	give	VERB
ma-301	52	13	by	by	ADP
ma-301	52	14	dqf	dqf	NOUN
ma-301	52	15	(	(	PUNCT
ma-301	52	16	x	x	NOUN
ma-301	52	17	)	)	PUNCT
ma-301	52	18	=	=	SYM
ma-301	52	19	f	f	PROPN
ma-301	52	20	(	(	PUNCT
ma-301	52	21	x)−	x)−	PROPN
ma-301	52	22	f	f	PROPN
ma-301	52	23	(	(	PUNCT
ma-301	52	24	qx	qx	PROPN
ma-301	52	25	)	)	PUNCT
ma-301	52	26	(	(	PUNCT
ma-301	52	27	1−	1−	NUM
ma-301	52	28	q)x	q)x	ADJ
ma-301	52	29	if	if	SCONJ
ma-301	52	30	x	x	PROPN
ma-301	52	31	6=	6=	ADP
ma-301	52	32	0.the	0.the	DET
ma-301	52	33	q	q	PROPN
ma-301	52	34	-	-	PUNCT
ma-301	52	35	jackson	jackson	PROPN
ma-301	52	36	’s	’s	PART
ma-301	52	37	integrals	integral	NOUN
ma-301	52	38	from	from	ADP
ma-301	52	39	0	0	NUM
ma-301	52	40	to	to	ADP
ma-301	52	41	a	a	PRON
ma-301	52	42	and	and	CCONJ
ma-301	52	43	from	from	ADP
ma-301	52	44	0	0	NUM
ma-301	52	45	to	to	ADP
ma-301	52	46	∞	∞	PROPN
ma-301	52	47	are	be	AUX
ma-301	52	48	defined	define	VERB
ma-301	52	49	by∫	by∫	PRON
ma-301	52	50	a	a	DET
ma-301	52	51	0	0	NUM
ma-301	52	52	f	f	NOUN
ma-301	52	53	(	(	PUNCT
ma-301	52	54	x)dqx	x)dqx	PROPN
ma-301	52	55	=	=	SYM
ma-301	52	56	(	(	PUNCT
ma-301	52	57	1−	1−	NUM
ma-301	52	58	q)a	q)a	NOUN
ma-301	52	59	∞∑	∞∑	NUM
ma-301	52	60	0	0	NUM
ma-301	52	61	f	f	NOUN
ma-301	52	62	(	(	PUNCT
ma-301	52	63	aqn	aqn	ADJ
ma-301	52	64	)	)	PUNCT
ma-301	52	65	qn	qn	PROPN
ma-301	52	66	,	,	PUNCT
ma-301	52	67	∫	∫	PROPN
ma-301	52	68	∞	∞	PROPN
ma-301	52	69	0	0	NUM
ma-301	53	1	f	f	PROPN
ma-301	53	2	(	(	PUNCT
ma-301	53	3	x)dqx	x)dqx	PROPN
ma-301	53	4	=	=	SYM
ma-301	53	5	(	(	PUNCT
ma-301	53	6	1−	1−	NUM
ma-301	53	7	q	q	NOUN
ma-301	53	8	)	)	PUNCT
ma-301	53	9	∞∑	∞∑	NUM
ma-301	53	10	n=−∞	n=−∞	NUM
ma-301	53	11	f	f	X
ma-301	53	12	(	(	PUNCT
ma-301	53	13	qn	qn	PROPN
ma-301	53	14	)	)	PUNCT
ma-301	53	15	qn	qn	NOUN
ma-301	53	16	.	.	PROPN
ma-301	53	17	provided	provide	VERB
ma-301	53	18	the	the	DET
ma-301	53	19	sums	sum	NOUN
ma-301	53	20	converge	converge	VERB
ma-301	53	21	absolutely	absolutely	ADV
ma-301	53	22	.	.	PUNCT
ma-301	54	1	the	the	DET
ma-301	54	2	normalized	normalize	VERB
ma-301	54	3	form	form	NOUN
ma-301	54	4	of	of	ADP
ma-301	54	5	the	the	DET
ma-301	54	6	q	q	ADJ
ma-301	54	7	-	-	PUNCT
ma-301	54	8	bessel	bessel	ADJ
ma-301	54	9	kernel	kernel	NOUN
ma-301	54	10	is	be	AUX
ma-301	54	11	defined	define	VERB
ma-301	54	12	in	in	ADP
ma-301	54	13	[	[	X
ma-301	54	14	14,17,28	14,17,28	NUM
ma-301	54	15	]	]	PUNCT
ma-301	54	16	by	by	ADP
ma-301	54	17	jα(x	jα(x	NOUN
ma-301	54	18	;	;	PUNCT
ma-301	54	19	q2	q2	NOUN
ma-301	54	20	)	)	PUNCT
ma-301	55	1	=	=	PUNCT
ma-301	56	1	∞∑	∞∑	NUM
ma-301	56	2	n=0	n=0	NUM
ma-301	56	3	(	(	PUNCT
ma-301	56	4	−1)n	−1)n	PROPN
ma-301	56	5	q	q	PROPN
ma-301	56	6	n(n+1	n(n+1	PROPN
ma-301	56	7	)	)	PUNCT
ma-301	56	8	2	2	NUM
ma-301	56	9	(	(	PUNCT
ma-301	56	10	qα+1	qα+1	NOUN
ma-301	56	11	;	;	PUNCT
ma-301	56	12	q)n	q)n	X
ma-301	56	13	(	(	PUNCT
ma-301	56	14	q	q	NOUN
ma-301	56	15	;	;	PUNCT
ma-301	56	16	q)n	q)n	X
ma-301	56	17	x2n	x2n	PROPN
ma-301	56	18	.	.	PUNCT
ma-301	57	1	(	(	PUNCT
ma-301	57	2	2.1	2.1	NUM
ma-301	57	3	)	)	PUNCT
ma-301	57	4	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	57	5	eur	eur	PROPN
ma-301	57	6	.	.	PUNCT
ma-301	58	1	j.	j.	PROPN
ma-301	58	2	math	math	PROPN
ma-301	58	3	.	.	PUNCT
ma-301	59	1	anal	anal	PROPN
ma-301	59	2	.	.	PUNCT
ma-301	60	1	10.28924	10.28924	NUM
ma-301	60	2	/	/	SYM
ma-301	60	3	ada	ada	PROPN
ma-301	60	4	/	/	SYM
ma-301	60	5	ma.5.8	ma.5.8	PROPN
ma-301	60	6	4it	4it	NOUN
ma-301	60	7	satisfies	satisfy	VERB
ma-301	60	8	the	the	DET
ma-301	60	9	following	follow	VERB
ma-301	60	10	estimate	estimate	NOUN
ma-301	60	11	[	[	X
ma-301	60	12	11	11	NUM
ma-301	60	13	]	]	PUNCT
ma-301	60	14	∀x	∀x	PUNCT
ma-301	60	15	∈	∈	PROPN
ma-301	60	16	r+q	r+q	PROPN
ma-301	60	17	,	,	PUNCT
ma-301	60	18	∣∣jα(x	∣∣jα(x	PROPN
ma-301	60	19	;	;	PUNCT
ma-301	60	20	q2	q2	PROPN
ma-301	60	21	)	)	PUNCT
ma-301	60	22	∣∣	∣∣	NUM
ma-301	60	23	≤	≤	NUM
ma-301	60	24	1	1	NUM
ma-301	60	25	.	.	PUNCT
ma-301	61	1	(	(	PUNCT
ma-301	61	2	2.2	2.2	NUM
ma-301	61	3	)	)	PUNCT
ma-301	61	4	2.2	2.2	NUM
ma-301	61	5	.	.	PUNCT
ma-301	62	1	the	the	DET
ma-301	62	2	q	q	ADJ
ma-301	62	3	-	-	PUNCT
ma-301	62	4	bessel	bessel	ADJ
ma-301	62	5	transform	transform	NOUN
ma-301	62	6	.	.	PUNCT
ma-301	63	1	in	in	ADP
ma-301	63	2	this	this	DET
ma-301	63	3	section	section	NOUN
ma-301	63	4	,	,	PUNCT
ma-301	63	5	we	we	PRON
ma-301	63	6	define	define	VERB
ma-301	63	7	and	and	CCONJ
ma-301	63	8	give	give	VERB
ma-301	63	9	some	some	DET
ma-301	63	10	basic	basic	ADJ
ma-301	63	11	properties	property	NOUN
ma-301	63	12	of	of	ADP
ma-301	63	13	q	q	NOUN
ma-301	63	14	-	-	PUNCT
ma-301	63	15	besseltransform	besseltransform	NOUN
ma-301	63	16	introduced	introduce	VERB
ma-301	63	17	in	in	ADP
ma-301	63	18	[	[	X
ma-301	63	19	11	11	NUM
ma-301	63	20	]	]	PUNCT
ma-301	63	21	.	.	PUNCT
ma-301	64	1	we	we	PRON
ma-301	64	2	first	first	ADV
ma-301	64	3	introduced	introduce	VERB
ma-301	64	4	the	the	DET
ma-301	64	5	following	follow	VERB
ma-301	64	6	spaces	space	NOUN
ma-301	64	7	and	and	CCONJ
ma-301	64	8	norms•	norms•	ADV
ma-301	64	9	c0,q(r+q	c0,q(r+q	PROPN
ma-301	64	10	)	)	PUNCT
ma-301	64	11	denotes	denote	VERB
ma-301	64	12	the	the	DET
ma-301	64	13	set	set	NOUN
ma-301	64	14	of	of	ADP
ma-301	64	15	all	all	DET
ma-301	64	16	functions	function	NOUN
ma-301	64	17	defined	define	VERB
ma-301	64	18	on	on	ADP
ma-301	64	19	r+q	r+q	PROPN
ma-301	64	20	continuous	continuous	ADJ
ma-301	64	21	at	at	ADP
ma-301	64	22	zero	zero	NUM
ma-301	64	23	and	and	CCONJ
ma-301	64	24	vanishing	vanish	VERB
ma-301	64	25	atinfinity	atinfinity	NOUN
ma-301	64	26	,	,	PUNCT
ma-301	64	27	equiped	equiped	ADJ
ma-301	64	28	with	with	ADP
ma-301	64	29	the	the	DET
ma-301	64	30	induced	induced	ADJ
ma-301	64	31	topology	topology	NOUN
ma-301	64	32	of	of	ADP
ma-301	64	33	uniforme	uniforme	ADJ
ma-301	64	34	convergence.•	convergence.•	NOUN
ma-301	64	35	lpα(r+q	lpα(r+q	PROPN
ma-301	64	36	)	)	PUNCT
ma-301	64	37	,	,	PUNCT
ma-301	64	38	1	1	NUM
ma-301	64	39	≤	≤	NOUN
ma-301	64	40	p	p	NOUN
ma-301	64	41	≤	≤	NUM
ma-301	64	42	∞	∞	PROPN
ma-301	64	43	,	,	PUNCT
ma-301	64	44	denotes	denote	VERB
ma-301	64	45	the	the	DET
ma-301	64	46	space	space	NOUN
ma-301	64	47	of	of	ADP
ma-301	64	48	measurable	measurable	ADJ
ma-301	64	49	functions	function	NOUN
ma-301	64	50	on	on	ADP
ma-301	64	51	r+q	r+q	PROPN
ma-301	64	52	,	,	PUNCT
ma-301	64	53	satisfying	satisfy	VERB
ma-301	64	54	‖f	‖f	ADP
ma-301	64	55	‖p	‖p	PROPN
ma-301	64	56	,	,	PUNCT
ma-301	64	57	q	q	X
ma-301	64	58	,	,	PUNCT
ma-301	64	59	α	α	PROPN
ma-301	64	60	=	=	NOUN
ma-301	64	61	:	:	PUNCT
ma-301	64	62	{	{	PUNCT
ma-301	64	63	(	(	PUNCT
ma-301	64	64	∫∞	∫∞	NOUN
ma-301	64	65	0	0	NUM
ma-301	64	66	|f	|f	PROPN
ma-301	64	67	(	(	PUNCT
ma-301	64	68	x)|pdµq	x)|pdµq	NOUN
ma-301	64	69	,	,	PUNCT
ma-301	64	70	α(x	α(x	NOUN
ma-301	64	71	)	)	PUNCT
ma-301	64	72	)	)	PUNCT
ma-301	65	1	1	1	X
ma-301	65	2	/	/	SYM
ma-301	65	3	p	p	X
ma-301	65	4	<	<	X
ma-301	65	5	∞	∞	PROPN
ma-301	65	6	,	,	PUNCT
ma-301	65	7	1	1	NUM
ma-301	65	8	≤	≤	NOUN
ma-301	66	1	p	p	X
ma-301	66	2	<	<	X
ma-301	66	3	∞	∞	PROPN
ma-301	66	4	,	,	PUNCT
ma-301	66	5	supx∈r+q	supx∈r+q	PROPN
ma-301	66	6	|f	|f	PROPN
ma-301	66	7	(	(	PUNCT
ma-301	66	8	x)|	x)|	PROPN
ma-301	66	9	<	<	X
ma-301	66	10	∞	∞	PROPN
ma-301	66	11	,	,	PUNCT
ma-301	66	12	p	p	PROPN
ma-301	66	13	=	=	ADJ
ma-301	66	14	∞.where	∞.where	NOUN
ma-301	66	15	dµq	dµq	PROPN
ma-301	66	16	,	,	PUNCT
ma-301	66	17	α(x	α(x	NOUN
ma-301	66	18	)	)	PUNCT
ma-301	66	19	=	=	PUNCT
ma-301	67	1	1	1	NUM
ma-301	67	2	1−	1−	NUM
ma-301	67	3	q	q	NOUN
ma-301	67	4	(	(	PUNCT
ma-301	67	5	q2α+2	q2α+2	ADP
ma-301	67	6	;	;	PUNCT
ma-301	67	7	q2	q2	NOUN
ma-301	67	8	)	)	PUNCT
ma-301	67	9	∞	∞	PROPN
ma-301	67	10	(	(	PUNCT
ma-301	67	11	q2	q2	NOUN
ma-301	67	12	;	;	PUNCT
ma-301	67	13	q2)∞	q2)∞	NOUN
ma-301	67	14	x2α+1dq(x	x2α+1dq(x	NUM
ma-301	67	15	)	)	PUNCT
ma-301	67	16	,	,	PUNCT
ma-301	67	17	definition	definition	NOUN
ma-301	67	18	2.1	2.1	NUM
ma-301	67	19	.	.	PUNCT
ma-301	68	1	(	(	PUNCT
ma-301	68	2	[	[	X
ma-301	68	3	11	11	NUM
ma-301	68	4	]	]	PUNCT
ma-301	68	5	)	)	PUNCT
ma-301	68	6	the	the	DET
ma-301	68	7	q	q	ADJ
ma-301	68	8	-	-	PUNCT
ma-301	68	9	bessel	bessel	ADJ
ma-301	68	10	transform	transform	NOUN
ma-301	68	11	hq	hq	NOUN
ma-301	68	12	,	,	PUNCT
ma-301	68	13	α	α	PROPN
ma-301	68	14	defined	define	VERB
ma-301	68	15	on	on	ADP
ma-301	68	16	l1α(r+q	l1α(r+q	PROPN
ma-301	68	17	)	)	PUNCT
ma-301	68	18	by	by	ADP
ma-301	68	19	hq	hq	PROPN
ma-301	68	20	,	,	PUNCT
ma-301	68	21	α(f	α(f	PROPN
ma-301	68	22	)	)	PUNCT
ma-301	68	23	(	(	PUNCT
ma-301	68	24	λ	λ	X
ma-301	68	25	)	)	PUNCT
ma-301	68	26	=	=	SYM
ma-301	69	1	∫	∫	PROPN
ma-301	70	1	∞	∞	NOUN
ma-301	70	2	0	0	NUM
ma-301	70	3	jα(λx	jα(λx	NOUN
ma-301	70	4	;	;	PUNCT
ma-301	70	5	q2)f	q2)f	NOUN
ma-301	70	6	(	(	PUNCT
ma-301	70	7	x)dµq	x)dµq	PROPN
ma-301	70	8	,	,	PUNCT
ma-301	70	9	α(x	α(x	NOUN
ma-301	70	10	)	)	PUNCT
ma-301	70	11	,	,	PUNCT
ma-301	70	12	for	for	ADP
ma-301	70	13	λ	λ	PROPN
ma-301	70	14	∈	∈	PROPN
ma-301	70	15	r+q	r+q	PROPN
ma-301	70	16	some	some	DET
ma-301	70	17	basic	basic	ADJ
ma-301	70	18	properties	property	NOUN
ma-301	70	19	of	of	ADP
ma-301	70	20	this	this	DET
ma-301	70	21	transform	transform	NOUN
ma-301	70	22	are	be	AUX
ma-301	70	23	as	as	SCONJ
ma-301	70	24	follows	follow	NOUN
ma-301	70	25	,	,	PUNCT
ma-301	70	26	for	for	ADP
ma-301	70	27	the	the	DET
ma-301	70	28	proofs	proof	NOUN
ma-301	70	29	,	,	PUNCT
ma-301	70	30	we	we	PRON
ma-301	70	31	refer	refer	VERB
ma-301	70	32	the	the	DET
ma-301	70	33	reader	reader	NOUN
ma-301	70	34	to	to	ADP
ma-301	70	35	[	[	X
ma-301	70	36	11,13,14,25	11,13,14,25	NUM
ma-301	70	37	]	]	PUNCT
ma-301	70	38	.	.	PUNCT
ma-301	71	1	proposition	proposition	NOUN
ma-301	71	2	2.1	2.1	NUM
ma-301	71	3	.	.	PUNCT
ma-301	72	1	(	(	PUNCT
ma-301	72	2	1	1	X
ma-301	72	3	)	)	PUNCT
ma-301	72	4	for	for	ADP
ma-301	72	5	every	every	DET
ma-301	72	6	f	f	PROPN
ma-301	72	7	∈	∈	PROPN
ma-301	72	8	l1α(r+q	l1α(r+q	PROPN
ma-301	72	9	)	)	PUNCT
ma-301	73	1	we	we	PRON
ma-301	73	2	have	have	VERB
ma-301	73	3	hq	hq	NOUN
ma-301	73	4	,	,	PUNCT
ma-301	73	5	α(f	α(f	NUM
ma-301	73	6	)	)	PUNCT
ma-301	74	1	∈	∈	PROPN
ma-301	74	2	c0,q(r+q	c0,q(r+q	PROPN
ma-301	74	3	)	)	PUNCT
ma-301	75	1	and	and	CCONJ
ma-301	75	2	we	we	PRON
ma-301	75	3	have	have	VERB
ma-301	75	4	‖hq	‖hq	NUM
ma-301	75	5	,	,	PUNCT
ma-301	75	6	α(f	α(f	PROPN
ma-301	75	7	)	)	PUNCT
ma-301	75	8	‖∞,q	‖∞,q	PROPN
ma-301	75	9	,	,	PUNCT
ma-301	75	10	α	α	PROPN
ma-301	75	11	≤	≤	PUNCT
ma-301	75	12	bq	bq	PROPN
ma-301	75	13	,	,	PUNCT
ma-301	75	14	α‖f	α‖f	ADJ
ma-301	75	15	‖1,q	‖1,q	NOUN
ma-301	75	16	,	,	PUNCT
ma-301	75	17	α	α	X
ma-301	75	18	.	.	PUNCT
ma-301	76	1	(	(	PUNCT
ma-301	76	2	2.3	2.3	NUM
ma-301	76	3	)	)	PUNCT
ma-301	77	1	where	where	SCONJ
ma-301	77	2	bq	bq	AUX
ma-301	77	3	,	,	PUNCT
ma-301	77	4	α	α	NOUN
ma-301	77	5	=	=	SYM
ma-301	77	6	1	1	NUM
ma-301	77	7	1−	1−	NUM
ma-301	77	8	q	q	NOUN
ma-301	77	9	(	(	PUNCT
ma-301	77	10	−q2α+2	−q2α+2	X
ma-301	77	11	;	;	PUNCT
ma-301	77	12	q2	q2	PROPN
ma-301	77	13	)	)	PUNCT
ma-301	77	14	∞	∞	PROPN
ma-301	77	15	(	(	PUNCT
ma-301	77	16	−q2	−q2	PROPN
ma-301	77	17	;	;	PUNCT
ma-301	77	18	q2	q2	NOUN
ma-301	77	19	)	)	PUNCT
ma-301	77	20	∞	∞	PROPN
ma-301	77	21	(	(	PUNCT
ma-301	77	22	q2	q2	NOUN
ma-301	77	23	;	;	PUNCT
ma-301	77	24	q2)∞	q2)∞	NOUN
ma-301	77	25	(	(	PUNCT
ma-301	77	26	2.4	2.4	NUM
ma-301	77	27	)	)	PUNCT
ma-301	77	28	(	(	PUNCT
ma-301	77	29	2)(q	2)(q	NUM
ma-301	77	30	-	-	PUNCT
ma-301	77	31	inversion	inversion	NOUN
ma-301	77	32	formula	formula	NOUN
ma-301	77	33	)	)	PUNCT
ma-301	77	34	for	for	ADP
ma-301	77	35	f	f	PROPN
ma-301	77	36	∈	∈	PROPN
ma-301	77	37	(	(	PUNCT
ma-301	77	38	l1α	l1α	PROPN
ma-301	77	39	∩	∩	PROPN
ma-301	77	40	l2α	l2α	X
ma-301	77	41	)	)	PUNCT
ma-301	77	42	(	(	PUNCT
ma-301	77	43	r+q	r+q	NUM
ma-301	77	44	)	)	PUNCT
ma-301	77	45	such	such	ADJ
ma-301	77	46	that	that	DET
ma-301	77	47	fα(f	fα(f	X
ma-301	77	48	)	)	PUNCT
ma-301	77	49	∈	∈	PROPN
ma-301	77	50	l1α(r+q	l1α(r+q	NOUN
ma-301	77	51	)	)	PUNCT
ma-301	78	1	we	we	PRON
ma-301	78	2	have	have	VERB
ma-301	78	3	f	f	PROPN
ma-301	78	4	(	(	PUNCT
ma-301	78	5	x	x	NOUN
ma-301	78	6	)	)	PUNCT
ma-301	78	7	=	=	SYM
ma-301	79	1	∫	∫	PROPN
ma-301	80	1	∞	∞	NOUN
ma-301	80	2	0	0	NUM
ma-301	81	1	jα(λx	jα(λx	NOUN
ma-301	81	2	;	;	PUNCT
ma-301	81	3	q2)hq	q2)hq	PROPN
ma-301	81	4	,	,	PUNCT
ma-301	81	5	α(f	α(f	PROPN
ma-301	81	6	)	)	PUNCT
ma-301	81	7	(	(	PUNCT
ma-301	81	8	λ)dµq	λ)dµq	PROPN
ma-301	81	9	,	,	PUNCT
ma-301	81	10	α(λ	α(λ	PROPN
ma-301	81	11	)	)	PUNCT
ma-301	81	12	,	,	PUNCT
ma-301	81	13	a.e	a.e	PROPN
ma-301	81	14	x	x	SYM
ma-301	81	15	∈	∈	PROPN
ma-301	81	16	r+q	r+q	PROPN
ma-301	81	17	.	.	PUNCT
ma-301	82	1	(	(	PUNCT
ma-301	82	2	2.5	2.5	NUM
ma-301	82	3	)	)	PUNCT
ma-301	82	4	(	(	PUNCT
ma-301	82	5	3	3	X
ma-301	82	6	)	)	PUNCT
ma-301	82	7	(	(	PUNCT
ma-301	82	8	q	q	ADJ
ma-301	82	9	-	-	PUNCT
ma-301	82	10	parseval	parseval	NOUN
ma-301	82	11	formula	formula	NOUN
ma-301	82	12	)	)	PUNCT
ma-301	82	13	for	for	ADP
ma-301	82	14	all	all	DET
ma-301	82	15	f	f	PROPN
ma-301	82	16	,	,	PUNCT
ma-301	82	17	g	g	PROPN
ma-301	82	18	∈	∈	PROPN
ma-301	82	19	l2α(r+q	l2α(r+q	PROPN
ma-301	82	20	)	)	PUNCT
ma-301	83	1	we	we	PRON
ma-301	83	2	have	have	VERB
ma-301	83	3	〈	〈	PROPN
ma-301	83	4	f	f	PROPN
ma-301	83	5	,	,	PUNCT
ma-301	83	6	g〉q	g〉q	PROPN
ma-301	83	7	=	=	SYM
ma-301	83	8	〈	〈	PROPN
ma-301	83	9	hq	hq	NOUN
ma-301	83	10	,	,	PUNCT
ma-301	83	11	α(f	α(f	PROPN
ma-301	83	12	)	)	PUNCT
ma-301	83	13	,	,	PUNCT
ma-301	83	14	hq	hq	INTJ
ma-301	83	15	,	,	PUNCT
ma-301	83	16	α(g)〉q	α(g)〉q	PROPN
ma-301	83	17	,	,	PUNCT
ma-301	83	18	(	(	PUNCT
ma-301	83	19	2.6	2.6	NUM
ma-301	83	20	)	)	PUNCT
ma-301	83	21	in	in	ADP
ma-301	83	22	particular	particular	ADJ
ma-301	83	23	we	we	PRON
ma-301	83	24	have	have	VERB
ma-301	83	25	‖f	‖f	ADP
ma-301	83	26	‖2,q	‖2,q	NOUN
ma-301	83	27	,	,	PUNCT
ma-301	83	28	α	α	NOUN
ma-301	83	29	=	=	SYM
ma-301	83	30	‖hq	‖hq	PROPN
ma-301	83	31	,	,	PUNCT
ma-301	83	32	α(f	α(f	PROPN
ma-301	83	33	)	)	PUNCT
ma-301	84	1	‖2,q	‖2,q	PROPN
ma-301	84	2	,	,	PUNCT
ma-301	84	3	α	α	NOUN
ma-301	84	4	.	.	PUNCT
ma-301	85	1	(	(	PUNCT
ma-301	85	2	2.7	2.7	NUM
ma-301	85	3	)	)	PUNCT
ma-301	85	4	(	(	PUNCT
ma-301	85	5	4	4	NUM
ma-301	85	6	)	)	PUNCT
ma-301	85	7	(	(	PUNCT
ma-301	85	8	q	q	ADJ
ma-301	85	9	-	-	PUNCT
ma-301	85	10	plancherel	plancherel	NOUN
ma-301	85	11	theorem	theorem	NOUN
ma-301	85	12	)	)	PUNCT
ma-301	85	13	the	the	DET
ma-301	85	14	q	q	ADJ
ma-301	85	15	-	-	PUNCT
ma-301	85	16	bessel	bessel	ADJ
ma-301	85	17	transform	transform	NOUN
ma-301	85	18	hq	hq	NOUN
ma-301	85	19	,	,	PUNCT
ma-301	85	20	α	α	PROPN
ma-301	85	21	can	can	AUX
ma-301	85	22	be	be	AUX
ma-301	85	23	extended	extend	VERB
ma-301	85	24	to	to	ADP
ma-301	85	25	an	an	DET
ma-301	85	26	isometric	isometric	ADJ
ma-301	85	27	isomorphism	isomorphism	NOUN
ma-301	85	28	from	from	ADP
ma-301	85	29	l2α(r+q	l2α(r+q	PROPN
ma-301	85	30	)	)	PUNCT
ma-301	85	31	into	into	ADP
ma-301	85	32	l2α(r+q	l2α(r+q	PROPN
ma-301	85	33	)	)	PUNCT
ma-301	85	34	.	.	PUNCT
ma-301	86	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	86	2	eur	eur	PROPN
ma-301	86	3	.	.	PUNCT
ma-301	87	1	j.	j.	PROPN
ma-301	87	2	math	math	PROPN
ma-301	87	3	.	.	PUNCT
ma-301	88	1	anal	anal	PROPN
ma-301	88	2	.	.	PUNCT
ma-301	89	1	10.28924	10.28924	NUM
ma-301	89	2	/	/	SYM
ma-301	89	3	ada	ada	PROPN
ma-301	89	4	/	/	PROPN
ma-301	89	5	ma.5.8	ma.5.8	PROPN
ma-301	89	6	52.3	52.3	NUM
ma-301	89	7	.	.	PUNCT
ma-301	90	1	the	the	DET
ma-301	90	2	translation	translation	NOUN
ma-301	90	3	operator	operator	NOUN
ma-301	90	4	associated	associate	VERB
ma-301	90	5	with	with	ADP
ma-301	90	6	the	the	DET
ma-301	90	7	q	q	ADJ
ma-301	90	8	-	-	PUNCT
ma-301	90	9	bessel	bessel	ADJ
ma-301	90	10	transform	transform	NOUN
ma-301	90	11	.	.	PUNCT
ma-301	91	1	definition	definition	NOUN
ma-301	91	2	2.2	2.2	NUM
ma-301	91	3	.	.	PUNCT
ma-301	92	1	(	(	PUNCT
ma-301	92	2	[	[	X
ma-301	92	3	13	13	NUM
ma-301	92	4	]	]	PUNCT
ma-301	92	5	)	)	PUNCT
ma-301	92	6	let	let	VERB
ma-301	92	7	x	x	PRON
ma-301	92	8	,	,	PUNCT
ma-301	92	9	y	y	PROPN
ma-301	92	10	∈	∈	PROPN
ma-301	92	11	r+q	r+q	PROPN
ma-301	92	12	and	and	CCONJ
ma-301	92	13	f	f	PROPN
ma-301	92	14	is	be	AUX
ma-301	92	15	a	a	DET
ma-301	92	16	measurable	measurable	ADJ
ma-301	92	17	function	function	NOUN
ma-301	92	18	on	on	ADP
ma-301	92	19	r+q	r+q	PROPN
ma-301	92	20	the	the	DET
ma-301	92	21	translation	translation	NOUN
ma-301	92	22	operator	operator	NOUN
ma-301	92	23	is	be	AUX
ma-301	92	24	defined	define	VERB
ma-301	92	25	by	by	ADP
ma-301	92	26	τxq	τxq	PROPN
ma-301	92	27	,	,	PUNCT
ma-301	92	28	αf	αf	ADP
ma-301	92	29	(	(	PUNCT
ma-301	92	30	y	y	NOUN
ma-301	92	31	)	)	PUNCT
ma-301	92	32	=	=	SYM
ma-301	93	1	∫	∫	PROPN
ma-301	94	1	∞	∞	NOUN
ma-301	94	2	0	0	NUM
ma-301	95	1	jα(λx	jα(λx	NOUN
ma-301	95	2	;	;	PUNCT
ma-301	95	3	q2)jα(λy	q2)jα(λy	NOUN
ma-301	95	4	;	;	PUNCT
ma-301	95	5	q2)hq	q2)hq	NOUN
ma-301	95	6	,	,	PUNCT
ma-301	95	7	α(f	α(f	PROPN
ma-301	95	8	)	)	PUNCT
ma-301	95	9	(	(	PUNCT
ma-301	95	10	λ)dµq	λ)dµq	PROPN
ma-301	95	11	,	,	PUNCT
ma-301	95	12	α(λ	α(λ	PROPN
ma-301	95	13	)	)	PUNCT
ma-301	95	14	,	,	PUNCT
ma-301	95	15	the	the	DET
ma-301	95	16	following	follow	VERB
ma-301	95	17	proposition	proposition	NOUN
ma-301	95	18	summarizes	summarize	VERB
ma-301	95	19	some	some	DET
ma-301	95	20	properties	property	NOUN
ma-301	95	21	of	of	ADP
ma-301	95	22	the	the	DET
ma-301	95	23	q	q	ADJ
ma-301	95	24	-	-	PUNCT
ma-301	95	25	bessel	bessel	ADJ
ma-301	95	26	translation	translation	NOUN
ma-301	95	27	operator	operator	NOUN
ma-301	95	28	see	see	VERB
ma-301	95	29	[	[	X
ma-301	95	30	13	13	NUM
ma-301	95	31	]	]	PUNCT
ma-301	95	32	.	.	PUNCT
ma-301	96	1	proposition	proposition	NOUN
ma-301	96	2	2.2	2.2	NUM
ma-301	96	3	.	.	PUNCT
ma-301	97	1	for	for	ADP
ma-301	97	2	all	all	DET
ma-301	97	3	x	x	NOUN
ma-301	97	4	,	,	PUNCT
ma-301	97	5	y	y	PROPN
ma-301	97	6	∈	∈	PROPN
ma-301	97	7	r+q	r+q	PROPN
ma-301	97	8	,	,	PUNCT
ma-301	97	9	we	we	PRON
ma-301	97	10	have	have	VERB
ma-301	97	11	:	:	PUNCT
ma-301	97	12	(	(	PUNCT
ma-301	97	13	1	1	X
ma-301	97	14	)	)	PUNCT
ma-301	97	15	τxq	τxq	NOUN
ma-301	97	16	,	,	PUNCT
ma-301	97	17	αf	αf	ADP
ma-301	97	18	(	(	PUNCT
ma-301	97	19	y	y	NOUN
ma-301	97	20	)	)	PUNCT
ma-301	97	21	=	=	PUNCT
ma-301	98	1	τyq	τyq	ADV
ma-301	98	2	,	,	PUNCT
ma-301	98	3	αf	αf	ADP
ma-301	98	4	(	(	PUNCT
ma-301	98	5	x	x	NOUN
ma-301	98	6	)	)	PUNCT
ma-301	98	7	.	.	PUNCT
ma-301	99	1	(	(	PUNCT
ma-301	99	2	2.8	2.8	NUM
ma-301	99	3	)	)	PUNCT
ma-301	99	4	(	(	PUNCT
ma-301	99	5	2	2	X
ma-301	99	6	)	)	PUNCT
ma-301	99	7	∫	∫	PROPN
ma-301	99	8	∞	∞	PROPN
ma-301	99	9	0	0	NUM
ma-301	99	10	τxq	τxq	NOUN
ma-301	99	11	,	,	PUNCT
ma-301	99	12	αf	αf	X
ma-301	99	13	(	(	PUNCT
ma-301	99	14	y)dµq	y)dµq	NOUN
ma-301	99	15	,	,	PUNCT
ma-301	99	16	α(y	α(y	NOUN
ma-301	99	17	)	)	PUNCT
ma-301	99	18	=	=	PUNCT
ma-301	100	1	∫	∫	PROPN
ma-301	101	1	∞	∞	NUM
ma-301	101	2	0	0	NUM
ma-301	102	1	f	f	PROPN
ma-301	102	2	(	(	PUNCT
ma-301	102	3	y)dµq	y)dµq	PROPN
ma-301	102	4	,	,	PUNCT
ma-301	102	5	α(y	α(y	NOUN
ma-301	102	6	)	)	PUNCT
ma-301	102	7	.	.	PUNCT
ma-301	103	1	(	(	PUNCT
ma-301	103	2	2.9	2.9	NUM
ma-301	103	3	)	)	PUNCT
ma-301	103	4	(	(	PUNCT
ma-301	103	5	3	3	X
ma-301	103	6	)	)	PUNCT
ma-301	103	7	for	for	ADP
ma-301	103	8	f	f	PROPN
ma-301	103	9	∈	∈	PROPN
ma-301	103	10	lpα(r+q	lpα(r+q	PROPN
ma-301	103	11	)	)	PUNCT
ma-301	103	12	with	with	ADP
ma-301	103	13	p	p	PROPN
ma-301	103	14	∈	∈	PROPN
ma-301	104	1	[	[	X
ma-301	104	2	1	1	NUM
ma-301	104	3	;	;	PUNCT
ma-301	104	4	+	+	NUM
ma-301	104	5	∞	∞	NOUN
ma-301	104	6	]	]	SYM
ma-301	104	7	τxq	τxq	NOUN
ma-301	104	8	,	,	PUNCT
ma-301	104	9	αf	αf	ADP
ma-301	104	10	∈	∈	NOUN
ma-301	104	11	l	l	NOUN
ma-301	104	12	p	p	X
ma-301	104	13	α(r+q	α(r+q	PROPN
ma-301	104	14	)	)	PUNCT
ma-301	105	1	and	and	CCONJ
ma-301	105	2	we	we	PRON
ma-301	105	3	have∥∥τxq	have∥∥τxq	PROPN
ma-301	105	4	,	,	PUNCT
ma-301	105	5	αf	αf	ADP
ma-301	105	6	∥∥p	∥∥p	ADJ
ma-301	105	7	,	,	PUNCT
ma-301	105	8	q	q	NOUN
ma-301	105	9	,	,	PUNCT
ma-301	105	10	α	α	NOUN
ma-301	105	11	≤	≤	NOUN
ma-301	105	12	‖f	‖f	ADP
ma-301	105	13	‖p	‖p	PROPN
ma-301	105	14	,	,	PUNCT
ma-301	105	15	q	q	X
ma-301	105	16	,	,	PUNCT
ma-301	105	17	α	α	NOUN
ma-301	105	18	,	,	PUNCT
ma-301	105	19	(	(	PUNCT
ma-301	105	20	2.10	2.10	NUM
ma-301	105	21	)	)	PUNCT
ma-301	105	22	(	(	PUNCT
ma-301	105	23	4	4	NUM
ma-301	105	24	)	)	PUNCT
ma-301	105	25	for	for	ADP
ma-301	105	26	f	f	PROPN
ma-301	105	27	∈	∈	PROPN
ma-301	105	28	l1α(r+q	l1α(r+q	PROPN
ma-301	105	29	)	)	PUNCT
ma-301	105	30	,	,	PUNCT
ma-301	105	31	τxq	τxq	NOUN
ma-301	105	32	,	,	PUNCT
ma-301	105	33	αf	αf	ADP
ma-301	105	34	∈	∈	NOUN
ma-301	105	35	l1α(r+q	l1α(r+q	NOUN
ma-301	105	36	)	)	PUNCT
ma-301	106	1	and	and	CCONJ
ma-301	106	2	we	we	PRON
ma-301	106	3	have	have	AUX
ma-301	106	4	hq	hq	PROPN
ma-301	106	5	,	,	PUNCT
ma-301	106	6	α	α	PROPN
ma-301	106	7	(	(	PUNCT
ma-301	106	8	τxq	τxq	NOUN
ma-301	106	9	,	,	PUNCT
ma-301	106	10	αf	αf	NOUN
ma-301	106	11	)	)	PUNCT
ma-301	106	12	(	(	PUNCT
ma-301	106	13	λ	λ	X
ma-301	106	14	)	)	PUNCT
ma-301	106	15	=	=	SYM
ma-301	107	1	jα(λx	jα(λx	PROPN
ma-301	107	2	;	;	PUNCT
ma-301	107	3	q2)hq	q2)hq	PROPN
ma-301	107	4	,	,	PUNCT
ma-301	107	5	α(f	α(f	PROPN
ma-301	107	6	)	)	PUNCT
ma-301	107	7	(	(	PUNCT
ma-301	107	8	λ	λ	NOUN
ma-301	107	9	)	)	PUNCT
ma-301	107	10	,	,	PUNCT
ma-301	107	11	∀λ	∀λ	X
ma-301	107	12	∈	∈	PROPN
ma-301	107	13	r+q	r+q	ADJ
ma-301	107	14	.	.	PUNCT
ma-301	108	1	(	(	PUNCT
ma-301	108	2	2.11	2.11	NUM
ma-301	108	3	)	)	PUNCT
ma-301	108	4	the	the	DET
ma-301	108	5	relation	relation	NOUN
ma-301	108	6	(	(	PUNCT
ma-301	108	7	2.11	2.11	NUM
ma-301	108	8	)	)	PUNCT
ma-301	108	9	shows	show	VERB
ma-301	108	10	that	that	SCONJ
ma-301	108	11	the	the	DET
ma-301	108	12	translation	translation	NOUN
ma-301	108	13	operator	operator	NOUN
ma-301	108	14	τxq	τxq	PROPN
ma-301	108	15	,	,	PUNCT
ma-301	108	16	α	α	PROPN
ma-301	108	17	is	be	AUX
ma-301	108	18	a	a	DET
ma-301	108	19	particular	particular	ADJ
ma-301	108	20	case	case	NOUN
ma-301	108	21	of	of	ADP
ma-301	108	22	the	the	DET
ma-301	108	23	q	q	ADJ
ma-301	108	24	-	-	PUNCT
ma-301	108	25	bessel	bessel	ADJ
ma-301	108	26	multiplier	multipli	ADJ
ma-301	108	27	operator	operator	NOUN
ma-301	108	28	(	(	PUNCT
ma-301	108	29	1.4	1.4	NUM
ma-301	108	30	)	)	PUNCT
ma-301	108	31	.	.	PUNCT
ma-301	109	1	by	by	ADP
ma-301	109	2	using	use	VERB
ma-301	109	3	the	the	DET
ma-301	109	4	q	q	ADJ
ma-301	109	5	-	-	PUNCT
ma-301	109	6	bessel	bessel	ADJ
ma-301	109	7	translation	translation	NOUN
ma-301	109	8	operator	operator	NOUN
ma-301	109	9	,	,	PUNCT
ma-301	109	10	we	we	PRON
ma-301	109	11	define	define	VERB
ma-301	109	12	the	the	DET
ma-301	109	13	generalized	generalized	ADJ
ma-301	109	14	convolution	convolution	NOUN
ma-301	109	15	product	product	NOUN
ma-301	109	16	of	of	ADP
ma-301	109	17	f	f	PROPN
ma-301	109	18	,	,	PUNCT
ma-301	109	19	g	g	PROPN
ma-301	109	20	by	by	ADP
ma-301	109	21	(	(	PUNCT
ma-301	109	22	f	f	PROPN
ma-301	109	23	∗q	∗q	PROPN
ma-301	109	24	g	g	PROPN
ma-301	109	25	)	)	PUNCT
ma-301	109	26	(	(	PUNCT
ma-301	109	27	x	x	X
ma-301	109	28	)	)	PUNCT
ma-301	109	29	=	=	SYM
ma-301	110	1	∫	∫	PROPN
ma-301	110	2	∞	∞	NUM
ma-301	110	3	0	0	NUM
ma-301	110	4	τxq	τxq	PROPN
ma-301	110	5	,	,	PUNCT
ma-301	110	6	α(f	α(f	PROPN
ma-301	110	7	)	)	PUNCT
ma-301	110	8	(	(	PUNCT
ma-301	110	9	y)g(y)dµq	y)g(y)dµq	NOUN
ma-301	110	10	,	,	PUNCT
ma-301	110	11	α(y	α(y	NOUN
ma-301	110	12	)	)	PUNCT
ma-301	110	13	.	.	PUNCT
ma-301	111	1	this	this	DET
ma-301	111	2	convolution	convolution	NOUN
ma-301	111	3	is	be	AUX
ma-301	111	4	commutative	commutative	ADJ
ma-301	111	5	,	,	PUNCT
ma-301	111	6	associative	associative	ADJ
ma-301	111	7	and	and	CCONJ
ma-301	111	8	its	its	PRON
ma-301	111	9	satisfies	satisfie	NOUN
ma-301	111	10	the	the	DET
ma-301	111	11	following	follow	VERB
ma-301	111	12	properties	property	NOUN
ma-301	111	13	see	see	VERB
ma-301	111	14	[	[	X
ma-301	111	15	11,13	11,13	NUM
ma-301	111	16	]	]	PUNCT
ma-301	111	17	.	.	PUNCT
ma-301	112	1	proposition	proposition	NOUN
ma-301	112	2	2.3	2.3	NUM
ma-301	112	3	.	.	PUNCT
ma-301	113	1	(	(	PUNCT
ma-301	113	2	1)(q	1)(q	NUM
ma-301	113	3	-	-	PUNCT
ma-301	113	4	young	young	ADJ
ma-301	113	5	’s	’s	PART
ma-301	113	6	inequality	inequality	NOUN
ma-301	113	7	)	)	PUNCT
ma-301	113	8	for	for	ADP
ma-301	113	9	all	all	DET
ma-301	113	10	p	p	NOUN
ma-301	113	11	,	,	PUNCT
ma-301	113	12	q	q	ADJ
ma-301	113	13	,	,	PUNCT
ma-301	113	14	r	r	NOUN
ma-301	113	15	∈	∈	PROPN
ma-301	114	1	[	[	X
ma-301	114	2	1	1	NUM
ma-301	114	3	;	;	PUNCT
ma-301	114	4	+	+	NUM
ma-301	114	5	∞	∞	NOUN
ma-301	114	6	]	]	X
ma-301	114	7	such	such	ADJ
ma-301	114	8	that	that	SCONJ
ma-301	114	9	:	:	PUNCT
ma-301	114	10	1	1	NUM
ma-301	114	11	p	p	NOUN
ma-301	114	12	+	+	NOUN
ma-301	114	13	1	1	NUM
ma-301	114	14	s	s	NOUN
ma-301	114	15	=	=	SYM
ma-301	114	16	1	1	NUM
ma-301	114	17	+	+	SYM
ma-301	114	18	1	1	NUM
ma-301	114	19	r	r	NOUN
ma-301	114	20	and	and	CCONJ
ma-301	114	21	for	for	ADP
ma-301	114	22	all	all	DET
ma-301	114	23	f	f	PROPN
ma-301	114	24	∈	∈	PROPN
ma-301	114	25	lpα(r+q	lpα(r+q	PROPN
ma-301	114	26	)	)	PUNCT
ma-301	114	27	,	,	PUNCT
ma-301	114	28	g	g	PROPN
ma-301	114	29	∈	∈	PROPN
ma-301	114	30	lsα(r+q	lsα(r+q	PROPN
ma-301	114	31	)	)	PUNCT
ma-301	114	32	the	the	DET
ma-301	114	33	function	function	NOUN
ma-301	114	34	f	f	PROPN
ma-301	114	35	∗α	∗α	PROPN
ma-301	114	36	g	g	PROPN
ma-301	114	37	belongs	belong	VERB
ma-301	114	38	to	to	ADP
ma-301	114	39	the	the	DET
ma-301	114	40	space	space	NOUN
ma-301	114	41	lrα(r+q	lrα(r+q	NOUN
ma-301	114	42	)	)	PUNCT
ma-301	114	43	and	and	CCONJ
ma-301	114	44	we	we	PRON
ma-301	114	45	have	have	VERB
ma-301	114	46	‖f	‖f	ADJ
ma-301	114	47	∗α	∗α	NOUN
ma-301	114	48	g‖r	g‖r	NOUN
ma-301	114	49	,	,	PUNCT
ma-301	114	50	q	q	NOUN
ma-301	114	51	,	,	PUNCT
ma-301	114	52	α	α	NOUN
ma-301	114	53	≤	≤	NOUN
ma-301	114	54	‖f	‖f	ADP
ma-301	114	55	‖p	‖p	PROPN
ma-301	114	56	,	,	PUNCT
ma-301	114	57	q	q	NOUN
ma-301	114	58	,	,	PUNCT
ma-301	114	59	α‖g‖s	α‖g‖s	ADJ
ma-301	114	60	,	,	PUNCT
ma-301	114	61	q	q	X
ma-301	114	62	,	,	PUNCT
ma-301	114	63	α	α	X
ma-301	114	64	(	(	PUNCT
ma-301	114	65	2.12	2.12	NUM
ma-301	114	66	)	)	PUNCT
ma-301	114	67	(	(	PUNCT
ma-301	114	68	2	2	NUM
ma-301	114	69	)	)	PUNCT
ma-301	114	70	for	for	ADP
ma-301	114	71	f	f	PROPN
ma-301	114	72	,	,	PUNCT
ma-301	114	73	g	g	PROPN
ma-301	114	74	∈	∈	PROPN
ma-301	114	75	l2α(r+q	l2α(r+q	PROPN
ma-301	114	76	)	)	PUNCT
ma-301	114	77	the	the	DET
ma-301	114	78	function	function	NOUN
ma-301	114	79	f	f	PROPN
ma-301	114	80	∗qg	∗qg	PROPN
ma-301	114	81	belongs	belong	VERB
ma-301	114	82	to	to	ADP
ma-301	114	83	l2α(r+q	l2α(r+q	PROPN
ma-301	114	84	)	)	PUNCT
ma-301	115	1	if	if	SCONJ
ma-301	115	2	and	and	CCONJ
ma-301	115	3	only	only	ADV
ma-301	115	4	if	if	SCONJ
ma-301	115	5	the	the	DET
ma-301	115	6	functionhq	functionhq	ADJ
ma-301	115	7	,	,	PUNCT
ma-301	115	8	α(f	α(f	PROPN
ma-301	115	9	)	)	PUNCT
ma-301	115	10	hq	hq	NOUN
ma-301	115	11	,	,	PUNCT
ma-301	115	12	α(g	α(g	NUM
ma-301	115	13	)	)	PUNCT
ma-301	115	14	belongs	belong	VERB
ma-301	115	15	to	to	ADP
ma-301	115	16	l2α(r+q	l2α(r+q	PROPN
ma-301	115	17	)	)	PUNCT
ma-301	115	18	and	and	CCONJ
ma-301	115	19	in	in	ADP
ma-301	115	20	this	this	DET
ma-301	115	21	case	case	NOUN
ma-301	115	22	we	we	PRON
ma-301	115	23	have	have	AUX
ma-301	115	24	hq	hq	PROPN
ma-301	115	25	,	,	PUNCT
ma-301	115	26	α	α	PROPN
ma-301	115	27	(	(	PUNCT
ma-301	115	28	f	f	PROPN
ma-301	115	29	∗q	∗q	PROPN
ma-301	115	30	g	g	PROPN
ma-301	115	31	)	)	PUNCT
ma-301	115	32	=	=	SYM
ma-301	115	33	hq	hq	PROPN
ma-301	115	34	,	,	PUNCT
ma-301	115	35	α(f	α(f	PROPN
ma-301	115	36	)	)	PUNCT
ma-301	115	37	hq	hq	NOUN
ma-301	115	38	,	,	PUNCT
ma-301	115	39	α(g	α(g	NUM
ma-301	115	40	)	)	PUNCT
ma-301	115	41	.	.	PUNCT
ma-301	116	1	(	(	PUNCT
ma-301	116	2	2.13	2.13	NUM
ma-301	116	3	)	)	PUNCT
ma-301	116	4	(	(	PUNCT
ma-301	116	5	3	3	X
ma-301	116	6	)	)	PUNCT
ma-301	116	7	for	for	ADP
ma-301	116	8	all	all	DET
ma-301	116	9	f	f	PROPN
ma-301	116	10	,	,	PUNCT
ma-301	116	11	g	g	PROPN
ma-301	116	12	∈	∈	PROPN
ma-301	116	13	l2α(r+q	l2α(r+q	PROPN
ma-301	116	14	)	)	PUNCT
ma-301	117	1	then	then	ADV
ma-301	117	2	we	we	PRON
ma-301	117	3	have∫	have∫	VERB
ma-301	117	4	∞	∞	PROPN
ma-301	117	5	0	0	NUM
ma-301	117	6	|f	|f	PROPN
ma-301	117	7	∗q	∗q	PROPN
ma-301	117	8	g(x	g(x	PROPN
ma-301	117	9	,	,	PUNCT
ma-301	117	10	t)|2	t)|2	PROPN
ma-301	117	11	dµq	dµq	PROPN
ma-301	117	12	,	,	PUNCT
ma-301	117	13	α(x	α(x	NOUN
ma-301	117	14	)	)	PUNCT
ma-301	117	15	=	=	SYM
ma-301	118	1	∫	∫	PROPN
ma-301	119	1	∞	∞	NUM
ma-301	119	2	0	0	NUM
ma-301	119	3	|hq	|hq	PROPN
ma-301	119	4	,	,	PUNCT
ma-301	119	5	α(f	α(f	PROPN
ma-301	119	6	)	)	PUNCT
ma-301	119	7	(	(	PUNCT
ma-301	119	8	λ)|2	λ)|2	PROPN
ma-301	119	9	|hq	|hq	NOUN
ma-301	119	10	,	,	PUNCT
ma-301	119	11	α(g)(λ)|2	α(g)(λ)|2	PROPN
ma-301	119	12	dµq	dµq	NOUN
ma-301	119	13	,	,	PUNCT
ma-301	119	14	α(λ	α(λ	PROPN
ma-301	119	15	)	)	PUNCT
ma-301	119	16	,	,	PUNCT
ma-301	119	17	(	(	PUNCT
ma-301	119	18	2.14	2.14	NUM
ma-301	119	19	)	)	PUNCT
ma-301	119	20	where	where	SCONJ
ma-301	119	21	both	both	DET
ma-301	119	22	integrals	integral	NOUN
ma-301	119	23	are	be	AUX
ma-301	119	24	simultaneously	simultaneously	ADV
ma-301	119	25	finite	finite	ADJ
ma-301	119	26	or	or	CCONJ
ma-301	119	27	infinite	infinite	VERB
ma-301	119	28	.	.	PUNCT
ma-301	120	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	120	2	eur	eur	PROPN
ma-301	120	3	.	.	PUNCT
ma-301	121	1	j.	j.	PROPN
ma-301	121	2	math	math	PROPN
ma-301	121	3	.	.	PUNCT
ma-301	122	1	anal	anal	PROPN
ma-301	122	2	.	.	PUNCT
ma-301	123	1	10.28924	10.28924	NUM
ma-301	123	2	/	/	SYM
ma-301	123	3	ada	ada	PROPN
ma-301	123	4	/	/	PROPN
ma-301	123	5	ma.5.8	ma.5.8	PROPN
ma-301	123	6	63	63	NUM
ma-301	123	7	.	.	PUNCT
ma-301	124	1	the	the	DET
ma-301	124	2	q	q	ADJ
ma-301	124	3	-	-	PUNCT
ma-301	124	4	bessel	bessel	ADJ
ma-301	124	5	l2α	l2α	NOUN
ma-301	124	6	-	-	ADJ
ma-301	124	7	multiplier	multipli	ADJ
ma-301	124	8	operators	operator	NOUN
ma-301	124	9	the	the	DET
ma-301	124	10	main	main	ADJ
ma-301	124	11	purpose	purpose	NOUN
ma-301	124	12	of	of	ADP
ma-301	124	13	this	this	DET
ma-301	124	14	section	section	NOUN
ma-301	124	15	is	be	AUX
ma-301	124	16	to	to	PART
ma-301	124	17	introduce	introduce	VERB
ma-301	124	18	the	the	DET
ma-301	124	19	q	q	ADJ
ma-301	124	20	-	-	PUNCT
ma-301	124	21	bessel	bessel	ADJ
ma-301	124	22	l2α	l2α	ADJ
ma-301	124	23	-	-	ADJ
ma-301	124	24	multiplier	multipli	ADJ
ma-301	124	25	operators	operator	NOUN
ma-301	124	26	on	on	ADP
ma-301	124	27	r+q	r+q	PROPN
ma-301	124	28	and	and	CCONJ
ma-301	124	29	to	to	PART
ma-301	124	30	establish	establish	VERB
ma-301	124	31	for	for	ADP
ma-301	124	32	them	they	PRON
ma-301	124	33	some	some	DET
ma-301	124	34	uncertainty	uncertainty	NOUN
ma-301	124	35	principles	principle	NOUN
ma-301	124	36	and	and	CCONJ
ma-301	124	37	calderon	calderon	NOUN
ma-301	124	38	’s	’s	PART
ma-301	124	39	reproducing	reproduce	VERB
ma-301	124	40	formulas	formula	NOUN
ma-301	124	41	.	.	PUNCT
ma-301	125	1	3.1	3.1	NUM
ma-301	125	2	.	.	PUNCT
ma-301	125	3	calderon	calderon	PROPN
ma-301	125	4	’s	’s	PART
ma-301	125	5	reproducing	reproduce	VERB
ma-301	125	6	formulas	formula	NOUN
ma-301	125	7	for	for	ADP
ma-301	125	8	the	the	DET
ma-301	125	9	q	q	ADJ
ma-301	125	10	-	-	ADJ
ma-301	125	11	bessel	bessel	ADJ
ma-301	125	12	l2α	l2α	NOUN
ma-301	125	13	-	-	ADJ
ma-301	125	14	multiplier	multipli	ADJ
ma-301	125	15	operators	operator	NOUN
ma-301	125	16	.	.	PUNCT
ma-301	126	1	definition	definition	NOUN
ma-301	126	2	3.1	3.1	NUM
ma-301	126	3	.	.	PUNCT
ma-301	127	1	let	let	VERB
ma-301	127	2	σ	σ	X
ma-301	127	3	∈	∈	PROPN
ma-301	127	4	l2α(r+q	l2α(r+q	PROPN
ma-301	127	5	)	)	PUNCT
ma-301	127	6	and	and	CCONJ
ma-301	127	7	β	β	X
ma-301	127	8	∈	∈	PROPN
ma-301	128	1	r+q	r+q	PROPN
ma-301	128	2	,	,	PUNCT
ma-301	128	3	the	the	DET
ma-301	128	4	q	q	ADJ
ma-301	128	5	-	-	PUNCT
ma-301	128	6	bessel	bessel	ADJ
ma-301	128	7	l2α	l2α	NOUN
ma-301	128	8	-	-	ADJ
ma-301	128	9	multiplier	multipli	ADJ
ma-301	128	10	operators	operator	NOUN
ma-301	128	11	are	be	AUX
ma-301	128	12	defined	define	VERB
ma-301	128	13	for	for	ADP
ma-301	128	14	smooth	smooth	ADJ
ma-301	128	15	function	function	NOUN
ma-301	128	16	f	f	PROPN
ma-301	128	17	on	on	ADP
ma-301	128	18	r+q	r+q	PROPN
ma-301	128	19	as	as	ADP
ma-301	128	20	mq	mq	PROPN
ma-301	128	21	,	,	PUNCT
ma-301	128	22	σ	σ	PROPN
ma-301	128	23	,	,	PUNCT
ma-301	128	24	β(f	β(f	PROPN
ma-301	128	25	)	)	PUNCT
ma-301	128	26	(	(	PUNCT
ma-301	128	27	x	x	X
ma-301	128	28	)	)	PUNCT
ma-301	128	29	:	:	PUNCT
ma-301	129	1	=	=	SYM
ma-301	129	2	h−1q	h−1q	PROPN
ma-301	129	3	,	,	PUNCT
ma-301	129	4	α	α	PROPN
ma-301	129	5	(	(	PUNCT
ma-301	129	6	σβhq	σβhq	PROPN
ma-301	129	7	,	,	PUNCT
ma-301	129	8	α(f	α(f	PROPN
ma-301	129	9	)	)	PUNCT
ma-301	129	10	)	)	PUNCT
ma-301	130	1	(	(	PUNCT
ma-301	130	2	x	x	X
ma-301	130	3	)	)	PUNCT
ma-301	130	4	,	,	PUNCT
ma-301	130	5	(	(	PUNCT
ma-301	130	6	3.1	3.1	NUM
ma-301	130	7	)	)	PUNCT
ma-301	130	8	where	where	SCONJ
ma-301	130	9	the	the	DET
ma-301	130	10	function	function	NOUN
ma-301	130	11	σβ	σβ	VERB
ma-301	130	12	is	be	AUX
ma-301	130	13	given	give	VERB
ma-301	130	14	by	by	ADP
ma-301	130	15	the	the	DET
ma-301	130	16	relation	relation	NOUN
ma-301	130	17	(	(	PUNCT
ma-301	130	18	1.5	1.5	NUM
ma-301	130	19	)	)	PUNCT
ma-301	130	20	and	and	CCONJ
ma-301	130	21	by	by	ADP
ma-301	130	22	a	a	DET
ma-301	130	23	simple	simple	ADJ
ma-301	130	24	change	change	NOUN
ma-301	130	25	of	of	ADP
ma-301	130	26	variable	variable	NOUN
ma-301	130	27	we	we	PRON
ma-301	130	28	find	find	VERB
ma-301	130	29	that	that	SCONJ
ma-301	130	30	for	for	ADP
ma-301	130	31	all	all	DET
ma-301	130	32	β	β	X
ma-301	130	33	∈	∈	PROPN
ma-301	130	34	r+q	r+q	PROPN
ma-301	130	35	,	,	PUNCT
ma-301	130	36	σβ	σβ	PROPN
ma-301	130	37	∈	∈	PROPN
ma-301	130	38	l2α(r+q	l2α(r+q	PROPN
ma-301	130	39	)	)	PUNCT
ma-301	131	1	and∥∥σβ∥∥2,q	and∥∥σβ∥∥2,q	PROPN
ma-301	131	2	,	,	PUNCT
ma-301	131	3	α	α	NOUN
ma-301	131	4	=	=	SYM
ma-301	131	5	1	1	NUM
ma-301	131	6	βα+1	βα+1	NUM
ma-301	131	7	‖σ‖2,q	‖σ‖2,q	PROPN
ma-301	131	8	,	,	PUNCT
ma-301	131	9	α	α	X
ma-301	131	10	.	.	PUNCT
ma-301	132	1	(	(	PUNCT
ma-301	132	2	3.2	3.2	NUM
ma-301	132	3	)	)	PUNCT
ma-301	132	4	remark	remark	NOUN
ma-301	132	5	3.1	3.1	NUM
ma-301	132	6	.	.	PUNCT
ma-301	133	1	according	accord	VERB
ma-301	133	2	to	to	ADP
ma-301	133	3	the	the	DET
ma-301	133	4	relation	relation	NOUN
ma-301	133	5	(	(	PUNCT
ma-301	133	6	2.13	2.13	NUM
ma-301	133	7	)	)	PUNCT
ma-301	133	8	we	we	PRON
ma-301	133	9	find	find	VERB
ma-301	133	10	that	that	SCONJ
ma-301	133	11	mq	mq	PROPN
ma-301	133	12	,	,	PUNCT
ma-301	133	13	σ	σ	PROPN
ma-301	133	14	,	,	PUNCT
ma-301	133	15	β(f	β(f	PROPN
ma-301	133	16	)	)	PUNCT
ma-301	133	17	(	(	PUNCT
ma-301	133	18	x	x	X
ma-301	133	19	)	)	PUNCT
ma-301	133	20	=	=	SYM
ma-301	133	21	(	(	PUNCT
ma-301	133	22	h−1q	h−1q	PROPN
ma-301	133	23	,	,	PUNCT
ma-301	133	24	α	α	PROPN
ma-301	133	25	(	(	PUNCT
ma-301	133	26	σβ	σβ	PROPN
ma-301	133	27	)	)	PUNCT
ma-301	133	28	∗α	∗α	PROPN
ma-301	133	29	f	f	PROPN
ma-301	133	30	)	)	PUNCT
ma-301	133	31	(	(	PUNCT
ma-301	133	32	x	x	NOUN
ma-301	133	33	)	)	PUNCT
ma-301	133	34	,	,	PUNCT
ma-301	133	35	(	(	PUNCT
ma-301	133	36	3.3	3.3	NUM
ma-301	133	37	)	)	PUNCT
ma-301	133	38	where	where	SCONJ
ma-301	133	39	h−1q	h−1q	NOUN
ma-301	133	40	,	,	PUNCT
ma-301	133	41	α	α	PROPN
ma-301	133	42	(	(	PUNCT
ma-301	133	43	σβ	σβ	NOUN
ma-301	133	44	)	)	PUNCT
ma-301	133	45	(	(	PUNCT
ma-301	133	46	x	x	X
ma-301	133	47	)	)	PUNCT
ma-301	133	48	=	=	SYM
ma-301	133	49	1	1	X
ma-301	133	50	β2α+2	β2α+2	PRON
ma-301	133	51	h−1q	h−1q	NOUN
ma-301	133	52	,	,	PUNCT
ma-301	133	53	α(σ	α(σ	NUM
ma-301	133	54	)	)	PUNCT
ma-301	133	55	(	(	PUNCT
ma-301	133	56	x	x	X
ma-301	133	57	β	β	X
ma-301	133	58	)	)	PUNCT
ma-301	133	59	.	.	PUNCT
ma-301	134	1	(	(	PUNCT
ma-301	134	2	3.4	3.4	NUM
ma-301	134	3	)	)	PUNCT
ma-301	134	4	we	we	PRON
ma-301	134	5	give	give	VERB
ma-301	134	6	some	some	DET
ma-301	134	7	properties	property	NOUN
ma-301	134	8	of	of	ADP
ma-301	134	9	the	the	DET
ma-301	134	10	q	q	ADJ
ma-301	134	11	-	-	PUNCT
ma-301	134	12	bessel	bessel	ADJ
ma-301	134	13	l2α	l2α	NOUN
ma-301	134	14	-	-	ADJ
ma-301	134	15	multiplier	multipli	ADJ
ma-301	134	16	operators	operator	NOUN
ma-301	134	17	.	.	PUNCT
ma-301	135	1	proposition	proposition	NOUN
ma-301	135	2	3.1	3.1	NUM
ma-301	135	3	.	.	PUNCT
ma-301	136	1	(	(	PUNCT
ma-301	136	2	i	i	NOUN
ma-301	136	3	)	)	PUNCT
ma-301	136	4	for	for	ADP
ma-301	136	5	every	every	DET
ma-301	136	6	σ	σ	PROPN
ma-301	136	7	∈	∈	PROPN
ma-301	136	8	l2α(r+q	l2α(r+q	PROPN
ma-301	136	9	)	)	PUNCT
ma-301	136	10	,	,	PUNCT
ma-301	136	11	and	and	CCONJ
ma-301	136	12	f	f	PROPN
ma-301	136	13	∈	∈	PROPN
ma-301	136	14	l1α(r+q	l1α(r+q	PROPN
ma-301	136	15	)	)	PUNCT
ma-301	136	16	,	,	PUNCT
ma-301	136	17	the	the	DET
ma-301	136	18	function	function	PROPN
ma-301	136	19	mq	mq	PROPN
ma-301	136	20	,	,	PUNCT
ma-301	136	21	σ	σ	PROPN
ma-301	136	22	,	,	PUNCT
ma-301	136	23	β(f	β(f	PROPN
ma-301	136	24	)	)	PUNCT
ma-301	136	25	belongs	belong	VERB
ma-301	136	26	to	to	ADP
ma-301	136	27	l2α(r+q	l2α(r+q	PROPN
ma-301	136	28	)	)	PUNCT
ma-301	136	29	,	,	PUNCT
ma-301	136	30	and	and	CCONJ
ma-301	136	31	we	we	PRON
ma-301	136	32	have	have	VERB
ma-301	136	33	∥∥mq	∥∥mq	NUM
ma-301	136	34	,	,	PUNCT
ma-301	136	35	σ	σ	PROPN
ma-301	136	36	,	,	PUNCT
ma-301	136	37	β(f	β(f	PROPN
ma-301	136	38	)	)	PUNCT
ma-301	137	1	∥∥	∥∥	PROPN
ma-301	137	2	2,q	2,q	NUM
ma-301	137	3	,	,	PUNCT
ma-301	137	4	α	α	PROPN
ma-301	137	5	≤	≤	NUM
ma-301	137	6	1	1	NUM
ma-301	137	7	βα+1	βα+1	PRON
ma-301	137	8	‖σ‖2,q	‖σ‖2,q	ADP
ma-301	137	9	,	,	PUNCT
ma-301	137	10	α‖f	α‖f	ADJ
ma-301	137	11	‖1,q	‖1,q	NOUN
ma-301	137	12	,	,	PUNCT
ma-301	137	13	α	α	X
ma-301	137	14	.	.	PUNCT
ma-301	137	15	(	(	PUNCT
ma-301	137	16	ii	ii	NOUN
ma-301	137	17	)	)	PUNCT
ma-301	137	18	for	for	ADP
ma-301	137	19	every	every	DET
ma-301	137	20	σ	σ	PROPN
ma-301	137	21	∈	∈	PROPN
ma-301	137	22	l∞α	l∞α	NOUN
ma-301	137	23	(	(	PUNCT
ma-301	137	24	r+q	r+q	PROPN
ma-301	137	25	)	)	PUNCT
ma-301	137	26	,	,	PUNCT
ma-301	137	27	and	and	CCONJ
ma-301	137	28	for	for	ADP
ma-301	137	29	every	every	DET
ma-301	137	30	f	f	PROPN
ma-301	137	31	∈	∈	PROPN
ma-301	137	32	l2α(r+q	l2α(r+q	PROPN
ma-301	137	33	)	)	PUNCT
ma-301	137	34	,	,	PUNCT
ma-301	137	35	the	the	DET
ma-301	137	36	function	function	PROPN
ma-301	137	37	mq	mq	PROPN
ma-301	137	38	,	,	PUNCT
ma-301	137	39	σ	σ	PROPN
ma-301	137	40	,	,	PUNCT
ma-301	137	41	β(f	β(f	PROPN
ma-301	137	42	)	)	PUNCT
ma-301	137	43	belongs	belong	VERB
ma-301	137	44	to	to	ADP
ma-301	137	45	l2α(r+q	l2α(r+q	PROPN
ma-301	137	46	)	)	PUNCT
ma-301	137	47	,	,	PUNCT
ma-301	137	48	and	and	CCONJ
ma-301	137	49	we	we	PRON
ma-301	137	50	have	have	VERB
ma-301	137	51	∥∥mq	∥∥mq	NUM
ma-301	137	52	,	,	PUNCT
ma-301	137	53	σ	σ	PROPN
ma-301	137	54	,	,	PUNCT
ma-301	137	55	β(f	β(f	PROPN
ma-301	137	56	)	)	PUNCT
ma-301	138	1	∥∥	∥∥	PROPN
ma-301	138	2	2,q	2,q	NUM
ma-301	138	3	,	,	PUNCT
ma-301	138	4	α	α	PROPN
ma-301	138	5	≤	≤	PUNCT
ma-301	138	6	‖σ‖∞,q	‖σ‖∞,q	PRON
ma-301	138	7	,	,	PUNCT
ma-301	138	8	α‖f	α‖f	ADJ
ma-301	138	9	‖2,q	‖2,q	NOUN
ma-301	138	10	,	,	PUNCT
ma-301	138	11	α	α	NOUN
ma-301	138	12	(	(	PUNCT
ma-301	138	13	3.5	3.5	NUM
ma-301	138	14	)	)	PUNCT
ma-301	138	15	(	(	PUNCT
ma-301	138	16	iii	iii	NOUN
ma-301	138	17	)	)	PUNCT
ma-301	138	18	for	for	ADP
ma-301	138	19	every	every	DET
ma-301	138	20	σ	σ	PROPN
ma-301	138	21	∈	∈	PROPN
ma-301	138	22	l2α(r+q	l2α(r+q	PROPN
ma-301	138	23	)	)	PUNCT
ma-301	138	24	,	,	PUNCT
ma-301	138	25	and	and	CCONJ
ma-301	138	26	for	for	ADP
ma-301	138	27	every	every	DET
ma-301	138	28	f	f	PROPN
ma-301	138	29	∈	∈	PROPN
ma-301	138	30	l2α(r+q	l2α(r+q	PROPN
ma-301	138	31	)	)	PUNCT
ma-301	138	32	,	,	PUNCT
ma-301	138	33	mq	mq	PROPN
ma-301	138	34	,	,	PUNCT
ma-301	138	35	σ	σ	PROPN
ma-301	138	36	,	,	PUNCT
ma-301	138	37	β(f	β(f	NUM
ma-301	138	38	)	)	PUNCT
ma-301	139	1	∈	∈	PROPN
ma-301	139	2	l∞α	l∞α	NOUN
ma-301	139	3	(	(	PUNCT
ma-301	139	4	r+q	r+q	PROPN
ma-301	139	5	)	)	PUNCT
ma-301	139	6	,	,	PUNCT
ma-301	139	7	and	and	CCONJ
ma-301	139	8	we	we	PRON
ma-301	139	9	have	have	VERB
ma-301	139	10	mq	mq	PROPN
ma-301	139	11	,	,	PUNCT
ma-301	139	12	σ	σ	PROPN
ma-301	139	13	,	,	PUNCT
ma-301	139	14	β(f	β(f	PROPN
ma-301	139	15	)	)	PUNCT
ma-301	139	16	(	(	PUNCT
ma-301	139	17	x	x	X
ma-301	139	18	)	)	PUNCT
ma-301	139	19	=	=	SYM
ma-301	139	20	∫	∫	PROPN
ma-301	139	21	∞	∞	NOUN
ma-301	139	22	0	0	NUM
ma-301	140	1	σ(βλ)jα(λx	σ(βλ)jα(λx	NOUN
ma-301	140	2	;	;	PUNCT
ma-301	140	3	q2)hq	q2)hq	NOUN
ma-301	140	4	,	,	PUNCT
ma-301	140	5	α(f	α(f	PROPN
ma-301	140	6	)	)	PUNCT
ma-301	140	7	(	(	PUNCT
ma-301	140	8	λ)dµq	λ)dµq	PROPN
ma-301	140	9	,	,	PUNCT
ma-301	140	10	α(λ	α(λ	PROPN
ma-301	140	11	)	)	PUNCT
ma-301	140	12	,	,	PUNCT
ma-301	140	13	a.e	a.e	PROPN
ma-301	140	14	x	x	SYM
ma-301	140	15	∈	∈	PROPN
ma-301	140	16	r+q	r+q	NUM
ma-301	140	17	(	(	PUNCT
ma-301	140	18	3.6	3.6	NUM
ma-301	140	19	)	)	PUNCT
ma-301	140	20	and	and	CCONJ
ma-301	140	21	∥∥mq	∥∥mq	PROPN
ma-301	140	22	,	,	PUNCT
ma-301	140	23	σ	σ	PROPN
ma-301	140	24	,	,	PUNCT
ma-301	140	25	β(f	β(f	PROPN
ma-301	140	26	)	)	PUNCT
ma-301	140	27	∥∥	∥∥	X
ma-301	140	28	∞,q	∞,q	NOUN
ma-301	140	29	,	,	PUNCT
ma-301	140	30	α	α	PROPN
ma-301	140	31	≤	≤	NUM
ma-301	140	32	1	1	NUM
ma-301	140	33	βα+1	βα+1	PRON
ma-301	140	34	‖σ‖2,q	‖σ‖2,q	PROPN
ma-301	140	35	,	,	PUNCT
ma-301	140	36	α‖f	α‖f	ADJ
ma-301	140	37	‖2,q	‖2,q	NOUN
ma-301	140	38	,	,	PUNCT
ma-301	140	39	α	α	NOUN
ma-301	140	40	.	.	PUNCT
ma-301	141	1	proof	proof	NOUN
ma-301	141	2	.	.	PUNCT
ma-301	142	1	(	(	PUNCT
ma-301	142	2	i	i	NOUN
ma-301	142	3	)	)	PUNCT
ma-301	142	4	by	by	ADP
ma-301	142	5	using	use	VERB
ma-301	142	6	the	the	DET
ma-301	142	7	relations	relation	NOUN
ma-301	142	8	(	(	PUNCT
ma-301	142	9	2.12),(3.3	2.12),(3.3	NUM
ma-301	142	10	)	)	PUNCT
ma-301	142	11	we	we	PRON
ma-301	142	12	find	find	VERB
ma-301	142	13	that∥∥mq	that∥∥mq	PROPN
ma-301	142	14	,	,	PUNCT
ma-301	142	15	σ	σ	PROPN
ma-301	142	16	,	,	PUNCT
ma-301	142	17	β(f	β(f	PROPN
ma-301	142	18	)	)	PUNCT
ma-301	142	19	∥∥2	∥∥2	PROPN
ma-301	143	1	2,q	2,q	NUM
ma-301	143	2	,	,	PUNCT
ma-301	143	3	α	α	NOUN
ma-301	143	4	=	=	SYM
ma-301	143	5	∥∥h−1q	∥∥h−1q	PROPN
ma-301	143	6	,	,	PUNCT
ma-301	143	7	α	α	PROPN
ma-301	143	8	(	(	PUNCT
ma-301	143	9	σβ	σβ	NOUN
ma-301	143	10	)	)	PUNCT
ma-301	143	11	∗q	∗q	PROPN
ma-301	143	12	f	f	PROPN
ma-301	143	13	∥∥22,q	∥∥22,q	PROPN
ma-301	143	14	,	,	PUNCT
ma-301	143	15	α	α	NOUN
ma-301	143	16	≤	≤	NOUN
ma-301	143	17	‖f	‖f	ADP
ma-301	143	18	‖21,q	‖21,q	NOUN
ma-301	143	19	,	,	PUNCT
ma-301	143	20	α	α	PROPN
ma-301	143	21	∥∥h−1q	∥∥h−1q	PROPN
ma-301	143	22	,	,	PUNCT
ma-301	143	23	α	α	X
ma-301	143	24	(	(	PUNCT
ma-301	143	25	σβ)∥∥22,q	σβ)∥∥22,q	NOUN
ma-301	143	26	,	,	PUNCT
ma-301	143	27	αplancherel	αplancherel	NOUN
ma-301	143	28	’s	’s	PART
ma-301	143	29	formula	formula	NOUN
ma-301	143	30	(	(	PUNCT
ma-301	143	31	2.7	2.7	NUM
ma-301	143	32	)	)	PUNCT
ma-301	143	33	and	and	CCONJ
ma-301	143	34	the	the	DET
ma-301	143	35	relation	relation	NOUN
ma-301	143	36	(	(	PUNCT
ma-301	143	37	3.2	3.2	NUM
ma-301	143	38	)	)	PUNCT
ma-301	143	39	gives	give	VERB
ma-301	143	40	the	the	DET
ma-301	143	41	desired	desire	VERB
ma-301	143	42	result.(ii	result.(ii	NOUN
ma-301	143	43	)	)	PUNCT
ma-301	143	44	is	be	AUX
ma-301	143	45	a	a	DET
ma-301	143	46	consequence	consequence	NOUN
ma-301	143	47	of	of	ADP
ma-301	143	48	plancherel	plancherel	NOUN
ma-301	143	49	’s	’s	PART
ma-301	143	50	formula	formula	NOUN
ma-301	143	51	(	(	PUNCT
ma-301	143	52	2.7	2.7	NUM
ma-301	143	53	)	)	PUNCT
ma-301	143	54	.	.	PUNCT
ma-301	144	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	144	2	eur	eur	PROPN
ma-301	144	3	.	.	PUNCT
ma-301	145	1	j.	j.	PROPN
ma-301	145	2	math	math	PROPN
ma-301	145	3	.	.	PUNCT
ma-301	146	1	anal	anal	PROPN
ma-301	146	2	.	.	PUNCT
ma-301	147	1	10.28924	10.28924	NUM
ma-301	147	2	/	/	SYM
ma-301	147	3	ada	ada	PROPN
ma-301	147	4	/	/	SYM
ma-301	147	5	ma.5.8	ma.5.8	PROPN
ma-301	147	6	7(iii	7(iii	NUM
ma-301	147	7	)	)	PUNCT
ma-301	147	8	is	be	AUX
ma-301	147	9	a	a	DET
ma-301	147	10	consequence	consequence	NOUN
ma-301	147	11	of	of	ADP
ma-301	147	12	the	the	DET
ma-301	147	13	relations	relation	NOUN
ma-301	147	14	(	(	PUNCT
ma-301	147	15	2.7),(2.12),(3.2	2.7),(2.12),(3.2	NUM
ma-301	147	16	)	)	PUNCT
ma-301	147	17	and	and	CCONJ
ma-301	147	18	(	(	PUNCT
ma-301	147	19	3.3	3.3	NUM
ma-301	147	20	)	)	PUNCT
ma-301	147	21	,	,	PUNCT
ma-301	147	22	on	on	ADP
ma-301	147	23	the	the	DET
ma-301	147	24	other	other	ADJ
ma-301	147	25	hand	hand	NOUN
ma-301	147	26	the	the	DET
ma-301	147	27	relation	relation	NOUN
ma-301	147	28	(	(	PUNCT
ma-301	147	29	3.6)follows	3.6)follows	NUM
ma-301	147	30	from	from	ADP
ma-301	147	31	inversion	inversion	NOUN
ma-301	147	32	formula	formula	NOUN
ma-301	147	33	(	(	PUNCT
ma-301	147	34	2.5	2.5	NUM
ma-301	147	35	)	)	PUNCT
ma-301	147	36	.	.	PUNCT
ma-301	148	1	�	�	PROPN
ma-301	148	2	in	in	ADP
ma-301	148	3	the	the	DET
ma-301	148	4	following	follow	VERB
ma-301	148	5	result	result	NOUN
ma-301	148	6	,	,	PUNCT
ma-301	148	7	we	we	PRON
ma-301	148	8	give	give	VERB
ma-301	148	9	plancherel	plancherel	NOUN
ma-301	148	10	’s	’s	PART
ma-301	148	11	and	and	CCONJ
ma-301	148	12	pointwise	pointwise	VERB
ma-301	148	13	reproducing	reproduce	VERB
ma-301	148	14	inversion	inversion	NOUN
ma-301	148	15	formula	formula	NOUN
ma-301	148	16	for	for	ADP
ma-301	148	17	the	the	DET
ma-301	148	18	q	q	ADJ
ma-301	148	19	-	-	ADJ
ma-301	148	20	bessel	bessel	ADJ
ma-301	148	21	l2α	l2α	NOUN
ma-301	148	22	-	-	ADJ
ma-301	148	23	multiplier	multipli	ADJ
ma-301	148	24	operators	operator	NOUN
ma-301	148	25	.	.	PUNCT
ma-301	149	1	theorem	theorem	VERB
ma-301	149	2	3.1	3.1	NUM
ma-301	149	3	.	.	PUNCT
ma-301	150	1	let	let	VERB
ma-301	150	2	σ	σ	X
ma-301	150	3	∈	∈	PROPN
ma-301	150	4	l2α(r+q	l2α(r+q	PROPN
ma-301	150	5	)	)	PUNCT
ma-301	151	1	satisfying	satisfy	VERB
ma-301	151	2	the	the	DET
ma-301	151	3	admissibility	admissibility	NOUN
ma-301	151	4	condition:∫	condition:∫	PROPN
ma-301	151	5	∞	∞	PROPN
ma-301	151	6	0	0	NUM
ma-301	152	1	∣∣σβ(λ	∣∣σβ(λ	NUM
ma-301	152	2	)	)	PUNCT
ma-301	152	3	∣∣2	∣∣2	PROPN
ma-301	152	4	dq(β	dq(β	NOUN
ma-301	152	5	)	)	PUNCT
ma-301	152	6	β	β	NOUN
ma-301	152	7	=	=	SYM
ma-301	152	8	1	1	NUM
ma-301	152	9	,	,	PUNCT
ma-301	152	10	λ	λ	PROPN
ma-301	152	11	∈	∈	PROPN
ma-301	152	12	r.	r.	PROPN
ma-301	152	13	(	(	PUNCT
ma-301	152	14	3.7	3.7	NUM
ma-301	152	15	)	)	PUNCT
ma-301	152	16	(	(	PUNCT
ma-301	152	17	i	i	NOUN
ma-301	152	18	)	)	PUNCT
ma-301	152	19	(	(	PUNCT
ma-301	152	20	plancherel	plancherel	NOUN
ma-301	152	21	formula	formula	NOUN
ma-301	152	22	)	)	PUNCT
ma-301	152	23	for	for	ADP
ma-301	152	24	all	all	DET
ma-301	152	25	f	f	PROPN
ma-301	152	26	in	in	ADP
ma-301	152	27	l2α(r+q	l2α(r+q	PROPN
ma-301	152	28	)	)	PUNCT
ma-301	152	29	,	,	PUNCT
ma-301	152	30	we	we	PRON
ma-301	152	31	have∫	have∫	VERB
ma-301	152	32	∞	∞	PROPN
ma-301	152	33	0	0	NUM
ma-301	152	34	|f	|f	PROPN
ma-301	152	35	(	(	PUNCT
ma-301	152	36	x)|2dµq	x)|2dµq	NOUN
ma-301	152	37	,	,	PUNCT
ma-301	152	38	α(x	α(x	NOUN
ma-301	152	39	)	)	PUNCT
ma-301	152	40	=	=	SYM
ma-301	153	1	∫	∫	PROPN
ma-301	154	1	∞	∞	PROPN
ma-301	154	2	0	0	NUM
ma-301	155	1	∥∥mq	∥∥mq	PROPN
ma-301	155	2	,	,	PUNCT
ma-301	155	3	σ	σ	PROPN
ma-301	155	4	,	,	PUNCT
ma-301	155	5	β(f	β(f	PROPN
ma-301	155	6	)	)	PUNCT
ma-301	155	7	∥∥2	∥∥2	PROPN
ma-301	156	1	2,q	2,q	NUM
ma-301	156	2	,	,	PUNCT
ma-301	156	3	α	α	NOUN
ma-301	156	4	dq(β	dq(β	NOUN
ma-301	156	5	)	)	PUNCT
ma-301	156	6	β	β	X
ma-301	156	7	.	.	PUNCT
ma-301	157	1	(	(	PUNCT
ma-301	157	2	3.8	3.8	NUM
ma-301	157	3	)	)	PUNCT
ma-301	157	4	(	(	PUNCT
ma-301	157	5	ii	ii	NOUN
ma-301	157	6	)	)	PUNCT
ma-301	157	7	(	(	PUNCT
ma-301	157	8	first	first	PROPN
ma-301	157	9	calderón	calderón	PROPN
ma-301	157	10	’s	’s	PART
ma-301	157	11	formula	formula	NOUN
ma-301	157	12	)	)	PUNCT
ma-301	157	13	let	let	VERB
ma-301	157	14	f	f	PROPN
ma-301	157	15	∈	∈	PROPN
ma-301	157	16	l1α(r+q	l1α(r+q	PROPN
ma-301	157	17	)	)	PUNCT
ma-301	157	18	such	such	ADJ
ma-301	157	19	that	that	DET
ma-301	157	20	hq	hq	NOUN
ma-301	157	21	,	,	PUNCT
ma-301	157	22	α(f	α(f	NUM
ma-301	157	23	)	)	PUNCT
ma-301	158	1	∈	∈	PROPN
ma-301	158	2	l1α(r+q	l1α(r+q	PROPN
ma-301	158	3	)	)	PUNCT
ma-301	159	1	then	then	ADV
ma-301	159	2	we	we	PRON
ma-301	159	3	have	have	VERB
ma-301	159	4	f	f	PROPN
ma-301	159	5	(	(	PUNCT
ma-301	159	6	x	x	NOUN
ma-301	159	7	)	)	PUNCT
ma-301	159	8	=	=	SYM
ma-301	160	1	∫	∫	PROPN
ma-301	160	2	∞	∞	PROPN
ma-301	160	3	0	0	NUM
ma-301	161	1	(	(	PUNCT
ma-301	161	2	mq	mq	PROPN
ma-301	161	3	,	,	PUNCT
ma-301	161	4	σ	σ	PROPN
ma-301	161	5	,	,	PUNCT
ma-301	161	6	β(f	β(f	NUM
ma-301	161	7	)	)	PUNCT
ma-301	161	8	∗α	∗α	PROPN
ma-301	161	9	h−1q	h−1q	PROPN
ma-301	161	10	,	,	PUNCT
ma-301	161	11	α	α	PROPN
ma-301	161	12	(	(	PUNCT
ma-301	161	13	σβ	σβ	NOUN
ma-301	161	14	)	)	PUNCT
ma-301	161	15	)	)	PUNCT
ma-301	161	16	(	(	PUNCT
ma-301	161	17	x	x	X
ma-301	161	18	)	)	PUNCT
ma-301	161	19	dβ	dβ	ADP
ma-301	161	20	β	β	X
ma-301	161	21	,	,	PUNCT
ma-301	161	22	a.e	a.e	PROPN
ma-301	161	23	.	.	PROPN
ma-301	161	24	x	x	PROPN
ma-301	161	25	∈	∈	PROPN
ma-301	161	26	r.	r.	NOUN
ma-301	161	27	proof	proof	NOUN
ma-301	161	28	.	.	PUNCT
ma-301	162	1	(	(	PUNCT
ma-301	162	2	i	i	NOUN
ma-301	162	3	)	)	PUNCT
ma-301	162	4	by	by	ADP
ma-301	162	5	using	use	VERB
ma-301	162	6	the	the	DET
ma-301	162	7	relations	relation	NOUN
ma-301	162	8	(	(	PUNCT
ma-301	162	9	2.14	2.14	NUM
ma-301	162	10	)	)	PUNCT
ma-301	162	11	and	and	CCONJ
ma-301	162	12	(	(	PUNCT
ma-301	162	13	3.3	3.3	NUM
ma-301	162	14	)	)	PUNCT
ma-301	162	15	we	we	PRON
ma-301	162	16	get∫	get∫	VERB
ma-301	162	17	∞	∞	PROPN
ma-301	162	18	0	0	NUM
ma-301	163	1	∥∥mq	∥∥mq	PROPN
ma-301	163	2	,	,	PUNCT
ma-301	163	3	σ	σ	PROPN
ma-301	163	4	,	,	PUNCT
ma-301	163	5	β(f	β(f	PROPN
ma-301	163	6	)	)	PUNCT
ma-301	164	1	∥∥2	∥∥2	PROPN
ma-301	165	1	2,q	2,q	NUM
ma-301	165	2	,	,	PUNCT
ma-301	165	3	α	α	NOUN
ma-301	165	4	dq(β	dq(β	X
ma-301	165	5	)	)	PUNCT
ma-301	165	6	β	β	X
ma-301	166	1	=	=	SYM
ma-301	167	1	∫	∫	PROPN
ma-301	168	1	∞	∞	NUM
ma-301	168	2	0	0	PUNCT
ma-301	169	1	[	[	X
ma-301	169	2	∫	∫	X
ma-301	169	3	∞	∞	PROPN
ma-301	169	4	0	0	NUM
ma-301	169	5	∣∣mq	∣∣mq	PROPN
ma-301	169	6	,	,	PUNCT
ma-301	169	7	σ	σ	PROPN
ma-301	169	8	,	,	PUNCT
ma-301	169	9	β(f	β(f	PROPN
ma-301	169	10	)	)	PUNCT
ma-301	169	11	(	(	PUNCT
ma-301	169	12	x	x	X
ma-301	169	13	)	)	PUNCT
ma-301	169	14	∣∣2	∣∣2	PROPN
ma-301	169	15	dµq	dµq	PROPN
ma-301	169	16	,	,	PUNCT
ma-301	169	17	α(x	α(x	NOUN
ma-301	169	18	)	)	PUNCT
ma-301	169	19	]	]	PUNCT
ma-301	170	1	dq(β	dq(β	X
ma-301	170	2	)	)	PUNCT
ma-301	170	3	β	β	X
ma-301	170	4	=	=	SYM
ma-301	170	5	∫	∫	PROPN
ma-301	170	6	∞	∞	NUM
ma-301	170	7	0	0	PUNCT
ma-301	171	1	[	[	X
ma-301	171	2	∫	∫	X
ma-301	171	3	∞	∞	NUM
ma-301	171	4	0	0	NUM
ma-301	171	5	|hq	|hq	PROPN
ma-301	171	6	,	,	PUNCT
ma-301	171	7	α(f	α(f	PROPN
ma-301	171	8	)	)	PUNCT
ma-301	171	9	(	(	PUNCT
ma-301	171	10	λ)|2	λ)|2	PROPN
ma-301	171	11	dµq	dµq	PROPN
ma-301	171	12	,	,	PUNCT
ma-301	171	13	α(λ	α(λ	PROPN
ma-301	171	14	)	)	PUNCT
ma-301	171	15	]	]	PUNCT
ma-301	171	16	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	171	17	)	)	PUNCT
ma-301	171	18	∣∣2	∣∣2	PROPN
ma-301	171	19	dq(β	dq(β	NOUN
ma-301	171	20	)	)	PUNCT
ma-301	171	21	βthe	βthe	ADJ
ma-301	171	22	admissibility	admissibility	NOUN
ma-301	171	23	condition	condition	NOUN
ma-301	171	24	(	(	PUNCT
ma-301	171	25	3.7	3.7	NUM
ma-301	171	26	)	)	PUNCT
ma-301	171	27	and	and	CCONJ
ma-301	171	28	plancherel	plancherel	PROPN
ma-301	171	29	’s	’s	PART
ma-301	171	30	formula	formula	NOUN
ma-301	171	31	(	(	PUNCT
ma-301	171	32	2.7	2.7	NUM
ma-301	171	33	)	)	PUNCT
ma-301	171	34	gives	give	VERB
ma-301	171	35	the	the	DET
ma-301	171	36	desired	desire	VERB
ma-301	171	37	result.(ii	result.(ii	NOUN
ma-301	171	38	)	)	PUNCT
ma-301	171	39	let	let	VERB
ma-301	171	40	f	f	PROPN
ma-301	171	41	∈	∈	PROPN
ma-301	171	42	l1α(r+q	l1α(r+q	PROPN
ma-301	171	43	)	)	PUNCT
ma-301	171	44	such	such	ADJ
ma-301	171	45	that	that	DET
ma-301	171	46	hq	hq	NOUN
ma-301	171	47	,	,	PUNCT
ma-301	171	48	α(f	α(f	NUM
ma-301	171	49	)	)	PUNCT
ma-301	172	1	∈	∈	PROPN
ma-301	172	2	l1α(r+q	l1α(r+q	PROPN
ma-301	172	3	)	)	PUNCT
ma-301	172	4	,	,	PUNCT
ma-301	172	5	by	by	ADP
ma-301	172	6	using	use	VERB
ma-301	172	7	the	the	DET
ma-301	172	8	relations	relation	NOUN
ma-301	172	9	(	(	PUNCT
ma-301	172	10	2.6),(2.11	2.6),(2.11	NUM
ma-301	172	11	)	)	PUNCT
ma-301	172	12	we	we	PRON
ma-301	172	13	find	find	VERB
ma-301	172	14	that∫	that∫	NOUN
ma-301	172	15	∞	∞	PROPN
ma-301	172	16	0	0	NUM
ma-301	173	1	(	(	PUNCT
ma-301	173	2	mq	mq	PROPN
ma-301	173	3	,	,	PUNCT
ma-301	173	4	σ	σ	PROPN
ma-301	173	5	,	,	PUNCT
ma-301	173	6	β(f	β(f	NUM
ma-301	173	7	)	)	PUNCT
ma-301	174	1	∗αh−1q	∗αh−1q	PROPN
ma-301	174	2	,	,	PUNCT
ma-301	174	3	α	α	PROPN
ma-301	174	4	(	(	PUNCT
ma-301	174	5	σβ	σβ	NOUN
ma-301	174	6	)	)	PUNCT
ma-301	174	7	)	)	PUNCT
ma-301	174	8	(	(	PUNCT
ma-301	174	9	x	x	X
ma-301	174	10	)	)	PUNCT
ma-301	174	11	dβ	dβ	ADP
ma-301	174	12	β	β	X
ma-301	174	13	=	=	SYM
ma-301	174	14	∫	∫	PROPN
ma-301	174	15	∞	∞	NUM
ma-301	174	16	0	0	PUNCT
ma-301	175	1	[	[	X
ma-301	175	2	∫	∫	X
ma-301	175	3	∞	∞	NOUN
ma-301	175	4	0	0	PUNCT
ma-301	175	5	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	175	6	)	)	PUNCT
ma-301	175	7	∣∣2hq	∣∣2hq	NOUN
ma-301	175	8	,	,	PUNCT
ma-301	175	9	α(f	α(f	PROPN
ma-301	175	10	)	)	PUNCT
ma-301	175	11	(	(	PUNCT
ma-301	175	12	λ)jα(λx	λ)jα(λx	X
ma-301	175	13	;	;	PUNCT
ma-301	175	14	q2)dµq	q2)dµq	NOUN
ma-301	175	15	,	,	PUNCT
ma-301	175	16	α(λ	α(λ	PROPN
ma-301	175	17	)	)	PUNCT
ma-301	175	18	]	]	PUNCT
ma-301	176	1	dq(β	dq(β	X
ma-301	176	2	)	)	PUNCT
ma-301	176	3	β	β	X
ma-301	176	4	=	=	SYM
ma-301	176	5	∫	∫	PROPN
ma-301	176	6	∞	∞	NUM
ma-301	176	7	0	0	PUNCT
ma-301	177	1	[	[	X
ma-301	177	2	∫	∫	X
ma-301	177	3	∞	∞	PROPN
ma-301	177	4	0	0	NUM
ma-301	177	5	hq	hq	PROPN
ma-301	177	6	,	,	PUNCT
ma-301	177	7	α(f	α(f	PROPN
ma-301	177	8	)	)	PUNCT
ma-301	177	9	(	(	PUNCT
ma-301	177	10	λ)jα(λx	λ)jα(λx	X
ma-301	177	11	;	;	PUNCT
ma-301	177	12	q2)dµq	q2)dµq	NOUN
ma-301	177	13	,	,	PUNCT
ma-301	177	14	α(λ	α(λ	PROPN
ma-301	177	15	)	)	PUNCT
ma-301	177	16	]	]	PUNCT
ma-301	178	1	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	178	2	)	)	PUNCT
ma-301	178	3	∣∣2	∣∣2	PROPN
ma-301	178	4	dq(β	dq(β	NOUN
ma-301	178	5	)	)	PUNCT
ma-301	178	6	βthe	βthe	ADJ
ma-301	178	7	admissibility	admissibility	NOUN
ma-301	178	8	condition	condition	NOUN
ma-301	178	9	(	(	PUNCT
ma-301	178	10	3.7),inversion	3.7),inversion	NUM
ma-301	178	11	formula	formula	NOUN
ma-301	178	12	(	(	PUNCT
ma-301	178	13	2.5	2.5	NUM
ma-301	178	14	)	)	PUNCT
ma-301	178	15	gives	give	VERB
ma-301	178	16	the	the	DET
ma-301	178	17	desired	desire	VERB
ma-301	178	18	result	result	NOUN
ma-301	178	19	.	.	PUNCT
ma-301	179	1	�	�	PROPN
ma-301	179	2	to	to	PART
ma-301	179	3	establish	establish	VERB
ma-301	179	4	the	the	DET
ma-301	179	5	second	second	ADJ
ma-301	179	6	calderon	calderon	NOUN
ma-301	179	7	’s	’s	PART
ma-301	179	8	reproducing	reproduce	VERB
ma-301	179	9	formula	formula	NOUN
ma-301	179	10	for	for	ADP
ma-301	179	11	the	the	DET
ma-301	179	12	q	q	ADJ
ma-301	179	13	-	-	ADJ
ma-301	179	14	bessel	bessel	ADJ
ma-301	179	15	l2α	l2α	ADJ
ma-301	179	16	-	-	ADJ
ma-301	179	17	multiplier	multipli	ADJ
ma-301	179	18	operators	operator	NOUN
ma-301	179	19	,	,	PUNCT
ma-301	179	20	we	we	PRON
ma-301	179	21	need	need	VERB
ma-301	179	22	the	the	DET
ma-301	179	23	following	follow	VERB
ma-301	179	24	technical	technical	ADJ
ma-301	179	25	result	result	NOUN
ma-301	179	26	.	.	PUNCT
ma-301	180	1	proposition	proposition	NOUN
ma-301	180	2	3.2	3.2	NUM
ma-301	180	3	.	.	PUNCT
ma-301	181	1	let	let	VERB
ma-301	181	2	σ	σ	X
ma-301	181	3	∈	∈	PROPN
ma-301	181	4	l2α(r+q	l2α(r+q	PROPN
ma-301	181	5	)	)	PUNCT
ma-301	181	6	∩	∩	ADJ
ma-301	181	7	l∞α	l∞α	NOUN
ma-301	181	8	(	(	PUNCT
ma-301	181	9	r+q	r+q	PROPN
ma-301	181	10	)	)	PUNCT
ma-301	181	11	satisfy	satisfy	VERB
ma-301	181	12	the	the	DET
ma-301	181	13	admissibility	admissibility	NOUN
ma-301	181	14	condition	condition	NOUN
ma-301	181	15	(	(	PUNCT
ma-301	181	16	3.7	3.7	NUM
ma-301	181	17	)	)	PUNCT
ma-301	181	18	then	then	ADV
ma-301	182	1	the	the	DET
ma-301	182	2	function	function	NOUN
ma-301	182	3	defined	define	VERB
ma-301	182	4	by	by	ADP
ma-301	182	5	φγ	φγ	NOUN
ma-301	182	6	,	,	PUNCT
ma-301	182	7	δ(λ	δ(λ	PROPN
ma-301	182	8	)	)	PUNCT
ma-301	182	9	=	=	SYM
ma-301	182	10	∫	∫	PROPN
ma-301	182	11	δ	δ	PROPN
ma-301	182	12	γ	γ	PROPN
ma-301	182	13	∣∣σβ(λ	∣∣σβ(λ	PROPN
ma-301	182	14	)	)	PUNCT
ma-301	182	15	∣∣2	∣∣2	PROPN
ma-301	182	16	dq(β	dq(β	NOUN
ma-301	182	17	)	)	PUNCT
ma-301	182	18	β	β	NOUN
ma-301	182	19	belongs	belong	VERB
ma-301	182	20	to	to	ADP
ma-301	182	21	l2α(r+q	l2α(r+q	PROPN
ma-301	182	22	)	)	PUNCT
ma-301	182	23	∩	∩	ADJ
ma-301	182	24	l∞α	l∞α	NOUN
ma-301	182	25	(	(	PUNCT
ma-301	182	26	r+q	r+q	PROPN
ma-301	182	27	)	)	PUNCT
ma-301	182	28	for	for	ADP
ma-301	182	29	all	all	PRON
ma-301	182	30	0	0	NUM
ma-301	182	31	<	<	X
ma-301	182	32	γ	γ	X
ma-301	182	33	<	<	X
ma-301	182	34	δ	δ	PROPN
ma-301	182	35	<	<	X
ma-301	182	36	∞.	∞.	PROPN
ma-301	182	37	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	182	38	eur	eur	PROPN
ma-301	182	39	.	.	PUNCT
ma-301	183	1	j.	j.	PROPN
ma-301	183	2	math	math	PROPN
ma-301	183	3	.	.	PUNCT
ma-301	184	1	anal	anal	PROPN
ma-301	184	2	.	.	PUNCT
ma-301	185	1	10.28924	10.28924	NUM
ma-301	185	2	/	/	SYM
ma-301	185	3	ada	ada	PROPN
ma-301	185	4	/	/	PROPN
ma-301	185	5	ma.5.8	ma.5.8	PROPN
ma-301	185	6	8	8	NUM
ma-301	185	7	proof	proof	NOUN
ma-301	185	8	.	.	PUNCT
ma-301	186	1	using	use	VERB
ma-301	186	2	hölder	hölder	PROPN
ma-301	186	3	’s	’s	PART
ma-301	186	4	inequality	inequality	NOUN
ma-301	186	5	for	for	ADP
ma-301	186	6	the	the	DET
ma-301	186	7	measure	measure	NOUN
ma-301	186	8	dq(β	dq(β	NOUN
ma-301	186	9	)	)	PUNCT
ma-301	186	10	β	β	NOUN
ma-301	186	11	and	and	CCONJ
ma-301	186	12	the	the	DET
ma-301	186	13	relation	relation	NOUN
ma-301	186	14	(	(	PUNCT
ma-301	186	15	3.2	3.2	NUM
ma-301	186	16	)	)	PUNCT
ma-301	186	17	we	we	PRON
ma-301	186	18	find	find	VERB
ma-301	186	19	that∥∥φγ	that∥∥φγ	PROPN
ma-301	186	20	,	,	PUNCT
ma-301	186	21	δ	δ	PROPN
ma-301	186	22	∥∥2	∥∥2	PROPN
ma-301	187	1	2,q	2,q	NUM
ma-301	187	2	,	,	PUNCT
ma-301	187	3	α	α	NOUN
ma-301	187	4	≤	≤	PUNCT
ma-301	187	5	log(δ	log(δ	PROPN
ma-301	187	6	/	/	SYM
ma-301	187	7	γ)‖σ‖22,q	γ)‖σ‖22,q	PROPN
ma-301	187	8	,	,	PUNCT
ma-301	187	9	α‖σ‖2∞,q	α‖σ‖2∞,q	PROPN
ma-301	187	10	,	,	PUNCT
ma-301	187	11	α	α	PRON
ma-301	187	12	∫	∫	PROPN
ma-301	187	13	δ	δ	PROPN
ma-301	187	14	γ	γ	NOUN
ma-301	187	15	dq(β	dq(β	NOUN
ma-301	187	16	)	)	PUNCT
ma-301	187	17	βα+2	βα+2	PUNCT
ma-301	188	1	<	<	X
ma-301	188	2	∞	∞	NUM
ma-301	188	3	so	so	ADV
ma-301	188	4	φγ	φγ	NOUN
ma-301	188	5	,	,	PUNCT
ma-301	188	6	δ	δ	PROPN
ma-301	188	7	∈	∈	PROPN
ma-301	188	8	l2α(r+q	l2α(r+q	PROPN
ma-301	188	9	)	)	PUNCT
ma-301	188	10	,	,	PUNCT
ma-301	188	11	furthermore	furthermore	ADV
ma-301	188	12	by	by	ADP
ma-301	188	13	using	use	VERB
ma-301	188	14	the	the	DET
ma-301	188	15	relation	relation	NOUN
ma-301	188	16	(	(	PUNCT
ma-301	188	17	3.7	3.7	NUM
ma-301	188	18	)	)	PUNCT
ma-301	188	19	we	we	PRON
ma-301	188	20	get	get	VERB
ma-301	188	21	∥∥φγ	∥∥φγ	NUM
ma-301	188	22	,	,	PUNCT
ma-301	188	23	δ	δ	PROPN
ma-301	188	24	∥∥	∥∥	X
ma-301	188	25	∞,q	∞,q	PROPN
ma-301	188	26	,	,	PUNCT
ma-301	188	27	α	α	X
ma-301	188	28	<	<	X
ma-301	188	29	∞	∞	PROPN
ma-301	188	30	therefore	therefore	ADV
ma-301	188	31	φγ	φγ	VERB
ma-301	188	32	,	,	PUNCT
ma-301	188	33	δ	δ	PROPN
ma-301	188	34	belongs	belong	VERB
ma-301	188	35	to	to	ADP
ma-301	188	36	l2α(r+q	l2α(r+q	PROPN
ma-301	188	37	)	)	PUNCT
ma-301	188	38	∩	∩	ADJ
ma-301	188	39	l∞α	l∞α	NOUN
ma-301	188	40	(	(	PUNCT
ma-301	188	41	r+q	r+q	PROPN
ma-301	188	42	)	)	PUNCT
ma-301	188	43	.	.	PUNCT
ma-301	189	1	�	�	PROPN
ma-301	189	2	theorem	theorem	VERB
ma-301	189	3	3.2	3.2	NUM
ma-301	189	4	.	.	PUNCT
ma-301	190	1	(	(	PUNCT
ma-301	190	2	second	second	ADJ
ma-301	190	3	calderón	calderón	NOUN
ma-301	190	4	’s	’s	PART
ma-301	190	5	formula	formula	NOUN
ma-301	190	6	)	)	PUNCT
ma-301	190	7	.	.	PUNCT
ma-301	191	1	let	let	VERB
ma-301	191	2	f	f	PROPN
ma-301	191	3	∈	∈	PROPN
ma-301	191	4	l2α(r+q	l2α(r+q	PROPN
ma-301	191	5	)	)	PUNCT
ma-301	191	6	and	and	CCONJ
ma-301	191	7	σ	σ	PROPN
ma-301	191	8	∈	∈	PROPN
ma-301	191	9	l2α(r+q	l2α(r+q	PROPN
ma-301	191	10	)	)	PUNCT
ma-301	191	11	∩	∩	ADJ
ma-301	191	12	l∞α	l∞α	NOUN
ma-301	191	13	(	(	PUNCT
ma-301	191	14	r+q	r+q	PROPN
ma-301	191	15	)	)	PUNCT
ma-301	191	16	satisfy	satisfy	VERB
ma-301	191	17	the	the	DET
ma-301	191	18	admissibility	admissibility	NOUN
ma-301	191	19	condition	condition	NOUN
ma-301	191	20	(	(	PUNCT
ma-301	191	21	3.7	3.7	NUM
ma-301	191	22	)	)	PUNCT
ma-301	191	23	and	and	CCONJ
ma-301	191	24	0	0	NUM
ma-301	191	25	<	<	X
ma-301	191	26	γ	γ	X
ma-301	191	27	<	<	X
ma-301	191	28	δ	δ	PROPN
ma-301	191	29	<	<	X
ma-301	191	30	∞.	∞.	PROPN
ma-301	191	31	then	then	ADV
ma-301	191	32	the	the	DET
ma-301	191	33	function	function	NOUN
ma-301	191	34	fγ	fγ	PROPN
ma-301	191	35	,	,	PUNCT
ma-301	191	36	δ(x	δ(x	ADJ
ma-301	191	37	)	)	PUNCT
ma-301	191	38	=	=	SYM
ma-301	192	1	∫	∫	PROPN
ma-301	192	2	δ	δ	PROPN
ma-301	192	3	γ	γ	PROPN
ma-301	192	4	(	(	PUNCT
ma-301	192	5	mq	mq	PROPN
ma-301	192	6	,	,	PUNCT
ma-301	192	7	σ	σ	PROPN
ma-301	192	8	,	,	PUNCT
ma-301	192	9	β(f	β(f	NUM
ma-301	192	10	)	)	PUNCT
ma-301	192	11	∗α	∗α	PROPN
ma-301	192	12	h−1q	h−1q	PROPN
ma-301	192	13	,	,	PUNCT
ma-301	192	14	α	α	PROPN
ma-301	192	15	(	(	PUNCT
ma-301	192	16	σβ	σβ	NOUN
ma-301	192	17	)	)	PUNCT
ma-301	192	18	)	)	PUNCT
ma-301	192	19	(	(	PUNCT
ma-301	192	20	x	x	X
ma-301	192	21	)	)	PUNCT
ma-301	192	22	dq(β	dq(β	ADP
ma-301	192	23	)	)	PUNCT
ma-301	192	24	β	β	NOUN
ma-301	192	25	,	,	PUNCT
ma-301	192	26	x	x	PUNCT
ma-301	192	27	∈	∈	PROPN
ma-301	192	28	r+q	r+q	PROPN
ma-301	192	29	belongs	belong	VERB
ma-301	192	30	to	to	ADP
ma-301	192	31	l2α(r+q	l2α(r+q	PROPN
ma-301	192	32	)	)	PUNCT
ma-301	192	33	and	and	CCONJ
ma-301	192	34	satisfies	satisfie	NOUN
ma-301	192	35	lim	lim	PROPN
ma-301	192	36	(	(	PUNCT
ma-301	192	37	γ	γ	PROPN
ma-301	192	38	,	,	PUNCT
ma-301	192	39	δ)→(0,∞	δ)→(0,∞	NOUN
ma-301	192	40	)	)	PUNCT
ma-301	193	1	∥∥fγ	∥∥fγ	PROPN
ma-301	193	2	,	,	PUNCT
ma-301	193	3	δ	δ	PROPN
ma-301	193	4	−	−	PROPN
ma-301	193	5	f	f	PROPN
ma-301	193	6	∥∥2,q	∥∥2,q	PROPN
ma-301	193	7	,	,	PUNCT
ma-301	193	8	α	α	NOUN
ma-301	193	9	=	=	SYM
ma-301	193	10	0	0	NUM
ma-301	193	11	(	(	PUNCT
ma-301	193	12	3.9	3.9	NUM
ma-301	193	13	)	)	PUNCT
ma-301	193	14	proof	proof	NOUN
ma-301	193	15	.	.	PUNCT
ma-301	194	1	by	by	ADP
ma-301	194	2	a	a	DET
ma-301	194	3	simple	simple	ADJ
ma-301	194	4	computation	computation	NOUN
ma-301	194	5	we	we	PRON
ma-301	194	6	find	find	VERB
ma-301	194	7	that	that	SCONJ
ma-301	194	8	fγ	fγ	NOUN
ma-301	194	9	,	,	PUNCT
ma-301	194	10	δ(x	δ(x	ADJ
ma-301	194	11	)	)	PUNCT
ma-301	194	12	=	=	SYM
ma-301	194	13	∫	∫	PROPN
ma-301	194	14	∞	∞	PROPN
ma-301	194	15	0	0	NUM
ma-301	194	16	φγ	φγ	PROPN
ma-301	194	17	,	,	PUNCT
ma-301	194	18	δ(λ)jα(λx	δ(λ)jα(λx	NOUN
ma-301	194	19	;	;	PUNCT
ma-301	194	20	q2)hq	q2)hq	PROPN
ma-301	194	21	,	,	PUNCT
ma-301	194	22	α(f	α(f	PROPN
ma-301	194	23	)	)	PUNCT
ma-301	194	24	(	(	PUNCT
ma-301	194	25	λ)dµq	λ)dµq	PROPN
ma-301	194	26	,	,	PUNCT
ma-301	194	27	α(λ	α(λ	PROPN
ma-301	194	28	)	)	PUNCT
ma-301	195	1	=	=	SYM
ma-301	195	2	h−1q	h−1q	PROPN
ma-301	195	3	,	,	PUNCT
ma-301	195	4	α	α	PROPN
ma-301	195	5	(	(	PUNCT
ma-301	195	6	φγ	φγ	ADP
ma-301	195	7	,	,	PUNCT
ma-301	195	8	δhq	δhq	NOUN
ma-301	195	9	,	,	PUNCT
ma-301	195	10	α(f	α(f	PROPN
ma-301	195	11	)	)	PUNCT
ma-301	195	12	)	)	PUNCT
ma-301	196	1	(	(	PUNCT
ma-301	196	2	x	x	X
ma-301	196	3	)	)	PUNCT
ma-301	196	4	,	,	PUNCT
ma-301	196	5	by	by	ADP
ma-301	196	6	using	use	VERB
ma-301	196	7	proposition	proposition	NOUN
ma-301	196	8	3.2	3.2	NUM
ma-301	196	9	we	we	PRON
ma-301	196	10	find	find	VERB
ma-301	196	11	that	that	SCONJ
ma-301	196	12	φγ	φγ	VERB
ma-301	196	13	,	,	PUNCT
ma-301	196	14	δ	δ	PROPN
ma-301	196	15	∈	∈	PROPN
ma-301	196	16	l∞α	l∞α	NOUN
ma-301	196	17	(	(	PUNCT
ma-301	196	18	r+q	r+q	PROPN
ma-301	196	19	)	)	PUNCT
ma-301	196	20	then	then	ADV
ma-301	196	21	we	we	PRON
ma-301	196	22	have	have	VERB
ma-301	196	23	fγ	fγ	PROPN
ma-301	196	24	,	,	PUNCT
ma-301	196	25	δ	δ	PROPN
ma-301	196	26	∈	∈	PROPN
ma-301	196	27	l2α(r+q	l2α(r+q	PROPN
ma-301	196	28	)	)	PUNCT
ma-301	196	29	and	and	CCONJ
ma-301	196	30	hq	hq	INTJ
ma-301	196	31	,	,	PUNCT
ma-301	196	32	α	α	PROPN
ma-301	196	33	(	(	PUNCT
ma-301	196	34	fγ	fγ	PROPN
ma-301	196	35	,	,	PUNCT
ma-301	196	36	δ	δ	PROPN
ma-301	196	37	)	)	PUNCT
ma-301	196	38	(	(	PUNCT
ma-301	196	39	λ	λ	X
ma-301	196	40	)	)	PUNCT
ma-301	196	41	=	=	SYM
ma-301	196	42	φγ	φγ	PROPN
ma-301	196	43	,	,	PUNCT
ma-301	196	44	δ(λ	δ(λ	PROPN
ma-301	196	45	,	,	PUNCT
ma-301	196	46	m)hq	m)hq	PROPN
ma-301	196	47	,	,	PUNCT
ma-301	196	48	α(f	α(f	PROPN
ma-301	196	49	)	)	PUNCT
ma-301	196	50	(	(	PUNCT
ma-301	196	51	λ	λ	X
ma-301	196	52	)	)	PUNCT
ma-301	196	53	on	on	ADP
ma-301	196	54	the	the	DET
ma-301	196	55	other	other	ADJ
ma-301	196	56	hand	hand	NOUN
ma-301	196	57	by	by	ADP
ma-301	196	58	using	use	VERB
ma-301	196	59	plancherel	plancherel	NOUN
ma-301	196	60	’s	’s	PART
ma-301	196	61	formula	formula	NOUN
ma-301	196	62	(	(	PUNCT
ma-301	196	63	2.7	2.7	NUM
ma-301	196	64	)	)	PUNCT
ma-301	196	65	we	we	PRON
ma-301	196	66	find	find	VERB
ma-301	196	67	that	that	SCONJ
ma-301	196	68	lim	lim	PROPN
ma-301	196	69	(	(	PUNCT
ma-301	196	70	γ	γ	PROPN
ma-301	196	71	,	,	PUNCT
ma-301	196	72	δ)→(0,∞	δ)→(0,∞	NOUN
ma-301	196	73	)	)	PUNCT
ma-301	196	74	∥∥fγ	∥∥fγ	PROPN
ma-301	196	75	,	,	PUNCT
ma-301	196	76	δ	δ	PROPN
ma-301	196	77	−	−	PROPN
ma-301	196	78	f	f	PROPN
ma-301	196	79	∥∥22,q	∥∥22,q	PROPN
ma-301	196	80	,	,	PUNCT
ma-301	196	81	α	α	PROPN
ma-301	196	82	=	=	SYM
ma-301	196	83	lim	lim	PROPN
ma-301	196	84	(	(	PUNCT
ma-301	196	85	γ	γ	PROPN
ma-301	196	86	,	,	PUNCT
ma-301	196	87	δ)→(0,∞	δ)→(0,∞	PROPN
ma-301	196	88	)	)	PUNCT
ma-301	196	89	∫	∫	PROPN
ma-301	196	90	∞	∞	PROPN
ma-301	196	91	0	0	NUM
ma-301	196	92	|hq	|hq	PROPN
ma-301	196	93	,	,	PUNCT
ma-301	196	94	α(f	α(f	PROPN
ma-301	196	95	)	)	PUNCT
ma-301	196	96	(	(	PUNCT
ma-301	196	97	λ)|2	λ)|2	PROPN
ma-301	196	98	(	(	PUNCT
ma-301	196	99	1−φγ	1−φγ	NUM
ma-301	196	100	,	,	PUNCT
ma-301	196	101	δ(λ	δ(λ	PROPN
ma-301	196	102	)	)	PUNCT
ma-301	196	103	)	)	PUNCT
ma-301	196	104	2	2	NUM
ma-301	196	105	dµq	dµq	NOUN
ma-301	196	106	,	,	PUNCT
ma-301	196	107	α(λ	α(λ	PROPN
ma-301	196	108	)	)	PUNCT
ma-301	196	109	by	by	ADP
ma-301	196	110	using	use	VERB
ma-301	196	111	the	the	DET
ma-301	196	112	admissibility	admissibility	NOUN
ma-301	196	113	condition	condition	NOUN
ma-301	196	114	(	(	PUNCT
ma-301	196	115	3.7	3.7	NUM
ma-301	196	116	)	)	PUNCT
ma-301	196	117	,	,	PUNCT
ma-301	196	118	the	the	DET
ma-301	196	119	relation	relation	NOUN
ma-301	196	120	(	(	PUNCT
ma-301	196	121	3.9	3.9	NUM
ma-301	196	122	)	)	PUNCT
ma-301	196	123	follows	follow	VERB
ma-301	196	124	from	from	ADP
ma-301	196	125	the	the	DET
ma-301	196	126	dominated	dominate	VERB
ma-301	196	127	convergencetheorem	convergencetheorem	PROPN
ma-301	196	128	.	.	PUNCT
ma-301	197	1	�	�	PROPN
ma-301	197	2	3.2	3.2	NUM
ma-301	197	3	.	.	PUNCT
ma-301	198	1	uncerainty	uncerainty	ADJ
ma-301	198	2	principles	principle	NOUN
ma-301	198	3	for	for	ADP
ma-301	198	4	the	the	DET
ma-301	198	5	q	q	ADJ
ma-301	198	6	-	-	ADJ
ma-301	198	7	bessel	bessel	ADJ
ma-301	198	8	l2α	l2α	NOUN
ma-301	198	9	-	-	ADJ
ma-301	198	10	multiplier	multipli	ADJ
ma-301	198	11	operators	operator	NOUN
ma-301	198	12	.	.	PUNCT
ma-301	199	1	the	the	DET
ma-301	199	2	main	main	ADJ
ma-301	199	3	purpose	purpose	NOUN
ma-301	199	4	of	of	ADP
ma-301	199	5	this	this	DET
ma-301	199	6	subsection	subsection	NOUN
ma-301	199	7	is	be	AUX
ma-301	199	8	to	to	PART
ma-301	199	9	establish	establish	VERB
ma-301	199	10	heisenberg	heisenberg	PROPN
ma-301	199	11	’s	’s	PART
ma-301	199	12	and	and	CCONJ
ma-301	199	13	donoho	donoho	PROPN
ma-301	199	14	-	-	PUNCT
ma-301	199	15	stark	stark	NOUN
ma-301	199	16	’s	’s	PART
ma-301	199	17	uncertainty	uncertainty	NOUN
ma-301	199	18	principles	principle	NOUN
ma-301	199	19	for	for	ADP
ma-301	199	20	the	the	DET
ma-301	199	21	q	q	ADJ
ma-301	199	22	-	-	ADJ
ma-301	199	23	bessel	bessel	ADJ
ma-301	199	24	l2α	l2α	NOUN
ma-301	199	25	-	-	ADJ
ma-301	199	26	multiplier	multipli	ADJ
ma-301	199	27	operators	operator	NOUN
ma-301	199	28	mq	mq	PROPN
ma-301	199	29	,	,	PUNCT
ma-301	199	30	σ	σ	PROPN
ma-301	199	31	,	,	PUNCT
ma-301	199	32	β	β	X
ma-301	199	33	.	.	PUNCT
ma-301	200	1	3.2.1	3.2.1	X
ma-301	200	2	.	.	PUNCT
ma-301	201	1	heisenberg	heisenberg	PROPN
ma-301	201	2	’s	’s	PART
ma-301	201	3	uncertainty	uncertainty	NOUN
ma-301	201	4	principle	principle	PROPN
ma-301	201	5	formq	formq	NOUN
ma-301	201	6	,	,	PUNCT
ma-301	201	7	σ	σ	PROPN
ma-301	201	8	,	,	PUNCT
ma-301	201	9	β	β	X
ma-301	201	10	.	.	PUNCT
ma-301	202	1	heisenberg	heisenberg	PROPN
ma-301	202	2	’s	’s	PART
ma-301	202	3	uncertainty	uncertainty	NOUN
ma-301	202	4	principle	principle	NOUN
ma-301	202	5	for	for	SCONJ
ma-301	202	6	the	the	DET
ma-301	202	7	qbessel	qbessel	NOUN
ma-301	202	8	fourier	fourier	NOUN
ma-301	202	9	transform	transform	VERB
ma-301	202	10	hq	hq	INTJ
ma-301	202	11	,	,	PUNCT
ma-301	202	12	α	α	PROPN
ma-301	202	13	has	have	AUX
ma-301	202	14	been	be	AUX
ma-301	202	15	established	establish	VERB
ma-301	202	16	in	in	ADP
ma-301	202	17	[	[	X
ma-301	202	18	9	9	NUM
ma-301	202	19	,	,	PUNCT
ma-301	202	20	11	11	NUM
ma-301	202	21	]	]	PUNCT
ma-301	202	22	as	as	SCONJ
ma-301	202	23	follows	follow	VERB
ma-301	202	24	,	,	PUNCT
ma-301	202	25	for	for	ADP
ma-301	202	26	all	all	DET
ma-301	202	27	f	f	PROPN
ma-301	202	28	∈	∈	PROPN
ma-301	202	29	l2α(r+q	l2α(r+q	PROPN
ma-301	202	30	)	)	PUNCT
ma-301	203	1	we	we	PRON
ma-301	203	2	have	have	VERB
ma-301	203	3	‖|x	‖|x	NOUN
ma-301	203	4	|f	|f	ADV
ma-301	203	5	‖2,q	‖2,q	NOUN
ma-301	203	6	,	,	PUNCT
ma-301	203	7	α	α	PROPN
ma-301	203	8	‖|λ|hq	‖|λ|hq	PROPN
ma-301	203	9	,	,	PUNCT
ma-301	203	10	α(f	α(f	PROPN
ma-301	203	11	)	)	PUNCT
ma-301	204	1	‖2,q	‖2,q	PROPN
ma-301	204	2	,	,	PUNCT
ma-301	204	3	α	α	PROPN
ma-301	204	4	≥	≥	NOUN
ma-301	204	5	kq	kq	NOUN
ma-301	204	6	,	,	PUNCT
ma-301	204	7	v‖f	v‖f	NOUN
ma-301	204	8	‖	‖	PROPN
ma-301	204	9	2	2	NUM
ma-301	204	10	2,q	2,q	PROPN
ma-301	204	11	,	,	PUNCT
ma-301	204	12	α	α	X
ma-301	204	13	,	,	PUNCT
ma-301	204	14	(	(	PUNCT
ma-301	204	15	3.10	3.10	NUM
ma-301	204	16	)	)	PUNCT
ma-301	204	17	where	where	SCONJ
ma-301	204	18	kq	kq	PROPN
ma-301	204	19	,	,	PUNCT
ma-301	204	20	α	α	X
ma-301	204	21	=	=	PUNCT
ma-301	205	1	[	[	X
ma-301	205	2	1	1	NUM
ma-301	205	3	+	+	NUM
ma-301	205	4	√	√	NOUN
ma-301	205	5	q×qα+1	q×qα+1	NOUN
ma-301	205	6	]	]	X
ma-301	205	7	1−q2(α+1	1−q2(α+1	NUM
ma-301	205	8	)	)	PUNCT
ma-301	205	9	.	.	PUNCT
ma-301	206	1	the	the	DET
ma-301	206	2	inequality	inequality	NOUN
ma-301	206	3	(	(	PUNCT
ma-301	206	4	3.10	3.10	NUM
ma-301	206	5	)	)	PUNCT
ma-301	206	6	says	say	VERB
ma-301	206	7	that	that	SCONJ
ma-301	206	8	if	if	SCONJ
ma-301	206	9	f	f	PROPN
ma-301	206	10	is	be	AUX
ma-301	206	11	highly	highly	ADV
ma-301	206	12	localized	localize	VERB
ma-301	206	13	,	,	PUNCT
ma-301	206	14	then	then	ADV
ma-301	206	15	hq	hq	VERB
ma-301	206	16	,	,	PUNCT
ma-301	206	17	α(f	α(f	PROPN
ma-301	206	18	)	)	PUNCT
ma-301	206	19	can	can	AUX
ma-301	206	20	not	not	PART
ma-301	206	21	be	be	AUX
ma-301	206	22	concentrated	concentrate	VERB
ma-301	206	23	near	near	ADP
ma-301	206	24	a	a	DET
ma-301	206	25	single	single	ADJ
ma-301	206	26	point	point	NOUN
ma-301	206	27	.	.	PUNCT
ma-301	207	1	we	we	PRON
ma-301	207	2	will	will	AUX
ma-301	207	3	generalize	generalize	VERB
ma-301	207	4	this	this	DET
ma-301	207	5	inequality	inequality	NOUN
ma-301	207	6	for	for	ADP
ma-301	207	7	mq	mq	PROPN
ma-301	207	8	,	,	PUNCT
ma-301	207	9	σ	σ	PROPN
ma-301	207	10	,	,	PUNCT
ma-301	207	11	β	β	PROPN
ma-301	207	12	,	,	PUNCT
ma-301	207	13	we	we	PRON
ma-301	207	14	have	have	VERB
ma-301	207	15	the	the	DET
ma-301	207	16	following	follow	VERB
ma-301	207	17	result	result	VERB
ma-301	207	18	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	207	19	eur	eur	PROPN
ma-301	207	20	.	.	PUNCT
ma-301	208	1	j.	j.	PROPN
ma-301	208	2	math	math	PROPN
ma-301	208	3	.	.	PUNCT
ma-301	209	1	anal	anal	PROPN
ma-301	209	2	.	.	PUNCT
ma-301	210	1	10.28924	10.28924	NUM
ma-301	210	2	/	/	SYM
ma-301	210	3	ada	ada	PROPN
ma-301	210	4	/	/	PROPN
ma-301	210	5	ma.5.8	ma.5.8	PROPN
ma-301	210	6	9	9	NUM
ma-301	210	7	theorem	theorem	VERB
ma-301	210	8	3.3	3.3	NUM
ma-301	210	9	.	.	PUNCT
ma-301	211	1	for	for	ADP
ma-301	211	2	all	all	DET
ma-301	211	3	f	f	PROPN
ma-301	211	4	∈	∈	PROPN
ma-301	211	5	l2α(r+q	l2α(r+q	PROPN
ma-301	211	6	)	)	PUNCT
ma-301	211	7	we	we	PRON
ma-301	211	8	have	have	VERB
ma-301	211	9	‖f	‖f	ADP
ma-301	211	10	‖22,q	‖22,q	PROPN
ma-301	211	11	,	,	PUNCT
ma-301	211	12	α	α	NOUN
ma-301	211	13	≤	≤	NOUN
ma-301	211	14	∥∥|λ|2hq	∥∥|λ|2hq	NUM
ma-301	211	15	,	,	PUNCT
ma-301	211	16	α(f	α(f	NUM
ma-301	211	17	)	)	PUNCT
ma-301	211	18	∥∥	∥∥	PROPN
ma-301	212	1	2,q	2,q	NUM
ma-301	212	2	,	,	PUNCT
ma-301	212	3	α	α	PROPN
ma-301	212	4	kq	kq	PROPN
ma-301	212	5	,	,	PUNCT
ma-301	212	6	α	α	PROPN
ma-301	213	1	[	[	X
ma-301	213	2	∫	∫	X
ma-301	213	3	∞	∞	NUM
ma-301	213	4	0	0	NUM
ma-301	214	1	∥∥|x	∥∥|x	PROPN
ma-301	214	2	|mq	|mq	ADP
ma-301	214	3	,	,	PUNCT
ma-301	214	4	σ	σ	PROPN
ma-301	214	5	,	,	PUNCT
ma-301	214	6	β(f	β(f	PROPN
ma-301	214	7	)	)	PUNCT
ma-301	214	8	∥∥2	∥∥2	PROPN
ma-301	215	1	2,q	2,q	NUM
ma-301	215	2	,	,	PUNCT
ma-301	215	3	α	α	NOUN
ma-301	215	4	dq(β	dq(β	X
ma-301	215	5	)	)	PUNCT
ma-301	215	6	β	β	X
ma-301	215	7	]	]	PUNCT
ma-301	215	8	1	1	NUM
ma-301	215	9	2	2	NUM
ma-301	215	10	proof	proof	NOUN
ma-301	215	11	.	.	PUNCT
ma-301	216	1	let	let	VERB
ma-301	216	2	us	we	PRON
ma-301	216	3	suppose	suppose	VERB
ma-301	216	4	that	that	SCONJ
ma-301	216	5	∥∥|λ|2hq	∥∥|λ|2hq	NUM
ma-301	216	6	,	,	PUNCT
ma-301	216	7	α(f	α(f	NUM
ma-301	216	8	)	)	PUNCT
ma-301	216	9	∥∥	∥∥	PROPN
ma-301	217	1	2,q	2,q	NUM
ma-301	217	2	,	,	PUNCT
ma-301	217	3	α	α	NOUN
ma-301	217	4	+	+	X
ma-301	218	1	[	[	X
ma-301	218	2	∫∞	∫∞	NOUN
ma-301	218	3	0	0	PUNCT
ma-301	219	1	∥∥|x	∥∥|x	PROPN
ma-301	219	2	|2mq	|2mq	PROPN
ma-301	219	3	,	,	PUNCT
ma-301	219	4	σ	σ	PROPN
ma-301	219	5	,	,	PUNCT
ma-301	219	6	β(f	β(f	NUM
ma-301	219	7	)	)	PUNCT
ma-301	219	8	∥∥2	∥∥2	PROPN
ma-301	220	1	2,q	2,q	NUM
ma-301	220	2	,	,	PUNCT
ma-301	220	3	α	α	NOUN
ma-301	220	4	dq(β	dq(β	X
ma-301	220	5	)	)	PUNCT
ma-301	220	6	β	β	X
ma-301	220	7	]	]	PUNCT
ma-301	220	8	<	<	X
ma-301	220	9	∞	∞	PROPN
ma-301	220	10	,	,	PUNCT
ma-301	220	11	by	by	ADP
ma-301	220	12	usingthe	usingthe	DET
ma-301	220	13	relation	relation	NOUN
ma-301	220	14	(	(	PUNCT
ma-301	220	15	3.10	3.10	NUM
ma-301	220	16	)	)	PUNCT
ma-301	220	17	we	we	PRON
ma-301	220	18	find	find	VERB
ma-301	220	19	that	that	SCONJ
ma-301	220	20	kq	kq	PROPN
ma-301	220	21	,	,	PUNCT
ma-301	220	22	α	α	PROPN
ma-301	220	23	∫	∫	PROPN
ma-301	220	24	∞	∞	PROPN
ma-301	220	25	0	0	NUM
ma-301	220	26	|mq	|mq	NUM
ma-301	220	27	,	,	PUNCT
ma-301	220	28	σ	σ	NOUN
ma-301	220	29	,	,	PUNCT
ma-301	220	30	β(f	β(f	PROPN
ma-301	220	31	)	)	PUNCT
ma-301	220	32	(	(	PUNCT
ma-301	220	33	x)|2dµq	x)|2dµq	NOUN
ma-301	220	34	,	,	PUNCT
ma-301	220	35	α(x	α(x	NOUN
ma-301	220	36	)	)	PUNCT
ma-301	220	37	≤	≤	NOUN
ma-301	221	1	∥∥|x	∥∥|x	PROPN
ma-301	221	2	|mq	|mq	ADP
ma-301	221	3	,	,	PUNCT
ma-301	221	4	σ	σ	PROPN
ma-301	221	5	,	,	PUNCT
ma-301	221	6	β(f	β(f	PROPN
ma-301	221	7	)	)	PUNCT
ma-301	221	8	∥∥	∥∥	PROPN
ma-301	221	9	2,q	2,q	NUM
ma-301	221	10	,	,	PUNCT
ma-301	221	11	α	α	NOUN
ma-301	221	12	∥∥|λ|σβhq	∥∥|λ|σβhq	NOUN
ma-301	221	13	,	,	PUNCT
ma-301	221	14	α(f	α(f	NUM
ma-301	221	15	)	)	PUNCT
ma-301	221	16	∥∥	∥∥	PROPN
ma-301	221	17	2,q	2,q	NUM
ma-301	221	18	,	,	PUNCT
ma-301	221	19	α	α	NOUN
ma-301	221	20	,	,	PUNCT
ma-301	221	21	integrating	integrate	VERB
ma-301	221	22	over	over	ADP
ma-301	221	23	]	]	X
ma-301	221	24	0,+∞	0,+∞	NUM
ma-301	221	25	[	[	PUNCT
ma-301	221	26	with	with	ADP
ma-301	221	27	respect	respect	NOUN
ma-301	221	28	to	to	ADP
ma-301	221	29	measure	measure	NOUN
ma-301	221	30	dq(β	dq(β	NOUN
ma-301	221	31	)	)	PUNCT
ma-301	221	32	β	β	NOUN
ma-301	221	33	and	and	CCONJ
ma-301	221	34	using	use	VERB
ma-301	221	35	plancherel	plancherel	NOUN
ma-301	221	36	’s	’s	PART
ma-301	221	37	formula	formula	NOUN
ma-301	221	38	(	(	PUNCT
ma-301	221	39	3.8	3.8	NUM
ma-301	221	40	)	)	PUNCT
ma-301	221	41	andschwartz	andschwartz	PROPN
ma-301	221	42	’s	’s	PART
ma-301	221	43	inequality	inequality	NOUN
ma-301	221	44	we	we	PRON
ma-301	221	45	get	get	VERB
ma-301	221	46	kq	kq	PROPN
ma-301	221	47	,	,	PUNCT
ma-301	221	48	α‖f	α‖f	VERB
ma-301	221	49	‖22,q	‖22,q	PROPN
ma-301	221	50	,	,	PUNCT
ma-301	221	51	α	α	PROPN
ma-301	221	52	≤	≤	PUNCT
ma-301	222	1	[	[	X
ma-301	222	2	∫	∫	X
ma-301	222	3	∞	∞	NUM
ma-301	222	4	0	0	NUM
ma-301	222	5	‖|x	‖|x	NOUN
ma-301	222	6	|mq	|mq	NOUN
ma-301	222	7	,	,	PUNCT
ma-301	222	8	σ	σ	NOUN
ma-301	222	9	,	,	PUNCT
ma-301	222	10	β(f	β(f	PROPN
ma-301	222	11	)	)	PUNCT
ma-301	222	12	‖22,q	‖22,q	PROPN
ma-301	222	13	,	,	PUNCT
ma-301	222	14	α	α	NOUN
ma-301	222	15	dq(β	dq(β	X
ma-301	222	16	)	)	PUNCT
ma-301	222	17	β	β	X
ma-301	222	18	]	]	PUNCT
ma-301	222	19	1	1	NUM
ma-301	222	20	2	2	NUM
ma-301	222	21	[	[	X
ma-301	222	22	∫	∫	X
ma-301	222	23	∞	∞	NUM
ma-301	222	24	0	0	PUNCT
ma-301	223	1	[	[	X
ma-301	223	2	∫	∫	X
ma-301	223	3	∞	∞	NUM
ma-301	223	4	0	0	NUM
ma-301	223	5	||λσβ(λ)|2	||λσβ(λ)|2	NOUN
ma-301	223	6	|hq	|hq	NUM
ma-301	223	7	,	,	PUNCT
ma-301	223	8	α(f	α(f	PROPN
ma-301	223	9	)	)	PUNCT
ma-301	223	10	(	(	PUNCT
ma-301	223	11	λ)|2|(λ)|dµq	λ)|2|(λ)|dµq	PROPN
ma-301	223	12	,	,	PUNCT
ma-301	223	13	α(λ	α(λ	PROPN
ma-301	223	14	)	)	PUNCT
ma-301	223	15	]	]	PUNCT
ma-301	224	1	dq(β	dq(β	X
ma-301	224	2	)	)	PUNCT
ma-301	224	3	β	β	X
ma-301	224	4	]	]	PUNCT
ma-301	224	5	1	1	NUM
ma-301	224	6	2	2	NUM
ma-301	224	7	the	the	DET
ma-301	224	8	admissibility	admissibility	NOUN
ma-301	224	9	condition	condition	NOUN
ma-301	224	10	(	(	PUNCT
ma-301	224	11	3.7	3.7	NUM
ma-301	224	12	)	)	PUNCT
ma-301	224	13	gives	give	VERB
ma-301	224	14	the	the	DET
ma-301	224	15	desired	desire	VERB
ma-301	224	16	result	result	NOUN
ma-301	224	17	.	.	PUNCT
ma-301	225	1	�	�	PROPN
ma-301	225	2	3.2.2	3.2.2	NUM
ma-301	225	3	.	.	PUNCT
ma-301	226	1	donoho	donoho	NOUN
ma-301	226	2	-	-	PUNCT
ma-301	226	3	stark	stark	PROPN
ma-301	226	4	’s	’s	PART
ma-301	226	5	uncertainty	uncertainty	NOUN
ma-301	226	6	principle	principle	PROPN
ma-301	226	7	formq	formq	NOUN
ma-301	226	8	,	,	PUNCT
ma-301	226	9	σ	σ	PROPN
ma-301	226	10	,	,	PUNCT
ma-301	226	11	β	β	X
ma-301	226	12	.	.	PUNCT
ma-301	227	1	building	build	VERB
ma-301	227	2	on	on	ADP
ma-301	227	3	the	the	DET
ma-301	227	4	ideas	idea	NOUN
ma-301	227	5	of	of	ADP
ma-301	227	6	donoho	donoho	NOUN
ma-301	227	7	and	and	CCONJ
ma-301	227	8	stark	stark	ADJ
ma-301	227	9	in	in	ADP
ma-301	227	10	[	[	X
ma-301	227	11	3	3	NUM
ma-301	227	12	]	]	PUNCT
ma-301	227	13	,	,	PUNCT
ma-301	227	14	the	the	DET
ma-301	227	15	main	main	ADJ
ma-301	227	16	purpose	purpose	NOUN
ma-301	227	17	of	of	ADP
ma-301	227	18	this	this	DET
ma-301	227	19	subsection	subsection	NOUN
ma-301	227	20	is	be	AUX
ma-301	227	21	to	to	PART
ma-301	227	22	give	give	VERB
ma-301	227	23	an	an	DET
ma-301	227	24	uncertainty	uncertainty	NOUN
ma-301	227	25	inequality	inequality	NOUN
ma-301	227	26	of	of	ADP
ma-301	227	27	concentration	concentration	NOUN
ma-301	227	28	type	type	NOUN
ma-301	227	29	in	in	ADP
ma-301	227	30	l2θ(r+q	l2θ(r+q	PROPN
ma-301	227	31	)	)	PUNCT
ma-301	227	32	where	where	SCONJ
ma-301	227	33	l2θ(r+q	l2θ(r+q	NOUN
ma-301	227	34	)	)	PUNCT
ma-301	227	35	the	the	DET
ma-301	227	36	space	space	NOUN
ma-301	227	37	of	of	ADP
ma-301	227	38	measurables	measurables	PROPN
ma-301	227	39	functions	function	NOUN
ma-301	227	40	on	on	ADP
ma-301	227	41	r+q	r+q	PROPN
ma-301	227	42	×	×	NOUN
ma-301	227	43	r+q	r+q	NOUN
ma-301	227	44	such	such	ADJ
ma-301	227	45	that	that	SCONJ
ma-301	227	46	‖f	‖f	ADP
ma-301	227	47	‖2,θα	‖2,θα	PUNCT
ma-301	228	1	=	=	SYM
ma-301	229	1	[	[	X
ma-301	229	2	∫	∫	X
ma-301	229	3	∞	∞	PROPN
ma-301	229	4	0	0	NUM
ma-301	230	1	‖f	‖f	PRON
ma-301	230	2	(	(	PUNCT
ma-301	230	3	β	β	X
ma-301	230	4	,	,	PUNCT
ma-301	230	5	.)‖22,q	.)‖22,q	PROPN
ma-301	230	6	,	,	PUNCT
ma-301	230	7	α	α	NOUN
ma-301	230	8	dq(β	dq(β	NOUN
ma-301	230	9	)	)	PUNCT
ma-301	230	10	β	β	X
ma-301	230	11	]	]	PUNCT
ma-301	230	12	1	1	NUM
ma-301	230	13	2	2	NUM
ma-301	230	14	.	.	PUNCT
ma-301	231	1	we	we	PRON
ma-301	231	2	denote	denote	VERB
ma-301	231	3	by	by	ADP
ma-301	231	4	θα	θα	ADP
ma-301	231	5	the	the	DET
ma-301	231	6	measure	measure	NOUN
ma-301	231	7	defined	define	VERB
ma-301	231	8	on	on	ADP
ma-301	231	9	r+q	r+q	PROPN
ma-301	231	10	×	×	NOUN
ma-301	231	11	r+q	r+q	NUM
ma-301	231	12	by	by	ADP
ma-301	231	13	dθα(β	dθα(β	PROPN
ma-301	231	14	,	,	PUNCT
ma-301	231	15	x	x	NOUN
ma-301	231	16	)	)	PUNCT
ma-301	231	17	=	=	SYM
ma-301	231	18	dµq	dµq	PROPN
ma-301	231	19	,	,	PUNCT
ma-301	231	20	α(x)⊗	α(x)⊗	PROPN
ma-301	231	21	dq(β	dq(β	NOUN
ma-301	231	22	)	)	PUNCT
ma-301	231	23	β	β	NOUN
ma-301	231	24	,	,	PUNCT
ma-301	231	25	definition	definition	NOUN
ma-301	231	26	3.2	3.2	NUM
ma-301	231	27	.	.	PUNCT
ma-301	232	1	[	[	X
ma-301	232	2	12	12	NUM
ma-301	232	3	]	]	PUNCT
ma-301	232	4	(	(	PUNCT
ma-301	232	5	i	i	NOUN
ma-301	232	6	)	)	PUNCT
ma-301	232	7	let	let	VERB
ma-301	232	8	e	e	PRON
ma-301	232	9	be	be	AUX
ma-301	232	10	a	a	DET
ma-301	232	11	measurable	measurable	ADJ
ma-301	232	12	subset	subset	NOUN
ma-301	232	13	of	of	ADP
ma-301	232	14	r+q	r+q	PROPN
ma-301	232	15	,	,	PUNCT
ma-301	232	16	we	we	PRON
ma-301	232	17	say	say	VERB
ma-301	232	18	that	that	SCONJ
ma-301	232	19	the	the	DET
ma-301	232	20	function	function	NOUN
ma-301	232	21	f	f	PROPN
ma-301	232	22	∈	∈	PROPN
ma-301	232	23	l2α(r+q	l2α(r+q	PROPN
ma-301	232	24	)	)	PUNCT
ma-301	232	25	is	be	AUX
ma-301	232	26	ε	ε	PROPN
ma-301	232	27	-	-	PUNCT
ma-301	232	28	concentrated	concentrate	VERB
ma-301	232	29	on	on	ADP
ma-301	232	30	e	e	NOUN
ma-301	232	31	if	if	SCONJ
ma-301	232	32	‖f	‖f	ADP
ma-301	232	33	−	−	NUM
ma-301	232	34	1ef	1ef	ADJ
ma-301	232	35	‖2,q	‖2,q	NOUN
ma-301	232	36	,	,	PUNCT
ma-301	232	37	α	α	NOUN
ma-301	232	38	≤	≤	NOUN
ma-301	232	39	ε‖f	ε‖f	NOUN
ma-301	232	40	‖2,q	‖2,q	NOUN
ma-301	232	41	,	,	PUNCT
ma-301	232	42	α	α	X
ma-301	232	43	,	,	PUNCT
ma-301	232	44	(	(	PUNCT
ma-301	232	45	3.11	3.11	NUM
ma-301	232	46	)	)	PUNCT
ma-301	232	47	where	where	SCONJ
ma-301	232	48	1e	1e	PROPN
ma-301	232	49	is	be	AUX
ma-301	232	50	the	the	DET
ma-301	232	51	indicator	indicator	NOUN
ma-301	232	52	function	function	NOUN
ma-301	232	53	of	of	ADP
ma-301	232	54	the	the	DET
ma-301	232	55	set	set	PROPN
ma-301	232	56	e.	e.	PROPN
ma-301	232	57	(	(	PUNCT
ma-301	232	58	ii	ii	PROPN
ma-301	232	59	)	)	PUNCT
ma-301	232	60	let	let	VERB
ma-301	232	61	f	f	PRON
ma-301	232	62	be	be	AUX
ma-301	232	63	a	a	DET
ma-301	232	64	measurable	measurable	ADJ
ma-301	232	65	subset	subset	NOUN
ma-301	232	66	of	of	ADP
ma-301	232	67	r+q	r+q	PROPN
ma-301	232	68	×	×	NOUN
ma-301	232	69	r+q	r+q	PROPN
ma-301	232	70	,	,	PUNCT
ma-301	232	71	we	we	PRON
ma-301	232	72	say	say	VERB
ma-301	232	73	that	that	SCONJ
ma-301	232	74	the	the	DET
ma-301	232	75	function	function	NOUN
ma-301	232	76	tσ	tσ	PROPN
ma-301	232	77	,	,	PUNCT
ma-301	232	78	β(f	β(f	NUM
ma-301	232	79	)	)	PUNCT
ma-301	232	80	is	be	AUX
ma-301	232	81	ρ	ρ	NOUN
ma-301	232	82	-	-	PUNCT
ma-301	232	83	concentrated	concentrated	ADJ
ma-301	232	84	on	on	ADP
ma-301	232	85	f	f	PROPN
ma-301	232	86	if	if	SCONJ
ma-301	232	87	‖mq	‖mq	PROPN
ma-301	232	88	,	,	PUNCT
ma-301	232	89	σ	σ	PROPN
ma-301	232	90	,	,	PUNCT
ma-301	232	91	β(f	β(f	PROPN
ma-301	232	92	)	)	PUNCT
ma-301	233	1	−	−	PROPN
ma-301	233	2	1fmq	1fmq	NUM
ma-301	233	3	,	,	PUNCT
ma-301	233	4	σ	σ	PROPN
ma-301	233	5	,	,	PUNCT
ma-301	233	6	β(f	β(f	NUM
ma-301	233	7	)	)	PUNCT
ma-301	233	8	‖2,θα	‖2,θα	PUNCT
ma-301	233	9	≤	≤	PROPN
ma-301	233	10	ρ‖mq	ρ‖mq	PROPN
ma-301	233	11	,	,	PUNCT
ma-301	233	12	σ	σ	PROPN
ma-301	233	13	,	,	PUNCT
ma-301	233	14	β(f	β(f	PROPN
ma-301	233	15	)	)	PUNCT
ma-301	233	16	‖2,θα	‖2,θα	PUNCT
ma-301	233	17	.	.	PUNCT
ma-301	234	1	(	(	PUNCT
ma-301	234	2	3.12	3.12	NUM
ma-301	234	3	)	)	PUNCT
ma-301	234	4	we	we	PRON
ma-301	234	5	have	have	VERB
ma-301	234	6	the	the	DET
ma-301	234	7	following	follow	VERB
ma-301	234	8	result	result	NOUN
ma-301	234	9	theorem	theorem	VERB
ma-301	234	10	3.4	3.4	NUM
ma-301	234	11	.	.	PUNCT
ma-301	235	1	let	let	VERB
ma-301	235	2	f	f	PROPN
ma-301	235	3	∈	∈	PROPN
ma-301	235	4	l2α(r+q	l2α(r+q	PROPN
ma-301	235	5	)	)	PUNCT
ma-301	235	6	and	and	CCONJ
ma-301	235	7	σ	σ	NUM
ma-301	235	8	∈	∈	PROPN
ma-301	235	9	σ	σ	X
ma-301	235	10	∈	∈	PROPN
ma-301	235	11	l2α(r+q	l2α(r+q	PROPN
ma-301	235	12	)	)	PUNCT
ma-301	235	13	)	)	PUNCT
ma-301	236	1	∩	∩	ADJ
ma-301	236	2	l∞α	l∞α	NOUN
ma-301	236	3	(	(	PUNCT
ma-301	236	4	r+q	r+q	PROPN
ma-301	236	5	)	)	PUNCT
ma-301	236	6	satisfying	satisfy	VERB
ma-301	236	7	the	the	DET
ma-301	236	8	admissibility	admissibility	NOUN
ma-301	236	9	condition	condition	NOUN
ma-301	236	10	(	(	PUNCT
ma-301	236	11	3.7	3.7	NUM
ma-301	236	12	)	)	PUNCT
ma-301	236	13	,	,	PUNCT
ma-301	236	14	if	if	SCONJ
ma-301	236	15	f	f	PROPN
ma-301	236	16	is	be	AUX
ma-301	236	17	ε	ε	PROPN
ma-301	236	18	-	-	PUNCT
ma-301	236	19	concentrated	concentrate	VERB
ma-301	236	20	on	on	ADP
ma-301	236	21	e	e	PROPN
ma-301	236	22	and	and	CCONJ
ma-301	236	23	mq	mq	PROPN
ma-301	236	24	,	,	PUNCT
ma-301	236	25	σ	σ	PROPN
ma-301	236	26	,	,	PUNCT
ma-301	236	27	β(f	β(f	NUM
ma-301	236	28	)	)	PUNCT
ma-301	236	29	is	be	AUX
ma-301	236	30	ρ	ρ	NOUN
ma-301	236	31	-	-	PUNCT
ma-301	236	32	concentrated	concentrated	ADJ
ma-301	236	33	on	on	ADP
ma-301	236	34	f	f	PROPN
ma-301	236	35	then	then	ADV
ma-301	236	36	we	we	PRON
ma-301	236	37	have	have	VERB
ma-301	236	38	‖σ‖2,q	‖σ‖2,q	PROPN
ma-301	236	39	,	,	PUNCT
ma-301	236	40	α(µα(e	α(µα(e	PROPN
ma-301	236	41	)	)	PUNCT
ma-301	236	42	)	)	PUNCT
ma-301	236	43	1	1	NUM
ma-301	236	44	2	2	NUM
ma-301	236	45	[	[	X
ma-301	236	46	∫	∫	X
ma-301	236	47	f	f	PROPN
ma-301	236	48	dθα(β	dθα(β	PROPN
ma-301	236	49	,	,	PUNCT
ma-301	236	50	x	x	X
ma-301	236	51	)	)	PUNCT
ma-301	236	52	β4α+2	β4α+2	VERB
ma-301	236	53	]	]	PUNCT
ma-301	236	54	1	1	NUM
ma-301	236	55	2	2	NUM
ma-301	236	56	≥	≥	NOUN
ma-301	236	57	1−	1−	NUM
ma-301	236	58	(	(	PUNCT
ma-301	236	59	ε+	ε+	NOUN
ma-301	236	60	ρ	ρ	NOUN
ma-301	236	61	)	)	PUNCT
ma-301	236	62	.	.	PUNCT
ma-301	237	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	237	2	eur	eur	PROPN
ma-301	237	3	.	.	PUNCT
ma-301	238	1	j.	j.	PROPN
ma-301	238	2	math	math	PROPN
ma-301	238	3	.	.	PUNCT
ma-301	239	1	anal	anal	PROPN
ma-301	239	2	.	.	PUNCT
ma-301	240	1	10.28924	10.28924	NUM
ma-301	240	2	/	/	SYM
ma-301	240	3	ada	ada	PROPN
ma-301	240	4	/	/	PROPN
ma-301	240	5	ma.5.8	ma.5.8	PROPN
ma-301	240	6	10	10	NUM
ma-301	240	7	proof	proof	NOUN
ma-301	240	8	.	.	PUNCT
ma-301	241	1	let	let	VERB
ma-301	241	2	f	f	PROPN
ma-301	241	3	∈	∈	PROPN
ma-301	241	4	l2α(r+q	l2α(r+q	PROPN
ma-301	241	5	)	)	PUNCT
ma-301	241	6	and	and	CCONJ
ma-301	241	7	σ	σ	PROPN
ma-301	241	8	∈	∈	PROPN
ma-301	241	9	l2α(r	l2α(r	PROPN
ma-301	241	10	)	)	PUNCT
ma-301	241	11	∩	∩	ADJ
ma-301	241	12	l∞α	l∞α	NOUN
ma-301	241	13	(	(	PUNCT
ma-301	241	14	r+q	r+q	PROPN
ma-301	241	15	)	)	PUNCT
ma-301	242	1	satisfying	satisfy	VERB
ma-301	242	2	(	(	PUNCT
ma-301	242	3	3.7	3.7	NUM
ma-301	242	4	)	)	PUNCT
ma-301	242	5	and	and	CCONJ
ma-301	242	6	assume	assume	VERB
ma-301	242	7	that	that	SCONJ
ma-301	242	8	µα(e	µα(e	PUNCT
ma-301	242	9	)	)	PUNCT
ma-301	242	10	<	<	X
ma-301	242	11	∞and	∞and	PROPN
ma-301	242	12	[	[	X
ma-301	242	13	∫f	∫f	PROPN
ma-301	242	14	dθα(β	dθα(β	PROPN
ma-301	242	15	,	,	PUNCT
ma-301	242	16	x	x	X
ma-301	242	17	)	)	PUNCT
ma-301	242	18	β4α+2	β4α+2	VERB
ma-301	242	19	]	]	PUNCT
ma-301	242	20	1	1	NUM
ma-301	242	21	2	2	NUM
ma-301	242	22	<	<	ADP
ma-301	242	23	∞.	∞.	PROPN
ma-301	242	24	according	accord	VERB
ma-301	242	25	to	to	ADP
ma-301	242	26	the	the	DET
ma-301	242	27	relations	relation	NOUN
ma-301	242	28	(	(	PUNCT
ma-301	242	29	3.11),(3.12	3.11),(3.12	NUM
ma-301	242	30	)	)	PUNCT
ma-301	242	31	we	we	PRON
ma-301	242	32	have	have	VERB
ma-301	242	33	‖mq	‖mq	NUM
ma-301	242	34	,	,	PUNCT
ma-301	242	35	σ	σ	PROPN
ma-301	242	36	,	,	PUNCT
ma-301	242	37	β(f	β(f	PROPN
ma-301	242	38	)	)	PUNCT
ma-301	243	1	−1fmq	−1fmq	NOUN
ma-301	243	2	,	,	PUNCT
ma-301	243	3	σ	σ	PROPN
ma-301	243	4	,	,	PUNCT
ma-301	243	5	β(1ef	β(1ef	NOUN
ma-301	243	6	)	)	PUNCT
ma-301	243	7	‖2,θα	‖2,θα	PUNCT
ma-301	243	8	≤	≤	NOUN
ma-301	243	9	‖mq	‖mq	NUM
ma-301	243	10	,	,	PUNCT
ma-301	243	11	σ	σ	PROPN
ma-301	243	12	,	,	PUNCT
ma-301	243	13	β(f	β(f	PROPN
ma-301	243	14	)	)	PUNCT
ma-301	243	15	−1fmq	−1fmq	NOUN
ma-301	243	16	,	,	PUNCT
ma-301	243	17	σ	σ	PROPN
ma-301	243	18	,	,	PUNCT
ma-301	243	19	β(f	β(f	NUM
ma-301	243	20	)	)	PUNCT
ma-301	243	21	‖2,θα	‖2,θα	PUNCT
ma-301	244	1	+	+	NUM
ma-301	244	2	‖1fmq	‖1fmq	NOUN
ma-301	244	3	,	,	PUNCT
ma-301	244	4	σ	σ	NOUN
ma-301	244	5	,	,	PUNCT
ma-301	244	6	β(f	β(f	PROPN
ma-301	244	7	−1ef	−1ef	NUM
ma-301	244	8	)	)	PUNCT
ma-301	244	9	‖2,θα	‖2,θα	PUNCT
ma-301	244	10	≤	≤	PROPN
ma-301	244	11	ρ‖mq	ρ‖mq	PROPN
ma-301	244	12	,	,	PUNCT
ma-301	244	13	σ	σ	PROPN
ma-301	244	14	,	,	PUNCT
ma-301	244	15	β(f	β(f	NUM
ma-301	244	16	)	)	PUNCT
ma-301	244	17	‖2,θα	‖2,θα	PUNCT
ma-301	245	1	+	+	PUNCT
ma-301	245	2	‖mq	‖mq	NUM
ma-301	245	3	,	,	PUNCT
ma-301	245	4	σ	σ	PROPN
ma-301	245	5	,	,	PUNCT
ma-301	245	6	β(f	β(f	PROPN
ma-301	245	7	−	−	NOUN
ma-301	245	8	1ef	1ef	ADJ
ma-301	245	9	)	)	PUNCT
ma-301	245	10	‖2,θα	‖2,θα	PUNCT
ma-301	245	11	,	,	PUNCT
ma-301	245	12	by	by	ADP
ma-301	245	13	using	use	VERB
ma-301	245	14	plancherel	plancherel	NOUN
ma-301	245	15	’s	’s	PART
ma-301	245	16	relation	relation	NOUN
ma-301	245	17	(	(	PUNCT
ma-301	245	18	3.8	3.8	NUM
ma-301	245	19	)	)	PUNCT
ma-301	245	20	we	we	PRON
ma-301	245	21	get	get	VERB
ma-301	245	22	‖mq	‖mq	NUM
ma-301	245	23	,	,	PUNCT
ma-301	245	24	σ	σ	PROPN
ma-301	245	25	,	,	PUNCT
ma-301	245	26	β(f	β(f	NUM
ma-301	245	27	)	)	PUNCT
ma-301	245	28	‖2,θα	‖2,θα	PUNCT
ma-301	245	29	≤	≤	NOUN
ma-301	245	30	‖mq	‖mq	NUM
ma-301	245	31	,	,	PUNCT
ma-301	245	32	σ	σ	PROPN
ma-301	245	33	,	,	PUNCT
ma-301	245	34	β(f	β(f	PROPN
ma-301	245	35	)	)	PUNCT
ma-301	245	36	−	−	PROPN
ma-301	246	1	1fmq	1fmq	NUM
ma-301	246	2	,	,	PUNCT
ma-301	246	3	σ	σ	PROPN
ma-301	246	4	,	,	PUNCT
ma-301	246	5	β(1ef	β(1ef	NOUN
ma-301	246	6	)	)	PUNCT
ma-301	246	7	‖2,θα	‖2,θα	PUNCT
ma-301	247	1	+	+	NUM
ma-301	247	2	‖1fmq	‖1fmq	NOUN
ma-301	247	3	,	,	PUNCT
ma-301	247	4	σ	σ	PROPN
ma-301	247	5	,	,	PUNCT
ma-301	247	6	β(1ef	β(1ef	NOUN
ma-301	247	7	)	)	PUNCT
ma-301	247	8	‖2,θα	‖2,θα	X
ma-301	247	9	≤	≤	NOUN
ma-301	247	10	(	(	PUNCT
ma-301	247	11	ε+	ε+	NOUN
ma-301	247	12	ρ)‖f	ρ)‖f	NOUN
ma-301	247	13	‖2,q	‖2,q	NOUN
ma-301	247	14	,	,	PUNCT
ma-301	247	15	α	α	NOUN
ma-301	247	16	+	+	X
ma-301	247	17	‖1fmq	‖1fmq	NOUN
ma-301	247	18	,	,	PUNCT
ma-301	247	19	σ	σ	PROPN
ma-301	247	20	,	,	PUNCT
ma-301	247	21	β(1ef	β(1ef	NOUN
ma-301	247	22	)	)	PUNCT
ma-301	247	23	‖2,θα	‖2,θα	PUNCT
ma-301	247	24	,	,	PUNCT
ma-301	247	25	(	(	PUNCT
ma-301	247	26	3.13)on	3.13)on	PROPN
ma-301	247	27	the	the	DET
ma-301	247	28	other	other	ADJ
ma-301	247	29	hand	hand	NOUN
ma-301	247	30	by	by	ADP
ma-301	247	31	using	use	VERB
ma-301	247	32	the	the	DET
ma-301	247	33	relation	relation	NOUN
ma-301	247	34	(	(	PUNCT
ma-301	247	35	3.6	3.6	NUM
ma-301	247	36	)	)	PUNCT
ma-301	247	37	and	and	CCONJ
ma-301	247	38	hölder	hölder	PROPN
ma-301	247	39	’s	’s	PART
ma-301	247	40	inequality	inequality	NOUN
ma-301	247	41	we	we	PRON
ma-301	247	42	find	find	VERB
ma-301	247	43	that	that	SCONJ
ma-301	247	44	‖1fmq	‖1fmq	NOUN
ma-301	247	45	,	,	PUNCT
ma-301	247	46	σ	σ	PROPN
ma-301	247	47	,	,	PUNCT
ma-301	247	48	β(1ef	β(1ef	NOUN
ma-301	247	49	)	)	PUNCT
ma-301	247	50	‖2,θα	‖2,θα	NUM
ma-301	247	51	≤	≤	NOUN
ma-301	247	52	‖f	‖f	ADP
ma-301	247	53	‖2,q	‖2,q	NOUN
ma-301	247	54	,	,	PUNCT
ma-301	247	55	α‖σ‖1,q	α‖σ‖1,q	NOUN
ma-301	247	56	,	,	PUNCT
ma-301	247	57	α(µ(e	α(µ(e	NUM
ma-301	247	58	)	)	PUNCT
ma-301	247	59	)	)	PUNCT
ma-301	247	60	1	1	NUM
ma-301	247	61	2	2	NUM
ma-301	247	62	[	[	X
ma-301	247	63	∫	∫	X
ma-301	247	64	f	f	PROPN
ma-301	247	65	dθα(β	dθα(β	PROPN
ma-301	247	66	,	,	PUNCT
ma-301	247	67	x	x	X
ma-301	247	68	)	)	PUNCT
ma-301	247	69	β4α+2	β4α+2	VERB
ma-301	247	70	]	]	PUNCT
ma-301	247	71	1	1	NUM
ma-301	247	72	2	2	NUM
ma-301	247	73	,	,	PUNCT
ma-301	247	74	(	(	PUNCT
ma-301	247	75	3.14	3.14	NUM
ma-301	247	76	)	)	PUNCT
ma-301	247	77	by	by	ADP
ma-301	247	78	the	the	DET
ma-301	247	79	relations	relation	NOUN
ma-301	247	80	(	(	PUNCT
ma-301	247	81	3.13),(3.14	3.13),(3.14	NOUN
ma-301	247	82	)	)	PUNCT
ma-301	247	83	we	we	PRON
ma-301	247	84	deduce	deduce	VERB
ma-301	247	85	that	that	SCONJ
ma-301	247	86	‖mq	‖mq	NUM
ma-301	247	87	,	,	PUNCT
ma-301	247	88	σ	σ	PROPN
ma-301	247	89	,	,	PUNCT
ma-301	247	90	β(f	β(f	NUM
ma-301	247	91	)	)	PUNCT
ma-301	247	92	‖2,θα	‖2,θα	PUNCT
ma-301	247	93	≤	≤	NOUN
ma-301	247	94	‖f	‖f	ADP
ma-301	247	95	‖2,q	‖2,q	NOUN
ma-301	247	96	,	,	PUNCT
ma-301	247	97	α	α	X
ma-301	247	98	[	[	PUNCT
ma-301	247	99	(	(	PUNCT
ma-301	247	100	ε+	ε+	NOUN
ma-301	247	101	ρ	ρ	NOUN
ma-301	247	102	)	)	PUNCT
ma-301	247	103	+	+	SYM
ma-301	247	104	‖σ‖1,q	‖σ‖1,q	PROPN
ma-301	247	105	,	,	PUNCT
ma-301	247	106	α(µα(e	α(µα(e	PROPN
ma-301	247	107	)	)	PUNCT
ma-301	247	108	)	)	PUNCT
ma-301	247	109	1	1	NUM
ma-301	247	110	2	2	NUM
ma-301	247	111	[	[	X
ma-301	247	112	∫	∫	X
ma-301	247	113	f	f	PROPN
ma-301	247	114	dθα(β	dθα(β	PROPN
ma-301	247	115	,	,	PUNCT
ma-301	247	116	x	x	PROPN
ma-301	247	117	β4α+2	β4α+2	PUNCT
ma-301	247	118	]	]	PUNCT
ma-301	247	119	1	1	NUM
ma-301	247	120	2	2	NUM
ma-301	247	121	]	]	PUNCT
ma-301	247	122	plancherel	plancherel	NOUN
ma-301	247	123	’s	’s	PART
ma-301	247	124	formula	formula	NOUN
ma-301	247	125	(	(	PUNCT
ma-301	247	126	3.8	3.8	NUM
ma-301	247	127	)	)	PUNCT
ma-301	247	128	for	for	ADP
ma-301	247	129	mσ	mσ	PROPN
ma-301	247	130	,	,	PUNCT
ma-301	247	131	β	β	PROPN
ma-301	247	132	gives	give	VERB
ma-301	247	133	the	the	DET
ma-301	247	134	desired	desire	VERB
ma-301	247	135	result	result	NOUN
ma-301	247	136	.	.	PUNCT
ma-301	248	1	�	�	PROPN
ma-301	248	2	4	4	NUM
ma-301	248	3	.	.	PUNCT
ma-301	248	4	extremal	extremal	ADJ
ma-301	248	5	functions	function	NOUN
ma-301	248	6	associated	associate	VERB
ma-301	248	7	with	with	ADP
ma-301	248	8	the	the	DET
ma-301	248	9	q	q	ADJ
ma-301	248	10	-	-	PUNCT
ma-301	248	11	bessel	bessel	ADJ
ma-301	248	12	l2α	l2α	ADJ
ma-301	248	13	-	-	ADJ
ma-301	248	14	multiplier	multipli	ADJ
ma-301	248	15	operators	operator	NOUN
ma-301	248	16	in	in	ADP
ma-301	248	17	the	the	DET
ma-301	248	18	following	following	NOUN
ma-301	248	19	,	,	PUNCT
ma-301	248	20	we	we	PRON
ma-301	248	21	study	study	VERB
ma-301	248	22	the	the	DET
ma-301	248	23	extremal	extremal	ADJ
ma-301	248	24	functions	function	NOUN
ma-301	248	25	associated	associate	VERB
ma-301	248	26	with	with	ADP
ma-301	248	27	the	the	DET
ma-301	248	28	the	the	DET
ma-301	248	29	q	q	ADJ
ma-301	248	30	-	-	PUNCT
ma-301	248	31	bessel	bessel	ADJ
ma-301	248	32	l2α	l2α	NOUN
ma-301	248	33	-	-	ADJ
ma-301	248	34	multiplier	multipli	ADJ
ma-301	248	35	operators	operator	NOUN
ma-301	248	36	.	.	PUNCT
ma-301	249	1	definition	definition	NOUN
ma-301	249	2	4.1	4.1	NUM
ma-301	249	3	.	.	PUNCT
ma-301	250	1	let	let	VERB
ma-301	250	2	ψ	ψ	PART
ma-301	250	3	be	be	AUX
ma-301	250	4	a	a	DET
ma-301	250	5	positive	positive	ADJ
ma-301	250	6	function	function	NOUN
ma-301	250	7	on	on	ADP
ma-301	250	8	r+q	r+q	PROPN
ma-301	250	9	satisfying	satisfy	VERB
ma-301	250	10	the	the	DET
ma-301	250	11	following	follow	VERB
ma-301	250	12	conditions	condition	NOUN
ma-301	250	13	1	1	NUM
ma-301	250	14	ψ	ψ	NOUN
ma-301	250	15	∈	∈	PROPN
ma-301	250	16	l1α(r+q	l1α(r+q	NOUN
ma-301	250	17	)	)	PUNCT
ma-301	250	18	(	(	PUNCT
ma-301	250	19	4.1	4.1	NUM
ma-301	250	20	)	)	PUNCT
ma-301	250	21	and	and	CCONJ
ma-301	250	22	ψ(λ	ψ(λ	PROPN
ma-301	250	23	)	)	PUNCT
ma-301	250	24	≥	≥	NOUN
ma-301	250	25	1	1	NUM
ma-301	250	26	,	,	PUNCT
ma-301	250	27	λ	λ	PROPN
ma-301	250	28	∈	∈	PROPN
ma-301	250	29	r+q	r+q	PROPN
ma-301	250	30	.	.	PUNCT
ma-301	251	1	(	(	PUNCT
ma-301	251	2	4.2	4.2	NUM
ma-301	251	3	)	)	PUNCT
ma-301	251	4	we	we	PRON
ma-301	251	5	define	define	VERB
ma-301	251	6	the	the	DET
ma-301	251	7	sobolev	sobolev	NOUN
ma-301	251	8	-	-	PUNCT
ma-301	251	9	type	type	NOUN
ma-301	251	10	space	space	NOUN
ma-301	251	11	sψ(r+q	sψ(r+q	PROPN
ma-301	251	12	)	)	PUNCT
ma-301	251	13	by	by	ADP
ma-301	251	14	sψ(r+q	sψ(r+q	PROPN
ma-301	251	15	)	)	PUNCT
ma-301	252	1	=	=	PRON
ma-301	252	2	{	{	PUNCT
ma-301	252	3	f	f	PROPN
ma-301	252	4	∈	∈	PROPN
ma-301	252	5	l2α(r+q	l2α(r+q	PROPN
ma-301	252	6	)	)	PUNCT
ma-301	252	7	:	:	PUNCT
ma-301	252	8	√	√	NUM
ma-301	252	9	ψhq	ψhq	DET
ma-301	252	10	,	,	PUNCT
ma-301	252	11	α(f	α(f	NUM
ma-301	252	12	)	)	PUNCT
ma-301	253	1	∈	∈	PROPN
ma-301	253	2	l2α(r+q	l2α(r+q	PROPN
ma-301	253	3	)	)	PUNCT
ma-301	253	4	}	}	PUNCT
ma-301	253	5	provided	provide	VERB
ma-301	253	6	with	with	ADP
ma-301	253	7	inner	inner	ADJ
ma-301	253	8	product	product	NOUN
ma-301	253	9	〈	〈	PROPN
ma-301	253	10	f	f	PROPN
ma-301	253	11	,	,	PUNCT
ma-301	253	12	g〉ψ	g〉ψ	PROPN
ma-301	253	13	=	=	SYM
ma-301	253	14	∫	∫	PROPN
ma-301	253	15	∞	∞	PROPN
ma-301	253	16	0	0	NUM
ma-301	254	1	ψ(λ	ψ(λ	PROPN
ma-301	254	2	,	,	PUNCT
ma-301	254	3	m)hq	m)hq	PROPN
ma-301	254	4	,	,	PUNCT
ma-301	254	5	α(f	α(f	PROPN
ma-301	254	6	)	)	PUNCT
ma-301	254	7	(	(	PUNCT
ma-301	254	8	λ)hq	λ)hq	PROPN
ma-301	254	9	,	,	PUNCT
ma-301	254	10	α(g)(λ)dµq	α(g)(λ)dµq	PROPN
ma-301	254	11	,	,	PUNCT
ma-301	254	12	α(λ	α(λ	PROPN
ma-301	254	13	)	)	PUNCT
ma-301	254	14	,	,	PUNCT
ma-301	254	15	and	and	CCONJ
ma-301	254	16	the	the	DET
ma-301	254	17	norm	norm	NOUN
ma-301	254	18	‖f	‖f	PUNCT
ma-301	254	19	‖ψ	‖ψ	NOUN
ma-301	254	20	=	=	PUNCT
ma-301	255	1	√	√	ADP
ma-301	255	2	〈	〈	PROPN
ma-301	255	3	f	f	PROPN
ma-301	255	4	,	,	PUNCT
ma-301	255	5	f	f	PROPN
ma-301	255	6	〉	〉	PROPN
ma-301	255	7	ψ	ψ	PROPN
ma-301	255	8	.	.	PUNCT
ma-301	256	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	256	2	eur	eur	PROPN
ma-301	256	3	.	.	PUNCT
ma-301	257	1	j.	j.	PROPN
ma-301	257	2	math	math	PROPN
ma-301	257	3	.	.	PUNCT
ma-301	258	1	anal	anal	PROPN
ma-301	258	2	.	.	PUNCT
ma-301	259	1	10.28924	10.28924	NUM
ma-301	259	2	/	/	SYM
ma-301	259	3	ada	ada	PROPN
ma-301	259	4	/	/	PROPN
ma-301	259	5	ma.5.8	ma.5.8	PROPN
ma-301	259	6	11	11	NUM
ma-301	259	7	proposition	proposition	NOUN
ma-301	259	8	4.1	4.1	NUM
ma-301	259	9	.	.	PUNCT
ma-301	260	1	let	let	VERB
ma-301	260	2	σ	σ	NOUN
ma-301	260	3	be	be	AUX
ma-301	260	4	a	a	DET
ma-301	260	5	function	function	NOUN
ma-301	260	6	in	in	ADP
ma-301	260	7	l∞α	l∞α	NOUN
ma-301	260	8	(	(	PUNCT
ma-301	260	9	r+q	r+q	PROPN
ma-301	260	10	)	)	PUNCT
ma-301	260	11	.	.	PUNCT
ma-301	261	1	then	then	ADV
ma-301	261	2	the	the	DET
ma-301	261	3	q	q	ADJ
ma-301	261	4	-	-	PUNCT
ma-301	261	5	bessel	bessel	ADJ
ma-301	261	6	l2α	l2α	NOUN
ma-301	261	7	-	-	ADJ
ma-301	261	8	multiplier	multipli	ADJ
ma-301	261	9	operatorsmq	operatorsmq	NOUN
ma-301	261	10	,	,	PUNCT
ma-301	261	11	σ	σ	PROPN
ma-301	261	12	,	,	PUNCT
ma-301	261	13	β	β	X
ma-301	261	14	are	be	AUX
ma-301	261	15	bounded	bound	VERB
ma-301	261	16	and	and	CCONJ
ma-301	261	17	linear	linear	ADJ
ma-301	261	18	from	from	ADP
ma-301	261	19	sψ(r+q	sψ(r+q	PROPN
ma-301	261	20	)	)	PUNCT
ma-301	261	21	into	into	ADP
ma-301	261	22	l2α(r+q	l2α(r+q	PROPN
ma-301	261	23	)	)	PUNCT
ma-301	261	24	and	and	CCONJ
ma-301	261	25	we	we	PRON
ma-301	261	26	have	have	VERB
ma-301	261	27	for	for	ADP
ma-301	261	28	all	all	DET
ma-301	261	29	f	f	PROPN
ma-301	261	30	∈	∈	PROPN
ma-301	262	1	sψ(r+q	sψ(r+q	PROPN
ma-301	262	2	)	)	PUNCT
ma-301	262	3	∥∥mq	∥∥mq	PROPN
ma-301	262	4	,	,	PUNCT
ma-301	262	5	σ	σ	PROPN
ma-301	262	6	,	,	PUNCT
ma-301	262	7	β(f	β(f	PROPN
ma-301	262	8	)	)	PUNCT
ma-301	262	9	∥∥	∥∥	PROPN
ma-301	262	10	2,q	2,q	NUM
ma-301	262	11	,	,	PUNCT
ma-301	262	12	α	α	PROPN
ma-301	262	13	≤	≤	PUNCT
ma-301	262	14	‖σ‖∞,q	‖σ‖∞,q	PRON
ma-301	262	15	,	,	PUNCT
ma-301	262	16	α‖f	α‖f	ADJ
ma-301	262	17	‖ψ	‖ψ	NOUN
ma-301	262	18	.	.	PUNCT
ma-301	263	1	(	(	PUNCT
ma-301	263	2	4.3	4.3	NUM
ma-301	263	3	)	)	PUNCT
ma-301	263	4	proof	proof	NOUN
ma-301	263	5	.	.	PUNCT
ma-301	264	1	by	by	ADP
ma-301	264	2	using	use	VERB
ma-301	264	3	the	the	DET
ma-301	264	4	relations	relation	NOUN
ma-301	264	5	(	(	PUNCT
ma-301	264	6	2.8),(3.5),(4.2	2.8),(3.5),(4.2	NUM
ma-301	264	7	)	)	PUNCT
ma-301	264	8	we	we	PRON
ma-301	264	9	get	get	VERB
ma-301	264	10	the	the	DET
ma-301	264	11	result	result	NOUN
ma-301	264	12	�	�	PROPN
ma-301	264	13	definition	definition	NOUN
ma-301	264	14	4.2	4.2	NUM
ma-301	264	15	.	.	PUNCT
ma-301	265	1	let	let	VERB
ma-301	265	2	η	η	PROPN
ma-301	265	3	>	>	X
ma-301	265	4	0	0	PUNCT
ma-301	266	1	and	and	CCONJ
ma-301	266	2	let	let	VERB
ma-301	266	3	σ	σ	NOUN
ma-301	266	4	be	be	AUX
ma-301	266	5	a	a	DET
ma-301	266	6	function	function	NOUN
ma-301	266	7	in	in	ADP
ma-301	266	8	l∞α	l∞α	NOUN
ma-301	266	9	(	(	PUNCT
ma-301	266	10	r+q	r+q	PROPN
ma-301	266	11	)	)	PUNCT
ma-301	266	12	.	.	PUNCT
ma-301	267	1	we	we	PRON
ma-301	267	2	denote	denote	VERB
ma-301	267	3	by	by	ADP
ma-301	267	4	〈	〈	PROPN
ma-301	267	5	f	f	PROPN
ma-301	267	6	,	,	PUNCT
ma-301	267	7	g〉ψ	g〉ψ	PROPN
ma-301	267	8	,	,	PUNCT
ma-301	267	9	η	η	PROPN
ma-301	267	10	the	the	DET
ma-301	267	11	inner	inner	ADJ
ma-301	267	12	product	product	NOUN
ma-301	267	13	defined	define	VERB
ma-301	267	14	on	on	ADP
ma-301	267	15	the	the	DET
ma-301	267	16	space	space	NOUN
ma-301	267	17	sψ(r+q	sψ(r+q	PROPN
ma-301	267	18	)	)	PUNCT
ma-301	267	19	by	by	ADP
ma-301	267	20	〈	〈	PROPN
ma-301	267	21	f	f	PROPN
ma-301	267	22	,	,	PUNCT
ma-301	267	23	g〉ψ	g〉ψ	PROPN
ma-301	267	24	,	,	PUNCT
ma-301	267	25	η	η	PROPN
ma-301	267	26	=	=	PROPN
ma-301	267	27	∫	∫	PROPN
ma-301	267	28	∞	∞	PROPN
ma-301	267	29	0	0	NUM
ma-301	268	1	(	(	PUNCT
ma-301	268	2	ηψ(λ	ηψ(λ	PROPN
ma-301	268	3	)	)	PUNCT
ma-301	269	1	+	+	CCONJ
ma-301	269	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	269	3	)	)	PUNCT
ma-301	269	4	∣∣2)hq	∣∣2)hq	NOUN
ma-301	269	5	,	,	PUNCT
ma-301	269	6	α(f	α(f	PROPN
ma-301	269	7	)	)	PUNCT
ma-301	269	8	(	(	PUNCT
ma-301	269	9	λ)hq	λ)hq	PROPN
ma-301	269	10	,	,	PUNCT
ma-301	269	11	α(g)(λ)dµq	α(g)(λ)dµq	PROPN
ma-301	269	12	,	,	PUNCT
ma-301	269	13	α(λ	α(λ	PROPN
ma-301	269	14	)	)	PUNCT
ma-301	269	15	,	,	PUNCT
ma-301	269	16	and	and	CCONJ
ma-301	269	17	the	the	DET
ma-301	269	18	norm	norm	NOUN
ma-301	269	19	‖f	‖f	ADP
ma-301	269	20	‖ψ	‖ψ	NOUN
ma-301	269	21	,	,	PUNCT
ma-301	269	22	η	η	PROPN
ma-301	269	23	=	=	PROPN
ma-301	269	24	√	√	PROPN
ma-301	269	25	〈	〈	PROPN
ma-301	269	26	f	f	PROPN
ma-301	269	27	,	,	PUNCT
ma-301	269	28	f	f	PROPN
ma-301	269	29	〉	〉	PROPN
ma-301	269	30	ψ	ψ	PROPN
ma-301	269	31	,	,	PUNCT
ma-301	269	32	η	η	PROPN
ma-301	269	33	theorem	theorem	VERB
ma-301	269	34	4.1	4.1	NUM
ma-301	269	35	.	.	PUNCT
ma-301	270	1	let	let	VERB
ma-301	270	2	σ	σ	NUM
ma-301	270	3	∈	∈	PROPN
ma-301	270	4	l∞α	l∞α	NOUN
ma-301	270	5	(	(	PUNCT
ma-301	270	6	r+q	r+q	PROPN
ma-301	270	7	)	)	PUNCT
ma-301	270	8	the	the	DET
ma-301	270	9	sobolev	sobolev	NOUN
ma-301	270	10	-	-	PUNCT
ma-301	270	11	type	type	NOUN
ma-301	270	12	space	space	NOUN
ma-301	270	13	(	(	PUNCT
ma-301	270	14	sψ(r+q	sψ(r+q	PROPN
ma-301	270	15	)	)	PUNCT
ma-301	270	16	,	,	PUNCT
ma-301	270	17	〈	〈	PROPN
ma-301	270	18	·	·	SYM
ma-301	270	19	,	,	PUNCT
ma-301	270	20	·	·	PUNCT
ma-301	270	21	〉	〉	PROPN
ma-301	270	22	ψ	ψ	PROPN
ma-301	270	23	,	,	PUNCT
ma-301	270	24	η	η	NOUN
ma-301	270	25	)	)	PUNCT
ma-301	270	26	is	be	AUX
ma-301	270	27	a	a	DET
ma-301	270	28	reproducing	reproduce	VERB
ma-301	270	29	kernel	kernel	NOUN
ma-301	270	30	hilbert	hilbert	PROPN
ma-301	270	31	space	space	NOUN
ma-301	270	32	with	with	ADP
ma-301	270	33	kernel	kernel	PROPN
ma-301	270	34	kq	kq	PROPN
ma-301	270	35	,	,	PUNCT
ma-301	270	36	ψ	ψ	PROPN
ma-301	270	37	,	,	PUNCT
ma-301	270	38	η(x	η(x	PROPN
ma-301	270	39	,	,	PUNCT
ma-301	270	40	y	y	NOUN
ma-301	270	41	)	)	PUNCT
ma-301	270	42	=	=	SYM
ma-301	271	1	∫	∫	PROPN
ma-301	272	1	∞	∞	NOUN
ma-301	272	2	0	0	NUM
ma-301	273	1	jα(λx	jα(λx	NOUN
ma-301	273	2	;	;	PUNCT
ma-301	273	3	q2)jα(λy	q2)jα(λy	NOUN
ma-301	273	4	;	;	PUNCT
ma-301	273	5	q2	q2	PROPN
ma-301	273	6	)	)	PUNCT
ma-301	273	7	ηψ(λ	ηψ(λ	PUNCT
ma-301	273	8	)	)	PUNCT
ma-301	274	1	+	+	CCONJ
ma-301	274	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	274	3	)	)	PUNCT
ma-301	274	4	∣∣2	∣∣2	PROPN
ma-301	274	5	dµq	dµq	PROPN
ma-301	274	6	,	,	PUNCT
ma-301	274	7	α(λ	α(λ	PROPN
ma-301	274	8	)	)	PUNCT
ma-301	274	9	,	,	PUNCT
ma-301	274	10	that	that	ADV
ma-301	274	11	is	is	ADV
ma-301	274	12	(	(	PUNCT
ma-301	274	13	i	i	NOUN
ma-301	274	14	)	)	PUNCT
ma-301	274	15	for	for	ADP
ma-301	274	16	all	all	DET
ma-301	274	17	y	y	PROPN
ma-301	274	18	∈	∈	PROPN
ma-301	274	19	r+q	r+q	PROPN
ma-301	274	20	,	,	PUNCT
ma-301	274	21	the	the	DET
ma-301	274	22	function	function	NOUN
ma-301	274	23	x	x	PROPN
ma-301	274	24	7→	7→	NUM
ma-301	274	25	kq	kq	PROPN
ma-301	274	26	,	,	PUNCT
ma-301	274	27	ψ	ψ	PROPN
ma-301	274	28	,	,	PUNCT
ma-301	274	29	η	η	PROPN
ma-301	274	30	(	(	PUNCT
ma-301	274	31	x	x	PROPN
ma-301	274	32	,	,	PUNCT
ma-301	274	33	y	y	PROPN
ma-301	274	34	)	)	PUNCT
ma-301	274	35	belongs	belong	VERB
ma-301	274	36	to	to	ADP
ma-301	274	37	sψ(r+q	sψ(r+q	PROPN
ma-301	274	38	)	)	PUNCT
ma-301	274	39	.	.	PUNCT
ma-301	275	1	(	(	PUNCT
ma-301	275	2	ii	ii	NOUN
ma-301	275	3	)	)	PUNCT
ma-301	275	4	for	for	ADP
ma-301	275	5	all	all	DET
ma-301	275	6	f	f	PROPN
ma-301	275	7	∈	∈	PROPN
ma-301	275	8	sψ(r+q	sψ(r+q	PROPN
ma-301	275	9	)	)	PUNCT
ma-301	275	10	and	and	CCONJ
ma-301	275	11	y	y	PROPN
ma-301	275	12	∈	∈	PROPN
ma-301	275	13	r+q	r+q	PROPN
ma-301	275	14	,	,	PUNCT
ma-301	275	15	we	we	PRON
ma-301	275	16	have	have	VERB
ma-301	275	17	the	the	DET
ma-301	275	18	reproducing	reproduce	VERB
ma-301	275	19	property	property	NOUN
ma-301	275	20	f	f	NOUN
ma-301	275	21	(	(	PUNCT
ma-301	275	22	y	y	NOUN
ma-301	275	23	)	)	PUNCT
ma-301	275	24	=	=	SYM
ma-301	276	1	〈	〈	PROPN
ma-301	276	2	f	f	PROPN
ma-301	276	3	,	,	PUNCT
ma-301	276	4	kq	kq	PROPN
ma-301	276	5	,	,	PUNCT
ma-301	276	6	ψ	ψ	PROPN
ma-301	276	7	,	,	PUNCT
ma-301	276	8	η	η	PROPN
ma-301	276	9	(	(	PUNCT
ma-301	276	10	·	·	PUNCT
ma-301	276	11	,	,	PUNCT
ma-301	276	12	(	(	PUNCT
ma-301	276	13	y	y	NOUN
ma-301	276	14	)	)	PUNCT
ma-301	276	15	)	)	PUNCT
ma-301	276	16	〉	〉	PROPN
ma-301	276	17	ψ	ψ	NOUN
ma-301	276	18	,	,	PUNCT
ma-301	276	19	η	η	PROPN
ma-301	276	20	.	.	PUNCT
ma-301	277	1	furthermore	furthermore	ADV
ma-301	277	2	the	the	DET
ma-301	277	3	kernel	kernel	PROPN
ma-301	277	4	kq	kq	PROPN
ma-301	277	5	,	,	PUNCT
ma-301	277	6	ψ	ψ	PROPN
ma-301	277	7	,	,	PUNCT
ma-301	277	8	η	η	PROPN
ma-301	277	9	is	be	AUX
ma-301	277	10	a	a	DET
ma-301	277	11	positive	positive	ADJ
ma-301	277	12	definite	definite	ADJ
ma-301	277	13	function	function	NOUN
ma-301	277	14	.	.	PUNCT
ma-301	278	1	proof	proof	NOUN
ma-301	278	2	.	.	PUNCT
ma-301	279	1	(	(	PUNCT
ma-301	279	2	i	i	NOUN
ma-301	279	3	)	)	PUNCT
ma-301	279	4	let	let	VERB
ma-301	279	5	y	y	PROPN
ma-301	279	6	∈	∈	PROPN
ma-301	279	7	r+q	r+q	PROPN
ma-301	279	8	,	,	PUNCT
ma-301	279	9	from	from	ADP
ma-301	279	10	the	the	DET
ma-301	279	11	relations	relation	NOUN
ma-301	279	12	(	(	PUNCT
ma-301	279	13	2.2),(4.1	2.2),(4.1	NUM
ma-301	279	14	)	)	PUNCT
ma-301	279	15	we	we	PRON
ma-301	279	16	have	have	VERB
ma-301	279	17	the	the	DET
ma-301	279	18	function	function	NOUN
ma-301	279	19	gy	gy	NOUN
ma-301	279	20	:	:	PUNCT
ma-301	279	21	λ	λ	NOUN
ma-301	279	22	−→	−→	NOUN
ma-301	279	23	jα(λy	jα(λy	PROPN
ma-301	279	24	;	;	PUNCT
ma-301	279	25	q2	q2	PROPN
ma-301	279	26	)	)	PUNCT
ma-301	279	27	ηψ(λ	ηψ(λ	PUNCT
ma-301	279	28	)	)	PUNCT
ma-301	280	1	+	+	CCONJ
ma-301	280	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	280	3	)	)	PUNCT
ma-301	280	4	∣∣2	∣∣2	PROPN
ma-301	280	5	belongs	belong	VERB
ma-301	280	6	to	to	ADP
ma-301	280	7	l1α(r+q	l1α(r+q	PROPN
ma-301	280	8	)	)	PUNCT
ma-301	280	9	∩	∩	PROPN
ma-301	280	10	l2α(r+q	l2α(r+q	PROPN
ma-301	280	11	)	)	PUNCT
ma-301	280	12	.	.	PUNCT
ma-301	281	1	hence	hence	ADV
ma-301	281	2	the	the	DET
ma-301	281	3	function	function	NOUN
ma-301	281	4	kq	kq	PROPN
ma-301	281	5	,	,	PUNCT
ma-301	281	6	ψ	ψ	PROPN
ma-301	281	7	,	,	PUNCT
ma-301	281	8	η	η	PROPN
ma-301	281	9	is	be	AUX
ma-301	281	10	well	well	ADV
ma-301	281	11	defined	define	VERB
ma-301	281	12	and	and	CCONJ
ma-301	281	13	by	by	ADP
ma-301	281	14	the	the	DET
ma-301	281	15	inversionformula	inversionformula	NOUN
ma-301	281	16	(	(	PUNCT
ma-301	281	17	2.5	2.5	NUM
ma-301	281	18	)	)	PUNCT
ma-301	281	19	,	,	PUNCT
ma-301	281	20	we	we	PRON
ma-301	281	21	get	get	VERB
ma-301	281	22	kq	kq	PROPN
ma-301	281	23	,	,	PUNCT
ma-301	281	24	ψ	ψ	PROPN
ma-301	281	25	,	,	PUNCT
ma-301	281	26	η(x	η(x	PROPN
ma-301	281	27	,	,	PUNCT
ma-301	281	28	y	y	NOUN
ma-301	281	29	)	)	PUNCT
ma-301	282	1	=	=	SYM
ma-301	282	2	h−1q	h−1q	NOUN
ma-301	282	3	,	,	PUNCT
ma-301	282	4	α(gy	α(gy	NOUN
ma-301	282	5	)	)	PUNCT
ma-301	282	6	(	(	PUNCT
ma-301	283	1	x)by	x)by	PROPN
ma-301	283	2	using	use	VERB
ma-301	283	3	plancherel	plancherel	NOUN
ma-301	283	4	’s	’s	PART
ma-301	283	5	theorem	theorem	NOUN
ma-301	283	6	for	for	ADP
ma-301	283	7	hq	hq	NOUN
ma-301	283	8	,	,	PUNCT
ma-301	283	9	α	α	PRON
ma-301	283	10	we	we	PRON
ma-301	283	11	find	find	VERB
ma-301	283	12	that	that	SCONJ
ma-301	283	13	kq	kq	PROPN
ma-301	283	14	,	,	PUNCT
ma-301	283	15	ψ	ψ	PROPN
ma-301	283	16	,	,	PUNCT
ma-301	283	17	η	η	PROPN
ma-301	283	18	(	(	PUNCT
ma-301	283	19	·	·	PUNCT
ma-301	283	20	,	,	PUNCT
ma-301	283	21	y	y	NOUN
ma-301	283	22	)	)	PUNCT
ma-301	283	23	belongs	belong	VERB
ma-301	283	24	to	to	ADP
ma-301	283	25	l2α(r+q	l2α(r+q	PROPN
ma-301	283	26	)	)	PUNCT
ma-301	283	27	and	and	CCONJ
ma-301	283	28	we	we	PRON
ma-301	283	29	have	have	VERB
ma-301	283	30	hq	hq	X
ma-301	283	31	,	,	PUNCT
ma-301	283	32	α(kq	α(kq	PROPN
ma-301	283	33	,	,	PUNCT
ma-301	283	34	ψ	ψ	NOUN
ma-301	283	35	,	,	PUNCT
ma-301	283	36	η	η	PROPN
ma-301	283	37	(	(	PUNCT
ma-301	283	38	·	·	PUNCT
ma-301	283	39	,	,	PUNCT
ma-301	283	40	y))(λ	y))(λ	NOUN
ma-301	283	41	)	)	PUNCT
ma-301	283	42	=	=	NOUN
ma-301	283	43	jα(λy	jα(λy	PROPN
ma-301	283	44	;	;	PUNCT
ma-301	283	45	q2	q2	PROPN
ma-301	283	46	)	)	PUNCT
ma-301	283	47	ηψ(λ	ηψ(λ	PUNCT
ma-301	283	48	)	)	PUNCT
ma-301	283	49	+	+	CCONJ
ma-301	283	50	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	283	51	)	)	PUNCT
ma-301	283	52	∣∣2	∣∣2	PROPN
ma-301	283	53	(	(	PUNCT
ma-301	283	54	4.4	4.4	NUM
ma-301	283	55	)	)	PUNCT
ma-301	283	56	by	by	ADP
ma-301	283	57	using	use	VERB
ma-301	283	58	the	the	DET
ma-301	283	59	relations	relation	NOUN
ma-301	283	60	(	(	PUNCT
ma-301	283	61	2.2),(4.1	2.2),(4.1	NUM
ma-301	283	62	)	)	PUNCT
ma-301	283	63	and	and	CCONJ
ma-301	283	64	(	(	PUNCT
ma-301	283	65	4.4	4.4	NUM
ma-301	283	66	)	)	PUNCT
ma-301	283	67	we	we	PRON
ma-301	283	68	find	find	VERB
ma-301	283	69	that	that	SCONJ
ma-301	283	70	‖	‖	ADJ
ma-301	283	71	√	√	NUM
ma-301	283	72	ψhq	ψhq	PRON
ma-301	283	73	,	,	PUNCT
ma-301	283	74	α(kq	α(kq	X
ma-301	283	75	,	,	PUNCT
ma-301	283	76	ψ	ψ	NOUN
ma-301	283	77	,	,	PUNCT
ma-301	283	78	η	η	PROPN
ma-301	283	79	(	(	PUNCT
ma-301	283	80	·	·	PUNCT
ma-301	283	81	,	,	PUNCT
ma-301	283	82	y))‖2,q	y))‖2,q	NOUN
ma-301	283	83	,	,	PUNCT
ma-301	283	84	α	α	NOUN
ma-301	283	85	≤	≤	NOUN
ma-301	283	86	1	1	NUM
ma-301	283	87	η2	η2	ADJ
ma-301	283	88	∥∥∥∥	∥∥∥∥	NUM
ma-301	283	89	1	1	NUM
ma-301	283	90	ψ	ψ	ADP
ma-301	283	91	∥∥∥∥	∥∥∥∥	PROPN
ma-301	283	92	1,q	1,q	NUM
ma-301	283	93	,	,	PUNCT
ma-301	283	94	α	α	NOUN
ma-301	283	95	<	<	X
ma-301	283	96	∞	∞	PROPN
ma-301	283	97	,	,	PUNCT
ma-301	283	98	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	283	99	eur	eur	PROPN
ma-301	283	100	.	.	PUNCT
ma-301	284	1	j.	j.	PROPN
ma-301	284	2	math	math	PROPN
ma-301	284	3	.	.	PUNCT
ma-301	285	1	anal	anal	PROPN
ma-301	285	2	.	.	PUNCT
ma-301	286	1	10.28924	10.28924	NUM
ma-301	286	2	/	/	SYM
ma-301	286	3	ada	ada	PROPN
ma-301	286	4	/	/	SYM
ma-301	286	5	ma.5.8	ma.5.8	PROPN
ma-301	286	6	12this	12this	PRON
ma-301	286	7	prove	prove	VERB
ma-301	286	8	that	that	SCONJ
ma-301	286	9	for	for	ADP
ma-301	286	10	every	every	DET
ma-301	286	11	y	y	PROPN
ma-301	286	12	∈	∈	PROPN
ma-301	286	13	r+q	r+q	PROPN
ma-301	286	14	the	the	DET
ma-301	286	15	function	function	NOUN
ma-301	286	16	x	x	PROPN
ma-301	286	17	7→	7→	NUM
ma-301	286	18	kq	kq	PROPN
ma-301	286	19	,	,	PUNCT
ma-301	286	20	ψ	ψ	PROPN
ma-301	286	21	,	,	PUNCT
ma-301	286	22	η	η	PROPN
ma-301	286	23	(	(	PUNCT
ma-301	286	24	x	x	PROPN
ma-301	286	25	,	,	PUNCT
ma-301	286	26	y	y	PROPN
ma-301	286	27	)	)	PUNCT
ma-301	286	28	belongs	belong	VERB
ma-301	286	29	to	to	ADP
ma-301	286	30	sψ(r+q	sψ(r+q	PROPN
ma-301	286	31	)	)	PUNCT
ma-301	287	1	.(ii	.(ii	PROPN
ma-301	287	2	)	)	PUNCT
ma-301	287	3	by	by	ADP
ma-301	287	4	using	use	VERB
ma-301	287	5	the	the	DET
ma-301	287	6	relation	relation	NOUN
ma-301	287	7	(	(	PUNCT
ma-301	287	8	4.4	4.4	NUM
ma-301	287	9	)	)	PUNCT
ma-301	287	10	we	we	PRON
ma-301	287	11	find	find	VERB
ma-301	287	12	that	that	SCONJ
ma-301	287	13	for	for	ADP
ma-301	287	14	all	all	DET
ma-301	287	15	f	f	PROPN
ma-301	287	16	∈	∈	PROPN
ma-301	287	17	hψ(r	hψ(r	PRON
ma-301	287	18	)	)	PUNCT
ma-301	287	19	,	,	PUNCT
ma-301	287	20	〈	〈	PROPN
ma-301	287	21	f	f	PROPN
ma-301	287	22	,	,	PUNCT
ma-301	287	23	kq	kq	PROPN
ma-301	287	24	,	,	PUNCT
ma-301	287	25	ψ	ψ	PROPN
ma-301	287	26	,	,	PUNCT
ma-301	287	27	η	η	PROPN
ma-301	287	28	(	(	PUNCT
ma-301	287	29	·	·	PROPN
ma-301	287	30	,	,	PUNCT
ma-301	287	31	y)〉ψ	y)〉ψ	PROPN
ma-301	287	32	,	,	PUNCT
ma-301	287	33	η	η	PROPN
ma-301	287	34	=	=	PROPN
ma-301	287	35	∫	∫	PROPN
ma-301	287	36	∞	∞	PROPN
ma-301	287	37	0	0	NUM
ma-301	287	38	(	(	PUNCT
ma-301	287	39	ηψ(λ	ηψ(λ	PROPN
ma-301	287	40	)	)	PUNCT
ma-301	288	1	+	+	CCONJ
ma-301	288	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	288	3	)	)	PUNCT
ma-301	288	4	∣∣2)hq	∣∣2)hq	NOUN
ma-301	288	5	,	,	PUNCT
ma-301	288	6	α(f	α(f	PROPN
ma-301	288	7	)	)	PUNCT
ma-301	288	8	(	(	PUNCT
ma-301	288	9	λ)hq	λ)hq	PROPN
ma-301	288	10	,	,	PUNCT
ma-301	288	11	α(kq	α(kq	X
ma-301	288	12	,	,	PUNCT
ma-301	288	13	ψ	ψ	NOUN
ma-301	288	14	,	,	PUNCT
ma-301	288	15	η	η	NOUN
ma-301	288	16	)	)	PUNCT
ma-301	288	17	(	(	PUNCT
ma-301	288	18	·	·	PUNCT
ma-301	288	19	,	,	PUNCT
ma-301	288	20	y))(λ)dµq	y))(λ)dµq	PROPN
ma-301	288	21	,	,	PUNCT
ma-301	288	22	α(λ	α(λ	PROPN
ma-301	288	23	)	)	PUNCT
ma-301	289	1	=	=	SYM
ma-301	290	1	∫	∫	PROPN
ma-301	291	1	∞	∞	NOUN
ma-301	291	2	0	0	NUM
ma-301	291	3	jα(λy	jα(λy	PROPN
ma-301	291	4	;	;	PUNCT
ma-301	291	5	q2)hq	q2)hq	NOUN
ma-301	291	6	,	,	PUNCT
ma-301	291	7	α(f	α(f	PROPN
ma-301	291	8	)	)	PUNCT
ma-301	291	9	(	(	PUNCT
ma-301	291	10	λ)dµq	λ)dµq	PROPN
ma-301	291	11	,	,	PUNCT
ma-301	291	12	α(λ	α(λ	PROPN
ma-301	291	13	)	)	PUNCT
ma-301	291	14	,	,	PUNCT
ma-301	291	15	inversion	inversion	NOUN
ma-301	291	16	formula	formula	NOUN
ma-301	291	17	(	(	PUNCT
ma-301	291	18	2.5	2.5	NUM
ma-301	291	19	)	)	PUNCT
ma-301	291	20	gives	give	VERB
ma-301	291	21	the	the	DET
ma-301	291	22	desired	desire	VERB
ma-301	291	23	result	result	NOUN
ma-301	291	24	.	.	PUNCT
ma-301	292	1	on	on	ADP
ma-301	292	2	the	the	DET
ma-301	292	3	other	other	ADJ
ma-301	292	4	hand	hand	NOUN
ma-301	292	5	since	since	SCONJ
ma-301	292	6	1ψ	1ψ	NUM
ma-301	292	7	is	be	AUX
ma-301	292	8	positive	positive	ADJ
ma-301	292	9	function	function	NOUN
ma-301	292	10	thenfor	thenfor	ADP
ma-301	292	11	all	all	DET
ma-301	292	12	z1	z1	NOUN
ma-301	292	13	,	,	PUNCT
ma-301	292	14	.	.	PUNCT
ma-301	292	15	.	.	PUNCT
ma-301	293	1	.	.	PUNCT
ma-301	294	1	.	.	PUNCT
ma-301	295	1	,	,	PUNCT
ma-301	295	2	zn	zn	PROPN
ma-301	295	3	complex	complex	ADJ
ma-301	295	4	numbers	number	NOUN
ma-301	295	5	and	and	CCONJ
ma-301	295	6	x1	x1	NUM
ma-301	295	7	,	,	PUNCT
ma-301	295	8	.	.	PUNCT
ma-301	295	9	.	.	PUNCT
ma-301	295	10	.	.	PUNCT
ma-301	295	11	.	.	PUNCT
ma-301	295	12	.	.	PUNCT
ma-301	296	1	.	.	PUNCT
ma-301	297	1	,	,	PUNCT
ma-301	297	2	xn	xn	PROPN
ma-301	297	3	in	in	ADP
ma-301	297	4	r+q	r+q	PROPN
ma-301	297	5	,	,	PUNCT
ma-301	297	6	we	we	PRON
ma-301	297	7	obtain	obtain	VERB
ma-301	297	8	n∑	n∑	ADJ
ma-301	297	9	r=1	r=1	NOUN
ma-301	297	10	n∑	n∑	PROPN
ma-301	297	11	l=1	l=1	PROPN
ma-301	297	12	zrzlkq	zrzlkq	NUM
ma-301	297	13	,	,	PUNCT
ma-301	297	14	ψ	ψ	X
ma-301	297	15	,	,	PUNCT
ma-301	297	16	η(xr	η(xr	PROPN
ma-301	297	17	,	,	PUNCT
ma-301	297	18	xl	xl	PROPN
ma-301	297	19	)	)	PUNCT
ma-301	298	1	=	=	SYM
ma-301	299	1	∫	∫	PROPN
ma-301	300	1	+	+	NUM
ma-301	300	2	∞	∞	NOUN
ma-301	300	3	0	0	NUM
ma-301	301	1	[	[	PUNCT
ma-301	301	2	n∑	n∑	NOUN
ma-301	301	3	r=1	r=1	PROPN
ma-301	301	4	n∑	n∑	PROPN
ma-301	301	5	l=1	l=1	PROPN
ma-301	301	6	zrzl	zrzl	NOUN
ma-301	301	7	jα	jα	X
ma-301	301	8	(	(	PUNCT
ma-301	301	9	xrλ	xrλ	NOUN
ma-301	301	10	;	;	PUNCT
ma-301	301	11	q2	q2	NOUN
ma-301	301	12	)	)	PUNCT
ma-301	302	1	jα	jα	PROPN
ma-301	302	2	(	(	PUNCT
ma-301	302	3	xlλ	xlλ	PROPN
ma-301	302	4	;	;	PUNCT
ma-301	302	5	q2	q2	PROPN
ma-301	302	6	)	)	PUNCT
ma-301	302	7	]	]	PUNCT
ma-301	302	8	1	1	NUM
ma-301	302	9	ψ	ψ	X
ma-301	302	10	(	(	PUNCT
ma-301	302	11	λ)dµq	λ)dµq	PROPN
ma-301	302	12	,	,	PUNCT
ma-301	302	13	α(λ	α(λ	PROPN
ma-301	302	14	)	)	PUNCT
ma-301	303	1	=	=	PUNCT
ma-301	303	2	∫	∫	PROPN
ma-301	304	1	+	+	NUM
ma-301	304	2	∞	∞	PROPN
ma-301	304	3	0	0	NUM
ma-301	304	4	∣∣∣∣∣	∣∣∣∣∣	SYM
ma-301	305	1	n∑	n∑	PROPN
ma-301	305	2	r=1	r=1	ADJ
ma-301	305	3	zr	zr	PROPN
ma-301	305	4	j	j	PROPN
ma-301	305	5	(	(	PUNCT
ma-301	305	6	xrλ	xrλ	NOUN
ma-301	305	7	;	;	PUNCT
ma-301	305	8	q2	q2	PROPN
ma-301	305	9	)	)	PUNCT
ma-301	305	10	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-301	305	11	2	2	NUM
ma-301	305	12	1	1	NUM
ma-301	305	13	ψ	ψ	X
ma-301	305	14	(	(	PUNCT
ma-301	305	15	λ)dµq	λ)dµq	PROPN
ma-301	305	16	,	,	PUNCT
ma-301	305	17	α(λ	α(λ	PROPN
ma-301	305	18	)	)	PUNCT
ma-301	305	19	≥	≥	NOUN
ma-301	305	20	0	0	NUM
ma-301	305	21	which	which	PRON
ma-301	305	22	proves	prove	VERB
ma-301	305	23	that	that	SCONJ
ma-301	305	24	the	the	DET
ma-301	305	25	kernel	kernel	PROPN
ma-301	305	26	kq	kq	PROPN
ma-301	305	27	,	,	PUNCT
ma-301	305	28	ψ	ψ	PROPN
ma-301	305	29	,	,	PUNCT
ma-301	305	30	η	η	PROPN
ma-301	305	31	is	be	AUX
ma-301	305	32	positive	positive	ADJ
ma-301	305	33	definite	definite	ADJ
ma-301	305	34	.	.	PUNCT
ma-301	306	1	�	�	PROPN
ma-301	306	2	the	the	DET
ma-301	306	3	main	main	ADJ
ma-301	306	4	result	result	NOUN
ma-301	306	5	of	of	ADP
ma-301	306	6	this	this	DET
ma-301	306	7	section	section	NOUN
ma-301	306	8	can	can	AUX
ma-301	306	9	be	be	AUX
ma-301	306	10	stated	state	VERB
ma-301	306	11	as	as	SCONJ
ma-301	306	12	follows	follow	NOUN
ma-301	306	13	theorem	theorem	VERB
ma-301	306	14	4.2	4.2	NUM
ma-301	306	15	.	.	PUNCT
ma-301	307	1	let	let	VERB
ma-301	307	2	σ	σ	NUM
ma-301	307	3	∈	∈	PROPN
ma-301	307	4	l∞α	l∞α	NOUN
ma-301	307	5	(	(	PUNCT
ma-301	307	6	r+q	r+q	PROPN
ma-301	307	7	)	)	PUNCT
ma-301	307	8	and	and	CCONJ
ma-301	307	9	β	β	X
ma-301	307	10	∈	∈	PROPN
ma-301	307	11	r+q	r+q	PROPN
ma-301	307	12	,	,	PUNCT
ma-301	307	13	for	for	ADP
ma-301	307	14	any	any	DET
ma-301	307	15	h	h	NOUN
ma-301	307	16	∈	∈	NOUN
ma-301	307	17	l2α	l2α	PROPN
ma-301	307	18	(	(	PUNCT
ma-301	307	19	r+q	r+q	PROPN
ma-301	307	20	)	)	PUNCT
ma-301	307	21	and	and	CCONJ
ma-301	307	22	for	for	ADP
ma-301	307	23	any	any	DET
ma-301	307	24	η	η	PROPN
ma-301	307	25	>	>	X
ma-301	307	26	0	0	PROPN
ma-301	307	27	,	,	PUNCT
ma-301	307	28	there	there	PRON
ma-301	307	29	exist	exist	VERB
ma-301	307	30	a	a	DET
ma-301	307	31	unique	unique	ADJ
ma-301	307	32	function	function	NOUN
ma-301	307	33	f	f	PROPN
ma-301	307	34	∗q	∗q	PROPN
ma-301	307	35	,	,	PUNCT
ma-301	307	36	η	η	PROPN
ma-301	307	37	,	,	PUNCT
ma-301	307	38	β	β	X
ma-301	307	39	,	,	PUNCT
ma-301	307	40	h	h	NOUN
ma-301	307	41	where	where	SCONJ
ma-301	307	42	the	the	DET
ma-301	307	43	infimum	infimum	ADJ
ma-301	307	44	inf	inf	NOUN
ma-301	307	45	f	f	PROPN
ma-301	307	46	∈sψ(r+q	∈sψ(r+q	PROPN
ma-301	307	47	)	)	PUNCT
ma-301	307	48	{	{	PUNCT
ma-301	307	49	η‖f	η‖f	VERB
ma-301	307	50	‖2ψ	‖2ψ	ADV
ma-301	307	51	+	+	CCONJ
ma-301	307	52	∥∥h	∥∥h	ADJ
ma-301	307	53	−mq	−mq	PROPN
ma-301	307	54	,	,	PUNCT
ma-301	307	55	σ	σ	PROPN
ma-301	307	56	,	,	PUNCT
ma-301	307	57	β(f	β(f	PROPN
ma-301	307	58	)	)	PUNCT
ma-301	307	59	∥∥2	∥∥2	PROPN
ma-301	308	1	2,q	2,q	NUM
ma-301	308	2	,	,	PUNCT
ma-301	308	3	α	α	NOUN
ma-301	308	4	}	}	PUNCT
ma-301	308	5	(	(	PUNCT
ma-301	308	6	4.5	4.5	NUM
ma-301	308	7	)	)	PUNCT
ma-301	308	8	is	be	AUX
ma-301	308	9	attained	attain	VERB
ma-301	308	10	.	.	PUNCT
ma-301	309	1	moreover	moreover	ADV
ma-301	309	2	the	the	DET
ma-301	309	3	extremal	extremal	ADJ
ma-301	309	4	function	function	NOUN
ma-301	309	5	f	f	PROPN
ma-301	309	6	∗q	∗q	PROPN
ma-301	309	7	,	,	PUNCT
ma-301	309	8	η	η	PROPN
ma-301	309	9	,	,	PUNCT
ma-301	309	10	β	β	X
ma-301	309	11	,	,	PUNCT
ma-301	309	12	h	h	PROPN
ma-301	309	13	is	be	AUX
ma-301	309	14	given	give	VERB
ma-301	309	15	by	by	ADP
ma-301	309	16	f	f	PROPN
ma-301	309	17	∗q	∗q	PROPN
ma-301	309	18	,	,	PUNCT
ma-301	309	19	η	η	PROPN
ma-301	309	20	,	,	PUNCT
ma-301	309	21	β	β	X
ma-301	309	22	,	,	PUNCT
ma-301	309	23	h(y	h(y	ADV
ma-301	309	24	)	)	PUNCT
ma-301	309	25	=	=	SYM
ma-301	310	1	∫	∫	PROPN
ma-301	310	2	∞	∞	PROPN
ma-301	310	3	0	0	NUM
ma-301	310	4	h(x)θq	h(x)θq	PROPN
ma-301	310	5	,	,	PUNCT
ma-301	310	6	η	η	NOUN
ma-301	310	7	,	,	PUNCT
ma-301	310	8	β(x	β(x	NOUN
ma-301	310	9	,	,	PUNCT
ma-301	310	10	y)dµq	y)dµq	PROPN
ma-301	310	11	,	,	PUNCT
ma-301	310	12	α(x	α(x	NOUN
ma-301	310	13	)	)	PUNCT
ma-301	310	14	,	,	PUNCT
ma-301	310	15	where	where	SCONJ
ma-301	310	16	θq	θq	ADP
ma-301	310	17	,	,	PUNCT
ma-301	310	18	η	η	PROPN
ma-301	310	19	,	,	PUNCT
ma-301	310	20	β	β	X
ma-301	310	21	is	be	AUX
ma-301	310	22	given	give	VERB
ma-301	310	23	by	by	ADP
ma-301	310	24	θq	θq	ADP
ma-301	310	25	,	,	PUNCT
ma-301	310	26	η	η	NOUN
ma-301	310	27	,	,	PUNCT
ma-301	310	28	β(x	β(x	PROPN
ma-301	310	29	,	,	PUNCT
ma-301	310	30	y	y	NOUN
ma-301	310	31	)	)	PUNCT
ma-301	310	32	=	=	SYM
ma-301	311	1	∫	∫	PROPN
ma-301	311	2	∞	∞	NUM
ma-301	311	3	0	0	PROPN
ma-301	311	4	σβ(λ)jα(λx	σβ(λ)jα(λx	NOUN
ma-301	311	5	;	;	PUNCT
ma-301	311	6	q2)jα(λy	q2)jα(λy	NOUN
ma-301	311	7	;	;	PUNCT
ma-301	311	8	q2	q2	PROPN
ma-301	311	9	)	)	PUNCT
ma-301	311	10	ηψ(λ	ηψ(λ	PUNCT
ma-301	311	11	)	)	PUNCT
ma-301	312	1	+	+	CCONJ
ma-301	312	2	|σβ(λ)|2	|σβ(λ)|2	PROPN
ma-301	312	3	dµq	dµq	PROPN
ma-301	312	4	,	,	PUNCT
ma-301	312	5	α(λ	α(λ	PROPN
ma-301	312	6	)	)	PUNCT
ma-301	312	7	proof	proof	NOUN
ma-301	312	8	.	.	PUNCT
ma-301	313	1	the	the	DET
ma-301	313	2	existence	existence	NOUN
ma-301	313	3	and	and	CCONJ
ma-301	313	4	the	the	DET
ma-301	313	5	unicity	unicity	NOUN
ma-301	313	6	of	of	ADP
ma-301	313	7	the	the	DET
ma-301	313	8	extremal	extremal	ADJ
ma-301	313	9	function	function	NOUN
ma-301	313	10	f	f	PROPN
ma-301	313	11	∗q	∗q	PROPN
ma-301	313	12	,	,	PUNCT
ma-301	313	13	η	η	PROPN
ma-301	313	14	,	,	PUNCT
ma-301	313	15	β	β	X
ma-301	313	16	,	,	PUNCT
ma-301	313	17	h	h	NOUN
ma-301	313	18	satisfying	satisfying	ADJ
ma-301	313	19	(	(	PUNCT
ma-301	313	20	4.5	4.5	NUM
ma-301	313	21	)	)	PUNCT
ma-301	313	22	is	be	AUX
ma-301	313	23	given	give	VERB
ma-301	313	24	in[23,30	in[23,30	PROPN
ma-301	313	25	]	]	X
ma-301	313	26	,	,	PUNCT
ma-301	313	27	furthermore	furthermore	ADV
ma-301	313	28	f	f	PROPN
ma-301	313	29	∗q	∗q	PROPN
ma-301	313	30	,	,	PUNCT
ma-301	313	31	η	η	PROPN
ma-301	313	32	,	,	PUNCT
ma-301	313	33	β	β	X
ma-301	313	34	,	,	PUNCT
ma-301	313	35	h	h	PROPN
ma-301	313	36	is	be	AUX
ma-301	313	37	given	give	VERB
ma-301	313	38	by	by	ADP
ma-301	313	39	f	f	PROPN
ma-301	313	40	∗q	∗q	PROPN
ma-301	313	41	,	,	PUNCT
ma-301	313	42	η	η	PROPN
ma-301	313	43	,	,	PUNCT
ma-301	313	44	β	β	X
ma-301	313	45	,	,	PUNCT
ma-301	313	46	h(y	h(y	ADV
ma-301	313	47	)	)	PUNCT
ma-301	313	48	=	=	PUNCT
ma-301	314	1	〈	〈	PROPN
ma-301	314	2	h	h	PROPN
ma-301	314	3	,	,	PUNCT
ma-301	314	4	mq	mq	PROPN
ma-301	314	5	,	,	PUNCT
ma-301	314	6	σ	σ	PROPN
ma-301	314	7	,	,	PUNCT
ma-301	314	8	β(kq	β(kq	PROPN
ma-301	314	9	,	,	PUNCT
ma-301	314	10	ψ	ψ	X
ma-301	314	11	,	,	PUNCT
ma-301	314	12	η	η	PROPN
ma-301	314	13	(	(	PUNCT
ma-301	314	14	·	·	PROPN
ma-301	314	15	,	,	PUNCT
ma-301	314	16	y))〉q	y))〉q	NOUN
ma-301	314	17	,	,	PUNCT
ma-301	314	18	by	by	ADP
ma-301	314	19	using	use	VERB
ma-301	314	20	inversion	inversion	NOUN
ma-301	314	21	formula	formula	NOUN
ma-301	314	22	(	(	PUNCT
ma-301	314	23	2.5	2.5	NUM
ma-301	314	24	)	)	PUNCT
ma-301	314	25	and	and	CCONJ
ma-301	314	26	the	the	DET
ma-301	314	27	relation	relation	NOUN
ma-301	314	28	(	(	PUNCT
ma-301	314	29	4.4	4.4	NUM
ma-301	314	30	)	)	PUNCT
ma-301	314	31	we	we	PRON
ma-301	314	32	get	get	VERB
ma-301	314	33	mq	mq	PROPN
ma-301	314	34	,	,	PUNCT
ma-301	314	35	σ	σ	PROPN
ma-301	314	36	,	,	PUNCT
ma-301	314	37	β(kq	β(kq	PROPN
ma-301	314	38	,	,	PUNCT
ma-301	314	39	ψ	ψ	X
ma-301	314	40	,	,	PUNCT
ma-301	314	41	η	η	PROPN
ma-301	314	42	(	(	PUNCT
ma-301	314	43	·	·	PROPN
ma-301	314	44	,	,	PUNCT
ma-301	314	45	y	y	PROPN
ma-301	314	46	)	)	PUNCT
ma-301	314	47	(	(	PUNCT
ma-301	314	48	x	x	X
ma-301	314	49	)	)	PUNCT
ma-301	314	50	=	=	SYM
ma-301	315	1	∫	∫	PROPN
ma-301	315	2	∞	∞	NUM
ma-301	315	3	0	0	PROPN
ma-301	315	4	σβ(λ)jα(λx	σβ(λ)jα(λx	NOUN
ma-301	315	5	;	;	PUNCT
ma-301	315	6	q2)jα(λy	q2)jα(λy	NOUN
ma-301	315	7	;	;	PUNCT
ma-301	315	8	q2	q2	PROPN
ma-301	315	9	)	)	PUNCT
ma-301	315	10	ηψ(λ	ηψ(λ	PUNCT
ma-301	315	11	)	)	PUNCT
ma-301	316	1	+	+	CCONJ
ma-301	316	2	|σβ(λ)|2	|σβ(λ)|2	PROPN
ma-301	316	3	dµq	dµq	NOUN
ma-301	316	4	,	,	PUNCT
ma-301	316	5	α(λ	α(λ	PROPN
ma-301	316	6	)	)	PUNCT
ma-301	316	7	=	=	SYM
ma-301	317	1	θq	θq	PROPN
ma-301	317	2	,	,	PUNCT
ma-301	317	3	η	η	NOUN
ma-301	317	4	,	,	PUNCT
ma-301	317	5	β(x	β(x	NOUN
ma-301	317	6	,	,	PUNCT
ma-301	317	7	y)and	y)and	NOUN
ma-301	317	8	the	the	DET
ma-301	317	9	proof	proof	NOUN
ma-301	317	10	is	be	AUX
ma-301	317	11	complete	complete	ADJ
ma-301	317	12	.	.	PUNCT
ma-301	318	1	�	�	PROPN
ma-301	318	2	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	318	3	eur	eur	PROPN
ma-301	318	4	.	.	PUNCT
ma-301	319	1	j.	j.	PROPN
ma-301	319	2	math	math	PROPN
ma-301	319	3	.	.	PUNCT
ma-301	320	1	anal	anal	PROPN
ma-301	320	2	.	.	PUNCT
ma-301	321	1	10.28924	10.28924	NUM
ma-301	321	2	/	/	SYM
ma-301	321	3	ada	ada	PROPN
ma-301	321	4	/	/	PROPN
ma-301	321	5	ma.5.8	ma.5.8	PROPN
ma-301	321	6	13	13	NUM
ma-301	321	7	theorem	theorem	VERB
ma-301	321	8	4.3	4.3	NUM
ma-301	321	9	.	.	PUNCT
ma-301	322	1	σ	σ	PROPN
ma-301	322	2	∈	∈	PROPN
ma-301	322	3	l∞α	l∞α	NOUN
ma-301	322	4	(	(	PUNCT
ma-301	322	5	r+q	r+q	PROPN
ma-301	322	6	)	)	PUNCT
ma-301	322	7	and	and	CCONJ
ma-301	322	8	h	h	NOUN
ma-301	322	9	∈	∈	PROPN
ma-301	322	10	l2α	l2α	PROPN
ma-301	322	11	(	(	PUNCT
ma-301	322	12	r+q	r+q	PROPN
ma-301	322	13	)	)	PUNCT
ma-301	322	14	then	then	ADV
ma-301	322	15	the	the	DET
ma-301	322	16	function	function	NOUN
ma-301	322	17	f	f	PROPN
ma-301	322	18	∗q	∗q	PROPN
ma-301	322	19	,	,	PUNCT
ma-301	322	20	η	η	PROPN
ma-301	322	21	,	,	PUNCT
ma-301	322	22	β	β	X
ma-301	322	23	,	,	PUNCT
ma-301	322	24	h	h	PROPN
ma-301	322	25	satisfies	satisfy	VERB
ma-301	322	26	the	the	DET
ma-301	322	27	following	follow	VERB
ma-301	322	28	properties	property	NOUN
ma-301	322	29	hq	hq	VERB
ma-301	322	30	,	,	PUNCT
ma-301	322	31	α(f	α(f	PROPN
ma-301	322	32	∗q	∗q	PROPN
ma-301	322	33	,	,	PUNCT
ma-301	322	34	η	η	PROPN
ma-301	322	35	,	,	PUNCT
ma-301	322	36	β	β	X
ma-301	322	37	,	,	PUNCT
ma-301	322	38	h)(λ	h)(λ	NUM
ma-301	322	39	)	)	PUNCT
ma-301	322	40	=	=	SYM
ma-301	322	41	σβ(λ	σβ(λ	X
ma-301	322	42	)	)	PUNCT
ma-301	322	43	ηψ(λ	ηψ(λ	PUNCT
ma-301	322	44	)	)	PUNCT
ma-301	323	1	+	+	CCONJ
ma-301	324	1	|σβ(λ)|2hq	|σβ(λ)|2hq	NUM
ma-301	324	2	,	,	PUNCT
ma-301	324	3	α(h)(λ	α(h)(λ	NUM
ma-301	324	4	)	)	PUNCT
ma-301	324	5	(	(	PUNCT
ma-301	324	6	4.6	4.6	NUM
ma-301	324	7	)	)	PUNCT
ma-301	324	8	and	and	CCONJ
ma-301	324	9	‖f	‖f	ADP
ma-301	324	10	∗q	∗q	PROPN
ma-301	324	11	,	,	PUNCT
ma-301	324	12	η	η	PROPN
ma-301	324	13	,	,	PUNCT
ma-301	324	14	β	β	NOUN
ma-301	324	15	,	,	PUNCT
ma-301	325	1	h‖ψ	h‖ψ	PROPN
ma-301	325	2	≤	≤	PROPN
ma-301	325	3	1√	1√	PROPN
ma-301	325	4	2η	2η	PROPN
ma-301	325	5	‖h‖2,q	‖h‖2,q	PROPN
ma-301	325	6	,	,	PUNCT
ma-301	325	7	α	α	X
ma-301	325	8	.	.	PUNCT
ma-301	326	1	proof	proof	NOUN
ma-301	326	2	.	.	PUNCT
ma-301	327	1	let	let	VERB
ma-301	328	1	y	y	PROPN
ma-301	328	2	∈	∈	PROPN
ma-301	328	3	r+q	r+q	PROPN
ma-301	329	1	then	then	ADV
ma-301	329	2	the	the	DET
ma-301	329	3	function	function	NOUN
ma-301	329	4	ky	ky	NOUN
ma-301	329	5	:	:	PUNCT
ma-301	329	6	λ	λ	X
ma-301	329	7	−→	−→	NOUN
ma-301	329	8	σβ(λ)jα(λy	σβ(λ)jα(λy	VERB
ma-301	329	9	;	;	PUNCT
ma-301	329	10	q2	q2	PROPN
ma-301	329	11	)	)	PUNCT
ma-301	329	12	ηψ(λ	ηψ(λ	PUNCT
ma-301	329	13	)	)	PUNCT
ma-301	330	1	+	+	CCONJ
ma-301	330	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	330	3	)	)	PUNCT
ma-301	330	4	∣∣2	∣∣2	PROPN
ma-301	330	5	belongs	belong	VERB
ma-301	330	6	to	to	ADP
ma-301	330	7	l2α(r+q	l2α(r+q	PROPN
ma-301	330	8	)	)	PUNCT
ma-301	330	9	∩	∩	PROPN
ma-301	330	10	l1α(r+q	l1α(r+q	PROPN
ma-301	330	11	)	)	PUNCT
ma-301	330	12	and	and	CCONJ
ma-301	330	13	by	by	ADP
ma-301	330	14	using	use	VERB
ma-301	330	15	inversion	inversion	NOUN
ma-301	330	16	formula	formula	NOUN
ma-301	330	17	(	(	PUNCT
ma-301	330	18	2.5	2.5	NUM
ma-301	330	19	)	)	PUNCT
ma-301	330	20	we	we	PRON
ma-301	330	21	get	get	VERB
ma-301	330	22	θq	θq	ADP
ma-301	330	23	,	,	PUNCT
ma-301	330	24	η	η	NOUN
ma-301	330	25	,	,	PUNCT
ma-301	330	26	β(x	β(x	PROPN
ma-301	330	27	,	,	PUNCT
ma-301	330	28	y	y	NOUN
ma-301	330	29	)	)	PUNCT
ma-301	330	30	=	=	SYM
ma-301	331	1	h−1q	h−1q	NOUN
ma-301	331	2	,	,	PUNCT
ma-301	331	3	α(ky	α(ky	NOUN
ma-301	331	4	)	)	PUNCT
ma-301	331	5	(	(	PUNCT
ma-301	331	6	x	x	X
ma-301	331	7	)	)	PUNCT
ma-301	331	8	using	use	VERB
ma-301	331	9	plancherel	plancherel	NOUN
ma-301	331	10	’s	’s	PART
ma-301	331	11	theorem	theorem	NOUN
ma-301	331	12	and	and	CCONJ
ma-301	331	13	parseval	parseval	NOUN
ma-301	331	14	’s	’s	PART
ma-301	331	15	relation	relation	NOUN
ma-301	331	16	(	(	PUNCT
ma-301	331	17	2.6	2.6	NUM
ma-301	331	18	)	)	PUNCT
ma-301	331	19	we	we	PRON
ma-301	331	20	find	find	VERB
ma-301	331	21	that	that	SCONJ
ma-301	331	22	θq	θq	ADP
ma-301	331	23	,	,	PUNCT
ma-301	331	24	η	η	PROPN
ma-301	331	25	,	,	PUNCT
ma-301	331	26	β	β	X
ma-301	331	27	(	(	PUNCT
ma-301	331	28	·	·	PUNCT
ma-301	331	29	,	,	PUNCT
ma-301	331	30	y	y	NOUN
ma-301	331	31	)	)	PUNCT
ma-301	331	32	∈	∈	PROPN
ma-301	331	33	l2α(r+q	l2α(r+q	PROPN
ma-301	331	34	)	)	PUNCT
ma-301	331	35	and	and	CCONJ
ma-301	331	36	f	f	PROPN
ma-301	331	37	∗q	∗q	PROPN
ma-301	331	38	,	,	PUNCT
ma-301	331	39	η	η	PROPN
ma-301	331	40	,	,	PUNCT
ma-301	331	41	β	β	X
ma-301	331	42	,	,	PUNCT
ma-301	331	43	h(y	h(y	ADV
ma-301	331	44	)	)	PUNCT
ma-301	331	45	=	=	SYM
ma-301	332	1	∫	∫	PROPN
ma-301	332	2	∞	∞	PROPN
ma-301	332	3	0	0	NUM
ma-301	333	1	hq	hq	PROPN
ma-301	333	2	,	,	PUNCT
ma-301	333	3	α(λ)ky	α(λ)ky	X
ma-301	333	4	(	(	PUNCT
ma-301	333	5	λ)dµq	λ)dµq	PROPN
ma-301	333	6	,	,	PUNCT
ma-301	333	7	α(λ	α(λ	PROPN
ma-301	333	8	)	)	PUNCT
ma-301	333	9	=	=	SYM
ma-301	334	1	∫	∫	PROPN
ma-301	334	2	∞	∞	NUM
ma-301	334	3	0	0	NUM
ma-301	334	4	σβ(λ	σβ(λ	NUM
ma-301	334	5	)	)	PUNCT
ma-301	334	6	ηψ(λ	ηψ(λ	PUNCT
ma-301	334	7	)	)	PUNCT
ma-301	335	1	+	+	CCONJ
ma-301	335	2	|σβ(λ)|2hq	|σβ(λ)|2hq	NUM
ma-301	335	3	,	,	PUNCT
ma-301	335	4	α(h)(λ)dµq	α(h)(λ)dµq	NOUN
ma-301	335	5	,	,	PUNCT
ma-301	335	6	α(λ	α(λ	PROPN
ma-301	335	7	)	)	PUNCT
ma-301	335	8	on	on	ADP
ma-301	335	9	the	the	DET
ma-301	335	10	other	other	ADJ
ma-301	335	11	hand	hand	NOUN
ma-301	335	12	the	the	DET
ma-301	335	13	function	function	NOUN
ma-301	335	14	f	f	NOUN
ma-301	335	15	:	:	PUNCT
ma-301	336	1	λ	λ	VERB
ma-301	336	2	−→	−→	NOUN
ma-301	336	3	σβ(λ)hq	σβ(λ)hq	NOUN
ma-301	336	4	,	,	PUNCT
ma-301	336	5	α(h)(λ	α(h)(λ	NOUN
ma-301	336	6	)	)	PUNCT
ma-301	336	7	ηψ(λ	ηψ(λ	PUNCT
ma-301	336	8	)	)	PUNCT
ma-301	337	1	+	+	CCONJ
ma-301	337	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	337	3	)	)	PUNCT
ma-301	337	4	∣∣2	∣∣2	PROPN
ma-301	337	5	belongs	belong	VERB
ma-301	337	6	to	to	ADP
ma-301	337	7	l1α(r+q	l1α(r+q	PROPN
ma-301	337	8	)	)	PUNCT
ma-301	337	9	∩	∩	PROPN
ma-301	337	10	l2α(r+q	l2α(r+q	PROPN
ma-301	337	11	)	)	PUNCT
ma-301	337	12	,	,	PUNCT
ma-301	337	13	by	by	ADP
ma-301	337	14	using	use	VERB
ma-301	337	15	inversion	inversion	NOUN
ma-301	337	16	formula	formula	NOUN
ma-301	337	17	(	(	PUNCT
ma-301	337	18	2.5	2.5	NUM
ma-301	337	19	)	)	PUNCT
ma-301	337	20	,	,	PUNCT
ma-301	337	21	plancherel	plancherel	PROPN
ma-301	337	22	’s	’s	PART
ma-301	337	23	theorem	theorem	NOUN
ma-301	337	24	we	we	PRON
ma-301	337	25	find	find	VERB
ma-301	337	26	that	that	SCONJ
ma-301	337	27	f	f	PROPN
ma-301	337	28	∗q	∗q	PROPN
ma-301	337	29	,	,	PUNCT
ma-301	337	30	η	η	PROPN
ma-301	337	31	,	,	PUNCT
ma-301	337	32	β	β	X
ma-301	337	33	,	,	PUNCT
ma-301	337	34	h	h	PROPN
ma-301	337	35	belongs	belong	VERB
ma-301	337	36	to	to	ADP
ma-301	337	37	l2α(r+q	l2α(r+q	PROPN
ma-301	337	38	)	)	PUNCT
ma-301	337	39	and	and	CCONJ
ma-301	337	40	hq	hq	VERB
ma-301	337	41	,	,	PUNCT
ma-301	337	42	α(f	α(f	PROPN
ma-301	337	43	∗q	∗q	PROPN
ma-301	337	44	,	,	PUNCT
ma-301	337	45	η	η	PROPN
ma-301	337	46	,	,	PUNCT
ma-301	337	47	β	β	X
ma-301	337	48	,	,	PUNCT
ma-301	337	49	h)(λ	h)(λ	NUM
ma-301	337	50	)	)	PUNCT
ma-301	338	1	=	=	SYM
ma-301	338	2	f	f	X
ma-301	338	3	(	(	PUNCT
ma-301	339	1	λ)on	λ)on	PROPN
ma-301	339	2	the	the	DET
ma-301	339	3	other	other	ADJ
ma-301	339	4	hand	hand	NOUN
ma-301	339	5	we	we	PRON
ma-301	339	6	have	have	VERB
ma-301	339	7	|hq	|hq	NUM
ma-301	339	8	,	,	PUNCT
ma-301	339	9	α(f	α(f	PROPN
ma-301	339	10	∗q	∗q	PROPN
ma-301	339	11	,	,	PUNCT
ma-301	339	12	η	η	PROPN
ma-301	339	13	,	,	PUNCT
ma-301	339	14	β	β	X
ma-301	339	15	,	,	PUNCT
ma-301	339	16	h)(λ)|2	h)(λ)|2	NOUN
ma-301	339	17	=	=	SYM
ma-301	339	18	∣∣σβ(λ	∣∣σβ(λ	PROPN
ma-301	339	19	)	)	PUNCT
ma-301	339	20	∣∣2	∣∣2	PROPN
ma-301	339	21	(	(	PUNCT
ma-301	339	22	ηψ(λ	ηψ(λ	NOUN
ma-301	339	23	)	)	PUNCT
ma-301	339	24	+	+	CCONJ
ma-301	339	25	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	339	26	)	)	PUNCT
ma-301	339	27	∣∣2)2	∣∣2)2	NOUN
ma-301	339	28	|hq	|hq	NUM
ma-301	339	29	,	,	PUNCT
ma-301	339	30	α(h)(λ)|2	α(h)(λ)|2	PROPN
ma-301	339	31	≤	≤	NUM
ma-301	339	32	1	1	NUM
ma-301	339	33	2ηψ(λ	2ηψ(λ	NUM
ma-301	339	34	)	)	PUNCT
ma-301	339	35	|hq	|hq	NOUN
ma-301	339	36	,	,	PUNCT
ma-301	339	37	α(h)(λ)|2	α(h)(λ)|2	PROPN
ma-301	339	38	by	by	ADP
ma-301	339	39	plancherel	plancherel	PROPN
ma-301	339	40	’s	’s	PART
ma-301	339	41	formula	formula	NOUN
ma-301	339	42	(	(	PUNCT
ma-301	339	43	2.7	2.7	NUM
ma-301	339	44	)	)	PUNCT
ma-301	339	45	we	we	PRON
ma-301	339	46	find	find	VERB
ma-301	339	47	that	that	SCONJ
ma-301	339	48	‖f	‖f	SCONJ
ma-301	339	49	∗q	∗q	PROPN
ma-301	339	50	,	,	PUNCT
ma-301	339	51	η	η	PROPN
ma-301	339	52	,	,	PUNCT
ma-301	339	53	β	β	NOUN
ma-301	339	54	,	,	PUNCT
ma-301	339	55	h‖ψ	h‖ψ	PROPN
ma-301	339	56	≤	≤	PROPN
ma-301	339	57	1√	1√	PROPN
ma-301	339	58	2η	2η	PROPN
ma-301	339	59	‖h‖2,q	‖h‖2,q	PROPN
ma-301	339	60	,	,	PUNCT
ma-301	339	61	α	α	PROPN
ma-301	339	62	.	.	PUNCT
ma-301	339	63	�	�	PROPN
ma-301	339	64	theorem	theorem	VERB
ma-301	339	65	4.4	4.4	NUM
ma-301	339	66	.	.	PUNCT
ma-301	340	1	(	(	PUNCT
ma-301	340	2	third	third	ADJ
ma-301	340	3	calderón	calderón	NOUN
ma-301	340	4	’s	’s	PART
ma-301	340	5	formula	formula	NOUN
ma-301	340	6	)	)	PUNCT
ma-301	340	7	let	let	VERB
ma-301	340	8	σ	σ	NUM
ma-301	340	9	∈	∈	PROPN
ma-301	340	10	l∞α	l∞α	NOUN
ma-301	340	11	(	(	PUNCT
ma-301	340	12	r+q	r+q	PROPN
ma-301	340	13	)	)	PUNCT
ma-301	340	14	and	and	CCONJ
ma-301	340	15	f	f	PROPN
ma-301	340	16	∈	∈	PROPN
ma-301	341	1	sψ(r+q	sψ(r+q	PROPN
ma-301	341	2	)	)	PUNCT
ma-301	342	1	then	then	ADV
ma-301	342	2	the	the	DET
ma-301	342	3	extremal	extremal	ADJ
ma-301	342	4	function	function	NOUN
ma-301	342	5	given	give	VERB
ma-301	342	6	by	by	ADP
ma-301	342	7	f	f	PROPN
ma-301	342	8	∗q	∗q	PROPN
ma-301	342	9	,	,	PUNCT
ma-301	342	10	η	η	PROPN
ma-301	342	11	,	,	PUNCT
ma-301	342	12	β(y	β(y	NUM
ma-301	342	13	)	)	PUNCT
ma-301	343	1	=	=	SYM
ma-301	343	2	∫	∫	PROPN
ma-301	344	1	∞	∞	PROPN
ma-301	344	2	0	0	NUM
ma-301	344	3	mq	mq	PROPN
ma-301	344	4	,	,	PUNCT
ma-301	344	5	σ	σ	PROPN
ma-301	344	6	,	,	PUNCT
ma-301	344	7	β(f	β(f	PROPN
ma-301	344	8	)	)	PUNCT
ma-301	344	9	(	(	PUNCT
ma-301	344	10	x)θq	x)θq	PROPN
ma-301	344	11	,	,	PUNCT
ma-301	344	12	η	η	PROPN
ma-301	344	13	,	,	PUNCT
ma-301	344	14	β(x	β(x	NOUN
ma-301	344	15	,	,	PUNCT
ma-301	344	16	y)dµq	y)dµq	PROPN
ma-301	344	17	,	,	PUNCT
ma-301	344	18	α(x	α(x	NOUN
ma-301	344	19	)	)	PUNCT
ma-301	344	20	,	,	PUNCT
ma-301	344	21	satisfies	satisfy	VERB
ma-301	344	22	lim	lim	PROPN
ma-301	344	23	η→0	η→0	PROPN
ma-301	344	24	+	+	CCONJ
ma-301	344	25	∥∥f	∥∥f	PROPN
ma-301	344	26	∗q	∗q	PROPN
ma-301	344	27	,	,	PUNCT
ma-301	344	28	η	η	PROPN
ma-301	344	29	,	,	PUNCT
ma-301	344	30	β	β	NOUN
ma-301	344	31	−	−	PROPN
ma-301	344	32	f	f	PROPN
ma-301	344	33	∥∥2,q	∥∥2,q	PROPN
ma-301	344	34	,	,	PUNCT
ma-301	344	35	α	α	NOUN
ma-301	344	36	=	=	SYM
ma-301	344	37	0	0	NUM
ma-301	344	38	(	(	PUNCT
ma-301	344	39	4.7	4.7	NUM
ma-301	344	40	)	)	PUNCT
ma-301	344	41	moreover	moreover	ADV
ma-301	344	42	we	we	PRON
ma-301	344	43	have	have	VERB
ma-301	344	44	f	f	PROPN
ma-301	344	45	∗q	∗q	PROPN
ma-301	344	46	,	,	PUNCT
ma-301	344	47	η	η	PROPN
ma-301	344	48	,	,	PUNCT
ma-301	344	49	β	β	X
ma-301	344	50	−→	−→	NOUN
ma-301	344	51	f	f	X
ma-301	344	52	uniformly	uniformly	ADV
ma-301	344	53	when	when	SCONJ
ma-301	344	54	η	η	PROPN
ma-301	344	55	−→	−→	NOUN
ma-301	344	56	0	0	NUM
ma-301	344	57	+	+	NOUN
ma-301	344	58	.	.	PUNCT
ma-301	344	59	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	344	60	eur	eur	PROPN
ma-301	344	61	.	.	PUNCT
ma-301	345	1	j.	j.	PROPN
ma-301	345	2	math	math	PROPN
ma-301	345	3	.	.	PUNCT
ma-301	346	1	anal	anal	PROPN
ma-301	346	2	.	.	PUNCT
ma-301	347	1	10.28924	10.28924	NUM
ma-301	347	2	/	/	SYM
ma-301	347	3	ada	ada	PROPN
ma-301	347	4	/	/	PROPN
ma-301	347	5	ma.5.8	ma.5.8	PROPN
ma-301	347	6	14	14	NUM
ma-301	347	7	proof	proof	NOUN
ma-301	347	8	.	.	PUNCT
ma-301	348	1	f	f	PROPN
ma-301	348	2	∈	∈	PROPN
ma-301	349	1	sψ(r+q	sψ(r+q	PROPN
ma-301	349	2	)	)	PUNCT
ma-301	349	3	,	,	PUNCT
ma-301	349	4	we	we	PRON
ma-301	349	5	put	put	VERB
ma-301	349	6	h	h	NOUN
ma-301	349	7	=	=	SYM
ma-301	349	8	mq	mq	PROPN
ma-301	349	9	,	,	PUNCT
ma-301	349	10	σ	σ	PROPN
ma-301	349	11	,	,	PUNCT
ma-301	349	12	β(f	β(f	PROPN
ma-301	349	13	)	)	PUNCT
ma-301	349	14	and	and	CCONJ
ma-301	349	15	f	f	PROPN
ma-301	349	16	∗q	∗q	PROPN
ma-301	349	17	,	,	PUNCT
ma-301	349	18	η	η	PROPN
ma-301	349	19	,	,	PUNCT
ma-301	349	20	β	β	X
ma-301	349	21	,	,	PUNCT
ma-301	349	22	h	h	NOUN
ma-301	349	23	=	=	SYM
ma-301	349	24	f	f	PROPN
ma-301	349	25	∗q	∗q	PROPN
ma-301	349	26	,	,	PUNCT
ma-301	349	27	η	η	PROPN
ma-301	349	28	,	,	PUNCT
ma-301	349	29	β	β	NOUN
ma-301	349	30	in	in	ADP
ma-301	349	31	the	the	DET
ma-301	349	32	relation	relation	NOUN
ma-301	349	33	(	(	PUNCT
ma-301	349	34	4.6	4.6	NUM
ma-301	349	35	)	)	PUNCT
ma-301	349	36	we	we	PRON
ma-301	349	37	find	find	VERB
ma-301	349	38	that	that	PRON
ma-301	349	39	hq	hq	NOUN
ma-301	349	40	,	,	PUNCT
ma-301	349	41	α(f	α(f	PROPN
ma-301	349	42	∗q	∗q	PROPN
ma-301	349	43	,	,	PUNCT
ma-301	349	44	η	η	PROPN
ma-301	349	45	,	,	PUNCT
ma-301	349	46	β	β	NOUN
ma-301	349	47	−	−	PROPN
ma-301	349	48	f	f	X
ma-301	349	49	)	)	PUNCT
ma-301	349	50	(	(	PUNCT
ma-301	349	51	λ	λ	NOUN
ma-301	349	52	)	)	PUNCT
ma-301	349	53	=	=	SYM
ma-301	349	54	−ηψ(λ)hq	−ηψ(λ)hq	NOUN
ma-301	349	55	,	,	PUNCT
ma-301	349	56	α(f	α(f	PROPN
ma-301	349	57	)	)	PUNCT
ma-301	349	58	(	(	PUNCT
ma-301	349	59	λ	λ	NOUN
ma-301	349	60	)	)	PUNCT
ma-301	349	61	ηψ(λ	ηψ(λ	PUNCT
ma-301	349	62	)	)	PUNCT
ma-301	349	63	+	+	CCONJ
ma-301	349	64	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	349	65	)	)	PUNCT
ma-301	349	66	∣∣2	∣∣2	PROPN
ma-301	349	67	(	(	PUNCT
ma-301	349	68	4.8	4.8	NUM
ma-301	349	69	)	)	PUNCT
ma-301	349	70	therefore	therefore	ADV
ma-301	349	71	∥∥f	∥∥f	PROPN
ma-301	349	72	∗q	∗q	PROPN
ma-301	349	73	,	,	PUNCT
ma-301	349	74	η	η	PROPN
ma-301	349	75	,	,	PUNCT
ma-301	349	76	β	β	NOUN
ma-301	349	77	−	−	NOUN
ma-301	349	78	f	f	X
ma-301	349	79	∥∥2ψ	∥∥2ψ	PUNCT
ma-301	350	1	=	=	SYM
ma-301	350	2	∫	∫	PROPN
ma-301	350	3	∞	∞	PROPN
ma-301	350	4	0	0	X
ma-301	350	5	η2	η2	PROPN
ma-301	350	6	(	(	PUNCT
ma-301	350	7	ψ(λ))3	ψ(λ))3	PROPN
ma-301	350	8	ηψ(λ	ηψ(λ	PUNCT
ma-301	350	9	)	)	PUNCT
ma-301	351	1	+	+	CCONJ
ma-301	351	2	|σβ(λ)|2	|σβ(λ)|2	PROPN
ma-301	351	3	|hq	|hq	NUM
ma-301	351	4	,	,	PUNCT
ma-301	351	5	α(f	α(f	PROPN
ma-301	351	6	)	)	PUNCT
ma-301	351	7	(	(	PUNCT
ma-301	351	8	λ)|2	λ)|2	PROPN
ma-301	351	9	dµq	dµq	PROPN
ma-301	351	10	,	,	PUNCT
ma-301	351	11	α(λ)on	α(λ)on	PROPN
ma-301	351	12	the	the	DET
ma-301	351	13	other	other	ADJ
ma-301	351	14	hand	hand	NOUN
ma-301	351	15	we	we	PRON
ma-301	351	16	have	have	VERB
ma-301	351	17	η2	η2	VERB
ma-301	351	18	(	(	PUNCT
ma-301	351	19	ψ(λ))3	ψ(λ))3	PROPN
ma-301	351	20	ηψ(λ	ηψ(λ	PUNCT
ma-301	351	21	)	)	PUNCT
ma-301	352	1	+	+	CCONJ
ma-301	352	2	|σβ(λ)|2	|σβ(λ)|2	PROPN
ma-301	352	3	|hq	|hq	NUM
ma-301	352	4	,	,	PUNCT
ma-301	352	5	α(f	α(f	PROPN
ma-301	352	6	)	)	PUNCT
ma-301	352	7	(	(	PUNCT
ma-301	352	8	λ)|2	λ)|2	PROPN
ma-301	352	9	≤	≤	NUM
ma-301	352	10	ψ(λ	ψ(λ	NOUN
ma-301	352	11	)	)	PUNCT
ma-301	352	12	|hq	|hq	NOUN
ma-301	352	13	,	,	PUNCT
ma-301	352	14	α(f	α(f	PROPN
ma-301	352	15	)	)	PUNCT
ma-301	352	16	(	(	PUNCT
ma-301	352	17	λ)|2	λ)|2	PROPN
ma-301	352	18	(	(	PUNCT
ma-301	352	19	4.9	4.9	NUM
ma-301	352	20	)	)	PUNCT
ma-301	352	21	the	the	DET
ma-301	352	22	result	result	NOUN
ma-301	352	23	(	(	PUNCT
ma-301	352	24	4.7	4.7	NUM
ma-301	352	25	)	)	PUNCT
ma-301	352	26	follows	follow	VERB
ma-301	352	27	from	from	ADP
ma-301	352	28	(	(	PUNCT
ma-301	352	29	4.9	4.9	NUM
ma-301	352	30	)	)	PUNCT
ma-301	352	31	and	and	CCONJ
ma-301	352	32	the	the	DET
ma-301	352	33	dominated	dominate	VERB
ma-301	352	34	convergence	convergence	NOUN
ma-301	352	35	theorem	theorem	VERB
ma-301	352	36	.	.	PUNCT
ma-301	353	1	now	now	ADV
ma-301	353	2	,	,	PUNCT
ma-301	353	3	for	for	ADP
ma-301	353	4	all	all	DET
ma-301	353	5	f	f	PROPN
ma-301	353	6	∈	∈	PROPN
ma-301	354	1	sψ(r+q	sψ(r+q	PROPN
ma-301	354	2	)	)	PUNCT
ma-301	354	3	we	we	PRON
ma-301	354	4	have	have	VERB
ma-301	354	5	hq	hq	NOUN
ma-301	354	6	,	,	PUNCT
ma-301	354	7	α(f	α(f	NUM
ma-301	354	8	)	)	PUNCT
ma-301	355	1	∈	∈	PROPN
ma-301	355	2	l2α(r+q	l2α(r+q	PROPN
ma-301	355	3	)	)	PUNCT
ma-301	355	4	∩	∩	PROPN
ma-301	355	5	l1α(r+q	l1α(r+q	PROPN
ma-301	355	6	)	)	PUNCT
ma-301	355	7	and	and	CCONJ
ma-301	355	8	by	by	ADP
ma-301	355	9	using	use	VERB
ma-301	355	10	the	the	DET
ma-301	355	11	relations	relation	NOUN
ma-301	355	12	(	(	PUNCT
ma-301	355	13	2.5	2.5	NUM
ma-301	355	14	)	)	PUNCT
ma-301	355	15	,	,	PUNCT
ma-301	355	16	(	(	PUNCT
ma-301	355	17	4.8	4.8	NUM
ma-301	355	18	)	)	PUNCT
ma-301	355	19	we	we	PRON
ma-301	355	20	find	find	VERB
ma-301	355	21	that	that	SCONJ
ma-301	355	22	f	f	PROPN
ma-301	355	23	∗q	∗q	PROPN
ma-301	355	24	,	,	PUNCT
ma-301	355	25	η	η	PROPN
ma-301	355	26	,	,	PUNCT
ma-301	355	27	β(y)−	β(y)−	ADJ
ma-301	355	28	f	f	X
ma-301	355	29	(	(	PUNCT
ma-301	355	30	y	y	NOUN
ma-301	355	31	)	)	PUNCT
ma-301	355	32	=	=	SYM
ma-301	356	1	∫	∫	PROPN
ma-301	356	2	∞	∞	NOUN
ma-301	356	3	0	0	NUM
ma-301	357	1	−ηψ(λ)hq	−ηψ(λ)hq	NOUN
ma-301	357	2	,	,	PUNCT
ma-301	357	3	α(f	α(f	PROPN
ma-301	357	4	)	)	PUNCT
ma-301	357	5	(	(	PUNCT
ma-301	357	6	λ	λ	NOUN
ma-301	357	7	)	)	PUNCT
ma-301	357	8	ηψ(λ	ηψ(λ	PUNCT
ma-301	357	9	)	)	PUNCT
ma-301	358	1	+	+	CCONJ
ma-301	358	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	358	3	)	)	PUNCT
ma-301	358	4	∣∣2	∣∣2	PROPN
ma-301	358	5	jα(λy	jα(λy	PROPN
ma-301	358	6	;	;	PUNCT
ma-301	358	7	q2)dµq	q2)dµq	NOUN
ma-301	358	8	,	,	PUNCT
ma-301	358	9	α(λ	α(λ	PROPN
ma-301	358	10	)	)	PUNCT
ma-301	358	11	and	and	CCONJ
ma-301	358	12	∣∣∣∣∣−ηψ(λ)hq	∣∣∣∣∣−ηψ(λ)hq	NOUN
ma-301	358	13	,	,	PUNCT
ma-301	358	14	α(f	α(f	PROPN
ma-301	358	15	)	)	PUNCT
ma-301	358	16	(	(	PUNCT
ma-301	358	17	λ	λ	NOUN
ma-301	358	18	)	)	PUNCT
ma-301	358	19	ηψ(λ	ηψ(λ	PUNCT
ma-301	358	20	)	)	PUNCT
ma-301	359	1	+	+	CCONJ
ma-301	359	2	∣∣σβ(λ	∣∣σβ(λ	NOUN
ma-301	359	3	)	)	PUNCT
ma-301	359	4	∣∣2	∣∣2	PROPN
ma-301	359	5	jα(λy	jα(λy	PROPN
ma-301	359	6	;	;	PUNCT
ma-301	359	7	q2	q2	NOUN
ma-301	359	8	)	)	PUNCT
ma-301	359	9	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-301	359	10	≤	≤	ADV
ma-301	359	11	|hq	|hq	NOUN
ma-301	359	12	,	,	PUNCT
ma-301	359	13	α(f	α(f	PROPN
ma-301	359	14	)	)	PUNCT
ma-301	359	15	(	(	PUNCT
ma-301	359	16	λ)|	λ)|	INTJ
ma-301	359	17	(	(	PUNCT
ma-301	359	18	4.10	4.10	NUM
ma-301	359	19	)	)	PUNCT
ma-301	359	20	by	by	ADP
ma-301	359	21	using	use	VERB
ma-301	359	22	the	the	DET
ma-301	359	23	relation	relation	NOUN
ma-301	359	24	(	(	PUNCT
ma-301	359	25	4.10	4.10	NUM
ma-301	359	26	)	)	PUNCT
ma-301	359	27	and	and	CCONJ
ma-301	359	28	the	the	DET
ma-301	359	29	dominated	dominate	VERB
ma-301	359	30	convergence	convergence	NOUN
ma-301	359	31	theorem	theorem	VERB
ma-301	359	32	we	we	PRON
ma-301	359	33	deduce	deduce	VERB
ma-301	359	34	that	that	SCONJ
ma-301	360	1	lim	lim	PROPN
ma-301	360	2	η→0	η→0	PROPN
ma-301	360	3	+	+	PROPN
ma-301	360	4	∣∣f	∣∣f	NOUN
ma-301	360	5	∗q	∗q	PROPN
ma-301	360	6	,	,	PUNCT
ma-301	360	7	η	η	PROPN
ma-301	360	8	,	,	PUNCT
ma-301	360	9	β(y)−	β(y)−	ADJ
ma-301	360	10	f	f	X
ma-301	360	11	(	(	PUNCT
ma-301	360	12	y	y	NOUN
ma-301	360	13	)	)	PUNCT
ma-301	360	14	∣∣	∣∣	X
ma-301	361	1	=	=	SYM
ma-301	361	2	0	0	NUM
ma-301	361	3	which	which	PRON
ma-301	361	4	complete	complete	VERB
ma-301	361	5	the	the	DET
ma-301	361	6	proof	proof	NOUN
ma-301	361	7	of	of	ADP
ma-301	361	8	the	the	DET
ma-301	361	9	theorem	theorem	PROPN
ma-301	361	10	.	.	PUNCT
ma-301	361	11	�	�	PROPN
ma-301	361	12	acknowledgments	acknowledgment	NOUN
ma-301	361	13	:	:	PUNCT
ma-301	361	14	the	the	DET
ma-301	361	15	authors	author	NOUN
ma-301	361	16	are	be	AUX
ma-301	361	17	deeply	deeply	ADV
ma-301	361	18	indebted	indebted	ADJ
ma-301	361	19	to	to	ADP
ma-301	361	20	the	the	DET
ma-301	361	21	referees	referee	NOUN
ma-301	361	22	for	for	ADP
ma-301	361	23	providing	provide	VERB
ma-301	361	24	constructivecomments	constructivecomment	NOUN
ma-301	361	25	and	and	CCONJ
ma-301	361	26	helps	help	VERB
ma-301	361	27	in	in	ADP
ma-301	361	28	improving	improve	VERB
ma-301	361	29	the	the	DET
ma-301	361	30	contents	content	NOUN
ma-301	361	31	of	of	ADP
ma-301	361	32	this	this	DET
ma-301	361	33	article	article	NOUN
ma-301	361	34	.	.	PUNCT
ma-301	362	1	authors	author	NOUN
ma-301	362	2	’	'	PUNCT
ma-301	362	3	contributions	contribution	NOUN
ma-301	362	4	:	:	PUNCT
ma-301	362	5	both	both	DET
ma-301	362	6	authors	author	NOUN
ma-301	362	7	contributed	contribute	VERB
ma-301	362	8	equally	equally	ADV
ma-301	362	9	to	to	ADP
ma-301	362	10	this	this	DET
ma-301	362	11	work	work	NOUN
ma-301	362	12	.	.	PUNCT
ma-301	363	1	references	reference	NOUN
ma-301	363	2	[	[	X
ma-301	363	3	1	1	NUM
ma-301	363	4	]	]	X
ma-301	363	5	l.d	l.d	PROPN
ma-301	363	6	.	.	PROPN
ma-301	363	7	abreu	abreu	PROPN
ma-301	363	8	,	,	PUNCT
ma-301	363	9	f.	f.	PROPN
ma-301	363	10	bouzeffour	bouzeffour	PROPN
ma-301	363	11	,	,	PUNCT
ma-301	363	12	a	a	DET
ma-301	363	13	paley	paley	ADJ
ma-301	363	14	-	-	PUNCT
ma-301	363	15	wiener	wiener	NOUN
ma-301	363	16	theorem	theorem	NOUN
ma-301	363	17	for	for	ADP
ma-301	363	18	the	the	DET
ma-301	363	19	askey	askey	NOUN
ma-301	363	20	-	-	PUNCT
ma-301	363	21	wilson	wilson	PROPN
ma-301	363	22	function	function	NOUN
ma-301	363	23	transform	transform	NOUN
ma-301	363	24	,	,	PUNCT
ma-301	363	25	proc	proc	NOUN
ma-301	363	26	.	.	PUNCT
ma-301	364	1	amer	amer	PROPN
ma-301	364	2	.	.	PUNCT
ma-301	365	1	math.soc	math.soc	X
ma-301	365	2	.	.	PROPN
ma-301	365	3	138	138	NUM
ma-301	365	4	(	(	PUNCT
ma-301	365	5	2010	2010	NUM
ma-301	365	6	)	)	PUNCT
ma-301	365	7	,	,	PUNCT
ma-301	366	1	2853–2853	2853–2853	NUM
ma-301	366	2	.	.	PUNCT
ma-301	366	3	https://doi.org/10.1090/s0002-9939-10-10327-x.[2	https://doi.org/10.1090/s0002-9939-10-10327-x.[2	NOUN
ma-301	366	4	]	]	X
ma-301	366	5	l.d	l.d	PROPN
ma-301	366	6	.	.	PROPN
ma-301	366	7	abreu	abreu	PROPN
ma-301	366	8	,	,	PUNCT
ma-301	366	9	sampling	sample	VERB
ma-301	366	10	theory	theory	NOUN
ma-301	366	11	associated	associate	VERB
ma-301	366	12	with	with	ADP
ma-301	366	13	q	q	PROPN
ma-301	366	14	-difference	-difference	NOUN
ma-301	366	15	equations	equation	NOUN
ma-301	366	16	of	of	ADP
ma-301	366	17	the	the	DET
ma-301	366	18	sturm	sturm	PROPN
ma-301	366	19	–	–	PUNCT
ma-301	366	20	liouville	liouville	NOUN
ma-301	366	21	type	type	NOUN
ma-301	366	22	,	,	PUNCT
ma-301	366	23	j.	j.	PROPN
ma-301	366	24	phys	phys	PROPN
ma-301	366	25	.	.	PUNCT
ma-301	367	1	a	a	DET
ma-301	367	2	:	:	PUNCT
ma-301	367	3	math.gen	math.gen	X
ma-301	367	4	.	.	NOUN
ma-301	367	5	38	38	NUM
ma-301	367	6	(	(	PUNCT
ma-301	367	7	2005	2005	NUM
ma-301	367	8	)	)	PUNCT
ma-301	367	9	,	,	PUNCT
ma-301	367	10	10311–10319	10311–10319	NUM
ma-301	367	11	.	.	PUNCT
ma-301	368	1	https://doi.org/10.1088/0305-4470/38/48/005.[3	https://doi.org/10.1088/0305-4470/38/48/005.[3	PROPN
ma-301	368	2	]	]	X
ma-301	368	3	n.	n.	PROPN
ma-301	368	4	akram	akram	PROPN
ma-301	368	5	,	,	PUNCT
ma-301	368	6	on	on	ADP
ma-301	368	7	a	a	DET
ma-301	368	8	q	q	ADJ
ma-301	368	9	-	-	PUNCT
ma-301	368	10	best	good	ADJ
ma-301	368	11	approximation	approximation	NOUN
ma-301	368	12	formulas	formula	NOUN
ma-301	368	13	for	for	ADP
ma-301	368	14	the	the	DET
ma-301	368	15	l2	l2	NOUN
ma-301	368	16	-	-	PUNCT
ma-301	368	17	multiplier	multipli	ADJ
ma-301	368	18	operators	operator	NOUN
ma-301	368	19	and	and	CCONJ
ma-301	368	20	applications	application	NOUN
ma-301	368	21	,	,	PUNCT
ma-301	368	22	res	re	NOUN
ma-301	368	23	.	.	PROPN
ma-301	368	24	math	math	NOUN
ma-301	368	25	.	.	PUNCT
ma-301	369	1	9(2022	9(2022	NUM
ma-301	369	2	)	)	PUNCT
ma-301	369	3	,	,	PUNCT
ma-301	369	4	2066812	2066812	NUM
ma-301	369	5	.	.	PUNCT
ma-301	370	1	https://doi.org/10.1080/27684830.2022.2066812.[4	https://doi.org/10.1080/27684830.2022.2066812.[4	NUM
ma-301	370	2	]	]	X
ma-301	370	3	n.	n.	NOUN
ma-301	370	4	aronszajn	aronszajn	PROPN
ma-301	370	5	,	,	PUNCT
ma-301	370	6	theory	theory	NOUN
ma-301	370	7	of	of	ADP
ma-301	370	8	reproducing	reproduce	VERB
ma-301	370	9	kernels	kernel	NOUN
ma-301	370	10	,	,	PUNCT
ma-301	370	11	trans	trans	PROPN
ma-301	370	12	.	.	PROPN
ma-301	371	1	amer	amer	PROPN
ma-301	371	2	.	.	PUNCT
ma-301	371	3	math	math	PROPN
ma-301	371	4	.	.	PUNCT
ma-301	372	1	soc	soc	PROPN
ma-301	372	2	.	.	PUNCT
ma-301	373	1	68	68	NUM
ma-301	373	2	(	(	PUNCT
ma-301	373	3	1950	1950	NUM
ma-301	373	4	)	)	PUNCT
ma-301	373	5	,	,	PUNCT
ma-301	373	6	337–404	337–404	NUM
ma-301	373	7	.	.	PUNCT
ma-301	374	1	https://doi.org/10	https://doi.org/10	PROPN
ma-301	374	2	.	.	PUNCT
ma-301	375	1	1090	1090	NUM
ma-301	375	2	/	/	SYM
ma-301	375	3	s0002	s0002	NOUN
ma-301	375	4	-	-	PUNCT
ma-301	375	5	9947	9947	NUM
ma-301	375	6	-	-	PUNCT
ma-301	375	7	1950	1950	NUM
ma-301	375	8	-	-	PUNCT
ma-301	375	9	0051437	0051437	NUM
ma-301	375	10	-	-	PUNCT
ma-301	375	11	7.[5	7.[5	NUM
ma-301	375	12	]	]	PUNCT
ma-301	375	13	r.	r.	NOUN
ma-301	375	14	bañuelos	bañuelos	PROPN
ma-301	375	15	,	,	PUNCT
ma-301	375	16	k.	k.	PROPN
ma-301	375	17	bogdan	bogdan	PROPN
ma-301	375	18	,	,	PUNCT
ma-301	375	19	lévy	lévy	ADJ
ma-301	375	20	processes	process	NOUN
ma-301	375	21	and	and	CCONJ
ma-301	375	22	fourier	fourier	NOUN
ma-301	375	23	multipliers	multiplier	NOUN
ma-301	375	24	,	,	PUNCT
ma-301	375	25	j.	j.	PROPN
ma-301	375	26	funct	funct	PROPN
ma-301	375	27	.	.	PUNCT
ma-301	376	1	anal	anal	PROPN
ma-301	376	2	.	.	PUNCT
ma-301	377	1	250	250	NUM
ma-301	377	2	(	(	PUNCT
ma-301	377	3	2007	2007	NUM
ma-301	377	4	)	)	PUNCT
ma-301	377	5	,	,	PUNCT
ma-301	378	1	197–213	197–213	NUM
ma-301	378	2	.	.	PUNCT
ma-301	378	3	https	https	NOUN
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ma-301	378	17	,	,	PUNCT
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ma-301	378	19	q	q	ADJ
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ma-301	378	24	,	,	PUNCT
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ma-301	379	1	6	6	NUM
ma-301	379	2	(	(	PUNCT
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ma-301	379	4	)	)	PUNCT
ma-301	379	5	,	,	PUNCT
ma-301	379	6	311	311	NUM
ma-301	379	7	-	-	SYM
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ma-301	379	9	]	]	PUNCT
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ma-301	379	17	-	-	PUNCT
ma-301	379	18	weierstrass	weierstrass	NOUN
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ma-301	379	22	the	the	DET
ma-301	379	23	q	q	ADJ
ma-301	379	24	-	-	PUNCT
ma-301	379	25	fourier	fourier	ADJ
ma-301	379	26	bessel	bessel	NOUN
ma-301	379	27	operator	operator	NOUN
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ma-301	379	29	le	le	X
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ma-301	379	33	(	(	PUNCT
ma-301	379	34	2016),81	2016),81	NUM
ma-301	379	35	-	-	SYM
ma-301	379	36	97	97	NUM
ma-301	379	37	.	.	PUNCT
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ma-301	381	2	https://doi.org/10.1090/s0002-9947-1950-0051437-7	https://doi.org/10.1090/s0002-9947-1950-0051437-7	ADV
ma-301	381	3	https://doi.org/10.1016/j.jfa.2007.05.013	https://doi.org/10.1016/j.jfa.2007.05.013	ADJ
ma-301	381	4	https://doi.org/10.1016/j.jfa.2007.05.013	https://doi.org/10.1016/j.jfa.2007.05.013	NOUN
ma-301	381	5	eur	eur	NOUN
ma-301	381	6	.	.	PUNCT
ma-301	382	1	j.	j.	PROPN
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ma-301	382	3	.	.	PUNCT
ma-301	383	1	anal	anal	PROPN
ma-301	383	2	.	.	PUNCT
ma-301	384	1	10.28924	10.28924	NUM
ma-301	384	2	/	/	SYM
ma-301	384	3	ada	ada	PROPN
ma-301	384	4	/	/	PROPN
ma-301	384	5	ma.5.8	ma.5.8	PROPN
ma-301	384	6	15	15	NUM
ma-301	384	7	[	[	SYM
ma-301	384	8	8	8	NUM
ma-301	384	9	]	]	PUNCT
ma-301	384	10	a.	a.	NOUN
ma-301	384	11	chana	chana	NOUN
ma-301	384	12	,	,	PUNCT
ma-301	384	13	a.	a.	PROPN
ma-301	384	14	akhlidj	akhlidj	PROPN
ma-301	384	15	,	,	PUNCT
ma-301	384	16	calderon	calderon	PROPN
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ma-301	384	18	formulas	formula	NOUN
ma-301	384	19	and	and	CCONJ
ma-301	384	20	uncertainty	uncertainty	NOUN
ma-301	384	21	principles	principle	NOUN
ma-301	384	22	for	for	ADP
ma-301	384	23	the	the	DET
ma-301	384	24	laguerre	laguerre	NOUN
ma-301	384	25	–	–	PUNCT
ma-301	384	26	bessel	bessel	ADJ
ma-301	384	27	lα2	lα2	ADJ
ma-301	384	28	-	-	PUNCT
ma-301	384	29	multiplier	multipli	ADJ
ma-301	384	30	op	op	NOUN
ma-301	384	31	-	-	PUNCT
ma-301	384	32	erators	erator	NOUN
ma-301	384	33	,	,	PUNCT
ma-301	384	34	integral	integral	ADJ
ma-301	384	35	transf	transf	NOUN
ma-301	384	36	.	.	PUNCT
ma-301	385	1	spec	spec	PROPN
ma-301	385	2	.	.	PUNCT
ma-301	386	1	funct	funct	PROPN
ma-301	386	2	.	.	PUNCT
ma-301	387	1	35	35	NUM
ma-301	387	2	(	(	PUNCT
ma-301	387	3	2024	2024	NUM
ma-301	387	4	)	)	PUNCT
ma-301	387	5	,	,	PUNCT
ma-301	387	6	747–764	747–764	NUM
ma-301	387	7	.	.	PUNCT
ma-301	388	1	https://doi.org/10.1080/10652469.2024.2382787.[9	https://doi.org/10.1080/10652469.2024.2382787.[9	NUM
ma-301	388	2	]	]	X
ma-301	388	3	s.	s.	PROPN
ma-301	388	4	chefai	chefai	PROPN
ma-301	388	5	,	,	PUNCT
ma-301	388	6	windowed	windowed	ADJ
ma-301	388	7	bessel	bessel	ADJ
ma-301	388	8	fourier	fourier	NOUN
ma-301	388	9	transform	transform	NOUN
ma-301	388	10	in	in	ADP
ma-301	388	11	quantum	quantum	ADJ
ma-301	388	12	calculus	calculus	NOUN
ma-301	388	13	and	and	CCONJ
ma-301	388	14	applications	application	NOUN
ma-301	388	15	,	,	PUNCT
ma-301	388	16	j.	j.	PROPN
ma-301	388	17	pseudo	pseudo	NOUN
ma-301	388	18	-	-	PUNCT
ma-301	388	19	differ	differ	VERB
ma-301	388	20	.	.	PUNCT
ma-301	389	1	oper	oper	NOUN
ma-301	389	2	.	.	PUNCT
ma-301	390	1	appl.8	appl.8	PROPN
ma-301	390	2	(	(	PUNCT
ma-301	390	3	2017	2017	NUM
ma-301	390	4	)	)	PUNCT
ma-301	390	5	,	,	PUNCT
ma-301	391	1	723–749	723–749	NUM
ma-301	391	2	.	.	PUNCT
ma-301	392	1	https://doi.org/10.1007/s11868-017-0215-y.[10	https://doi.org/10.1007/s11868-017-0215-y.[10	PRON
ma-301	392	2	]	]	PUNCT
ma-301	392	3	s.	s.	PROPN
ma-301	392	4	chefai	chefai	PROPN
ma-301	392	5	,	,	PUNCT
ma-301	392	6	l.	l.	PROPN
ma-301	392	7	dhaouadi	dhaouadi	PROPN
ma-301	392	8	,	,	PUNCT
ma-301	392	9	a.	a.	NOUN
ma-301	392	10	fitouhi	fitouhi	PROPN
ma-301	392	11	,	,	PUNCT
ma-301	392	12	inverse	inverse	NOUN
ma-301	392	13	problems	problem	NOUN
ma-301	392	14	and	and	CCONJ
ma-301	392	15	approximations	approximation	NOUN
ma-301	392	16	in	in	ADP
ma-301	392	17	quantum	quantum	NOUN
ma-301	392	18	calculus	calculus	NOUN
ma-301	392	19	,	,	PUNCT
ma-301	392	20	afr	afr	PROPN
ma-301	392	21	.	.	PUNCT
ma-301	393	1	diaspora	diaspora	PROPN
ma-301	393	2	j.math	j.math	PROPN
ma-301	393	3	.	.	PROPN
ma-301	394	1	17	17	NUM
ma-301	394	2	(	(	PUNCT
ma-301	394	3	2014	2014	NUM
ma-301	394	4	)	)	PUNCT
ma-301	394	5	,	,	PUNCT
ma-301	394	6	75–84.[11	75–84.[11	NUM
ma-301	394	7	]	]	X
ma-301	395	1	l.	l.	PROPN
ma-301	395	2	dhaoudi	dhaoudi	PROPN
ma-301	395	3	,	,	PUNCT
ma-301	395	4	on	on	ADP
ma-301	395	5	the	the	DET
ma-301	395	6	q	q	ADJ
ma-301	395	7	-	-	PUNCT
ma-301	395	8	bessel	bessel	ADJ
ma-301	395	9	fourier	fourier	NOUN
ma-301	395	10	transform	transform	NOUN
ma-301	395	11	,	,	PUNCT
ma-301	395	12	bull	bull	NOUN
ma-301	395	13	.	.	PUNCT
ma-301	396	1	math	math	NOUN
ma-301	396	2	.	.	PUNCT
ma-301	397	1	anal	anal	PROPN
ma-301	397	2	.	.	PUNCT
ma-301	397	3	appl	appl	PROPN
ma-301	397	4	.	.	PROPN
ma-301	397	5	5	5	NUM
ma-301	397	6	(	(	PUNCT
ma-301	397	7	2013	2013	NUM
ma-301	397	8	)	)	PUNCT
ma-301	397	9	,	,	PUNCT
ma-301	397	10	42–60.[12	42–60.[12	NUM
ma-301	397	11	]	]	X
ma-301	397	12	d.l	d.l	PROPN
ma-301	397	13	.	.	PROPN
ma-301	397	14	donoho	donoho	PROPN
ma-301	397	15	,	,	PUNCT
ma-301	397	16	p.b	p.b	PROPN
ma-301	397	17	.	.	PROPN
ma-301	397	18	stark	stark	PROPN
ma-301	397	19	,	,	PUNCT
ma-301	397	20	uncertainty	uncertainty	NOUN
ma-301	397	21	principles	principle	NOUN
ma-301	397	22	and	and	CCONJ
ma-301	397	23	signal	signal	NOUN
ma-301	397	24	recovery	recovery	NOUN
ma-301	397	25	,	,	PUNCT
ma-301	398	1	siam	siam	PROPN
ma-301	398	2	j.	j.	PROPN
ma-301	398	3	appl	appl	PROPN
ma-301	398	4	.	.	PROPN
ma-301	398	5	math	math	PROPN
ma-301	398	6	.	.	PUNCT
ma-301	399	1	49	49	NUM
ma-301	399	2	(	(	PUNCT
ma-301	399	3	1989	1989	NUM
ma-301	399	4	)	)	PUNCT
ma-301	399	5	,	,	PUNCT
ma-301	399	6	906–931	906–931	NUM
ma-301	399	7	.	.	PUNCT
ma-301	400	1	https://doi.org/10.1137/0149053.[13	https://doi.org/10.1137/0149053.[13	PROPN
ma-301	400	2	]	]	PUNCT
ma-301	400	3	a.	a.	NOUN
ma-301	400	4	fitouhi	fitouhi	PROPN
ma-301	400	5	,	,	PUNCT
ma-301	400	6	l.	l.	PROPN
ma-301	400	7	dhaouadi	dhaouadi	PROPN
ma-301	400	8	,	,	PUNCT
ma-301	400	9	positivity	positivity	NOUN
ma-301	400	10	of	of	ADP
ma-301	400	11	the	the	DET
ma-301	400	12	generalized	generalized	ADJ
ma-301	400	13	translation	translation	NOUN
ma-301	400	14	associated	associate	VERB
ma-301	400	15	with	with	ADP
ma-301	400	16	the	the	DET
ma-301	400	17	q	q	NOUN
ma-301	400	18	-	-	PUNCT
ma-301	400	19	hankel	hankel	NOUN
ma-301	400	20	transform	transform	NOUN
ma-301	400	21	,	,	PUNCT
ma-301	400	22	construct.approx	construct.approx	NOUN
ma-301	400	23	.	.	PUNCT
ma-301	400	24	34	34	NUM
ma-301	400	25	(	(	PUNCT
ma-301	400	26	2011	2011	NUM
ma-301	400	27	)	)	PUNCT
ma-301	400	28	,	,	PUNCT
ma-301	400	29	453–472	453–472	NUM
ma-301	400	30	.	.	PUNCT
ma-301	401	1	https://doi.org/10.1007/s00365-011-9132-0.[14	https://doi.org/10.1007/s00365-011-9132-0.[14	ADJ
ma-301	401	2	]	]	PUNCT
ma-301	401	3	a.	a.	NOUN
ma-301	401	4	fitouhi	fitouhi	PROPN
ma-301	401	5	,	,	PUNCT
ma-301	401	6	m.m	m.m	PROPN
ma-301	401	7	.	.	PROPN
ma-301	401	8	hamza	hamza	PROPN
ma-301	401	9	,	,	PUNCT
ma-301	401	10	f.	f.	PROPN
ma-301	401	11	bouzeffour	bouzeffour	PROPN
ma-301	401	12	,	,	PUNCT
ma-301	401	13	the	the	DET
ma-301	401	14	q	q	X
ma-301	401	15	−	−	NOUN
ma-301	401	16	jα	jα	PROPN
ma-301	401	17	bessel	bessel	NOUN
ma-301	401	18	function	function	NOUN
ma-301	401	19	,	,	PUNCT
ma-301	401	20	j.	j.	PROPN
ma-301	401	21	approx	approx	PROPN
ma-301	401	22	.	.	PUNCT
ma-301	402	1	theory	theory	NOUN
ma-301	402	2	115	115	NUM
ma-301	402	3	(	(	PUNCT
ma-301	402	4	2002	2002	NUM
ma-301	402	5	)	)	PUNCT
ma-301	402	6	,	,	PUNCT
ma-301	402	7	144–166	144–166	NUM
ma-301	402	8	.	.	PUNCT
ma-301	403	1	https://doi.org/10.1006/jath.2001.3645.[15	https://doi.org/10.1006/jath.2001.3645.[15	NOUN
ma-301	403	2	]	]	PUNCT
ma-301	404	1	a.	a.	NOUN
ma-301	404	2	fitouhi	fitouhi	PROPN
ma-301	404	3	,	,	PUNCT
ma-301	404	4	n.	n.	NOUN
ma-301	404	5	bettaibi	bettaibi	NOUN
ma-301	404	6	,	,	PUNCT
ma-301	404	7	wavelet	wavelet	NOUN
ma-301	404	8	transforms	transform	VERB
ma-301	404	9	in	in	ADP
ma-301	404	10	quantum	quantum	ADJ
ma-301	404	11	calculus	calculus	NOUN
ma-301	404	12	,	,	PUNCT
ma-301	404	13	j.	j.	PROPN
ma-301	404	14	nonlinear	nonlinear	PROPN
ma-301	404	15	math	math	PROPN
ma-301	404	16	.	.	PUNCT
ma-301	405	1	phys	phy	NOUN
ma-301	405	2	.	.	PUNCT
ma-301	406	1	13	13	NUM
ma-301	406	2	(	(	PUNCT
ma-301	406	3	2006	2006	NUM
ma-301	406	4	)	)	PUNCT
ma-301	407	1	,	,	PUNCT
ma-301	407	2	492	492	NUM
ma-301	407	3	.	.	PUNCT
ma-301	407	4	https	https	NOUN
ma-301	407	5	:	:	PUNCT
ma-301	407	6	//doi.org/10.2991	//doi.org/10.2991	PROPN
ma-301	407	7	/	/	SYM
ma-301	407	8	jnmp.2006.13.4.4.[16	jnmp.2006.13.4.4.[16	PROPN
ma-301	407	9	]	]	PUNCT
ma-301	407	10	a.	a.	NOUN
ma-301	407	11	fitouhi	fitouhi	PROPN
ma-301	407	12	,	,	PUNCT
ma-301	407	13	k.	k.	PROPN
ma-301	407	14	brahim	brahim	PROPN
ma-301	407	15	,	,	PUNCT
ma-301	407	16	n.	n.	NOUN
ma-301	407	17	bettaibi	bettaibi	NOUN
ma-301	407	18	,	,	PUNCT
ma-301	407	19	on	on	ADP
ma-301	407	20	some	some	DET
ma-301	407	21	q	q	NOUN
ma-301	407	22	-	-	PUNCT
ma-301	407	23	versions	version	NOUN
ma-301	407	24	of	of	ADP
ma-301	407	25	the	the	DET
ma-301	407	26	ramanujan	ramanujan	PROPN
ma-301	407	27	master	master	PROPN
ma-301	407	28	theorem	theorem	PROPN
ma-301	407	29	,	,	PUNCT
ma-301	407	30	ramanujan	ramanujan	NOUN
ma-301	407	31	j.	j.	PROPN
ma-301	407	32	50	50	NUM
ma-301	407	33	(	(	PUNCT
ma-301	407	34	2019),433–458	2019),433–458	NOUN
ma-301	407	35	.	.	PUNCT
ma-301	408	1	https://doi.org/10.1007/s11139-019-00141-4.[17	https://doi.org/10.1007/s11139-019-00141-4.[17	PROPN
ma-301	408	2	]	]	PUNCT
ma-301	408	3	g.	g.	PROPN
ma-301	408	4	gasper	gasper	PROPN
ma-301	408	5	,	,	PUNCT
ma-301	408	6	m.	m.	PROPN
ma-301	408	7	rahman	rahman	PROPN
ma-301	408	8	,	,	PUNCT
ma-301	408	9	basic	basic	ADJ
ma-301	408	10	hypergeometric	hypergeometric	ADJ
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ma-301	408	12	,	,	PUNCT
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ma-301	408	16	,	,	PUNCT
ma-301	408	17	(	(	PUNCT
ma-301	408	18	2011).[18	2011).[18	NUM
ma-301	408	19	]	]	PUNCT
ma-301	408	20	l.	l.	PROPN
ma-301	408	21	hörmander	hörmander	PROPN
ma-301	408	22	,	,	PUNCT
ma-301	408	23	estimates	estimate	VERB
ma-301	408	24	for	for	ADP
ma-301	408	25	translation	translation	NOUN
ma-301	408	26	invariant	invariant	ADJ
ma-301	408	27	operators	operator	NOUN
ma-301	408	28	in	in	ADP
ma-301	408	29	lp	lp	ADJ
ma-301	408	30	spaces	space	NOUN
ma-301	408	31	,	,	PUNCT
ma-301	408	32	acta	acta	PROPN
ma-301	408	33	math	math	PROPN
ma-301	408	34	.	.	PUNCT
ma-301	409	1	104	104	NUM
ma-301	409	2	(	(	PUNCT
ma-301	409	3	1960	1960	NUM
ma-301	409	4	)	)	PUNCT
ma-301	409	5	,	,	PUNCT
ma-301	410	1	93–140	93–140	NUM
ma-301	410	2	.	.	PUNCT
ma-301	410	3	https	https	NOUN
ma-301	410	4	:	:	PUNCT
ma-301	410	5	//doi.org/10.1007	//doi.org/10.1007	NOUN
ma-301	410	6	/	/	SYM
ma-301	410	7	bf02547187.[19	bf02547187.[19	PROPN
ma-301	410	8	]	]	X
ma-301	410	9	f.h	f.h	PROPN
ma-301	410	10	.	.	PROPN
ma-301	410	11	jackson	jackson	PROPN
ma-301	410	12	,	,	PUNCT
ma-301	410	13	on	on	ADP
ma-301	410	14	q	q	ADJ
ma-301	410	15	-	-	ADJ
ma-301	410	16	definite	definite	ADJ
ma-301	410	17	integrals	integral	NOUN
ma-301	410	18	,	,	PUNCT
ma-301	410	19	quart	quart	NOUN
ma-301	410	20	.	.	PUNCT
ma-301	411	1	j.	j.	PROPN
ma-301	411	2	pure	pure	PROPN
ma-301	411	3	appl	appl	PROPN
ma-301	411	4	.	.	PUNCT
ma-301	411	5	math	math	NOUN
ma-301	411	6	.	.	PUNCT
ma-301	412	1	41	41	NUM
ma-301	412	2	(	(	PUNCT
ma-301	412	3	1910	1910	NUM
ma-301	412	4	)	)	PUNCT
ma-301	412	5	,	,	PUNCT
ma-301	412	6	193	193	NUM
ma-301	412	7	-	-	SYM
ma-301	412	8	203.[20	203.[20	NUM
ma-301	412	9	]	]	X
ma-301	412	10	v.g	v.g	PROPN
ma-301	412	11	.	.	PROPN
ma-301	412	12	kac	kac	PROPN
ma-301	412	13	,	,	PUNCT
ma-301	412	14	p.	p.	PROPN
ma-301	412	15	cheung	cheung	PROPN
ma-301	412	16	,	,	PUNCT
ma-301	412	17	quantum	quantum	NOUN
ma-301	412	18	calculus	calculus	NOUN
ma-301	412	19	,	,	PUNCT
ma-301	412	20	new	new	PROPN
ma-301	412	21	york	york	PROPN
ma-301	412	22	:	:	PUNCT
ma-301	412	23	springer	springer	NOUN
ma-301	412	24	,	,	PUNCT
ma-301	412	25	(	(	PUNCT
ma-301	412	26	2002).[21	2002).[21	X
ma-301	412	27	]	]	X
ma-301	412	28	t.h	t.h	PROPN
ma-301	412	29	.	.	PROPN
ma-301	412	30	koornwinder	koornwinder	PROPN
ma-301	412	31	,	,	PUNCT
ma-301	412	32	r.f	r.f	PROPN
ma-301	412	33	.	.	PROPN
ma-301	412	34	swarttouw	swarttouw	PROPN
ma-301	412	35	,	,	PUNCT
ma-301	412	36	on	on	ADP
ma-301	412	37	q	q	NOUN
ma-301	412	38	-	-	PUNCT
ma-301	412	39	analogues	analogue	NOUN
ma-301	412	40	of	of	ADP
ma-301	412	41	the	the	DET
ma-301	412	42	fourier	fourier	NOUN
ma-301	412	43	and	and	CCONJ
ma-301	412	44	hankel	hankel	NOUN
ma-301	412	45	transforms	transform	VERB
ma-301	412	46	,	,	PUNCT
ma-301	412	47	trans	trans	PROPN
ma-301	412	48	.	.	PROPN
ma-301	413	1	amer	amer	PROPN
ma-301	413	2	.	.	PUNCT
ma-301	413	3	math	math	PROPN
ma-301	413	4	.	.	PUNCT
ma-301	414	1	soc.333	soc.333	NOUN
ma-301	414	2	(	(	PUNCT
ma-301	414	3	1992	1992	NUM
ma-301	414	4	)	)	PUNCT
ma-301	414	5	,	,	PUNCT
ma-301	414	6	445–461	445–461	NUM
ma-301	414	7	.	.	PUNCT
ma-301	415	1	https://doi.org/10.1090/s0002-9947-1992-1069750-0.[22	https://doi.org/10.1090/s0002-9947-1992-1069750-0.[22	PROPN
ma-301	415	2	]	]	X
ma-301	415	3	v.	v.	PROPN
ma-301	415	4	kumar	kumar	PROPN
ma-301	415	5	,	,	PUNCT
ma-301	415	6	m.	m.	NOUN
ma-301	415	7	ruzhansky	ruzhansky	PROPN
ma-301	415	8	,	,	PUNCT
ma-301	415	9	lp	lp	ADJ
ma-301	415	10	-	-	PUNCT
ma-301	415	11	lq	lq	ADV
ma-301	415	12	boundedness	boundedness	NOUN
ma-301	415	13	of	of	ADP
ma-301	415	14	(	(	PUNCT
ma-301	415	15	k	k	X
ma-301	415	16	,	,	PUNCT
ma-301	415	17	a)-fourier	a)-fouri	ADJ
ma-301	415	18	multipliers	multiplier	NOUN
ma-301	415	19	with	with	ADP
ma-301	415	20	applications	application	NOUN
ma-301	415	21	to	to	ADP
ma-301	415	22	nonlinear	nonlinear	ADJ
ma-301	415	23	equations	equation	NOUN
ma-301	415	24	,	,	PUNCT
ma-301	415	25	int	int	NOUN
ma-301	415	26	.	.	PUNCT
ma-301	416	1	math	math	NOUN
ma-301	416	2	.	.	PUNCT
ma-301	417	1	res	re	NOUN
ma-301	417	2	.	.	PUNCT
ma-301	417	3	notices	notice	NOUN
ma-301	417	4	,	,	PUNCT
ma-301	417	5	2023	2023	NUM
ma-301	417	6	(	(	PUNCT
ma-301	417	7	2023	2023	NUM
ma-301	417	8	)	)	PUNCT
ma-301	417	9	,	,	PUNCT
ma-301	417	10	1073	1073	NUM
ma-301	417	11	-	-	SYM
ma-301	417	12	1093	1093	NUM
ma-301	417	13	.	.	PUNCT
ma-301	418	1	https://doi.org/10.1093/imrn/rnab256.[23	https://doi.org/10.1093/imrn/rnab256.[23	PROPN
ma-301	418	2	]	]	PUNCT
ma-301	418	3	t.	t.	PROPN
ma-301	418	4	matsuura	matsuura	PROPN
ma-301	418	5	,	,	PUNCT
ma-301	418	6	s.	s.	PROPN
ma-301	418	7	saitoh	saitoh	PROPN
ma-301	418	8	,	,	PUNCT
ma-301	418	9	d.d	d.d	PROPN
ma-301	418	10	.	.	PROPN
ma-301	418	11	trong	trong	PROPN
ma-301	418	12	,	,	PUNCT
ma-301	418	13	approximate	approximate	ADJ
ma-301	418	14	and	and	CCONJ
ma-301	418	15	analytical	analytical	ADJ
ma-301	418	16	inversion	inversion	NOUN
ma-301	418	17	formulas	formula	NOUN
ma-301	418	18	in	in	ADP
ma-301	418	19	heat	heat	NOUN
ma-301	418	20	conductionon	conductionon	NOUN
ma-301	418	21	multidimensional	multidimensional	ADJ
ma-301	418	22	spaces	space	NOUN
ma-301	418	23	,	,	PUNCT
ma-301	418	24	j.	j.	PROPN
ma-301	418	25	inverse	inverse	PROPN
ma-301	418	26	ill	ill	ADV
ma-301	418	27	-	-	PUNCT
ma-301	418	28	posed	pose	VERB
ma-301	418	29	probl	probl	NOUN
ma-301	418	30	.	.	PUNCT
ma-301	419	1	13	13	NUM
ma-301	419	2	(	(	PUNCT
ma-301	419	3	2005	2005	NUM
ma-301	419	4	)	)	PUNCT
ma-301	419	5	,	,	PUNCT
ma-301	419	6	479–493	479–493	NUM
ma-301	419	7	.	.	PUNCT
ma-301	420	1	https://doi.org/10.1515/	https://doi.org/10.1515/	PROPN
ma-301	420	2	156939405775297452.[24	156939405775297452.[24	NUM
ma-301	420	3	]	]	SYM
ma-301	420	4	t.r	t.r	PROPN
ma-301	420	5	.	.	PROPN
ma-301	420	6	mcconnell	mcconnell	PROPN
ma-301	420	7	,	,	PUNCT
ma-301	420	8	on	on	ADP
ma-301	420	9	fourier	fourier	NOUN
ma-301	420	10	multiplier	multipli	ADJ
ma-301	420	11	transformations	transformation	NOUN
ma-301	420	12	of	of	ADP
ma-301	420	13	banach	banach	ADV
ma-301	420	14	-	-	PUNCT
ma-301	420	15	valued	value	VERB
ma-301	420	16	functions	function	NOUN
ma-301	420	17	,	,	PUNCT
ma-301	420	18	trans	trans	PROPN
ma-301	420	19	.	.	PROPN
ma-301	421	1	amer	amer	PROPN
ma-301	421	2	.	.	PUNCT
ma-301	421	3	math	math	PROPN
ma-301	421	4	.	.	PUNCT
ma-301	422	1	soc	soc	PROPN
ma-301	422	2	.	.	PUNCT
ma-301	423	1	285(1984	285(1984	NUM
ma-301	423	2	)	)	PUNCT
ma-301	423	3	,	,	PUNCT
ma-301	423	4	739	739	NUM
ma-301	423	5	-	-	SYM
ma-301	423	6	757.[25	757.[25	PROPN
ma-301	423	7	]	]	X
ma-301	423	8	s.g	s.g	PROPN
ma-301	423	9	.	.	PROPN
ma-301	423	10	mikhlin	mikhlin	PROPN
ma-301	423	11	,	,	PUNCT
ma-301	423	12	on	on	ADP
ma-301	423	13	the	the	DET
ma-301	423	14	multipliers	multiplier	NOUN
ma-301	423	15	of	of	ADP
ma-301	423	16	fourier	fourier	NOUN
ma-301	423	17	integrals	integral	NOUN
ma-301	423	18	,	,	PUNCT
ma-301	423	19	dokl	dokl	NOUN
ma-301	423	20	.	.	PUNCT
ma-301	424	1	akad	akad	PROPN
ma-301	424	2	.	.	PUNCT
ma-301	425	1	nauk	nauk	PROPN
ma-301	425	2	sssr	sssr	NOUN
ma-301	425	3	109	109	NUM
ma-301	425	4	(	(	PUNCT
ma-301	425	5	1956	1956	NUM
ma-301	425	6	)	)	PUNCT
ma-301	425	7	,	,	PUNCT
ma-301	425	8	701	701	NUM
ma-301	425	9	-	-	SYM
ma-301	425	10	703.[26	703.[26	PROPN
ma-301	425	11	]	]	PUNCT
ma-301	425	12	a.	a.	NOUN
ma-301	425	13	nemri	nemri	PROPN
ma-301	425	14	,	,	PUNCT
ma-301	425	15	f.	f.	PROPN
ma-301	425	16	soltani	soltani	PROPN
ma-301	425	17	,	,	PUNCT
ma-301	425	18	analytical	analytical	ADJ
ma-301	425	19	approximation	approximation	NOUN
ma-301	425	20	formulas	formula	NOUN
ma-301	425	21	in	in	ADP
ma-301	425	22	quantum	quantum	NOUN
ma-301	425	23	calculus	calculus	NOUN
ma-301	425	24	,	,	PUNCT
ma-301	425	25	math	math	NOUN
ma-301	425	26	.	.	PUNCT
ma-301	426	1	mech	mech	PROPN
ma-301	426	2	.	.	PUNCT
ma-301	427	1	solids	solid	NOUN
ma-301	427	2	,	,	PUNCT
ma-301	427	3	22	22	NUM
ma-301	427	4	(	(	PUNCT
ma-301	427	5	2017),2075	2017),2075	NUM
ma-301	427	6	-	-	SYM
ma-301	427	7	2090.[27	2090.[27	NUM
ma-301	427	8	]	]	PUNCT
ma-301	427	9	a.	a.	PROPN
ma-301	427	10	nemri	nemri	PROPN
ma-301	427	11	,	,	PUNCT
ma-301	427	12	b.	b.	PROPN
ma-301	427	13	selmi	selmi	PROPN
ma-301	427	14	,	,	PUNCT
ma-301	427	15	sobolev	sobolev	ADJ
ma-301	427	16	type	type	NOUN
ma-301	427	17	spaces	space	NOUN
ma-301	427	18	in	in	ADP
ma-301	427	19	quantum	quantum	NOUN
ma-301	427	20	calculus	calculus	NOUN
ma-301	427	21	,	,	PUNCT
ma-301	427	22	j.	j.	PROPN
ma-301	427	23	math	math	PROPN
ma-301	427	24	.	.	PUNCT
ma-301	428	1	anal	anal	PROPN
ma-301	428	2	.	.	PUNCT
ma-301	428	3	appl	appl	PROPN
ma-301	428	4	.	.	PUNCT
ma-301	429	1	359	359	NUM
ma-301	429	2	(	(	PUNCT
ma-301	429	3	2009	2009	NUM
ma-301	429	4	)	)	PUNCT
ma-301	429	5	,	,	PUNCT
ma-301	429	6	588–601	588–601	NUM
ma-301	429	7	.	.	PUNCT
ma-301	429	8	https	https	NOUN
ma-301	429	9	:	:	PUNCT
ma-301	430	1	//doi.org/10.1016	//doi.org/10.1016	PROPN
ma-301	430	2	/	/	SYM
ma-301	430	3	j.jmaa.2009.06.008.[28	j.jmaa.2009.06.008.[28	PROPN
ma-301	430	4	]	]	X
ma-301	430	5	r.l	r.l	PROPN
ma-301	430	6	.	.	PROPN
ma-301	430	7	rubin	rubin	PROPN
ma-301	430	8	,	,	PUNCT
ma-301	430	9	a	a	DET
ma-301	430	10	q2	q2	NOUN
ma-301	430	11	-	-	PUNCT
ma-301	430	12	analogue	analogue	NOUN
ma-301	430	13	operator	operator	NOUN
ma-301	430	14	for	for	ADP
ma-301	430	15	q2	q2	NOUN
ma-301	430	16	-	-	PUNCT
ma-301	430	17	analogue	analogue	NOUN
ma-301	430	18	fourier	fourier	NOUN
ma-301	430	19	analysis	analysis	NOUN
ma-301	430	20	,	,	PUNCT
ma-301	430	21	j.	j.	PROPN
ma-301	430	22	math	math	PROPN
ma-301	430	23	.	.	PUNCT
ma-301	431	1	anal	anal	PROPN
ma-301	431	2	.	.	PUNCT
ma-301	432	1	appl	appl	PROPN
ma-301	432	2	.	.	PROPN
ma-301	433	1	212	212	NUM
ma-301	433	2	(	(	PUNCT
ma-301	433	3	1997	1997	NUM
ma-301	433	4	)	)	PUNCT
ma-301	433	5	,	,	PUNCT
ma-301	434	1	571	571	NUM
ma-301	434	2	-	-	SYM
ma-301	434	3	582.[29	582.[29	NUM
ma-301	434	4	]	]	X
ma-301	434	5	r.l	r.l	PROPN
ma-301	434	6	.	.	PROPN
ma-301	434	7	rubin	rubin	PROPN
ma-301	434	8	,	,	PUNCT
ma-301	434	9	duhamel	duhamel	VERB
ma-301	434	10	solutions	solution	NOUN
ma-301	434	11	of	of	ADP
ma-301	434	12	non	non	ADJ
ma-301	434	13	-	-	ADJ
ma-301	434	14	homogeneous	homogeneous	ADJ
ma-301	434	15	q2	q2	NOUN
ma-301	434	16	-	-	PUNCT
ma-301	434	17	analogue	analogue	NOUN
ma-301	434	18	wave	wave	NOUN
ma-301	434	19	equations	equation	NOUN
ma-301	434	20	,	,	PUNCT
ma-301	434	21	proc	proc	NOUN
ma-301	434	22	.	.	PUNCT
ma-301	435	1	amer	amer	PROPN
ma-301	435	2	.	.	PUNCT
ma-301	435	3	math	math	PROPN
ma-301	435	4	.	.	PUNCT
ma-301	436	1	soc	soc	PROPN
ma-301	436	2	.	.	PUNCT
ma-301	437	1	135(2006	135(2006	NUM
ma-301	437	2	)	)	PUNCT
ma-301	437	3	,	,	PUNCT
ma-301	438	1	777–785	777–785	NUM
ma-301	438	2	.	.	PUNCT
ma-301	438	3	https://doi.org/10.1090/s0002-9939-06-08525-x.[30	https://doi.org/10.1090/s0002-9939-06-08525-x.[30	X
ma-301	438	4	]	]	X
ma-301	438	5	t.	t.	PROPN
ma-301	438	6	matsuura	matsuura	PROPN
ma-301	438	7	,	,	PUNCT
ma-301	438	8	s.	s.	PROPN
ma-301	438	9	saitoh	saitoh	PROPN
ma-301	438	10	,	,	PUNCT
ma-301	438	11	analytical	analytical	ADJ
ma-301	438	12	and	and	CCONJ
ma-301	438	13	numerical	numerical	ADJ
ma-301	438	14	inversion	inversion	NOUN
ma-301	438	15	formulas	formula	NOUN
ma-301	438	16	in	in	ADP
ma-301	438	17	the	the	DET
ma-301	438	18	gaussian	gaussian	ADJ
ma-301	438	19	convolution	convolution	NOUN
ma-301	438	20	by	by	ADP
ma-301	438	21	using	use	VERB
ma-301	438	22	thepaley	thepaley	NOUN
ma-301	438	23	–	–	PUNCT
ma-301	438	24	wiener	wiener	NOUN
ma-301	438	25	spaces	space	NOUN
ma-301	438	26	,	,	PUNCT
ma-301	438	27	appl	appl	PROPN
ma-301	438	28	.	.	PROPN
ma-301	439	1	anal	anal	PROPN
ma-301	439	2	.	.	PUNCT
ma-301	439	3	85	85	NUM
ma-301	439	4	(	(	PUNCT
ma-301	439	5	2006	2006	NUM
ma-301	439	6	)	)	PUNCT
ma-301	439	7	,	,	PUNCT
ma-301	439	8	901–915	901–915	NUM
ma-301	439	9	.	.	PUNCT
ma-301	440	1	https://doi.org/10.1080/00036810600643662.[31	https://doi.org/10.1080/00036810600643662.[31	PROPN
ma-301	440	2	]	]	X
ma-301	440	3	a.	a.	NOUN
ma-301	440	4	saoudi	saoudi	PROPN
ma-301	440	5	,	,	PUNCT
ma-301	440	6	uncertainty	uncertainty	NOUN
ma-301	440	7	principle	principle	NOUN
ma-301	440	8	for	for	ADP
ma-301	440	9	the	the	DET
ma-301	440	10	fourier	fourier	ADJ
ma-301	440	11	-	-	PUNCT
ma-301	440	12	like	like	ADJ
ma-301	440	13	multipliers	multiplier	NOUN
ma-301	440	14	operators	operator	NOUN
ma-301	440	15	in	in	ADP
ma-301	440	16	q	q	ADJ
ma-301	440	17	-	-	PUNCT
ma-301	440	18	rubin	rubin	PROPN
ma-301	440	19	setting	setting	NOUN
ma-301	440	20	,	,	PUNCT
ma-301	440	21	bull	bull	NOUN
ma-301	440	22	.	.	PUNCT
ma-301	441	1	math	math	NOUN
ma-301	441	2	.	.	PUNCT
ma-301	442	1	anal.appl	anal.appl	PROPN
ma-301	442	2	.	.	PROPN
ma-301	442	3	14	14	NUM
ma-301	442	4	(	(	PUNCT
ma-301	442	5	2022	2022	NUM
ma-301	442	6	)	)	PUNCT
ma-301	442	7	,	,	PUNCT
ma-301	442	8	1	1	NUM
ma-301	442	9	-	-	SYM
ma-301	442	10	10	10	NUM
ma-301	442	11	.	.	PUNCT
ma-301	443	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	443	2	https://doi.org/10.1080/10652469.2024.2382787	https://doi.org/10.1080/10652469.2024.2382787	PROPN
ma-301	444	1	https://doi.org/10.1007/s11868-017-0215-y	https://doi.org/10.1007/s11868-017-0215-y	PROPN
ma-301	444	2	https://doi.org/10.1137/0149053	https://doi.org/10.1137/0149053	NOUN
ma-301	444	3	https://doi.org/10.1007/s00365-011-9132-0	https://doi.org/10.1007/s00365-011-9132-0	PROPN
ma-301	445	1	https://doi.org/10.1006/jath.2001.3645	https://doi.org/10.1006/jath.2001.3645	PROPN
ma-301	445	2	https://doi.org/10.2991/jnmp.2006.13.4.4	https://doi.org/10.2991/jnmp.2006.13.4.4	PRON
ma-301	445	3	https://doi.org/10.2991/jnmp.2006.13.4.4	https://doi.org/10.2991/jnmp.2006.13.4.4	NUM
ma-301	445	4	https://doi.org/10.1007/s11139-019-00141-4	https://doi.org/10.1007/s11139-019-00141-4	NUM
ma-301	445	5	https://doi.org/10.1007/bf02547187	https://doi.org/10.1007/bf02547187	X
ma-301	445	6	https://doi.org/10.1007/bf02547187	https://doi.org/10.1007/bf02547187	PART
ma-301	445	7	https://doi.org/10.1090/s0002-9947-1992-1069750-0	https://doi.org/10.1090/s0002-9947-1992-1069750-0	PROPN
ma-301	445	8	https://doi.org/10.1093/imrn/rnab256	https://doi.org/10.1093/imrn/rnab256	ADP
ma-301	445	9	https://doi.org/10.1515/156939405775297452	https://doi.org/10.1515/156939405775297452	NOUN
ma-301	445	10	https://doi.org/10.1515/156939405775297452	https://doi.org/10.1515/156939405775297452	NOUN
ma-301	445	11	https://doi.org/10.1016/j.jmaa.2009.06.008	https://doi.org/10.1016/j.jmaa.2009.06.008	NOUN
ma-301	445	12	https://doi.org/10.1016/j.jmaa.2009.06.008	https://doi.org/10.1016/j.jmaa.2009.06.008	NOUN
ma-301	445	13	https://doi.org/10.1090/s0002-9939-06-08525-x	https://doi.org/10.1090/s0002-9939-06-08525-x	PROPN
ma-301	445	14	https://doi.org/10.1080/00036810600643662	https://doi.org/10.1080/00036810600643662	X
ma-301	445	15	eur	eur	NOUN
ma-301	445	16	.	.	PUNCT
ma-301	446	1	j.	j.	PROPN
ma-301	446	2	math	math	PROPN
ma-301	446	3	.	.	PUNCT
ma-301	447	1	anal	anal	PROPN
ma-301	447	2	.	.	PUNCT
ma-301	448	1	10.28924	10.28924	NUM
ma-301	448	2	/	/	SYM
ma-301	448	3	ada	ada	PROPN
ma-301	448	4	/	/	SYM
ma-301	448	5	ma.5.8	ma.5.8	PROPN
ma-301	448	6	16	16	NUM
ma-301	449	1	[	[	X
ma-301	449	2	32	32	NUM
ma-301	449	3	]	]	PUNCT
ma-301	449	4	a.	a.	NOUN
ma-301	449	5	saoudi	saoudi	PROPN
ma-301	449	6	,	,	PUNCT
ma-301	449	7	reproducing	reproduce	VERB
ma-301	449	8	formulas	formula	NOUN
ma-301	449	9	for	for	ADP
ma-301	449	10	the	the	DET
ma-301	449	11	fourier	fourier	ADJ
ma-301	449	12	-	-	PUNCT
ma-301	449	13	like	like	ADJ
ma-301	449	14	multipliers	multiplier	NOUN
ma-301	449	15	operators	operator	NOUN
ma-301	449	16	in	in	ADP
ma-301	449	17	q	q	ADJ
ma-301	449	18	-	-	PUNCT
ma-301	449	19	rubin	rubin	PROPN
ma-301	449	20	setting	setting	NOUN
ma-301	449	21	,	,	PUNCT
ma-301	449	22	int	int	NOUN
ma-301	449	23	.	.	PUNCT
ma-301	450	1	j.	j.	PROPN
ma-301	450	2	anal	anal	PROPN
ma-301	450	3	.	.	PUNCT
ma-301	451	1	appl.18	appl.18	PROPN
ma-301	451	2	(	(	PUNCT
ma-301	451	3	2020	2020	NUM
ma-301	451	4	)	)	PUNCT
ma-301	451	5	,	,	PUNCT
ma-301	451	6	366	366	NUM
ma-301	451	7	-	-	SYM
ma-301	451	8	380.[33	380.[33	NUM
ma-301	451	9	]	]	PUNCT
ma-301	451	10	f.	f.	PROPN
ma-301	451	11	soltani	soltani	PROPN
ma-301	451	12	,	,	PUNCT
ma-301	451	13	i.	i.	PROPN
ma-301	451	14	maktouf	maktouf	PROPN
ma-301	451	15	,	,	PUNCT
ma-301	451	16	dunkl	dunkl	PROPN
ma-301	451	17	–	–	PUNCT
ma-301	451	18	weinstein	weinstein	PROPN
ma-301	451	19	multiplier	multiplier	PROPN
ma-301	451	20	operators	operator	NOUN
ma-301	451	21	and	and	CCONJ
ma-301	451	22	applications	application	NOUN
ma-301	451	23	to	to	ADP
ma-301	451	24	reproducing	reproduce	VERB
ma-301	451	25	kernel	kernel	PROPN
ma-301	451	26	theory	theory	PROPN
ma-301	451	27	,	,	PUNCT
ma-301	451	28	mediterranean	mediterranean	PROPN
ma-301	451	29	j.	j.	PROPN
ma-301	451	30	math	math	PROPN
ma-301	451	31	.	.	PUNCT
ma-301	452	1	21	21	NUM
ma-301	452	2	(	(	PUNCT
ma-301	452	3	2024	2024	NUM
ma-301	452	4	)	)	PUNCT
ma-301	452	5	,	,	PUNCT
ma-301	452	6	80	80	NUM
ma-301	452	7	.	.	PUNCT
ma-301	452	8	https://doi.org/10.1007/s00009-024-02623-2	https://doi.org/10.1007/s00009-024-02623-2	PROPN
ma-301	452	9	.	.	PUNCT
ma-301	453	1	https://doi.org/10.28924/ada/ma.5.8	https://doi.org/10.28924/ada/ma.5.8	PROPN
ma-301	453	2	https://doi.org/10.1007/s00009-024-02623-2	https://doi.org/10.1007/s00009-024-02623-2	NUM
ma-301	453	3	1	1	NUM
ma-301	453	4	.	.	PUNCT
ma-301	453	5	introduction	introduction	NOUN
ma-301	453	6	2	2	NUM
ma-301	453	7	.	.	PUNCT
ma-301	453	8	harmonic	harmonic	ADJ
ma-301	453	9	analysis	analysis	NOUN
ma-301	453	10	associated	associate	VERB
ma-301	453	11	with	with	ADP
ma-301	453	12	the	the	DET
ma-301	453	13	q	q	ADJ
ma-301	453	14	-	-	PUNCT
ma-301	453	15	bessel	bessel	NOUN
ma-301	453	16	transform	transform	VERB
ma-301	453	17	2.1	2.1	NUM
ma-301	453	18	.	.	PUNCT
ma-301	453	19	notations	notation	NOUN
ma-301	453	20	and	and	CCONJ
ma-301	453	21	preliminaries	preliminary	NOUN
ma-301	453	22	2.2	2.2	NUM
ma-301	453	23	.	.	PUNCT
ma-301	454	1	the	the	DET
ma-301	454	2	q	q	ADJ
ma-301	454	3	-	-	PUNCT
ma-301	454	4	bessel	bessel	NOUN
ma-301	454	5	transform	transform	VERB
ma-301	454	6	2.3	2.3	NUM
ma-301	454	7	.	.	PUNCT
ma-301	455	1	the	the	DET
ma-301	455	2	translation	translation	NOUN
ma-301	455	3	operator	operator	NOUN
ma-301	455	4	associated	associate	VERB
ma-301	455	5	with	with	ADP
ma-301	455	6	the	the	DET
ma-301	455	7	q	q	ADJ
ma-301	455	8	-	-	PUNCT
ma-301	455	9	bessel	bessel	ADJ
ma-301	455	10	transform	transform	VERB
ma-301	455	11	3	3	NUM
ma-301	455	12	.	.	PUNCT
ma-301	456	1	the	the	DET
ma-301	456	2	q	q	ADJ
ma-301	456	3	-	-	PUNCT
ma-301	456	4	bessel	bessel	ADJ
ma-301	456	5	l2	l2	NOUN
ma-301	456	6	-	-	PUNCT
ma-301	456	7	multiplier	multipli	ADJ
ma-301	456	8	operators	operator	NOUN
ma-301	456	9	3.1	3.1	NUM
ma-301	456	10	.	.	PUNCT
ma-301	456	11	calderon	calderon	PROPN
ma-301	456	12	's	's	PART
ma-301	456	13	reproducing	reproduce	VERB
ma-301	456	14	formulas	formula	NOUN
ma-301	456	15	for	for	ADP
ma-301	456	16	the	the	DET
ma-301	456	17	q	q	ADJ
ma-301	456	18	-	-	PUNCT
ma-301	456	19	bessel	bessel	ADJ
ma-301	456	20	l2	l2	NOUN
ma-301	456	21	-	-	PUNCT
ma-301	456	22	multiplier	multipli	ADJ
ma-301	456	23	operators	operator	NOUN
ma-301	456	24	3.2	3.2	NUM
ma-301	456	25	.	.	PUNCT
ma-301	457	1	uncerainty	uncerainty	ADJ
ma-301	457	2	principles	principle	NOUN
ma-301	457	3	for	for	ADP
ma-301	457	4	the	the	DET
ma-301	457	5	q	q	ADJ
ma-301	457	6	-	-	PUNCT
ma-301	457	7	bessel	bessel	ADJ
ma-301	457	8	l2	l2	NOUN
ma-301	457	9	-	-	PUNCT
ma-301	457	10	multiplier	multipli	ADJ
ma-301	457	11	operators	operator	NOUN
ma-301	457	12	4	4	NUM
ma-301	457	13	.	.	PUNCT
ma-301	457	14	extremal	extremal	ADJ
ma-301	457	15	functions	function	NOUN
ma-301	457	16	associated	associate	VERB
ma-301	457	17	with	with	ADP
ma-301	457	18	the	the	DET
ma-301	457	19	q	q	ADJ
ma-301	457	20	-	-	PUNCT
ma-301	457	21	bessel	bessel	ADJ
ma-301	457	22	l2	l2	NOUN
ma-301	457	23	-	-	PUNCT
ma-301	457	24	multiplier	multipli	ADJ
ma-301	457	25	operators	operator	NOUN
ma-301	457	26	acknowledgments	acknowledgment	NOUN
ma-301	457	27	:	:	PUNCT
ma-301	457	28	authors	author	NOUN
ma-301	457	29	'	'	PART
ma-301	457	30	contributions	contribution	NOUN
ma-301	457	31	:	:	PUNCT
ma-301	457	32	references	reference	NOUN