id	sid	tid	token	lemma	pos
ma-267	1	1	2024	2024	NUM
ma-267	1	2	ada	ada	PROPN
ma-267	1	3	academica	academica	PROPN
ma-267	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-267	1	5	.	.	PUNCT
ma-267	2	1	j.	j.	PROPN
ma-267	2	2	math	math	PROPN
ma-267	2	3	.	.	PUNCT
ma-267	3	1	anal	anal	ADJ
ma-267	3	2	.	.	PUNCT
ma-267	4	1	4	4	NUM
ma-267	4	2	(	(	PUNCT
ma-267	4	3	2024	2024	NUM
ma-267	4	4	)	)	PUNCT
ma-267	4	5	20doi	20doi	NOUN
ma-267	4	6	:	:	PUNCT
ma-267	4	7	10.28924	10.28924	NUM
ma-267	4	8	/	/	SYM
ma-267	4	9	ada	ada	PROPN
ma-267	4	10	/	/	SYM
ma-267	4	11	ma.4.20	ma.4.20	PROPN
ma-267	4	12	extremal	extremal	ADJ
ma-267	4	13	functions	function	NOUN
ma-267	4	14	and	and	CCONJ
ma-267	4	15	calderon	calderon	NOUN
ma-267	4	16	’s	’s	PART
ma-267	4	17	formulas	formula	NOUN
ma-267	4	18	for	for	ADP
ma-267	4	19	the	the	DET
ma-267	4	20	riemann	riemann	PROPN
ma-267	4	21	-	-	PUNCT
ma-267	4	22	liouville	liouville	VERB
ma-267	4	23	two	two	NUM
ma-267	4	24	-	-	PUNCT
ma-267	4	25	wavelet	wavelet	NOUN
ma-267	4	26	transform	transform	NOUN
ma-267	4	27	ahmed	ahmed	PROPN
ma-267	4	28	chana∗	chana∗	PROPN
ma-267	4	29	,	,	PUNCT
ma-267	4	30	abdellatif	abdellatif	NOUN
ma-267	4	31	akhlidj	akhlidj	VERB
ma-267	4	32	laboratory	laboratory	NOUN
ma-267	4	33	of	of	ADP
ma-267	4	34	fundamental	fundamental	ADJ
ma-267	4	35	and	and	CCONJ
ma-267	4	36	applied	applied	ADJ
ma-267	4	37	mathematics	mathematic	NOUN
ma-267	4	38	,	,	PUNCT
ma-267	4	39	department	department	NOUN
ma-267	4	40	of	of	ADP
ma-267	4	41	mathematics	mathematics	PROPN
ma-267	4	42	and	and	CCONJ
ma-267	4	43	informatics	informatic	NOUN
ma-267	4	44	,	,	PUNCT
ma-267	4	45	facultyof	facultyof	PROPN
ma-267	4	46	sciences	sciences	PROPN
ma-267	4	47	ain	ain	PROPN
ma-267	4	48	chock	chock	NOUN
ma-267	4	49	,	,	PUNCT
ma-267	4	50	university	university	NOUN
ma-267	4	51	of	of	ADP
ma-267	4	52	hassan	hassan	PROPN
ma-267	4	53	ii	ii	PROPN
ma-267	4	54	,	,	PUNCT
ma-267	4	55	b.p	b.p	PROPN
ma-267	4	56	5366	5366	NUM
ma-267	4	57	maarif	maarif	PROPN
ma-267	4	58	,	,	PUNCT
ma-267	4	59	casablanca	casablanca	PROPN
ma-267	4	60	,	,	PUNCT
ma-267	4	61	moroccomaths.chana@gmail.com	moroccomaths.chana@gmail.com	PROPN
ma-267	4	62	,	,	PUNCT
ma-267	4	63	akhlidj@hotmail.fr	akhlidj@hotmail.fr	PROPN
ma-267	4	64	∗correspondence	∗correspondence	NOUN
ma-267	4	65	:	:	PUNCT
ma-267	4	66	maths.chana@gmail.com	maths.chana@gmail.com	X
ma-267	4	67	abstract	abstract	NOUN
ma-267	4	68	.	.	PUNCT
ma-267	5	1	the	the	DET
ma-267	5	2	riemann	riemann	PROPN
ma-267	5	3	-	-	PUNCT
ma-267	5	4	liouville	liouville	NOUN
ma-267	5	5	operator	operator	NOUN
ma-267	5	6	has	have	AUX
ma-267	5	7	been	be	AUX
ma-267	5	8	extensively	extensively	ADV
ma-267	5	9	investigated	investigate	VERB
ma-267	5	10	and	and	CCONJ
ma-267	5	11	his	his	PRON
ma-267	5	12	witnessed	witness	VERB
ma-267	5	13	aremarkable	aremarkable	ADJ
ma-267	5	14	development	development	NOUN
ma-267	5	15	in	in	ADP
ma-267	5	16	numerous	numerous	ADJ
ma-267	5	17	fields	field	NOUN
ma-267	5	18	of	of	ADP
ma-267	5	19	harmonic	harmonic	ADJ
ma-267	5	20	analysis	analysis	NOUN
ma-267	5	21	.	.	PUNCT
ma-267	6	1	knowing	know	VERB
ma-267	6	2	the	the	DET
ma-267	6	3	fact	fact	NOUN
ma-267	6	4	of	of	ADP
ma-267	6	5	the	the	DET
ma-267	6	6	study	study	NOUN
ma-267	6	7	ofthe	ofthe	ADJ
ma-267	6	8	time	time	NOUN
ma-267	6	9	-	-	PUNCT
ma-267	6	10	frequency	frequency	NOUN
ma-267	6	11	analysis	analysis	NOUN
ma-267	6	12	are	be	AUX
ma-267	6	13	both	both	PRON
ma-267	6	14	theoritically	theoritically	ADV
ma-267	6	15	interesting	interesting	ADJ
ma-267	6	16	and	and	CCONJ
ma-267	6	17	pratically	pratically	ADV
ma-267	6	18	useful	useful	ADJ
ma-267	6	19	,	,	PUNCT
ma-267	6	20	we	we	PRON
ma-267	6	21	investigatedseveral	investigatedseveral	ADJ
ma-267	6	22	problems	problem	NOUN
ma-267	6	23	for	for	ADP
ma-267	6	24	this	this	DET
ma-267	6	25	subject	subject	NOUN
ma-267	6	26	on	on	ADP
ma-267	6	27	the	the	DET
ma-267	6	28	setting	setting	NOUN
ma-267	6	29	of	of	ADP
ma-267	6	30	the	the	DET
ma-267	6	31	riemann	riemann	PROPN
ma-267	6	32	-	-	PUNCT
ma-267	6	33	liouville	liouville	VERB
ma-267	6	34	wavelet	wavelet	NOUN
ma-267	6	35	transform	transform	NOUN
ma-267	6	36	.	.	PUNCT
ma-267	7	1	firstly	firstly	ADV
ma-267	7	2	,	,	PUNCT
ma-267	7	3	we	we	PRON
ma-267	7	4	introduce	introduce	VERB
ma-267	7	5	the	the	DET
ma-267	7	6	notion	notion	NOUN
ma-267	7	7	of	of	ADP
ma-267	7	8	riemann	riemann	PROPN
ma-267	7	9	-	-	PUNCT
ma-267	7	10	liouville	liouville	VERB
ma-267	7	11	two	two	NUM
ma-267	7	12	-	-	PUNCT
ma-267	7	13	wavelet	wavelet	NOUN
ma-267	8	1	and	and	CCONJ
ma-267	8	2	we	we	PRON
ma-267	8	3	present	present	VERB
ma-267	8	4	generalized	generalized	ADJ
ma-267	8	5	version	version	NOUN
ma-267	8	6	ofparseval	ofparseval	PROPN
ma-267	8	7	’s	’s	PART
ma-267	8	8	,	,	PUNCT
ma-267	8	9	plancherel	plancherel	PROPN
ma-267	8	10	’s	’s	PART
ma-267	8	11	,	,	PUNCT
ma-267	8	12	inversion	inversion	NOUN
ma-267	8	13	and	and	CCONJ
ma-267	8	14	calderon	calderon	NOUN
ma-267	8	15	’s	’s	PART
ma-267	8	16	reproducing	reproduce	VERB
ma-267	8	17	formulas	formula	NOUN
ma-267	8	18	.	.	PUNCT
ma-267	9	1	next	next	ADV
ma-267	9	2	,	,	PUNCT
ma-267	9	3	using	use	VERB
ma-267	9	4	the	the	DET
ma-267	9	5	theory	theory	NOUN
ma-267	9	6	ofreproducing	ofreproduce	VERB
ma-267	9	7	kernels	kernel	NOUN
ma-267	9	8	,	,	PUNCT
ma-267	9	9	we	we	PRON
ma-267	9	10	give	give	VERB
ma-267	9	11	best	good	ADJ
ma-267	9	12	estimates	estimate	NOUN
ma-267	9	13	and	and	CCONJ
ma-267	9	14	an	an	DET
ma-267	9	15	integral	integral	ADJ
ma-267	9	16	representation	representation	NOUN
ma-267	9	17	of	of	ADP
ma-267	9	18	the	the	DET
ma-267	9	19	extremal	extremal	NOUN
ma-267	9	20	functionsrelated	functionsrelate	VERB
ma-267	9	21	to	to	ADP
ma-267	9	22	the	the	DET
ma-267	9	23	riemann	riemann	PROPN
ma-267	9	24	-	-	PUNCT
ma-267	9	25	liouville	liouville	VERB
ma-267	9	26	wavelet	wavelet	NOUN
ma-267	9	27	transform	transform	NOUN
ma-267	9	28	on	on	ADP
ma-267	9	29	weighted	weight	VERB
ma-267	9	30	sobolev	sobolev	NOUN
ma-267	9	31	spaces	space	NOUN
ma-267	9	32	.	.	PUNCT
ma-267	10	1	1	1	X
ma-267	10	2	.	.	X
ma-267	10	3	introduction	introduction	NOUN
ma-267	10	4	the	the	DET
ma-267	10	5	mean	mean	ADJ
ma-267	10	6	operator	operator	NOUN
ma-267	10	7	is	be	AUX
ma-267	10	8	defined	define	VERB
ma-267	10	9	for	for	ADP
ma-267	10	10	a	a	DET
ma-267	10	11	continuous	continuous	ADJ
ma-267	10	12	function	function	NOUN
ma-267	10	13	on	on	ADP
ma-267	10	14	r2	r2	NOUN
ma-267	10	15	,	,	PUNCT
ma-267	10	16	even	even	ADV
ma-267	10	17	with	with	ADP
ma-267	10	18	respect	respect	NOUN
ma-267	10	19	to	to	ADP
ma-267	10	20	the	the	DET
ma-267	10	21	firstvariable	firstvariable	NOUN
ma-267	10	22	by	by	ADP
ma-267	10	23	r0(f	r0(f	PROPN
ma-267	10	24	)	)	PUNCT
ma-267	10	25	(	(	PUNCT
ma-267	10	26	x	x	X
ma-267	10	27	,	,	PUNCT
ma-267	10	28	t	t	PROPN
ma-267	10	29	)	)	PUNCT
ma-267	10	30	=	=	SYM
ma-267	11	1	1	1	NUM
ma-267	11	2	2π	2π	NUM
ma-267	11	3	∫	∫	PROPN
ma-267	11	4	2π	2π	PROPN
ma-267	11	5	0	0	PUNCT
ma-267	12	1	f	f	X
ma-267	12	2	(	(	PUNCT
ma-267	12	3	x	x	SYM
ma-267	12	4	sin	sin	PROPN
ma-267	12	5	θ	θ	PROPN
ma-267	12	6	,	,	PUNCT
ma-267	12	7	t	t	PROPN
ma-267	13	1	+	+	CCONJ
ma-267	13	2	x	x	AUX
ma-267	13	3	cos	cos	PROPN
ma-267	13	4	θ)dθ.which	θ)dθ.which	NOUN
ma-267	13	5	means	mean	VERB
ma-267	13	6	that	that	SCONJ
ma-267	13	7	r0(f	r0(f	PROPN
ma-267	13	8	)	)	PUNCT
ma-267	13	9	(	(	PUNCT
ma-267	13	10	x	x	X
ma-267	13	11	,	,	PUNCT
ma-267	13	12	t	t	PROPN
ma-267	13	13	)	)	PUNCT
ma-267	13	14	is	be	AUX
ma-267	13	15	the	the	DET
ma-267	13	16	mean	mean	ADJ
ma-267	13	17	value	value	NOUN
ma-267	13	18	of	of	ADP
ma-267	13	19	f	f	PROPN
ma-267	13	20	on	on	ADP
ma-267	13	21	the	the	DET
ma-267	13	22	circle	circle	NOUN
ma-267	13	23	centered	center	VERB
ma-267	13	24	at	at	ADP
ma-267	13	25	(	(	PUNCT
ma-267	13	26	0	0	NUM
ma-267	13	27	,	,	PUNCT
ma-267	13	28	t	t	PROPN
ma-267	13	29	)	)	PUNCT
ma-267	13	30	and	and	CCONJ
ma-267	14	1	radius	radius	NOUN
ma-267	14	2	x	x	X
ma-267	14	3	.	.	PUNCT
ma-267	15	1	the	the	DET
ma-267	15	2	operators	operator	NOUN
ma-267	15	3	r0	r0	NOUN
ma-267	15	4	play	play	VERB
ma-267	15	5	an	an	DET
ma-267	15	6	mportant	mportant	ADJ
ma-267	15	7	role	role	NOUN
ma-267	15	8	and	and	CCONJ
ma-267	15	9	has	have	VERB
ma-267	15	10	many	many	ADJ
ma-267	15	11	applications	application	NOUN
ma-267	15	12	,	,	PUNCT
ma-267	15	13	for	for	ADP
ma-267	15	14	example	example	NOUN
ma-267	15	15	,	,	PUNCT
ma-267	15	16	in	in	ADP
ma-267	15	17	imageprocessing	imageprocesse	VERB
ma-267	15	18	of	of	ADP
ma-267	15	19	so	so	ADV
ma-267	15	20	-	-	PUNCT
ma-267	15	21	called	call	VERB
ma-267	15	22	synthetic	synthetic	ADJ
ma-267	15	23	aperture	aperture	NOUN
ma-267	15	24	radar	radar	NOUN
ma-267	15	25	(	(	PUNCT
ma-267	15	26	sar	sar	PROPN
ma-267	15	27	)	)	PUNCT
ma-267	15	28	data	datum	NOUN
ma-267	15	29	see	see	VERB
ma-267	15	30	[	[	X
ma-267	15	31	10,11	10,11	NOUN
ma-267	15	32	]	]	X
ma-267	15	33	,	,	PUNCT
ma-267	15	34	or	or	CCONJ
ma-267	15	35	in	in	ADP
ma-267	15	36	the	the	DET
ma-267	15	37	linearized	linearize	VERB
ma-267	15	38	inversescattering	inversescattering	NOUN
ma-267	15	39	problem	problem	NOUN
ma-267	15	40	in	in	ADP
ma-267	15	41	acoustics	acoustic	NOUN
ma-267	15	42	see	see	VERB
ma-267	15	43	[	[	X
ma-267	15	44	5	5	NUM
ma-267	15	45	,	,	PUNCT
ma-267	15	46	7].in	7].in	NUM
ma-267	16	1	[	[	X
ma-267	16	2	3	3	NUM
ma-267	16	3	]	]	PUNCT
ma-267	16	4	,	,	PUNCT
ma-267	16	5	the	the	DET
ma-267	16	6	authors	author	NOUN
ma-267	16	7	have	have	AUX
ma-267	16	8	generalized	generalize	VERB
ma-267	16	9	r0	r0	NOUN
ma-267	16	10	and	and	CCONJ
ma-267	16	11	its	its	PRON
ma-267	16	12	dual	dual	ADJ
ma-267	16	13	tr0	tr0	NOUN
ma-267	16	14	by	by	ADP
ma-267	16	15	introducing	introduce	VERB
ma-267	16	16	the	the	DET
ma-267	16	17	so	so	ADV
ma-267	16	18	-	-	PUNCT
ma-267	16	19	called	call	VERB
ma-267	16	20	riemann	riemann	PROPN
ma-267	16	21	-	-	PUNCT
ma-267	16	22	liouville	liouville	NOUN
ma-267	16	23	operator	operator	NOUN
ma-267	16	24	defined	define	VERB
ma-267	16	25	on	on	ADP
ma-267	16	26	the	the	DET
ma-267	16	27	space	space	NOUN
ma-267	16	28	of	of	ADP
ma-267	16	29	continuous	continuous	ADJ
ma-267	16	30	functions	function	NOUN
ma-267	16	31	on	on	ADP
ma-267	16	32	r2	r2	NOUN
ma-267	16	33	,	,	PUNCT
ma-267	16	34	even	even	ADV
ma-267	16	35	with	with	ADP
ma-267	16	36	the	the	DET
ma-267	16	37	respect	respect	NOUN
ma-267	16	38	to	to	ADP
ma-267	16	39	the	the	DET
ma-267	16	40	received	received	NOUN
ma-267	16	41	:	:	PUNCT
ma-267	16	42	17	17	NUM
ma-267	16	43	aug	aug	PROPN
ma-267	16	44	2024.key	2024.key	NOUN
ma-267	16	45	words	word	NOUN
ma-267	16	46	and	and	CCONJ
ma-267	16	47	phrases	phrase	NOUN
ma-267	16	48	.	.	PUNCT
ma-267	17	1	riemann	riemann	PROPN
ma-267	17	2	-	-	PUNCT
ma-267	17	3	liouville	liouville	NOUN
ma-267	17	4	operator	operator	NOUN
ma-267	17	5	,	,	PUNCT
ma-267	17	6	wavelet	wavelet	NOUN
ma-267	17	7	transform	transform	NOUN
ma-267	17	8	,	,	PUNCT
ma-267	17	9	time	time	NOUN
ma-267	17	10	-	-	PUNCT
ma-267	17	11	frequency	frequency	NOUN
ma-267	17	12	analysis	analysis	NOUN
ma-267	17	13	,	,	PUNCT
ma-267	17	14	extremal	extremal	ADJ
ma-267	17	15	func	func	NOUN
ma-267	17	16	-	-	PUNCT
ma-267	17	17	tions	tion	NOUN
ma-267	17	18	,	,	PUNCT
ma-267	17	19	weighted	weight	VERB
ma-267	17	20	sobolev	sobolev	NOUN
ma-267	17	21	spaces	space	NOUN
ma-267	17	22	.	.	PUNCT
ma-267	18	1	1	1	NUM
ma-267	18	2	https://adac.ee	https://adac.ee	PROPN
ma-267	18	3	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	18	4	eur	eur	PROPN
ma-267	18	5	.	.	PUNCT
ma-267	19	1	j.	j.	PROPN
ma-267	19	2	math	math	PROPN
ma-267	19	3	.	.	PUNCT
ma-267	20	1	anal	anal	PROPN
ma-267	20	2	.	.	PUNCT
ma-267	21	1	10.28924	10.28924	NUM
ma-267	21	2	/	/	SYM
ma-267	21	3	ada	ada	PROPN
ma-267	21	4	/	/	SYM
ma-267	21	5	ma.4.20	ma.4.20	NOUN
ma-267	22	1	2first	2first	NUM
ma-267	22	2	variable	variable	NOUN
ma-267	22	3	by	by	ADP
ma-267	22	4	rα(f	rα(f	PUNCT
ma-267	22	5	)	)	PUNCT
ma-267	22	6	(	(	PUNCT
ma-267	22	7	x	x	X
ma-267	22	8	,	,	PUNCT
ma-267	22	9	t	t	PROPN
ma-267	22	10	)	)	PUNCT
ma-267	22	11	:	:	PUNCT
ma-267	23	1	=	=	PUNCT
ma-267	24	1			NOUN
ma-267	24	2	α	α	NOUN
ma-267	24	3	π	π	NOUN
ma-267	24	4	∫	∫	PROPN
ma-267	24	5	1	1	NUM
ma-267	24	6	−1	−1	NOUN
ma-267	24	7	∫	∫	NOUN
ma-267	24	8	1	1	NUM
ma-267	24	9	−1	−1	NOUN
ma-267	24	10	f	f	PROPN
ma-267	24	11	(	(	PUNCT
ma-267	24	12	xs	xs	PROPN
ma-267	24	13	√	√	PROPN
ma-267	24	14	1−	1−	NUM
ma-267	24	15	y2	y2	PROPN
ma-267	24	16	,	,	PUNCT
ma-267	24	17	t	t	PROPN
ma-267	24	18	+	+	X
ma-267	24	19	xy	xy	NOUN
ma-267	24	20	)	)	PUNCT
ma-267	25	1	(	(	PUNCT
ma-267	25	2	1−	1−	NUM
ma-267	25	3	y2	y2	INTJ
ma-267	25	4	)	)	PUNCT
ma-267	25	5	α−	α−	ADP
ma-267	25	6	1	1	NUM
ma-267	25	7	2	2	NUM
ma-267	25	8	(	(	PUNCT
ma-267	25	9	1−	1−	NUM
ma-267	25	10	s2	s2	NOUN
ma-267	25	11	)	)	PUNCT
ma-267	25	12	α−1	α−1	PROPN
ma-267	25	13	dyds	dyds	NOUN
ma-267	25	14	if	if	SCONJ
ma-267	25	15	α	α	PROPN
ma-267	25	16	>	>	X
ma-267	25	17	0	0	NUM
ma-267	25	18	,	,	PUNCT
ma-267	25	19	1	1	NUM
ma-267	25	20	π	π	NOUN
ma-267	25	21	∫	∫	PROPN
ma-267	25	22	1	1	NUM
ma-267	25	23	−1	−1	NOUN
ma-267	25	24	f	f	PROPN
ma-267	25	25	(	(	PUNCT
ma-267	25	26	r	r	NOUN
ma-267	25	27	√	√	PROPN
ma-267	25	28	1−	1−	NUM
ma-267	25	29	y2	y2	PROPN
ma-267	25	30	,	,	PUNCT
ma-267	25	31	t	t	PROPN
ma-267	25	32	+	+	CCONJ
ma-267	25	33	xy	xy	PROPN
ma-267	25	34	)	)	PUNCT
ma-267	25	35	dt√	dt√	PROPN
ma-267	25	36	1−t2	1−t2	NUM
ma-267	25	37	if	if	SCONJ
ma-267	25	38	α	α	NOUN
ma-267	25	39	=	=	NOUN
ma-267	25	40	0,(1.1)many	0,(1.1)many	NUM
ma-267	25	41	harmonic	harmonic	VERB
ma-267	25	42	analysis	analysis	NOUN
ma-267	25	43	results	result	NOUN
ma-267	25	44	related	relate	VERB
ma-267	25	45	to	to	ADP
ma-267	25	46	the	the	DET
ma-267	25	47	riemann	riemann	PROPN
ma-267	25	48	-	-	PUNCT
ma-267	25	49	liouville	liouville	NOUN
ma-267	25	50	operator	operator	NOUN
ma-267	25	51	(	(	PUNCT
ma-267	25	52	1.1	1.1	NUM
ma-267	25	53	)	)	PUNCT
ma-267	25	54	have	have	AUX
ma-267	25	55	been	be	AUX
ma-267	25	56	estab	estab	NOUN
ma-267	25	57	-	-	PUNCT
ma-267	25	58	lished	lishe	VERB
ma-267	25	59	see	see	NOUN
ma-267	25	60	[	[	X
ma-267	25	61	1–4	1–4	NOUN
ma-267	25	62	]	]	PUNCT
ma-267	25	63	and	and	CCONJ
ma-267	25	64	the	the	DET
ma-267	25	65	references	reference	NOUN
ma-267	25	66	therein	therein	ADV
ma-267	25	67	.	.	PUNCT
ma-267	26	1	the	the	DET
ma-267	26	2	wavelet	wavelet	NOUN
ma-267	26	3	transform	transform	NOUN
ma-267	26	4	has	have	VERB
ma-267	26	5	a	a	DET
ma-267	26	6	long	long	ADJ
ma-267	26	7	story	story	NOUN
ma-267	26	8	which	which	PRON
ma-267	26	9	stared	stare	VERB
ma-267	26	10	in1984	in1984	PRON
ma-267	26	11	with	with	ADP
ma-267	26	12	mortel	mortel	PROPN
ma-267	26	13	,	,	PUNCT
ma-267	26	14	a	a	DET
ma-267	26	15	french	french	ADJ
ma-267	26	16	petroleum	petroleum	NOUN
ma-267	26	17	engineer	engineer	NOUN
ma-267	26	18	in	in	ADP
ma-267	26	19	connection	connection	NOUN
ma-267	26	20	with	with	ADP
ma-267	26	21	his	his	PRON
ma-267	26	22	study	study	NOUN
ma-267	26	23	of	of	ADP
ma-267	26	24	seismic	seismic	ADJ
ma-267	26	25	traces	trace	NOUN
ma-267	26	26	,	,	PUNCT
ma-267	26	27	themathematical	themathematical	ADJ
ma-267	26	28	foundations	foundation	NOUN
ma-267	26	29	were	be	AUX
ma-267	26	30	given	give	VERB
ma-267	26	31	by	by	ADP
ma-267	26	32	grossman	grossman	NOUN
ma-267	26	33	and	and	CCONJ
ma-267	26	34	mortel	mortel	PROPN
ma-267	26	35	in	in	ADP
ma-267	26	36	[	[	X
ma-267	26	37	8	8	NUM
ma-267	26	38	,	,	PUNCT
ma-267	26	39	9	9	NUM
ma-267	26	40	]	]	PUNCT
ma-267	26	41	.	.	PUNCT
ma-267	27	1	mortel	mortel	PROPN
ma-267	27	2	defined	define	VERB
ma-267	27	3	a	a	DET
ma-267	27	4	waveletas	waveleta	NOUN
ma-267	27	5	a	a	DET
ma-267	27	6	collection	collection	NOUN
ma-267	27	7	of	of	ADP
ma-267	27	8	functions	function	NOUN
ma-267	27	9	constructed	construct	VERB
ma-267	27	10	by	by	ADP
ma-267	27	11	using	use	VERB
ma-267	27	12	translation	translation	NOUN
ma-267	27	13	and	and	CCONJ
ma-267	27	14	dilatation	dilatation	NOUN
ma-267	27	15	of	of	ADP
ma-267	27	16	a	a	DET
ma-267	27	17	single	single	ADJ
ma-267	27	18	function	function	NOUN
ma-267	27	19	ψ	ψ	X
ma-267	27	20	∈	∈	PROPN
ma-267	27	21	l2(r	l2(r	NOUN
ma-267	27	22	)	)	PUNCT
ma-267	27	23	called	call	VERB
ma-267	27	24	the	the	DET
ma-267	27	25	mother	mother	NOUN
ma-267	27	26	wavelet	wavelet	NOUN
ma-267	27	27	by	by	ADP
ma-267	27	28	:	:	PUNCT
ma-267	27	29	ψb	ψb	ADJ
ma-267	27	30	,	,	PUNCT
ma-267	27	31	a(t	a(t	NOUN
ma-267	27	32	)	)	PUNCT
ma-267	28	1	=	=	PUNCT
ma-267	28	2	1√	1√	NOUN
ma-267	28	3	a	a	DET
ma-267	28	4	ψ	ψ	X
ma-267	28	5	(	(	PUNCT
ma-267	28	6	t	t	PROPN
ma-267	28	7	−	−	PROPN
ma-267	28	8	b	b	PROPN
ma-267	28	9	a	a	NOUN
ma-267	28	10	)	)	PUNCT
ma-267	28	11	,	,	PUNCT
ma-267	28	12	where	where	SCONJ
ma-267	28	13	a	a	DET
ma-267	28	14	>	>	X
ma-267	28	15	0	0	NUM
ma-267	28	16	is	be	AUX
ma-267	28	17	called	call	VERB
ma-267	28	18	the	the	DET
ma-267	28	19	scaling	scaling	ADJ
ma-267	28	20	parameter	parameter	NOUN
ma-267	28	21	,	,	PUNCT
ma-267	28	22	wich	wich	PRON
ma-267	28	23	measure	measure	VERB
ma-267	28	24	the	the	DET
ma-267	28	25	degree	degree	NOUN
ma-267	28	26	of	of	ADP
ma-267	28	27	compression	compression	NOUN
ma-267	28	28	and	and	CCONJ
ma-267	28	29	b	b	X
ma-267	28	30	∈	∈	NOUN
ma-267	28	31	ris	ris	X
ma-267	28	32	a	a	DET
ma-267	28	33	translation	translation	NOUN
ma-267	28	34	parameter	parameter	NOUN
ma-267	28	35	wich	wich	PRON
ma-267	28	36	determines	determine	VERB
ma-267	28	37	the	the	DET
ma-267	28	38	time	time	NOUN
ma-267	28	39	location	location	NOUN
ma-267	28	40	of	of	ADP
ma-267	28	41	the	the	DET
ma-267	28	42	wavelet	wavelet	NOUN
ma-267	28	43	.	.	PUNCT
ma-267	29	1	the	the	DET
ma-267	29	2	theory	theory	NOUN
ma-267	29	3	of	of	ADP
ma-267	29	4	wavelethave	wavelethave	ADJ
ma-267	29	5	applications	application	NOUN
ma-267	29	6	in	in	ADP
ma-267	29	7	several	several	ADJ
ma-267	29	8	research	research	NOUN
ma-267	29	9	area	area	NOUN
ma-267	29	10	as	as	ADP
ma-267	29	11	signal	signal	NOUN
ma-267	29	12	theory	theory	NOUN
ma-267	29	13	,	,	PUNCT
ma-267	29	14	time	time	NOUN
ma-267	29	15	frequency	frequency	NOUN
ma-267	29	16	analysis	analysis	NOUN
ma-267	29	17	,	,	PUNCT
ma-267	29	18	geophysicsand	geophysicsand	NOUN
ma-267	29	19	medicine	medicine	NOUN
ma-267	29	20	see	see	VERB
ma-267	29	21	[	[	X
ma-267	29	22	6	6	NUM
ma-267	29	23	,	,	PUNCT
ma-267	29	24	9].a	9].a	NUM
ma-267	29	25	lot	lot	NOUN
ma-267	29	26	of	of	ADP
ma-267	29	27	attention	attention	NOUN
ma-267	29	28	has	have	AUX
ma-267	29	29	been	be	AUX
ma-267	29	30	given	give	VERB
ma-267	29	31	to	to	ADP
ma-267	29	32	various	various	ADJ
ma-267	29	33	generalization	generalization	NOUN
ma-267	29	34	of	of	ADP
ma-267	29	35	the	the	DET
ma-267	29	36	classical	classical	ADJ
ma-267	29	37	fourier	fourier	NOUN
ma-267	29	38	transform	transform	NOUN
ma-267	29	39	,	,	PUNCT
ma-267	29	40	this	this	DET
ma-267	29	41	paper	paper	NOUN
ma-267	29	42	focuses	focus	VERB
ma-267	29	43	on	on	ADP
ma-267	29	44	the	the	DET
ma-267	29	45	generalized	generalized	ADJ
ma-267	29	46	fourier	fourier	NOUN
ma-267	29	47	transform	transform	NOUN
ma-267	29	48	associated	associate	VERB
ma-267	29	49	with	with	ADP
ma-267	29	50	the	the	DET
ma-267	29	51	riemann	riemann	PROPN
ma-267	29	52	-	-	PUNCT
ma-267	29	53	liouvilleoperator	liouvilleoperator	NOUN
ma-267	29	54	(	(	PUNCT
ma-267	29	55	1.1	1.1	NUM
ma-267	29	56	)	)	PUNCT
ma-267	29	57	called	call	VERB
ma-267	29	58	the	the	DET
ma-267	29	59	riemann	riemann	PROPN
ma-267	29	60	-	-	PUNCT
ma-267	29	61	liouville	liouville	NOUN
ma-267	29	62	transform	transform	NOUN
ma-267	29	63	,	,	PUNCT
ma-267	29	64	more	more	ADV
ma-267	29	65	precisely	precisely	ADV
ma-267	29	66	we	we	PRON
ma-267	29	67	consider	consider	VERB
ma-267	29	68	a	a	DET
ma-267	29	69	system	system	NOUN
ma-267	29	70	ofpartial	ofpartial	ADJ
ma-267	29	71	differential	differential	NOUN
ma-267	29	72	operator	operator	NOUN
ma-267	29	73	∆1	∆1	PUNCT
ma-267	29	74	and	and	CCONJ
ma-267	29	75	∆2	∆2	PROPN
ma-267	29	76	defined	define	VERB
ma-267	29	77	by	by	PROPN
ma-267	29	78	∆1	∆1	NUM
ma-267	29	79	:	:	PUNCT
ma-267	29	80	=	=	SYM
ma-267	29	81	∂	∂	NUM
ma-267	29	82	∂x	∂x	NOUN
ma-267	29	83	;	;	PUNCT
ma-267	29	84	α	α	PRON
ma-267	29	85	≥	≥	NOUN
ma-267	29	86	0	0	NUM
ma-267	29	87	,	,	PUNCT
ma-267	29	88	t	t	X
ma-267	29	89	>	>	X
ma-267	29	90	0	0	PROPN
ma-267	29	91	,	,	PUNCT
ma-267	29	92	∆2	∆2	X
ma-267	29	93	:	:	PUNCT
ma-267	29	94	=	=	SYM
ma-267	29	95	∂2	∂2	NOUN
ma-267	29	96	∂t2	∂t2	NOUN
ma-267	29	97	+	+	CCONJ
ma-267	29	98	2α+	2α+	NUM
ma-267	29	99	1	1	NUM
ma-267	29	100	t	t	NOUN
ma-267	29	101	∂	∂	NOUN
ma-267	29	102	∂t	∂t	PROPN
ma-267	29	103	−	−	PROPN
ma-267	29	104	∂2	∂2	PROPN
ma-267	29	105	∂x2	∂x2	NOUN
ma-267	29	106	,	,	PUNCT
ma-267	29	107	α	α	PRON
ma-267	29	108	≥	≥	NOUN
ma-267	29	109	0	0	NUM
ma-267	29	110	.	.	PUNCT
ma-267	30	1	from	from	ADP
ma-267	30	2	[	[	X
ma-267	30	3	3	3	NUM
ma-267	30	4	]	]	PUNCT
ma-267	30	5	,	,	PUNCT
ma-267	30	6	the	the	DET
ma-267	30	7	authors	author	NOUN
ma-267	30	8	gives	give	VERB
ma-267	30	9	the	the	DET
ma-267	30	10	connection	connection	NOUN
ma-267	30	11	between	between	ADP
ma-267	30	12	the	the	DET
ma-267	30	13	eigenfunctions	eigenfunction	NOUN
ma-267	30	14	of	of	ADP
ma-267	30	15	this	this	DET
ma-267	30	16	system	system	NOUN
ma-267	30	17	denoted	denote	VERB
ma-267	30	18	by	by	ADP
ma-267	30	19	ϕµ,λ	ϕµ,λ	PUNCT
ma-267	30	20	with	with	ADP
ma-267	30	21	(	(	PUNCT
ma-267	30	22	µ	µ	NUM
ma-267	30	23	,	,	PUNCT
ma-267	30	24	λ	λ	NOUN
ma-267	30	25	)	)	PUNCT
ma-267	30	26	∈	∈	PROPN
ma-267	30	27	c2	c2	PROPN
ma-267	30	28	and	and	CCONJ
ma-267	30	29	the	the	DET
ma-267	30	30	riemann	riemann	PROPN
ma-267	30	31	-	-	PUNCT
ma-267	30	32	liouville	liouville	NOUN
ma-267	30	33	operator	operator	NOUN
ma-267	30	34	(	(	PUNCT
ma-267	30	35	1.1	1.1	NUM
ma-267	30	36	)	)	PUNCT
ma-267	30	37	as	as	SCONJ
ma-267	30	38	follows	follow	VERB
ma-267	30	39	:	:	PUNCT
ma-267	30	40	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	30	41	,	,	PUNCT
ma-267	30	42	t	t	PROPN
ma-267	30	43	)	)	PUNCT
ma-267	30	44	=	=	SYM
ma-267	31	1	rα(cos(µ.	rα(cos(µ.	NUM
ma-267	31	2	)	)	PUNCT
ma-267	31	3	exp(−iλ·))(x	exp(−iλ·))(x	PROPN
ma-267	31	4	,	,	PUNCT
ma-267	31	5	t	t	PROPN
ma-267	31	6	)	)	PUNCT
ma-267	31	7	.	.	PUNCT
ma-267	32	1	(	(	PUNCT
ma-267	32	2	1.2	1.2	NUM
ma-267	32	3	)	)	PUNCT
ma-267	32	4	wavelet	wavelet	NOUN
ma-267	32	5	analysis	analysis	NOUN
ma-267	32	6	has	have	AUX
ma-267	32	7	attracted	attract	VERB
ma-267	32	8	attention	attention	NOUN
ma-267	32	9	for	for	ADP
ma-267	32	10	its	its	PRON
ma-267	32	11	ability	ability	NOUN
ma-267	32	12	to	to	PART
ma-267	32	13	analyse	analyse	VERB
ma-267	32	14	rapidly	rapidly	ADV
ma-267	32	15	changing	change	VERB
ma-267	32	16	transient	transient	ADJ
ma-267	32	17	signals	signal	NOUN
ma-267	32	18	,	,	PUNCT
ma-267	32	19	any	any	DET
ma-267	32	20	application	application	NOUN
ma-267	32	21	using	use	VERB
ma-267	32	22	the	the	DET
ma-267	32	23	fourier	fourier	NOUN
ma-267	32	24	like	like	ADP
ma-267	32	25	transform	transform	NOUN
ma-267	32	26	can	can	AUX
ma-267	32	27	be	be	AUX
ma-267	32	28	formulated	formulate	VERB
ma-267	32	29	using	use	VERB
ma-267	32	30	wavelets	wavelet	NOUN
ma-267	32	31	to	to	PART
ma-267	32	32	provide	provide	VERB
ma-267	32	33	moretime	moretime	NOUN
ma-267	32	34	and	and	CCONJ
ma-267	32	35	frequency	frequency	VERB
ma-267	32	36	information.the	information.the	DET
ma-267	32	37	reason	reason	NOUN
ma-267	32	38	for	for	ADP
ma-267	32	39	the	the	DET
ma-267	32	40	extension	extension	NOUN
ma-267	32	41	from	from	ADP
ma-267	32	42	one	one	NUM
ma-267	32	43	wavelet	wavelet	NOUN
ma-267	32	44	to	to	ADP
ma-267	32	45	two	two	NUM
ma-267	32	46	-	-	PUNCT
ma-267	32	47	wavelet	wavelet	NOUN
ma-267	32	48	comes	come	VERB
ma-267	32	49	from	from	ADP
ma-267	32	50	the	the	DET
ma-267	32	51	extra	extra	ADJ
ma-267	32	52	degree	degree	NOUN
ma-267	32	53	offlexibility	offlexibility	NOUN
ma-267	32	54	in	in	ADP
ma-267	32	55	signal	signal	ADJ
ma-267	32	56	analysis	analysis	NOUN
ma-267	32	57	and	and	CCONJ
ma-267	32	58	imaging	imaging	NOUN
ma-267	32	59	when	when	SCONJ
ma-267	32	60	the	the	DET
ma-267	32	61	localization	localization	NOUN
ma-267	32	62	operators	operator	NOUN
ma-267	32	63	are	be	AUX
ma-267	32	64	used	use	VERB
ma-267	32	65	as	as	ADP
ma-267	32	66	time	time	NOUN
ma-267	32	67	-	-	PUNCT
ma-267	32	68	varingfilters	varingfilter	NOUN
ma-267	32	69	.	.	PUNCT
ma-267	33	1	this	this	DET
ma-267	33	2	paper	paper	NOUN
ma-267	33	3	is	be	AUX
ma-267	33	4	an	an	DET
ma-267	33	5	attempt	attempt	NOUN
ma-267	33	6	to	to	PART
ma-267	33	7	fill	fill	VERB
ma-267	33	8	this	this	DET
ma-267	33	9	gap	gap	NOUN
ma-267	33	10	by	by	ADP
ma-267	33	11	extending	extend	VERB
ma-267	33	12	one	one	NUM
ma-267	33	13	wavelet	wavelet	NOUN
ma-267	33	14	to	to	ADP
ma-267	33	15	two	two	NUM
ma-267	33	16	wavelets	wavelet	NOUN
ma-267	33	17	in	in	ADP
ma-267	33	18	theriemann	theriemann	NOUN
ma-267	33	19	-	-	PUNCT
ma-267	33	20	liouville	liouville	NOUN
ma-267	33	21	setting	setting	NOUN
ma-267	33	22	.	.	PUNCT
ma-267	34	1	the	the	DET
ma-267	34	2	remainder	remainder	NOUN
ma-267	34	3	of	of	ADP
ma-267	34	4	this	this	DET
ma-267	34	5	paper	paper	NOUN
ma-267	34	6	is	be	AUX
ma-267	34	7	arranged	arrange	VERB
ma-267	34	8	as	as	SCONJ
ma-267	34	9	follows	follow	VERB
ma-267	34	10	,	,	PUNCT
ma-267	34	11	in	in	ADP
ma-267	34	12	section	section	NOUN
ma-267	34	13	2	2	NUM
ma-267	34	14	werecall	werecall	NOUN
ma-267	34	15	the	the	DET
ma-267	34	16	main	main	ADJ
ma-267	34	17	results	result	NOUN
ma-267	34	18	concerning	concern	VERB
ma-267	34	19	the	the	DET
ma-267	34	20	harmonic	harmonic	ADJ
ma-267	34	21	analysis	analysis	NOUN
ma-267	34	22	associated	associate	VERB
ma-267	34	23	with	with	ADP
ma-267	34	24	the	the	DET
ma-267	34	25	riemann	riemann	PROPN
ma-267	34	26	-	-	PUNCT
ma-267	34	27	liouvilletransform	liouvilletransform	NOUN
ma-267	34	28	,	,	PUNCT
ma-267	34	29	in	in	ADP
ma-267	34	30	section	section	NOUN
ma-267	34	31	3	3	NUM
ma-267	34	32	we	we	PRON
ma-267	34	33	introduce	introduce	VERB
ma-267	34	34	the	the	DET
ma-267	34	35	notion	notion	NOUN
ma-267	34	36	of	of	ADP
ma-267	34	37	riemann	riemann	PROPN
ma-267	34	38	-	-	PUNCT
ma-267	34	39	liouville	liouville	VERB
ma-267	34	40	two	two	NUM
ma-267	34	41	-	-	PUNCT
ma-267	34	42	wavelet	wavelet	NOUN
ma-267	35	1	and	and	CCONJ
ma-267	35	2	we	we	PRON
ma-267	35	3	give	give	VERB
ma-267	35	4	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	35	5	eur	eur	PROPN
ma-267	35	6	.	.	PUNCT
ma-267	36	1	j.	j.	PROPN
ma-267	36	2	math	math	PROPN
ma-267	36	3	.	.	PUNCT
ma-267	37	1	anal	anal	PROPN
ma-267	37	2	.	.	PUNCT
ma-267	38	1	10.28924	10.28924	NUM
ma-267	38	2	/	/	SYM
ma-267	38	3	ada	ada	PROPN
ma-267	38	4	/	/	SYM
ma-267	38	5	ma.4.20	ma.4.20	PROPN
ma-267	38	6	3a	3a	NUM
ma-267	38	7	generalized	generalize	VERB
ma-267	38	8	version	version	NOUN
ma-267	38	9	of	of	ADP
ma-267	38	10	parseval	parseval	NOUN
ma-267	38	11	’s	’s	PART
ma-267	38	12	,	,	PUNCT
ma-267	38	13	plancherel	plancherel	PROPN
ma-267	38	14	’s	’s	PART
ma-267	38	15	,	,	PUNCT
ma-267	38	16	inversion	inversion	NOUN
ma-267	38	17	and	and	CCONJ
ma-267	38	18	calderon	calderon	NOUN
ma-267	38	19	’s	’s	PART
ma-267	38	20	reproducing	reproducing	NOUN
ma-267	38	21	formulasrelated	formulasrelate	VERB
ma-267	38	22	to	to	ADP
ma-267	38	23	this	this	DET
ma-267	38	24	transform	transform	NOUN
ma-267	38	25	,	,	PUNCT
ma-267	38	26	the	the	DET
ma-267	38	27	last	last	ADJ
ma-267	38	28	section	section	NOUN
ma-267	38	29	is	be	AUX
ma-267	38	30	devoted	devote	VERB
ma-267	38	31	to	to	PART
ma-267	38	32	give	give	VERB
ma-267	38	33	an	an	DET
ma-267	38	34	integral	integral	ADJ
ma-267	38	35	representation	representation	NOUN
ma-267	38	36	and	and	CCONJ
ma-267	38	37	bestestimates	bestestimate	NOUN
ma-267	38	38	of	of	ADP
ma-267	38	39	extremal	extremal	ADJ
ma-267	38	40	functions	function	NOUN
ma-267	38	41	related	relate	VERB
ma-267	38	42	to	to	ADP
ma-267	38	43	the	the	DET
ma-267	38	44	riemann	riemann	PROPN
ma-267	38	45	-	-	PUNCT
ma-267	38	46	liouville	liouville	VERB
ma-267	38	47	wavelet	wavelet	NOUN
ma-267	38	48	transform	transform	NOUN
ma-267	38	49	on	on	ADP
ma-267	38	50	weightedsobolev	weightedsobolev	ADJ
ma-267	38	51	spaces	space	NOUN
ma-267	38	52	.	.	PUNCT
ma-267	39	1	2	2	X
ma-267	39	2	.	.	X
ma-267	39	3	harmonic	harmonic	ADJ
ma-267	39	4	analysis	analysis	NOUN
ma-267	39	5	associated	associate	VERB
ma-267	39	6	with	with	ADP
ma-267	39	7	the	the	DET
ma-267	39	8	riemann	riemann	PROPN
ma-267	39	9	-	-	PUNCT
ma-267	39	10	liouville	liouville	NOUN
ma-267	39	11	operator	operator	NOUN
ma-267	39	12	in	in	ADP
ma-267	39	13	this	this	DET
ma-267	39	14	section	section	NOUN
ma-267	39	15	we	we	PRON
ma-267	39	16	set	set	VERB
ma-267	39	17	some	some	DET
ma-267	39	18	notations	notation	NOUN
ma-267	39	19	and	and	CCONJ
ma-267	39	20	we	we	PRON
ma-267	39	21	recall	recall	VERB
ma-267	39	22	some	some	DET
ma-267	39	23	results	result	NOUN
ma-267	39	24	in	in	ADP
ma-267	39	25	harmonic	harmonic	ADJ
ma-267	39	26	analysis	analysis	NOUN
ma-267	39	27	relatedto	relatedto	ADJ
ma-267	39	28	the	the	DET
ma-267	39	29	riemann	riemann	PROPN
ma-267	39	30	-	-	PUNCT
ma-267	39	31	liouville	liouville	NOUN
ma-267	39	32	operator	operator	NOUN
ma-267	39	33	(	(	PUNCT
ma-267	39	34	1.1	1.1	NUM
ma-267	39	35	)	)	PUNCT
ma-267	39	36	,	,	PUNCT
ma-267	39	37	for	for	ADP
ma-267	39	38	more	more	ADJ
ma-267	39	39	details	detail	NOUN
ma-267	39	40	we	we	PRON
ma-267	39	41	refer	refer	VERB
ma-267	39	42	the	the	DET
ma-267	39	43	reader	reader	NOUN
ma-267	39	44	to	to	ADP
ma-267	39	45	[	[	X
ma-267	39	46	1–4	1–4	PROPN
ma-267	39	47	,	,	PUNCT
ma-267	39	48	15	15	NUM
ma-267	39	49	]	]	PUNCT
ma-267	39	50	.	.	PUNCT
ma-267	40	1	in	in	ADP
ma-267	40	2	thefollowing	thefollowing	NOUN
ma-267	40	3	we	we	PRON
ma-267	40	4	denote	denote	VERB
ma-267	40	5	by•	by•	NOUN
ma-267	40	6	k	k	PROPN
ma-267	40	7	:	:	PUNCT
ma-267	41	1	=]	=]	PROPN
ma-267	41	2	0,+∞[×r	0,+∞[×r	PROPN
ma-267	41	3	equipped	equip	VERB
ma-267	41	4	with	with	ADP
ma-267	41	5	the	the	DET
ma-267	41	6	weighted	weight	VERB
ma-267	41	7	lebesgue	lebesgue	NOUN
ma-267	41	8	measure	measure	NOUN
ma-267	41	9	µα	µα	AUX
ma-267	41	10	given	give	VERB
ma-267	41	11	by	by	ADP
ma-267	41	12	dµα(x	dµα(x	PROPN
ma-267	41	13	,	,	PUNCT
ma-267	41	14	t	t	PROPN
ma-267	41	15	)	)	PUNCT
ma-267	41	16	:	:	PUNCT
ma-267	42	1	=	=	NOUN
ma-267	42	2	x2α+1	x2α+1	PUNCT
ma-267	43	1	2αγ(α+	2αγ(α+	NUM
ma-267	43	2	1	1	NUM
ma-267	43	3	)	)	PUNCT
ma-267	43	4	√	√	PROPN
ma-267	43	5	2π	2π	PROPN
ma-267	43	6	dx	dx	PROPN
ma-267	43	7	⊗	⊗	PROPN
ma-267	43	8	dt	dt	PROPN
ma-267	43	9	,	,	PUNCT
ma-267	43	10	,	,	PUNCT
ma-267	43	11	α	α	PRON
ma-267	43	12	≥	≥	NOUN
ma-267	43	13	0	0	NUM
ma-267	43	14	,	,	PUNCT
ma-267	43	15	where	where	SCONJ
ma-267	43	16	γ	γ	PROPN
ma-267	43	17	is	be	AUX
ma-267	43	18	the	the	DET
ma-267	43	19	gamma	gamma	NOUN
ma-267	43	20	function.•	function.•	PROPN
ma-267	43	21	lpα(k	lpα(k	PROPN
ma-267	43	22	)	)	PUNCT
ma-267	43	23	,	,	PUNCT
ma-267	43	24	1	1	NUM
ma-267	43	25	≤	≤	NOUN
ma-267	43	26	p	p	NOUN
ma-267	43	27	≤	≤	NUM
ma-267	43	28	∞	∞	PROPN
ma-267	43	29	,	,	PUNCT
ma-267	43	30	the	the	DET
ma-267	43	31	space	space	NOUN
ma-267	43	32	of	of	ADP
ma-267	43	33	measurable	measurable	ADJ
ma-267	43	34	functions	function	NOUN
ma-267	43	35	on	on	ADP
ma-267	43	36	k	k	NOUN
ma-267	43	37	,	,	PUNCT
ma-267	43	38	satisfying	satisfy	VERB
ma-267	43	39	‖f	‖f	ADP
ma-267	43	40	‖p,µα	‖p,µα	NUM
ma-267	43	41	:	:	PUNCT
ma-267	43	42	=	=	SYM
ma-267	43	43			PUNCT
ma-267	43	44	(	(	PUNCT
ma-267	43	45	∫	∫	PROPN
ma-267	43	46	k	k	PROPN
ma-267	43	47	|f	|f	PROPN
ma-267	43	48	(	(	PUNCT
ma-267	43	49	x	x	PROPN
ma-267	43	50	,	,	PUNCT
ma-267	43	51	t)|pdµα(x	t)|pdµα(x	NOUN
ma-267	43	52	,	,	PUNCT
ma-267	43	53	t	t	PROPN
ma-267	43	54	)	)	PUNCT
ma-267	43	55	)	)	PUNCT
ma-267	43	56	1	1	X
ma-267	43	57	/	/	SYM
ma-267	43	58	p	p	X
ma-267	43	59	<	<	X
ma-267	43	60	∞	∞	PROPN
ma-267	43	61	,	,	PUNCT
ma-267	43	62	if	if	SCONJ
ma-267	43	63	p	p	X
ma-267	43	64	∈	∈	PROPN
ma-267	43	65	[	[	X
ma-267	43	66	1,+∞	1,+∞	NUM
ma-267	43	67	[	[	X
ma-267	43	68	,	,	PUNCT
ma-267	43	69	ess	ess	PROPN
ma-267	43	70	sup(x	sup(x	PROPN
ma-267	43	71	,	,	PUNCT
ma-267	43	72	t)∈k	t)∈k	NUM
ma-267	43	73	|f	|f	PROPN
ma-267	43	74	(	(	PUNCT
ma-267	43	75	x	x	X
ma-267	43	76	,	,	PUNCT
ma-267	43	77	t)|	t)|	NOUN
ma-267	43	78	<	<	X
ma-267	43	79	∞	∞	PROPN
ma-267	43	80	,	,	PUNCT
ma-267	43	81	if	if	SCONJ
ma-267	43	82	p	p	X
ma-267	43	83	=	=	SYM
ma-267	44	1	+	+	NOUN
ma-267	44	2	∞.	∞.	PROPN
ma-267	44	3	•	•	NOUN
ma-267	44	4	k̂	k̂	NOUN
ma-267	44	5	:	:	PUNCT
ma-267	45	1	=	=	PUNCT
ma-267	46	1	[	[	X
ma-267	46	2	0,+∞[×r	0,+∞[×r	NUM
ma-267	46	3	∪	∪	ADJ
ma-267	46	4	{	{	PUNCT
ma-267	46	5	(	(	PUNCT
ma-267	46	6	i	i	NOUN
ma-267	46	7	s	s	PROPN
ma-267	46	8	,	,	PUNCT
ma-267	46	9	y	y	PROPN
ma-267	46	10	)	)	PUNCT
ma-267	46	11	;	;	PUNCT
ma-267	46	12	(	(	PUNCT
ma-267	46	13	s	s	X
ma-267	46	14	,	,	PUNCT
ma-267	46	15	y	y	NOUN
ma-267	46	16	)	)	PUNCT
ma-267	46	17	∈	∈	PROPN
ma-267	47	1	[	[	X
ma-267	47	2	0,+∞[×r	0,+∞[×r	NUM
ma-267	47	3	;	;	PUNCT
ma-267	47	4	s	s	NUM
ma-267	47	5	6	6	NUM
ma-267	47	6	|y	|y	NOUN
ma-267	47	7	|}.•	|}.•	NOUN
ma-267	47	8	bk̂	bk̂	PROPN
ma-267	47	9	the	the	DET
ma-267	47	10	σ	σ	PROPN
ma-267	47	11	-	-	PUNCT
ma-267	47	12	algebra	algebra	NOUN
ma-267	47	13	defined	define	VERB
ma-267	47	14	on	on	ADP
ma-267	47	15	k̂	k̂	NUM
ma-267	47	16	by	by	ADP
ma-267	47	17	bk̂	bk̂	PROPN
ma-267	47	18	=	=	SYM
ma-267	47	19	{	{	PUNCT
ma-267	47	20	θ−1(b	θ−1(b	PROPN
ma-267	47	21	)	)	PUNCT
ma-267	47	22	,	,	PUNCT
ma-267	47	23	b	b	PROPN
ma-267	47	24	∈	∈	PROPN
ma-267	47	25	b([0,+∞[×r	b([0,+∞[×r	PROPN
ma-267	47	26	)	)	PUNCT
ma-267	47	27	}	}	PUNCT
ma-267	47	28	,	,	PUNCT
ma-267	47	29	where	where	SCONJ
ma-267	47	30	θ	θ	PROPN
ma-267	47	31	is	be	AUX
ma-267	47	32	the	the	DET
ma-267	47	33	bijective	bijective	ADJ
ma-267	47	34	function	function	NOUN
ma-267	47	35	given	give	VERB
ma-267	47	36	by	by	ADP
ma-267	47	37	θ(s	θ(s	PROPN
ma-267	47	38	,	,	PUNCT
ma-267	47	39	y	y	NOUN
ma-267	47	40	)	)	PUNCT
ma-267	47	41	=	=	SYM
ma-267	47	42	(	(	PUNCT
ma-267	47	43	√	√	NUM
ma-267	47	44	s2	s2	NOUN
ma-267	47	45	+	+	CCONJ
ma-267	47	46	y2	y2	PROPN
ma-267	47	47	,	,	PUNCT
ma-267	47	48	y	y	PROPN
ma-267	47	49	)	)	PUNCT
ma-267	47	50	.	.	PUNCT
ma-267	48	1	•	•	NUM
ma-267	48	2	dγα	dγα	VERB
ma-267	48	3	the	the	DET
ma-267	48	4	measure	measure	NOUN
ma-267	48	5	defined	define	VERB
ma-267	48	6	on	on	ADP
ma-267	48	7	bk̂	bk̂	PROPN
ma-267	48	8	by	by	ADP
ma-267	48	9	∀a	∀a	NOUN
ma-267	48	10	∈	∈	PROPN
ma-267	48	11	bk̂	bk̂	PROPN
ma-267	48	12	;	;	PUNCT
ma-267	48	13	γα(a	γα(a	X
ma-267	48	14	)	)	PUNCT
ma-267	48	15	=	=	SYM
ma-267	48	16	µα(θ(a	µα(θ(a	NOUN
ma-267	48	17	)	)	PUNCT
ma-267	48	18	)	)	PUNCT
ma-267	48	19	.	.	PUNCT
ma-267	49	1	and	and	CCONJ
ma-267	49	2	for	for	ADP
ma-267	49	3	all	all	DET
ma-267	49	4	non	non	ADJ
ma-267	49	5	-	-	ADJ
ma-267	49	6	negative	negative	ADJ
ma-267	49	7	measurable	measurable	ADJ
ma-267	49	8	function	function	NOUN
ma-267	49	9	on	on	ADP
ma-267	49	10	k̂	k̂	NOUN
ma-267	49	11	we	we	PRON
ma-267	49	12	have∫	have∫	VERB
ma-267	49	13	k̂	k̂	PROPN
ma-267	50	1	g(µ	g(µ	PROPN
ma-267	50	2	,	,	PUNCT
ma-267	50	3	λ)dγα(µ	λ)dγα(µ	PROPN
ma-267	50	4	,	,	PUNCT
ma-267	50	5	λ	λ	NOUN
ma-267	50	6	)	)	PUNCT
ma-267	50	7	=	=	SYM
ma-267	50	8	1	1	NUM
ma-267	50	9	2αγ(α+	2αγ(α+	NUM
ma-267	50	10	1	1	NUM
ma-267	50	11	)	)	PUNCT
ma-267	50	12	√	√	PROPN
ma-267	50	13	2π	2π	PROPN
ma-267	50	14	(	(	PUNCT
ma-267	50	15	∫	∫	PROPN
ma-267	51	1	+	+	NOUN
ma-267	51	2	∞	∞	PROPN
ma-267	51	3	0	0	NUM
ma-267	51	4	∫	∫	PROPN
ma-267	51	5	r	r	PROPN
ma-267	51	6	g(µ	g(µ	PROPN
ma-267	51	7	,	,	PUNCT
ma-267	51	8	λ	λ	PROPN
ma-267	51	9	)	)	PUNCT
ma-267	51	10	(	(	PUNCT
ma-267	51	11	µ2	µ2	PROPN
ma-267	51	12	+	+	CCONJ
ma-267	51	13	λ2	λ2	NOUN
ma-267	51	14	)	)	PUNCT
ma-267	51	15	α	α	NOUN
ma-267	51	16	µdµdλ	µdµdλ	NOUN
ma-267	51	17	+	+	CCONJ
ma-267	51	18	∫	∫	PROPN
ma-267	51	19	r	r	NOUN
ma-267	51	20	∫	∫	PROPN
ma-267	51	21	|λ|	|λ|	NOUN
ma-267	51	22	0	0	NUM
ma-267	51	23	g(iµ	g(iµ	NOUN
ma-267	51	24	,	,	PUNCT
ma-267	51	25	λ	λ	X
ma-267	51	26	)	)	PUNCT
ma-267	51	27	(	(	PUNCT
ma-267	51	28	λ2	λ2	NOUN
ma-267	51	29	−	−	PROPN
ma-267	51	30	µ2	µ2	PROPN
ma-267	51	31	)	)	PUNCT
ma-267	51	32	α	α	NOUN
ma-267	51	33	µdµdλ	µdµdλ	NOUN
ma-267	51	34	)	)	PUNCT
ma-267	51	35	.	.	PUNCT
ma-267	52	1	(	(	PUNCT
ma-267	52	2	2.1	2.1	NUM
ma-267	52	3	)	)	PUNCT
ma-267	52	4	•	•	NUM
ma-267	52	5	lpα(k̂	lpα(k̂	NOUN
ma-267	52	6	)	)	PUNCT
ma-267	52	7	with	with	ADP
ma-267	52	8	p	p	PROPN
ma-267	52	9	∈	∈	PROPN
ma-267	53	1	[	[	X
ma-267	53	2	1,+∞	1,+∞	NUM
ma-267	53	3	]	]	X
ma-267	53	4	the	the	DET
ma-267	53	5	space	space	NOUN
ma-267	53	6	of	of	ADP
ma-267	53	7	measurable	measurable	ADJ
ma-267	53	8	functions	function	NOUN
ma-267	53	9	on	on	ADP
ma-267	53	10	k̂	k̂	PROPN
ma-267	53	11	satisfying	satisfy	VERB
ma-267	53	12	‖g‖p	‖g‖p	NOUN
ma-267	53	13	,	,	PUNCT
ma-267	53	14	γα	γα	X
ma-267	53	15	:	:	PUNCT
ma-267	53	16	=	=	PUNCT
ma-267	53	17			PUNCT
ma-267	53	18	(	(	PUNCT
ma-267	53	19	∫	∫	PROPN
ma-267	53	20	k̂	k̂	PROPN
ma-267	53	21	|g(λ	|g(λ	PROPN
ma-267	53	22	,	,	PUNCT
ma-267	53	23	m)|pdγα(λ	m)|pdγα(λ	NOUN
ma-267	53	24	,	,	PUNCT
ma-267	53	25	m	m	PROPN
ma-267	53	26	)	)	PUNCT
ma-267	53	27	)	)	PUNCT
ma-267	54	1	1	1	NUM
ma-267	54	2	p	p	NOUN
ma-267	54	3	<	<	X
ma-267	54	4	∞	∞	NOUN
ma-267	54	5	if	if	SCONJ
ma-267	54	6	p	p	PROPN
ma-267	54	7	∈	∈	PROPN
ma-267	55	1	[	[	X
ma-267	55	2	1,+∞[,ess	1,+∞[,ess	PROPN
ma-267	55	3	sup(λ	sup(λ	PROPN
ma-267	55	4	,	,	PUNCT
ma-267	55	5	m)∈k̂	m)∈k̂	PROPN
ma-267	55	6	|g(λ	|g(λ	PROPN
ma-267	55	7	,	,	PUNCT
ma-267	55	8	m)|	m)|	NOUN
ma-267	55	9	<	<	X
ma-267	55	10	∞	∞	PROPN
ma-267	55	11	,	,	PUNCT
ma-267	55	12	if	if	SCONJ
ma-267	55	13	p	p	X
ma-267	55	14	=	=	SYM
ma-267	55	15	+	+	NUM
ma-267	55	16	∞.	∞.	PROPN
ma-267	55	17	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	55	18	eur	eur	PROPN
ma-267	55	19	.	.	PUNCT
ma-267	56	1	j.	j.	PROPN
ma-267	56	2	math	math	PROPN
ma-267	56	3	.	.	PUNCT
ma-267	57	1	anal	anal	PROPN
ma-267	57	2	.	.	PUNCT
ma-267	58	1	10.28924	10.28924	NUM
ma-267	58	2	/	/	SYM
ma-267	58	3	ada	ada	PROPN
ma-267	58	4	/	/	SYM
ma-267	58	5	ma.4.20	ma.4.20	NOUN
ma-267	58	6	42.1	42.1	NUM
ma-267	58	7	.	.	PUNCT
ma-267	59	1	the	the	DET
ma-267	59	2	eigenfunctions	eigenfunction	NOUN
ma-267	59	3	of	of	ADP
ma-267	59	4	the	the	DET
ma-267	59	5	partial	partial	ADJ
ma-267	59	6	differential	differential	NOUN
ma-267	59	7	operators	operator	NOUN
ma-267	59	8	∆1	∆1	PROPN
ma-267	59	9	and	and	CCONJ
ma-267	59	10	∆2	∆2	PROPN
ma-267	59	11	.	.	PUNCT
ma-267	60	1	for	for	ADP
ma-267	60	2	(	(	PUNCT
ma-267	60	3	µ	µ	X
ma-267	60	4	,	,	PUNCT
ma-267	60	5	λ	λ	NOUN
ma-267	60	6	)	)	PUNCT
ma-267	60	7	∈	∈	PROPN
ma-267	60	8	k̂	k̂	NOUN
ma-267	60	9	weconsider	weconsider	NOUN
ma-267	60	10	the	the	DET
ma-267	60	11	following	follow	VERB
ma-267	60	12	cauchy	cauchy	PROPN
ma-267	60	13	problem	problem	NOUN
ma-267	60	14	(	(	PUNCT
ma-267	60	15	s	s	NOUN
ma-267	60	16	)	)	PUNCT
ma-267	60	17	:	:	PUNCT
ma-267	60	18			PUNCT
ma-267	60	19	∆1(u)(x	∆1(u)(x	PROPN
ma-267	60	20	,	,	PUNCT
ma-267	60	21	t	t	PROPN
ma-267	60	22	)	)	PUNCT
ma-267	60	23	=	=	SYM
ma-267	61	1	−λ2u(x	−λ2u(x	PROPN
ma-267	61	2	,	,	PUNCT
ma-267	61	3	t	t	PROPN
ma-267	61	4	)	)	PUNCT
ma-267	61	5	,	,	PUNCT
ma-267	61	6	∆2(u)(x	∆2(u)(x	PROPN
ma-267	61	7	,	,	PUNCT
ma-267	61	8	t	t	PROPN
ma-267	61	9	)	)	PUNCT
ma-267	61	10	=	=	SYM
ma-267	62	1	−µ2u(x	−µ2u(x	PROPN
ma-267	62	2	,	,	PUNCT
ma-267	62	3	t	t	PROPN
ma-267	62	4	)	)	PUNCT
ma-267	62	5	u(0	u(0	PROPN
ma-267	62	6	,	,	PUNCT
ma-267	62	7	0	0	NUM
ma-267	62	8	)	)	PUNCT
ma-267	62	9	=	=	SYM
ma-267	62	10	1	1	NUM
ma-267	62	11	;	;	PUNCT
ma-267	62	12	∂u∂x	∂u∂x	NOUN
ma-267	62	13	(	(	PUNCT
ma-267	62	14	0	0	NUM
ma-267	62	15	,	,	PUNCT
ma-267	62	16	t	t	PROPN
ma-267	62	17	)	)	PUNCT
ma-267	62	18	=	=	VERB
ma-267	63	1	0.from	0.from	NUM
ma-267	64	1	[	[	X
ma-267	64	2	3	3	NUM
ma-267	64	3	]	]	PUNCT
ma-267	64	4	,	,	PUNCT
ma-267	64	5	the	the	DET
ma-267	64	6	cauchy	cauchy	ADJ
ma-267	64	7	problem	problem	NOUN
ma-267	64	8	(	(	PUNCT
ma-267	64	9	s	s	X
ma-267	64	10	)	)	PUNCT
ma-267	64	11	admits	admit	VERB
ma-267	64	12	a	a	DET
ma-267	64	13	unique	unique	ADJ
ma-267	64	14	solution	solution	NOUN
ma-267	64	15	ϕµ,λ	ϕµ,λ	PUNCT
ma-267	64	16	given	give	VERB
ma-267	64	17	by	by	ADP
ma-267	64	18	:	:	PUNCT
ma-267	64	19	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	64	20	,	,	PUNCT
ma-267	64	21	t	t	PROPN
ma-267	64	22	)	)	PUNCT
ma-267	65	1	=	=	SYM
ma-267	66	1	jα	jα	NOUN
ma-267	66	2	(	(	PUNCT
ma-267	66	3	x	x	SYM
ma-267	66	4	√	√	PUNCT
ma-267	66	5	µ2	µ2	PROPN
ma-267	66	6	+	+	CCONJ
ma-267	66	7	λ2	λ2	NOUN
ma-267	66	8	)	)	PUNCT
ma-267	66	9	exp(−iλt	exp(−iλt	NUM
ma-267	66	10	)	)	PUNCT
ma-267	66	11	,	,	PUNCT
ma-267	66	12	(	(	PUNCT
ma-267	66	13	2.2	2.2	NUM
ma-267	66	14	)	)	PUNCT
ma-267	66	15	where	where	SCONJ
ma-267	66	16	jα	jα	PROPN
ma-267	66	17	is	be	AUX
ma-267	66	18	the	the	DET
ma-267	66	19	spherical	spherical	ADJ
ma-267	66	20	bessel	bessel	NOUN
ma-267	66	21	function	function	NOUN
ma-267	66	22	of	of	ADP
ma-267	66	23	index	index	NOUN
ma-267	66	24	α	α	NOUN
ma-267	66	25	see	see	VERB
ma-267	66	26	[	[	X
ma-267	66	27	16	16	NUM
ma-267	66	28	]	]	PUNCT
ma-267	66	29	for	for	ADP
ma-267	66	30	more	more	ADJ
ma-267	66	31	information	information	NOUN
ma-267	66	32	about	about	ADP
ma-267	66	33	the	the	DET
ma-267	66	34	besselfunctions	besselfunction	NOUN
ma-267	66	35	.	.	PUNCT
ma-267	67	1	the	the	DET
ma-267	67	2	function	function	NOUN
ma-267	67	3	ϕµ,λ	ϕµ,λ	PUNCT
ma-267	67	4	is	be	AUX
ma-267	67	5	infinitely	infinitely	ADV
ma-267	67	6	differentiable	differentiable	ADJ
ma-267	67	7	on	on	ADP
ma-267	67	8	r2	r2	PROPN
ma-267	67	9	,	,	PUNCT
ma-267	67	10	even	even	ADV
ma-267	67	11	with	with	ADP
ma-267	67	12	respect	respect	NOUN
ma-267	67	13	to	to	ADP
ma-267	67	14	each	each	DET
ma-267	67	15	variableand	variableand	NOUN
ma-267	67	16	we	we	PRON
ma-267	67	17	have	have	VERB
ma-267	67	18	the	the	DET
ma-267	67	19	following	follow	VERB
ma-267	67	20	important	important	ADJ
ma-267	67	21	result	result	NOUN
ma-267	67	22	:	:	PUNCT
ma-267	67	23	sup	sup	NOUN
ma-267	67	24	(	(	PUNCT
ma-267	67	25	x	x	NOUN
ma-267	67	26	,	,	PUNCT
ma-267	67	27	t)∈r2	t)∈r2	X
ma-267	67	28	∣∣ϕµ,λ(x	∣∣ϕµ,λ(x	PROPN
ma-267	67	29	,	,	PUNCT
ma-267	67	30	t	t	PROPN
ma-267	67	31	)	)	PUNCT
ma-267	67	32	∣∣	∣∣	X
ma-267	68	1	=	=	SYM
ma-267	68	2	1	1	X
ma-267	68	3	.	.	X
ma-267	69	1	for	for	ADP
ma-267	69	2	(	(	PUNCT
ma-267	69	3	µ	µ	X
ma-267	69	4	,	,	PUNCT
ma-267	69	5	λ	λ	NOUN
ma-267	69	6	)	)	PUNCT
ma-267	69	7	∈	∈	PROPN
ma-267	69	8	k̂.	k̂.	NOUN
ma-267	69	9	(	(	PUNCT
ma-267	69	10	2.3	2.3	NUM
ma-267	69	11	)	)	PUNCT
ma-267	69	12	2.2	2.2	NUM
ma-267	69	13	.	.	PUNCT
ma-267	70	1	the	the	DET
ma-267	70	2	riemann	riemann	PROPN
ma-267	70	3	-	-	PUNCT
ma-267	70	4	liouville	liouville	NOUN
ma-267	70	5	transform	transform	NOUN
ma-267	70	6	.	.	PUNCT
ma-267	71	1	definition	definition	NOUN
ma-267	71	2	2.1	2.1	NUM
ma-267	71	3	.	.	PUNCT
ma-267	72	1	the	the	DET
ma-267	72	2	generalized	generalized	ADJ
ma-267	72	3	fourier	fourier	NOUN
ma-267	72	4	transform	transform	NOUN
ma-267	72	5	fα	fα	SCONJ
ma-267	72	6	associated	associate	VERB
ma-267	72	7	with	with	ADP
ma-267	72	8	the	the	DET
ma-267	72	9	riemann	riemann	PROPN
ma-267	72	10	-	-	PUNCT
ma-267	72	11	liouville	liouville	VERB
ma-267	72	12	op	op	NOUN
ma-267	72	13	-	-	PUNCT
ma-267	72	14	erator	erator	NOUN
ma-267	72	15	(	(	PUNCT
ma-267	72	16	1.1	1.1	NUM
ma-267	72	17	)	)	PUNCT
ma-267	72	18	is	be	AUX
ma-267	72	19	defined	define	VERB
ma-267	72	20	on	on	ADP
ma-267	72	21	l1α(k	l1α(k	NOUN
ma-267	72	22	)	)	PUNCT
ma-267	72	23	by	by	ADP
ma-267	72	24	fα(f	fα(f	NOUN
ma-267	72	25	)	)	PUNCT
ma-267	72	26	(	(	PUNCT
ma-267	72	27	µ	µ	X
ma-267	72	28	,	,	PUNCT
ma-267	72	29	λ	λ	NOUN
ma-267	72	30	)	)	PUNCT
ma-267	72	31	=	=	SYM
ma-267	73	1	∫	∫	PROPN
ma-267	73	2	k	k	PROPN
ma-267	73	3	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	73	4	,	,	PUNCT
ma-267	73	5	t)f	t)f	PUNCT
ma-267	73	6	(	(	PUNCT
ma-267	73	7	x	x	X
ma-267	73	8	,	,	PUNCT
ma-267	73	9	t)dµα(x	t)dµα(x	NUM
ma-267	73	10	,	,	PUNCT
ma-267	73	11	t	t	PROPN
ma-267	73	12	)	)	PUNCT
ma-267	73	13	,	,	PUNCT
ma-267	73	14	for	for	ADP
ma-267	73	15	(	(	PUNCT
ma-267	73	16	µ	µ	X
ma-267	73	17	,	,	PUNCT
ma-267	73	18	λ	λ	NOUN
ma-267	73	19	)	)	PUNCT
ma-267	73	20	∈	∈	PROPN
ma-267	73	21	k̂.	k̂.	VERB
ma-267	73	22	some	some	DET
ma-267	73	23	basic	basic	ADJ
ma-267	73	24	properties	property	NOUN
ma-267	73	25	of	of	ADP
ma-267	73	26	this	this	DET
ma-267	73	27	transform	transform	NOUN
ma-267	73	28	are	be	AUX
ma-267	73	29	as	as	SCONJ
ma-267	73	30	follows	follow	NOUN
ma-267	73	31	,	,	PUNCT
ma-267	73	32	for	for	ADP
ma-267	73	33	the	the	DET
ma-267	73	34	proofs	proof	NOUN
ma-267	73	35	one	one	PRON
ma-267	73	36	can	can	AUX
ma-267	73	37	see	see	VERB
ma-267	73	38	[	[	X
ma-267	73	39	2–4	2–4	NUM
ma-267	73	40	]	]	PUNCT
ma-267	73	41	.	.	PUNCT
ma-267	74	1	proposition	proposition	NOUN
ma-267	74	2	2.1.(1	2.1.(1	NUM
ma-267	74	3	)	)	PUNCT
ma-267	74	4	for	for	ADP
ma-267	74	5	every	every	DET
ma-267	74	6	f	f	PROPN
ma-267	74	7	∈	∈	PROPN
ma-267	74	8	l1α(k	l1α(k	NOUN
ma-267	74	9	)	)	PUNCT
ma-267	75	1	,	,	PUNCT
ma-267	75	2	we	we	PRON
ma-267	75	3	have	have	VERB
ma-267	75	4	‖fα(f	‖fα(f	PUNCT
ma-267	75	5	)	)	PUNCT
ma-267	75	6	‖∞,γα	‖∞,γα	PROPN
ma-267	75	7	≤	≤	NOUN
ma-267	75	8	‖f	‖f	PRON
ma-267	75	9	‖1,µα	‖1,µα	PUNCT
ma-267	75	10	.	.	PUNCT
ma-267	76	1	(	(	PUNCT
ma-267	76	2	2.4	2.4	NUM
ma-267	76	3	)	)	PUNCT
ma-267	76	4	(	(	PUNCT
ma-267	76	5	2)(inversion	2)(inversion	NOUN
ma-267	76	6	formula	formula	NOUN
ma-267	76	7	)	)	PUNCT
ma-267	76	8	for	for	ADP
ma-267	76	9	f	f	PROPN
ma-267	76	10	∈	∈	PROPN
ma-267	76	11	(	(	PUNCT
ma-267	76	12	l1α	l1α	PROPN
ma-267	76	13	∩	∩	ADJ
ma-267	76	14	l2α	l2α	PROPN
ma-267	76	15	)	)	PUNCT
ma-267	76	16	(	(	PUNCT
ma-267	76	17	k	k	X
ma-267	76	18	)	)	PUNCT
ma-267	76	19	such	such	ADJ
ma-267	76	20	that	that	PRON
ma-267	76	21	fα(f	fα(f	X
ma-267	76	22	)	)	PUNCT
ma-267	76	23	∈	∈	PROPN
ma-267	77	1	l1α(k̂	l1α(k̂	PROPN
ma-267	77	2	)	)	PUNCT
ma-267	77	3	we	we	PRON
ma-267	77	4	have	have	VERB
ma-267	77	5	f	f	PROPN
ma-267	77	6	(	(	PUNCT
ma-267	77	7	x	x	PROPN
ma-267	77	8	,	,	PUNCT
ma-267	77	9	t	t	PROPN
ma-267	77	10	)	)	PUNCT
ma-267	77	11	=	=	SYM
ma-267	78	1	∫	∫	PROPN
ma-267	78	2	k̂	k̂	PROPN
ma-267	78	3	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	78	4	,	,	PUNCT
ma-267	78	5	t)fα(f	t)fα(f	NUM
ma-267	78	6	)	)	PUNCT
ma-267	78	7	(	(	PUNCT
ma-267	78	8	µ	µ	NUM
ma-267	78	9	,	,	PUNCT
ma-267	78	10	λ)dγα(µ	λ)dγα(µ	PROPN
ma-267	78	11	,	,	PUNCT
ma-267	78	12	λ	λ	NOUN
ma-267	78	13	)	)	PUNCT
ma-267	78	14	,	,	PUNCT
ma-267	78	15	a.e	a.e	PROPN
ma-267	78	16	(	(	PUNCT
ma-267	78	17	x	x	NOUN
ma-267	78	18	,	,	PUNCT
ma-267	78	19	t	t	PROPN
ma-267	78	20	)	)	PUNCT
ma-267	78	21	∈	∈	PROPN
ma-267	78	22	k.	k.	PROPN
ma-267	78	23	(	(	PUNCT
ma-267	78	24	2.5	2.5	NUM
ma-267	78	25	)	)	PUNCT
ma-267	78	26	(	(	PUNCT
ma-267	78	27	3	3	X
ma-267	78	28	)	)	PUNCT
ma-267	78	29	(	(	PUNCT
ma-267	78	30	parseval	parseval	NOUN
ma-267	78	31	formula	formula	NOUN
ma-267	78	32	)	)	PUNCT
ma-267	78	33	for	for	ADP
ma-267	78	34	all	all	DET
ma-267	78	35	f	f	PROPN
ma-267	78	36	,	,	PUNCT
ma-267	78	37	g	g	PROPN
ma-267	78	38	∈	∈	PROPN
ma-267	78	39	l2α(k	l2α(k	PROPN
ma-267	78	40	)	)	PUNCT
ma-267	78	41	we	we	PRON
ma-267	78	42	have∫	have∫	VERB
ma-267	78	43	k	k	PROPN
ma-267	78	44	f	f	X
ma-267	78	45	(	(	PUNCT
ma-267	78	46	x	x	NOUN
ma-267	78	47	,	,	PUNCT
ma-267	78	48	t)g(x	t)g(x	ADJ
ma-267	78	49	,	,	PUNCT
ma-267	78	50	t)dµα(x	t)dµα(x	NUM
ma-267	78	51	,	,	PUNCT
ma-267	78	52	t	t	PROPN
ma-267	78	53	)	)	PUNCT
ma-267	78	54	=	=	SYM
ma-267	79	1	∫	∫	PROPN
ma-267	79	2	k̂	k̂	PROPN
ma-267	79	3	fα(f	fα(f	PUNCT
ma-267	79	4	)	)	PUNCT
ma-267	79	5	(	(	PUNCT
ma-267	79	6	µ	µ	NOUN
ma-267	79	7	,	,	PUNCT
ma-267	79	8	λ)fα(g)(µ	λ)fα(g)(µ	NOUN
ma-267	79	9	,	,	PUNCT
ma-267	79	10	λ)dγα(µ	λ)dγα(µ	X
ma-267	79	11	,	,	PUNCT
ma-267	79	12	λ	λ	NOUN
ma-267	79	13	)	)	PUNCT
ma-267	79	14	,	,	PUNCT
ma-267	79	15	(	(	PUNCT
ma-267	79	16	2.6	2.6	NUM
ma-267	79	17	)	)	PUNCT
ma-267	79	18	in	in	ADP
ma-267	79	19	particular	particular	ADJ
ma-267	79	20	we	we	PRON
ma-267	79	21	have	have	VERB
ma-267	79	22	‖f	‖f	PRON
ma-267	79	23	‖2,µα	‖2,µα	NOUN
ma-267	79	24	=	=	SYM
ma-267	79	25	‖fα(f	‖fα(f	PUNCT
ma-267	79	26	)	)	PUNCT
ma-267	79	27	‖2,γα	‖2,γα	PUNCT
ma-267	79	28	.	.	PUNCT
ma-267	80	1	(	(	PUNCT
ma-267	80	2	2.7)(4	2.7)(4	NUM
ma-267	80	3	)	)	PUNCT
ma-267	80	4	(	(	PUNCT
ma-267	80	5	plancherel	plancherel	PROPN
ma-267	80	6	’s	’s	PART
ma-267	80	7	theorem	theorem	NOUN
ma-267	80	8	)	)	PUNCT
ma-267	80	9	the	the	DET
ma-267	80	10	reimann	reimann	NOUN
ma-267	80	11	-	-	PUNCT
ma-267	80	12	liouville	liouville	NOUN
ma-267	80	13	transform	transform	NOUN
ma-267	80	14	fα	fα	NOUN
ma-267	80	15	can	can	AUX
ma-267	80	16	be	be	AUX
ma-267	80	17	extended	extend	VERB
ma-267	80	18	to	to	ADP
ma-267	80	19	an	an	DET
ma-267	80	20	isometricisomorphism	isometricisomorphism	NOUN
ma-267	80	21	from	from	ADP
ma-267	80	22	l2α(k	l2α(k	PROPN
ma-267	80	23	)	)	PUNCT
ma-267	80	24	into	into	ADP
ma-267	80	25	l2α(k̂	l2α(k̂	NOUN
ma-267	80	26	)	)	PUNCT
ma-267	80	27	.	.	PUNCT
ma-267	81	1	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	81	2	eur	eur	PROPN
ma-267	81	3	.	.	PUNCT
ma-267	82	1	j.	j.	PROPN
ma-267	82	2	math	math	PROPN
ma-267	82	3	.	.	PUNCT
ma-267	83	1	anal	anal	PROPN
ma-267	83	2	.	.	PUNCT
ma-267	84	1	10.28924	10.28924	NUM
ma-267	84	2	/	/	SYM
ma-267	84	3	ada	ada	PROPN
ma-267	84	4	/	/	SYM
ma-267	84	5	ma.4.20	ma.4.20	NOUN
ma-267	84	6	52.3	52.3	NUM
ma-267	84	7	.	.	PUNCT
ma-267	85	1	generalized	generalized	ADJ
ma-267	85	2	translation	translation	NOUN
ma-267	85	3	operator	operator	NOUN
ma-267	85	4	associated	associate	VERB
ma-267	85	5	with	with	ADP
ma-267	85	6	the	the	DET
ma-267	85	7	riemann	riemann	PROPN
ma-267	85	8	-	-	PUNCT
ma-267	85	9	liouville	liouville	NOUN
ma-267	85	10	operator	operator	NOUN
ma-267	85	11	.	.	PUNCT
ma-267	86	1	definition	definition	NOUN
ma-267	86	2	2.2	2.2	NUM
ma-267	86	3	.	.	PUNCT
ma-267	87	1	the	the	DET
ma-267	87	2	translation	translation	NOUN
ma-267	87	3	operator	operator	NOUN
ma-267	87	4	associated	associate	VERB
ma-267	87	5	with	with	ADP
ma-267	87	6	riemann	riemann	PROPN
ma-267	87	7	-	-	PUNCT
ma-267	87	8	liouville	liouville	VERB
ma-267	87	9	transform	transform	NOUN
ma-267	87	10	is	be	AUX
ma-267	87	11	defined	define	VERB
ma-267	87	12	on	on	ADP
ma-267	87	13	lpα(k	lpα(k	NOUN
ma-267	87	14	)	)	PUNCT
ma-267	87	15	,	,	PUNCT
ma-267	87	16	for	for	ADP
ma-267	87	17	all	all	PRON
ma-267	87	18	(	(	PUNCT
ma-267	87	19	x	x	NOUN
ma-267	87	20	,	,	PUNCT
ma-267	87	21	t	t	PROPN
ma-267	87	22	)	)	PUNCT
ma-267	87	23	,	,	PUNCT
ma-267	87	24	(	(	PUNCT
ma-267	87	25	y	y	PROPN
ma-267	87	26	,	,	PUNCT
ma-267	87	27	s	s	X
ma-267	87	28	)	)	PUNCT
ma-267	87	29	∈	∈	PROPN
ma-267	87	30	k	k	NOUN
ma-267	87	31	,	,	PUNCT
ma-267	87	32	by	by	ADP
ma-267	87	33	τ	τ	PROPN
ma-267	87	34	(	(	PUNCT
ma-267	87	35	x	x	PROPN
ma-267	87	36	,	,	PUNCT
ma-267	87	37	t	t	PROPN
ma-267	87	38	)	)	PUNCT
ma-267	87	39	α	α	PROPN
ma-267	87	40	(	(	PUNCT
ma-267	87	41	f	f	PROPN
ma-267	87	42	)	)	PUNCT
ma-267	87	43	(	(	PUNCT
ma-267	87	44	y	y	PROPN
ma-267	87	45	,	,	PUNCT
ma-267	87	46	s	s	X
ma-267	87	47	)	)	PUNCT
ma-267	87	48	=	=	PUNCT
ma-267	87	49	γ(α+	γ(α+	PROPN
ma-267	87	50	1)√	1)√	NUM
ma-267	87	51	πγ(α+	πγ(α+	PROPN
ma-267	87	52	1/2	1/2	NUM
ma-267	87	53	)	)	PUNCT
ma-267	87	54	∫	∫	PROPN
ma-267	88	1	π	π	NOUN
ma-267	88	2	0	0	PUNCT
ma-267	88	3	f	f	PROPN
ma-267	88	4	(	(	PUNCT
ma-267	88	5	√	√	PROPN
ma-267	88	6	x2	x2	PROPN
ma-267	89	1	+	+	CCONJ
ma-267	89	2	y2	y2	PROPN
ma-267	89	3	+	+	CCONJ
ma-267	89	4	2xy	2xy	ADJ
ma-267	89	5	cos	cos	PROPN
ma-267	89	6	θ	θ	PROPN
ma-267	89	7	,	,	PUNCT
ma-267	89	8	t	t	PROPN
ma-267	89	9	+	+	SYM
ma-267	89	10	s	s	PART
ma-267	89	11	)	)	PUNCT
ma-267	89	12	sin2α	sin2α	PROPN
ma-267	89	13	θdθ	θdθ	NOUN
ma-267	89	14	.	.	PUNCT
ma-267	90	1	the	the	DET
ma-267	90	2	following	follow	VERB
ma-267	90	3	proposition	proposition	NOUN
ma-267	90	4	summarizes	summarize	VERB
ma-267	90	5	some	some	DET
ma-267	90	6	properties	property	NOUN
ma-267	90	7	of	of	ADP
ma-267	90	8	the	the	DET
ma-267	90	9	riemann	riemann	PROPN
ma-267	90	10	-	-	PUNCT
ma-267	90	11	liouville	liouville	VERB
ma-267	90	12	translation	translation	NOUN
ma-267	90	13	operatorsee	operatorsee	VERB
ma-267	91	1	[	[	X
ma-267	91	2	2–4	2–4	NUM
ma-267	91	3	]	]	PUNCT
ma-267	91	4	.	.	PUNCT
ma-267	92	1	proposition	proposition	NOUN
ma-267	92	2	2.2	2.2	NUM
ma-267	92	3	.	.	PUNCT
ma-267	93	1	for	for	ADP
ma-267	93	2	all	all	PRON
ma-267	93	3	(	(	PUNCT
ma-267	93	4	x	x	NOUN
ma-267	93	5	,	,	PUNCT
ma-267	93	6	t	t	PROPN
ma-267	93	7	)	)	PUNCT
ma-267	93	8	,	,	PUNCT
ma-267	93	9	(	(	PUNCT
ma-267	93	10	y	y	PROPN
ma-267	93	11	,	,	PUNCT
ma-267	93	12	s	s	X
ma-267	93	13	)	)	PUNCT
ma-267	93	14	∈	∈	PROPN
ma-267	93	15	k	k	PROPN
ma-267	93	16	,	,	PUNCT
ma-267	93	17	f	f	PROPN
ma-267	93	18	∈	∈	PROPN
ma-267	93	19	lpα(k	lpα(k	PROPN
ma-267	93	20	)	)	PUNCT
ma-267	93	21	we	we	PRON
ma-267	93	22	have:(1	have:(1	PROPN
ma-267	93	23	)	)	PUNCT
ma-267	93	24	∫	∫	PROPN
ma-267	94	1	k	k	PROPN
ma-267	94	2	τ	τ	PROPN
ma-267	94	3	(	(	PUNCT
ma-267	94	4	x	x	PROPN
ma-267	94	5	,	,	PUNCT
ma-267	94	6	t	t	PROPN
ma-267	94	7	)	)	PUNCT
ma-267	94	8	α	α	PROPN
ma-267	94	9	(	(	PUNCT
ma-267	94	10	f	f	PROPN
ma-267	94	11	)	)	PUNCT
ma-267	94	12	(	(	PUNCT
ma-267	94	13	y	y	PROPN
ma-267	94	14	,	,	PUNCT
ma-267	94	15	s)dµα(y	s)dµα(y	X
ma-267	94	16	,	,	PUNCT
ma-267	94	17	s	s	X
ma-267	94	18	)	)	PUNCT
ma-267	94	19	=	=	SYM
ma-267	95	1	∫	∫	PROPN
ma-267	95	2	k	k	PROPN
ma-267	95	3	f	f	PROPN
ma-267	95	4	(	(	PUNCT
ma-267	95	5	y	y	PROPN
ma-267	95	6	,	,	PUNCT
ma-267	95	7	s)dµα(y	s)dµα(y	X
ma-267	95	8	,	,	PUNCT
ma-267	95	9	s	s	NOUN
ma-267	95	10	)	)	PUNCT
ma-267	95	11	.	.	PUNCT
ma-267	96	1	(	(	PUNCT
ma-267	96	2	2.8	2.8	NUM
ma-267	96	3	)	)	PUNCT
ma-267	96	4	(	(	PUNCT
ma-267	96	5	2	2	NUM
ma-267	96	6	)	)	PUNCT
ma-267	96	7	for	for	ADP
ma-267	96	8	f	f	PROPN
ma-267	96	9	∈	∈	PROPN
ma-267	96	10	lpα(k	lpα(k	PROPN
ma-267	96	11	)	)	PUNCT
ma-267	96	12	with	with	ADP
ma-267	96	13	p	p	PROPN
ma-267	96	14	∈	∈	PROPN
ma-267	97	1	[	[	X
ma-267	97	2	1	1	NUM
ma-267	97	3	;	;	PUNCT
ma-267	97	4	+	+	NUM
ma-267	97	5	∞	∞	NOUN
ma-267	97	6	]	]	X
ma-267	97	7	τ	τ	PROPN
ma-267	97	8	(	(	PUNCT
ma-267	97	9	x	x	PROPN
ma-267	97	10	,	,	PUNCT
ma-267	97	11	t	t	PROPN
ma-267	97	12	)	)	PUNCT
ma-267	97	13	α	α	PROPN
ma-267	97	14	(	(	PUNCT
ma-267	97	15	f	f	X
ma-267	97	16	)	)	PUNCT
ma-267	97	17	∈	∈	PROPN
ma-267	97	18	lpα(k	lpα(k	NOUN
ma-267	97	19	)	)	PUNCT
ma-267	97	20	and	and	CCONJ
ma-267	97	21	we	we	PRON
ma-267	97	22	have∥∥∥τ	have∥∥∥τ	PROPN
ma-267	97	23	(	(	PUNCT
ma-267	97	24	x	x	NOUN
ma-267	97	25	,	,	PUNCT
ma-267	97	26	t)α	t)α	PUNCT
ma-267	97	27	(	(	PUNCT
ma-267	97	28	f	f	PROPN
ma-267	97	29	)	)	PUNCT
ma-267	97	30	∥∥∥	∥∥∥	PROPN
ma-267	97	31	p,µα	p,µα	NUM
ma-267	97	32	≤	≤	NOUN
ma-267	97	33	‖f	‖f	PRON
ma-267	97	34	‖p,µα	‖p,µα	NUM
ma-267	97	35	.	.	PUNCT
ma-267	98	1	(	(	PUNCT
ma-267	98	2	2.9	2.9	NUM
ma-267	98	3	)	)	PUNCT
ma-267	98	4	(	(	PUNCT
ma-267	98	5	3	3	X
ma-267	98	6	)	)	PUNCT
ma-267	98	7	for	for	ADP
ma-267	98	8	f	f	PROPN
ma-267	98	9	∈	∈	PROPN
ma-267	98	10	l1α(k	l1α(k	PROPN
ma-267	98	11	)	)	PUNCT
ma-267	98	12	,	,	PUNCT
ma-267	98	13	τ	τ	PROPN
ma-267	98	14	(	(	PUNCT
ma-267	98	15	x,−t	x,−t	PROPN
ma-267	98	16	)	)	PUNCT
ma-267	98	17	α	α	NOUN
ma-267	98	18	(	(	PUNCT
ma-267	98	19	f	f	X
ma-267	98	20	)	)	PUNCT
ma-267	98	21	∈	∈	PROPN
ma-267	98	22	l1α(k	l1α(k	NOUN
ma-267	98	23	)	)	PUNCT
ma-267	98	24	and	and	CCONJ
ma-267	98	25	we	we	PRON
ma-267	98	26	have	have	VERB
ma-267	98	27	fα	fα	PART
ma-267	98	28	(	(	PUNCT
ma-267	98	29	τ	τ	PROPN
ma-267	98	30	(	(	PUNCT
ma-267	98	31	x,−t	x,−t	PROPN
ma-267	98	32	)	)	PUNCT
ma-267	98	33	α	α	PROPN
ma-267	98	34	(	(	PUNCT
ma-267	98	35	f	f	PROPN
ma-267	98	36	)	)	PUNCT
ma-267	98	37	)	)	PUNCT
ma-267	98	38	(	(	PUNCT
ma-267	98	39	µ	µ	X
ma-267	98	40	,	,	PUNCT
ma-267	98	41	λ	λ	NOUN
ma-267	98	42	)	)	PUNCT
ma-267	98	43	=	=	SYM
ma-267	98	44	ϕµ,λ(x	ϕµ,λ(x	NOUN
ma-267	98	45	,	,	PUNCT
ma-267	98	46	t)fα(f	t)fα(f	NUM
ma-267	98	47	)	)	PUNCT
ma-267	98	48	(	(	PUNCT
ma-267	98	49	µ	µ	X
ma-267	98	50	,	,	PUNCT
ma-267	98	51	λ	λ	NOUN
ma-267	98	52	)	)	PUNCT
ma-267	98	53	,	,	PUNCT
ma-267	98	54	∀(µ	∀(µ	NUM
ma-267	98	55	,	,	PUNCT
ma-267	98	56	λ	λ	NOUN
ma-267	98	57	)	)	PUNCT
ma-267	98	58	∈	∈	PROPN
ma-267	98	59	k̂.	k̂.	NOUN
ma-267	98	60	(	(	PUNCT
ma-267	98	61	2.10	2.10	NUM
ma-267	98	62	)	)	PUNCT
ma-267	98	63	by	by	ADP
ma-267	98	64	using	use	VERB
ma-267	98	65	the	the	DET
ma-267	98	66	generalized	generalized	ADJ
ma-267	98	67	translation	translation	NOUN
ma-267	98	68	,	,	PUNCT
ma-267	98	69	we	we	PRON
ma-267	98	70	define	define	VERB
ma-267	98	71	the	the	DET
ma-267	98	72	generalized	generalized	ADJ
ma-267	98	73	convolution	convolution	NOUN
ma-267	98	74	product	product	NOUN
ma-267	98	75	of	of	ADP
ma-267	98	76	f	f	PROPN
ma-267	98	77	,	,	PUNCT
ma-267	98	78	g	g	PROPN
ma-267	98	79	by	by	ADP
ma-267	98	80	(	(	PUNCT
ma-267	98	81	f	f	PROPN
ma-267	98	82	∗α	∗α	PROPN
ma-267	98	83	g	g	PROPN
ma-267	98	84	)	)	PUNCT
ma-267	98	85	(	(	PUNCT
ma-267	98	86	x	x	X
ma-267	98	87	,	,	PUNCT
ma-267	98	88	t	t	PROPN
ma-267	98	89	)	)	PUNCT
ma-267	98	90	=	=	SYM
ma-267	98	91	∫	∫	PROPN
ma-267	99	1	k	k	PROPN
ma-267	99	2	τ	τ	PROPN
ma-267	99	3	(	(	PUNCT
ma-267	99	4	x,−t	x,−t	PROPN
ma-267	99	5	)	)	PUNCT
ma-267	99	6	α	α	PROPN
ma-267	99	7	(	(	PUNCT
ma-267	99	8	f̌	f̌	PROPN
ma-267	99	9	)	)	PUNCT
ma-267	99	10	(	(	PUNCT
ma-267	99	11	y	y	NOUN
ma-267	99	12	,	,	PUNCT
ma-267	99	13	s)g(y	s)g(y	NOUN
ma-267	99	14	,	,	PUNCT
ma-267	99	15	s)dµα(y	s)dµα(y	X
ma-267	99	16	,	,	PUNCT
ma-267	99	17	s	s	PROPN
ma-267	99	18	)	)	PUNCT
ma-267	99	19	.	.	PUNCT
ma-267	100	1	where	where	SCONJ
ma-267	100	2	f̌	f̌	PROPN
ma-267	100	3	(	(	PUNCT
ma-267	100	4	y	y	PROPN
ma-267	100	5	,	,	PUNCT
ma-267	100	6	s	s	X
ma-267	100	7	)	)	PUNCT
ma-267	100	8	=	=	SYM
ma-267	100	9	f	f	PROPN
ma-267	100	10	(	(	PUNCT
ma-267	100	11	y	y	PROPN
ma-267	100	12	,	,	PUNCT
ma-267	100	13	−s).with	−s).with	PROPN
ma-267	100	14	this	this	DET
ma-267	100	15	convolution	convolution	NOUN
ma-267	100	16	product	product	NOUN
ma-267	100	17	(	(	PUNCT
ma-267	100	18	k	k	NOUN
ma-267	100	19	,	,	PUNCT
ma-267	100	20	∗α	∗α	NOUN
ma-267	100	21	)	)	PUNCT
ma-267	100	22	is	be	AUX
ma-267	100	23	a	a	DET
ma-267	100	24	hypergroup	hypergroup	NOUN
ma-267	100	25	in	in	ADP
ma-267	100	26	the	the	DET
ma-267	100	27	sense	sense	NOUN
ma-267	100	28	of	of	ADP
ma-267	100	29	jewett	jewett	PROPN
ma-267	100	30	[	[	X
ma-267	100	31	13].we	13].we	NUM
ma-267	100	32	have	have	VERB
ma-267	100	33	the	the	DET
ma-267	100	34	following	follow	VERB
ma-267	100	35	results	result	NOUN
ma-267	100	36	for	for	ADP
ma-267	100	37	the	the	DET
ma-267	100	38	proofs	proof	NOUN
ma-267	100	39	,	,	PUNCT
ma-267	100	40	we	we	PRON
ma-267	100	41	refer	refer	VERB
ma-267	100	42	the	the	DET
ma-267	100	43	reader	reader	NOUN
ma-267	100	44	to	to	ADP
ma-267	100	45	[	[	X
ma-267	100	46	2–4	2–4	NUM
ma-267	100	47	]	]	PUNCT
ma-267	100	48	.	.	PUNCT
ma-267	101	1	proposition	proposition	NOUN
ma-267	101	2	2.3.(1)(young	2.3.(1)(young	NUM
ma-267	101	3	’s	’s	PART
ma-267	101	4	inequality	inequality	NOUN
ma-267	101	5	)	)	PUNCT
ma-267	101	6	for	for	ADP
ma-267	101	7	all	all	DET
ma-267	101	8	p	p	NOUN
ma-267	101	9	,	,	PUNCT
ma-267	101	10	q	q	ADJ
ma-267	101	11	,	,	PUNCT
ma-267	101	12	r	r	NOUN
ma-267	101	13	∈	∈	PROPN
ma-267	102	1	[	[	X
ma-267	102	2	1	1	NUM
ma-267	102	3	;	;	PUNCT
ma-267	102	4	+	+	NUM
ma-267	102	5	∞	∞	NOUN
ma-267	102	6	]	]	X
ma-267	102	7	such	such	ADJ
ma-267	102	8	that	that	SCONJ
ma-267	102	9	:	:	PUNCT
ma-267	102	10	1p	1p	NUM
ma-267	102	11	+	+	CCONJ
ma-267	102	12	1	1	NUM
ma-267	102	13	q	q	NOUN
ma-267	102	14	=	=	SYM
ma-267	102	15	1	1	NUM
ma-267	102	16	+	+	SYM
ma-267	102	17	1	1	NUM
ma-267	102	18	r	r	NOUN
ma-267	102	19	and	and	CCONJ
ma-267	102	20	for	for	ADP
ma-267	102	21	all	all	DET
ma-267	102	22	f	f	PROPN
ma-267	102	23	∈	∈	PROPN
ma-267	102	24	lpα(k	lpα(k	PROPN
ma-267	102	25	)	)	PUNCT
ma-267	102	26	,	,	PUNCT
ma-267	102	27	g	g	PROPN
ma-267	102	28	∈	∈	PROPN
ma-267	102	29	lqα(k	lqα(k	PROPN
ma-267	102	30	)	)	PUNCT
ma-267	102	31	the	the	DET
ma-267	102	32	function	function	NOUN
ma-267	102	33	f	f	PROPN
ma-267	102	34	∗α	∗α	PROPN
ma-267	102	35	g	g	PROPN
ma-267	102	36	belongs	belong	VERB
ma-267	102	37	to	to	ADP
ma-267	102	38	the	the	DET
ma-267	102	39	space	space	NOUN
ma-267	102	40	lrα(k	lrα(k	NOUN
ma-267	102	41	)	)	PUNCT
ma-267	102	42	and	and	CCONJ
ma-267	102	43	we	we	PRON
ma-267	102	44	have	have	VERB
ma-267	102	45	‖f	‖f	DET
ma-267	102	46	∗α	∗α	NOUN
ma-267	102	47	g‖r,µα	g‖r,µα	VERB
ma-267	102	48	≤	≤	NOUN
ma-267	102	49	‖f	‖f	ADP
ma-267	102	50	‖p,µα‖g‖q,µα	‖p,µα‖g‖q,µα	NUM
ma-267	102	51	(	(	PUNCT
ma-267	102	52	2.11	2.11	NUM
ma-267	102	53	)	)	PUNCT
ma-267	102	54	(	(	PUNCT
ma-267	102	55	2	2	NUM
ma-267	102	56	)	)	PUNCT
ma-267	102	57	for	for	ADP
ma-267	102	58	f	f	PROPN
ma-267	102	59	,	,	PUNCT
ma-267	102	60	g	g	PROPN
ma-267	102	61	∈	∈	PROPN
ma-267	102	62	l2α(k	l2α(k	PROPN
ma-267	102	63	)	)	PUNCT
ma-267	102	64	the	the	DET
ma-267	102	65	function	function	NOUN
ma-267	102	66	f	f	PROPN
ma-267	102	67	∗α	∗α	PROPN
ma-267	102	68	g	g	PROPN
ma-267	102	69	belongs	belong	VERB
ma-267	102	70	to	to	ADP
ma-267	102	71	l2α(k	l2α(k	PROPN
ma-267	102	72	)	)	PUNCT
ma-267	103	1	if	if	SCONJ
ma-267	103	2	and	and	CCONJ
ma-267	103	3	only	only	ADV
ma-267	103	4	if	if	SCONJ
ma-267	103	5	the	the	DET
ma-267	103	6	function	function	NOUN
ma-267	103	7	fα(f	fα(f	NOUN
ma-267	103	8	)	)	PUNCT
ma-267	103	9	fα(g)belongs	fα(g)belong	NOUN
ma-267	103	10	to	to	ADP
ma-267	103	11	l2α(k̂	l2α(k̂	NOUN
ma-267	103	12	)	)	PUNCT
ma-267	103	13	and	and	CCONJ
ma-267	103	14	in	in	ADP
ma-267	103	15	this	this	DET
ma-267	103	16	case	case	NOUN
ma-267	103	17	we	we	PRON
ma-267	103	18	have	have	VERB
ma-267	103	19	fα	fα	PART
ma-267	103	20	(	(	PUNCT
ma-267	103	21	f	f	PROPN
ma-267	103	22	∗α	∗α	PROPN
ma-267	103	23	g	g	NOUN
ma-267	103	24	)	)	PUNCT
ma-267	103	25	=	=	SYM
ma-267	103	26	fα(f	fα(f	X
ma-267	103	27	)	)	PUNCT
ma-267	103	28	fα(g	fα(g	PUNCT
ma-267	103	29	)	)	PUNCT
ma-267	103	30	.	.	PUNCT
ma-267	104	1	(	(	PUNCT
ma-267	104	2	2.12	2.12	NUM
ma-267	104	3	)	)	PUNCT
ma-267	104	4	(	(	PUNCT
ma-267	104	5	3	3	X
ma-267	104	6	)	)	PUNCT
ma-267	104	7	for	for	ADP
ma-267	104	8	f	f	PROPN
ma-267	104	9	,	,	PUNCT
ma-267	104	10	g	g	PROPN
ma-267	104	11	∈	∈	PROPN
ma-267	104	12	l2α(k	l2α(k	PROPN
ma-267	104	13	)	)	PUNCT
ma-267	104	14	then	then	ADV
ma-267	104	15	we	we	PRON
ma-267	104	16	have∫	have∫	VERB
ma-267	104	17	k	k	PROPN
ma-267	104	18	|f	|f	PROPN
ma-267	104	19	∗α	∗α	PROPN
ma-267	104	20	g(x	g(x	PROPN
ma-267	104	21	,	,	PUNCT
ma-267	104	22	t)|2	t)|2	PROPN
ma-267	104	23	dµα(x	dµα(x	PROPN
ma-267	104	24	,	,	PUNCT
ma-267	104	25	t	t	PROPN
ma-267	104	26	)	)	PUNCT
ma-267	104	27	=	=	SYM
ma-267	105	1	∫	∫	PROPN
ma-267	105	2	k̂	k̂	PROPN
ma-267	105	3	|fα(f	|fα(f	X
ma-267	105	4	)	)	PUNCT
ma-267	105	5	(	(	PUNCT
ma-267	105	6	µ	µ	NOUN
ma-267	105	7	,	,	PUNCT
ma-267	105	8	λ)|2	λ)|2	PROPN
ma-267	105	9	|fα(g)(µ	|fα(g)(µ	PROPN
ma-267	105	10	,	,	PUNCT
ma-267	105	11	λ)|2	λ)|2	PROPN
ma-267	105	12	dγα(µ	dγα(µ	PROPN
ma-267	105	13	,	,	PUNCT
ma-267	105	14	λ	λ	PROPN
ma-267	105	15	)	)	PUNCT
ma-267	105	16	,	,	PUNCT
ma-267	105	17	(	(	PUNCT
ma-267	105	18	2.13	2.13	NUM
ma-267	105	19	)	)	PUNCT
ma-267	105	20	where	where	SCONJ
ma-267	105	21	both	both	DET
ma-267	105	22	integrals	integral	NOUN
ma-267	105	23	are	be	AUX
ma-267	105	24	simultaneously	simultaneously	ADV
ma-267	105	25	finite	finite	ADJ
ma-267	105	26	or	or	CCONJ
ma-267	105	27	infinite	infinite	VERB
ma-267	105	28	.	.	PUNCT
ma-267	106	1	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	106	2	eur	eur	PROPN
ma-267	106	3	.	.	PUNCT
ma-267	107	1	j.	j.	PROPN
ma-267	107	2	math	math	PROPN
ma-267	107	3	.	.	PUNCT
ma-267	108	1	anal	anal	PROPN
ma-267	108	2	.	.	PUNCT
ma-267	109	1	10.28924	10.28924	NUM
ma-267	109	2	/	/	SYM
ma-267	109	3	ada	ada	PROPN
ma-267	109	4	/	/	SYM
ma-267	109	5	ma.4.20	ma.4.20	PROPN
ma-267	109	6	63	63	NUM
ma-267	109	7	.	.	PUNCT
ma-267	110	1	calderón	calderón	PROPN
ma-267	110	2	’s	’s	PART
ma-267	110	3	reproducing	reproduce	VERB
ma-267	110	4	formula	formula	NOUN
ma-267	110	5	for	for	ADP
ma-267	110	6	the	the	DET
ma-267	110	7	riemann	riemann	PROPN
ma-267	110	8	-	-	PUNCT
ma-267	110	9	liouville	liouville	VERB
ma-267	110	10	two	two	NUM
ma-267	110	11	-	-	PUNCT
ma-267	110	12	wavelet	wavelet	NOUN
ma-267	110	13	transform	transform	NOUN
ma-267	110	14	using	use	VERB
ma-267	110	15	the	the	DET
ma-267	110	16	harmonic	harmonic	ADJ
ma-267	110	17	analysis	analysis	NOUN
ma-267	110	18	associated	associate	VERB
ma-267	110	19	with	with	ADP
ma-267	110	20	the	the	DET
ma-267	110	21	riemann	riemann	PROPN
ma-267	110	22	-	-	PUNCT
ma-267	110	23	liouville	liouville	NOUN
ma-267	110	24	transform	transform	NOUN
ma-267	110	25	,	,	PUNCT
ma-267	110	26	the	the	DET
ma-267	110	27	main	main	ADJ
ma-267	110	28	purposeof	purposeof	NOUN
ma-267	110	29	this	this	DET
ma-267	110	30	section	section	NOUN
ma-267	110	31	is	be	AUX
ma-267	110	32	to	to	PART
ma-267	110	33	define	define	VERB
ma-267	110	34	the	the	DET
ma-267	110	35	wavelet	wavelet	NOUN
ma-267	110	36	transform	transform	NOUN
ma-267	110	37	associated	associate	VERB
ma-267	110	38	with	with	ADP
ma-267	110	39	the	the	DET
ma-267	110	40	riemann	riemann	PROPN
ma-267	110	41	-	-	PUNCT
ma-267	110	42	liouville	liouville	VERB
ma-267	110	43	operatorand	operatorand	NOUN
ma-267	110	44	to	to	PART
ma-267	110	45	give	give	VERB
ma-267	110	46	generalized	generalized	ADJ
ma-267	110	47	parseval	parseval	NOUN
ma-267	110	48	’s	’s	PART
ma-267	110	49	,	,	PUNCT
ma-267	110	50	plancherel	plancherel	PROPN
ma-267	110	51	’s	’s	PART
ma-267	110	52	,	,	PUNCT
ma-267	110	53	inversion	inversion	NOUN
ma-267	110	54	and	and	CCONJ
ma-267	110	55	calderon	calderon	NOUN
ma-267	110	56	’s	’s	PART
ma-267	110	57	reproducing	reproducing	NOUN
ma-267	110	58	formulasrelated	formulasrelate	VERB
ma-267	110	59	to	to	ADP
ma-267	110	60	this	this	DET
ma-267	110	61	transform	transform	NOUN
ma-267	110	62	which	which	PRON
ma-267	110	63	generalizes	generalize	VERB
ma-267	110	64	all	all	DET
ma-267	110	65	the	the	DET
ma-267	110	66	results	result	NOUN
ma-267	110	67	proved	prove	VERB
ma-267	110	68	in	in	ADP
ma-267	110	69	[	[	X
ma-267	110	70	4	4	NUM
ma-267	110	71	]	]	PUNCT
ma-267	110	72	.	.	PUNCT
ma-267	111	1	notation	notation	NOUN
ma-267	111	2	:	:	PUNCT
ma-267	111	3	we	we	PRON
ma-267	111	4	denote	denote	VERB
ma-267	111	5	by•lpα(r+	by•lpα(r+	PROPN
ma-267	111	6	×k),1	×k),1	PROPN
ma-267	111	7	≤	≤	PROPN
ma-267	111	8	p	p	NOUN
ma-267	111	9	≤	≤	NUM
ma-267	112	1	+	+	PROPN
ma-267	112	2	∞	∞	NUM
ma-267	112	3	the	the	DET
ma-267	112	4	space	space	NOUN
ma-267	112	5	of	of	ADP
ma-267	112	6	measurable	measurable	ADJ
ma-267	112	7	functions	function	NOUN
ma-267	112	8	on	on	ADP
ma-267	112	9	r+	r+	PUNCT
ma-267	112	10	×ksatisfying	×ksatisfye	VERB
ma-267	112	11	‖f	‖f	ADP
ma-267	112	12	‖p	‖p	NOUN
ma-267	112	13	,	,	PUNCT
ma-267	112	14	θα	θα	ADP
ma-267	112	15	:	:	PUNCT
ma-267	112	16	=	=	SYM
ma-267	112	17			PROPN
ma-267	112	18	(	(	PUNCT
ma-267	112	19	∫	∫	PROPN
ma-267	113	1	+	+	NOUN
ma-267	113	2	∞	∞	PROPN
ma-267	113	3	0	0	NUM
ma-267	113	4	∫	∫	PROPN
ma-267	113	5	k	k	PROPN
ma-267	113	6	|f	|f	PROPN
ma-267	113	7	(	(	PUNCT
ma-267	113	8	a	a	PRON
ma-267	113	9	,	,	PUNCT
ma-267	113	10	x	x	NOUN
ma-267	113	11	,	,	PUNCT
ma-267	113	12	t)|pdθα(a	t)|pdθα(a	NOUN
ma-267	113	13	,	,	PUNCT
ma-267	113	14	x	x	NOUN
ma-267	113	15	,	,	PUNCT
ma-267	113	16	t	t	PROPN
ma-267	113	17	)	)	PUNCT
ma-267	113	18	)	)	PUNCT
ma-267	113	19	1	1	NUM
ma-267	114	1	p	p	NOUN
ma-267	114	2	<	<	X
ma-267	114	3	∞	∞	PROPN
ma-267	114	4	,	,	PUNCT
ma-267	114	5	if	if	SCONJ
ma-267	114	6	p	p	X
ma-267	114	7	∈	∈	PROPN
ma-267	115	1	[	[	X
ma-267	115	2	1,+∞	1,+∞	NUM
ma-267	115	3	[	[	X
ma-267	115	4	,	,	PUNCT
ma-267	115	5	ess	ess	PROPN
ma-267	115	6	sup	sup	PROPN
ma-267	115	7	|f	|f	PROPN
ma-267	115	8	(	(	PUNCT
ma-267	115	9	a	a	PRON
ma-267	115	10	,	,	PUNCT
ma-267	115	11	x	x	PROPN
ma-267	115	12	,	,	PUNCT
ma-267	115	13	t	t	PROPN
ma-267	115	14	)	)	PUNCT
ma-267	115	15	(	(	PUNCT
ma-267	115	16	a	a	DET
ma-267	115	17	,	,	PUNCT
ma-267	115	18	x	x	NOUN
ma-267	115	19	,	,	PUNCT
ma-267	115	20	t)∈r+×k	t)∈r+×k	PROPN
ma-267	115	21	|	|	CCONJ
ma-267	115	22	<	<	X
ma-267	115	23	∞	∞	PROPN
ma-267	115	24	,	,	PUNCT
ma-267	115	25	if	if	SCONJ
ma-267	115	26	p	p	X
ma-267	115	27	=	=	X
ma-267	115	28	+	+	NOUN
ma-267	115	29	∞	∞	PROPN
ma-267	115	30	..	..	PUNCT
ma-267	115	31	where	where	SCONJ
ma-267	115	32	θα	θα	NOUN
ma-267	115	33	is	be	AUX
ma-267	115	34	the	the	DET
ma-267	115	35	measure	measure	NOUN
ma-267	115	36	defined	define	VERB
ma-267	115	37	on	on	ADP
ma-267	115	38	r+	r+	PUNCT
ma-267	115	39	×k	×k	VERB
ma-267	115	40	by	by	ADP
ma-267	115	41	dθα(a	dθα(a	PROPN
ma-267	115	42	,	,	PUNCT
ma-267	115	43	x	x	PROPN
ma-267	115	44	,	,	PUNCT
ma-267	115	45	t	t	PROPN
ma-267	115	46	)	)	PUNCT
ma-267	115	47	:	:	PUNCT
ma-267	116	1	=	=	SYM
ma-267	116	2	a2α+2da	a2α+2da	PROPN
ma-267	116	3	⊗	⊗	PROPN
ma-267	116	4	dµα(x	dµα(x	PROPN
ma-267	116	5	,	,	PUNCT
ma-267	116	6	t	t	PROPN
ma-267	116	7	)	)	PUNCT
ma-267	116	8	.	.	PUNCT
ma-267	117	1	definition	definition	NOUN
ma-267	117	2	3.1	3.1	NUM
ma-267	117	3	.	.	PUNCT
ma-267	118	1	let	let	VERB
ma-267	118	2	ψ1	ψ1	NOUN
ma-267	118	3	,	,	PUNCT
ma-267	118	4	ψ2	ψ2	NOUN
ma-267	118	5	∈	∈	PROPN
ma-267	118	6	l2α(k	l2α(k	PROPN
ma-267	118	7	)	)	PUNCT
ma-267	118	8	,	,	PUNCT
ma-267	118	9	the	the	DET
ma-267	118	10	pair	pair	NOUN
ma-267	118	11	(	(	PUNCT
ma-267	118	12	ψ1	ψ1	NOUN
ma-267	118	13	,	,	PUNCT
ma-267	118	14	ψ2	ψ2	NOUN
ma-267	118	15	)	)	PUNCT
ma-267	118	16	is	be	AUX
ma-267	118	17	said	say	VERB
ma-267	118	18	to	to	PART
ma-267	118	19	be	be	AUX
ma-267	118	20	a	a	DET
ma-267	118	21	rieman	rieman	NOUN
ma-267	118	22	-	-	PUNCT
ma-267	118	23	liouville	liouville	NOUN
ma-267	118	24	two	two	NUM
ma-267	118	25	-	-	PUNCT
ma-267	118	26	waveleton	waveleton	NOUN
ma-267	118	27	k	k	PROPN
ma-267	119	1	if	if	SCONJ
ma-267	119	2	for	for	ADP
ma-267	119	3	almost	almost	ADV
ma-267	119	4	all	all	PRON
ma-267	119	5	(	(	PUNCT
ma-267	119	6	µ	µ	X
ma-267	119	7	,	,	PUNCT
ma-267	119	8	λ	λ	NOUN
ma-267	119	9	)	)	PUNCT
ma-267	119	10	∈	∈	PROPN
ma-267	119	11	k̂	k̂	NOUN
ma-267	119	12	we	we	PRON
ma-267	119	13	have	have	VERB
ma-267	119	14	0	0	NUM
ma-267	119	15	<	<	X
ma-267	119	16	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	119	17	:	:	PUNCT
ma-267	120	1	=	=	SYM
ma-267	120	2	∫	∫	PROPN
ma-267	120	3	∞	∞	NUM
ma-267	120	4	0	0	NUM
ma-267	120	5	fα(ψ1	fα(ψ1	NOUN
ma-267	120	6	)	)	PUNCT
ma-267	120	7	(	(	PUNCT
ma-267	120	8	µ	µ	X
ma-267	120	9	a	a	PRON
ma-267	120	10	,	,	PUNCT
ma-267	120	11	λ	λ	X
ma-267	120	12	a	a	PRON
ma-267	120	13	)	)	PUNCT
ma-267	120	14	fα(ψ2	fα(ψ2	NOUN
ma-267	120	15	)	)	PUNCT
ma-267	120	16	(	(	PUNCT
ma-267	120	17	µ	µ	X
ma-267	120	18	a	a	PRON
ma-267	120	19	,	,	PUNCT
ma-267	120	20	λ	λ	X
ma-267	120	21	a	a	PRON
ma-267	120	22	)	)	PUNCT
ma-267	120	23	da	da	NOUN
ma-267	120	24	a	a	DET
ma-267	120	25	<	<	X
ma-267	120	26	+	+	PROPN
ma-267	120	27	∞.	∞.	PROPN
ma-267	120	28	(	(	PUNCT
ma-267	120	29	3.1	3.1	NUM
ma-267	120	30	)	)	PUNCT
ma-267	120	31	remark	remark	NOUN
ma-267	120	32	3.1	3.1	NUM
ma-267	120	33	.	.	PUNCT
ma-267	121	1	its	its	PRON
ma-267	121	2	clear	clear	ADJ
ma-267	121	3	that	that	SCONJ
ma-267	121	4	if	if	SCONJ
ma-267	121	5	ψ	ψ	ADP
ma-267	121	6	=	=	SYM
ma-267	121	7	ψ1	ψ1	NOUN
ma-267	121	8	=	=	PUNCT
ma-267	121	9	ψ2	ψ2	NOUN
ma-267	121	10	,	,	PUNCT
ma-267	121	11	we	we	PRON
ma-267	121	12	have	have	AUX
ma-267	121	13	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	121	14	=	=	SYM
ma-267	121	15	cψ	cψ	NOUN
ma-267	121	16	:	:	PUNCT
ma-267	121	17	=	=	SYM
ma-267	121	18	∫	∫	PROPN
ma-267	121	19	∞	∞	PROPN
ma-267	121	20	0	0	NUM
ma-267	121	21	∣∣∣∣fα(ψ	∣∣∣∣fα(ψ	PROPN
ma-267	121	22	)	)	PUNCT
ma-267	121	23	(	(	PUNCT
ma-267	121	24	µ	µ	X
ma-267	121	25	a	a	PRON
ma-267	121	26	,	,	PUNCT
ma-267	121	27	λ	λ	NOUN
ma-267	121	28	a	a	DET
ma-267	121	29	)	)	PUNCT
ma-267	121	30	∣∣∣∣2	∣∣∣∣2	NOUN
ma-267	121	31	daa	daa	NOUN
ma-267	121	32	<	<	X
ma-267	121	33	+	+	PROPN
ma-267	121	34	∞	∞	PROPN
ma-267	121	35	,	,	PUNCT
ma-267	121	36	(	(	PUNCT
ma-267	121	37	3.2	3.2	NUM
ma-267	121	38	)	)	PUNCT
ma-267	121	39	in	in	ADP
ma-267	121	40	this	this	DET
ma-267	121	41	case	case	NOUN
ma-267	121	42	we	we	PRON
ma-267	121	43	say	say	VERB
ma-267	121	44	that	that	SCONJ
ma-267	121	45	ψ	ψ	NOUN
ma-267	121	46	is	be	AUX
ma-267	121	47	a	a	DET
ma-267	121	48	riemann	riemann	PROPN
ma-267	121	49	-	-	PUNCT
ma-267	121	50	liouville	liouville	VERB
ma-267	121	51	wavelet	wavelet	NOUN
ma-267	121	52	in	in	ADP
ma-267	121	53	l2α(k).let	l2α(k).let	PROPN
ma-267	121	54	a	a	DET
ma-267	121	55	>	>	X
ma-267	121	56	0	0	NUM
ma-267	121	57	,	,	PUNCT
ma-267	121	58	we	we	PRON
ma-267	121	59	define	define	VERB
ma-267	121	60	the	the	DET
ma-267	121	61	dilatation	dilatation	NOUN
ma-267	121	62	operator	operator	NOUN
ma-267	121	63	da	da	NOUN
ma-267	121	64	of	of	ADP
ma-267	121	65	a	a	DET
ma-267	121	66	measurable	measurable	ADJ
ma-267	121	67	function	function	NOUN
ma-267	121	68	ψ	ψ	NOUN
ma-267	121	69	on	on	ADP
ma-267	121	70	c2	c2	PROPN
ma-267	121	71	by	by	ADP
ma-267	121	72	da(ψ)(x	da(ψ)(x	PROPN
ma-267	121	73	,	,	PUNCT
ma-267	121	74	t	t	PROPN
ma-267	121	75	)	)	PUNCT
ma-267	121	76	=	=	SYM
ma-267	122	1	aα+3/2ψ(ax	aα+3/2ψ(ax	NOUN
ma-267	122	2	,	,	PUNCT
ma-267	122	3	at	at	ADP
ma-267	122	4	)	)	PUNCT
ma-267	122	5	,	,	PUNCT
ma-267	122	6	(	(	PUNCT
ma-267	122	7	x	x	X
ma-267	122	8	,	,	PUNCT
ma-267	122	9	t	t	PROPN
ma-267	122	10	)	)	PUNCT
ma-267	122	11	∈	∈	PROPN
ma-267	122	12	c2	c2	PROPN
ma-267	122	13	.	.	PUNCT
ma-267	123	1	the	the	DET
ma-267	123	2	dilatation	dilatation	NOUN
ma-267	123	3	operator	operator	NOUN
ma-267	123	4	da	da	NOUN
ma-267	123	5	satisfies	satisfy	VERB
ma-267	123	6	the	the	DET
ma-267	123	7	following	follow	VERB
ma-267	123	8	properties•	properties•	NOUN
ma-267	123	9	for	for	ADP
ma-267	123	10	all	all	DET
ma-267	123	11	ψ	ψ	PRON
ma-267	123	12	∈	∈	PROPN
ma-267	123	13	lpα(k	lpα(k	NOUN
ma-267	123	14	)	)	PUNCT
ma-267	123	15	we	we	PRON
ma-267	123	16	have	have	VERB
ma-267	123	17	da(ψ	da(ψ	NOUN
ma-267	123	18	)	)	PUNCT
ma-267	123	19	∈	∈	PROPN
ma-267	123	20	lpα(k	lpα(k	NOUN
ma-267	123	21	)	)	PUNCT
ma-267	123	22	and	and	CCONJ
ma-267	123	23	‖da(ψ)‖p,µα	‖da(ψ)‖p,µα	VERB
ma-267	124	1	=	=	SYM
ma-267	124	2	a	a	PRON
ma-267	124	3	(	(	PUNCT
ma-267	124	4	1	1	NUM
ma-267	124	5	2	2	NUM
ma-267	124	6	−	−	NOUN
ma-267	124	7	1	1	NUM
ma-267	124	8	p	p	NOUN
ma-267	124	9	)	)	PUNCT
ma-267	124	10	(	(	PUNCT
ma-267	124	11	2α+3)‖ψ‖p,µα	2α+3)‖ψ‖p,µα	NUM
ma-267	124	12	.	.	PUNCT
ma-267	125	1	(	(	PUNCT
ma-267	125	2	3.3	3.3	NUM
ma-267	125	3	)	)	PUNCT
ma-267	125	4	•	•	NOUN
ma-267	125	5	for	for	ADP
ma-267	125	6	all	all	DET
ma-267	125	7	ψ	ψ	ADP
ma-267	125	8	∈	∈	PROPN
ma-267	125	9	l2(k	l2(k	PROPN
ma-267	125	10	)	)	PUNCT
ma-267	125	11	we	we	PRON
ma-267	125	12	have	have	VERB
ma-267	125	13	fα(da(ψ))(µ	fα(da(ψ))(µ	NOUN
ma-267	125	14	,	,	PUNCT
ma-267	125	15	λ	λ	NOUN
ma-267	125	16	)	)	PUNCT
ma-267	125	17	=	=	SYM
ma-267	125	18	1	1	NUM
ma-267	125	19	aα+3/2	aα+3/2	PROPN
ma-267	125	20	fα(ψ	fα(ψ	NUM
ma-267	125	21	)	)	PUNCT
ma-267	125	22	(	(	PUNCT
ma-267	125	23	µ	µ	X
ma-267	125	24	a	a	PRON
ma-267	125	25	,	,	PUNCT
ma-267	125	26	λ	λ	X
ma-267	125	27	a	a	PRON
ma-267	125	28	)	)	PUNCT
ma-267	125	29	.	.	PUNCT
ma-267	126	1	(	(	PUNCT
ma-267	126	2	3.4	3.4	NUM
ma-267	126	3	)	)	PUNCT
ma-267	126	4	let	let	VERB
ma-267	126	5	ψ	ψ	PART
ma-267	126	6	be	be	AUX
ma-267	126	7	a	a	DET
ma-267	126	8	riemann	riemann	PROPN
ma-267	126	9	-	-	PUNCT
ma-267	126	10	liouville	liouville	NOUN
ma-267	126	11	wavelet	wavelet	NOUN
ma-267	126	12	on	on	ADP
ma-267	126	13	k	k	PROPN
ma-267	126	14	in	in	ADP
ma-267	126	15	lp(k	lp(k	NOUN
ma-267	126	16	)	)	PUNCT
ma-267	126	17	with	with	ADP
ma-267	126	18	1	1	NUM
ma-267	126	19	≤	≤	NOUN
ma-267	126	20	p	p	NOUN
ma-267	126	21	≤	≤	NUM
ma-267	126	22	∞	∞	PROPN
ma-267	126	23	,	,	PUNCT
ma-267	126	24	for	for	ADP
ma-267	126	25	all	all	DET
ma-267	126	26	a	a	DET
ma-267	126	27	>	>	X
ma-267	126	28	0,(x	0,(x	PROPN
ma-267	126	29	,	,	PUNCT
ma-267	126	30	t	t	PROPN
ma-267	126	31	)	)	PUNCT
ma-267	126	32	∈	∈	PROPN
ma-267	127	1	k	k	NOUN
ma-267	127	2	we	we	PRON
ma-267	127	3	define	define	VERB
ma-267	127	4	the	the	DET
ma-267	127	5	function	function	NOUN
ma-267	127	6	ψa	ψa	ADP
ma-267	127	7	,	,	PUNCT
ma-267	127	8	x	x	INTJ
ma-267	127	9	,	,	PUNCT
ma-267	127	10	t(y	t(y	PROPN
ma-267	127	11	,	,	PUNCT
ma-267	127	12	s	s	X
ma-267	127	13	)	)	PUNCT
ma-267	127	14	=	=	SYM
ma-267	127	15	τ	τ	PROPN
ma-267	127	16	(	(	PUNCT
ma-267	127	17	x,−t	x,−t	PROPN
ma-267	127	18	)	)	PUNCT
ma-267	127	19	α	α	PROPN
ma-267	127	20	(	(	PUNCT
ma-267	127	21	da(ψ))(y	da(ψ))(y	NOUN
ma-267	127	22	,	,	PUNCT
ma-267	127	23	s	s	NOUN
ma-267	127	24	)	)	PUNCT
ma-267	127	25	.	.	PUNCT
ma-267	128	1	(	(	PUNCT
ma-267	128	2	3.5	3.5	NUM
ma-267	128	3	)	)	PUNCT
ma-267	128	4	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	128	5	eur	eur	PROPN
ma-267	128	6	.	.	PUNCT
ma-267	129	1	j.	j.	PROPN
ma-267	129	2	math	math	PROPN
ma-267	129	3	.	.	PUNCT
ma-267	130	1	anal	anal	PROPN
ma-267	130	2	.	.	PUNCT
ma-267	131	1	10.28924	10.28924	NUM
ma-267	131	2	/	/	SYM
ma-267	131	3	ada	ada	PROPN
ma-267	131	4	/	/	SYM
ma-267	131	5	ma.4.20	ma.4.20	PROPN
ma-267	131	6	7by	7by	PROPN
ma-267	131	7	using	use	VERB
ma-267	131	8	the	the	DET
ma-267	131	9	relations	relation	NOUN
ma-267	131	10	(	(	PUNCT
ma-267	131	11	2.9	2.9	NUM
ma-267	131	12	)	)	PUNCT
ma-267	131	13	and	and	CCONJ
ma-267	131	14	(	(	PUNCT
ma-267	131	15	3.3	3.3	NUM
ma-267	131	16	)	)	PUNCT
ma-267	131	17	we	we	PRON
ma-267	131	18	find	find	VERB
ma-267	131	19	that	that	SCONJ
ma-267	131	20	ψa	ψa	ADP
ma-267	131	21	,	,	PUNCT
ma-267	131	22	x	x	PROPN
ma-267	131	23	,	,	PUNCT
ma-267	131	24	t	t	PROPN
ma-267	131	25	∈	∈	PROPN
ma-267	131	26	lpα(k	lpα(k	NOUN
ma-267	131	27	)	)	PUNCT
ma-267	131	28	and	and	CCONJ
ma-267	131	29	‖ψa	‖ψa	NUM
ma-267	131	30	,	,	PUNCT
ma-267	131	31	x	x	PRON
ma-267	131	32	,	,	PUNCT
ma-267	131	33	t‖p,µα	t‖p,µα	VERB
ma-267	131	34	≤	≤	DET
ma-267	131	35	a	a	DET
ma-267	131	36	(	(	PUNCT
ma-267	131	37	1	1	NUM
ma-267	131	38	2	2	NUM
ma-267	131	39	−	−	NOUN
ma-267	131	40	1	1	NUM
ma-267	131	41	p	p	NOUN
ma-267	131	42	)	)	PUNCT
ma-267	131	43	(	(	PUNCT
ma-267	131	44	2α+3)‖ψ‖p,µα	2α+3)‖ψ‖p,µα	NUM
ma-267	131	45	..	..	PUNCT
ma-267	131	46	(	(	PUNCT
ma-267	131	47	3.6	3.6	NUM
ma-267	131	48	)	)	PUNCT
ma-267	131	49	definition	definition	NOUN
ma-267	131	50	3.2	3.2	NUM
ma-267	131	51	.	.	PUNCT
ma-267	132	1	(	(	PUNCT
ma-267	132	2	[	[	X
ma-267	132	3	4	4	NUM
ma-267	132	4	]	]	PUNCT
ma-267	132	5	)	)	PUNCT
ma-267	132	6	let	let	VERB
ma-267	132	7	ψ	ψ	PART
ma-267	132	8	be	be	AUX
ma-267	132	9	a	a	DET
ma-267	132	10	riemann	riemann	PROPN
ma-267	132	11	-	-	PUNCT
ma-267	132	12	liouville	liouville	NOUN
ma-267	132	13	wavelet	wavelet	NOUN
ma-267	132	14	on	on	ADP
ma-267	132	15	k	k	PROPN
ma-267	132	16	in	in	ADP
ma-267	132	17	l2α(k	l2α(k	PROPN
ma-267	132	18	)	)	PUNCT
ma-267	132	19	the	the	DET
ma-267	132	20	continuous	continuous	ADJ
ma-267	132	21	wavelettransform	wavelettransform	NOUN
ma-267	132	22	sαψ	sαψ	NOUN
ma-267	132	23	associated	associate	VERB
ma-267	132	24	with	with	ADP
ma-267	132	25	the	the	DET
ma-267	132	26	riemann	riemann	PROPN
ma-267	132	27	-	-	PUNCT
ma-267	132	28	liouville	liouville	NOUN
ma-267	132	29	operator	operator	NOUN
ma-267	132	30	is	be	AUX
ma-267	132	31	defined	define	VERB
ma-267	132	32	for	for	ADP
ma-267	132	33	a	a	DET
ma-267	132	34	function	function	NOUN
ma-267	132	35	f	f	PROPN
ma-267	132	36	∈	∈	PROPN
ma-267	132	37	l2α(k)and	l2α(k)and	PROPN
ma-267	132	38	(	(	PUNCT
ma-267	132	39	a	a	PRON
ma-267	132	40	,	,	PUNCT
ma-267	132	41	x	x	PROPN
ma-267	132	42	,	,	PUNCT
ma-267	132	43	t	t	PROPN
ma-267	132	44	)	)	PUNCT
ma-267	132	45	∈	∈	PROPN
ma-267	132	46	r+	r+	PUNCT
ma-267	132	47	×k	×k	VERB
ma-267	132	48	by	by	ADP
ma-267	132	49	sαψ(f	sαψ(f	PROPN
ma-267	132	50	)	)	PUNCT
ma-267	132	51	(	(	PUNCT
ma-267	132	52	a	a	PRON
ma-267	132	53	,	,	PUNCT
ma-267	132	54	x	x	PROPN
ma-267	132	55	,	,	PUNCT
ma-267	132	56	t	t	PROPN
ma-267	132	57	)	)	PUNCT
ma-267	132	58	:	:	PUNCT
ma-267	133	1	=	=	PUNCT
ma-267	133	2	∫	∫	PROPN
ma-267	134	1	k	k	PROPN
ma-267	134	2	f	f	PROPN
ma-267	134	3	(	(	PUNCT
ma-267	134	4	y	y	PROPN
ma-267	134	5	,	,	PUNCT
ma-267	134	6	s)ψa	s)ψa	PROPN
ma-267	134	7	,	,	PUNCT
ma-267	134	8	x	x	NOUN
ma-267	134	9	,	,	PUNCT
ma-267	134	10	t(y	t(y	ADV
ma-267	134	11	,	,	PUNCT
ma-267	134	12	s)dµα(y	s)dµα(y	X
ma-267	134	13	,	,	PUNCT
ma-267	134	14	s	s	NOUN
ma-267	134	15	)	)	PUNCT
ma-267	134	16	.	.	PUNCT
ma-267	135	1	(	(	PUNCT
ma-267	135	2	3.7	3.7	NUM
ma-267	135	3	)	)	PUNCT
ma-267	135	4	remark	remark	NOUN
ma-267	135	5	3.2	3.2	NUM
ma-267	135	6	.	.	PUNCT
ma-267	136	1	the	the	DET
ma-267	136	2	riemann	riemann	PROPN
ma-267	136	3	-	-	PUNCT
ma-267	136	4	liouville	liouville	VERB
ma-267	136	5	wavelet	wavelet	NOUN
ma-267	136	6	transform	transform	NOUN
ma-267	136	7	(	(	PUNCT
ma-267	136	8	3.7	3.7	NUM
ma-267	136	9	)	)	PUNCT
ma-267	136	10	can	can	AUX
ma-267	136	11	be	be	AUX
ma-267	136	12	written	write	VERB
ma-267	136	13	as	as	ADP
ma-267	136	14	sαψ(f	sαψ(f	PROPN
ma-267	136	15	)	)	PUNCT
ma-267	136	16	(	(	PUNCT
ma-267	136	17	a	a	PRON
ma-267	136	18	,	,	PUNCT
ma-267	136	19	x	x	PROPN
ma-267	136	20	,	,	PUNCT
ma-267	136	21	t	t	PROPN
ma-267	136	22	)	)	PUNCT
ma-267	136	23	=	=	SYM
ma-267	136	24	(	(	PUNCT
ma-267	136	25	da(ψ̌	da(ψ̌	PROPN
ma-267	136	26	)	)	PUNCT
ma-267	136	27	∗α	∗α	PROPN
ma-267	136	28	f	f	PROPN
ma-267	136	29	)	)	PUNCT
ma-267	136	30	(	(	PUNCT
ma-267	136	31	x	x	X
ma-267	136	32	,	,	PUNCT
ma-267	136	33	t	t	PROPN
ma-267	136	34	)	)	PUNCT
ma-267	136	35	=	=	PUNCT
ma-267	137	1	〈	〈	PROPN
ma-267	137	2	f	f	X
ma-267	137	3	,	,	PUNCT
ma-267	137	4	ψa	ψa	PROPN
ma-267	137	5	,	,	PUNCT
ma-267	137	6	x	x	NOUN
ma-267	137	7	,	,	PUNCT
ma-267	137	8	t〉α	t〉α	NOUN
ma-267	137	9	.	.	PUNCT
ma-267	138	1	(	(	PUNCT
ma-267	138	2	3.8	3.8	NUM
ma-267	138	3	)	)	PUNCT
ma-267	138	4	the	the	DET
ma-267	138	5	following	following	ADJ
ma-267	138	6	result	result	NOUN
ma-267	138	7	gives	give	VERB
ma-267	138	8	the	the	DET
ma-267	138	9	relation	relation	NOUN
ma-267	138	10	between	between	ADP
ma-267	138	11	the	the	DET
ma-267	138	12	riemann	riemann	PROPN
ma-267	138	13	-	-	PUNCT
ma-267	138	14	liouville	liouville	VERB
ma-267	138	15	transform	transform	NOUN
ma-267	138	16	fα	fα	ADP
ma-267	138	17	and	and	CCONJ
ma-267	138	18	theriemann	theriemann	NOUN
ma-267	138	19	-	-	PUNCT
ma-267	138	20	liouville	liouville	VERB
ma-267	138	21	wavelet	wavelet	NOUN
ma-267	138	22	transform	transform	NOUN
ma-267	138	23	sαψ	sαψ	NOUN
ma-267	138	24	.	.	PUNCT
ma-267	139	1	proposition	proposition	NOUN
ma-267	139	2	3.1	3.1	NUM
ma-267	139	3	.	.	PUNCT
ma-267	140	1	let	let	VERB
ma-267	140	2	ψ	ψ	PART
ma-267	140	3	be	be	AUX
ma-267	140	4	a	a	DET
ma-267	140	5	riemann	riemann	PROPN
ma-267	140	6	-	-	PUNCT
ma-267	140	7	liouville	liouville	NOUN
ma-267	140	8	wavelet	wavelet	NOUN
ma-267	140	9	for	for	ADP
ma-267	140	10	all	all	DET
ma-267	140	11	f	f	PROPN
ma-267	140	12	∈	∈	PROPN
ma-267	140	13	l2(k	l2(k	PROPN
ma-267	140	14	)	)	PUNCT
ma-267	140	15	we	we	PRON
ma-267	140	16	have	have	VERB
ma-267	140	17	fα	fα	VERB
ma-267	140	18	[	[	PUNCT
ma-267	140	19	sαψ(f	sαψ(f	PROPN
ma-267	140	20	)	)	PUNCT
ma-267	140	21	(	(	PUNCT
ma-267	140	22	a	a	PRON
ma-267	140	23	,	,	PUNCT
ma-267	140	24	.	.	PUNCT
ma-267	140	25	)	)	PUNCT
ma-267	141	1	]	]	PUNCT
ma-267	141	2	(	(	PUNCT
ma-267	141	3	µ	µ	X
ma-267	141	4	,	,	PUNCT
ma-267	141	5	λ	λ	NOUN
ma-267	141	6	)	)	PUNCT
ma-267	141	7	=	=	SYM
ma-267	141	8	1	1	NUM
ma-267	141	9	aα+3/2	aα+3/2	ADV
ma-267	141	10	fα	fα	ADP
ma-267	141	11	(	(	PUNCT
ma-267	141	12	f	f	PROPN
ma-267	141	13	)	)	PUNCT
ma-267	141	14	(	(	PUNCT
ma-267	141	15	µ	µ	NOUN
ma-267	141	16	,	,	PUNCT
ma-267	141	17	λ)fα(ψ	λ)fα(ψ	PROPN
ma-267	141	18	)	)	PUNCT
ma-267	141	19	(	(	PUNCT
ma-267	141	20	µ	µ	X
ma-267	141	21	a	a	PRON
ma-267	141	22	,	,	PUNCT
ma-267	141	23	λ	λ	X
ma-267	141	24	a	a	PRON
ma-267	141	25	)	)	PUNCT
ma-267	141	26	.	.	PUNCT
ma-267	142	1	(	(	PUNCT
ma-267	142	2	3.9	3.9	NUM
ma-267	142	3	)	)	PUNCT
ma-267	142	4	proof	proof	NOUN
ma-267	142	5	.	.	PUNCT
ma-267	143	1	is	be	AUX
ma-267	143	2	a	a	DET
ma-267	143	3	consequence	consequence	NOUN
ma-267	143	4	of	of	ADP
ma-267	143	5	the	the	DET
ma-267	143	6	convolution	convolution	NOUN
ma-267	143	7	theorem	theorem	NOUN
ma-267	143	8	(	(	PUNCT
ma-267	143	9	2.12	2.12	NUM
ma-267	143	10	)	)	PUNCT
ma-267	143	11	and	and	CCONJ
ma-267	143	12	the	the	DET
ma-267	143	13	relations	relation	NOUN
ma-267	143	14	(	(	PUNCT
ma-267	143	15	3.4),(3.8	3.4),(3.8	NUM
ma-267	143	16	)	)	PUNCT
ma-267	143	17	.	.	PUNCT
ma-267	144	1	�	�	PROPN
ma-267	145	1	the	the	DET
ma-267	145	2	following	follow	VERB
ma-267	145	3	theorem	theorem	NOUN
ma-267	145	4	generalizes	generalize	VERB
ma-267	145	5	the	the	DET
ma-267	145	6	parseval	parseval	NOUN
ma-267	145	7	’s	’s	PART
ma-267	145	8	formula	formula	NOUN
ma-267	145	9	for	for	ADP
ma-267	145	10	the	the	DET
ma-267	145	11	riemann	riemann	PROPN
ma-267	145	12	-	-	PUNCT
ma-267	145	13	liouville	liouville	VERB
ma-267	145	14	wavelettransform	wavelettransform	NOUN
ma-267	145	15	sαψ(f	sαψ(f	PROPN
ma-267	145	16	)	)	PUNCT
ma-267	145	17	proved	prove	VERB
ma-267	145	18	in	in	ADP
ma-267	145	19	[	[	X
ma-267	145	20	4	4	NUM
ma-267	145	21	]	]	PUNCT
ma-267	145	22	.	.	PUNCT
ma-267	146	1	theorem	theorem	VERB
ma-267	146	2	3.1	3.1	NUM
ma-267	146	3	.	.	PUNCT
ma-267	147	1	let	let	VERB
ma-267	147	2	(	(	PUNCT
ma-267	147	3	ψ1	ψ1	VERB
ma-267	147	4	,	,	PUNCT
ma-267	147	5	ψ2	ψ2	NOUN
ma-267	147	6	)	)	PUNCT
ma-267	147	7	be	be	VERB
ma-267	147	8	a	a	DET
ma-267	147	9	rieman	rieman	NOUN
ma-267	147	10	-	-	PUNCT
ma-267	147	11	liouville	liouville	NOUN
ma-267	147	12	two	two	NUM
ma-267	147	13	-	-	PUNCT
ma-267	147	14	wavelet	wavelet	NOUN
ma-267	147	15	on	on	ADP
ma-267	147	16	k	k	PROPN
ma-267	147	17	for	for	ADP
ma-267	147	18	all	all	DET
ma-267	147	19	f	f	PROPN
ma-267	147	20	,	,	PUNCT
ma-267	148	1	g	g	PROPN
ma-267	148	2	∈	∈	PROPN
ma-267	148	3	l2(k	l2(k	PROPN
ma-267	148	4	)	)	PUNCT
ma-267	148	5	we	we	PRON
ma-267	148	6	have∫	have∫	VERB
ma-267	149	1	+	+	NOUN
ma-267	149	2	∞	∞	NOUN
ma-267	149	3	0	0	NUM
ma-267	149	4	∫	∫	PROPN
ma-267	149	5	k	k	PROPN
ma-267	149	6	sαψ1(f	sαψ1(f	PROPN
ma-267	149	7	)	)	PUNCT
ma-267	149	8	(	(	PUNCT
ma-267	149	9	a	a	DET
ma-267	149	10	,	,	PUNCT
ma-267	149	11	x	x	NOUN
ma-267	149	12	,	,	PUNCT
ma-267	149	13	t)sαψ2(g)(a	t)sαψ2(g)(a	PROPN
ma-267	149	14	,	,	PUNCT
ma-267	149	15	x	x	X
ma-267	149	16	,	,	PUNCT
ma-267	149	17	t)dθα(a	t)dθα(a	NOUN
ma-267	149	18	,	,	PUNCT
ma-267	149	19	x	x	PROPN
ma-267	149	20	,	,	PUNCT
ma-267	149	21	t	t	PROPN
ma-267	149	22	)	)	PUNCT
ma-267	150	1	=	=	SYM
ma-267	150	2	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	150	3	∫	∫	PROPN
ma-267	150	4	k	k	PROPN
ma-267	150	5	f	f	PROPN
ma-267	150	6	(	(	PUNCT
ma-267	150	7	y	y	PROPN
ma-267	150	8	,	,	PUNCT
ma-267	150	9	s)g(y	s)g(y	NOUN
ma-267	150	10	,	,	PUNCT
ma-267	150	11	s)dµα(y	s)dµα(y	X
ma-267	150	12	,	,	PUNCT
ma-267	150	13	s	s	X
ma-267	150	14	)	)	PUNCT
ma-267	150	15	,	,	PUNCT
ma-267	150	16	(	(	PUNCT
ma-267	150	17	3.10	3.10	NUM
ma-267	150	18	)	)	PUNCT
ma-267	150	19	where	where	SCONJ
ma-267	150	20	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	150	21	is	be	AUX
ma-267	150	22	the	the	DET
ma-267	150	23	constant	constant	ADJ
ma-267	150	24	given	give	VERB
ma-267	150	25	by	by	ADP
ma-267	150	26	the	the	DET
ma-267	150	27	relation	relation	NOUN
ma-267	150	28	(	(	PUNCT
ma-267	150	29	3.1	3.1	NUM
ma-267	150	30	)	)	PUNCT
ma-267	150	31	.	.	PUNCT
ma-267	151	1	proof	proof	NOUN
ma-267	151	2	.	.	PUNCT
ma-267	152	1	by	by	ADP
ma-267	152	2	using	use	VERB
ma-267	152	3	the	the	DET
ma-267	152	4	relations	relation	NOUN
ma-267	152	5	(	(	PUNCT
ma-267	152	6	2.5),(2.12),(3.4	2.5),(2.12),(3.4	NOUN
ma-267	152	7	)	)	PUNCT
ma-267	152	8	,	,	PUNCT
ma-267	152	9	(	(	PUNCT
ma-267	152	10	3.8	3.8	NUM
ma-267	152	11	)	)	PUNCT
ma-267	152	12	and	and	CCONJ
ma-267	152	13	fubini	fubini	NOUN
ma-267	152	14	’s	’s	PART
ma-267	152	15	theorem	theorem	NOUN
ma-267	152	16	we	we	PRON
ma-267	152	17	get∫	get∫	PROPN
ma-267	152	18	+	+	PROPN
ma-267	152	19	∞	∞	PROPN
ma-267	152	20	0	0	NUM
ma-267	152	21	∫	∫	PROPN
ma-267	152	22	k	k	PROPN
ma-267	152	23	sαψ1(f	sαψ1(f	PROPN
ma-267	152	24	)	)	PUNCT
ma-267	152	25	(	(	PUNCT
ma-267	152	26	a	a	PRON
ma-267	152	27	,	,	PUNCT
ma-267	152	28	x	x	NOUN
ma-267	152	29	,	,	PUNCT
ma-267	152	30	t)sαψ1(g)(a	t)sαψ1(g)(a	PROPN
ma-267	152	31	,	,	PUNCT
ma-267	152	32	x	x	PRON
ma-267	152	33	,	,	PUNCT
ma-267	152	34	t)dθα(a	t)dθα(a	NOUN
ma-267	152	35	,	,	PUNCT
ma-267	152	36	x	x	PROPN
ma-267	152	37	,	,	PUNCT
ma-267	152	38	t	t	PROPN
ma-267	152	39	)	)	PUNCT
ma-267	152	40	=	=	SYM
ma-267	152	41	∫	∫	PROPN
ma-267	153	1	+	+	NUM
ma-267	153	2	∞	∞	PROPN
ma-267	153	3	0	0	NUM
ma-267	154	1	[	[	PUNCT
ma-267	154	2	∫	∫	X
ma-267	154	3	k	k	X
ma-267	154	4	(	(	PUNCT
ma-267	154	5	da(ψ̌1	da(ψ̌1	NOUN
ma-267	154	6	)	)	PUNCT
ma-267	154	7	∗α	∗α	PROPN
ma-267	154	8	f	f	PROPN
ma-267	154	9	)	)	PUNCT
ma-267	154	10	(	(	PUNCT
ma-267	154	11	x	x	NOUN
ma-267	154	12	,	,	PUNCT
ma-267	154	13	t)(da(ψ̌2	t)(da(ψ̌2	NOUN
ma-267	154	14	)	)	PUNCT
ma-267	154	15	∗α	∗α	PROPN
ma-267	154	16	g)(x	g)(x	PROPN
ma-267	154	17	,	,	PUNCT
ma-267	154	18	t)dµα(x	t)dµα(x	NUM
ma-267	154	19	,	,	PUNCT
ma-267	154	20	t)]a2α+2da	t)]a2α+2da	NOUN
ma-267	154	21	=	=	SYM
ma-267	155	1	∫	∫	PROPN
ma-267	156	1	+	+	NUM
ma-267	156	2	∞	∞	PROPN
ma-267	156	3	0	0	NUM
ma-267	157	1	[	[	PUNCT
ma-267	157	2	∫	∫	PROPN
ma-267	157	3	k̂	k̂	PROPN
ma-267	157	4	fα(da(ψ̌1))(µ	fα(da(ψ̌1))(µ	PROPN
ma-267	157	5	,	,	PUNCT
ma-267	157	6	λ)fα(f	λ)fα(f	X
ma-267	157	7	)	)	PUNCT
ma-267	157	8	(	(	PUNCT
ma-267	157	9	µ	µ	NOUN
ma-267	157	10	,	,	PUNCT
ma-267	157	11	λ)fα(da(ψ̌2))(µ	λ)fα(da(ψ̌2))(µ	NUM
ma-267	157	12	,	,	PUNCT
ma-267	157	13	λ)fα(g)(µ	λ)fα(g)(µ	NOUN
ma-267	157	14	,	,	PUNCT
ma-267	157	15	λ)dγα(λ	λ)dγα(λ	PROPN
ma-267	157	16	,	,	PUNCT
ma-267	157	17	m)]a2α+2da	m)]a2α+2da	X
ma-267	157	18	=	=	SYM
ma-267	157	19	cψ1,ψ2	cψ1,ψ2	PROPN
ma-267	157	20	∫	∫	PROPN
ma-267	157	21	k̂	k̂	PROPN
ma-267	157	22	fα(f	fα(f	PROPN
ma-267	157	23	)	)	PUNCT
ma-267	157	24	(	(	PUNCT
ma-267	157	25	µ	µ	NOUN
ma-267	157	26	,	,	PUNCT
ma-267	157	27	λ)fα(g)(µ	λ)fα(g)(µ	NOUN
ma-267	157	28	,	,	PUNCT
ma-267	157	29	λ)dγα(µ	λ)dγα(µ	X
ma-267	157	30	,	,	PUNCT
ma-267	157	31	λ	λ	NOUN
ma-267	157	32	)	)	PUNCT
ma-267	157	33	,	,	PUNCT
ma-267	157	34	by	by	ADP
ma-267	157	35	using	use	VERB
ma-267	157	36	parseval	parseval	NOUN
ma-267	157	37	’s	’s	PART
ma-267	157	38	formula	formula	NOUN
ma-267	157	39	for	for	ADP
ma-267	157	40	the	the	DET
ma-267	157	41	riemann	riemann	PROPN
ma-267	157	42	-	-	PUNCT
ma-267	157	43	liouville	liouville	VERB
ma-267	157	44	transform	transform	NOUN
ma-267	157	45	(	(	PUNCT
ma-267	157	46	2.5	2.5	NUM
ma-267	157	47	)	)	PUNCT
ma-267	157	48	we	we	PRON
ma-267	157	49	find	find	VERB
ma-267	157	50	the	the	DET
ma-267	157	51	disered	disere	VERB
ma-267	157	52	result	result	NOUN
ma-267	157	53	.	.	PUNCT
ma-267	158	1	�	�	PROPN
ma-267	158	2	in	in	ADP
ma-267	158	3	the	the	DET
ma-267	158	4	following	following	NOUN
ma-267	158	5	we	we	PRON
ma-267	158	6	establish	establish	VERB
ma-267	158	7	an	an	DET
ma-267	158	8	inversion	inversion	NOUN
ma-267	158	9	formula	formula	NOUN
ma-267	158	10	for	for	ADP
ma-267	158	11	the	the	DET
ma-267	158	12	riemann	riemann	PROPN
ma-267	158	13	-	-	PUNCT
ma-267	158	14	liouville	liouville	VERB
ma-267	158	15	two	two	NUM
ma-267	158	16	-	-	PUNCT
ma-267	158	17	wavelet	wavelet	NOUN
ma-267	158	18	trans	trans	NOUN
ma-267	158	19	-	-	NOUN
ma-267	158	20	form	form	NOUN
ma-267	158	21	.	.	PUNCT
ma-267	159	1	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	159	2	eur	eur	PROPN
ma-267	159	3	.	.	PUNCT
ma-267	160	1	j.	j.	PROPN
ma-267	160	2	math	math	PROPN
ma-267	160	3	.	.	PUNCT
ma-267	161	1	anal	anal	PROPN
ma-267	161	2	.	.	PUNCT
ma-267	162	1	10.28924	10.28924	NUM
ma-267	162	2	/	/	SYM
ma-267	162	3	ada	ada	PROPN
ma-267	162	4	/	/	SYM
ma-267	162	5	ma.4.20	ma.4.20	NOUN
ma-267	162	6	8	8	NUM
ma-267	162	7	theorem	theorem	VERB
ma-267	162	8	3.2	3.2	NUM
ma-267	162	9	.	.	PUNCT
ma-267	163	1	let	let	VERB
ma-267	163	2	(	(	PUNCT
ma-267	163	3	ψ1	ψ1	VERB
ma-267	163	4	,	,	PUNCT
ma-267	163	5	ψ2	ψ2	NOUN
ma-267	163	6	)	)	PUNCT
ma-267	163	7	be	be	VERB
ma-267	163	8	a	a	DET
ma-267	163	9	riemann	riemann	PROPN
ma-267	163	10	-	-	PUNCT
ma-267	163	11	liouville	liouville	VERB
ma-267	163	12	two	two	NUM
ma-267	163	13	-	-	PUNCT
ma-267	163	14	wavelet	wavelet	NOUN
ma-267	163	15	such	such	ADJ
ma-267	163	16	that	that	SCONJ
ma-267	163	17	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	163	18	6=	6=	ADP
ma-267	163	19	0	0	NUM
ma-267	163	20	for	for	ADP
ma-267	163	21	all	all	DET
ma-267	163	22	f	f	PROPN
ma-267	163	23	∈	∈	PROPN
ma-267	163	24	l1α(k	l1α(k	NOUN
ma-267	163	25	)	)	PUNCT
ma-267	164	1	such	such	ADJ
ma-267	164	2	that	that	PRON
ma-267	164	3	fα(f	fα(f	X
ma-267	164	4	)	)	PUNCT
ma-267	164	5	∈	∈	PROPN
ma-267	164	6	l1α(k̂	l1α(k̂	PROPN
ma-267	164	7	)	)	PUNCT
ma-267	164	8	∩	∩	ADJ
ma-267	164	9	l∞α	l∞α	NOUN
ma-267	164	10	(	(	PUNCT
ma-267	164	11	k̂	k̂	NOUN
ma-267	164	12	)	)	PUNCT
ma-267	164	13	we	we	PRON
ma-267	164	14	have	have	VERB
ma-267	164	15	f	f	PROPN
ma-267	164	16	(	(	PUNCT
ma-267	164	17	·	·	PUNCT
ma-267	164	18	)	)	PUNCT
ma-267	164	19	=	=	SYM
ma-267	164	20	1	1	NUM
ma-267	164	21	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	164	22	∫	∫	PROPN
ma-267	164	23	∞	∞	PROPN
ma-267	164	24	0	0	NUM
ma-267	165	1	(	(	PUNCT
ma-267	165	2	∫	∫	PROPN
ma-267	165	3	k	k	PROPN
ma-267	165	4	sαψ1(f	sαψ1(f	PROPN
ma-267	165	5	)	)	PUNCT
ma-267	165	6	(	(	PUNCT
ma-267	165	7	a	a	PRON
ma-267	165	8	,	,	PUNCT
ma-267	165	9	x	x	NOUN
ma-267	165	10	,	,	PUNCT
ma-267	165	11	t)ψ2,a	t)ψ2,a	PROPN
ma-267	165	12	,	,	PUNCT
ma-267	165	13	x	x	NOUN
ma-267	165	14	,	,	PUNCT
ma-267	165	15	t(·)dµα(x	t(·)dµα(x	NOUN
ma-267	165	16	,	,	PUNCT
ma-267	165	17	t	t	PROPN
ma-267	165	18	)	)	PUNCT
ma-267	165	19	)	)	PUNCT
ma-267	165	20	a2α+2da	a2α+2da	PROPN
ma-267	165	21	.	.	PUNCT
ma-267	166	1	proof	proof	NOUN
ma-267	166	2	.	.	PUNCT
ma-267	167	1	let	let	VERB
ma-267	167	2	f	f	NOUN
ma-267	167	3	,	,	PUNCT
ma-267	167	4	g	g	PROPN
ma-267	167	5	∈	∈	PROPN
ma-267	167	6	l2α(k	l2α(k	PROPN
ma-267	167	7	)	)	PUNCT
ma-267	167	8	,	,	PUNCT
ma-267	167	9	by	by	ADP
ma-267	167	10	using	use	VERB
ma-267	167	11	the	the	DET
ma-267	167	12	relation	relation	NOUN
ma-267	167	13	(	(	PUNCT
ma-267	167	14	3.10	3.10	NUM
ma-267	167	15	)	)	PUNCT
ma-267	167	16	,	,	PUNCT
ma-267	167	17	fubini	fubini	PROPN
ma-267	167	18	’s	’s	PART
ma-267	167	19	theorem	theorem	NOUN
ma-267	167	20	we	we	PRON
ma-267	167	21	find	find	VERB
ma-267	167	22	that	that	SCONJ
ma-267	167	23	∫	∫	PROPN
ma-267	168	1	k	k	PROPN
ma-267	168	2	f	f	PROPN
ma-267	168	3	(	(	PUNCT
ma-267	168	4	y	y	PROPN
ma-267	168	5	,	,	PUNCT
ma-267	168	6	s)g(y	s)g(y	NOUN
ma-267	168	7	,	,	PUNCT
ma-267	168	8	s)dµα(y	s)dµα(y	X
ma-267	168	9	,	,	PUNCT
ma-267	168	10	s	s	X
ma-267	168	11	)	)	PUNCT
ma-267	168	12	=	=	SYM
ma-267	168	13	1	1	NUM
ma-267	168	14	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	168	15	∫	∫	PROPN
ma-267	169	1	+	+	PROPN
ma-267	169	2	∞	∞	PROPN
ma-267	169	3	0	0	NUM
ma-267	169	4	∫	∫	PROPN
ma-267	169	5	k	k	PROPN
ma-267	169	6	sαψ1(f	sαψ1(f	PROPN
ma-267	169	7	)	)	PUNCT
ma-267	169	8	(	(	PUNCT
ma-267	169	9	a	a	DET
ma-267	169	10	,	,	PUNCT
ma-267	169	11	x	x	NOUN
ma-267	169	12	,	,	PUNCT
ma-267	169	13	t)sαψ2(g)(a	t)sαψ2(g)(a	PROPN
ma-267	169	14	,	,	PUNCT
ma-267	169	15	x	x	X
ma-267	169	16	,	,	PUNCT
ma-267	169	17	t)dθα(a	t)dθα(a	NOUN
ma-267	169	18	,	,	PUNCT
ma-267	169	19	x	x	PROPN
ma-267	169	20	,	,	PUNCT
ma-267	169	21	t	t	PROPN
ma-267	169	22	)	)	PUNCT
ma-267	169	23	=	=	SYM
ma-267	169	24	1	1	NUM
ma-267	169	25	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	169	26	∫	∫	PROPN
ma-267	170	1	k	k	PROPN
ma-267	170	2	[	[	PUNCT
ma-267	170	3	∫	∫	PROPN
ma-267	170	4	∞	∞	PROPN
ma-267	170	5	0	0	NUM
ma-267	171	1	(	(	PUNCT
ma-267	171	2	∫	∫	PROPN
ma-267	171	3	k	k	PROPN
ma-267	171	4	sαψ1(f	sαψ1(f	PROPN
ma-267	171	5	)	)	PUNCT
ma-267	171	6	(	(	PUNCT
ma-267	171	7	a	a	PRON
ma-267	171	8	,	,	PUNCT
ma-267	171	9	x	x	NOUN
ma-267	171	10	,	,	PUNCT
ma-267	171	11	t)ψ2,a	t)ψ2,a	PROPN
ma-267	171	12	,	,	PUNCT
ma-267	171	13	x	x	INTJ
ma-267	171	14	,	,	PUNCT
ma-267	171	15	t(y	t(y	PROPN
ma-267	171	16	,	,	PUNCT
ma-267	171	17	s)dµα(x	s)dµα(x	PROPN
ma-267	171	18	,	,	PUNCT
ma-267	171	19	t	t	PROPN
ma-267	171	20	)	)	PUNCT
ma-267	171	21	)	)	PUNCT
ma-267	171	22	a2α+2da]g(y	a2α+2da]g(y	NOUN
ma-267	171	23	,	,	PUNCT
ma-267	171	24	s)dµα(y	s)dµα(y	X
ma-267	171	25	,	,	PUNCT
ma-267	171	26	s)which	s)which	PRON
ma-267	171	27	gives	give	VERB
ma-267	171	28	the	the	DET
ma-267	171	29	result	result	NOUN
ma-267	171	30	.	.	PUNCT
ma-267	172	1	�	�	PROPN
ma-267	172	2	the	the	DET
ma-267	172	3	rest	rest	NOUN
ma-267	172	4	of	of	ADP
ma-267	172	5	this	this	DET
ma-267	172	6	subsection	subsection	NOUN
ma-267	172	7	is	be	AUX
ma-267	172	8	devoted	devote	VERB
ma-267	172	9	to	to	PART
ma-267	172	10	give	give	VERB
ma-267	172	11	a	a	DET
ma-267	172	12	calderón	calderón	NOUN
ma-267	172	13	’s	’s	PART
ma-267	172	14	reproducing	reproduce	VERB
ma-267	172	15	formula	formula	NOUN
ma-267	172	16	for	for	ADP
ma-267	172	17	the	the	DET
ma-267	172	18	reimann	reimann	NOUN
ma-267	172	19	-	-	PUNCT
ma-267	172	20	liouville	liouville	NOUN
ma-267	172	21	two	two	NUM
ma-267	172	22	-	-	PUNCT
ma-267	172	23	wavelet	wavelet	NOUN
ma-267	172	24	(	(	PUNCT
ma-267	172	25	ψ1	ψ1	NOUN
ma-267	172	26	,	,	PUNCT
ma-267	172	27	ψ2	ψ2	NOUN
ma-267	172	28	)	)	PUNCT
ma-267	172	29	under	under	ADP
ma-267	172	30	the	the	DET
ma-267	172	31	following	follow	VERB
ma-267	172	32	condition	condition	NOUN
ma-267	172	33	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	172	34	6=	6=	ADP
ma-267	172	35	0	0	NUM
ma-267	172	36	and	and	CCONJ
ma-267	172	37	fα(da(ψ1)),fα(da(ψ2	fα(da(ψ1)),fα(da(ψ2	PROPN
ma-267	172	38	)	)	PUNCT
ma-267	172	39	)	)	PUNCT
ma-267	173	1	∈	∈	PROPN
ma-267	173	2	l∞α	l∞α	NOUN
ma-267	173	3	(	(	PUNCT
ma-267	173	4	k̂	k̂	PROPN
ma-267	173	5	)	)	PUNCT
ma-267	173	6	,	,	PUNCT
ma-267	173	7	(	(	PUNCT
ma-267	173	8	3.11	3.11	NUM
ma-267	173	9	)	)	PUNCT
ma-267	173	10	proposition	proposition	NOUN
ma-267	173	11	3.2	3.2	NUM
ma-267	173	12	.	.	PUNCT
ma-267	174	1	for	for	ADP
ma-267	174	2	0	0	NUM
ma-267	174	3	<	<	X
ma-267	174	4	ε	ε	PROPN
ma-267	174	5	<	<	X
ma-267	174	6	δ	δ	PROPN
ma-267	174	7	<	<	X
ma-267	174	8	∞	∞	PROPN
ma-267	174	9	,	,	PUNCT
ma-267	174	10	we	we	PRON
ma-267	174	11	put	put	VERB
ma-267	174	12	gε	gε	NOUN
ma-267	174	13	,	,	PUNCT
ma-267	174	14	δ(x	δ(x	PROPN
ma-267	174	15	,	,	PUNCT
ma-267	174	16	t	t	PROPN
ma-267	174	17	)	)	PUNCT
ma-267	174	18	:	:	PUNCT
ma-267	175	1	=	=	SYM
ma-267	175	2	1	1	NUM
ma-267	175	3	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	175	4	∫	∫	PROPN
ma-267	175	5	δ	δ	PROPN
ma-267	175	6	ε	ε	PROPN
ma-267	175	7	(	(	PUNCT
ma-267	175	8	da(ψ̌2	da(ψ̌2	ADJ
ma-267	175	9	)	)	PUNCT
ma-267	175	10	∗α	∗α	NOUN
ma-267	175	11	da(ψ1	da(ψ1	NOUN
ma-267	175	12	)	)	PUNCT
ma-267	175	13	)	)	PUNCT
ma-267	175	14	(	(	PUNCT
ma-267	175	15	x	x	X
ma-267	175	16	,	,	PUNCT
ma-267	175	17	t)a2α+2da	t)a2α+2da	NOUN
ma-267	175	18	and	and	CCONJ
ma-267	175	19	kε	kε	PROPN
ma-267	175	20	,	,	PUNCT
ma-267	175	21	δ(λ	δ(λ	PROPN
ma-267	175	22	,	,	PUNCT
ma-267	175	23	m	m	PROPN
ma-267	175	24	)	)	PUNCT
ma-267	175	25	:	:	PUNCT
ma-267	175	26	=	=	SYM
ma-267	175	27	1	1	NUM
ma-267	175	28	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	175	29	∫	∫	PROPN
ma-267	175	30	δ	δ	PROPN
ma-267	175	31	ε	ε	PROPN
ma-267	175	32	fα(ψ1	fα(ψ1	NOUN
ma-267	175	33	)	)	PUNCT
ma-267	175	34	(	(	PUNCT
ma-267	175	35	µ	µ	X
ma-267	175	36	a	a	PRON
ma-267	175	37	,	,	PUNCT
ma-267	175	38	λ	λ	X
ma-267	175	39	a	a	PRON
ma-267	175	40	)	)	PUNCT
ma-267	175	41	fα(ψ2	fα(ψ2	NOUN
ma-267	175	42	)	)	PUNCT
ma-267	175	43	(	(	PUNCT
ma-267	175	44	µ	µ	X
ma-267	175	45	a	a	PRON
ma-267	175	46	,	,	PUNCT
ma-267	175	47	λ	λ	X
ma-267	175	48	a	a	PRON
ma-267	175	49	)	)	PUNCT
ma-267	175	50	da	da	NOUN
ma-267	175	51	a	a	PRON
ma-267	175	52	.under	.under	PUNCT
ma-267	175	53	the	the	DET
ma-267	175	54	condition	condition	NOUN
ma-267	175	55	(	(	PUNCT
ma-267	175	56	3.11	3.11	NUM
ma-267	175	57	)	)	PUNCT
ma-267	175	58	we	we	PRON
ma-267	175	59	have	have	VERB
ma-267	175	60	gε	gε	PROPN
ma-267	175	61	,	,	PUNCT
ma-267	175	62	δ	δ	PROPN
ma-267	175	63	∈	∈	PROPN
ma-267	175	64	l2α(k	l2α(k	PROPN
ma-267	175	65	)	)	PUNCT
ma-267	175	66	,	,	PUNCT
ma-267	175	67	kε	kε	PROPN
ma-267	175	68	,	,	PUNCT
ma-267	175	69	δ	δ	PROPN
ma-267	175	70	∈	∈	PROPN
ma-267	175	71	l1α(k̂	l1α(k̂	PROPN
ma-267	175	72	)	)	PUNCT
ma-267	175	73	∩	∩	ADJ
ma-267	175	74	l∞α	l∞α	NOUN
ma-267	175	75	(	(	PUNCT
ma-267	175	76	k̂	k̂	NOUN
ma-267	175	77	)	)	PUNCT
ma-267	175	78	and	and	CCONJ
ma-267	175	79	fα(gε	fα(gε	PROPN
ma-267	175	80	,	,	PUNCT
ma-267	175	81	δ)(λ	δ)(λ	NOUN
ma-267	175	82	,	,	PUNCT
ma-267	175	83	m	m	NOUN
ma-267	175	84	)	)	PUNCT
ma-267	175	85	=	=	SYM
ma-267	175	86	kε	kε	PROPN
ma-267	175	87	,	,	PUNCT
ma-267	175	88	δ(λ	δ(λ	PROPN
ma-267	175	89	,	,	PUNCT
ma-267	175	90	m	m	PROPN
ma-267	175	91	)	)	PUNCT
ma-267	175	92	(	(	PUNCT
ma-267	175	93	3.12	3.12	NUM
ma-267	175	94	)	)	PUNCT
ma-267	175	95	proof	proof	NOUN
ma-267	175	96	.	.	PUNCT
ma-267	176	1	by	by	ADP
ma-267	176	2	using	use	VERB
ma-267	176	3	hölder	hölder	PROPN
ma-267	176	4	’s	’s	PART
ma-267	176	5	inequality	inequality	NOUN
ma-267	176	6	for	for	ADP
ma-267	176	7	the	the	DET
ma-267	176	8	measure	measure	NOUN
ma-267	176	9	a2α+2da	a2α+2da	VERB
ma-267	176	10	we	we	PRON
ma-267	176	11	obtain	obtain	VERB
ma-267	176	12	‖gε	‖gε	NUM
ma-267	176	13	,	,	PUNCT
ma-267	176	14	δ‖22,µα	δ‖22,µα	PROPN
ma-267	176	15	≤	≤	PROPN
ma-267	176	16	δ2α+3	δ2α+3	NOUN
ma-267	176	17	−	−	PUNCT
ma-267	176	18	ε2α+3	ε2α+3	PROPN
ma-267	176	19	c2ψ1,ψ2	c2ψ1,ψ2	PROPN
ma-267	176	20	∫	∫	PROPN
ma-267	176	21	δ	δ	PROPN
ma-267	176	22	ε	ε	PROPN
ma-267	176	23	(	(	PUNCT
ma-267	176	24	∫	∫	PROPN
ma-267	176	25	k	k	PROPN
ma-267	176	26	∣∣∣(da(ψ̌2	∣∣∣(da(ψ̌2	PROPN
ma-267	176	27	)	)	PUNCT
ma-267	176	28	∗α	∗α	NOUN
ma-267	176	29	da(ψ1	da(ψ1	NOUN
ma-267	176	30	)	)	PUNCT
ma-267	176	31	)	)	PUNCT
ma-267	176	32	(	(	PUNCT
ma-267	176	33	x	x	X
ma-267	176	34	,	,	PUNCT
ma-267	176	35	t	t	PROPN
ma-267	176	36	)	)	PUNCT
ma-267	176	37	∣∣∣2	∣∣∣2	NOUN
ma-267	177	1	dµα(x	dµα(x	PROPN
ma-267	177	2	,	,	PUNCT
ma-267	177	3	t	t	PROPN
ma-267	177	4	)	)	PUNCT
ma-267	177	5	)	)	PUNCT
ma-267	177	6	a2α+2da	a2α+2da	PROPN
ma-267	177	7	,	,	PUNCT
ma-267	177	8	by	by	ADP
ma-267	177	9	using	use	VERB
ma-267	177	10	the	the	DET
ma-267	177	11	relations	relation	NOUN
ma-267	177	12	(	(	PUNCT
ma-267	177	13	2.13	2.13	NUM
ma-267	177	14	)	)	PUNCT
ma-267	177	15	and	and	CCONJ
ma-267	178	1	(	(	PUNCT
ma-267	178	2	3.4	3.4	NUM
ma-267	178	3	)	)	PUNCT
ma-267	178	4	we	we	PRON
ma-267	178	5	find	find	VERB
ma-267	178	6	that	that	SCONJ
ma-267	178	7	‖gε	‖gε	PROPN
ma-267	178	8	,	,	PUNCT
ma-267	178	9	δ‖22,µα	δ‖22,µα	PROPN
ma-267	178	10	≤	≤	PROPN
ma-267	178	11	δ2α+3	δ2α+3	NOUN
ma-267	178	12	−	−	PRON
ma-267	178	13	ε2α+3	ε2α+3	PROPN
ma-267	178	14	c2ψ1,ψ2	c2ψ1,ψ2	PROPN
ma-267	178	15	‖fα(da(ψ2))‖2∞,γα‖ψ1‖	‖fα(da(ψ2))‖2∞,γα‖ψ1‖	VERB
ma-267	178	16	2	2	NUM
ma-267	178	17	2,µα	2,µα	NUM
ma-267	178	18	∫	∫	PROPN
ma-267	178	19	δ	δ	PROPN
ma-267	178	20	ε	ε	PROPN
ma-267	178	21	da	da	PROPN
ma-267	178	22	a	a	DET
ma-267	178	23	<	<	X
ma-267	178	24	∞.	∞.	PROPN
ma-267	178	25	which	which	PRON
ma-267	178	26	prove	prove	VERB
ma-267	178	27	that	that	SCONJ
ma-267	178	28	gε	gε	NOUN
ma-267	178	29	,	,	PUNCT
ma-267	178	30	δ	δ	PROPN
ma-267	178	31	∈	∈	PROPN
ma-267	178	32	l2α(k	l2α(k	PROPN
ma-267	178	33	)	)	PUNCT
ma-267	178	34	,	,	PUNCT
ma-267	178	35	the	the	DET
ma-267	178	36	result	result	NOUN
ma-267	178	37	kε	kε	PROPN
ma-267	178	38	,	,	PUNCT
ma-267	178	39	δ	δ	PROPN
ma-267	178	40	∈	∈	PROPN
ma-267	178	41	l1α(k̂	l1α(k̂	PROPN
ma-267	178	42	)	)	PUNCT
ma-267	178	43	∩	∩	ADJ
ma-267	178	44	l∞α	l∞α	NOUN
ma-267	178	45	(	(	PUNCT
ma-267	178	46	k̂	k̂	NOUN
ma-267	178	47	)	)	PUNCT
ma-267	178	48	can	can	AUX
ma-267	178	49	be	be	AUX
ma-267	178	50	easily	easily	ADV
ma-267	178	51	checked	check	VERB
ma-267	178	52	,	,	PUNCT
ma-267	178	53	on	on	ADP
ma-267	178	54	theother	theother	ADJ
ma-267	178	55	hand	hand	NOUN
ma-267	178	56	by	by	ADP
ma-267	178	57	using	use	VERB
ma-267	178	58	the	the	DET
ma-267	178	59	relations	relation	NOUN
ma-267	178	60	(	(	PUNCT
ma-267	178	61	2.5	2.5	NUM
ma-267	178	62	)	)	PUNCT
ma-267	178	63	,	,	PUNCT
ma-267	178	64	(	(	PUNCT
ma-267	178	65	2.10	2.10	NUM
ma-267	178	66	)	)	PUNCT
ma-267	178	67	,	,	PUNCT
ma-267	178	68	(	(	PUNCT
ma-267	178	69	3.4	3.4	NUM
ma-267	178	70	)	)	PUNCT
ma-267	178	71	and	and	CCONJ
ma-267	178	72	fubini	fubini	NOUN
ma-267	178	73	’s	’s	PART
ma-267	178	74	theorem	theorem	NOUN
ma-267	178	75	we	we	PRON
ma-267	178	76	find	find	VERB
ma-267	178	77	that	that	SCONJ
ma-267	178	78	gε	gε	NOUN
ma-267	178	79	,	,	PUNCT
ma-267	178	80	δ(x	δ(x	PROPN
ma-267	178	81	,	,	PUNCT
ma-267	178	82	t	t	PROPN
ma-267	178	83	)	)	PUNCT
ma-267	178	84	=	=	SYM
ma-267	179	1	∫	∫	PROPN
ma-267	179	2	k̂	k̂	PROPN
ma-267	179	3	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	179	4	,	,	PUNCT
ma-267	179	5	t)kε	t)kε	PROPN
ma-267	179	6	,	,	PUNCT
ma-267	179	7	δ(µ	δ(µ	NOUN
ma-267	179	8	,	,	PUNCT
ma-267	179	9	λ)dγα(µ	λ)dγα(µ	PROPN
ma-267	179	10	,	,	PUNCT
ma-267	179	11	λ	λ	NOUN
ma-267	179	12	)	)	PUNCT
ma-267	179	13	,	,	PUNCT
ma-267	179	14	inversion	inversion	NOUN
ma-267	179	15	formula	formula	NOUN
ma-267	179	16	(	(	PUNCT
ma-267	179	17	2.5	2.5	NUM
ma-267	179	18	)	)	PUNCT
ma-267	179	19	gives	give	VERB
ma-267	179	20	the	the	DET
ma-267	179	21	relation	relation	NOUN
ma-267	179	22	(	(	PUNCT
ma-267	179	23	3.12	3.12	NUM
ma-267	179	24	)	)	PUNCT
ma-267	179	25	.	.	PUNCT
ma-267	180	1	�	�	PROPN
ma-267	180	2	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	180	3	eur	eur	PROPN
ma-267	180	4	.	.	PUNCT
ma-267	181	1	j.	j.	PROPN
ma-267	181	2	math	math	PROPN
ma-267	181	3	.	.	PUNCT
ma-267	182	1	anal	anal	PROPN
ma-267	182	2	.	.	PUNCT
ma-267	183	1	10.28924	10.28924	NUM
ma-267	183	2	/	/	SYM
ma-267	183	3	ada	ada	PROPN
ma-267	183	4	/	/	SYM
ma-267	183	5	ma.4.20	ma.4.20	PROPN
ma-267	183	6	9we	9we	PROPN
ma-267	183	7	can	can	AUX
ma-267	183	8	now	now	ADV
ma-267	183	9	state	state	VERB
ma-267	183	10	the	the	DET
ma-267	183	11	main	main	ADJ
ma-267	183	12	result	result	NOUN
ma-267	183	13	of	of	ADP
ma-267	183	14	this	this	DET
ma-267	183	15	section	section	NOUN
ma-267	183	16	theorem	theorem	VERB
ma-267	183	17	3.3	3.3	NUM
ma-267	183	18	.	.	PUNCT
ma-267	184	1	(	(	PUNCT
ma-267	184	2	first	first	PROPN
ma-267	184	3	calderón	calderón	PROPN
ma-267	184	4	’s	’s	PART
ma-267	184	5	reproducing	reproduce	VERB
ma-267	184	6	formula)let	formula)let	NOUN
ma-267	184	7	(	(	PUNCT
ma-267	184	8	ψ1	ψ1	NOUN
ma-267	184	9	,	,	PUNCT
ma-267	184	10	ψ2	ψ2	NOUN
ma-267	184	11	)	)	PUNCT
ma-267	184	12	be	be	VERB
ma-267	184	13	a	a	DET
ma-267	184	14	reimann	reimann	NOUN
ma-267	184	15	-	-	PUNCT
ma-267	184	16	liouville	liouville	NOUN
ma-267	184	17	two	two	NUM
ma-267	184	18	-	-	PUNCT
ma-267	184	19	wavelet	wavelet	NOUN
ma-267	184	20	satisfying	satisfy	VERB
ma-267	184	21	the	the	DET
ma-267	184	22	condition	condition	NOUN
ma-267	184	23	(	(	PUNCT
ma-267	184	24	3.11	3.11	NUM
ma-267	184	25	)	)	PUNCT
ma-267	184	26	and	and	CCONJ
ma-267	184	27	let	let	VERB
ma-267	184	28	0	0	NUM
ma-267	184	29	<	<	X
ma-267	184	30	ε	ε	PROPN
ma-267	184	31	<	<	X
ma-267	184	32	δ	δ	X
ma-267	184	33	<	<	X
ma-267	184	34	∞	∞	PROPN
ma-267	184	35	then	then	ADV
ma-267	184	36	for	for	ADP
ma-267	184	37	all	all	DET
ma-267	184	38	f	f	PROPN
ma-267	184	39	∈	∈	PROPN
ma-267	184	40	l2α(k	l2α(k	PROPN
ma-267	184	41	)	)	PUNCT
ma-267	184	42	,	,	PUNCT
ma-267	184	43	the	the	DET
ma-267	184	44	function	function	NOUN
ma-267	184	45	fε	fε	NOUN
ma-267	184	46	,	,	PUNCT
ma-267	184	47	δ	δ	PROPN
ma-267	184	48	given	give	VERB
ma-267	184	49	by	by	ADP
ma-267	184	50	fε	fε	NOUN
ma-267	184	51	,	,	PUNCT
ma-267	184	52	δ(x	δ(x	PROPN
ma-267	184	53	,	,	PUNCT
ma-267	184	54	t	t	PROPN
ma-267	184	55	)	)	PUNCT
ma-267	184	56	=	=	SYM
ma-267	184	57	1	1	NUM
ma-267	184	58	cψ1,ψ2	cψ1,ψ2	NOUN
ma-267	184	59	∫	∫	PROPN
ma-267	184	60	δ	δ	PROPN
ma-267	184	61	ε	ε	PROPN
ma-267	184	62	(	(	PUNCT
ma-267	184	63	∫	∫	PROPN
ma-267	184	64	k	k	PROPN
ma-267	184	65	sαψ1(f	sαψ1(f	PROPN
ma-267	184	66	)	)	PUNCT
ma-267	184	67	(	(	PUNCT
ma-267	184	68	a	a	X
ma-267	184	69	,	,	PUNCT
ma-267	184	70	y	y	PROPN
ma-267	184	71	,	,	PUNCT
ma-267	184	72	s)ψ2,a	s)ψ2,a	PROPN
ma-267	184	73	,	,	PUNCT
ma-267	184	74	x	x	NOUN
ma-267	184	75	,	,	PUNCT
ma-267	184	76	t(y	t(y	ADV
ma-267	184	77	,	,	PUNCT
ma-267	184	78	s)dµα(y	s)dµα(y	X
ma-267	184	79	,	,	PUNCT
ma-267	184	80	s	s	NOUN
ma-267	184	81	)	)	PUNCT
ma-267	184	82	)	)	PUNCT
ma-267	185	1	a2α+2da	a2α+2da	PROPN
ma-267	185	2	,	,	PUNCT
ma-267	185	3	belongs	belong	VERB
ma-267	185	4	to	to	ADP
ma-267	185	5	l2α(k	l2α(k	PROPN
ma-267	185	6	)	)	PUNCT
ma-267	185	7	and	and	CCONJ
ma-267	185	8	satisfies	satisfy	VERB
ma-267	185	9	lim	lim	PROPN
ma-267	185	10	ε→0,δ→∞	ε→0,δ→∞	PROPN
ma-267	185	11	‖fε	‖fε	PROPN
ma-267	185	12	,	,	PUNCT
ma-267	185	13	δ	δ	PROPN
ma-267	185	14	−	−	PROPN
ma-267	185	15	f	f	PROPN
ma-267	185	16	‖2,µα	‖2,µα	NUM
ma-267	185	17	=	=	SYM
ma-267	185	18	0	0	X
ma-267	185	19	.	.	PUNCT
ma-267	186	1	(	(	PUNCT
ma-267	186	2	3.13	3.13	NUM
ma-267	186	3	)	)	PUNCT
ma-267	186	4	proof	proof	NOUN
ma-267	186	5	.	.	PUNCT
ma-267	187	1	it	it	PRON
ma-267	187	2	is	be	AUX
ma-267	187	3	easy	easy	ADJ
ma-267	187	4	to	to	PART
ma-267	187	5	see	see	VERB
ma-267	187	6	that	that	SCONJ
ma-267	187	7	fε	fε	NOUN
ma-267	187	8	,	,	PUNCT
ma-267	187	9	δ	δ	PROPN
ma-267	187	10	=	=	SYM
ma-267	187	11	f	f	PROPN
ma-267	187	12	∗α	∗α	PROPN
ma-267	187	13	gε	gε	PROPN
ma-267	187	14	,	,	PUNCT
ma-267	187	15	δthen	δthen	ADV
ma-267	187	16	by	by	ADP
ma-267	187	17	using	use	VERB
ma-267	187	18	the	the	DET
ma-267	187	19	relations	relation	NOUN
ma-267	187	20	(	(	PUNCT
ma-267	187	21	2.7	2.7	NUM
ma-267	187	22	)	)	PUNCT
ma-267	187	23	and	and	CCONJ
ma-267	187	24	(	(	PUNCT
ma-267	187	25	3.12	3.12	NUM
ma-267	187	26	)	)	PUNCT
ma-267	187	27	we	we	PRON
ma-267	187	28	find	find	VERB
ma-267	187	29	that	that	SCONJ
ma-267	187	30	‖fε	‖fε	NOUN
ma-267	187	31	,	,	PUNCT
ma-267	187	32	δ	δ	PROPN
ma-267	187	33	−	−	PROPN
ma-267	187	34	f	f	PROPN
ma-267	187	35	‖22,µα	‖22,µα	PROPN
ma-267	187	36	=	=	SYM
ma-267	187	37	∫	∫	PROPN
ma-267	187	38	k̂	k̂	PROPN
ma-267	187	39	|fα(f	|fα(f	X
ma-267	187	40	)	)	PUNCT
ma-267	187	41	(	(	PUNCT
ma-267	187	42	µ	µ	NUM
ma-267	187	43	,	,	PUNCT
ma-267	187	44	λ)|2(1−kε	λ)|2(1−kε	NOUN
ma-267	187	45	,	,	PUNCT
ma-267	187	46	δ(µ	δ(µ	NOUN
ma-267	187	47	,	,	PUNCT
ma-267	187	48	λ))2dγα(µ	λ))2dγα(µ	PUNCT
ma-267	187	49	,	,	PUNCT
ma-267	187	50	λ	λ	PROPN
ma-267	187	51	)	)	PUNCT
ma-267	187	52	,	,	PUNCT
ma-267	187	53	the	the	DET
ma-267	187	54	relation	relation	NOUN
ma-267	187	55	(	(	PUNCT
ma-267	187	56	3.13	3.13	NUM
ma-267	187	57	)	)	PUNCT
ma-267	187	58	follows	follow	VERB
ma-267	187	59	from	from	ADP
ma-267	187	60	the	the	DET
ma-267	187	61	admissibility	admissibility	NOUN
ma-267	187	62	condition	condition	NOUN
ma-267	187	63	(	(	PUNCT
ma-267	187	64	3.1	3.1	NUM
ma-267	187	65	)	)	PUNCT
ma-267	187	66	and	and	CCONJ
ma-267	187	67	the	the	DET
ma-267	187	68	dominated	dominate	VERB
ma-267	187	69	convergencetheorem	convergencetheorem	PROPN
ma-267	187	70	.	.	PUNCT
ma-267	188	1	�	�	PROPN
ma-267	188	2	4	4	NUM
ma-267	188	3	.	.	PUNCT
ma-267	188	4	extremal	extremal	ADJ
ma-267	188	5	functions	function	NOUN
ma-267	188	6	associated	associate	VERB
ma-267	188	7	with	with	ADP
ma-267	188	8	the	the	DET
ma-267	188	9	riemann	riemann	PROPN
ma-267	188	10	-	-	PUNCT
ma-267	188	11	liouville	liouville	VERB
ma-267	188	12	wavelet	wavelet	NOUN
ma-267	188	13	transform	transform	NOUN
ma-267	188	14	by	by	ADP
ma-267	188	15	using	use	VERB
ma-267	188	16	the	the	DET
ma-267	188	17	theory	theory	NOUN
ma-267	188	18	of	of	ADP
ma-267	188	19	reproducing	reproduce	VERB
ma-267	188	20	kernels	kernel	NOUN
ma-267	188	21	[	[	X
ma-267	188	22	18,19	18,19	NUM
ma-267	188	23	]	]	PUNCT
ma-267	188	24	,	,	PUNCT
ma-267	188	25	the	the	DET
ma-267	188	26	main	main	ADJ
ma-267	188	27	purpose	purpose	NOUN
ma-267	188	28	of	of	ADP
ma-267	188	29	this	this	DET
ma-267	188	30	section	section	NOUN
ma-267	188	31	is	be	AUX
ma-267	188	32	to	to	ADP
ma-267	188	33	studythe	studythe	ADJ
ma-267	188	34	extremal	extremal	ADJ
ma-267	188	35	functions	function	NOUN
ma-267	188	36	associated	associate	VERB
ma-267	188	37	with	with	ADP
ma-267	188	38	the	the	DET
ma-267	188	39	riemann	riemann	PROPN
ma-267	188	40	-	-	PUNCT
ma-267	188	41	liouville	liouville	VERB
ma-267	188	42	wavelet	wavelet	NOUN
ma-267	188	43	transform	transform	NOUN
ma-267	188	44	and	and	CCONJ
ma-267	188	45	to	to	PART
ma-267	188	46	give	give	VERB
ma-267	188	47	anintegral	anintegral	ADJ
ma-267	188	48	representation	representation	NOUN
ma-267	188	49	and	and	CCONJ
ma-267	188	50	best	good	ADJ
ma-267	188	51	estimate	estimate	NOUN
ma-267	188	52	of	of	ADP
ma-267	188	53	these	these	DET
ma-267	188	54	functions	function	NOUN
ma-267	188	55	on	on	ADP
ma-267	188	56	weighted	weight	VERB
ma-267	188	57	sobolev	sobolev	NOUN
ma-267	188	58	spaces	space	NOUN
ma-267	188	59	.	.	PUNCT
ma-267	189	1	4.1	4.1	NUM
ma-267	189	2	.	.	PUNCT
ma-267	189	3	sobolev	sobolev	ADJ
ma-267	189	4	type	type	NOUN
ma-267	189	5	spaces	space	NOUN
ma-267	189	6	associated	associate	VERB
ma-267	189	7	with	with	ADP
ma-267	189	8	the	the	DET
ma-267	189	9	riemann	riemann	PROPN
ma-267	189	10	-	-	PUNCT
ma-267	189	11	liouville	liouville	NOUN
ma-267	189	12	transform	transform	NOUN
ma-267	189	13	.	.	PUNCT
ma-267	190	1	let	let	VERB
ma-267	190	2	s	s	PRON
ma-267	190	3	>	>	X
ma-267	190	4	0	0	NUM
ma-267	190	5	,	,	PUNCT
ma-267	190	6	we	we	PRON
ma-267	190	7	definethe	definethe	VERB
ma-267	190	8	sobolev	sobolev	NOUN
ma-267	190	9	spaces	space	NOUN
ma-267	190	10	associated	associate	VERB
ma-267	190	11	with	with	ADP
ma-267	190	12	the	the	DET
ma-267	190	13	riemann	riemann	PROPN
ma-267	190	14	-	-	PUNCT
ma-267	190	15	liouville	liouville	VERB
ma-267	190	16	transform	transform	NOUN
ma-267	190	17	as	as	ADP
ma-267	190	18	hsα(k	hsα(k	PROPN
ma-267	190	19	)	)	PUNCT
ma-267	190	20	:	:	PUNCT
ma-267	191	1	=	=	X
ma-267	191	2	{	{	PUNCT
ma-267	191	3	f	f	PROPN
ma-267	191	4	∈	∈	PROPN
ma-267	191	5	l2α(k)/	l2α(k)/	PUNCT
ma-267	191	6	(	(	PUNCT
ma-267	191	7	1	1	NUM
ma-267	191	8	+	+	CCONJ
ma-267	191	9	µ2	µ2	PROPN
ma-267	191	10	+	+	CCONJ
ma-267	191	11	2λ2	2λ2	NUM
ma-267	191	12	)	)	PUNCT
ma-267	191	13	s/2	s/2	NOUN
ma-267	191	14	fα(f	fα(f	NUM
ma-267	191	15	)	)	PUNCT
ma-267	191	16	∈	∈	PROPN
ma-267	191	17	l2α(k̂	l2α(k̂	PROPN
ma-267	191	18	)	)	PUNCT
ma-267	191	19	}	}	PUNCT
ma-267	191	20	.	.	PUNCT
ma-267	192	1	the	the	DET
ma-267	192	2	space	space	NOUN
ma-267	192	3	hsα(k	hsα(k	NOUN
ma-267	192	4	)	)	PUNCT
ma-267	192	5	provided	provide	VERB
ma-267	192	6	with	with	ADP
ma-267	192	7	the	the	DET
ma-267	192	8	inner	inner	ADJ
ma-267	192	9	product	product	NOUN
ma-267	192	10	〈	〈	PROPN
ma-267	192	11	f	f	X
ma-267	192	12	,	,	PUNCT
ma-267	192	13	g〉hsα	g〉hsα	PUNCT
ma-267	192	14	:	:	PUNCT
ma-267	192	15	=	=	SYM
ma-267	192	16	∫	∫	PROPN
ma-267	192	17	k̂	k̂	PROPN
ma-267	193	1	(	(	PUNCT
ma-267	193	2	1	1	X
ma-267	193	3	+	+	CCONJ
ma-267	193	4	µ2	µ2	PROPN
ma-267	193	5	+	+	CCONJ
ma-267	193	6	2λ2	2λ2	NUM
ma-267	193	7	)	)	PUNCT
ma-267	193	8	s	s	PART
ma-267	193	9	fα(f	fα(f	NOUN
ma-267	193	10	)	)	PUNCT
ma-267	193	11	(	(	PUNCT
ma-267	193	12	µ	µ	NOUN
ma-267	193	13	,	,	PUNCT
ma-267	193	14	λ)fα(g)(µ	λ)fα(g)(µ	NOUN
ma-267	193	15	,	,	PUNCT
ma-267	193	16	λ)dγα(µ	λ)dγα(µ	X
ma-267	193	17	,	,	PUNCT
ma-267	193	18	λ	λ	NOUN
ma-267	193	19	)	)	PUNCT
ma-267	193	20	,	,	PUNCT
ma-267	193	21	(	(	PUNCT
ma-267	193	22	4.1	4.1	NUM
ma-267	193	23	)	)	PUNCT
ma-267	193	24	and	and	CCONJ
ma-267	193	25	the	the	DET
ma-267	193	26	norm	norm	NOUN
ma-267	193	27	‖f	‖f	ADP
ma-267	193	28	‖2hsα	‖2hsα	NOUN
ma-267	193	29	:	:	PUNCT
ma-267	193	30	=	=	PUNCT
ma-267	193	31	〈	〈	PROPN
ma-267	193	32	f	f	PROPN
ma-267	193	33	,	,	PUNCT
ma-267	193	34	f	f	PROPN
ma-267	193	35	〉	〉	PROPN
ma-267	193	36	hsα	hsα	NOUN
ma-267	193	37	=	=	SYM
ma-267	193	38	∫	∫	PROPN
ma-267	193	39	k̂	k̂	PROPN
ma-267	194	1	(	(	PUNCT
ma-267	194	2	1	1	X
ma-267	194	3	+	+	CCONJ
ma-267	194	4	µ2	µ2	PROPN
ma-267	194	5	+	+	CCONJ
ma-267	194	6	2λ2	2λ2	NUM
ma-267	194	7	)	)	PUNCT
ma-267	194	8	s	s	PART
ma-267	194	9	|fα(f	|fα(f	X
ma-267	194	10	)	)	PUNCT
ma-267	194	11	(	(	PUNCT
ma-267	194	12	µ	µ	NOUN
ma-267	194	13	,	,	PUNCT
ma-267	194	14	λ)|2dγα(µ	λ)|2dγα(µ	ADV
ma-267	194	15	,	,	PUNCT
ma-267	194	16	λ	λ	PROPN
ma-267	194	17	)	)	PUNCT
ma-267	194	18	,	,	PUNCT
ma-267	194	19	(	(	PUNCT
ma-267	194	20	4.2	4.2	NUM
ma-267	194	21	)	)	PUNCT
ma-267	194	22	is	be	AUX
ma-267	194	23	a	a	DET
ma-267	194	24	hilbert	hilbert	NOUN
ma-267	194	25	space	space	NOUN
ma-267	194	26	.	.	PUNCT
ma-267	195	1	definition	definition	NOUN
ma-267	195	2	4.1	4.1	NUM
ma-267	195	3	.	.	PUNCT
ma-267	196	1	let	let	VERB
ma-267	196	2	ψ	ψ	PART
ma-267	196	3	be	be	AUX
ma-267	196	4	a	a	DET
ma-267	196	5	riemann	riemann	PROPN
ma-267	196	6	-	-	PUNCT
ma-267	196	7	liouville	liouville	NOUN
ma-267	196	8	wavelet	wavelet	NOUN
ma-267	196	9	on	on	ADP
ma-267	196	10	k	k	PROPN
ma-267	196	11	in	in	ADP
ma-267	196	12	l2α(k	l2α(k	PROPN
ma-267	196	13	)	)	PUNCT
ma-267	196	14	,	,	PUNCT
ma-267	196	15	we	we	PRON
ma-267	196	16	introduce	introduce	VERB
ma-267	196	17	the	the	DET
ma-267	196	18	innerproduct	innerproduct	NOUN
ma-267	196	19	in	in	ADP
ma-267	196	20	the	the	DET
ma-267	196	21	hilbert	hilbert	NOUN
ma-267	196	22	space	space	NOUN
ma-267	196	23	hsα(k	hsα(k	PROPN
ma-267	196	24	)	)	PUNCT
ma-267	196	25	for	for	ADP
ma-267	196	26	any	any	DET
ma-267	196	27	fixed	fixed	ADJ
ma-267	196	28	β	β	X
ma-267	196	29	>	>	X
ma-267	196	30	0	0	PUNCT
ma-267	197	1	by	by	ADP
ma-267	197	2	〈	〈	PROPN
ma-267	197	3	f	f	PROPN
ma-267	197	4	,	,	PUNCT
ma-267	197	5	g〉hsψ	g〉hsψ	PROPN
ma-267	197	6	,	,	PUNCT
ma-267	197	7	β	β	X
ma-267	197	8	:	:	PUNCT
ma-267	197	9	=	=	SYM
ma-267	197	10	β〈f	β〈f	PRON
ma-267	197	11	,	,	PUNCT
ma-267	197	12	g〉hsα	g〉hsα	PROPN
ma-267	197	13	+	+	CCONJ
ma-267	198	1	〈	〈	PROPN
ma-267	198	2	sαψ(f	sαψ(f	NOUN
ma-267	198	3	)	)	PUNCT
ma-267	198	4	,	,	PUNCT
ma-267	198	5	sαψ(g)〉θα	sαψ(g)〉θα	PROPN
ma-267	198	6	,	,	PUNCT
ma-267	198	7	(	(	PUNCT
ma-267	198	8	4.3	4.3	NUM
ma-267	198	9	)	)	PUNCT
ma-267	198	10	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	198	11	eur	eur	PROPN
ma-267	198	12	.	.	PUNCT
ma-267	199	1	j.	j.	PROPN
ma-267	199	2	math	math	PROPN
ma-267	199	3	.	.	PUNCT
ma-267	200	1	anal	anal	PROPN
ma-267	200	2	.	.	PUNCT
ma-267	201	1	10.28924	10.28924	NUM
ma-267	201	2	/	/	SYM
ma-267	201	3	ada	ada	PROPN
ma-267	201	4	/	/	SYM
ma-267	201	5	ma.4.20	ma.4.20	PROPN
ma-267	201	6	10the	10the	DET
ma-267	201	7	norm	norm	NOUN
ma-267	201	8	associated	associate	VERB
ma-267	201	9	to	to	ADP
ma-267	201	10	this	this	DET
ma-267	201	11	inner	inner	ADJ
ma-267	201	12	product	product	NOUN
ma-267	201	13	is	be	AUX
ma-267	201	14	defined	define	VERB
ma-267	201	15	by	by	ADP
ma-267	201	16	‖f	‖f	ADJ
ma-267	201	17	‖2hsψ	‖2hsψ	NOUN
ma-267	201	18	,	,	PUNCT
ma-267	201	19	β	β	X
ma-267	201	20	:	:	PUNCT
ma-267	201	21	=	=	SYM
ma-267	201	22	β‖f	β‖f	ADJ
ma-267	201	23	‖2hsα	‖2hsα	NOUN
ma-267	201	24	+	+	CCONJ
ma-267	201	25	‖sαψ1(f	‖sαψ1(f	NUM
ma-267	201	26	)	)	PUNCT
ma-267	201	27	‖22,θα	‖22,θα	PROPN
ma-267	201	28	.	.	PUNCT
ma-267	202	1	(	(	PUNCT
ma-267	202	2	4.4	4.4	NUM
ma-267	202	3	)	)	PUNCT
ma-267	202	4	we	we	PRON
ma-267	202	5	have	have	VERB
ma-267	202	6	the	the	DET
ma-267	202	7	following	follow	VERB
ma-267	202	8	result	result	NOUN
ma-267	202	9	proposition	proposition	NOUN
ma-267	202	10	4.1	4.1	NUM
ma-267	202	11	.	.	PUNCT
ma-267	203	1	let	let	VERB
ma-267	203	2	s	s	PRON
ma-267	203	3	>	>	X
ma-267	203	4	2α+3	2α+3	PROPN
ma-267	203	5	2	2	NUM
ma-267	203	6	,	,	PUNCT
ma-267	203	7	ψ	ψ	AUX
ma-267	203	8	be	be	AUX
ma-267	203	9	ariemann	ariemann	ADJ
ma-267	203	10	-	-	PUNCT
ma-267	203	11	liouville	liouville	NOUN
ma-267	203	12	wavelet	wavelet	NOUN
ma-267	203	13	on	on	ADP
ma-267	203	14	k	k	PROPN
ma-267	203	15	in	in	ADP
ma-267	203	16	l2α(k	l2α(k	PROPN
ma-267	203	17	)	)	PUNCT
ma-267	203	18	and	and	CCONJ
ma-267	203	19	β	β	X
ma-267	203	20	>	>	X
ma-267	203	21	0	0	PUNCT
ma-267	204	1	thenwe	thenwe	PROPN
ma-267	204	2	have	have	VERB
ma-267	204	3	f	f	PROPN
ma-267	204	4	∈	∈	PROPN
ma-267	204	5	hsψ	hsψ	NOUN
ma-267	204	6	,	,	PUNCT
ma-267	204	7	β(k)⇒	β(k)⇒	NOUN
ma-267	204	8	fα(f	fα(f	NOUN
ma-267	204	9	)	)	PUNCT
ma-267	205	1	∈	∈	PROPN
ma-267	205	2	l1α(k̂	l1α(k̂	PROPN
ma-267	205	3	)	)	PUNCT
ma-267	205	4	(	(	PUNCT
ma-267	205	5	4.5	4.5	X
ma-267	205	6	)	)	PUNCT
ma-267	205	7	proof	proof	NOUN
ma-267	205	8	.	.	PUNCT
ma-267	206	1	let	let	VERB
ma-267	206	2	f	f	PROPN
ma-267	206	3	∈	∈	PROPN
ma-267	206	4	hsψ	hsψ	PROPN
ma-267	206	5	,	,	PUNCT
ma-267	206	6	β(k	β(k	PROPN
ma-267	206	7	)	)	PUNCT
ma-267	206	8	,	,	PUNCT
ma-267	206	9	by	by	ADP
ma-267	206	10	using	use	VERB
ma-267	206	11	the	the	DET
ma-267	206	12	relations	relation	NOUN
ma-267	206	13	(	(	PUNCT
ma-267	206	14	2.9	2.9	NUM
ma-267	206	15	)	)	PUNCT
ma-267	206	16	,	,	PUNCT
ma-267	206	17	(	(	PUNCT
ma-267	206	18	3.9	3.9	NUM
ma-267	206	19	)	)	PUNCT
ma-267	206	20	,	,	PUNCT
ma-267	206	21	(	(	PUNCT
ma-267	206	22	4.2	4.2	NUM
ma-267	206	23	)	)	PUNCT
ma-267	206	24	and	and	CCONJ
ma-267	206	25	(	(	PUNCT
ma-267	206	26	4.4	4.4	NUM
ma-267	206	27	)	)	PUNCT
ma-267	206	28	we	we	PRON
ma-267	206	29	find	find	VERB
ma-267	206	30	that	that	SCONJ
ma-267	206	31	‖f	‖f	ADJ
ma-267	206	32	‖2hsψ	‖2hsψ	NOUN
ma-267	206	33	,	,	PUNCT
ma-267	206	34	β	β	X
ma-267	206	35	=	=	SYM
ma-267	206	36	∫	∫	PROPN
ma-267	207	1	k̂	k̂	X
ma-267	208	1	[	[	PUNCT
ma-267	208	2	β	β	X
ma-267	208	3	(	(	PUNCT
ma-267	208	4	1	1	NUM
ma-267	208	5	+	+	CCONJ
ma-267	208	6	µ2	µ2	PROPN
ma-267	208	7	+	+	CCONJ
ma-267	208	8	2λ2	2λ2	NUM
ma-267	208	9	)	)	PUNCT
ma-267	208	10	s	s	PART
ma-267	209	1	+	+	X
ma-267	209	2	cψ	cψ	X
ma-267	209	3	]	]	PUNCT
ma-267	209	4	|fα(f	|fα(f	X
ma-267	209	5	)	)	PUNCT
ma-267	209	6	(	(	PUNCT
ma-267	209	7	µ	µ	NOUN
ma-267	209	8	,	,	PUNCT
ma-267	209	9	λ)|2dγα(µ	λ)|2dγα(µ	ADV
ma-267	209	10	,	,	PUNCT
ma-267	209	11	λ	λ	X
ma-267	209	12	)	)	PUNCT
ma-267	209	13	(	(	PUNCT
ma-267	209	14	4.6	4.6	NUM
ma-267	209	15	)	)	PUNCT
ma-267	209	16	by	by	ADP
ma-267	209	17	using	use	VERB
ma-267	209	18	hölder	hölder	PROPN
ma-267	209	19	’s	’s	PART
ma-267	209	20	inequality	inequality	NOUN
ma-267	209	21	,	,	PUNCT
ma-267	209	22	the	the	DET
ma-267	209	23	relation	relation	NOUN
ma-267	209	24	(	(	PUNCT
ma-267	209	25	2.1	2.1	NUM
ma-267	209	26	)	)	PUNCT
ma-267	209	27	and	and	CCONJ
ma-267	209	28	the	the	DET
ma-267	209	29	fact	fact	NOUN
ma-267	209	30	that	that	SCONJ
ma-267	209	31	s	s	VERB
ma-267	209	32	>	>	X
ma-267	209	33	2α+3	2α+3	PROPN
ma-267	209	34	3	3	NUM
ma-267	209	35	we	we	PRON
ma-267	209	36	find	find	VERB
ma-267	209	37	that	that	PRON
ma-267	209	38	‖fα(f	‖fα(f	PUNCT
ma-267	209	39	)	)	PUNCT
ma-267	209	40	‖1,γα	‖1,γα	PUNCT
ma-267	209	41	≤	≤	NOUN
ma-267	209	42	‖f	‖f	ADP
ma-267	209	43	‖hsψ	‖hsψ	NOUN
ma-267	209	44	,	,	PUNCT
ma-267	209	45	β	β	X
ma-267	209	46	(	(	PUNCT
ma-267	209	47	∫	∫	PROPN
ma-267	209	48	k̂	k̂	PROPN
ma-267	209	49	dγα(µ	dγα(µ	PROPN
ma-267	209	50	,	,	PUNCT
ma-267	209	51	λ	λ	PROPN
ma-267	209	52	)	)	PUNCT
ma-267	209	53	β	β	NOUN
ma-267	209	54	(	(	PUNCT
ma-267	209	55	1	1	NUM
ma-267	209	56	+	+	CCONJ
ma-267	209	57	µ2	µ2	PROPN
ma-267	209	58	+	+	CCONJ
ma-267	209	59	2λ2)s	2λ2)s	NUM
ma-267	209	60	+	+	CCONJ
ma-267	209	61	cψ	cψ	NOUN
ma-267	209	62	)	)	PUNCT
ma-267	209	63	1	1	NUM
ma-267	209	64	2	2	NUM
ma-267	209	65	<	<	X
ma-267	209	66	∞	∞	NUM
ma-267	209	67	wich	wich	PRON
ma-267	209	68	give	give	VERB
ma-267	209	69	the	the	DET
ma-267	209	70	result	result	NOUN
ma-267	209	71	.	.	PUNCT
ma-267	210	1	�	�	PROPN
ma-267	210	2	theorem	theorem	VERB
ma-267	210	3	4.1	4.1	NUM
ma-267	210	4	.	.	PUNCT
ma-267	211	1	let	let	VERB
ma-267	211	2	s	s	PRON
ma-267	211	3	>	>	X
ma-267	211	4	2α+3	2α+3	PROPN
ma-267	211	5	2	2	NUM
ma-267	211	6	,	,	PUNCT
ma-267	211	7	ψ	ψ	AUX
ma-267	211	8	be	be	AUX
ma-267	211	9	a	a	DET
ma-267	211	10	riemann	riemann	PROPN
ma-267	211	11	-	-	PUNCT
ma-267	211	12	liouville	liouville	NOUN
ma-267	211	13	wavelet	wavelet	NOUN
ma-267	211	14	on	on	ADP
ma-267	211	15	k	k	PROPN
ma-267	211	16	in	in	ADP
ma-267	211	17	l2α(k	l2α(k	PROPN
ma-267	211	18	)	)	PUNCT
ma-267	211	19	and	and	CCONJ
ma-267	211	20	β	β	X
ma-267	211	21	>	>	X
ma-267	211	22	0	0	PUNCT
ma-267	212	1	then	then	ADV
ma-267	212	2	thespace	thespace	NOUN
ma-267	212	3	(	(	PUNCT
ma-267	212	4	hsψ	hsψ	NOUN
ma-267	212	5	,	,	PUNCT
ma-267	212	6	β(k	β(k	PROPN
ma-267	212	7	)	)	PUNCT
ma-267	212	8	,	,	PUNCT
ma-267	212	9	〈	〈	PROPN
ma-267	212	10	,	,	PUNCT
ma-267	212	11	〉	〉	NOUN
ma-267	212	12	hsψ	hsψ	NOUN
ma-267	212	13	,	,	PUNCT
ma-267	212	14	β	β	X
ma-267	212	15	)	)	PUNCT
ma-267	212	16	is	be	AUX
ma-267	212	17	a	a	DET
ma-267	212	18	reproducing	reproduce	VERB
ma-267	212	19	kernel	kernel	NOUN
ma-267	212	20	hilbert	hilbert	PROPN
ma-267	212	21	space	space	NOUN
ma-267	212	22	with	with	ADP
ma-267	212	23	kernel	kernel	PROPN
ma-267	212	24	function	function	NOUN
ma-267	212	25	given	give	VERB
ma-267	212	26	by	by	ADP
ma-267	212	27	kψ	kψ	PROPN
ma-267	212	28	,	,	PUNCT
ma-267	212	29	β[(x	β[(x	PROPN
ma-267	212	30	,	,	PUNCT
ma-267	212	31	t	t	PROPN
ma-267	212	32	)	)	PUNCT
ma-267	212	33	,	,	PUNCT
ma-267	212	34	(	(	PUNCT
ma-267	212	35	y	y	PROPN
ma-267	212	36	,	,	PUNCT
ma-267	212	37	z	z	NOUN
ma-267	212	38	)	)	PUNCT
ma-267	212	39	]	]	PUNCT
ma-267	213	1	=	=	PUNCT
ma-267	213	2	∫	∫	PROPN
ma-267	213	3	k̂	k̂	PROPN
ma-267	213	4	ϕµ,−λ(x	ϕµ,−λ(x	PROPN
ma-267	213	5	,	,	PUNCT
ma-267	213	6	t)ϕµ,λ(y	t)ϕµ,λ(y	PROPN
ma-267	213	7	,	,	PUNCT
ma-267	213	8	z	z	X
ma-267	213	9	)	)	PUNCT
ma-267	213	10	β	β	NOUN
ma-267	213	11	(	(	PUNCT
ma-267	213	12	1	1	NUM
ma-267	213	13	+	+	CCONJ
ma-267	213	14	µ2	µ2	PROPN
ma-267	213	15	+	+	CCONJ
ma-267	213	16	2λ2)s	2λ2)s	NUM
ma-267	213	17	+	+	CCONJ
ma-267	213	18	cψ	cψ	PROPN
ma-267	213	19	dγα(µ	dγα(µ	PROPN
ma-267	213	20	,	,	PUNCT
ma-267	213	21	λ	λ	PROPN
ma-267	213	22	)	)	PUNCT
ma-267	213	23	(	(	PUNCT
ma-267	213	24	4.7	4.7	NUM
ma-267	213	25	)	)	PUNCT
ma-267	213	26	that	that	PRON
ma-267	213	27	is	be	AUX
ma-267	213	28	for	for	ADP
ma-267	213	29	every	every	DET
ma-267	213	30	(	(	PUNCT
ma-267	213	31	y	y	PROPN
ma-267	213	32	,	,	PUNCT
ma-267	213	33	z	z	X
ma-267	213	34	)	)	PUNCT
ma-267	213	35	∈	∈	PROPN
ma-267	213	36	k,(1	k,(1	PROPN
ma-267	213	37	)	)	PUNCT
ma-267	213	38	the	the	DET
ma-267	213	39	function	function	NOUN
ma-267	213	40	(	(	PUNCT
ma-267	213	41	x	x	X
ma-267	213	42	,	,	PUNCT
ma-267	213	43	t)→	t)→	PROPN
ma-267	213	44	kr	kr	PROPN
ma-267	213	45	,	,	PUNCT
ma-267	213	46	h[(x	h[(x	PROPN
ma-267	213	47	,	,	PUNCT
ma-267	213	48	t	t	PROPN
ma-267	213	49	)	)	PUNCT
ma-267	213	50	,	,	PUNCT
ma-267	213	51	(	(	PUNCT
ma-267	213	52	y	y	PROPN
ma-267	213	53	,	,	PUNCT
ma-267	213	54	z	z	NOUN
ma-267	213	55	)	)	PUNCT
ma-267	213	56	]	]	PUNCT
ma-267	213	57	∈	∈	PROPN
ma-267	213	58	hsψ	hsψ	NOUN
ma-267	213	59	,	,	PUNCT
ma-267	213	60	β(k).(2	β(k).(2	NUM
ma-267	213	61	)	)	PUNCT
ma-267	213	62	for	for	ADP
ma-267	213	63	every	every	DET
ma-267	213	64	f	f	PROPN
ma-267	213	65	∈	∈	PROPN
ma-267	213	66	hsψ	hsψ	NOUN
ma-267	213	67	,	,	PUNCT
ma-267	213	68	β(k	β(k	PROPN
ma-267	213	69	)	)	PUNCT
ma-267	213	70	and	and	CCONJ
ma-267	213	71	(	(	PUNCT
ma-267	213	72	y	y	PROPN
ma-267	213	73	,	,	PUNCT
ma-267	213	74	z	z	X
ma-267	213	75	)	)	PUNCT
ma-267	213	76	∈	∈	PROPN
ma-267	214	1	k	k	NOUN
ma-267	214	2	we	we	PRON
ma-267	214	3	have	have	VERB
ma-267	214	4	f	f	PROPN
ma-267	214	5	(	(	PUNCT
ma-267	214	6	y	y	PROPN
ma-267	214	7	,	,	PUNCT
ma-267	214	8	z	z	NOUN
ma-267	214	9	)	)	PUNCT
ma-267	214	10	=	=	PUNCT
ma-267	214	11	〈	〈	PROPN
ma-267	214	12	f	f	PROPN
ma-267	214	13	,	,	PUNCT
ma-267	214	14	kψ	kψ	PROPN
ma-267	214	15	,	,	PUNCT
ma-267	214	16	β	β	X
ma-267	214	17	[	[	X
ma-267	214	18	·	·	PUNCT
ma-267	214	19	,	,	PUNCT
ma-267	214	20	(	(	PUNCT
ma-267	214	21	y	y	PROPN
ma-267	214	22	,	,	PUNCT
ma-267	214	23	z	z	NOUN
ma-267	214	24	)	)	PUNCT
ma-267	214	25	]	]	PUNCT
ma-267	214	26	〉	〉	NUM
ma-267	214	27	hsψ	hsψ	NOUN
ma-267	214	28	,	,	PUNCT
ma-267	214	29	β	β	X
ma-267	214	30	.	.	PUNCT
ma-267	215	1	proof	proof	NOUN
ma-267	215	2	.	.	PUNCT
ma-267	216	1	let	let	VERB
ma-267	216	2	(	(	PUNCT
ma-267	216	3	y	y	NOUN
ma-267	216	4	,	,	PUNCT
ma-267	216	5	z	z	X
ma-267	216	6	)	)	PUNCT
ma-267	216	7	∈	∈	PROPN
ma-267	217	1	k	k	NOUN
ma-267	217	2	,	,	PUNCT
ma-267	217	3	by	by	ADP
ma-267	217	4	using	use	VERB
ma-267	217	5	the	the	DET
ma-267	217	6	fact	fact	NOUN
ma-267	217	7	that	that	SCONJ
ma-267	217	8	s	s	VERB
ma-267	217	9	>	>	X
ma-267	217	10	2α+3	2α+3	PROPN
ma-267	217	11	2	2	NUM
ma-267	217	12	and	and	CCONJ
ma-267	217	13	the	the	DET
ma-267	217	14	relation	relation	NOUN
ma-267	217	15	(	(	PUNCT
ma-267	217	16	2.3	2.3	NUM
ma-267	217	17	)	)	PUNCT
ma-267	217	18	we	we	PRON
ma-267	217	19	find	find	VERB
ma-267	217	20	that	that	SCONJ
ma-267	217	21	thefunction	thefunction	NOUN
ma-267	217	22	(	(	PUNCT
ma-267	217	23	µ	µ	NOUN
ma-267	217	24	,	,	PUNCT
ma-267	217	25	λ)→	λ)→	PROPN
ma-267	217	26	ϕµ,λ(y	ϕµ,λ(y	NOUN
ma-267	217	27	,	,	PUNCT
ma-267	217	28	z	z	X
ma-267	217	29	)	)	PUNCT
ma-267	217	30	β	β	NOUN
ma-267	217	31	(	(	PUNCT
ma-267	217	32	1	1	NUM
ma-267	217	33	+	+	CCONJ
ma-267	217	34	µ2	µ2	PROPN
ma-267	217	35	+	+	CCONJ
ma-267	217	36	2λ2)s	2λ2)s	NUM
ma-267	217	37	+	+	NUM
ma-267	217	38	cψbelongs	cψbelong	NOUN
ma-267	217	39	to	to	ADP
ma-267	217	40	l1α(k̂)∩l2α(k̂	l1α(k̂)∩l2α(k̂	NUM
ma-267	217	41	)	)	PUNCT
ma-267	217	42	,	,	PUNCT
ma-267	217	43	by	by	ADP
ma-267	217	44	using	use	VERB
ma-267	217	45	plancherel	plancherel	NOUN
ma-267	217	46	’s	’s	PART
ma-267	217	47	theorem	theorem	NOUN
ma-267	217	48	for	for	ADP
ma-267	217	49	the	the	DET
ma-267	217	50	riemann	riemann	PROPN
ma-267	217	51	-	-	PUNCT
ma-267	217	52	liouville	liouville	VERB
ma-267	217	53	transform	transform	NOUN
ma-267	217	54	thereexist	thereexist	VERB
ma-267	217	55	a	a	DET
ma-267	217	56	unique	unique	ADJ
ma-267	217	57	function	function	NOUN
ma-267	217	58	in	in	ADP
ma-267	217	59	l2α(k	l2α(k	PROPN
ma-267	217	60	)	)	PUNCT
ma-267	217	61	,	,	PUNCT
ma-267	217	62	wich	wich	PRON
ma-267	217	63	we	we	PRON
ma-267	217	64	denote	denote	VERB
ma-267	217	65	by	by	ADP
ma-267	217	66	kψ	kψ	PROPN
ma-267	217	67	,	,	PUNCT
ma-267	217	68	β	β	X
ma-267	217	69	[	[	X
ma-267	217	70	·	·	PUNCT
ma-267	217	71	,	,	PUNCT
ma-267	217	72	(	(	PUNCT
ma-267	217	73	y	y	PROPN
ma-267	217	74	,	,	PUNCT
ma-267	217	75	z	z	PROPN
ma-267	217	76	)	)	PUNCT
ma-267	217	77	]	]	PUNCT
ma-267	217	78	such	such	ADJ
ma-267	217	79	that	that	SCONJ
ma-267	217	80	fα	fα	NOUN
ma-267	217	81	(	(	PUNCT
ma-267	217	82	kψ	kψ	NOUN
ma-267	217	83	,	,	PUNCT
ma-267	217	84	β	β	X
ma-267	217	85	[	[	X
ma-267	217	86	·	·	PUNCT
ma-267	217	87	,	,	PUNCT
ma-267	217	88	(	(	PUNCT
ma-267	217	89	y	y	PROPN
ma-267	217	90	,	,	PUNCT
ma-267	217	91	z	z	PROPN
ma-267	217	92	)	)	PUNCT
ma-267	217	93	]	]	PUNCT
ma-267	217	94	)	)	PUNCT
ma-267	218	1	=	=	SYM
ma-267	218	2	ϕµ,λ(y	ϕµ,λ(y	NOUN
ma-267	218	3	,	,	PUNCT
ma-267	218	4	z	z	X
ma-267	218	5	)	)	PUNCT
ma-267	218	6	r	r	NOUN
ma-267	219	1	[	[	X
ma-267	219	2	1	1	NUM
ma-267	219	3	+	+	NUM
ma-267	219	4	λ2	λ2	NOUN
ma-267	219	5	(	(	PUNCT
ma-267	219	6	1	1	NUM
ma-267	219	7	+	+	ADV
ma-267	219	8	m2)]s	m2)]s	ADJ
ma-267	219	9	+	+	CCONJ
ma-267	219	10	ch	ch	NOUN
ma-267	219	11	,	,	PUNCT
ma-267	219	12	(	(	PUNCT
ma-267	219	13	4.8	4.8	NUM
ma-267	219	14	)	)	PUNCT
ma-267	219	15	by	by	ADP
ma-267	219	16	using	use	VERB
ma-267	219	17	the	the	DET
ma-267	219	18	relation	relation	NOUN
ma-267	219	19	(	(	PUNCT
ma-267	219	20	2.5	2.5	NUM
ma-267	219	21	)	)	PUNCT
ma-267	219	22	we	we	PRON
ma-267	219	23	find	find	VERB
ma-267	219	24	that	that	SCONJ
ma-267	219	25	kψ	kψ	NOUN
ma-267	219	26	,	,	PUNCT
ma-267	219	27	β[(x	β[(x	PROPN
ma-267	219	28	,	,	PUNCT
ma-267	219	29	t	t	PROPN
ma-267	219	30	)	)	PUNCT
ma-267	219	31	,	,	PUNCT
ma-267	219	32	(	(	PUNCT
ma-267	219	33	y	y	PROPN
ma-267	219	34	,	,	PUNCT
ma-267	219	35	z	z	NOUN
ma-267	219	36	)	)	PUNCT
ma-267	219	37	]	]	PUNCT
ma-267	220	1	=	=	PUNCT
ma-267	220	2	∫	∫	PROPN
ma-267	220	3	k̂	k̂	PROPN
ma-267	220	4	ϕµ,−λ(x	ϕµ,−λ(x	PROPN
ma-267	220	5	,	,	PUNCT
ma-267	220	6	t)ϕµ,λ(y	t)ϕµ,λ(y	PROPN
ma-267	220	7	,	,	PUNCT
ma-267	220	8	z	z	X
ma-267	220	9	)	)	PUNCT
ma-267	220	10	β	β	NOUN
ma-267	220	11	(	(	PUNCT
ma-267	220	12	1	1	NUM
ma-267	220	13	+	+	CCONJ
ma-267	220	14	µ2	µ2	PROPN
ma-267	220	15	+	+	CCONJ
ma-267	220	16	2λ2)s	2λ2)s	NUM
ma-267	220	17	+	+	CCONJ
ma-267	220	18	cψ	cψ	PROPN
ma-267	220	19	dγα(µ	dγα(µ	PROPN
ma-267	220	20	,	,	PUNCT
ma-267	220	21	λ	λ	PROPN
ma-267	220	22	)	)	PUNCT
ma-267	220	23	,	,	PUNCT
ma-267	220	24	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	220	25	eur	eur	PROPN
ma-267	220	26	.	.	PUNCT
ma-267	221	1	j.	j.	PROPN
ma-267	221	2	math	math	PROPN
ma-267	221	3	.	.	PUNCT
ma-267	222	1	anal	anal	PROPN
ma-267	222	2	.	.	PUNCT
ma-267	223	1	10.28924	10.28924	NUM
ma-267	223	2	/	/	SYM
ma-267	223	3	ada	ada	PROPN
ma-267	223	4	/	/	SYM
ma-267	223	5	ma.4.20	ma.4.20	PROPN
ma-267	223	6	11furthermore	11furthermore	NUM
ma-267	223	7	by	by	ADP
ma-267	223	8	using	use	VERB
ma-267	223	9	the	the	DET
ma-267	223	10	relations	relation	NOUN
ma-267	223	11	(	(	PUNCT
ma-267	223	12	2.3	2.3	NUM
ma-267	223	13	)	)	PUNCT
ma-267	223	14	,	,	PUNCT
ma-267	223	15	(	(	PUNCT
ma-267	223	16	4.6	4.6	NUM
ma-267	223	17	)	)	PUNCT
ma-267	223	18	and	and	CCONJ
ma-267	223	19	(	(	PUNCT
ma-267	223	20	4.8	4.8	NUM
ma-267	223	21	)	)	PUNCT
ma-267	223	22	we	we	PRON
ma-267	223	23	find	find	VERB
ma-267	223	24	that∥∥kψ	that∥∥kψ	PROPN
ma-267	223	25	,	,	PUNCT
ma-267	223	26	β	β	X
ma-267	223	27	[	[	X
ma-267	223	28	·	·	PUNCT
ma-267	223	29	,	,	PUNCT
ma-267	223	30	(	(	PUNCT
ma-267	223	31	y	y	PROPN
ma-267	223	32	,	,	PUNCT
ma-267	223	33	z	z	PROPN
ma-267	223	34	)	)	PUNCT
ma-267	223	35	]	]	PUNCT
ma-267	224	1	∥∥2	∥∥2	PROPN
ma-267	224	2	hsψ	hsψ	NOUN
ma-267	224	3	,	,	PUNCT
ma-267	224	4	β	β	X
ma-267	224	5	≤	≤	NUM
ma-267	224	6	∫	∫	PROPN
ma-267	224	7	k̂	k̂	PROPN
ma-267	224	8	dγα(µ	dγα(µ	PROPN
ma-267	224	9	,	,	PUNCT
ma-267	224	10	λ	λ	PROPN
ma-267	224	11	)	)	PUNCT
ma-267	224	12	β	β	NOUN
ma-267	224	13	(	(	PUNCT
ma-267	224	14	1	1	NUM
ma-267	224	15	+	+	CCONJ
ma-267	224	16	µ2	µ2	PROPN
ma-267	224	17	+	+	CCONJ
ma-267	224	18	2λ2)s	2λ2)s	NUM
ma-267	224	19	+	+	CCONJ
ma-267	224	20	cψ	cψ	PRON
ma-267	224	21	<	<	X
ma-267	224	22	∞.	∞.	PROPN
ma-267	224	23	wich	wich	PRON
ma-267	224	24	proves	prove	VERB
ma-267	224	25	that	that	SCONJ
ma-267	224	26	kψ	kψ	NOUN
ma-267	224	27	,	,	PUNCT
ma-267	224	28	β	β	X
ma-267	224	29	[	[	X
ma-267	224	30	·	·	PUNCT
ma-267	224	31	,	,	PUNCT
ma-267	224	32	(	(	PUNCT
ma-267	224	33	y	y	PROPN
ma-267	224	34	,	,	PUNCT
ma-267	224	35	z	z	NOUN
ma-267	224	36	)	)	PUNCT
ma-267	224	37	]	]	PUNCT
ma-267	224	38	∈	∈	PROPN
ma-267	224	39	hsψ	hsψ	NOUN
ma-267	224	40	,	,	PUNCT
ma-267	224	41	β(k	β(k	PROPN
ma-267	224	42	)	)	PUNCT
ma-267	224	43	.	.	PUNCT
ma-267	225	1	let	let	VERB
ma-267	225	2	f	f	PROPN
ma-267	225	3	∈	∈	PROPN
ma-267	225	4	hsψ	hsψ	PROPN
ma-267	225	5	,	,	PUNCT
ma-267	225	6	β(k	β(k	PROPN
ma-267	225	7	)	)	PUNCT
ma-267	225	8	,	,	PUNCT
ma-267	225	9	by	by	ADP
ma-267	225	10	using	use	VERB
ma-267	225	11	the	the	DET
ma-267	225	12	relations	relation	NOUN
ma-267	225	13	the	the	DET
ma-267	225	14	relations(3.10),(4.1),(4.3	relations(3.10),(4.1),(4.3	NOUN
ma-267	225	15	)	)	PUNCT
ma-267	225	16	,	,	PUNCT
ma-267	225	17	(	(	PUNCT
ma-267	225	18	and	and	CCONJ
ma-267	225	19	(	(	PUNCT
ma-267	225	20	4.8	4.8	NUM
ma-267	225	21	)	)	PUNCT
ma-267	225	22	we	we	PRON
ma-267	225	23	find	find	VERB
ma-267	225	24	that	that	SCONJ
ma-267	225	25	〈	〈	PROPN
ma-267	225	26	f	f	PROPN
ma-267	225	27	,	,	PUNCT
ma-267	225	28	kψ	kψ	PROPN
ma-267	225	29	,	,	PUNCT
ma-267	225	30	β	β	X
ma-267	225	31	[	[	X
ma-267	225	32	·	·	PUNCT
ma-267	225	33	,	,	PUNCT
ma-267	225	34	(	(	PUNCT
ma-267	225	35	y	y	PROPN
ma-267	225	36	,	,	PUNCT
ma-267	225	37	z	z	NOUN
ma-267	225	38	)	)	PUNCT
ma-267	225	39	]	]	PUNCT
ma-267	225	40	〉	〉	PROPN
ma-267	225	41	hsr	hsr	PROPN
ma-267	225	42	,	,	PUNCT
ma-267	225	43	h	h	NOUN
ma-267	225	44	=	=	SYM
ma-267	225	45	∫	∫	PROPN
ma-267	225	46	k̂	k̂	PROPN
ma-267	225	47	ϕµ,λ(y	ϕµ,λ(y	PROPN
ma-267	225	48	,	,	PUNCT
ma-267	225	49	z)fα(f	z)fα(f	NUM
ma-267	225	50	)	)	PUNCT
ma-267	225	51	(	(	PUNCT
ma-267	225	52	µ	µ	NUM
ma-267	225	53	,	,	PUNCT
ma-267	225	54	λ)dγα(µ	λ)dγα(µ	PROPN
ma-267	225	55	,	,	PUNCT
ma-267	225	56	λ	λ	NOUN
ma-267	225	57	)	)	PUNCT
ma-267	225	58	,	,	PUNCT
ma-267	225	59	inversion	inversion	NOUN
ma-267	225	60	formula	formula	NOUN
ma-267	225	61	(	(	PUNCT
ma-267	225	62	2.5	2.5	NUM
ma-267	225	63	)	)	PUNCT
ma-267	225	64	gives	give	VERB
ma-267	225	65	the	the	DET
ma-267	225	66	disered	disered	ADJ
ma-267	225	67	result	result	NOUN
ma-267	225	68	.	.	PUNCT
ma-267	226	1	�	�	PROPN
ma-267	226	2	in	in	ADP
ma-267	226	3	the	the	DET
ma-267	226	4	following	following	NOUN
ma-267	226	5	we	we	PRON
ma-267	226	6	give	give	VERB
ma-267	226	7	the	the	DET
ma-267	226	8	main	main	ADJ
ma-267	226	9	result	result	NOUN
ma-267	226	10	of	of	ADP
ma-267	226	11	this	this	DET
ma-267	226	12	section	section	NOUN
ma-267	226	13	.	.	PUNCT
ma-267	227	1	theorem	theorem	VERB
ma-267	227	2	4.2	4.2	NUM
ma-267	227	3	.	.	PUNCT
ma-267	228	1	let	let	VERB
ma-267	228	2	ψ	ψ	PART
ma-267	228	3	be	be	AUX
ma-267	228	4	a	a	DET
ma-267	228	5	riemann	riemann	PROPN
ma-267	228	6	-	-	PUNCT
ma-267	228	7	liouville	liouville	VERB
ma-267	228	8	wavelet	wavelet	NOUN
ma-267	228	9	in	in	ADP
ma-267	228	10	l2α(k	l2α(k	PROPN
ma-267	228	11	)	)	PUNCT
ma-267	228	12	,	,	PUNCT
ma-267	228	13	s	s	VERB
ma-267	228	14	>	>	X
ma-267	228	15	2α+3	2α+3	PROPN
ma-267	228	16	2	2	NUM
ma-267	228	17	,	,	PUNCT
ma-267	228	18	g	g	PROPN
ma-267	228	19	∈	∈	PROPN
ma-267	228	20	l2α	l2α	X
ma-267	228	21	(	(	PUNCT
ma-267	228	22	r+	r+	X
ma-267	228	23	×k	×k	NOUN
ma-267	228	24	)	)	PUNCT
ma-267	228	25	and	and	CCONJ
ma-267	228	26	β	β	X
ma-267	228	27	>	>	X
ma-267	228	28	0	0	PUNCT
ma-267	229	1	then	then	ADV
ma-267	229	2	the	the	DET
ma-267	229	3	infimum	infimum	ADJ
ma-267	229	4	inf	inf	NOUN
ma-267	229	5	f	f	PROPN
ma-267	229	6	∈hsα(k	∈hsα(k	NOUN
ma-267	229	7	)	)	PUNCT
ma-267	229	8	{	{	PUNCT
ma-267	229	9	β‖f	β‖f	ADV
ma-267	229	10	‖2hsα	‖2hsα	NOUN
ma-267	229	11	+	+	CCONJ
ma-267	229	12	∥∥sαψ(f	∥∥sαψ(f	NOUN
ma-267	229	13	)	)	PUNCT
ma-267	229	14	−	−	PROPN
ma-267	229	15	g	g	PROPN
ma-267	229	16	∥∥2	∥∥2	PROPN
ma-267	229	17	2,θα	2,θα	NUM
ma-267	229	18	}	}	PUNCT
ma-267	229	19	(	(	PUNCT
ma-267	229	20	4.9	4.9	NUM
ma-267	229	21	)	)	PUNCT
ma-267	229	22	is	be	AUX
ma-267	229	23	attained	attain	VERB
ma-267	229	24	by	by	ADP
ma-267	229	25	a	a	DET
ma-267	229	26	unique	unique	ADJ
ma-267	229	27	function	function	NOUN
ma-267	229	28	f	f	PROPN
ma-267	229	29	∗g	∗g	PROPN
ma-267	229	30	,	,	PUNCT
ma-267	229	31	ψ	ψ	X
ma-267	229	32	,	,	PUNCT
ma-267	229	33	β	β	PROPN
ma-267	229	34	given	give	VERB
ma-267	229	35	explicitly	explicitly	ADV
ma-267	229	36	by	by	ADP
ma-267	229	37	f	f	PROPN
ma-267	229	38	∗g	∗g	PROPN
ma-267	229	39	,	,	PUNCT
ma-267	229	40	ψ	ψ	SYM
ma-267	229	41	,	,	PUNCT
ma-267	229	42	β(x	β(x	PROPN
ma-267	229	43	,	,	PUNCT
ma-267	229	44	t	t	NOUN
ma-267	229	45	)	)	PUNCT
ma-267	229	46	=	=	SYM
ma-267	230	1	∫	∫	PROPN
ma-267	231	1	+	+	NUM
ma-267	231	2	∞	∞	PROPN
ma-267	231	3	0	0	NUM
ma-267	231	4	∫	∫	PROPN
ma-267	231	5	k	k	PROPN
ma-267	231	6	g(a	g(a	PROPN
ma-267	231	7	,	,	PUNCT
ma-267	231	8	y	y	PROPN
ma-267	231	9	,	,	PUNCT
ma-267	231	10	z)φψ	z)φψ	PROPN
ma-267	231	11	,	,	PUNCT
ma-267	231	12	β	β	X
ma-267	231	13	(	(	PUNCT
ma-267	231	14	a	a	PROPN
ma-267	231	15	,	,	PUNCT
ma-267	231	16	(	(	PUNCT
ma-267	231	17	y	y	PROPN
ma-267	231	18	,	,	PUNCT
ma-267	231	19	z	z	PROPN
ma-267	231	20	)	)	PUNCT
ma-267	231	21	,	,	PUNCT
ma-267	231	22	(	(	PUNCT
ma-267	231	23	x	x	X
ma-267	231	24	,	,	PUNCT
ma-267	231	25	t	t	PROPN
ma-267	231	26	)	)	PUNCT
ma-267	231	27	)	)	PUNCT
ma-267	232	1	dθα(a	dθα(a	PROPN
ma-267	232	2	,	,	PUNCT
ma-267	232	3	y	y	PROPN
ma-267	232	4	,	,	PUNCT
ma-267	232	5	z	z	PROPN
ma-267	232	6	)	)	PUNCT
ma-267	232	7	,	,	PUNCT
ma-267	232	8	(	(	PUNCT
ma-267	232	9	4.10	4.10	NUM
ma-267	232	10	)	)	PUNCT
ma-267	232	11	where	where	SCONJ
ma-267	232	12	φψ	φψ	NOUN
ma-267	232	13	,	,	PUNCT
ma-267	232	14	β	β	X
ma-267	232	15	is	be	AUX
ma-267	232	16	given	give	VERB
ma-267	232	17	by	by	ADP
ma-267	232	18	φψ	φψ	NOUN
ma-267	232	19	,	,	PUNCT
ma-267	232	20	β	β	X
ma-267	232	21	(	(	PUNCT
ma-267	232	22	a	a	PROPN
ma-267	232	23	,	,	PUNCT
ma-267	232	24	(	(	PUNCT
ma-267	232	25	y	y	PROPN
ma-267	232	26	,	,	PUNCT
ma-267	232	27	z	z	PROPN
ma-267	232	28	)	)	PUNCT
ma-267	232	29	,	,	PUNCT
ma-267	232	30	(	(	PUNCT
ma-267	232	31	x	x	X
ma-267	232	32	,	,	PUNCT
ma-267	232	33	t	t	PROPN
ma-267	232	34	)	)	PUNCT
ma-267	232	35	)	)	PUNCT
ma-267	233	1	=	=	SYM
ma-267	233	2	a−	a−	PROPN
ma-267	233	3	2α+3	2α+3	PROPN
ma-267	233	4	2	2	NUM
ma-267	233	5	∫	∫	PROPN
ma-267	233	6	k̂	k̂	PROPN
ma-267	233	7	ϕµ,−λ(x	ϕµ,−λ(x	PROPN
ma-267	233	8	,	,	PUNCT
ma-267	233	9	t)ϕµ,λ(y	t)ϕµ,λ(y	PROPN
ma-267	233	10	,	,	PUNCT
ma-267	233	11	z)fα(ψ	z)fα(ψ	NUM
ma-267	233	12	)	)	PUNCT
ma-267	233	13	(	(	PUNCT
ma-267	233	14	µ	µ	X
ma-267	233	15	a	a	PRON
ma-267	233	16	,	,	PUNCT
ma-267	233	17	λ	λ	X
ma-267	233	18	a	a	NOUN
ma-267	233	19	)	)	PUNCT
ma-267	233	20	β	β	NOUN
ma-267	233	21	(	(	PUNCT
ma-267	233	22	1	1	NUM
ma-267	233	23	+	+	CCONJ
ma-267	233	24	µ2	µ2	PROPN
ma-267	233	25	+	+	CCONJ
ma-267	233	26	2λ2)s	2λ2)s	NUM
ma-267	233	27	+	+	CCONJ
ma-267	233	28	cψ	cψ	PROPN
ma-267	233	29	dγα(µ	dγα(µ	PROPN
ma-267	233	30	,	,	PUNCT
ma-267	233	31	λ	λ	NOUN
ma-267	233	32	)	)	PUNCT
ma-267	233	33	.	.	PUNCT
ma-267	234	1	(	(	PUNCT
ma-267	234	2	4.11	4.11	NUM
ma-267	234	3	)	)	PUNCT
ma-267	234	4	proof	proof	NOUN
ma-267	234	5	.	.	PUNCT
ma-267	235	1	the	the	DET
ma-267	235	2	existence	existence	NOUN
ma-267	235	3	and	and	CCONJ
ma-267	235	4	unicity	unicity	NOUN
ma-267	235	5	of	of	ADP
ma-267	235	6	the	the	DET
ma-267	235	7	extremal	extremal	ADJ
ma-267	235	8	function	function	NOUN
ma-267	235	9	f	f	PROPN
ma-267	235	10	∗g	∗g	PROPN
ma-267	235	11	,	,	PUNCT
ma-267	235	12	ψ	ψ	X
ma-267	235	13	,	,	PUNCT
ma-267	235	14	β	β	X
ma-267	235	15	solution	solution	NOUN
ma-267	235	16	of	of	ADP
ma-267	235	17	the	the	DET
ma-267	235	18	problem	problem	NOUN
ma-267	235	19	(	(	PUNCT
ma-267	235	20	4.9	4.9	NUM
ma-267	235	21	)	)	PUNCT
ma-267	235	22	isassured	isassure	VERB
ma-267	235	23	in	in	ADP
ma-267	235	24	[	[	X
ma-267	235	25	18,19	18,19	NUM
ma-267	235	26	]	]	X
ma-267	235	27	,	,	PUNCT
ma-267	235	28	moreover	moreover	ADV
ma-267	235	29	this	this	DET
ma-267	235	30	solution	solution	NOUN
ma-267	235	31	is	be	AUX
ma-267	235	32	given	give	VERB
ma-267	235	33	by	by	ADP
ma-267	235	34	f	f	PROPN
ma-267	235	35	∗g	∗g	PROPN
ma-267	235	36	,	,	PUNCT
ma-267	235	37	ψ	ψ	SYM
ma-267	235	38	,	,	PUNCT
ma-267	235	39	β(x	β(x	PROPN
ma-267	235	40	,	,	PUNCT
ma-267	235	41	t	t	NOUN
ma-267	235	42	)	)	PUNCT
ma-267	236	1	=	=	PUNCT
ma-267	236	2	〈	〈	PROPN
ma-267	236	3	g	g	NOUN
ma-267	236	4	,	,	PUNCT
ma-267	236	5	sαψ	sαψ	INTJ
ma-267	236	6	(	(	PUNCT
ma-267	236	7	kψ	kψ	NOUN
ma-267	236	8	,	,	PUNCT
ma-267	236	9	β	β	X
ma-267	236	10	[	[	X
ma-267	236	11	·	·	PUNCT
ma-267	236	12	,	,	PUNCT
ma-267	236	13	(	(	PUNCT
ma-267	236	14	x	x	NOUN
ma-267	236	15	,	,	PUNCT
ma-267	236	16	t	t	PROPN
ma-267	236	17	)	)	PUNCT
ma-267	236	18	]	]	PUNCT
ma-267	236	19	)	)	PUNCT
ma-267	236	20	〉	〉	NOUN
ma-267	236	21	θα	θα	NOUN
ma-267	236	22	,	,	PUNCT
ma-267	236	23	(	(	PUNCT
ma-267	236	24	4.12	4.12	NUM
ma-267	236	25	)	)	PUNCT
ma-267	236	26	where	where	SCONJ
ma-267	236	27	kψ	kψ	NOUN
ma-267	236	28	,	,	PUNCT
ma-267	236	29	β	β	X
ma-267	236	30	is	be	AUX
ma-267	236	31	the	the	DET
ma-267	236	32	kernel	kernel	NOUN
ma-267	236	33	given	give	VERB
ma-267	236	34	by	by	ADP
ma-267	236	35	(	(	PUNCT
ma-267	236	36	4.8	4.8	NUM
ma-267	236	37	)	)	PUNCT
ma-267	236	38	,	,	PUNCT
ma-267	236	39	by	by	ADP
ma-267	236	40	using	use	VERB
ma-267	236	41	the	the	DET
ma-267	236	42	relations	relation	NOUN
ma-267	236	43	(	(	PUNCT
ma-267	236	44	2.6	2.6	NUM
ma-267	236	45	)	)	PUNCT
ma-267	236	46	,	,	PUNCT
ma-267	236	47	(	(	PUNCT
ma-267	236	48	2.10	2.10	NUM
ma-267	236	49	)	)	PUNCT
ma-267	236	50	,	,	PUNCT
ma-267	236	51	(	(	PUNCT
ma-267	236	52	3.4	3.4	NUM
ma-267	236	53	)	)	PUNCT
ma-267	236	54	and	and	CCONJ
ma-267	236	55	(	(	PUNCT
ma-267	236	56	4.8	4.8	NUM
ma-267	236	57	)	)	PUNCT
ma-267	237	1	we	we	PRON
ma-267	237	2	findthat	findthat	INTJ
ma-267	238	1	sαψ	sαψ	INTJ
ma-267	238	2	(	(	PUNCT
ma-267	238	3	kψ	kψ	NOUN
ma-267	238	4	,	,	PUNCT
ma-267	238	5	β	β	X
ma-267	238	6	[	[	X
ma-267	238	7	·	·	PUNCT
ma-267	238	8	,	,	PUNCT
ma-267	238	9	(	(	PUNCT
ma-267	238	10	x	x	NOUN
ma-267	238	11	,	,	PUNCT
ma-267	238	12	t	t	PROPN
ma-267	238	13	)	)	PUNCT
ma-267	238	14	]	]	PUNCT
ma-267	238	15	)	)	PUNCT
ma-267	239	1	(	(	PUNCT
ma-267	239	2	a	a	X
ma-267	239	3	,	,	PUNCT
ma-267	239	4	y	y	PROPN
ma-267	239	5	,	,	PUNCT
ma-267	239	6	z	z	NOUN
ma-267	239	7	)	)	PUNCT
ma-267	239	8	=	=	SYM
ma-267	239	9	a−	a−	PROPN
ma-267	239	10	2α+3	2α+3	PROPN
ma-267	239	11	2	2	NUM
ma-267	239	12	∫	∫	PROPN
ma-267	239	13	k̂	k̂	PROPN
ma-267	239	14	ϕµ,λ(x	ϕµ,λ(x	PROPN
ma-267	239	15	,	,	PUNCT
ma-267	239	16	t)ϕµ,−λ(y	t)ϕµ,−λ(y	VERB
ma-267	239	17	,	,	PUNCT
ma-267	239	18	z)fα(ψ	z)fα(ψ	NUM
ma-267	239	19	)	)	PUNCT
ma-267	239	20	(	(	PUNCT
ma-267	239	21	µ	µ	X
ma-267	239	22	a	a	PRON
ma-267	239	23	,	,	PUNCT
ma-267	239	24	λ	λ	X
ma-267	239	25	a	a	NOUN
ma-267	239	26	)	)	PUNCT
ma-267	239	27	β	β	NOUN
ma-267	239	28	(	(	PUNCT
ma-267	239	29	1	1	NUM
ma-267	239	30	+	+	CCONJ
ma-267	239	31	µ2	µ2	PROPN
ma-267	239	32	+	+	CCONJ
ma-267	239	33	2λ2)s	2λ2)s	NUM
ma-267	239	34	+	+	CCONJ
ma-267	239	35	cψ	cψ	PROPN
ma-267	239	36	dγα(µ	dγα(µ	PROPN
ma-267	239	37	,	,	PUNCT
ma-267	239	38	λ	λ	PROPN
ma-267	239	39	)	)	PUNCT
ma-267	239	40	,	,	PUNCT
ma-267	239	41	(	(	PUNCT
ma-267	239	42	4.13	4.13	NUM
ma-267	239	43	)	)	PUNCT
ma-267	239	44	by	by	ADP
ma-267	239	45	using	use	VERB
ma-267	239	46	the	the	DET
ma-267	239	47	relations	relation	NOUN
ma-267	239	48	(	(	PUNCT
ma-267	239	49	4.12	4.12	NUM
ma-267	239	50	)	)	PUNCT
ma-267	239	51	and	and	CCONJ
ma-267	239	52	(	(	PUNCT
ma-267	239	53	4.13	4.13	X
ma-267	239	54	)	)	PUNCT
ma-267	239	55	we	we	PRON
ma-267	239	56	find	find	VERB
ma-267	239	57	the	the	DET
ma-267	239	58	desired	desire	VERB
ma-267	239	59	result	result	NOUN
ma-267	239	60	.	.	PUNCT
ma-267	240	1	�	�	PROPN
ma-267	240	2	we	we	PRON
ma-267	240	3	have	have	VERB
ma-267	240	4	the	the	DET
ma-267	240	5	following	follow	VERB
ma-267	240	6	results	result	NOUN
ma-267	240	7	theorem	theorem	VERB
ma-267	240	8	4.3	4.3	NUM
ma-267	240	9	.	.	PUNCT
ma-267	241	1	let	let	VERB
ma-267	241	2	s	s	PRON
ma-267	241	3	>	>	X
ma-267	241	4	2α+3	2α+3	PROPN
ma-267	241	5	2	2	NUM
ma-267	241	6	,	,	PUNCT
ma-267	241	7	ψ	ψ	AUX
ma-267	241	8	be	be	AUX
ma-267	241	9	a	a	DET
ma-267	241	10	riemann	riemann	PROPN
ma-267	241	11	-	-	PUNCT
ma-267	241	12	liouville	liouville	NOUN
ma-267	241	13	wavelet	wavelet	NOUN
ma-267	241	14	in	in	ADP
ma-267	241	15	on	on	ADP
ma-267	241	16	k	k	PROPN
ma-267	241	17	in	in	ADP
ma-267	241	18	l2α(k	l2α(k	PROPN
ma-267	241	19	)	)	PUNCT
ma-267	241	20	,	,	PUNCT
ma-267	241	21	,	,	PUNCT
ma-267	241	22	and	and	CCONJ
ma-267	241	23	g	g	PROPN
ma-267	241	24	∈	∈	PROPN
ma-267	241	25	l2α	l2α	X
ma-267	241	26	(	(	PUNCT
ma-267	241	27	r+	r+	X
ma-267	241	28	×k	×k	PROPN
ma-267	241	29	)	)	PUNCT
ma-267	241	30	,	,	PUNCT
ma-267	241	31	β	β	X
ma-267	241	32	>	>	X
ma-267	241	33	0	0	PUNCT
ma-267	242	1	then	then	ADV
ma-267	242	2	we	we	PRON
ma-267	242	3	have	have	VERB
ma-267	242	4	(	(	PUNCT
ma-267	242	5	i	i	NOUN
ma-267	242	6	)	)	PUNCT
ma-267	242	7	f	f	PROPN
ma-267	243	1	∗g	∗g	PROPN
ma-267	243	2	,	,	PUNCT
ma-267	243	3	ψ	ψ	SYM
ma-267	243	4	,	,	PUNCT
ma-267	243	5	β(x	β(x	PROPN
ma-267	243	6	,	,	PUNCT
ma-267	243	7	t	t	NOUN
ma-267	243	8	)	)	PUNCT
ma-267	243	9	=	=	SYM
ma-267	244	1	∫	∫	PROPN
ma-267	245	1	+	+	NUM
ma-267	245	2	∞	∞	PROPN
ma-267	245	3	0	0	NUM
ma-267	245	4	∫	∫	PROPN
ma-267	245	5	k̂	k̂	PROPN
ma-267	245	6	ϕµ,−λ(x	ϕµ,−λ(x	PROPN
ma-267	245	7	,	,	PUNCT
ma-267	245	8	t)fα(ψ	t)fα(ψ	NUM
ma-267	245	9	)	)	PUNCT
ma-267	245	10	(	(	PUNCT
ma-267	245	11	µ	µ	X
ma-267	245	12	a	a	PRON
ma-267	245	13	,	,	PUNCT
ma-267	245	14	λ	λ	X
ma-267	245	15	a	a	PRON
ma-267	245	16	)	)	PUNCT
ma-267	245	17	fα(g(a	fα(g(a	NOUN
ma-267	245	18	,	,	PUNCT
ma-267	245	19	.))(µ	.))(µ	PROPN
ma-267	245	20	,	,	PUNCT
ma-267	245	21	λ	λ	X
ma-267	245	22	)	)	PUNCT
ma-267	245	23	β	β	NOUN
ma-267	245	24	(	(	PUNCT
ma-267	245	25	1	1	NUM
ma-267	245	26	+	+	CCONJ
ma-267	245	27	µ2	µ2	PROPN
ma-267	245	28	+	+	CCONJ
ma-267	245	29	2λ2)s	2λ2)s	NUM
ma-267	245	30	+	+	CCONJ
ma-267	245	31	cψ	cψ	VERB
ma-267	245	32	a	a	DET
ma-267	245	33	2α+1	2α+1	PROPN
ma-267	245	34	2	2	NUM
ma-267	245	35	da	da	PROPN
ma-267	245	36	⊗	⊗	PROPN
ma-267	245	37	dγα(µ	dγα(µ	PROPN
ma-267	245	38	,	,	PUNCT
ma-267	245	39	λ).(4.14	λ).(4.14	NOUN
ma-267	245	40	)	)	PUNCT
ma-267	245	41	(	(	PUNCT
ma-267	245	42	i	i	PRON
ma-267	245	43	i	i	PROPN
ma-267	245	44	)	)	PUNCT
ma-267	245	45	fα(f	fα(f	PUNCT
ma-267	246	1	∗g	∗g	PROPN
ma-267	246	2	,	,	PUNCT
ma-267	246	3	ψ	ψ	X
ma-267	246	4	,	,	PUNCT
ma-267	246	5	β)(µ	β)(µ	PRON
ma-267	246	6	,	,	PUNCT
ma-267	246	7	λ	λ	X
ma-267	246	8	)	)	PUNCT
ma-267	246	9	=	=	SYM
ma-267	246	10	∫	∫	PROPN
ma-267	247	1	+	+	NUM
ma-267	247	2	∞	∞	PROPN
ma-267	247	3	0	0	NUM
ma-267	247	4	fα(ψ	fα(ψ	NUM
ma-267	247	5	)	)	PUNCT
ma-267	247	6	(	(	PUNCT
ma-267	247	7	µ	µ	X
ma-267	247	8	a	a	PRON
ma-267	247	9	,	,	PUNCT
ma-267	247	10	λ	λ	X
ma-267	247	11	a	a	PRON
ma-267	247	12	)	)	PUNCT
ma-267	247	13	fα(g(a	fα(g(a	NOUN
ma-267	247	14	,	,	PUNCT
ma-267	247	15	.))(µ	.))(µ	PROPN
ma-267	247	16	,	,	PUNCT
ma-267	247	17	λ	λ	X
ma-267	247	18	)	)	PUNCT
ma-267	247	19	β	β	NOUN
ma-267	247	20	(	(	PUNCT
ma-267	247	21	1	1	NUM
ma-267	247	22	+	+	CCONJ
ma-267	247	23	µ2	µ2	PROPN
ma-267	247	24	+	+	CCONJ
ma-267	247	25	2λ2)s	2λ2)s	NUM
ma-267	247	26	+	+	CCONJ
ma-267	247	27	cψ	cψ	VERB
ma-267	247	28	a	a	DET
ma-267	247	29	2α+1	2α+1	PROPN
ma-267	247	30	2	2	NUM
ma-267	247	31	da	da	NOUN
ma-267	247	32	(	(	PUNCT
ma-267	247	33	4.15	4.15	NUM
ma-267	247	34	)	)	PUNCT
ma-267	247	35	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	247	36	eur	eur	PROPN
ma-267	247	37	.	.	PUNCT
ma-267	248	1	j.	j.	PROPN
ma-267	248	2	math	math	PROPN
ma-267	248	3	.	.	PUNCT
ma-267	249	1	anal	anal	PROPN
ma-267	249	2	.	.	PUNCT
ma-267	250	1	10.28924	10.28924	NUM
ma-267	250	2	/	/	SYM
ma-267	250	3	ada	ada	PROPN
ma-267	250	4	/	/	SYM
ma-267	250	5	ma.4.20	ma.4.20	PROPN
ma-267	250	6	12	12	NUM
ma-267	250	7	(	(	PUNCT
ma-267	250	8	i	i	PRON
ma-267	250	9	i	i	PROPN
ma-267	250	10	i	i	PROPN
ma-267	250	11	)	)	PUNCT
ma-267	250	12	‖f	‖f	ADP
ma-267	251	1	∗g	∗g	PROPN
ma-267	251	2	,	,	PUNCT
ma-267	251	3	ψ	ψ	X
ma-267	251	4	,	,	PUNCT
ma-267	251	5	β‖hsα	β‖hsα	PUNCT
ma-267	251	6	≤	≤	NUM
ma-267	251	7	‖g‖2,θα√	‖g‖2,θα√	NUM
ma-267	251	8	2β	2β	NOUN
ma-267	251	9	.	.	PUNCT
ma-267	252	1	(	(	PUNCT
ma-267	252	2	4.16	4.16	NUM
ma-267	252	3	)	)	PUNCT
ma-267	252	4	proof	proof	NOUN
ma-267	252	5	.	.	PUNCT
ma-267	253	1	(	(	PUNCT
ma-267	253	2	i	i	NOUN
ma-267	253	3	)	)	PUNCT
ma-267	253	4	is	be	AUX
ma-267	253	5	a	a	DET
ma-267	253	6	consequence	consequence	NOUN
ma-267	253	7	of	of	ADP
ma-267	253	8	(	(	PUNCT
ma-267	253	9	4.10	4.10	NUM
ma-267	253	10	)	)	PUNCT
ma-267	253	11	,	,	PUNCT
ma-267	253	12	(	(	PUNCT
ma-267	253	13	4.11	4.11	NUM
ma-267	253	14	)	)	PUNCT
ma-267	253	15	and	and	CCONJ
ma-267	253	16	fubini	fubini	NOUN
ma-267	253	17	’s	’s	PART
ma-267	253	18	theorem.(ii	theorem.(ii	NOUN
ma-267	253	19	)	)	PUNCT
ma-267	254	1	is	be	AUX
ma-267	254	2	a	a	DET
ma-267	254	3	consquence	consquence	NOUN
ma-267	254	4	of	of	ADP
ma-267	254	5	fubini	fubini	NOUN
ma-267	254	6	’s	’s	PART
ma-267	254	7	theorem	theorem	NOUN
ma-267	254	8	and	and	CCONJ
ma-267	254	9	the	the	DET
ma-267	254	10	relations	relation	NOUN
ma-267	254	11	(	(	PUNCT
ma-267	254	12	2.5),(4.10	2.5),(4.10	NUM
ma-267	254	13	)	)	PUNCT
ma-267	254	14	and	and	CCONJ
ma-267	254	15	(	(	PUNCT
ma-267	254	16	4.11).(iii)by	4.11).(iii)by	NOUN
ma-267	254	17	using	use	VERB
ma-267	254	18	the	the	DET
ma-267	254	19	relation	relation	NOUN
ma-267	254	20	(	(	PUNCT
ma-267	254	21	4.2	4.2	NUM
ma-267	254	22	)	)	PUNCT
ma-267	254	23	we	we	PRON
ma-267	254	24	find	find	VERB
ma-267	254	25	that	that	SCONJ
ma-267	254	26	‖f	‖f	ADP
ma-267	254	27	∗g	∗g	NOUN
ma-267	254	28	,	,	PUNCT
ma-267	254	29	ψ	ψ	X
ma-267	254	30	,	,	PUNCT
ma-267	254	31	β‖2hsα	β‖2hsα	X
ma-267	254	32	=	=	SYM
ma-267	254	33	∫	∫	PROPN
ma-267	254	34	k̂	k̂	PROPN
ma-267	255	1	(	(	PUNCT
ma-267	255	2	1	1	X
ma-267	255	3	+	+	CCONJ
ma-267	255	4	µ2	µ2	PROPN
ma-267	255	5	+	+	CCONJ
ma-267	255	6	2λ2	2λ2	NUM
ma-267	255	7	)	)	PUNCT
ma-267	255	8	s	s	PART
ma-267	255	9	|fα(f	|fα(f	NOUN
ma-267	255	10	∗g	∗g	PROPN
ma-267	255	11	,	,	PUNCT
ma-267	255	12	ψ	ψ	X
ma-267	255	13	,	,	PUNCT
ma-267	255	14	β)(µ	β)(µ	PRON
ma-267	255	15	,	,	PUNCT
ma-267	255	16	λ)|2dγα(µ	λ)|2dγα(µ	ADV
ma-267	255	17	,	,	PUNCT
ma-267	255	18	λ	λ	PROPN
ma-267	255	19	)	)	PUNCT
ma-267	255	20	,	,	PUNCT
ma-267	255	21	by	by	ADP
ma-267	255	22	using	use	VERB
ma-267	255	23	holder	holder	NOUN
ma-267	255	24	’s	’s	PART
ma-267	255	25	inequality	inequality	NOUN
ma-267	255	26	and	and	CCONJ
ma-267	255	27	the	the	DET
ma-267	255	28	relations	relation	NOUN
ma-267	255	29	(	(	PUNCT
ma-267	255	30	3.2),(4.15	3.2),(4.15	X
ma-267	255	31	)	)	PUNCT
ma-267	255	32	we	we	PRON
ma-267	255	33	find	find	VERB
ma-267	255	34	that	that	SCONJ
ma-267	255	35	‖f	‖f	ADP
ma-267	255	36	∗g	∗g	NOUN
ma-267	255	37	,	,	PUNCT
ma-267	255	38	ψ	ψ	SYM
ma-267	255	39	,	,	PUNCT
ma-267	255	40	β‖2hsα	β‖2hsα	NUM
ma-267	255	41	≤	≤	NUM
ma-267	255	42	1	1	NUM
ma-267	255	43	2β	2β	NUM
ma-267	255	44	∫	∫	NOUN
ma-267	255	45	k̂	k̂	PROPN
ma-267	255	46	(	(	PUNCT
ma-267	255	47	∫	∫	PROPN
ma-267	256	1	+	+	NOUN
ma-267	256	2	∞	∞	PROPN
ma-267	256	3	0	0	SYM
ma-267	256	4	|fα(g(a	|fα(g(a	NOUN
ma-267	256	5	,	,	PUNCT
ma-267	256	6	.))(µ	.))(µ	NOUN
ma-267	256	7	,	,	PUNCT
ma-267	256	8	λ)|2a2α+2da	λ)|2a2α+2da	X
ma-267	256	9	)	)	PUNCT
ma-267	256	10	dγα(µ	dγα(µ	PROPN
ma-267	256	11	,	,	PUNCT
ma-267	256	12	λ	λ	NOUN
ma-267	256	13	)	)	PUNCT
ma-267	256	14	,	,	PUNCT
ma-267	256	15	by	by	ADP
ma-267	256	16	using	use	VERB
ma-267	256	17	fubini	fubini	NOUN
ma-267	256	18	’s	’s	PART
ma-267	256	19	theorem	theorem	NOUN
ma-267	256	20	and	and	CCONJ
ma-267	256	21	plancherel	plancherel	NOUN
ma-267	256	22	’s	’s	PART
ma-267	256	23	formula	formula	NOUN
ma-267	256	24	(	(	PUNCT
ma-267	256	25	2.7	2.7	NUM
ma-267	256	26	)	)	PUNCT
ma-267	256	27	we	we	PRON
ma-267	256	28	find	find	VERB
ma-267	256	29	that	that	SCONJ
ma-267	256	30	‖f	‖f	ADP
ma-267	256	31	∗g	∗g	NOUN
ma-267	256	32	,	,	PUNCT
ma-267	256	33	ψ	ψ	SYM
ma-267	256	34	,	,	PUNCT
ma-267	256	35	β‖2hsα	β‖2hsα	NUM
ma-267	256	36	≤	≤	NUM
ma-267	256	37	1	1	NUM
ma-267	256	38	2β	2β	NUM
ma-267	256	39	‖g‖22,θαwhich	‖g‖22,θαwhich	PRON
ma-267	256	40	gives	give	VERB
ma-267	256	41	the	the	DET
ma-267	256	42	result	result	NOUN
ma-267	256	43	.	.	PUNCT
ma-267	257	1	�	�	PROPN
ma-267	257	2	corollary	corollary	NOUN
ma-267	257	3	4.1	4.1	NUM
ma-267	257	4	.	.	PUNCT
ma-267	258	1	let	let	VERB
ma-267	258	2	s	s	PRON
ma-267	258	3	>	>	X
ma-267	258	4	2α+3	2α+3	PROPN
ma-267	258	5	2	2	NUM
ma-267	258	6	,	,	PUNCT
ma-267	258	7	ψ	ψ	AUX
ma-267	258	8	be	be	AUX
ma-267	258	9	a	a	DET
ma-267	258	10	riemann	riemann	PROPN
ma-267	258	11	-	-	PUNCT
ma-267	258	12	liouville	liouville	NOUN
ma-267	258	13	wavelet	wavelet	NOUN
ma-267	258	14	on	on	ADP
ma-267	258	15	k	k	PROPN
ma-267	258	16	in	in	ADP
ma-267	258	17	l2α(k	l2α(k	PROPN
ma-267	258	18	)	)	PUNCT
ma-267	258	19	,	,	PUNCT
ma-267	258	20	,	,	PUNCT
ma-267	258	21	and	and	CCONJ
ma-267	258	22	β	β	X
ma-267	258	23	>	>	X
ma-267	258	24	0	0	NUM
ma-267	258	25	,	,	PUNCT
ma-267	258	26	for	for	ADP
ma-267	258	27	all	all	DET
ma-267	258	28	f	f	PROPN
ma-267	258	29	∈	∈	PROPN
ma-267	258	30	hsα(k	hsα(k	PROPN
ma-267	258	31	)	)	PUNCT
ma-267	258	32	and	and	CCONJ
ma-267	258	33	g	g	NOUN
ma-267	258	34	=	=	SYM
ma-267	258	35	sαψ(f	sαψ(f	PROPN
ma-267	258	36	)	)	PUNCT
ma-267	258	37	,	,	PUNCT
ma-267	258	38	the	the	DET
ma-267	258	39	extremal	extremal	ADJ
ma-267	258	40	function	function	NOUN
ma-267	258	41	f	f	PROPN
ma-267	258	42	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	258	43	)	)	PUNCT
ma-267	258	44	,	,	PUNCT
ma-267	258	45	ψ	ψ	X
ma-267	258	46	,	,	PUNCT
ma-267	258	47	β	β	X
ma-267	258	48	satisfies	satisfy	VERB
ma-267	258	49	the	the	DET
ma-267	258	50	following	follow	VERB
ma-267	258	51	properties	property	NOUN
ma-267	258	52	(	(	PUNCT
ma-267	258	53	i	i	NOUN
ma-267	258	54	)	)	PUNCT
ma-267	258	55	fα(f	fα(f	X
ma-267	258	56	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	258	57	)	)	PUNCT
ma-267	258	58	,	,	PUNCT
ma-267	258	59	ψ	ψ	X
ma-267	258	60	,	,	PUNCT
ma-267	258	61	β	β	X
ma-267	258	62	)	)	PUNCT
ma-267	258	63	(	(	PUNCT
ma-267	258	64	µ	µ	X
ma-267	258	65	,	,	PUNCT
ma-267	258	66	λ	λ	NOUN
ma-267	258	67	)	)	PUNCT
ma-267	258	68	=	=	SYM
ma-267	258	69	cψfα(f	cψfα(f	PROPN
ma-267	258	70	)	)	PUNCT
ma-267	258	71	(	(	PUNCT
ma-267	258	72	µ	µ	X
ma-267	258	73	,	,	PUNCT
ma-267	258	74	λ	λ	NOUN
ma-267	258	75	)	)	PUNCT
ma-267	258	76	β	β	NOUN
ma-267	258	77	(	(	PUNCT
ma-267	258	78	1	1	NUM
ma-267	258	79	+	+	CCONJ
ma-267	258	80	µ2	µ2	PROPN
ma-267	258	81	+	+	CCONJ
ma-267	258	82	2λ2)s	2λ2)s	NUM
ma-267	258	83	+	+	CCONJ
ma-267	258	84	cψ	cψ	NOUN
ma-267	258	85	.	.	PUNCT
ma-267	259	1	(	(	PUNCT
ma-267	259	2	4.17	4.17	NUM
ma-267	259	3	)	)	PUNCT
ma-267	259	4	(	(	PUNCT
ma-267	260	1	i	i	PRON
ma-267	260	2	i	i	PROPN
ma-267	260	3	)	)	PUNCT
ma-267	260	4	‖f	‖f	PRON
ma-267	260	5	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	260	6	)	)	PUNCT
ma-267	260	7	,	,	PUNCT
ma-267	260	8	ψ	ψ	PROPN
ma-267	260	9	,	,	PUNCT
ma-267	260	10	β‖hsα	β‖hsα	PUNCT
ma-267	260	11	≤	≤	NUM
ma-267	260	12	√	√	ADP
ma-267	260	13	cψ	cψ	NOUN
ma-267	260	14	2β	2β	NOUN
ma-267	260	15	‖f	‖f	ADP
ma-267	260	16	‖2,µα	‖2,µα	NUM
ma-267	260	17	(	(	PUNCT
ma-267	260	18	4.18	4.18	NUM
ma-267	260	19	)	)	PUNCT
ma-267	260	20	proof	proof	NOUN
ma-267	260	21	.	.	PUNCT
ma-267	261	1	(	(	PUNCT
ma-267	261	2	i	i	NOUN
ma-267	261	3	)	)	PUNCT
ma-267	261	4	by	by	ADP
ma-267	261	5	using	use	VERB
ma-267	261	6	the	the	DET
ma-267	261	7	relations	relation	NOUN
ma-267	261	8	(	(	PUNCT
ma-267	261	9	3.2	3.2	NUM
ma-267	261	10	)	)	PUNCT
ma-267	261	11	,	,	PUNCT
ma-267	261	12	(	(	PUNCT
ma-267	261	13	3.9	3.9	NUM
ma-267	261	14	)	)	PUNCT
ma-267	261	15	and	and	CCONJ
ma-267	261	16	(	(	PUNCT
ma-267	261	17	4.15	4.15	NUM
ma-267	261	18	)	)	PUNCT
ma-267	261	19	we	we	PRON
ma-267	261	20	find	find	VERB
ma-267	261	21	the	the	DET
ma-267	261	22	result.(ii	result.(ii	NOUN
ma-267	261	23	)	)	PUNCT
ma-267	261	24	is	be	AUX
ma-267	261	25	a	a	DET
ma-267	261	26	consequence	consequence	NOUN
ma-267	261	27	of	of	ADP
ma-267	261	28	(	(	PUNCT
ma-267	261	29	3.9	3.9	NUM
ma-267	261	30	)	)	PUNCT
ma-267	261	31	and	and	CCONJ
ma-267	261	32	(	(	PUNCT
ma-267	261	33	4.16	4.16	NUM
ma-267	261	34	)	)	PUNCT
ma-267	261	35	.	.	PUNCT
ma-267	262	1	�	�	PROPN
ma-267	262	2	theorem	theorem	VERB
ma-267	262	3	4.4	4.4	NUM
ma-267	262	4	.	.	PUNCT
ma-267	263	1	(	(	PUNCT
ma-267	263	2	second	second	ADJ
ma-267	263	3	calderon	calderon	NOUN
ma-267	263	4	’s	’s	PART
ma-267	263	5	reproducing	reproduce	VERB
ma-267	263	6	formula)let	formula)let	NOUN
ma-267	263	7	s	s	PART
ma-267	263	8	>	>	X
ma-267	263	9	2α+3	2α+3	PROPN
ma-267	263	10	2	2	NUM
ma-267	263	11	,	,	PUNCT
ma-267	263	12	ψ	ψ	AUX
ma-267	263	13	be	be	AUX
ma-267	263	14	a	a	DET
ma-267	263	15	riemann	riemann	PROPN
ma-267	263	16	-	-	PUNCT
ma-267	263	17	liouville	liouville	NOUN
ma-267	263	18	wavelet	wavelet	NOUN
ma-267	263	19	in	in	ADP
ma-267	263	20	on	on	ADP
ma-267	263	21	k	k	PROPN
ma-267	263	22	in	in	ADP
ma-267	263	23	l2α(k	l2α(k	PROPN
ma-267	263	24	)	)	PUNCT
ma-267	263	25	,	,	PUNCT
ma-267	263	26	,	,	PUNCT
ma-267	263	27	and	and	CCONJ
ma-267	263	28	β	β	X
ma-267	263	29	>	>	X
ma-267	263	30	0	0	NUM
ma-267	263	31	,	,	PUNCT
ma-267	263	32	for	for	ADP
ma-267	263	33	all	all	DET
ma-267	263	34	f	f	PROPN
ma-267	263	35	∈	∈	PROPN
ma-267	263	36	hsα(k)and	hsα(k)and	NOUN
ma-267	263	37	g	g	NOUN
ma-267	263	38	=	=	PROPN
ma-267	263	39	sαψ(f	sαψ(f	PROPN
ma-267	263	40	)	)	PUNCT
ma-267	263	41	,	,	PUNCT
ma-267	263	42	the	the	DET
ma-267	263	43	extremal	extremal	ADJ
ma-267	263	44	function	function	NOUN
ma-267	263	45	f	f	PROPN
ma-267	263	46	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	263	47	)	)	PUNCT
ma-267	263	48	,	,	PUNCT
ma-267	263	49	ψ	ψ	X
ma-267	263	50	,	,	PUNCT
ma-267	263	51	β	β	X
ma-267	263	52	satisfies	satisfie	NOUN
ma-267	263	53	lim	lim	PROPN
ma-267	263	54	β→0	β→0	PROPN
ma-267	263	55	+	+	NUM
ma-267	263	56	∥∥∥f	∥∥∥f	NOUN
ma-267	263	57	∗sαψ(f	∗sαψ(f	NUM
ma-267	263	58	)	)	PUNCT
ma-267	263	59	,	,	PUNCT
ma-267	263	60	ψ	ψ	PROPN
ma-267	263	61	,	,	PUNCT
ma-267	263	62	β	β	X
ma-267	263	63	−	−	NOUN
ma-267	263	64	f	f	X
ma-267	263	65	∥∥∥hsα	∥∥∥hsα	PUNCT
ma-267	264	1	=	=	NOUN
ma-267	264	2	0	0	X
ma-267	264	3	.	.	PUNCT
ma-267	265	1	moreover	moreover	ADV
ma-267	265	2	we	we	PRON
ma-267	265	3	have	have	VERB
ma-267	265	4	f	f	NOUN
ma-267	265	5	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	265	6	)	)	PUNCT
ma-267	265	7	,	,	PUNCT
ma-267	265	8	ψ	ψ	X
ma-267	265	9	,	,	PUNCT
ma-267	265	10	β	β	X
ma-267	265	11	−→	−→	NOUN
ma-267	265	12	f	f	X
ma-267	265	13	uniformly	uniformly	ADV
ma-267	265	14	when	when	SCONJ
ma-267	265	15	β	β	X
ma-267	265	16	−→	−→	NOUN
ma-267	265	17	0	0	NUM
ma-267	265	18	+	+	NOUN
ma-267	265	19	.	.	PUNCT
ma-267	265	20	proof	proof	NOUN
ma-267	265	21	.	.	PUNCT
ma-267	266	1	by	by	ADP
ma-267	266	2	using	use	VERB
ma-267	266	3	the	the	DET
ma-267	266	4	relation	relation	NOUN
ma-267	266	5	(	(	PUNCT
ma-267	266	6	4.17	4.17	NUM
ma-267	266	7	)	)	PUNCT
ma-267	266	8	we	we	PRON
ma-267	266	9	find	find	VERB
ma-267	266	10	that	that	SCONJ
ma-267	266	11	fα(f	fα(f	NOUN
ma-267	266	12	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	266	13	)	)	PUNCT
ma-267	266	14	,	,	PUNCT
ma-267	266	15	ψ	ψ	PROPN
ma-267	266	16	,	,	PUNCT
ma-267	266	17	β	β	NOUN
ma-267	266	18	−	−	PROPN
ma-267	266	19	f	f	PROPN
ma-267	266	20	)	)	PUNCT
ma-267	266	21	(	(	PUNCT
ma-267	266	22	µ	µ	X
ma-267	266	23	,	,	PUNCT
ma-267	266	24	λ	λ	NOUN
ma-267	266	25	)	)	PUNCT
ma-267	266	26	=	=	SYM
ma-267	267	1	−β	−β	NOUN
ma-267	267	2	(	(	PUNCT
ma-267	267	3	1	1	NUM
ma-267	267	4	+	+	CCONJ
ma-267	267	5	µ2	µ2	PROPN
ma-267	267	6	+	+	CCONJ
ma-267	267	7	2λ2	2λ2	NUM
ma-267	267	8	)	)	PUNCT
ma-267	267	9	s	s	PART
ma-267	267	10	fα(f	fα(f	NOUN
ma-267	267	11	)	)	PUNCT
ma-267	267	12	(	(	PUNCT
ma-267	267	13	µ	µ	X
ma-267	267	14	,	,	PUNCT
ma-267	267	15	λ	λ	NOUN
ma-267	267	16	)	)	PUNCT
ma-267	267	17	β	β	NOUN
ma-267	267	18	(	(	PUNCT
ma-267	267	19	1	1	NUM
ma-267	267	20	+	+	CCONJ
ma-267	267	21	µ2	µ2	PROPN
ma-267	267	22	+	+	CCONJ
ma-267	267	23	2λ2)s	2λ2)s	NUM
ma-267	267	24	+	+	CCONJ
ma-267	267	25	cψ	cψ	NOUN
ma-267	267	26	(	(	PUNCT
ma-267	267	27	4.19	4.19	NUM
ma-267	267	28	)	)	PUNCT
ma-267	267	29	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	PROPN
ma-267	267	30	eur	eur	PROPN
ma-267	267	31	.	.	PUNCT
ma-267	268	1	j.	j.	PROPN
ma-267	268	2	math	math	PROPN
ma-267	268	3	.	.	PUNCT
ma-267	269	1	anal	anal	PROPN
ma-267	269	2	.	.	PUNCT
ma-267	270	1	10.28924	10.28924	NUM
ma-267	270	2	/	/	SYM
ma-267	270	3	ada	ada	PROPN
ma-267	270	4	/	/	SYM
ma-267	270	5	ma.4.20	ma.4.20	PROPN
ma-267	271	1	13consequently	13consequently	ADV
ma-267	271	2	we	we	PRON
ma-267	271	3	find	find	VERB
ma-267	271	4	that∥∥∥f	that∥∥∥f	NOUN
ma-267	271	5	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	271	6	)	)	PUNCT
ma-267	271	7	,	,	PUNCT
ma-267	271	8	ψ	ψ	PROPN
ma-267	271	9	,	,	PUNCT
ma-267	271	10	β	β	X
ma-267	271	11	−	−	NOUN
ma-267	271	12	f	f	X
ma-267	271	13	∥∥∥hsα	∥∥∥hsα	PUNCT
ma-267	272	1	=	=	SYM
ma-267	272	2	∫	∫	PROPN
ma-267	272	3	k̂	k̂	PROPN
ma-267	272	4	β2	β2	PROPN
ma-267	272	5	(	(	PUNCT
ma-267	272	6	1	1	NUM
ma-267	272	7	+	+	CCONJ
ma-267	272	8	µ2	µ2	PROPN
ma-267	272	9	+	+	CCONJ
ma-267	272	10	2λ2	2λ2	NUM
ma-267	272	11	)	)	PUNCT
ma-267	272	12	3s	3s	NUM
ma-267	272	13	|fα(f	|fα(f	PRON
ma-267	272	14	)	)	PUNCT
ma-267	272	15	(	(	PUNCT
ma-267	272	16	µ	µ	NOUN
ma-267	272	17	,	,	PUNCT
ma-267	272	18	λ)|2	λ)|2	PRON
ma-267	272	19	β	β	X
ma-267	272	20	(	(	PUNCT
ma-267	272	21	1	1	NUM
ma-267	272	22	+	+	CCONJ
ma-267	272	23	µ2	µ2	PROPN
ma-267	272	24	+	+	CCONJ
ma-267	272	25	2λ2)s	2λ2)s	NUM
ma-267	272	26	+	+	CCONJ
ma-267	272	27	cψ	cψ	PROPN
ma-267	272	28	dγα(µ	dγα(µ	PROPN
ma-267	272	29	,	,	PUNCT
ma-267	272	30	λ	λ	NOUN
ma-267	272	31	)	)	PUNCT
ma-267	272	32	by	by	ADP
ma-267	272	33	using	use	VERB
ma-267	272	34	the	the	DET
ma-267	272	35	dominated	dominate	VERB
ma-267	272	36	convergence	convergence	NOUN
ma-267	272	37	theorem	theorem	VERB
ma-267	272	38	and	and	CCONJ
ma-267	272	39	the	the	DET
ma-267	272	40	fact	fact	NOUN
ma-267	272	41	that	that	SCONJ
ma-267	272	42	β2	β2	NOUN
ma-267	272	43	(	(	PUNCT
ma-267	272	44	1	1	NUM
ma-267	272	45	+	+	CCONJ
ma-267	272	46	µ2	µ2	PROPN
ma-267	272	47	+	+	CCONJ
ma-267	272	48	2λ2	2λ2	NUM
ma-267	272	49	)	)	PUNCT
ma-267	272	50	3s	3s	NUM
ma-267	272	51	|fα(f	|fα(f	PRON
ma-267	272	52	)	)	PUNCT
ma-267	272	53	(	(	PUNCT
ma-267	272	54	µ	µ	NOUN
ma-267	272	55	,	,	PUNCT
ma-267	272	56	λ)|2	λ)|2	PRON
ma-267	272	57	β	β	X
ma-267	272	58	(	(	PUNCT
ma-267	272	59	1	1	NUM
ma-267	272	60	+	+	CCONJ
ma-267	272	61	µ2	µ2	PROPN
ma-267	272	62	+	+	CCONJ
ma-267	272	63	2λ2)s	2λ2)s	NUM
ma-267	272	64	+	+	CCONJ
ma-267	272	65	cψ	cψ	PRON
ma-267	272	66	≤	≤	NOUN
ma-267	272	67	(	(	PUNCT
ma-267	272	68	1	1	NUM
ma-267	272	69	+	+	CCONJ
ma-267	272	70	µ2	µ2	PROPN
ma-267	272	71	+	+	CCONJ
ma-267	272	72	2λ2	2λ2	NUM
ma-267	272	73	)	)	PUNCT
ma-267	272	74	s	s	PART
ma-267	272	75	|fα(f	|fα(f	X
ma-267	272	76	)	)	PUNCT
ma-267	272	77	(	(	PUNCT
ma-267	272	78	µ	µ	NOUN
ma-267	272	79	,	,	PUNCT
ma-267	272	80	λ)|2	λ)|2	NOUN
ma-267	272	81	,	,	PUNCT
ma-267	272	82	we	we	PRON
ma-267	272	83	deduce	deduce	VERB
ma-267	272	84	that	that	SCONJ
ma-267	272	85	lim	lim	PROPN
ma-267	272	86	β→0	β→0	PUNCT
ma-267	272	87	+	+	NUM
ma-267	272	88	∥∥∥f	∥∥∥f	NOUN
ma-267	272	89	∗sαψ(f	∗sαψ(f	NUM
ma-267	272	90	)	)	PUNCT
ma-267	272	91	,	,	PUNCT
ma-267	272	92	ψ	ψ	PROPN
ma-267	272	93	,	,	PUNCT
ma-267	272	94	β	β	X
ma-267	272	95	−	−	NOUN
ma-267	272	96	f	f	X
ma-267	272	97	∥∥∥hsα	∥∥∥hsα	PUNCT
ma-267	273	1	=	=	SYM
ma-267	273	2	0on	0on	NOUN
ma-267	273	3	the	the	DET
ma-267	273	4	other	other	ADJ
ma-267	273	5	hand	hand	NOUN
ma-267	273	6	by	by	ADP
ma-267	273	7	using	use	VERB
ma-267	273	8	inversion	inversion	NOUN
ma-267	273	9	formula	formula	NOUN
ma-267	273	10	(	(	PUNCT
ma-267	273	11	2.5	2.5	NUM
ma-267	273	12	)	)	PUNCT
ma-267	273	13	and	and	CCONJ
ma-267	273	14	the	the	DET
ma-267	273	15	relation	relation	NOUN
ma-267	273	16	(	(	PUNCT
ma-267	273	17	4.19	4.19	NUM
ma-267	273	18	)	)	PUNCT
ma-267	273	19	we	we	PRON
ma-267	273	20	find	find	VERB
ma-267	273	21	that	that	SCONJ
ma-267	273	22	f	f	PROPN
ma-267	273	23	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	273	24	)	)	PUNCT
ma-267	273	25	,	,	PUNCT
ma-267	273	26	ψ	ψ	X
ma-267	273	27	,	,	PUNCT
ma-267	273	28	β	β	X
ma-267	273	29	(	(	PUNCT
ma-267	273	30	y	y	PROPN
ma-267	273	31	,	,	PUNCT
ma-267	273	32	v)−	v)−	PROPN
ma-267	273	33	f	f	X
ma-267	273	34	(	(	PUNCT
ma-267	273	35	y	y	PROPN
ma-267	273	36	,	,	PUNCT
ma-267	273	37	v	v	NOUN
ma-267	273	38	)	)	PUNCT
ma-267	273	39	=	=	SYM
ma-267	274	1	∫	∫	PROPN
ma-267	274	2	k̂	k̂	PROPN
ma-267	274	3	fα(f	fα(f	X
ma-267	274	4	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	274	5	)	)	PUNCT
ma-267	274	6	,	,	PUNCT
ma-267	274	7	ψ	ψ	PROPN
ma-267	274	8	,	,	PUNCT
ma-267	274	9	β	β	NOUN
ma-267	274	10	−	−	PROPN
ma-267	274	11	f	f	PROPN
ma-267	274	12	)	)	PUNCT
ma-267	274	13	(	(	PUNCT
ma-267	274	14	µ	µ	NOUN
ma-267	274	15	,	,	PUNCT
ma-267	274	16	λ)ϕµ,λ(y	λ)ϕµ,λ(y	PROPN
ma-267	274	17	,	,	PUNCT
ma-267	274	18	v)dγα(µ	v)dγα(µ	PROPN
ma-267	274	19	,	,	PUNCT
ma-267	274	20	λ	λ	NOUN
ma-267	274	21	)	)	PUNCT
ma-267	274	22	,	,	PUNCT
ma-267	274	23	=	=	SYM
ma-267	274	24	∫	∫	PROPN
ma-267	274	25	k̂	k̂	PROPN
ma-267	274	26	−β	−β	PROPN
ma-267	274	27	(	(	PUNCT
ma-267	274	28	1	1	NUM
ma-267	274	29	+	+	CCONJ
ma-267	274	30	µ2	µ2	PROPN
ma-267	274	31	+	+	CCONJ
ma-267	274	32	2λ2	2λ2	NUM
ma-267	274	33	)	)	PUNCT
ma-267	274	34	s	s	PART
ma-267	274	35	fα(f	fα(f	NOUN
ma-267	274	36	)	)	PUNCT
ma-267	274	37	(	(	PUNCT
ma-267	274	38	λ)ϕµ,λ(y	λ)ϕµ,λ(y	PROPN
ma-267	274	39	,	,	PUNCT
ma-267	274	40	v	v	NOUN
ma-267	274	41	)	)	PUNCT
ma-267	274	42	β	β	NOUN
ma-267	274	43	(	(	PUNCT
ma-267	274	44	1	1	NUM
ma-267	274	45	+	+	CCONJ
ma-267	274	46	µ2	µ2	PROPN
ma-267	274	47	+	+	CCONJ
ma-267	274	48	2λ2)s	2λ2)s	NUM
ma-267	274	49	+	+	CCONJ
ma-267	274	50	cψ	cψ	PROPN
ma-267	274	51	dγα(µ	dγα(µ	PROPN
ma-267	274	52	,	,	PUNCT
ma-267	274	53	λ	λ	NOUN
ma-267	274	54	)	)	PUNCT
ma-267	274	55	again	again	ADV
ma-267	274	56	by	by	ADP
ma-267	274	57	dominated	dominated	ADJ
ma-267	274	58	convergence	convergence	NOUN
ma-267	274	59	theorem	theorem	NOUN
ma-267	274	60	and	and	CCONJ
ma-267	274	61	the	the	DET
ma-267	274	62	fact	fact	NOUN
ma-267	274	63	that∣∣∣∣∣−β	that∣∣∣∣∣−β	PROPN
ma-267	274	64	(	(	PUNCT
ma-267	274	65	1	1	NUM
ma-267	274	66	+	+	CCONJ
ma-267	274	67	µ2	µ2	PROPN
ma-267	274	68	+	+	CCONJ
ma-267	274	69	2λ2	2λ2	NUM
ma-267	274	70	)	)	PUNCT
ma-267	274	71	s	s	PART
ma-267	274	72	fα(f	fα(f	NOUN
ma-267	274	73	)	)	PUNCT
ma-267	274	74	(	(	PUNCT
ma-267	274	75	λ)ϕµ,λ(y	λ)ϕµ,λ(y	PROPN
ma-267	274	76	,	,	PUNCT
ma-267	274	77	s	s	PROPN
ma-267	274	78	)	)	PUNCT
ma-267	274	79	β	β	NOUN
ma-267	274	80	(	(	PUNCT
ma-267	274	81	1	1	NUM
ma-267	274	82	+	+	CCONJ
ma-267	274	83	µ2	µ2	PROPN
ma-267	274	84	+	+	CCONJ
ma-267	274	85	2λ2)s	2λ2)s	NUM
ma-267	274	86	+	+	CCONJ
ma-267	274	87	cψ	cψ	PRON
ma-267	274	88	∣∣∣∣∣	∣∣∣∣∣	ADJ
ma-267	274	89	≤	≤	NOUN
ma-267	274	90	|fα(f	|fα(f	X
ma-267	274	91	)	)	PUNCT
ma-267	274	92	(	(	PUNCT
ma-267	274	93	µ	µ	NOUN
ma-267	274	94	,	,	PUNCT
ma-267	274	95	λ)|	λ)|	INTJ
ma-267	274	96	we	we	PRON
ma-267	274	97	deduce	deduce	VERB
ma-267	274	98	that	that	SCONJ
ma-267	274	99	lim	lim	PROPN
ma-267	274	100	β→0	β→0	PUNCT
ma-267	274	101	+	+	NUM
ma-267	274	102	∥∥∥f	∥∥∥f	NOUN
ma-267	274	103	∗sαψ(f	∗sαψ(f	NUM
ma-267	274	104	)	)	PUNCT
ma-267	274	105	,	,	PUNCT
ma-267	274	106	ψ	ψ	PROPN
ma-267	274	107	,	,	PUNCT
ma-267	274	108	β	β	X
ma-267	274	109	−	−	NOUN
ma-267	274	110	f	f	PROPN
ma-267	274	111	∥∥∥∞,µα	∥∥∥∞,µα	PROPN
ma-267	274	112	=	=	SYM
ma-267	274	113	0	0	NUM
ma-267	274	114	which	which	PRON
ma-267	274	115	proves	prove	VERB
ma-267	274	116	that	that	SCONJ
ma-267	274	117	f	f	PROPN
ma-267	274	118	∗sαψ(f	∗sαψ(f	PUNCT
ma-267	274	119	)	)	PUNCT
ma-267	274	120	,	,	PUNCT
ma-267	274	121	ψ	ψ	X
ma-267	274	122	,	,	PUNCT
ma-267	274	123	β	β	X
ma-267	274	124	−→	−→	NOUN
ma-267	274	125	f	f	X
ma-267	274	126	uniformly	uniformly	ADV
ma-267	274	127	when	when	SCONJ
ma-267	274	128	β	β	X
ma-267	274	129	−→	−→	NOUN
ma-267	274	130	0	0	NUM
ma-267	274	131	+	+	NOUN
ma-267	274	132	.	.	PUNCT
ma-267	274	133	�	�	PROPN
ma-267	274	134	authors	authors	PROPN
ma-267	274	135	’	'	PUNCT
ma-267	274	136	contribution	contribution	NOUN
ma-267	274	137	the	the	DET
ma-267	274	138	authors	author	NOUN
ma-267	274	139	contributed	contribute	VERB
ma-267	274	140	equally	equally	ADV
ma-267	274	141	to	to	ADP
ma-267	274	142	this	this	DET
ma-267	274	143	work	work	NOUN
ma-267	274	144	.	.	PUNCT
ma-267	275	1	references	reference	NOUN
ma-267	275	2	[	[	X
ma-267	275	3	1	1	NUM
ma-267	275	4	]	]	PUNCT
ma-267	275	5	b.	b.	PROPN
ma-267	275	6	amri	amri	PROPN
ma-267	275	7	,	,	PUNCT
ma-267	275	8	l.	l.	PROPN
ma-267	275	9	t.	t.	PROPN
ma-267	275	10	rachdi	rachdi	PROPN
ma-267	275	11	,	,	PUNCT
ma-267	275	12	beckner	beckner	ADJ
ma-267	275	13	logarithmic	logarithmic	ADJ
ma-267	275	14	uncertainty	uncertainty	NOUN
ma-267	275	15	principle	principle	NOUN
ma-267	275	16	for	for	ADP
ma-267	275	17	the	the	DET
ma-267	275	18	riemann	riemann	PROPN
ma-267	275	19	–	–	PUNCT
ma-267	275	20	liouville	liouville	NOUN
ma-267	275	21	operator	operator	NOUN
ma-267	275	22	.	.	PUNCT
ma-267	276	1	int	int	NOUN
ma-267	276	2	.	.	PUNCT
ma-267	277	1	j.	j.	PROPN
ma-267	277	2	math	math	PROPN
ma-267	277	3	.	.	PUNCT
ma-267	278	1	24(2013	24(2013	NUM
ma-267	278	2	)	)	PUNCT
ma-267	278	3	,	,	PUNCT
ma-267	278	4	1350070	1350070	NUM
ma-267	278	5	.	.	PUNCT
ma-267	279	1	https://doi.org/10.1142/s0129167x13500705.[2	https://doi.org/10.1142/s0129167x13500705.[2	ADP
ma-267	279	2	]	]	X
ma-267	279	3	b.	b.	PROPN
ma-267	279	4	amri	amri	PROPN
ma-267	279	5	,	,	PUNCT
ma-267	279	6	l.	l.	PROPN
ma-267	279	7	t.	t.	PROPN
ma-267	279	8	rachdi	rachdi	PROPN
ma-267	279	9	,	,	PUNCT
ma-267	279	10	uncertainty	uncertainty	NOUN
ma-267	279	11	principle	principle	NOUN
ma-267	279	12	in	in	ADP
ma-267	279	13	terms	term	NOUN
ma-267	279	14	of	of	ADP
ma-267	279	15	entropy	entropy	NOUN
ma-267	279	16	for	for	ADP
ma-267	279	17	the	the	DET
ma-267	279	18	riemann	riemann	PROPN
ma-267	279	19	–	–	PUNCT
ma-267	279	20	liouville	liouville	NOUN
ma-267	279	21	operator	operator	NOUN
ma-267	279	22	.	.	PUNCT
ma-267	280	1	bull	bull	NOUN
ma-267	280	2	.	.	PUNCT
ma-267	281	1	malays.math	malays.math	PROPN
ma-267	281	2	.	.	PUNCT
ma-267	282	1	sci	sci	PROPN
ma-267	282	2	.	.	PROPN
ma-267	282	3	soc	soc	PROPN
ma-267	282	4	.	.	PUNCT
ma-267	283	1	39	39	NUM
ma-267	283	2	(	(	PUNCT
ma-267	283	3	2016	2016	NUM
ma-267	283	4	)	)	PUNCT
ma-267	283	5	,	,	PUNCT
ma-267	283	6	457	457	NUM
ma-267	283	7	-	-	SYM
ma-267	283	8	481	481	NUM
ma-267	283	9	.	.	PUNCT
ma-267	284	1	https://doi.org/10.1007/s40840-015-0121-5.[3	https://doi.org/10.1007/s40840-015-0121-5.[3	ADJ
ma-267	284	2	]	]	X
ma-267	284	3	c.	c.	PROPN
ma-267	284	4	baccar	baccar	NOUN
ma-267	284	5	,	,	PUNCT
ma-267	284	6	n.	n.	PROPN
ma-267	284	7	b.	b.	PROPN
ma-267	284	8	hamadi	hamadi	PROPN
ma-267	284	9	,	,	PUNCT
ma-267	284	10	l.	l.	PROPN
ma-267	284	11	t.	t.	PROPN
ma-267	284	12	rachdi	rachdi	NOUN
ma-267	284	13	,	,	PUNCT
ma-267	284	14	inversion	inversion	NOUN
ma-267	284	15	formulas	formula	NOUN
ma-267	284	16	for	for	ADP
ma-267	284	17	riemann	riemann	PROPN
ma-267	284	18	-	-	PUNCT
ma-267	284	19	liouville	liouville	VERB
ma-267	284	20	transform	transform	NOUN
ma-267	284	21	and	and	CCONJ
ma-267	284	22	its	its	PRON
ma-267	284	23	dual	dual	ADJ
ma-267	284	24	associatedwith	associatedwith	PRON
ma-267	284	25	singular	singular	ADJ
ma-267	284	26	partial	partial	ADJ
ma-267	284	27	differential	differential	NOUN
ma-267	284	28	operators	operator	NOUN
ma-267	284	29	.	.	PUNCT
ma-267	285	1	international	international	ADJ
ma-267	285	2	journal	journal	PROPN
ma-267	285	3	of	of	ADP
ma-267	285	4	mathematics	mathematics	PROPN
ma-267	285	5	and	and	CCONJ
ma-267	285	6	mathematical	mathematical	ADJ
ma-267	285	7	sciences	science	NOUN
ma-267	285	8	,	,	PUNCT
ma-267	285	9	2006(2006	2006(2006	NUM
ma-267	285	10	)	)	PUNCT
ma-267	285	11	,	,	PUNCT
ma-267	285	12	086238	086238	NUM
ma-267	285	13	.	.	PUNCT
ma-267	286	1	https://doi.org/10.1155/ijmms/2006/86238.[4	https://doi.org/10.1155/ijmms/2006/86238.[4	PROPN
ma-267	286	2	]	]	X
ma-267	286	3	c.	c.	NOUN
ma-267	286	4	baccar	baccar	NOUN
ma-267	286	5	,	,	PUNCT
ma-267	286	6	n.	n.	PROPN
ma-267	286	7	b.	b.	PROPN
ma-267	286	8	hamadi	hamadi	PROPN
ma-267	286	9	,	,	PUNCT
ma-267	286	10	localization	localization	NOUN
ma-267	286	11	operators	operator	NOUN
ma-267	286	12	of	of	ADP
ma-267	286	13	the	the	DET
ma-267	286	14	wavelet	wavelet	NOUN
ma-267	286	15	transform	transform	NOUN
ma-267	286	16	associated	associate	VERB
ma-267	286	17	to	to	ADP
ma-267	286	18	the	the	DET
ma-267	286	19	riemann	riemann	PROPN
ma-267	286	20	–	–	PUNCT
ma-267	286	21	liouvilleoperator	liouvilleoperator	NOUN
ma-267	286	22	.	.	PUNCT
ma-267	287	1	int	int	NOUN
ma-267	287	2	.	.	PUNCT
ma-267	288	1	j.	j.	PROPN
ma-267	288	2	math	math	PROPN
ma-267	288	3	.	.	PUNCT
ma-267	289	1	27	27	NUM
ma-267	289	2	(	(	PUNCT
ma-267	289	3	2016	2016	NUM
ma-267	289	4	)	)	PUNCT
ma-267	289	5	,	,	PUNCT
ma-267	289	6	1650036	1650036	NUM
ma-267	289	7	.	.	PUNCT
ma-267	290	1	https://doi.org/10.1142/s0129167x16500361.[5	https://doi.org/10.1142/s0129167x16500361.[5	X
ma-267	290	2	]	]	X
ma-267	290	3	j.	j.	PROPN
ma-267	290	4	k.	k.	PROPN
ma-267	290	5	cohen	cohen	PROPN
ma-267	290	6	,	,	PUNCT
ma-267	290	7	n.	n.	PROPN
ma-267	290	8	bleistein	bleistein	NOUN
ma-267	290	9	,	,	PUNCT
ma-267	290	10	velocity	velocity	NOUN
ma-267	290	11	inversion	inversion	NOUN
ma-267	290	12	procedure	procedure	NOUN
ma-267	290	13	for	for	ADP
ma-267	290	14	acoustic	acoustic	ADJ
ma-267	290	15	waves	wave	NOUN
ma-267	290	16	.	.	PUNCT
ma-267	291	1	geophysics	geophysic	NOUN
ma-267	291	2	,	,	PUNCT
ma-267	291	3	44	44	NUM
ma-267	291	4	(	(	PUNCT
ma-267	291	5	1979	1979	NUM
ma-267	291	6	)	)	PUNCT
ma-267	291	7	,	,	PUNCT
ma-267	291	8	1077	1077	NUM
ma-267	291	9	-	-	SYM
ma-267	291	10	1087	1087	NUM
ma-267	291	11	.	.	PUNCT
ma-267	292	1	https://doi.org/10.1190/1.1440996.[6	https://doi.org/10.1190/1.1440996.[6	X
ma-267	292	2	]	]	X
ma-267	292	3	i.	i.	PROPN
ma-267	292	4	daubechies	daubechies	PROPN
ma-267	292	5	,	,	PUNCT
ma-267	292	6	ten	ten	NUM
ma-267	292	7	lectures	lecture	NOUN
ma-267	292	8	on	on	ADP
ma-267	292	9	wavelets	wavelet	NOUN
ma-267	292	10	,	,	PUNCT
ma-267	292	11	siam	siam	NOUN
ma-267	292	12	,	,	PUNCT
ma-267	292	13	(	(	PUNCT
ma-267	292	14	1992).[7	1992).[7	X
ma-267	292	15	]	]	X
ma-267	292	16	j.a	j.a	PROPN
ma-267	292	17	.	.	PROPN
ma-267	292	18	fawcett	fawcett	PROPN
ma-267	292	19	,	,	PUNCT
ma-267	292	20	inversion	inversion	NOUN
ma-267	292	21	of	of	ADP
ma-267	292	22	n	n	CCONJ
ma-267	292	23	-	-	PUNCT
ma-267	292	24	dimensional	dimensional	ADJ
ma-267	292	25	spherical	spherical	ADJ
ma-267	292	26	averages	average	NOUN
ma-267	292	27	,	,	PUNCT
ma-267	292	28	siam	siam	PROPN
ma-267	292	29	j.	j.	PROPN
ma-267	292	30	appl	appl	PROPN
ma-267	292	31	.	.	PROPN
ma-267	292	32	math	math	PROPN
ma-267	292	33	.	.	PUNCT
ma-267	293	1	45	45	NUM
ma-267	293	2	(	(	PUNCT
ma-267	293	3	1985	1985	NUM
ma-267	293	4	)	)	PUNCT
ma-267	293	5	,	,	PUNCT
ma-267	293	6	336–341	336–341	NUM
ma-267	293	7	.	.	PUNCT
ma-267	293	8	https	https	NOUN
ma-267	293	9	:	:	PUNCT
ma-267	293	10	//doi.org/10.1137/0145018	//doi.org/10.1137/0145018	PROPN
ma-267	293	11	.	.	PUNCT
ma-267	294	1	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	NOUN
ma-267	294	2	https://doi.org/10.1142/s0129167x13500705	https://doi.org/10.1142/s0129167x13500705	PROPN
ma-267	294	3	https://doi.org/10.1007/s40840-015-0121-5	https://doi.org/10.1007/s40840-015-0121-5	NUM
ma-267	294	4	https://doi.org/10.1155/ijmms/2006/86238	https://doi.org/10.1155/ijmms/2006/86238	VERB
ma-267	294	5	https://doi.org/10.1142/s0129167x16500361	https://doi.org/10.1142/s0129167x16500361	NOUN
ma-267	294	6	https://doi.org/10.1190/1.1440996	https://doi.org/10.1190/1.1440996	X
ma-267	294	7	https://doi.org/10.1137/0145018	https://doi.org/10.1137/0145018	PROPN
ma-267	294	8	https://doi.org/10.1137/0145018	https://doi.org/10.1137/0145018	PROPN
ma-267	294	9	eur	eur	PROPN
ma-267	294	10	.	.	PUNCT
ma-267	295	1	j.	j.	PROPN
ma-267	295	2	math	math	PROPN
ma-267	295	3	.	.	PUNCT
ma-267	296	1	anal	anal	PROPN
ma-267	296	2	.	.	PUNCT
ma-267	297	1	10.28924	10.28924	NUM
ma-267	297	2	/	/	SYM
ma-267	297	3	ada	ada	PROPN
ma-267	297	4	/	/	SYM
ma-267	297	5	ma.4.20	ma.4.20	NOUN
ma-267	297	6	14	14	NUM
ma-267	297	7	[	[	X
ma-267	297	8	8	8	NUM
ma-267	297	9	]	]	PUNCT
ma-267	297	10	a.	a.	NOUN
ma-267	297	11	grossmann	grossmann	PROPN
ma-267	297	12	,	,	PUNCT
ma-267	297	13	j.	j.	PROPN
ma-267	297	14	morlet	morlet	PROPN
ma-267	297	15	,	,	PUNCT
ma-267	297	16	decomposition	decomposition	NOUN
ma-267	297	17	of	of	ADP
ma-267	297	18	hardy	hardy	ADJ
ma-267	297	19	functions	function	NOUN
ma-267	297	20	into	into	ADP
ma-267	297	21	square	square	ADJ
ma-267	297	22	integrable	integrable	ADJ
ma-267	297	23	wavelets	wavelet	NOUN
ma-267	297	24	of	of	ADP
ma-267	297	25	constant	constant	ADJ
ma-267	297	26	shape	shape	NOUN
ma-267	297	27	,	,	PUNCT
ma-267	297	28	siamj	siamj	NOUN
ma-267	297	29	.	.	PUNCT
ma-267	298	1	math	math	NOUN
ma-267	298	2	.	.	PUNCT
ma-267	299	1	anal	anal	PROPN
ma-267	299	2	.	.	PUNCT
ma-267	300	1	15	15	NUM
ma-267	300	2	(	(	PUNCT
ma-267	300	3	1984	1984	NUM
ma-267	300	4	)	)	PUNCT
ma-267	300	5	,	,	PUNCT
ma-267	301	1	723–736	723–736	NUM
ma-267	301	2	.	.	PUNCT
ma-267	302	1	https://doi.org/10.1137/0515056.[9	https://doi.org/10.1137/0515056.[9	NUM
ma-267	302	2	]	]	X
ma-267	302	3	p.	p.	NOUN
ma-267	302	4	goupillaud	goupillaud	PROPN
ma-267	302	5	,	,	PUNCT
ma-267	302	6	a.	a.	NOUN
ma-267	302	7	grossmann	grossmann	PROPN
ma-267	302	8	,	,	PUNCT
ma-267	302	9	j.	j.	PROPN
ma-267	302	10	morlet	morlet	PROPN
ma-267	302	11	,	,	PUNCT
ma-267	302	12	cycle	cycle	NOUN
ma-267	302	13	-	-	PUNCT
ma-267	302	14	octave	octave	NOUN
ma-267	302	15	and	and	CCONJ
ma-267	302	16	related	related	ADJ
ma-267	302	17	transforms	transform	NOUN
ma-267	302	18	in	in	ADP
ma-267	302	19	seismic	seismic	ADJ
ma-267	302	20	signal	signal	NOUN
ma-267	302	21	analysis	analysis	NOUN
ma-267	302	22	,	,	PUNCT
ma-267	302	23	geoexplo	geoexplo	NOUN
ma-267	302	24	-	-	PUNCT
ma-267	302	25	ration	ration	NOUN
ma-267	302	26	23	23	NUM
ma-267	302	27	(	(	PUNCT
ma-267	302	28	1984	1984	NUM
ma-267	302	29	)	)	PUNCT
ma-267	302	30	,	,	PUNCT
ma-267	302	31	85–102	85–102	NUM
ma-267	302	32	.	.	PUNCT
ma-267	302	33	https://doi.org/10.1016/0016-7142(84)90025-5.[10	https://doi.org/10.1016/0016-7142(84)90025-5.[10	ADV
ma-267	302	34	]	]	PUNCT
ma-267	302	35	h.	h.	PROPN
ma-267	302	36	hellsten	hellsten	PROPN
ma-267	302	37	,	,	PUNCT
ma-267	302	38	l.e	l.e	PROPN
ma-267	302	39	.	.	PROPN
ma-267	302	40	andersson	andersson	PROPN
ma-267	302	41	,	,	PUNCT
ma-267	302	42	an	an	DET
ma-267	302	43	inverse	inverse	NOUN
ma-267	302	44	method	method	NOUN
ma-267	302	45	for	for	ADP
ma-267	302	46	the	the	DET
ma-267	302	47	processing	processing	NOUN
ma-267	302	48	of	of	ADP
ma-267	302	49	synthetic	synthetic	ADJ
ma-267	302	50	aperture	aperture	NOUN
ma-267	302	51	radar	radar	NOUN
ma-267	302	52	data	datum	NOUN
ma-267	302	53	,	,	PUNCT
ma-267	302	54	inverse	inverse	NOUN
ma-267	302	55	probl.3	probl.3	NOUN
ma-267	302	56	(	(	PUNCT
ma-267	302	57	1987	1987	NUM
ma-267	302	58	)	)	PUNCT
ma-267	302	59	,	,	PUNCT
ma-267	302	60	111–124	111–124	NUM
ma-267	302	61	.	.	PUNCT
ma-267	303	1	https://doi.org/10.1088/0266-5611/3/1/013.[11	https://doi.org/10.1088/0266-5611/3/1/013.[11	PROPN
ma-267	303	2	]	]	PUNCT
ma-267	303	3	l.h	l.h	PROPN
ma-267	303	4	.	.	PROPN
ma-267	303	5	herbertson	herbertson	PROPN
ma-267	303	6	,	,	PUNCT
ma-267	303	7	evaluation	evaluation	NOUN
ma-267	303	8	of	of	ADP
ma-267	303	9	fluid	fluid	ADJ
ma-267	303	10	mechanics	mechanic	NOUN
ma-267	303	11	and	and	CCONJ
ma-267	303	12	cavitation	cavitation	NOUN
ma-267	303	13	generated	generate	VERB
ma-267	303	14	by	by	ADP
ma-267	303	15	mechanical	mechanical	ADJ
ma-267	303	16	heart	heart	NOUN
ma-267	303	17	valves	valve	NOUN
ma-267	303	18	during	during	ADP
ma-267	303	19	theclosing	theclosing	NOUN
ma-267	303	20	phase	phase	NOUN
ma-267	303	21	.	.	PUNCT
ma-267	304	1	the	the	DET
ma-267	304	2	pennsylvania	pennsylvania	PROPN
ma-267	304	3	state	state	PROPN
ma-267	304	4	university	university	PROPN
ma-267	304	5	,	,	PUNCT
ma-267	304	6	(	(	PUNCT
ma-267	304	7	2009).[12	2009).[12	NOUN
ma-267	304	8	]	]	X
ma-267	304	9	n.b	n.b	PROPN
ma-267	304	10	.	.	PROPN
ma-267	304	11	hamadi	hamadi	PROPN
ma-267	304	12	,	,	PUNCT
ma-267	304	13	l.t	l.t	PROPN
ma-267	304	14	.	.	PROPN
ma-267	304	15	rachdi	rachdi	NOUN
ma-267	304	16	,	,	PUNCT
ma-267	304	17	weyl	weyl	PROPN
ma-267	304	18	transforms	transform	VERB
ma-267	304	19	associated	associate	VERB
ma-267	304	20	with	with	ADP
ma-267	304	21	the	the	DET
ma-267	304	22	riemann	riemann	PROPN
ma-267	304	23	-	-	PUNCT
ma-267	304	24	liouville	liouville	NOUN
ma-267	304	25	operator	operator	NOUN
ma-267	304	26	,	,	PUNCT
ma-267	304	27	int	int	NOUN
ma-267	304	28	.	.	PUNCT
ma-267	305	1	j.	j.	PROPN
ma-267	305	2	math	math	PROPN
ma-267	305	3	.	.	PUNCT
ma-267	306	1	math	math	NOUN
ma-267	306	2	.	.	PUNCT
ma-267	307	1	sci.2006	sci.2006	NUM
ma-267	307	2	(	(	PUNCT
ma-267	307	3	2006	2006	NUM
ma-267	307	4	)	)	PUNCT
ma-267	307	5	,	,	PUNCT
ma-267	307	6	094768	094768	NUM
ma-267	307	7	.	.	PUNCT
ma-267	308	1	https://doi.org/10.1155/ijmms/2006/94768.[13	https://doi.org/10.1155/ijmms/2006/94768.[13	PROPN
ma-267	308	2	]	]	X
ma-267	308	3	r.i	r.i	PROPN
ma-267	308	4	.	.	PROPN
ma-267	308	5	jewett	jewett	PROPN
ma-267	308	6	,	,	PUNCT
ma-267	308	7	spaces	space	VERB
ma-267	308	8	with	with	ADP
ma-267	308	9	an	an	DET
ma-267	308	10	abstract	abstract	ADJ
ma-267	308	11	convolution	convolution	NOUN
ma-267	308	12	of	of	ADP
ma-267	308	13	measures	measure	NOUN
ma-267	308	14	,	,	PUNCT
ma-267	308	15	adv	adv	PROPN
ma-267	308	16	.	.	PUNCT
ma-267	308	17	math	math	PROPN
ma-267	308	18	.	.	PUNCT
ma-267	309	1	18	18	NUM
ma-267	309	2	(	(	PUNCT
ma-267	309	3	1975	1975	NUM
ma-267	309	4	)	)	PUNCT
ma-267	309	5	,	,	PUNCT
ma-267	309	6	1–101	1–101	NUM
ma-267	309	7	.	.	PUNCT
ma-267	310	1	https://doi.org/10	https://doi.org/10	PROPN
ma-267	310	2	.	.	PUNCT
ma-267	311	1	1016/0001	1016/0001	NUM
ma-267	311	2	-	-	PUNCT
ma-267	311	3	8708(75)90002	8708(75)90002	NUM
ma-267	311	4	-	-	SYM
ma-267	311	5	x.[14	x.[14	PROPN
ma-267	311	6	]	]	PUNCT
ma-267	311	7	h.	h.	PROPN
ma-267	311	8	mejjaoli	mejjaoli	PROPN
ma-267	311	9	,	,	PUNCT
ma-267	311	10	dunkl	dunkl	VERB
ma-267	311	11	two	two	NUM
ma-267	311	12	-	-	PUNCT
ma-267	311	13	wavelet	wavelet	NOUN
ma-267	311	14	theory	theory	NOUN
ma-267	311	15	and	and	CCONJ
ma-267	311	16	localization	localization	NOUN
ma-267	311	17	operators	operator	NOUN
ma-267	311	18	,	,	PUNCT
ma-267	311	19	j.	j.	PROPN
ma-267	311	20	pseudo	pseudo	NOUN
ma-267	311	21	-	-	PUNCT
ma-267	311	22	differ	differ	VERB
ma-267	311	23	.	.	PUNCT
ma-267	312	1	oper	oper	PROPN
ma-267	312	2	.	.	PUNCT
ma-267	312	3	appl	appl	PROPN
ma-267	312	4	.	.	PROPN
ma-267	313	1	8	8	NUM
ma-267	313	2	(	(	PUNCT
ma-267	313	3	2017	2017	NUM
ma-267	313	4	)	)	PUNCT
ma-267	313	5	,	,	PUNCT
ma-267	313	6	349–387	349–387	NUM
ma-267	313	7	.	.	PUNCT
ma-267	314	1	https://doi.org/10.1007/s11868-017-0196-x.[15	https://doi.org/10.1007/s11868-017-0196-x.[15	X
ma-267	314	2	]	]	PUNCT
ma-267	314	3	s.	s.	PROPN
ma-267	314	4	omri	omri	PROPN
ma-267	314	5	,	,	PUNCT
ma-267	314	6	l.t	l.t	PROPN
ma-267	314	7	.	.	PROPN
ma-267	314	8	rachdi	rachdi	PROPN
ma-267	314	9	,	,	PUNCT
ma-267	314	10	heisenberg	heisenberg	PROPN
ma-267	314	11	-	-	PUNCT
ma-267	314	12	pauli	pauli	PROPN
ma-267	314	13	-	-	PUNCT
ma-267	314	14	weyl	weyl	VERB
ma-267	314	15	uncertainty	uncertainty	NOUN
ma-267	314	16	principle	principle	NOUN
ma-267	314	17	for	for	ADP
ma-267	314	18	the	the	DET
ma-267	314	19	riemann	riemann	PROPN
ma-267	314	20	-	-	PUNCT
ma-267	314	21	liouville	liouville	NOUN
ma-267	314	22	operator	operator	NOUN
ma-267	314	23	.	.	PUNCT
ma-267	315	1	j.	j.	PROPN
ma-267	315	2	inequal.pure	inequal.pure	PROPN
ma-267	315	3	appl	appl	PROPN
ma-267	315	4	.	.	PUNCT
ma-267	316	1	math	math	NOUN
ma-267	316	2	.	.	PUNCT
ma-267	317	1	9	9	NUM
ma-267	317	2	(	(	PUNCT
ma-267	317	3	2008	2008	NUM
ma-267	317	4	)	)	PUNCT
ma-267	317	5	,	,	PUNCT
ma-267	317	6	1	1	NUM
ma-267	317	7	-	-	SYM
ma-267	317	8	23.[16	23.[16	NUM
ma-267	317	9	]	]	PUNCT
ma-267	317	10	m.	m.	NOUN
ma-267	317	11	rosler	rosler	NOUN
ma-267	317	12	,	,	PUNCT
ma-267	317	13	bessel	bessel	NOUN
ma-267	317	14	-	-	PUNCT
ma-267	317	15	type	type	NOUN
ma-267	317	16	signed	sign	VERB
ma-267	317	17	hypergroups	hypergroup	NOUN
ma-267	317	18	on	on	ADP
ma-267	317	19	r.	r.	PROPN
ma-267	317	20	probability	probability	NOUN
ma-267	317	21	measures	measure	NOUN
ma-267	317	22	on	on	ADP
ma-267	317	23	groups	group	NOUN
ma-267	317	24	and	and	CCONJ
ma-267	317	25	related	related	ADJ
ma-267	317	26	structures	structure	NOUN
ma-267	317	27	xi	xi	INTJ
ma-267	317	28	.	.	PUNCT
ma-267	318	1	pro	pro	ADJ
ma-267	318	2	-	-	NOUN
ma-267	318	3	ceedings	ceeding	NOUN
ma-267	318	4	,	,	PUNCT
ma-267	318	5	oberwolfach	oberwolfach	ADV
ma-267	318	6	,	,	PUNCT
ma-267	318	7	292	292	NUM
ma-267	318	8	-	-	SYM
ma-267	318	9	304	304	NUM
ma-267	318	10	,	,	PUNCT
ma-267	318	11	1994.[17	1994.[17	NUM
ma-267	318	12	]	]	PUNCT
ma-267	318	13	a.	a.	NOUN
ma-267	318	14	saoudi	saoudi	PROPN
ma-267	318	15	,	,	PUNCT
ma-267	318	16	time	time	NOUN
ma-267	318	17	-	-	PUNCT
ma-267	318	18	scale	scale	NOUN
ma-267	318	19	localization	localization	NOUN
ma-267	318	20	operators	operator	NOUN
ma-267	318	21	in	in	ADP
ma-267	318	22	the	the	DET
ma-267	318	23	weinstein	weinstein	PROPN
ma-267	318	24	setting	setting	NOUN
ma-267	318	25	,	,	PUNCT
ma-267	318	26	results	result	VERB
ma-267	318	27	math	math	NOUN
ma-267	318	28	.	.	PUNCT
ma-267	319	1	78	78	NUM
ma-267	319	2	(	(	PUNCT
ma-267	319	3	2023	2023	NUM
ma-267	319	4	)	)	PUNCT
ma-267	319	5	,	,	PUNCT
ma-267	319	6	14	14	NUM
ma-267	319	7	.	.	PUNCT
ma-267	320	1	https://doi	https://doi	X
ma-267	320	2	.	.	PUNCT
ma-267	320	3	org/10.1007	org/10.1007	PROPN
ma-267	320	4	/	/	SYM
ma-267	320	5	s00025	s00025	PROPN
ma-267	320	6	-	-	PUNCT
ma-267	320	7	022	022	NUM
ma-267	320	8	-	-	PUNCT
ma-267	320	9	01792	01792	NUM
ma-267	320	10	-	-	SYM
ma-267	320	11	4.[18	4.[18	NUM
ma-267	320	12	]	]	PUNCT
ma-267	321	1	s.	s.	PROPN
ma-267	321	2	saitoh	saitoh	PROPN
ma-267	321	3	,	,	PUNCT
ma-267	321	4	y.	y.	PROPN
ma-267	321	5	sawano	sawano	PROPN
ma-267	321	6	,	,	PUNCT
ma-267	321	7	theory	theory	NOUN
ma-267	321	8	of	of	ADP
ma-267	321	9	reproducing	reproduce	VERB
ma-267	321	10	kernels	kernel	NOUN
ma-267	321	11	and	and	CCONJ
ma-267	321	12	applications	application	NOUN
ma-267	321	13	,	,	PUNCT
ma-267	321	14	singapore	singapore	PROPN
ma-267	321	15	:	:	PUNCT
ma-267	321	16	springer	springer	PROPN
ma-267	321	17	singapore	singapore	PROPN
ma-267	321	18	,	,	PUNCT
ma-267	321	19	2016.[19	2016.[19	NUM
ma-267	321	20	]	]	X
ma-267	321	21	s.	s.	PROPN
ma-267	321	22	saitoh	saitoh	PROPN
ma-267	321	23	,	,	PUNCT
ma-267	321	24	applications	application	NOUN
ma-267	321	25	of	of	ADP
ma-267	321	26	tikhonov	tikhonov	NOUN
ma-267	321	27	regularization	regularization	NOUN
ma-267	321	28	to	to	ADP
ma-267	321	29	inverse	inverse	NOUN
ma-267	321	30	problems	problem	NOUN
ma-267	321	31	using	use	VERB
ma-267	321	32	reproducing	reproduce	VERB
ma-267	321	33	kernels	kernel	NOUN
ma-267	321	34	,	,	PUNCT
ma-267	321	35	j.	j.	PROPN
ma-267	321	36	phys	phys	PROPN
ma-267	321	37	.	.	PUNCT
ma-267	321	38	:	:	PUNCT
ma-267	321	39	conf	conf	NOUN
ma-267	321	40	.	.	PUNCT
ma-267	322	1	ser.73	ser.73	PUNCT
ma-267	322	2	(	(	PUNCT
ma-267	322	3	2007	2007	NUM
ma-267	322	4	)	)	PUNCT
ma-267	322	5	,	,	PUNCT
ma-267	322	6	012019	012019	NUM
ma-267	322	7	.	.	PUNCT
ma-267	323	1	https://doi.org/10.1088/1742-6596/73/1/012019	https://doi.org/10.1088/1742-6596/73/1/012019	X
ma-267	323	2	.	.	PUNCT
ma-267	324	1	https://doi.org/10.28924/ada/ma.4.20	https://doi.org/10.28924/ada/ma.4.20	NOUN
ma-267	324	2	https://doi.org/10.1137/0515056	https://doi.org/10.1137/0515056	VERB
ma-267	324	3	https://doi.org/10.1016/0016-7142(84)90025-5	https://doi.org/10.1016/0016-7142(84)90025-5	NOUN
ma-267	324	4	https://doi.org/10.1088/0266-5611/3/1/013	https://doi.org/10.1088/0266-5611/3/1/013	PROPN
ma-267	324	5	https://doi.org/10.1155/ijmms/2006/94768	https://doi.org/10.1155/ijmms/2006/94768	NOUN
ma-267	324	6	https://doi.org/10.1016/0001-8708(75)90002-x	https://doi.org/10.1016/0001-8708(75)90002-x	NOUN
ma-267	324	7	https://doi.org/10.1016/0001-8708(75)90002-x	https://doi.org/10.1016/0001-8708(75)90002-x	PROPN
ma-267	324	8	https://doi.org/10.1007/s11868-017-0196-x	https://doi.org/10.1007/s11868-017-0196-x	PROPN
ma-267	324	9	https://doi.org/10.1007/s00025-022-01792-4	https://doi.org/10.1007/s00025-022-01792-4	NUM
ma-267	324	10	https://doi.org/10.1007/s00025-022-01792-4	https://doi.org/10.1007/s00025-022-01792-4	NUM
ma-267	325	1	https://doi.org/10.1088/1742-6596/73/1/012019	https://doi.org/10.1088/1742-6596/73/1/012019	PROPN
ma-267	325	2	1	1	NUM
ma-267	325	3	.	.	PUNCT
ma-267	325	4	introduction	introduction	NOUN
ma-267	325	5	2	2	NUM
ma-267	325	6	.	.	PUNCT
ma-267	325	7	harmonic	harmonic	ADJ
ma-267	325	8	analysis	analysis	NOUN
ma-267	325	9	associated	associate	VERB
ma-267	325	10	with	with	ADP
ma-267	325	11	the	the	DET
ma-267	325	12	riemann	riemann	PROPN
ma-267	325	13	-	-	PUNCT
ma-267	325	14	liouville	liouville	NOUN
ma-267	325	15	operator	operator	NOUN
ma-267	325	16	2.1	2.1	NUM
ma-267	325	17	.	.	PUNCT
ma-267	326	1	the	the	DET
ma-267	326	2	eigenfunctions	eigenfunction	NOUN
ma-267	326	3	of	of	ADP
ma-267	326	4	the	the	DET
ma-267	326	5	partial	partial	ADJ
ma-267	326	6	differential	differential	NOUN
ma-267	326	7	operators	operator	NOUN
ma-267	326	8	1	1	NUM
ma-267	326	9	and	and	CCONJ
ma-267	326	10	2	2	NUM
ma-267	326	11	2.2	2.2	NUM
ma-267	326	12	.	.	PUNCT
ma-267	327	1	the	the	DET
ma-267	327	2	riemann	riemann	PROPN
ma-267	327	3	-	-	PUNCT
ma-267	327	4	liouville	liouville	NOUN
ma-267	327	5	transform	transform	NOUN
ma-267	327	6	2.3	2.3	NUM
ma-267	327	7	.	.	PUNCT
ma-267	328	1	generalized	generalize	VERB
ma-267	328	2	translation	translation	NOUN
ma-267	328	3	operator	operator	NOUN
ma-267	328	4	associated	associate	VERB
ma-267	328	5	with	with	ADP
ma-267	328	6	the	the	DET
ma-267	328	7	riemann	riemann	PROPN
ma-267	328	8	-	-	PUNCT
ma-267	328	9	liouville	liouville	NOUN
ma-267	328	10	operator	operator	NOUN
ma-267	328	11	3	3	NUM
ma-267	328	12	.	.	PUNCT
ma-267	329	1	calderón	calderón	PROPN
ma-267	329	2	's	's	PART
ma-267	329	3	reproducing	reproduce	VERB
ma-267	329	4	formula	formula	NOUN
ma-267	329	5	for	for	ADP
ma-267	329	6	the	the	DET
ma-267	329	7	riemann	riemann	PROPN
ma-267	329	8	-	-	PUNCT
ma-267	329	9	liouville	liouville	VERB
ma-267	329	10	two	two	NUM
ma-267	329	11	-	-	PUNCT
ma-267	329	12	wavelet	wavelet	NOUN
ma-267	329	13	transform	transform	NOUN
ma-267	329	14	4	4	NUM
ma-267	329	15	.	.	PUNCT
ma-267	329	16	extremal	extremal	ADJ
ma-267	329	17	functions	function	NOUN
ma-267	329	18	associated	associate	VERB
ma-267	329	19	with	with	ADP
ma-267	329	20	the	the	DET
ma-267	329	21	riemann	riemann	PROPN
ma-267	329	22	-	-	PUNCT
ma-267	329	23	liouville	liouville	VERB
ma-267	329	24	wavelet	wavelet	NOUN
ma-267	329	25	transform	transform	VERB
ma-267	329	26	4.1	4.1	NUM
ma-267	329	27	.	.	PUNCT
ma-267	329	28	sobolev	sobolev	NOUN
ma-267	329	29	type	type	NOUN
ma-267	329	30	spaces	space	NOUN
ma-267	329	31	associated	associate	VERB
ma-267	329	32	with	with	ADP
ma-267	329	33	the	the	DET
ma-267	329	34	riemann	riemann	PROPN
ma-267	329	35	-	-	PUNCT
ma-267	329	36	liouville	liouville	VERB
ma-267	329	37	transform	transform	NOUN
ma-267	329	38	authors	author	NOUN
ma-267	329	39	'	'	PART
ma-267	329	40	contribution	contribution	NOUN
ma-267	329	41	references	reference	NOUN