id	sid	tid	token	lemma	pos
ma-182	1	1	2023	2023	NUM
ma-182	1	2	ada	ada	PROPN
ma-182	1	3	academica	academica	PROPN
ma-182	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-182	1	5	.	.	PUNCT
ma-182	2	1	j.	j.	PROPN
ma-182	2	2	math	math	PROPN
ma-182	2	3	.	.	PUNCT
ma-182	3	1	anal	anal	ADJ
ma-182	3	2	.	.	PUNCT
ma-182	4	1	3	3	NUM
ma-182	4	2	(	(	PUNCT
ma-182	4	3	2023	2023	NUM
ma-182	4	4	)	)	PUNCT
ma-182	4	5	27doi	27doi	NOUN
ma-182	4	6	:	:	PUNCT
ma-182	4	7	10.28924	10.28924	NUM
ma-182	4	8	/	/	SYM
ma-182	4	9	ada	ada	PROPN
ma-182	4	10	/	/	SYM
ma-182	4	11	ma.3.27	ma.3.27	PROPN
ma-182	5	1	on	on	ADP
ma-182	5	2	a	a	DET
ma-182	5	3	generalization	generalization	NOUN
ma-182	5	4	of	of	ADP
ma-182	5	5	(	(	PUNCT
ma-182	5	6	l1ω	l1ω	PROPN
ma-182	5	7	,	,	PUNCT
ma-182	5	8	l	l	NOUN
ma-182	5	9	p	p	NOUN
ma-182	5	10	ω)-multipliers	ω)-multipliers	PROPN
ma-182	5	11	yaovi	yaovi	PROPN
ma-182	5	12	a.	a.	NOUN
ma-182	5	13	tissinam1	tissinam1	PROPN
ma-182	5	14	,	,	PUNCT
ma-182	5	15	abudulaï	abudulaï	PROPN
ma-182	5	16	issa1	issa1	PROPN
ma-182	5	17	,	,	PUNCT
ma-182	6	1	yaogan	yaogan	PROPN
ma-182	6	2	mensah1,2,∗	mensah1,2,∗	PROPN
ma-182	6	3	1department	1department	NUM
ma-182	6	4	of	of	ADP
ma-182	6	5	mathematics	mathematic	NOUN
ma-182	6	6	,	,	PUNCT
ma-182	6	7	university	university	NOUN
ma-182	6	8	of	of	ADP
ma-182	6	9	lomé	lomé	NOUN
ma-182	6	10	,	,	PUNCT
ma-182	6	11	togo	togo	PROPN
ma-182	6	12	asseketis@gmail.com	asseketis@gmail.com	PROPN
ma-182	6	13	,	,	PUNCT
ma-182	6	14	issaabudulai13@gmail.com	issaabudulai13@gmail.com	PROPN
ma-182	6	15	,	,	PUNCT
ma-182	6	16	mensahyaogan2@gmail.com	mensahyaogan2@gmail.com	NOUN
ma-182	6	17	2icmpa	2icmpa	NUM
ma-182	6	18	,	,	PUNCT
ma-182	6	19	university	university	NOUN
ma-182	6	20	of	of	ADP
ma-182	6	21	abomey	abomey	NOUN
ma-182	6	22	-	-	PUNCT
ma-182	6	23	calavi	calavi	NOUN
ma-182	6	24	,	,	PUNCT
ma-182	6	25	benin	benin	PROPN
ma-182	6	26	∗correspondence	∗correspondence	NOUN
ma-182	6	27	:	:	PUNCT
ma-182	6	28	mensahyaogan2@gmail.com	mensahyaogan2@gmail.com	X
ma-182	6	29	,	,	PUNCT
ma-182	6	30	ymensah@univ-lome.tg	ymensah@univ-lome.tg	X
ma-182	6	31	abstract	abstract	ADJ
ma-182	6	32	.	.	PUNCT
ma-182	7	1	this	this	DET
ma-182	7	2	paper	paper	NOUN
ma-182	7	3	deals	deal	VERB
ma-182	7	4	with	with	ADP
ma-182	7	5	a	a	DET
ma-182	7	6	generalized	generalized	ADJ
ma-182	7	7	aspect	aspect	NOUN
ma-182	7	8	of	of	ADP
ma-182	7	9	multipliers	multiplier	NOUN
ma-182	7	10	for	for	ADP
ma-182	7	11	the	the	DET
ma-182	7	12	pair	pair	NOUN
ma-182	7	13	(	(	PUNCT
ma-182	7	14	l1ω	l1ω	ADJ
ma-182	7	15	,	,	PUNCT
ma-182	7	16	lpω	lpω	PROPN
ma-182	7	17	)	)	PUNCT
ma-182	7	18	of	of	ADP
ma-182	7	19	beurlingspaces	beurlingspace	NOUN
ma-182	7	20	.	.	PUNCT
ma-182	8	1	using	use	VERB
ma-182	8	2	the	the	DET
ma-182	8	3	fourier	fourier	NOUN
ma-182	8	4	transform	transform	NOUN
ma-182	8	5	related	relate	VERB
ma-182	8	6	to	to	ADP
ma-182	8	7	a	a	DET
ma-182	8	8	beurling	beurling	ADJ
ma-182	8	9	weight	weight	NOUN
ma-182	8	10	,	,	PUNCT
ma-182	8	11	we	we	PRON
ma-182	8	12	give	give	VERB
ma-182	8	13	a	a	DET
ma-182	8	14	characterization	characterization	NOUN
ma-182	8	15	of	of	ADP
ma-182	8	16	theaforementioned	theaforementione	VERB
ma-182	8	17	multipliers	multiplier	NOUN
ma-182	8	18	.	.	PUNCT
ma-182	9	1	we	we	PRON
ma-182	9	2	also	also	ADV
ma-182	9	3	prove	prove	VERB
ma-182	9	4	the	the	DET
ma-182	9	5	identification	identification	NOUN
ma-182	9	6	of	of	ADP
ma-182	9	7	the	the	DET
ma-182	9	8	space	space	NOUN
ma-182	9	9	of	of	ADP
ma-182	9	10	the	the	DET
ma-182	9	11	multipliers	multiplier	NOUN
ma-182	9	12	for	for	ADP
ma-182	9	13	thepair	thepair	NOUN
ma-182	9	14	(	(	PUNCT
ma-182	9	15	l1ω	l1ω	ADJ
ma-182	9	16	,	,	PUNCT
ma-182	9	17	lpω	lpω	PROPN
ma-182	9	18	)	)	PUNCT
ma-182	9	19	with	with	ADP
ma-182	9	20	the	the	DET
ma-182	9	21	beurling	beurling	ADJ
ma-182	9	22	space	space	NOUN
ma-182	9	23	lpω	lpω	VERB
ma-182	9	24	when	when	SCONJ
ma-182	9	25	1	1	NUM
ma-182	9	26	<	<	X
ma-182	9	27	p	p	X
ma-182	9	28	<	<	X
ma-182	9	29	∞.	∞.	PROPN
ma-182	9	30	1	1	NUM
ma-182	9	31	.	.	PUNCT
ma-182	10	1	introduction	introduction	NOUN
ma-182	10	2	multipliers	multiplier	NOUN
ma-182	10	3	are	be	AUX
ma-182	10	4	intensively	intensively	ADV
ma-182	10	5	studied	study	VERB
ma-182	10	6	by	by	ADP
ma-182	10	7	many	many	ADJ
ma-182	10	8	researchers	researcher	NOUN
ma-182	10	9	.	.	PUNCT
ma-182	11	1	they	they	PRON
ma-182	11	2	appear	appear	VERB
ma-182	11	3	in	in	ADP
ma-182	11	4	several	several	ADJ
ma-182	11	5	fields	field	NOUN
ma-182	11	6	of	of	ADP
ma-182	11	7	math	math	NOUN
ma-182	11	8	-	-	PUNCT
ma-182	11	9	ematics	ematic	NOUN
ma-182	11	10	and	and	CCONJ
ma-182	11	11	in	in	ADP
ma-182	11	12	various	various	ADJ
ma-182	11	13	contexts	contexts	NOUN
ma-182	11	14	,	,	PUNCT
ma-182	11	15	namely	namely	ADV
ma-182	11	16	:	:	PUNCT
ma-182	11	17	mobile	mobile	ADJ
ma-182	11	18	communication	communication	NOUN
ma-182	11	19	,	,	PUNCT
ma-182	11	20	signal	signal	ADJ
ma-182	11	21	processing	processing	NOUN
ma-182	11	22	,	,	PUNCT
ma-182	11	23	stochasticprocess	stochasticprocess	NOUN
ma-182	11	24	,	,	PUNCT
ma-182	11	25	partial	partial	ADJ
ma-182	11	26	differential	differential	NOUN
ma-182	11	27	equation	equation	NOUN
ma-182	11	28	etc	etc	X
ma-182	11	29	.	.	X
ma-182	11	30	from	from	ADP
ma-182	11	31	a	a	DET
ma-182	11	32	theoretical	theoretical	ADJ
ma-182	11	33	point	point	NOUN
ma-182	11	34	of	of	ADP
ma-182	11	35	view	view	NOUN
ma-182	11	36	,	,	PUNCT
ma-182	11	37	we	we	PRON
ma-182	11	38	refer	refer	VERB
ma-182	11	39	to	to	ADP
ma-182	11	40	the	the	DET
ma-182	11	41	source	source	NOUN
ma-182	12	1	[	[	X
ma-182	12	2	9]for	9]for	NUM
ma-182	12	3	more	more	ADJ
ma-182	12	4	details	detail	NOUN
ma-182	12	5	about	about	ADP
ma-182	12	6	multipliers	multiplier	NOUN
ma-182	12	7	for	for	ADP
ma-182	12	8	commutative	commutative	ADJ
ma-182	12	9	banach	banach	NOUN
ma-182	12	10	algebras.like	algebras.like	ADV
ma-182	12	11	in	in	ADP
ma-182	12	12	[	[	X
ma-182	12	13	4	4	NUM
ma-182	12	14	]	]	PUNCT
ma-182	12	15	,	,	PUNCT
ma-182	12	16	we	we	PRON
ma-182	12	17	are	be	AUX
ma-182	12	18	interested	interested	ADJ
ma-182	12	19	in	in	ADP
ma-182	12	20	the	the	DET
ma-182	12	21	multipliers	multiplier	NOUN
ma-182	12	22	on	on	ADP
ma-182	12	23	a	a	DET
ma-182	12	24	certain	certain	ADJ
ma-182	12	25	large	large	ADJ
ma-182	12	26	class	class	NOUN
ma-182	12	27	of	of	ADP
ma-182	12	28	banach	banach	NOUN
ma-182	12	29	spaces	space	NOUN
ma-182	12	30	related	relate	VERB
ma-182	12	31	to	to	ADP
ma-182	12	32	alocally	alocally	ADV
ma-182	12	33	compact	compact	ADJ
ma-182	12	34	abelian	abelian	ADJ
ma-182	12	35	group	group	NOUN
ma-182	12	36	.	.	PUNCT
ma-182	13	1	namely	namely	ADV
ma-182	13	2	,	,	PUNCT
ma-182	13	3	multipliers	multiplier	NOUN
ma-182	13	4	of	of	ADP
ma-182	13	5	beurling	beurling	ADJ
ma-182	13	6	spaces	space	NOUN
ma-182	13	7	are	be	AUX
ma-182	13	8	concerned	concern	VERB
ma-182	13	9	.	.	PUNCT
ma-182	14	1	some	some	DET
ma-182	14	2	inter	inter	ADJ
ma-182	14	3	-	-	ADJ
ma-182	14	4	esting	esting	ADJ
ma-182	14	5	publications	publication	NOUN
ma-182	14	6	about	about	ADP
ma-182	14	7	multipliers	multiplier	NOUN
ma-182	14	8	associated	associate	VERB
ma-182	14	9	with	with	ADP
ma-182	14	10	locally	locally	ADV
ma-182	14	11	compact	compact	ADJ
ma-182	14	12	groups	group	NOUN
ma-182	14	13	are	be	AUX
ma-182	14	14	[	[	X
ma-182	14	15	1,6,11,12,14,18].in	1,6,11,12,14,18].in	NUM
ma-182	14	16	[	[	X
ma-182	14	17	4	4	NUM
ma-182	14	18	]	]	PUNCT
ma-182	14	19	,	,	PUNCT
ma-182	14	20	we	we	PRON
ma-182	14	21	study	study	VERB
ma-182	14	22	the	the	DET
ma-182	14	23	multipliers	multiplier	NOUN
ma-182	14	24	on	on	ADP
ma-182	14	25	the	the	DET
ma-182	14	26	weighted	weight	VERB
ma-182	14	27	group	group	NOUN
ma-182	14	28	algebra	algebra	PROPN
ma-182	14	29	l1ω(g	l1ω(g	PROPN
ma-182	14	30	)	)	PUNCT
ma-182	14	31	which	which	PRON
ma-182	14	32	is	be	AUX
ma-182	14	33	the	the	DET
ma-182	14	34	banach	banach	NOUN
ma-182	14	35	space	space	NOUN
ma-182	14	36	l1ω(g	l1ω(g	NOUN
ma-182	14	37	)	)	PUNCT
ma-182	14	38	endowed	endow	VERB
ma-182	14	39	with	with	ADP
ma-182	14	40	a	a	DET
ma-182	14	41	generalized	generalized	ADJ
ma-182	14	42	convolution	convolution	NOUN
ma-182	14	43	product	product	NOUN
ma-182	14	44	∗ω	∗ω	NOUN
ma-182	14	45	which	which	PRON
ma-182	14	46	depends	depend	VERB
ma-182	14	47	on	on	ADP
ma-182	14	48	the	the	DET
ma-182	14	49	weight	weight	NOUN
ma-182	14	50	ω	ω	PROPN
ma-182	14	51	.	.	PUNCT
ma-182	15	1	thisgeneralized	thisgeneralize	VERB
ma-182	15	2	convolution	convolution	NOUN
ma-182	15	3	product	product	NOUN
ma-182	15	4	first	first	ADV
ma-182	15	5	appeared	appear	VERB
ma-182	15	6	in	in	ADP
ma-182	15	7	[	[	X
ma-182	15	8	10	10	NUM
ma-182	15	9	]	]	PUNCT
ma-182	15	10	.	.	PUNCT
ma-182	16	1	the	the	DET
ma-182	16	2	authors	author	NOUN
ma-182	16	3	in	in	ADP
ma-182	16	4	[	[	X
ma-182	16	5	4	4	NUM
ma-182	16	6	]	]	PUNCT
ma-182	16	7	characterized	characterize	VERB
ma-182	16	8	the	the	DET
ma-182	16	9	multi	multi	NOUN
ma-182	16	10	-	-	NOUN
ma-182	16	11	pliers	plier	NOUN
ma-182	16	12	on	on	ADP
ma-182	16	13	this	this	DET
ma-182	16	14	weighted	weight	VERB
ma-182	16	15	group	group	NOUN
ma-182	16	16	algebra.the	algebra.the	DET
ma-182	16	17	present	present	ADJ
ma-182	16	18	paper	paper	NOUN
ma-182	16	19	is	be	AUX
ma-182	16	20	the	the	DET
ma-182	16	21	continuation	continuation	NOUN
ma-182	16	22	of	of	ADP
ma-182	16	23	the	the	DET
ma-182	16	24	study	study	NOUN
ma-182	16	25	started	start	VERB
ma-182	16	26	in	in	ADP
ma-182	16	27	[	[	X
ma-182	16	28	4	4	NUM
ma-182	16	29	]	]	PUNCT
ma-182	16	30	.	.	PUNCT
ma-182	17	1	we	we	PRON
ma-182	17	2	consider	consider	VERB
ma-182	17	3	a	a	DET
ma-182	17	4	generalization	generalization	NOUN
ma-182	17	5	ofthe	ofthe	NOUN
ma-182	17	6	multipliers	multiplier	NOUN
ma-182	17	7	for	for	ADP
ma-182	17	8	the	the	DET
ma-182	17	9	pair	pair	NOUN
ma-182	17	10	(	(	PUNCT
ma-182	17	11	l1ω(g	l1ω(g	PROPN
ma-182	17	12	)	)	PUNCT
ma-182	17	13	,	,	PUNCT
ma-182	17	14	lpω(g	lpω(g	PROPN
ma-182	17	15	)	)	PUNCT
ma-182	17	16	)	)	PUNCT
ma-182	17	17	.	.	PUNCT
ma-182	18	1	that	that	PRON
ma-182	18	2	is	be	AUX
ma-182	18	3	,	,	PUNCT
ma-182	18	4	the	the	DET
ma-182	18	5	linear	linear	PROPN
ma-182	18	6	maps	maps	PROPN
ma-182	18	7	t	t	NOUN
ma-182	18	8	:	:	PUNCT
ma-182	18	9	l1ω(g	l1ω(g	NOUN
ma-182	18	10	)	)	PUNCT
ma-182	18	11	−→	−→	NOUN
ma-182	18	12	lpω(g	lpω(g	NOUN
ma-182	18	13	)	)	PUNCT
ma-182	18	14	)	)	PUNCT
ma-182	18	15	thatcommute	thatcommute	VERB
ma-182	18	16	with	with	ADP
ma-182	18	17	a	a	DET
ma-182	18	18	certain	certain	ADJ
ma-182	18	19	class	class	NOUN
ma-182	18	20	of	of	ADP
ma-182	18	21	generalized	generalized	ADJ
ma-182	18	22	translation	translation	NOUN
ma-182	18	23	operators	operator	NOUN
ma-182	18	24	denoted	denote	VERB
ma-182	18	25	here	here	ADV
ma-182	18	26	by	by	ADP
ma-182	18	27	γsω	γsω	NOUN
ma-182	18	28	.	.	PUNCT
ma-182	19	1	if	if	SCONJ
ma-182	19	2	ω	ω	PROPN
ma-182	19	3	≡	≡	PROPN
ma-182	19	4	1,then	1,then	NUM
ma-182	19	5	we	we	PRON
ma-182	19	6	recover	recover	VERB
ma-182	19	7	the	the	DET
ma-182	19	8	classical	classical	ADJ
ma-182	19	9	concept	concept	NOUN
ma-182	19	10	of	of	ADP
ma-182	19	11	multipliers	multiplier	NOUN
ma-182	19	12	.	.	PUNCT
ma-182	20	1	via	via	ADP
ma-182	20	2	the	the	DET
ma-182	20	3	weight	weight	NOUN
ma-182	20	4	fourier	fourier	NOUN
ma-182	20	5	transform	transform	NOUN
ma-182	20	6	,	,	PUNCT
ma-182	20	7	we	we	PRON
ma-182	20	8	obtain	obtain	VERB
ma-182	20	9	,	,	PUNCT
ma-182	20	10	among	among	ADP
ma-182	20	11	other	other	ADJ
ma-182	20	12	results	result	NOUN
ma-182	20	13	,	,	PUNCT
ma-182	20	14	a	a	DET
ma-182	20	15	characterization	characterization	NOUN
ma-182	20	16	of	of	ADP
ma-182	20	17	the	the	DET
ma-182	20	18	multipliers	multiplier	NOUN
ma-182	20	19	for	for	ADP
ma-182	20	20	the	the	DET
ma-182	20	21	pair	pair	NOUN
ma-182	20	22	(	(	PUNCT
ma-182	20	23	l1ω(g	l1ω(g	PROPN
ma-182	20	24	)	)	PUNCT
ma-182	20	25	,	,	PUNCT
ma-182	20	26	lpω(g	lpω(g	PROPN
ma-182	20	27	)	)	PUNCT
ma-182	20	28	)	)	PUNCT
ma-182	20	29	.	.	PUNCT
ma-182	21	1	received	receive	VERB
ma-182	21	2	:	:	PUNCT
ma-182	21	3	14	14	NUM
ma-182	21	4	jul	jul	PROPN
ma-182	21	5	2023	2023	NUM
ma-182	21	6	.	.	PUNCT
ma-182	22	1	key	key	ADJ
ma-182	22	2	words	word	NOUN
ma-182	22	3	and	and	CCONJ
ma-182	22	4	phrases	phrase	NOUN
ma-182	22	5	.	.	PUNCT
ma-182	23	1	weight	weight	NOUN
ma-182	23	2	,	,	PUNCT
ma-182	23	3	convolution	convolution	NOUN
ma-182	23	4	,	,	PUNCT
ma-182	23	5	multiplier	multipli	ADJ
ma-182	23	6	,	,	PUNCT
ma-182	23	7	group	group	NOUN
ma-182	23	8	algebra	algebra	NOUN
ma-182	23	9	,	,	PUNCT
ma-182	23	10	fourier	fourier	NOUN
ma-182	23	11	transform	transform	NOUN
ma-182	23	12	,	,	PUNCT
ma-182	23	13	measure.1	measure.1	PROPN
ma-182	23	14	https://adac.ee	https://adac.ee	PROPN
ma-182	23	15	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	23	16	eur	eur	PROPN
ma-182	23	17	.	.	PUNCT
ma-182	24	1	j.	j.	PROPN
ma-182	24	2	math	math	PROPN
ma-182	24	3	.	.	PUNCT
ma-182	25	1	anal	anal	PROPN
ma-182	25	2	.	.	PUNCT
ma-182	26	1	10.28924	10.28924	NUM
ma-182	26	2	/	/	SYM
ma-182	26	3	ada	ada	PROPN
ma-182	26	4	/	/	SYM
ma-182	26	5	ma.3.27	ma.3.27	PROPN
ma-182	26	6	2the	2the	DET
ma-182	26	7	paper	paper	NOUN
ma-182	26	8	is	be	AUX
ma-182	26	9	organized	organize	VERB
ma-182	26	10	as	as	SCONJ
ma-182	26	11	follows	follow	VERB
ma-182	26	12	.	.	PUNCT
ma-182	27	1	in	in	ADP
ma-182	27	2	section	section	NOUN
ma-182	27	3	2	2	NUM
ma-182	27	4	,	,	PUNCT
ma-182	27	5	the	the	DET
ma-182	27	6	definition	definition	NOUN
ma-182	27	7	of	of	ADP
ma-182	27	8	beurling	beurling	ADJ
ma-182	27	9	spaces	space	NOUN
ma-182	27	10	and	and	CCONJ
ma-182	27	11	someresults	someresult	NOUN
ma-182	27	12	from	from	ADP
ma-182	27	13	[	[	X
ma-182	27	14	4	4	NUM
ma-182	27	15	,	,	PUNCT
ma-182	27	16	7	7	NUM
ma-182	27	17	,	,	PUNCT
ma-182	27	18	10	10	NUM
ma-182	27	19	]	]	PUNCT
ma-182	27	20	are	be	AUX
ma-182	27	21	recalled	recall	VERB
ma-182	27	22	.	.	PUNCT
ma-182	28	1	in	in	ADP
ma-182	28	2	section	section	NOUN
ma-182	28	3	3	3	NUM
ma-182	28	4	,	,	PUNCT
ma-182	28	5	we	we	PRON
ma-182	28	6	state	state	VERB
ma-182	28	7	our	our	PRON
ma-182	28	8	main	main	ADJ
ma-182	28	9	results	result	NOUN
ma-182	28	10	.	.	PUNCT
ma-182	29	1	2	2	X
ma-182	29	2	.	.	NUM
ma-182	29	3	preliminaries	preliminary	NOUN
ma-182	29	4	2.1	2.1	NUM
ma-182	29	5	.	.	PUNCT
ma-182	30	1	the	the	DET
ma-182	30	2	beurling	beurling	NOUN
ma-182	30	3	spaces	space	VERB
ma-182	30	4	.	.	PUNCT
ma-182	31	1	let	let	VERB
ma-182	31	2	g	g	PRON
ma-182	31	3	be	be	AUX
ma-182	31	4	a	a	DET
ma-182	31	5	group	group	NOUN
ma-182	31	6	whose	whose	DET
ma-182	31	7	neutral	neutral	ADJ
ma-182	31	8	element	element	NOUN
ma-182	31	9	is	be	AUX
ma-182	31	10	denoted	denote	VERB
ma-182	31	11	by	by	ADP
ma-182	31	12	e	e	X
ma-182	31	13	.	.	PUNCT
ma-182	32	1	a	a	DET
ma-182	32	2	beurling	beurling	ADJ
ma-182	32	3	weight	weight	NOUN
ma-182	32	4	on	on	ADP
ma-182	32	5	g	g	PROPN
ma-182	32	6	is	be	AUX
ma-182	32	7	a	a	DET
ma-182	32	8	continuous	continuous	ADJ
ma-182	32	9	fonction	fonction	NOUN
ma-182	32	10	ω	ω	NOUN
ma-182	32	11	:	:	PUNCT
ma-182	32	12	g	g	NOUN
ma-182	32	13	→	→	SYM
ma-182	32	14	(	(	PUNCT
ma-182	32	15	0,∞	0,∞	NOUN
ma-182	32	16	)	)	PUNCT
ma-182	33	1	such	such	ADJ
ma-182	33	2	that	that	SCONJ
ma-182	33	3	∀x	∀x	NUM
ma-182	33	4	,	,	PUNCT
ma-182	33	5	y	y	PROPN
ma-182	33	6	∈	∈	PROPN
ma-182	33	7	g	g	PROPN
ma-182	33	8	,	,	PUNCT
ma-182	33	9	ω(xy	ω(xy	NUM
ma-182	33	10	)	)	PUNCT
ma-182	33	11	6	6	NUM
ma-182	33	12	ω(x)ω(y	ω(x)ω(y	NOUN
ma-182	33	13	)	)	PUNCT
ma-182	33	14	,	,	PUNCT
ma-182	33	15	ω(x	ω(x	X
ma-182	33	16	)	)	PUNCT
ma-182	33	17	>	>	X
ma-182	33	18	1	1	NUM
ma-182	33	19	,	,	PUNCT
ma-182	33	20	ω(e	ω(e	NOUN
ma-182	33	21	)	)	PUNCT
ma-182	33	22	=	=	SYM
ma-182	34	1	1	1	X
ma-182	34	2	.	.	X
ma-182	34	3	for	for	ADP
ma-182	34	4	instance	instance	NOUN
ma-182	34	5	,	,	PUNCT
ma-182	34	6	for	for	ADP
ma-182	34	7	each	each	DET
ma-182	34	8	α	α	PROPN
ma-182	34	9	≥	≥	NOUN
ma-182	34	10	0	0	NUM
ma-182	34	11	,	,	PUNCT
ma-182	34	12	the	the	DET
ma-182	34	13	function	function	NOUN
ma-182	34	14	ωα	ωα	VERB
ma-182	34	15	defined	define	VERB
ma-182	34	16	by	by	ADP
ma-182	34	17	ωα(x	ωα(x	NOUN
ma-182	34	18	)	)	PUNCT
ma-182	34	19	=	=	PUNCT
ma-182	34	20	(	(	PUNCT
ma-182	34	21	1	1	NUM
ma-182	34	22	+	+	CCONJ
ma-182	34	23	‖x‖)α	‖x‖)α	NOUN
ma-182	34	24	,	,	PUNCT
ma-182	34	25	where	where	SCONJ
ma-182	34	26	x	x	PUNCT
ma-182	34	27	=	=	PRON
ma-182	34	28	(	(	PUNCT
ma-182	34	29	x1	x1	PROPN
ma-182	34	30	,	,	PUNCT
ma-182	34	31	·	·	PUNCT
ma-182	34	32	·	·	PUNCT
ma-182	34	33	·	·	PUNCT
ma-182	34	34	,	,	PUNCT
ma-182	34	35	xn	xn	X
ma-182	34	36	)	)	PUNCT
ma-182	34	37	∈	∈	PROPN
ma-182	34	38	rn	rn	PROPN
ma-182	34	39	and	and	CCONJ
ma-182	34	40	‖x‖	‖x‖	PROPN
ma-182	34	41	=	=	PUNCT
ma-182	35	1	√	√	NUM
ma-182	35	2	n∑	n∑	NOUN
ma-182	35	3	i=1	i=1	PROPN
ma-182	35	4	x2i	x2i	PROPN
ma-182	35	5	,	,	PUNCT
ma-182	35	6	is	be	AUX
ma-182	35	7	a	a	DET
ma-182	35	8	beurling	beurling	ADJ
ma-182	35	9	weight	weight	NOUN
ma-182	35	10	on	on	ADP
ma-182	35	11	(	(	PUNCT
ma-182	35	12	rn,+	rn,+	X
ma-182	35	13	)	)	PUNCT
ma-182	35	14	.	.	PUNCT
ma-182	36	1	integration	integration	NOUN
ma-182	36	2	on	on	ADP
ma-182	36	3	g	g	PROPN
ma-182	36	4	is	be	AUX
ma-182	36	5	taken	take	VERB
ma-182	36	6	with	with	ADP
ma-182	36	7	respect	respect	NOUN
ma-182	36	8	to	to	ADP
ma-182	36	9	a	a	DET
ma-182	36	10	left	left	ADJ
ma-182	36	11	haar	haar	NOUN
ma-182	36	12	measure	measure	NOUN
ma-182	36	13	.	.	PUNCT
ma-182	37	1	beurling	beurle	VERB
ma-182	37	2	spaces	space	NOUN
ma-182	37	3	are	be	AUX
ma-182	37	4	defined	define	VERB
ma-182	37	5	tobe	tobe	ADJ
ma-182	37	6	lpω(g	lpω(g	NOUN
ma-182	37	7	)	)	PUNCT
ma-182	37	8	=	=	PRON
ma-182	38	1	{	{	PUNCT
ma-182	38	2	f	f	X
ma-182	38	3	:	:	PUNCT
ma-182	38	4	g	g	PROPN
ma-182	38	5	→	→	SYM
ma-182	38	6	c	c	PROPN
ma-182	38	7	:	:	PUNCT
ma-182	38	8	∫	∫	PROPN
ma-182	38	9	g	g	PROPN
ma-182	38	10	|f	|f	PROPN
ma-182	39	1	(	(	PUNCT
ma-182	39	2	x)|pω(x)dx	x)|pω(x)dx	PROPN
ma-182	39	3	<	<	X
ma-182	39	4	∞	∞	NUM
ma-182	39	5	}	}	PUNCT
ma-182	39	6	,	,	PUNCT
ma-182	39	7	1	1	NUM
ma-182	39	8	6	6	NUM
ma-182	39	9	p	p	NOUN
ma-182	39	10	<	<	X
ma-182	39	11	+	+	NOUN
ma-182	39	12	∞.	∞.	PROPN
ma-182	39	13	the	the	DET
ma-182	39	14	case	case	NOUN
ma-182	39	15	where	where	SCONJ
ma-182	39	16	p	p	NOUN
ma-182	39	17	=	=	NOUN
ma-182	39	18	∞	∞	PROPN
ma-182	39	19	is	be	AUX
ma-182	39	20	defined	define	VERB
ma-182	39	21	in	in	ADP
ma-182	39	22	an	an	DET
ma-182	39	23	obvious	obvious	ADJ
ma-182	39	24	way	way	NOUN
ma-182	39	25	by	by	ADP
ma-182	39	26	essential	essential	ADJ
ma-182	39	27	boundedness	boundedness	NOUN
ma-182	39	28	.	.	PUNCT
ma-182	40	1	the	the	DET
ma-182	40	2	mapping	mapping	NOUN
ma-182	40	3	f	f	PROPN
ma-182	40	4	7−→	7−→	PROPN
ma-182	40	5	‖f	‖f	ADJ
ma-182	40	6	‖p	‖p	PROPN
ma-182	40	7	,	,	PUNCT
ma-182	40	8	ω	ω	PROPN
ma-182	40	9	=	=	SYM
ma-182	40	10	(	(	PUNCT
ma-182	40	11	∫	∫	PROPN
ma-182	40	12	g	g	PROPN
ma-182	40	13	|f	|f	PROPN
ma-182	40	14	(	(	PUNCT
ma-182	40	15	x)|pω(x)dx	x)|pω(x)dx	PROPN
ma-182	40	16	)	)	PUNCT
ma-182	40	17	1	1	NUM
ma-182	41	1	p	p	NOUN
ma-182	41	2	is	be	AUX
ma-182	41	3	a	a	DET
ma-182	41	4	norm	norm	NOUN
ma-182	41	5	on	on	ADP
ma-182	41	6	lpω(g).it	lpω(g).it	PROPN
ma-182	41	7	is	be	AUX
ma-182	41	8	well	well	ADV
ma-182	41	9	-	-	PUNCT
ma-182	41	10	known	know	VERB
ma-182	41	11	in	in	ADP
ma-182	41	12	the	the	DET
ma-182	41	13	mathematical	mathematical	ADJ
ma-182	41	14	litterature	litterature	NOUN
ma-182	41	15	that	that	PRON
ma-182	41	16	l1ω(g	l1ω(g	NOUN
ma-182	41	17	)	)	PUNCT
ma-182	41	18	is	be	AUX
ma-182	41	19	a	a	DET
ma-182	41	20	banach	banach	NOUN
ma-182	41	21	algebra	algebra	NOUN
ma-182	41	22	under	under	ADP
ma-182	41	23	theconvolution	theconvolution	NOUN
ma-182	41	24	product	product	NOUN
ma-182	41	25	∗	∗	NOUN
ma-182	41	26	defined	define	VERB
ma-182	41	27	by	by	ADP
ma-182	41	28	(	(	PUNCT
ma-182	41	29	f	f	PROPN
ma-182	41	30	∗	∗	PROPN
ma-182	41	31	g)(x	g)(x	PROPN
ma-182	41	32	)	)	PUNCT
ma-182	42	1	=	=	SYM
ma-182	43	1	∫	∫	PROPN
ma-182	43	2	g	g	PROPN
ma-182	43	3	f	f	PROPN
ma-182	43	4	(	(	PUNCT
ma-182	43	5	y)g(y−1x)dy	y)g(y−1x)dy	PROPN
ma-182	43	6	.	.	PUNCT
ma-182	44	1	the	the	DET
ma-182	44	2	following	follow	VERB
ma-182	44	3	sufficient	sufficient	ADJ
ma-182	44	4	condition	condition	NOUN
ma-182	44	5	for	for	ADP
ma-182	44	6	lpω(g	lpω(g	NOUN
ma-182	44	7	)	)	PUNCT
ma-182	44	8	,	,	PUNCT
ma-182	44	9	1	1	NUM
ma-182	44	10	<	<	X
ma-182	44	11	p	p	X
ma-182	44	12	<	<	X
ma-182	44	13	∞	∞	PROPN
ma-182	44	14	,	,	PUNCT
ma-182	44	15	to	to	PART
ma-182	44	16	be	be	AUX
ma-182	44	17	a	a	DET
ma-182	44	18	banach	banach	NOUN
ma-182	44	19	algebra	algebra	NOUN
ma-182	44	20	under	under	ADP
ma-182	44	21	theconvolution	theconvolution	NOUN
ma-182	44	22	product	product	NOUN
ma-182	44	23	∗	∗	NOUN
ma-182	44	24	can	can	AUX
ma-182	44	25	be	be	AUX
ma-182	44	26	found	find	VERB
ma-182	44	27	in	in	ADP
ma-182	44	28	[	[	X
ma-182	44	29	7	7	NUM
ma-182	44	30	]	]	PUNCT
ma-182	44	31	:	:	PUNCT
ma-182	44	32	the	the	DET
ma-182	44	33	space	space	NOUN
ma-182	44	34	lpω(g	lpω(g	PROPN
ma-182	44	35	)	)	PUNCT
ma-182	44	36	,	,	PUNCT
ma-182	44	37	1	1	NUM
ma-182	44	38	<	<	X
ma-182	44	39	p	p	X
ma-182	44	40	<	<	X
ma-182	44	41	∞	∞	PROPN
ma-182	44	42	is	be	AUX
ma-182	44	43	banach	banach	NOUN
ma-182	44	44	algebra	algebra	VERB
ma-182	44	45	underthe	underthe	ADJ
ma-182	44	46	convolution	convolution	NOUN
ma-182	44	47	product	product	NOUN
ma-182	44	48	∗	∗	NOUN
ma-182	44	49	if	if	SCONJ
ma-182	44	50	ω	ω	PROPN
ma-182	44	51	1	1	NUM
ma-182	44	52	1−p	1−p	NUM
ma-182	44	53	∗ω	∗ω	PROPN
ma-182	44	54	1	1	NUM
ma-182	44	55	1−p	1−p	NUM
ma-182	44	56	6	6	NUM
ma-182	44	57	ω	ω	NUM
ma-182	44	58	1	1	NUM
ma-182	44	59	1−p	1−p	NUM
ma-182	44	60	.	.	PUNCT
ma-182	45	1	for	for	ADP
ma-182	45	2	a	a	DET
ma-182	45	3	general	general	ADJ
ma-182	45	4	background	background	NOUN
ma-182	45	5	and	and	CCONJ
ma-182	45	6	history	history	NOUN
ma-182	45	7	on	on	ADP
ma-182	45	8	beurlingspaces	beurlingspace	NOUN
ma-182	45	9	,	,	PUNCT
ma-182	45	10	we	we	PRON
ma-182	45	11	refer	refer	VERB
ma-182	45	12	to	to	ADP
ma-182	45	13	[	[	X
ma-182	45	14	13,15	13,15	NUM
ma-182	45	15	]	]	X
ma-182	45	16	.	.	PUNCT
ma-182	46	1	2.2	2.2	NUM
ma-182	46	2	.	.	PUNCT
ma-182	47	1	a	a	DET
ma-182	47	2	generalized	generalized	ADJ
ma-182	47	3	convolution	convolution	NOUN
ma-182	47	4	product	product	NOUN
ma-182	47	5	.	.	PUNCT
ma-182	48	1	in	in	ADP
ma-182	48	2	[	[	X
ma-182	48	3	10	10	NUM
ma-182	48	4	]	]	PUNCT
ma-182	48	5	,	,	PUNCT
ma-182	48	6	the	the	DET
ma-182	48	7	author	author	NOUN
ma-182	48	8	introduced	introduce	VERB
ma-182	48	9	a	a	DET
ma-182	48	10	new	new	ADJ
ma-182	48	11	convolution	convolution	NOUN
ma-182	48	12	producton	producton	PROPN
ma-182	48	13	l1ω(g	l1ω(g	PROPN
ma-182	48	14	)	)	PUNCT
ma-182	48	15	which	which	PRON
ma-182	48	16	has	have	VERB
ma-182	48	17	the	the	DET
ma-182	48	18	particularity	particularity	NOUN
ma-182	48	19	to	to	PART
ma-182	48	20	depend	depend	VERB
ma-182	48	21	of	of	ADP
ma-182	48	22	the	the	DET
ma-182	48	23	weight	weight	NOUN
ma-182	48	24	ω	ω	NOUN
ma-182	48	25	.	.	PUNCT
ma-182	49	1	that	that	PRON
ma-182	49	2	is	be	AUX
ma-182	49	3	,	,	PUNCT
ma-182	49	4	f	f	PROPN
ma-182	49	5	∗ω	∗ω	PROPN
ma-182	49	6	g(x	g(x	PROPN
ma-182	49	7	)	)	PUNCT
ma-182	50	1	=	=	SYM
ma-182	51	1	∫	∫	PROPN
ma-182	51	2	g	g	PROPN
ma-182	51	3	f	f	PROPN
ma-182	51	4	(	(	PUNCT
ma-182	51	5	y)g(y−1x	y)g(y−1x	NOUN
ma-182	51	6	)	)	PUNCT
ma-182	51	7	ω(y)ω(y−1x	ω(y)ω(y−1x	NOUN
ma-182	51	8	)	)	PUNCT
ma-182	51	9	ω(x	ω(x	NOUN
ma-182	51	10	)	)	PUNCT
ma-182	51	11	dy	dy	NOUN
ma-182	51	12	.	.	PUNCT
ma-182	52	1	if	if	SCONJ
ma-182	52	2	ω	ω	PROPN
ma-182	52	3	≡	≡	PROPN
ma-182	52	4	1	1	NUM
ma-182	52	5	,	,	PUNCT
ma-182	52	6	then	then	ADV
ma-182	52	7	one	one	PRON
ma-182	52	8	recovers	recover	VERB
ma-182	52	9	the	the	DET
ma-182	52	10	usual	usual	ADJ
ma-182	52	11	convolution	convolution	NOUN
ma-182	52	12	(	(	PUNCT
ma-182	52	13	f	f	PROPN
ma-182	52	14	∗	∗	PROPN
ma-182	52	15	g)(x	g)(x	PROPN
ma-182	52	16	)	)	PUNCT
ma-182	53	1	=	=	SYM
ma-182	54	1	∫	∫	PROPN
ma-182	54	2	g	g	PROPN
ma-182	54	3	f	f	PROPN
ma-182	54	4	(	(	PUNCT
ma-182	54	5	y)g(y−1x)dy	y)g(y−1x)dy	PROPN
ma-182	54	6	.	.	PROPN
ma-182	55	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	55	2	eur	eur	PROPN
ma-182	55	3	.	.	PUNCT
ma-182	56	1	j.	j.	PROPN
ma-182	56	2	math	math	PROPN
ma-182	56	3	.	.	PUNCT
ma-182	57	1	anal	anal	PROPN
ma-182	57	2	.	.	PUNCT
ma-182	58	1	10.28924	10.28924	NUM
ma-182	58	2	/	/	SYM
ma-182	58	3	ada	ada	PROPN
ma-182	58	4	/	/	SYM
ma-182	58	5	ma.3.27	ma.3.27	PROPN
ma-182	59	1	3hence	3hence	NUM
ma-182	59	2	,	,	PUNCT
ma-182	59	3	the	the	DET
ma-182	59	4	convolution	convolution	NOUN
ma-182	59	5	product	product	NOUN
ma-182	59	6	∗ω	∗ω	PROPN
ma-182	59	7	is	be	AUX
ma-182	59	8	a	a	DET
ma-182	59	9	generalization	generalization	NOUN
ma-182	59	10	of	of	ADP
ma-182	59	11	the	the	DET
ma-182	59	12	usual	usual	ADJ
ma-182	59	13	convolution	convolution	NOUN
ma-182	59	14	product	product	NOUN
ma-182	59	15	.	.	PUNCT
ma-182	60	1	it	it	PRON
ma-182	60	2	wasshown	wasshown	VERB
ma-182	60	3	that	that	SCONJ
ma-182	60	4	l1ω(g	l1ω(g	NOUN
ma-182	60	5	)	)	PUNCT
ma-182	60	6	is	be	AUX
ma-182	60	7	a	a	DET
ma-182	60	8	banach	banach	NOUN
ma-182	60	9	algebra	algebra	NOUN
ma-182	60	10	under	under	ADP
ma-182	60	11	this	this	DET
ma-182	60	12	new	new	ADJ
ma-182	60	13	convolution	convolution	NOUN
ma-182	60	14	product	product	NOUN
ma-182	60	15	[	[	X
ma-182	60	16	10	10	NUM
ma-182	60	17	]	]	PUNCT
ma-182	60	18	.	.	PUNCT
ma-182	61	1	we	we	PRON
ma-182	61	2	denote	denote	VERB
ma-182	61	3	by	by	ADP
ma-182	61	4	l1ω(g	l1ω(g	NOUN
ma-182	61	5	)	)	PUNCT
ma-182	61	6	this	this	DET
ma-182	61	7	new	new	ADJ
ma-182	61	8	banach	banach	NOUN
ma-182	61	9	algebra	algebra	NOUN
ma-182	61	10	;	;	PUNCT
ma-182	61	11	in	in	ADP
ma-182	61	12	other	other	ADJ
ma-182	61	13	words	word	NOUN
ma-182	61	14	l1ω(g	l1ω(g	NOUN
ma-182	61	15	)	)	PUNCT
ma-182	61	16	=	=	SYM
ma-182	61	17	(	(	PUNCT
ma-182	61	18	l1ω(g	l1ω(g	PROPN
ma-182	61	19	)	)	PUNCT
ma-182	61	20	,	,	PUNCT
ma-182	61	21	‖	‖	PROPN
ma-182	61	22	·	·	PUNCT
ma-182	61	23	‖1,ω	‖1,ω	NOUN
ma-182	61	24	,	,	PUNCT
ma-182	61	25	∗ω).for	∗ω).for	PROPN
ma-182	61	26	s	s	PROPN
ma-182	61	27	∈	∈	PROPN
ma-182	61	28	g	g	NOUN
ma-182	61	29	,	,	PUNCT
ma-182	61	30	define	define	VERB
ma-182	61	31	the	the	DET
ma-182	61	32	operator	operator	NOUN
ma-182	61	33	γsω	γsω	VERB
ma-182	61	34	by	by	ADP
ma-182	61	35	γsωf	γsωf	NOUN
ma-182	61	36	(	(	PUNCT
ma-182	61	37	x	x	X
ma-182	61	38	)	)	PUNCT
ma-182	61	39	=	=	SYM
ma-182	62	1	τsmωf	τsmωf	NOUN
ma-182	62	2	(	(	PUNCT
ma-182	62	3	x	x	NOUN
ma-182	62	4	)	)	PUNCT
ma-182	62	5	ω(x	ω(x	NOUN
ma-182	62	6	)	)	PUNCT
ma-182	62	7	,	,	PUNCT
ma-182	62	8	f	f	PROPN
ma-182	62	9	∈	∈	PROPN
ma-182	62	10	l1ω(g	l1ω(g	PROPN
ma-182	62	11	)	)	PUNCT
ma-182	62	12	,	,	PUNCT
ma-182	62	13	where	where	SCONJ
ma-182	62	14	mω	mω	NOUN
ma-182	62	15	is	be	AUX
ma-182	62	16	the	the	DET
ma-182	62	17	multiplication	multiplication	NOUN
ma-182	62	18	operator	operator	NOUN
ma-182	62	19	defined	define	VERB
ma-182	62	20	by	by	ADP
ma-182	62	21	(	(	PUNCT
ma-182	62	22	mωf	mωf	NOUN
ma-182	62	23	)	)	PUNCT
ma-182	62	24	(	(	PUNCT
ma-182	62	25	x	x	X
ma-182	62	26	)	)	PUNCT
ma-182	62	27	=	=	SYM
ma-182	62	28	ω(x)f	ω(x)f	PROPN
ma-182	62	29	(	(	PUNCT
ma-182	62	30	x	x	NOUN
ma-182	62	31	)	)	PUNCT
ma-182	62	32	and	and	CCONJ
ma-182	62	33	τs	τs	X
ma-182	62	34	is	be	AUX
ma-182	62	35	the	the	DET
ma-182	62	36	translation	translation	NOUN
ma-182	62	37	operator	operator	NOUN
ma-182	62	38	defined	define	VERB
ma-182	62	39	by	by	ADP
ma-182	62	40	(	(	PUNCT
ma-182	62	41	τs	τs	PROPN
ma-182	62	42	f	f	PROPN
ma-182	62	43	)	)	PUNCT
ma-182	62	44	(	(	PUNCT
ma-182	62	45	x	x	X
ma-182	62	46	)	)	PUNCT
ma-182	62	47	=	=	SYM
ma-182	62	48	f	f	PROPN
ma-182	62	49	(	(	PUNCT
ma-182	62	50	s−1x	s−1x	NOUN
ma-182	62	51	)	)	PUNCT
ma-182	62	52	.	.	PUNCT
ma-182	63	1	the	the	DET
ma-182	63	2	operator	operator	NOUN
ma-182	63	3	γsω	γsω	ADV
ma-182	63	4	appears	appear	VERB
ma-182	63	5	first	first	ADV
ma-182	63	6	in	in	ADP
ma-182	63	7	[	[	X
ma-182	63	8	4	4	NUM
ma-182	63	9	]	]	PUNCT
ma-182	63	10	for	for	ADP
ma-182	63	11	the	the	DET
ma-182	63	12	study	study	NOUN
ma-182	63	13	of	of	ADP
ma-182	63	14	the	the	DET
ma-182	63	15	multipliers	multiplier	NOUN
ma-182	63	16	for	for	ADP
ma-182	63	17	the	the	DET
ma-182	63	18	algebra	algebra	PROPN
ma-182	63	19	l1ω(g	l1ω(g	NOUN
ma-182	63	20	)	)	PUNCT
ma-182	63	21	.	.	PUNCT
ma-182	64	1	alinear	alinear	PROPN
ma-182	64	2	map	map	VERB
ma-182	64	3	t	t	NOUN
ma-182	64	4	:	:	PUNCT
ma-182	64	5	l1ω(g)→	l1ω(g)→	PROPN
ma-182	64	6	l1ω(g	l1ω(g	NOUN
ma-182	64	7	)	)	PUNCT
ma-182	64	8	is	be	AUX
ma-182	64	9	called	call	VERB
ma-182	64	10	a	a	DET
ma-182	64	11	multiplier	multipli	ADJ
ma-182	64	12	if	if	SCONJ
ma-182	64	13	t	t	NOUN
ma-182	64	14	commutes	commute	VERB
ma-182	64	15	with	with	ADP
ma-182	64	16	the	the	DET
ma-182	64	17	operators	operator	NOUN
ma-182	64	18	γsω	γsω	VERB
ma-182	64	19	for	for	ADP
ma-182	64	20	all	all	DET
ma-182	64	21	s	s	PROPN
ma-182	64	22	∈	∈	PROPN
ma-182	64	23	g.	g.	NOUN
ma-182	64	24	since	since	SCONJ
ma-182	64	25	the	the	DET
ma-182	64	26	operator	operator	NOUN
ma-182	64	27	γsω	γsω	VERB
ma-182	64	28	is	be	AUX
ma-182	64	29	a	a	DET
ma-182	64	30	generalization	generalization	NOUN
ma-182	64	31	of	of	ADP
ma-182	64	32	the	the	DET
ma-182	64	33	translation	translation	NOUN
ma-182	64	34	operator	operator	NOUN
ma-182	64	35	τs	τs	ADP
ma-182	64	36	,	,	PUNCT
ma-182	64	37	the	the	DET
ma-182	64	38	latter	latter	ADJ
ma-182	64	39	notion	notion	NOUN
ma-182	64	40	ofmultiplier	ofmultiplier	ADV
ma-182	64	41	covers	cover	VERB
ma-182	64	42	the	the	DET
ma-182	64	43	classical	classical	ADJ
ma-182	64	44	one	one	NOUN
ma-182	64	45	related	relate	VERB
ma-182	64	46	to	to	ADP
ma-182	64	47	commutation	commutation	NOUN
ma-182	64	48	with	with	ADP
ma-182	64	49	translations.the	translations.the	DET
ma-182	64	50	natural	natural	ADJ
ma-182	64	51	next	next	ADJ
ma-182	64	52	step	step	NOUN
ma-182	64	53	is	be	AUX
ma-182	64	54	to	to	PART
ma-182	64	55	investigate	investigate	VERB
ma-182	64	56	the	the	DET
ma-182	64	57	multipliers	multiplier	NOUN
ma-182	64	58	for	for	ADP
ma-182	64	59	the	the	DET
ma-182	64	60	pair	pair	NOUN
ma-182	64	61	(	(	PUNCT
ma-182	64	62	l1ω(g	l1ω(g	PROPN
ma-182	64	63	)	)	PUNCT
ma-182	64	64	,	,	PUNCT
ma-182	64	65	lpω(g	lpω(g	PROPN
ma-182	64	66	)	)	PUNCT
ma-182	64	67	)	)	PUNCT
ma-182	64	68	.	.	PUNCT
ma-182	65	1	this	this	PRON
ma-182	65	2	is	be	AUX
ma-182	65	3	themain	themain	ADJ
ma-182	65	4	purpose	purpose	NOUN
ma-182	65	5	of	of	ADP
ma-182	65	6	the	the	DET
ma-182	65	7	present	present	ADJ
ma-182	65	8	article.we	article.we	NUM
ma-182	65	9	denote	denote	NOUN
ma-182	65	10	by	by	ADP
ma-182	65	11	m1ω(g	m1ω(g	PROPN
ma-182	65	12	)	)	PUNCT
ma-182	65	13	the	the	DET
ma-182	65	14	banach	banach	NOUN
ma-182	65	15	space	space	NOUN
ma-182	65	16	of	of	ADP
ma-182	65	17	all	all	DET
ma-182	65	18	complex	complex	NOUN
ma-182	65	19	bounded	bounded	ADJ
ma-182	65	20	regular	regular	ADJ
ma-182	65	21	borel	borel	NOUN
ma-182	65	22	measures	measure	NOUN
ma-182	65	23	µ	µ	X
ma-182	65	24	on	on	ADP
ma-182	65	25	gsuch	gsuch	NOUN
ma-182	65	26	that	that	SCONJ
ma-182	65	27	‖µ‖ω	‖µ‖ω	VERB
ma-182	65	28	=	=	PUNCT
ma-182	65	29	∫	∫	NOUN
ma-182	65	30	g	g	PROPN
ma-182	65	31	ω(x)d	ω(x)d	PROPN
ma-182	65	32	|µ|(x	|µ|(x	NOUN
ma-182	65	33	)	)	PUNCT
ma-182	66	1	<	<	X
ma-182	66	2	∞.	∞.	PROPN
ma-182	66	3	(	(	PUNCT
ma-182	66	4	1	1	X
ma-182	66	5	)	)	PUNCT
ma-182	66	6	we	we	PRON
ma-182	66	7	write	write	VERB
ma-182	66	8	m1(g	m1(g	NOUN
ma-182	66	9	)	)	PUNCT
ma-182	66	10	in	in	ADP
ma-182	66	11	the	the	DET
ma-182	66	12	case	case	NOUN
ma-182	66	13	where	where	SCONJ
ma-182	66	14	ω	ω	PROPN
ma-182	66	15	≡	≡	PROPN
ma-182	66	16	1	1	NUM
ma-182	66	17	.	.	PUNCT
ma-182	67	1	for	for	ADP
ma-182	67	2	µ	µ	NUM
ma-182	67	3	,	,	PUNCT
ma-182	67	4	ν	ν	PROPN
ma-182	67	5	∈	∈	PROPN
ma-182	67	6	m1ω(g	m1ω(g	PROPN
ma-182	67	7	)	)	PUNCT
ma-182	67	8	,	,	PUNCT
ma-182	67	9	define	define	VERB
ma-182	67	10	µ	µ	DET
ma-182	67	11	∗ω	∗ω	NOUN
ma-182	67	12	ν	ν	NOUN
ma-182	67	13	by	by	ADP
ma-182	67	14	µ	µ	PROPN
ma-182	67	15	∗ω	∗ω	PROPN
ma-182	67	16	ν(f	ν(f	PROPN
ma-182	67	17	)	)	PUNCT
ma-182	68	1	=	=	PUNCT
ma-182	68	2	∫	∫	PROPN
ma-182	69	1	g	g	PROPN
ma-182	69	2	∫	∫	PROPN
ma-182	69	3	g	g	PROPN
ma-182	69	4	f	f	PROPN
ma-182	69	5	(	(	PUNCT
ma-182	69	6	xy	xy	PROPN
ma-182	69	7	)	)	PUNCT
ma-182	69	8	ω(x)ω(y	ω(x)ω(y	NOUN
ma-182	69	9	)	)	PUNCT
ma-182	69	10	ω(xy	ω(xy	NUM
ma-182	69	11	)	)	PUNCT
ma-182	69	12	dµ(x)dν(y	dµ(x)dν(y	PROPN
ma-182	69	13	)	)	PUNCT
ma-182	69	14	,	,	PUNCT
ma-182	69	15	f	f	PROPN
ma-182	69	16	∈	∈	PROPN
ma-182	69	17	cc(g	cc(g	NOUN
ma-182	69	18	,	,	PUNCT
ma-182	69	19	ω−1	ω−1	NOUN
ma-182	69	20	)	)	PUNCT
ma-182	69	21	where	where	SCONJ
ma-182	69	22	cc(g	cc(g	NOUN
ma-182	69	23	,	,	PUNCT
ma-182	69	24	ω−1	ω−1	NOUN
ma-182	69	25	)	)	PUNCT
ma-182	69	26	is	be	AUX
ma-182	69	27	the	the	DET
ma-182	69	28	set	set	NOUN
ma-182	69	29	of	of	ADP
ma-182	69	30	complex	complex	ADJ
ma-182	69	31	functions	function	NOUN
ma-182	69	32	f	f	PRON
ma-182	69	33	defined	define	VERB
ma-182	69	34	on	on	ADP
ma-182	69	35	g	g	PROPN
ma-182	70	1	such	such	ADJ
ma-182	70	2	that	that	SCONJ
ma-182	70	3	f	f	PROPN
ma-182	71	1	ω−1	ω−1	NOUN
ma-182	71	2	is	be	AUX
ma-182	71	3	of	of	ADP
ma-182	71	4	compactsupport	compactsupport	NOUN
ma-182	71	5	.	.	PUNCT
ma-182	72	1	also	also	ADV
ma-182	72	2	,	,	PUNCT
ma-182	72	3	define	define	VERB
ma-182	72	4	µ	µ	PRON
ma-182	72	5	∗ω	∗ω	PROPN
ma-182	72	6	f	f	X
ma-182	72	7	(	(	PUNCT
ma-182	72	8	x	x	X
ma-182	72	9	)	)	PUNCT
ma-182	72	10	=	=	SYM
ma-182	73	1	∫	∫	PROPN
ma-182	73	2	g	g	PROPN
ma-182	73	3	f	f	PROPN
ma-182	73	4	(	(	PUNCT
ma-182	73	5	y−1x	y−1x	NOUN
ma-182	73	6	)	)	PUNCT
ma-182	73	7	ω(y)ω(y−1x	ω(y)ω(y−1x	NOUN
ma-182	73	8	)	)	PUNCT
ma-182	73	9	ω(x	ω(x	NOUN
ma-182	73	10	)	)	PUNCT
ma-182	73	11	dµ(y	dµ(y	NUM
ma-182	73	12	)	)	PUNCT
ma-182	74	1	for	for	ADP
ma-182	74	2	f	f	PROPN
ma-182	74	3	∈	∈	PROPN
ma-182	74	4	l1ω(g	l1ω(g	PROPN
ma-182	74	5	)	)	PUNCT
ma-182	74	6	and	and	CCONJ
ma-182	74	7	µ	µ	PRON
ma-182	74	8	∈	∈	PROPN
ma-182	74	9	m1ω(g	m1ω(g	PROPN
ma-182	74	10	)	)	PUNCT
ma-182	74	11	.	.	PUNCT
ma-182	75	1	then	then	ADV
ma-182	75	2	,	,	PUNCT
ma-182	75	3	the	the	DET
ma-182	75	4	banach	banach	NOUN
ma-182	75	5	space	space	NOUN
ma-182	75	6	m1ω(g	m1ω(g	PROPN
ma-182	75	7	)	)	PUNCT
ma-182	75	8	is	be	AUX
ma-182	75	9	a	a	DET
ma-182	75	10	unital	unital	ADJ
ma-182	75	11	banach	banach	NOUN
ma-182	75	12	algebra	algebra	NOUN
ma-182	75	13	withrespect	withrespect	ADJ
ma-182	75	14	to	to	ADP
ma-182	75	15	the	the	DET
ma-182	75	16	convolution	convolution	NOUN
ma-182	75	17	product	product	NOUN
ma-182	75	18	∗ω	∗ω	PROPN
ma-182	75	19	and	and	CCONJ
ma-182	75	20	l1ω(g	l1ω(g	PROPN
ma-182	75	21	)	)	PUNCT
ma-182	75	22	is	be	AUX
ma-182	75	23	a	a	DET
ma-182	75	24	closed	closed	ADJ
ma-182	75	25	ideal	ideal	NOUN
ma-182	75	26	of	of	ADP
ma-182	75	27	m1ω(g	m1ω(g	PROPN
ma-182	75	28	)	)	PUNCT
ma-182	76	1	[	[	X
ma-182	76	2	10	10	NUM
ma-182	76	3	,	,	PUNCT
ma-182	76	4	theorem	theorem	VERB
ma-182	76	5	5.1	5.1	NUM
ma-182	76	6	]	]	PUNCT
ma-182	76	7	.	.	PUNCT
ma-182	77	1	2.3	2.3	NUM
ma-182	77	2	.	.	PUNCT
ma-182	78	1	some	some	DET
ma-182	78	2	useful	useful	ADJ
ma-182	78	3	facts	fact	NOUN
ma-182	78	4	.	.	PUNCT
ma-182	79	1	let	let	VERB
ma-182	79	2	g	g	PRON
ma-182	79	3	be	be	AUX
ma-182	79	4	a	a	DET
ma-182	79	5	locally	locally	ADV
ma-182	79	6	compact	compact	ADJ
ma-182	79	7	abelian	abelian	ADJ
ma-182	79	8	group	group	NOUN
ma-182	79	9	with	with	ADP
ma-182	79	10	pontryagin	pontryagin	NOUN
ma-182	79	11	dual	dual	ADJ
ma-182	79	12	group	group	NOUN
ma-182	79	13	ĝ.we	ĝ.we	NOUN
ma-182	79	14	denote	denote	VERB
ma-182	79	15	by	by	ADP
ma-182	79	16	m̂1(g	m̂1(g	PROPN
ma-182	79	17	)	)	PUNCT
ma-182	79	18	the	the	DET
ma-182	79	19	collection	collection	NOUN
ma-182	79	20	of	of	ADP
ma-182	79	21	all	all	DET
ma-182	79	22	the	the	DET
ma-182	79	23	fourier	fourier	NOUN
ma-182	79	24	-	-	PUNCT
ma-182	79	25	stieltjes	stieltjes	NOUN
ma-182	79	26	transforms	transform	VERB
ma-182	79	27	of	of	ADP
ma-182	79	28	elements	element	NOUN
ma-182	79	29	of	of	ADP
ma-182	79	30	m1(g).that	m1(g).that	PROPN
ma-182	79	31	is	be	AUX
ma-182	79	32	,	,	PUNCT
ma-182	79	33	m̂1(g	m̂1(g	PROPN
ma-182	79	34	)	)	PUNCT
ma-182	79	35	=	=	SYM
ma-182	79	36	{	{	PUNCT
ma-182	79	37	µ̂	µ̂	NOUN
ma-182	79	38	:	:	PUNCT
ma-182	80	1	µ	µ	X
ma-182	80	2	∈	∈	NOUN
ma-182	80	3	m1(g)}where	m1(g)}where	ADV
ma-182	80	4	µ̂	µ̂	PRON
ma-182	80	5	is	be	AUX
ma-182	80	6	defined	define	VERB
ma-182	80	7	by	by	ADP
ma-182	80	8	µ̂(γ	µ̂(γ	NOUN
ma-182	80	9	)	)	PUNCT
ma-182	80	10	=	=	SYM
ma-182	80	11	∫	∫	PROPN
ma-182	80	12	g	g	PROPN
ma-182	80	13	γ(x)dµ(x	γ(x)dµ(x	PROPN
ma-182	80	14	)	)	PUNCT
ma-182	80	15	,	,	PUNCT
ma-182	80	16	γ	γ	PROPN
ma-182	80	17	∈	∈	PROPN
ma-182	80	18	ĝ.	ĝ.	PROPN
ma-182	80	19	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	80	20	eur	eur	PROPN
ma-182	80	21	.	.	PUNCT
ma-182	81	1	j.	j.	PROPN
ma-182	81	2	math	math	PROPN
ma-182	81	3	.	.	PUNCT
ma-182	82	1	anal	anal	PROPN
ma-182	82	2	.	.	PUNCT
ma-182	83	1	10.28924	10.28924	NUM
ma-182	83	2	/	/	SYM
ma-182	83	3	ada	ada	PROPN
ma-182	83	4	/	/	SYM
ma-182	83	5	ma.3.27	ma.3.27	PROPN
ma-182	83	6	4	4	NUM
ma-182	83	7	for	for	ADP
ma-182	83	8	a	a	DET
ma-182	83	9	function	function	NOUN
ma-182	83	10	f	f	PROPN
ma-182	83	11	∈	∈	PROPN
ma-182	83	12	l1ω(g	l1ω(g	PROPN
ma-182	83	13	)	)	PUNCT
ma-182	83	14	,	,	PUNCT
ma-182	83	15	the	the	DET
ma-182	83	16	fourier	fourier	NOUN
ma-182	83	17	transform	transform	NOUN
ma-182	83	18	of	of	ADP
ma-182	83	19	f	f	PROPN
ma-182	83	20	,	,	PUNCT
ma-182	83	21	denoted	denote	VERB
ma-182	83	22	f	f	PROPN
ma-182	83	23	f	f	PROPN
ma-182	83	24	or	or	CCONJ
ma-182	83	25	f̂	f̂	NUM
ma-182	83	26	,	,	PUNCT
ma-182	83	27	is	be	AUX
ma-182	83	28	defined	define	VERB
ma-182	83	29	by	by	ADP
ma-182	83	30	(	(	PUNCT
ma-182	83	31	f	f	PROPN
ma-182	83	32	f	f	PROPN
ma-182	83	33	)	)	PUNCT
ma-182	83	34	(	(	PUNCT
ma-182	83	35	γ	γ	X
ma-182	83	36	)	)	PUNCT
ma-182	83	37	:	:	PUNCT
ma-182	83	38	=	=	SYM
ma-182	83	39	f̂	f̂	X
ma-182	83	40	(	(	PUNCT
ma-182	83	41	γ	γ	NOUN
ma-182	83	42	)	)	PUNCT
ma-182	83	43	=	=	SYM
ma-182	83	44	∫	∫	PROPN
ma-182	83	45	g	g	PROPN
ma-182	83	46	f	f	PROPN
ma-182	83	47	(	(	PUNCT
ma-182	83	48	x)γ(x)dx	x)γ(x)dx	VERB
ma-182	83	49	the	the	DET
ma-182	83	50	following	follow	VERB
ma-182	83	51	theorems	theorem	NOUN
ma-182	83	52	will	will	AUX
ma-182	83	53	play	play	VERB
ma-182	83	54	an	an	DET
ma-182	83	55	important	important	ADJ
ma-182	83	56	role	role	NOUN
ma-182	83	57	.	.	PUNCT
ma-182	84	1	theorem	theorem	VERB
ma-182	84	2	2.1	2.1	NUM
ma-182	84	3	(	(	PUNCT
ma-182	84	4	[	[	X
ma-182	84	5	3	3	NUM
ma-182	84	6	]	]	PUNCT
ma-182	84	7	or	or	CCONJ
ma-182	84	8	[	[	X
ma-182	84	9	17	17	NUM
ma-182	84	10	]	]	PUNCT
ma-182	84	11	)	)	PUNCT
ma-182	84	12	.	.	PUNCT
ma-182	85	1	let	let	VERB
ma-182	85	2	g	g	PRON
ma-182	85	3	be	be	AUX
ma-182	85	4	a	a	DET
ma-182	85	5	locally	locally	ADV
ma-182	85	6	compact	compact	ADJ
ma-182	85	7	abelian	abelian	NOUN
ma-182	85	8	group	group	NOUN
ma-182	85	9	and	and	CCONJ
ma-182	85	10	let	let	VERB
ma-182	85	11	ϕ	ϕ	NOUN
ma-182	85	12	be	be	AUX
ma-182	85	13	a	a	DET
ma-182	85	14	complex	complex	ADJ
ma-182	85	15	function	function	NOUN
ma-182	85	16	on	on	ADP
ma-182	85	17	ĝ.	ĝ.	PROPN
ma-182	85	18	then	then	ADV
ma-182	85	19	,	,	PUNCT
ma-182	85	20	the	the	DET
ma-182	85	21	following	follow	VERB
ma-182	85	22	assertions	assertion	NOUN
ma-182	85	23	are	be	AUX
ma-182	85	24	equivalent.(1	equivalent.(1	PROPN
ma-182	85	25	)	)	PUNCT
ma-182	85	26	ϕ	ϕ	PROPN
ma-182	85	27	∈	∈	PROPN
ma-182	85	28	m̂1(g	m̂1(g	PROPN
ma-182	85	29	)	)	PUNCT
ma-182	85	30	and	and	CCONJ
ma-182	85	31	‖ϕ‖∞	‖ϕ‖∞	NOUN
ma-182	85	32	6	6	NUM
ma-182	85	33	c.(2	c.(2	NOUN
ma-182	85	34	)	)	PUNCT
ma-182	85	35	ϕ	ϕ	NOUN
ma-182	85	36	is	be	AUX
ma-182	85	37	continuous	continuous	ADJ
ma-182	85	38	and	and	CCONJ
ma-182	85	39	there	there	PRON
ma-182	85	40	exists	exist	VERB
ma-182	85	41	a	a	DET
ma-182	85	42	constant	constant	ADJ
ma-182	85	43	c	c	NOUN
ma-182	85	44	>	>	X
ma-182	85	45	0	0	NUM
ma-182	85	46	such	such	ADJ
ma-182	85	47	that∣∣∣∣∣	that∣∣∣∣∣	PROPN
ma-182	85	48	n∑	n∑	PROPN
ma-182	85	49	i=1	i=1	PROPN
ma-182	85	50	ciϕ(γi	ciϕ(γi	NOUN
ma-182	85	51	)	)	PUNCT
ma-182	85	52	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ma-182	86	1	<	<	X
ma-182	86	2	c	c	X
ma-182	86	3	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	86	4	n∑	n∑	PROPN
ma-182	86	5	i=1	i=1	PROPN
ma-182	86	6	ciγi	ciγi	PROPN
ma-182	86	7	(	(	PUNCT
ma-182	86	8	·	·	PUNCT
ma-182	86	9	)	)	PUNCT
ma-182	86	10	∥∥∥∥∥	∥∥∥∥∥	X
ma-182	87	1	∞	∞	NUM
ma-182	87	2	(	(	PUNCT
ma-182	87	3	2	2	NUM
ma-182	87	4	)	)	PUNCT
ma-182	87	5	for	for	ADP
ma-182	87	6	all	all	DET
ma-182	87	7	positive	positive	ADJ
ma-182	87	8	integer	integer	NOUN
ma-182	87	9	n	n	NOUN
ma-182	87	10	and	and	CCONJ
ma-182	87	11	all	all	DET
ma-182	87	12	choices	choice	NOUN
ma-182	87	13	of	of	ADP
ma-182	87	14	ci	ci	NOUN
ma-182	87	15	∈	∈	PROPN
ma-182	87	16	c	c	PROPN
ma-182	87	17	and	and	CCONJ
ma-182	87	18	γi	γi	PROPN
ma-182	87	19	∈	∈	PROPN
ma-182	87	20	ĝ	ĝ	NOUN
ma-182	87	21	,	,	PUNCT
ma-182	87	22	i	i	PRON
ma-182	87	23	=	=	NOUN
ma-182	87	24	1	1	NUM
ma-182	87	25	,	,	PUNCT
ma-182	87	26	2	2	NUM
ma-182	87	27	,	,	PUNCT
ma-182	87	28	·	·	PUNCT
ma-182	87	29	·	·	PUNCT
ma-182	87	30	·	·	PUNCT
ma-182	87	31	,	,	PUNCT
ma-182	87	32	n.	n.	PROPN
ma-182	87	33	moreover	moreover	ADV
ma-182	87	34	,	,	PUNCT
ma-182	87	35	if	if	SCONJ
ma-182	87	36	ϕ	ϕ	NOUN
ma-182	87	37	=	=	SYM
ma-182	87	38	µ̂	µ̂	X
ma-182	87	39	,	,	PUNCT
ma-182	87	40	then	then	ADV
ma-182	87	41	‖µ‖	‖µ‖	PROPN
ma-182	87	42	is	be	AUX
ma-182	87	43	the	the	DET
ma-182	87	44	smallest	small	ADJ
ma-182	87	45	constant	constant	ADJ
ma-182	87	46	c	c	NOUN
ma-182	87	47	for	for	ADP
ma-182	87	48	which	which	PRON
ma-182	87	49	(	(	PUNCT
ma-182	87	50	2	2	X
ma-182	87	51	)	)	PUNCT
ma-182	87	52	holds	hold	NOUN
ma-182	87	53	.	.	PUNCT
ma-182	87	54	theorem	theorem	VERB
ma-182	87	55	2.2	2.2	NUM
ma-182	87	56	.	.	PUNCT
ma-182	88	1	(	(	PUNCT
ma-182	88	2	[	[	X
ma-182	88	3	9	9	NUM
ma-182	88	4	,	,	PUNCT
ma-182	88	5	page	page	NOUN
ma-182	88	6	252	252	NUM
ma-182	88	7	]	]	PUNCT
ma-182	88	8	)	)	PUNCT
ma-182	88	9	let	let	VERB
ma-182	88	10	g	g	PRON
ma-182	88	11	be	be	AUX
ma-182	88	12	a	a	DET
ma-182	88	13	locally	locally	ADV
ma-182	88	14	compact	compact	ADJ
ma-182	88	15	abelian	abelian	ADJ
ma-182	88	16	group	group	NOUN
ma-182	88	17	.	.	PUNCT
ma-182	89	1	then	then	ADV
ma-182	89	2	,	,	PUNCT
ma-182	89	3	for	for	ADP
ma-182	89	4	each	each	DET
ma-182	89	5	compact	compact	PROPN
ma-182	89	6	k	k	PROPN
ma-182	89	7	⊂	⊂	X
ma-182	89	8	ĝ	ĝ	PROPN
ma-182	89	9	and	and	CCONJ
ma-182	89	10	ε	ε	PROPN
ma-182	89	11	>	>	X
ma-182	89	12	0	0	PROPN
ma-182	89	13	,	,	PUNCT
ma-182	89	14	given	give	VERB
ma-182	89	15	an	an	DET
ma-182	89	16	open	open	ADJ
ma-182	89	17	set	set	NOUN
ma-182	89	18	u	u	NOUN
ma-182	89	19	containing	contain	VERB
ma-182	89	20	k	k	X
ma-182	89	21	,	,	PUNCT
ma-182	89	22	there	there	PRON
ma-182	89	23	exists	exist	VERB
ma-182	89	24	a	a	DET
ma-182	89	25	function	function	NOUN
ma-182	89	26	f	f	PROPN
ma-182	89	27	∈	∈	PROPN
ma-182	89	28	l1(g	l1(g	PROPN
ma-182	89	29	)	)	PUNCT
ma-182	89	30	such	such	ADJ
ma-182	89	31	that	that	SCONJ
ma-182	89	32	0	0	NUM
ma-182	89	33	6	6	NUM
ma-182	89	34	f̂	f̂	NUM
ma-182	89	35	(	(	PUNCT
ma-182	89	36	γ	γ	NOUN
ma-182	89	37	)	)	PUNCT
ma-182	89	38	6	6	NUM
ma-182	89	39	1	1	NUM
ma-182	89	40	if	if	SCONJ
ma-182	89	41	γ	γ	PROPN
ma-182	89	42	∈	∈	PROPN
ma-182	89	43	ĝ	ĝ	PROPN
ma-182	89	44	,	,	PUNCT
ma-182	89	45	f̂	f̂	X
ma-182	89	46	(	(	PUNCT
ma-182	89	47	γ	γ	NOUN
ma-182	89	48	)	)	PUNCT
ma-182	89	49	=	=	SYM
ma-182	89	50	1	1	NUM
ma-182	89	51	if	if	SCONJ
ma-182	89	52	γ	γ	PROPN
ma-182	89	53	∈	∈	PROPN
ma-182	89	54	k	k	PROPN
ma-182	89	55	,	,	PUNCT
ma-182	89	56	f̂	f̂	X
ma-182	89	57	(	(	PUNCT
ma-182	89	58	γ	γ	NOUN
ma-182	89	59	)	)	PUNCT
ma-182	89	60	=	=	SYM
ma-182	89	61	0	0	PUNCT
ma-182	90	1	if	if	SCONJ
ma-182	90	2	γ	γ	X
ma-182	90	3	/∈	/∈	PUNCT
ma-182	90	4	u	u	NOUN
ma-182	90	5	and	and	CCONJ
ma-182	90	6	‖f	‖f	PRON
ma-182	90	7	‖	‖	PROPN
ma-182	90	8	6	6	NUM
ma-182	90	9	ε	ε	NOUN
ma-182	90	10	+	+	PROPN
ma-182	90	11	1	1	X
ma-182	90	12	.	.	PUNCT
ma-182	91	1	in	in	ADP
ma-182	91	2	particular	particular	ADJ
ma-182	91	3	,	,	PUNCT
ma-182	91	4	given	give	VERB
ma-182	91	5	any	any	DET
ma-182	91	6	open	open	ADJ
ma-182	91	7	set	set	NOUN
ma-182	91	8	u	u	PROPN
ma-182	91	9	⊂	⊂	PROPN
ma-182	91	10	ĝ	ĝ	X
ma-182	91	11	with	with	ADP
ma-182	91	12	compact	compact	ADJ
ma-182	91	13	closure	closure	NOUN
ma-182	91	14	,	,	PUNCT
ma-182	91	15	it	it	PRON
ma-182	91	16	is	be	AUX
ma-182	91	17	possible	possible	ADJ
ma-182	91	18	to	to	PART
ma-182	91	19	find	find	VERB
ma-182	91	20	f	f	X
ma-182	91	21	∈	∈	PROPN
ma-182	91	22	l1(g	l1(g	PROPN
ma-182	91	23	)	)	PUNCT
ma-182	91	24	such	such	ADJ
ma-182	91	25	that	that	SCONJ
ma-182	91	26	f̂	f̂	PROPN
ma-182	91	27	(	(	PUNCT
ma-182	91	28	γ	γ	NOUN
ma-182	91	29	)	)	PUNCT
ma-182	91	30	=	=	SYM
ma-182	91	31	1	1	NUM
ma-182	91	32	if	if	SCONJ
ma-182	91	33	γ	γ	PROPN
ma-182	91	34	∈	∈	PROPN
ma-182	91	35	u.	u.	PROPN
ma-182	91	36	theorem	theorem	VERB
ma-182	91	37	2.3	2.3	NUM
ma-182	91	38	.	.	PUNCT
ma-182	92	1	(	(	PUNCT
ma-182	92	2	[	[	X
ma-182	92	3	5	5	NUM
ma-182	92	4	,	,	PUNCT
ma-182	92	5	theorem	theorem	VERB
ma-182	92	6	3.2	3.2	NUM
ma-182	92	7	]	]	PUNCT
ma-182	92	8	)	)	PUNCT
ma-182	92	9	let	let	VERB
ma-182	92	10	g	g	PRON
ma-182	92	11	be	be	AUX
ma-182	92	12	a	a	DET
ma-182	92	13	locally	locally	ADV
ma-182	92	14	compact	compact	ADJ
ma-182	92	15	group	group	NOUN
ma-182	92	16	.	.	PUNCT
ma-182	93	1	let	let	VERB
ma-182	93	2	f	f	PRON
ma-182	93	3	∈	∈	PROPN
ma-182	93	4	lpω(g	lpω(g	PROPN
ma-182	93	5	)	)	PUNCT
ma-182	93	6	,	,	PUNCT
ma-182	93	7	1	1	NUM
ma-182	93	8	6	6	NUM
ma-182	93	9	p	p	NOUN
ma-182	93	10	<	<	X
ma-182	93	11	∞.	∞.	PROPN
ma-182	93	12	then	then	ADV
ma-182	93	13	,	,	PUNCT
ma-182	93	14	∀s	∀s	PROPN
ma-182	93	15	∈	∈	PROPN
ma-182	93	16	g	g	PROPN
ma-182	93	17	,	,	PUNCT
ma-182	93	18	[	[	X
ma-182	93	19	ω(s	ω(s	NOUN
ma-182	93	20	)	)	PUNCT
ma-182	93	21	]	]	PUNCT
ma-182	94	1	1−p	1−p	PROPN
ma-182	94	2	p	p	X
ma-182	94	3	‖f	‖f	ADJ
ma-182	94	4	‖p	‖p	PROPN
ma-182	94	5	,	,	PUNCT
ma-182	94	6	ω	ω	PROPN
ma-182	94	7	6	6	NUM
ma-182	94	8	‖γsωf	‖γsωf	ADJ
ma-182	94	9	‖p	‖p	PROPN
ma-182	94	10	,	,	PUNCT
ma-182	94	11	ω	ω	PROPN
ma-182	94	12	6	6	NUM
ma-182	94	13	[	[	PUNCT
ma-182	94	14	ω(s−1	ω(s−1	NOUN
ma-182	94	15	)	)	PUNCT
ma-182	94	16	]	]	PUNCT
ma-182	95	1	p−1	p−1	PROPN
ma-182	95	2	p	p	PROPN
ma-182	95	3	‖f	‖f	PROPN
ma-182	95	4	‖p	‖p	PROPN
ma-182	95	5	,	,	PUNCT
ma-182	95	6	ω	ω	NOUN
ma-182	95	7	.	.	PUNCT
ma-182	96	1	(	(	PUNCT
ma-182	96	2	3	3	NUM
ma-182	96	3	)	)	PUNCT
ma-182	96	4	3	3	NUM
ma-182	96	5	.	.	PUNCT
ma-182	96	6	multipliers	multiplier	NOUN
ma-182	96	7	for	for	ADP
ma-182	96	8	the	the	DET
ma-182	96	9	pair	pair	NOUN
ma-182	96	10	(	(	PUNCT
ma-182	96	11	l1ω(g	l1ω(g	PROPN
ma-182	96	12	)	)	PUNCT
ma-182	96	13	,	,	PUNCT
ma-182	96	14	lpω(g	lpω(g	PROPN
ma-182	96	15	)	)	PUNCT
ma-182	96	16	)	)	PUNCT
ma-182	96	17	in	in	ADP
ma-182	96	18	this	this	DET
ma-182	96	19	section	section	NOUN
ma-182	96	20	,	,	PUNCT
ma-182	96	21	we	we	PRON
ma-182	96	22	study	study	VERB
ma-182	96	23	a	a	DET
ma-182	96	24	generalization	generalization	NOUN
ma-182	96	25	of	of	ADP
ma-182	96	26	the	the	DET
ma-182	96	27	concept	concept	NOUN
ma-182	96	28	of	of	ADP
ma-182	96	29	multipliers	multiplier	NOUN
ma-182	96	30	.	.	PUNCT
ma-182	97	1	here	here	ADV
ma-182	97	2	,	,	PUNCT
ma-182	97	3	the	the	DET
ma-182	97	4	multipliersare	multipliersare	NOUN
ma-182	97	5	defined	define	VERB
ma-182	97	6	with	with	ADP
ma-182	97	7	respect	respect	NOUN
ma-182	97	8	to	to	ADP
ma-182	97	9	the	the	DET
ma-182	97	10	generalized	generalize	VERB
ma-182	97	11	translation	translation	NOUN
ma-182	97	12	operators	operator	NOUN
ma-182	97	13	γsω	γsω	VERB
ma-182	97	14	.	.	PUNCT
ma-182	98	1	throughout	throughout	ADP
ma-182	98	2	this	this	DET
ma-182	98	3	section	section	NOUN
ma-182	98	4	,	,	PUNCT
ma-182	98	5	weassume	weassume	VERB
ma-182	98	6	that	that	SCONJ
ma-182	98	7	g	g	PROPN
ma-182	98	8	is	be	AUX
ma-182	98	9	a	a	DET
ma-182	98	10	locally	locally	ADV
ma-182	98	11	compact	compact	ADJ
ma-182	98	12	abelian	abelian	NOUN
ma-182	98	13	group	group	NOUN
ma-182	98	14	.	.	PUNCT
ma-182	99	1	a	a	DET
ma-182	99	2	look	look	NOUN
ma-182	99	3	at	at	ADP
ma-182	99	4	theorem	theorem	ADJ
ma-182	99	5	2.3	2.3	NUM
ma-182	99	6	shows	show	VERB
ma-182	99	7	that	that	SCONJ
ma-182	99	8	f	f	PROPN
ma-182	99	9	∈	∈	PROPN
ma-182	99	10	lpω(g	lpω(g	PROPN
ma-182	99	11	)	)	PUNCT
ma-182	99	12	ifand	ifand	NOUN
ma-182	99	13	only	only	ADV
ma-182	99	14	if	if	SCONJ
ma-182	99	15	γsωf	γsωf	ADJ
ma-182	99	16	∈	∈	NOUN
ma-182	99	17	l	l	NOUN
ma-182	99	18	p	p	NOUN
ma-182	99	19	ω(g	ω(g	NOUN
ma-182	99	20	)	)	PUNCT
ma-182	99	21	.	.	PUNCT
ma-182	100	1	that	that	PRON
ma-182	100	2	is	be	AUX
ma-182	100	3	,	,	PUNCT
ma-182	100	4	the	the	DET
ma-182	100	5	spaces	space	NOUN
ma-182	100	6	lpω(g	lpω(g	NOUN
ma-182	100	7	)	)	PUNCT
ma-182	100	8	are	be	AUX
ma-182	100	9	stable	stable	ADJ
ma-182	100	10	under	under	ADP
ma-182	100	11	the	the	DET
ma-182	100	12	action	action	NOUN
ma-182	100	13	of	of	ADP
ma-182	100	14	the	the	DET
ma-182	100	15	operators	operator	NOUN
ma-182	100	16	γsω	γsω	VERB
ma-182	100	17	.	.	PUNCT
ma-182	101	1	therefore	therefore	ADV
ma-182	101	2	,	,	PUNCT
ma-182	101	3	we	we	PRON
ma-182	101	4	are	be	AUX
ma-182	101	5	able	able	ADJ
ma-182	101	6	to	to	PART
ma-182	101	7	define	define	VERB
ma-182	101	8	a	a	DET
ma-182	101	9	concept	concept	NOUN
ma-182	101	10	of	of	ADP
ma-182	101	11	multiplier	multipli	ADJ
ma-182	101	12	in	in	ADP
ma-182	101	13	the	the	DET
ma-182	101	14	framework	framework	NOUN
ma-182	101	15	of	of	ADP
ma-182	101	16	this	this	DET
ma-182	101	17	study	study	NOUN
ma-182	101	18	.	.	PUNCT
ma-182	102	1	definition	definition	NOUN
ma-182	102	2	3.1	3.1	NUM
ma-182	102	3	.	.	PUNCT
ma-182	103	1	a	a	DET
ma-182	103	2	linear	linear	ADJ
ma-182	103	3	operator	operator	NOUN
ma-182	103	4	t	t	NOUN
ma-182	103	5	:	:	PUNCT
ma-182	103	6	l1ω(g	l1ω(g	PROPN
ma-182	103	7	)	)	PUNCT
ma-182	103	8	−→	−→	NOUN
ma-182	103	9	lpω(g	lpω(g	NOUN
ma-182	103	10	)	)	PUNCT
ma-182	103	11	is	be	AUX
ma-182	103	12	said	say	VERB
ma-182	103	13	to	to	PART
ma-182	103	14	be	be	AUX
ma-182	103	15	a	a	DET
ma-182	103	16	multiplier	multipli	ADJ
ma-182	103	17	if	if	SCONJ
ma-182	103	18	t	t	NOUN
ma-182	103	19	commutes	commute	VERB
ma-182	103	20	with	with	ADP
ma-182	103	21	all	all	DET
ma-182	103	22	the	the	DET
ma-182	103	23	operators	operator	NOUN
ma-182	103	24	γsω	γsω	VERB
ma-182	103	25	,	,	PUNCT
ma-182	103	26	s	s	PROPN
ma-182	103	27	∈	∈	PROPN
ma-182	103	28	g.	g.	NOUN
ma-182	103	29	that	that	PRON
ma-182	103	30	is	be	AUX
ma-182	103	31	,	,	PUNCT
ma-182	103	32	∀s	∀s	PROPN
ma-182	103	33	∈	∈	PROPN
ma-182	103	34	g	g	PROPN
ma-182	103	35	,	,	PUNCT
ma-182	103	36	tγsω	tγsω	NOUN
ma-182	103	37	=	=	SYM
ma-182	103	38	γsωt	γsωt	NOUN
ma-182	103	39	.	.	PUNCT
ma-182	104	1	we	we	PRON
ma-182	104	2	denote	denote	VERB
ma-182	104	3	by	by	ADP
ma-182	104	4	m1,p	m1,p	PROPN
ma-182	104	5	ω	ω	NOUN
ma-182	104	6	(	(	PUNCT
ma-182	104	7	g	g	NOUN
ma-182	104	8	)	)	PUNCT
ma-182	104	9	the	the	DET
ma-182	104	10	set	set	NOUN
ma-182	104	11	of	of	ADP
ma-182	104	12	such	such	ADJ
ma-182	104	13	multipliers	multiplier	NOUN
ma-182	104	14	.	.	PUNCT
ma-182	105	1	we	we	PRON
ma-182	105	2	denote	denote	VERB
ma-182	105	3	by	by	ADP
ma-182	105	4	‖t‖	‖t‖	PROPN
ma-182	105	5	the	the	DET
ma-182	105	6	operator	operator	NOUN
ma-182	105	7	norm	norm	NOUN
ma-182	105	8	of	of	ADP
ma-182	105	9	t	t	PROPN
ma-182	105	10	∈m1,p	∈m1,p	PROPN
ma-182	105	11	ω	ω	PROPN
ma-182	105	12	(	(	PUNCT
ma-182	105	13	g	g	NOUN
ma-182	105	14	)	)	PUNCT
ma-182	105	15	.	.	PUNCT
ma-182	106	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	106	2	eur	eur	PROPN
ma-182	106	3	.	.	PUNCT
ma-182	107	1	j.	j.	PROPN
ma-182	107	2	math	math	PROPN
ma-182	107	3	.	.	PUNCT
ma-182	108	1	anal	anal	PROPN
ma-182	108	2	.	.	PUNCT
ma-182	109	1	10.28924	10.28924	NUM
ma-182	109	2	/	/	SYM
ma-182	109	3	ada	ada	PROPN
ma-182	109	4	/	/	SYM
ma-182	109	5	ma.3.27	ma.3.27	PROPN
ma-182	109	6	5we	5we	NOUN
ma-182	109	7	will	will	AUX
ma-182	109	8	use	use	VERB
ma-182	109	9	the	the	DET
ma-182	109	10	fact	fact	NOUN
ma-182	109	11	that	that	SCONJ
ma-182	109	12	for	for	ADP
ma-182	109	13	1	1	NUM
ma-182	109	14	<	<	X
ma-182	109	15	p	p	X
ma-182	109	16	<	<	X
ma-182	109	17	∞	∞	PROPN
ma-182	109	18	,	,	PUNCT
ma-182	109	19	the	the	DET
ma-182	109	20	following	follow	VERB
ma-182	109	21	identification	identification	NOUN
ma-182	109	22	holds	hold	VERB
ma-182	109	23	[	[	X
ma-182	109	24	7	7	NUM
ma-182	109	25	]	]	PUNCT
ma-182	109	26	:	:	PUNCT
ma-182	109	27	(	(	PUNCT
ma-182	109	28	lpω(g))′	lpω(g))′	PROPN
ma-182	109	29	=	=	SYM
ma-182	109	30	lqw	lqw	ADJ
ma-182	109	31	(	(	PUNCT
ma-182	109	32	g	g	NOUN
ma-182	109	33	)	)	PUNCT
ma-182	109	34	with	with	ADP
ma-182	109	35	1	1	NUM
ma-182	109	36	p	p	NOUN
ma-182	109	37	+	+	NOUN
ma-182	109	38	1	1	NUM
ma-182	109	39	q	q	NOUN
ma-182	109	40	=	=	SYM
ma-182	109	41	1	1	NUM
ma-182	109	42	and	and	CCONJ
ma-182	109	43	w	w	NOUN
ma-182	109	44	=	=	PUNCT
ma-182	109	45	ω−	ω−	PROPN
ma-182	109	46	q	q	PROPN
ma-182	110	1	p	p	NOUN
ma-182	110	2	.	.	PUNCT
ma-182	111	1	from	from	ADP
ma-182	111	2	that	that	PRON
ma-182	111	3	,	,	PUNCT
ma-182	111	4	one	one	PRON
ma-182	111	5	may	may	AUX
ma-182	111	6	deduce	deduce	VERB
ma-182	111	7	that	that	PRON
ma-182	111	8	for	for	ADP
ma-182	111	9	1	1	NUM
ma-182	111	10	<	<	X
ma-182	111	11	p	p	X
ma-182	111	12	<	<	X
ma-182	111	13	∞	∞	PROPN
ma-182	111	14	,	,	PUNCT
ma-182	111	15	the	the	DET
ma-182	111	16	space	space	NOUN
ma-182	111	17	lpω(g)is	lpω(g)i	VERB
ma-182	111	18	a	a	DET
ma-182	111	19	reflexive	reflexive	ADJ
ma-182	111	20	space.for	space.for	ADP
ma-182	111	21	f	f	PROPN
ma-182	111	22	∈	∈	PROPN
ma-182	111	23	l1ω(g	l1ω(g	PROPN
ma-182	111	24	)	)	PUNCT
ma-182	111	25	,	,	PUNCT
ma-182	111	26	define	define	VERB
ma-182	111	27	the	the	DET
ma-182	111	28	fourier	fourier	ADJ
ma-182	111	29	transform	transform	NOUN
ma-182	111	30	of	of	ADP
ma-182	111	31	f	f	PROPN
ma-182	111	32	by	by	ADP
ma-182	111	33	fω(f	fω(f	NOUN
ma-182	111	34	)	)	PUNCT
ma-182	111	35	(	(	PUNCT
ma-182	111	36	γ	γ	X
ma-182	111	37	)	)	PUNCT
ma-182	111	38	=	=	SYM
ma-182	112	1	∫	∫	PROPN
ma-182	112	2	g	g	PROPN
ma-182	112	3	f	f	PROPN
ma-182	112	4	(	(	PUNCT
ma-182	112	5	x)γ(x)ω(x)dx	x)γ(x)ω(x)dx	PROPN
ma-182	112	6	,	,	PUNCT
ma-182	112	7	γ	γ	PROPN
ma-182	112	8	∈	∈	PROPN
ma-182	112	9	ĝ.	ĝ.	NOUN
ma-182	112	10	in	in	ADP
ma-182	112	11	[	[	X
ma-182	112	12	4	4	NUM
ma-182	112	13	]	]	PUNCT
ma-182	112	14	,	,	PUNCT
ma-182	112	15	the	the	DET
ma-182	112	16	following	follow	VERB
ma-182	112	17	convolution	convolution	NOUN
ma-182	112	18	result	result	NOUN
ma-182	112	19	was	be	AUX
ma-182	112	20	proved	prove	VERB
ma-182	112	21	.	.	PUNCT
ma-182	113	1	∀f	∀f	PROPN
ma-182	113	2	,	,	PUNCT
ma-182	113	3	g	g	PROPN
ma-182	113	4	∈	∈	PROPN
ma-182	113	5	l1ω(g	l1ω(g	NOUN
ma-182	113	6	)	)	PUNCT
ma-182	113	7	,	,	PUNCT
ma-182	113	8	fω(f	fω(f	VERB
ma-182	113	9	∗ω	∗ω	PROPN
ma-182	113	10	g	g	NOUN
ma-182	113	11	)	)	PUNCT
ma-182	113	12	=	=	SYM
ma-182	113	13	fω(f	fω(f	X
ma-182	113	14	)	)	PUNCT
ma-182	113	15	fω(g	fω(g	X
ma-182	113	16	)	)	PUNCT
ma-182	113	17	.	.	PUNCT
ma-182	114	1	set	set	VERB
ma-182	114	2	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	114	3	)	)	PUNCT
ma-182	114	4	)	)	PUNCT
ma-182	115	1	=	=	PRON
ma-182	115	2	{	{	PUNCT
ma-182	115	3	fω(f	fω(f	NOUN
ma-182	115	4	)	)	PUNCT
ma-182	115	5	:	:	PUNCT
ma-182	116	1	f	f	PROPN
ma-182	116	2	∈	∈	PROPN
ma-182	116	3	l1ω(g	l1ω(g	PROPN
ma-182	116	4	)	)	PUNCT
ma-182	116	5	}	}	PUNCT
ma-182	116	6	.	.	PUNCT
ma-182	117	1	let	let	VERB
ma-182	117	2	us	we	PRON
ma-182	117	3	remark	remark	VERB
ma-182	117	4	that	that	SCONJ
ma-182	117	5	functions	function	NOUN
ma-182	117	6	in	in	ADP
ma-182	117	7	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	117	8	)	)	PUNCT
ma-182	117	9	)	)	PUNCT
ma-182	117	10	are	be	AUX
ma-182	117	11	continuous	continuous	ADJ
ma-182	117	12	and	and	CCONJ
ma-182	117	13	vanished	vanish	VERB
ma-182	117	14	at	at	ADP
ma-182	117	15	infinity	infinity	NOUN
ma-182	117	16	by	by	ADP
ma-182	117	17	the	the	DET
ma-182	117	18	riemann	riemann	PROPN
ma-182	117	19	-	-	PUNCT
ma-182	117	20	lebesgue	lebesgue	NOUN
ma-182	117	21	theorem	theorem	NOUN
ma-182	117	22	.	.	PUNCT
ma-182	118	1	we	we	PRON
ma-182	118	2	fit	fit	VERB
ma-182	118	3	out	out	ADP
ma-182	118	4	the	the	DET
ma-182	118	5	space	space	NOUN
ma-182	118	6	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	118	7	)	)	PUNCT
ma-182	118	8	)	)	PUNCT
ma-182	118	9	with	with	ADP
ma-182	118	10	the	the	DET
ma-182	118	11	norm	norm	NOUN
ma-182	118	12	defined	define	VERB
ma-182	118	13	by	by	ADP
ma-182	118	14	‖fω(f	‖fω(f	PROPN
ma-182	118	15	)	)	PUNCT
ma-182	118	16	‖	‖	PROPN
ma-182	118	17	=	=	NOUN
ma-182	118	18	‖f	‖f	ADJ
ma-182	118	19	‖1,ω	‖1,ω	NOUN
ma-182	118	20	,	,	PUNCT
ma-182	118	21	f	f	PROPN
ma-182	118	22	∈	∈	PROPN
ma-182	118	23	l1ω(g	l1ω(g	PROPN
ma-182	118	24	)	)	PUNCT
ma-182	118	25	.	.	PUNCT
ma-182	119	1	then	then	ADV
ma-182	119	2	,	,	PUNCT
ma-182	119	3	we	we	PRON
ma-182	119	4	have	have	VERB
ma-182	119	5	the	the	DET
ma-182	119	6	following	follow	VERB
ma-182	119	7	result	result	NOUN
ma-182	119	8	.	.	PUNCT
ma-182	120	1	theorem	theorem	ADJ
ma-182	120	2	3.2	3.2	NUM
ma-182	120	3	.	.	PUNCT
ma-182	121	1	the	the	DET
ma-182	121	2	space	space	NOUN
ma-182	121	3	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	121	4	)	)	PUNCT
ma-182	121	5	)	)	PUNCT
ma-182	121	6	is	be	AUX
ma-182	121	7	a	a	DET
ma-182	121	8	banach	banach	NOUN
ma-182	121	9	algebra	algebra	NOUN
ma-182	121	10	for	for	ADP
ma-182	121	11	the	the	DET
ma-182	121	12	pointwise	pointwise	ADJ
ma-182	121	13	multiplication	multiplication	NOUN
ma-182	121	14	.	.	PUNCT
ma-182	122	1	proof	proof	NOUN
ma-182	122	2	.	.	PUNCT
ma-182	123	1	let	let	VERB
ma-182	123	2	(	(	PUNCT
ma-182	123	3	fω(fn	fω(fn	PROPN
ma-182	123	4	)	)	PUNCT
ma-182	123	5	)	)	PUNCT
ma-182	124	1	be	be	AUX
ma-182	124	2	a	a	DET
ma-182	124	3	cauchy	cauchy	ADJ
ma-182	124	4	sequence	sequence	NOUN
ma-182	124	5	in	in	ADP
ma-182	124	6	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	124	7	)	)	PUNCT
ma-182	124	8	)	)	PUNCT
ma-182	124	9	.	.	PUNCT
ma-182	125	1	let	let	VERB
ma-182	125	2	p	p	PRON
ma-182	125	3	,	,	PUNCT
ma-182	125	4	q	q	PROPN
ma-182	125	5	∈	∈	PROPN
ma-182	125	6	n.	n.	NOUN
ma-182	125	7	the	the	DET
ma-182	125	8	equality	equality	NOUN
ma-182	125	9	‖fω(fp)−fω(fq)‖	‖fω(fp)−fω(fq)‖	PROPN
ma-182	125	10	=	=	SYM
ma-182	126	1	‖fp	‖fp	NUM
ma-182	126	2	−	−	NUM
ma-182	126	3	fq‖1,ω	fq‖1,ω	NOUN
ma-182	126	4	and	and	CCONJ
ma-182	126	5	the	the	DET
ma-182	126	6	fact	fact	NOUN
ma-182	126	7	that	that	SCONJ
ma-182	126	8	(	(	PUNCT
ma-182	126	9	l1ω(g	l1ω(g	PROPN
ma-182	126	10	)	)	PUNCT
ma-182	126	11	,	,	PUNCT
ma-182	126	12	‖	‖	PROPN
ma-182	126	13	·	·	PUNCT
ma-182	126	14	‖1,ω	‖1,ω	NOUN
ma-182	126	15	)	)	PUNCT
ma-182	126	16	is	be	AUX
ma-182	126	17	a	a	DET
ma-182	126	18	banach	banach	NOUN
ma-182	126	19	space	space	NOUN
ma-182	126	20	show	show	NOUN
ma-182	126	21	that	that	SCONJ
ma-182	126	22	there	there	PRON
ma-182	126	23	exists	exist	VERB
ma-182	126	24	f	f	PROPN
ma-182	126	25	∈	∈	PROPN
ma-182	126	26	l1ω(g	l1ω(g	PROPN
ma-182	126	27	)	)	PUNCT
ma-182	126	28	such	such	ADJ
ma-182	126	29	that	that	SCONJ
ma-182	126	30	(	(	PUNCT
ma-182	126	31	fn	fn	NOUN
ma-182	126	32	)	)	PUNCT
ma-182	126	33	converges	converge	NOUN
ma-182	126	34	to	to	ADP
ma-182	126	35	f	f	PROPN
ma-182	126	36	in	in	ADP
ma-182	126	37	l1ω(g	l1ω(g	PROPN
ma-182	126	38	)	)	PUNCT
ma-182	126	39	.	.	PUNCT
ma-182	127	1	now	now	ADV
ma-182	127	2	,	,	PUNCT
ma-182	127	3	‖fω(fn)−fω(f	‖fω(fn)−fω(f	X
ma-182	127	4	)	)	PUNCT
ma-182	127	5	‖	‖	PROPN
ma-182	127	6	=	=	SYM
ma-182	128	1	‖fn	‖fn	NUM
ma-182	128	2	−	−	NOUN
ma-182	128	3	f	f	PROPN
ma-182	128	4	‖1,ω	‖1,ω	NOUN
ma-182	128	5	.	.	PUNCT
ma-182	129	1	thus	thus	ADV
ma-182	129	2	,	,	PUNCT
ma-182	129	3	(	(	PUNCT
ma-182	129	4	fω(fn	fω(fn	NOUN
ma-182	129	5	)	)	PUNCT
ma-182	129	6	)	)	PUNCT
ma-182	129	7	converges	converge	VERB
ma-182	129	8	to	to	ADP
ma-182	129	9	(	(	PUNCT
ma-182	129	10	fω(f	fω(f	X
ma-182	129	11	)	)	PUNCT
ma-182	129	12	)	)	PUNCT
ma-182	129	13	in	in	ADP
ma-182	129	14	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	129	15	)	)	PUNCT
ma-182	129	16	)	)	PUNCT
ma-182	129	17	.	.	PUNCT
ma-182	130	1	thus	thus	ADV
ma-182	130	2	,	,	PUNCT
ma-182	130	3	the	the	DET
ma-182	130	4	space	space	NOUN
ma-182	130	5	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	130	6	)	)	PUNCT
ma-182	130	7	)	)	PUNCT
ma-182	130	8	is	be	AUX
ma-182	130	9	a	a	DET
ma-182	130	10	banach	banach	NOUN
ma-182	130	11	space	space	NOUN
ma-182	130	12	.	.	PUNCT
ma-182	131	1	moreover	moreover	ADV
ma-182	131	2	,	,	PUNCT
ma-182	131	3	‖fω(f	‖fω(f	PROPN
ma-182	131	4	)	)	PUNCT
ma-182	131	5	fω(g)‖	fω(g)‖	NOUN
ma-182	131	6	=	=	SYM
ma-182	132	1	‖fω(f	‖fω(f	PROPN
ma-182	132	2	∗ω	∗ω	PROPN
ma-182	132	3	g)‖	g)‖	NOUN
ma-182	132	4	=	=	SYM
ma-182	132	5	‖f	‖f	ADP
ma-182	132	6	∗ω	∗ω	NOUN
ma-182	132	7	g‖1,ω	g‖1,ω	PROPN
ma-182	132	8	6	6	NUM
ma-182	132	9	‖f	‖f	ADP
ma-182	132	10	‖1,ω‖g‖1,ω	‖1,ω‖g‖1,ω	PROPN
ma-182	132	11	=	=	SYM
ma-182	132	12	‖fω(f	‖fω(f	PROPN
ma-182	132	13	)	)	PUNCT
ma-182	132	14	‖‖fω(g)‖.	‖‖fω(g)‖.	VERB
ma-182	132	15	thus	thus	ADV
ma-182	132	16	,	,	PUNCT
ma-182	132	17	the	the	DET
ma-182	132	18	space	space	NOUN
ma-182	132	19	(	(	PUNCT
ma-182	132	20	fω(l1ω(g	fω(l1ω(g	PROPN
ma-182	132	21	)	)	PUNCT
ma-182	132	22	)	)	PUNCT
ma-182	132	23	)	)	PUNCT
ma-182	132	24	,	,	PUNCT
ma-182	132	25	·	·	PUNCT
ma-182	132	26	,	,	PUNCT
ma-182	132	27	‖·‖1,ω	‖·‖1,ω	NUM
ma-182	132	28	)	)	PUNCT
ma-182	132	29	is	be	AUX
ma-182	132	30	a	a	DET
ma-182	132	31	banach	banach	NOUN
ma-182	132	32	algebra	algebra	NOUN
ma-182	132	33	.	.	PUNCT
ma-182	133	1	�	�	PROPN
ma-182	133	2	for	for	ADP
ma-182	133	3	f	f	PROPN
ma-182	133	4	∈	∈	PROPN
ma-182	133	5	lpω(g	lpω(g	X
ma-182	133	6	)	)	PUNCT
ma-182	133	7	and	and	CCONJ
ma-182	133	8	h	h	NOUN
ma-182	133	9	∈	∈	PROPN
ma-182	133	10	lqw	lqw	ADJ
ma-182	133	11	(	(	PUNCT
ma-182	133	12	g	g	NOUN
ma-182	133	13	)	)	PUNCT
ma-182	133	14	with	with	ADP
ma-182	133	15	1	1	NUM
ma-182	133	16	p	p	NOUN
ma-182	134	1	+	+	NOUN
ma-182	134	2	1	1	NUM
ma-182	134	3	q	q	NOUN
ma-182	134	4	=	=	NOUN
ma-182	134	5	1	1	NUM
ma-182	134	6	,	,	PUNCT
ma-182	134	7	we	we	PRON
ma-182	134	8	set	set	VERB
ma-182	134	9	〈	〈	PROPN
ma-182	134	10	f	f	PROPN
ma-182	134	11	,	,	PUNCT
ma-182	134	12	h〉ω	h〉ω	VERB
ma-182	134	13	=	=	SYM
ma-182	134	14	∫	∫	PROPN
ma-182	134	15	g	g	PROPN
ma-182	134	16	f	f	PROPN
ma-182	134	17	(	(	PUNCT
ma-182	134	18	x)h(x−1)ω(x)dx	x)h(x−1)ω(x)dx	PROPN
ma-182	134	19	.	.	PUNCT
ma-182	135	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	135	2	eur	eur	PROPN
ma-182	135	3	.	.	PUNCT
ma-182	136	1	j.	j.	PROPN
ma-182	136	2	math	math	PROPN
ma-182	136	3	.	.	PUNCT
ma-182	137	1	anal	anal	PROPN
ma-182	137	2	.	.	PUNCT
ma-182	138	1	10.28924	10.28924	NUM
ma-182	138	2	/	/	SYM
ma-182	138	3	ada	ada	PROPN
ma-182	138	4	/	/	SYM
ma-182	138	5	ma.3.27	ma.3.27	PROPN
ma-182	138	6	6	6	NUM
ma-182	138	7	theorem	theorem	VERB
ma-182	138	8	3.3	3.3	NUM
ma-182	138	9	.	.	PUNCT
ma-182	139	1	let	let	VERB
ma-182	139	2	g	g	PRON
ma-182	139	3	be	be	AUX
ma-182	139	4	a	a	DET
ma-182	139	5	locally	locally	ADV
ma-182	139	6	compact	compact	ADJ
ma-182	139	7	abelian	abelian	NOUN
ma-182	139	8	group	group	NOUN
ma-182	139	9	.	.	PUNCT
ma-182	140	1	let	let	VERB
ma-182	140	2	t	t	NOUN
ma-182	140	3	:	:	PUNCT
ma-182	140	4	l1ω(g	l1ω(g	PROPN
ma-182	140	5	)	)	PUNCT
ma-182	140	6	−→	−→	NOUN
ma-182	140	7	lpω(g	lpω(g	NOUN
ma-182	140	8	)	)	PUNCT
ma-182	140	9	be	be	VERB
ma-182	140	10	a	a	DET
ma-182	140	11	bounded	bounded	ADJ
ma-182	140	12	linear	linear	ADJ
ma-182	140	13	transformation	transformation	NOUN
ma-182	140	14	.	.	PUNCT
ma-182	141	1	then	then	ADV
ma-182	141	2	,	,	PUNCT
ma-182	141	3	t	t	PROPN
ma-182	141	4	∈m1,p	∈m1,p	PROPN
ma-182	141	5	ω	ω	PROPN
ma-182	141	6	(	(	PUNCT
ma-182	141	7	g	g	NOUN
ma-182	141	8	)	)	PUNCT
ma-182	141	9	if	if	SCONJ
ma-182	141	10	and	and	CCONJ
ma-182	141	11	only	only	ADV
ma-182	141	12	if	if	SCONJ
ma-182	141	13	there	there	PRON
ma-182	141	14	exists	exist	VERB
ma-182	141	15	a	a	DET
ma-182	141	16	unique	unique	ADJ
ma-182	141	17	element	element	NOUN
ma-182	141	18	ϕ	ϕ	NOUN
ma-182	141	19	such	such	ADJ
ma-182	141	20	that	that	SCONJ
ma-182	141	21	tg	tg	PROPN
ma-182	141	22	=	=	SYM
ma-182	141	23	ϕ	ϕ	PROPN
ma-182	141	24	∗ω	∗ω	PROPN
ma-182	141	25	g	g	PROPN
ma-182	141	26	for	for	ADP
ma-182	141	27	all	all	DET
ma-182	141	28	g	g	PROPN
ma-182	141	29	∈	∈	PROPN
ma-182	141	30	l1ω(g	l1ω(g	NOUN
ma-182	141	31	)	)	PUNCT
ma-182	141	32	,	,	PUNCT
ma-182	141	33	where	where	SCONJ
ma-182	141	34	ϕ	ϕ	PROPN
ma-182	141	35	∈	∈	PROPN
ma-182	141	36	m1ω(g	m1ω(g	PROPN
ma-182	141	37	)	)	PUNCT
ma-182	141	38	if	if	SCONJ
ma-182	141	39	p	p	NOUN
ma-182	141	40	=	=	NOUN
ma-182	141	41	1	1	NUM
ma-182	141	42	and	and	CCONJ
ma-182	141	43	ϕ	ϕ	NOUN
ma-182	141	44	∈	∈	PROPN
ma-182	141	45	lpω(g	lpω(g	X
ma-182	141	46	)	)	PUNCT
ma-182	141	47	if	if	SCONJ
ma-182	141	48	1	1	NUM
ma-182	141	49	<	<	X
ma-182	141	50	p	p	X
ma-182	141	51	<	<	X
ma-182	141	52	∞.	∞.	PROPN
ma-182	141	53	proof	proof	NOUN
ma-182	141	54	.	.	PUNCT
ma-182	142	1	(	(	PUNCT
ma-182	142	2	1	1	X
ma-182	142	3	)	)	PUNCT
ma-182	142	4	suppose	suppose	VERB
ma-182	142	5	p	p	X
ma-182	142	6	=	=	NOUN
ma-182	142	7	1	1	X
ma-182	142	8	.	.	PUNCT
ma-182	143	1	let	let	VERB
ma-182	143	2	t	t	PROPN
ma-182	143	3	∈	∈	PROPN
ma-182	143	4	m1,1	m1,1	PROPN
ma-182	143	5	ω	ω	X
ma-182	143	6	(	(	PUNCT
ma-182	143	7	g	g	NOUN
ma-182	143	8	)	)	PUNCT
ma-182	143	9	.	.	PUNCT
ma-182	144	1	in	in	ADP
ma-182	144	2	[	[	X
ma-182	144	3	4	4	NUM
ma-182	144	4	,	,	PUNCT
ma-182	144	5	proposition	proposition	NOUN
ma-182	144	6	5.4	5.4	NUM
ma-182	144	7	]	]	PUNCT
ma-182	144	8	,	,	PUNCT
ma-182	144	9	it	it	PRON
ma-182	144	10	was	be	AUX
ma-182	144	11	shown	show	VERB
ma-182	144	12	that	that	SCONJ
ma-182	144	13	t	t	PROPN
ma-182	144	14	∈	∈	PROPN
ma-182	144	15	m1,1	m1,1	PROPN
ma-182	144	16	ω	ω	X
ma-182	144	17	(	(	PUNCT
ma-182	144	18	g	g	NOUN
ma-182	144	19	)	)	PUNCT
ma-182	144	20	if	if	SCONJ
ma-182	144	21	and	and	CCONJ
ma-182	144	22	only	only	ADV
ma-182	144	23	if	if	SCONJ
ma-182	144	24	there	there	PRON
ma-182	144	25	exists	exist	VERB
ma-182	144	26	a	a	DET
ma-182	144	27	unique	unique	ADJ
ma-182	144	28	function	function	NOUN
ma-182	144	29	b	b	NOUN
ma-182	144	30	defined	define	VERB
ma-182	144	31	on	on	ADP
ma-182	144	32	ĝ	ĝ	NOUN
ma-182	144	33	such	such	ADJ
ma-182	144	34	that	that	SCONJ
ma-182	144	35	fω(t	fω(t	NOUN
ma-182	144	36	f	f	NOUN
ma-182	144	37	)	)	PUNCT
ma-182	144	38	=	=	SYM
ma-182	144	39	bfω(f	bfω(f	PROPN
ma-182	144	40	)	)	PUNCT
ma-182	144	41	for	for	ADP
ma-182	144	42	all	all	DET
ma-182	144	43	f	f	PROPN
ma-182	144	44	∈	∈	PROPN
ma-182	144	45	l1ω(g	l1ω(g	NOUN
ma-182	144	46	)	)	PUNCT
ma-182	144	47	.	.	PUNCT
ma-182	145	1	clearly	clearly	ADV
ma-182	145	2	,	,	PUNCT
ma-182	145	3	bfω(f	bfω(f	PROPN
ma-182	145	4	)	)	PUNCT
ma-182	145	5	∈	∈	PROPN
ma-182	145	6	fω(l1ω(g	fω(l1ω(g	NOUN
ma-182	145	7	)	)	PUNCT
ma-182	145	8	)	)	PUNCT
ma-182	145	9	.	.	PUNCT
ma-182	146	1	therefore	therefore	ADV
ma-182	146	2	,	,	PUNCT
ma-182	146	3	thefunction	thefunction	NOUN
ma-182	146	4	bfω(f	bfω(f	PROPN
ma-182	146	5	)	)	PUNCT
ma-182	146	6	is	be	AUX
ma-182	146	7	continuous	continuous	ADJ
ma-182	146	8	for	for	ADP
ma-182	146	9	all	all	DET
ma-182	146	10	f	f	PROPN
ma-182	146	11	∈	∈	PROPN
ma-182	146	12	l1ω(g	l1ω(g	PROPN
ma-182	146	13	)	)	PUNCT
ma-182	146	14	(	(	PUNCT
ma-182	146	15	the	the	DET
ma-182	146	16	fourier	fourier	NOUN
ma-182	146	17	transform	transform	NOUN
ma-182	146	18	of	of	ADP
ma-182	146	19	a	a	DET
ma-182	146	20	function	function	NOUN
ma-182	146	21	is	be	AUX
ma-182	146	22	acontinuous	acontinuous	ADJ
ma-182	146	23	function	function	NOUN
ma-182	146	24	)	)	PUNCT
ma-182	146	25	.	.	PUNCT
ma-182	147	1	moreover	moreover	ADV
ma-182	147	2	,	,	PUNCT
ma-182	147	3	for	for	ADP
ma-182	147	4	each	each	DET
ma-182	147	5	open	open	ADJ
ma-182	147	6	set	set	VERB
ma-182	147	7	in	in	ADP
ma-182	147	8	ĝ	ĝ	NOUN
ma-182	147	9	with	with	ADP
ma-182	147	10	compact	compact	ADJ
ma-182	147	11	closure	closure	NOUN
ma-182	147	12	,	,	PUNCT
ma-182	147	13	there	there	PRON
ma-182	147	14	exists	exist	VERB
ma-182	147	15	afunction	afunction	NOUN
ma-182	147	16	f	f	PROPN
ma-182	147	17	∈	∈	PROPN
ma-182	147	18	l1ω(g	l1ω(g	PROPN
ma-182	147	19	)	)	PUNCT
ma-182	147	20	such	such	ADJ
ma-182	147	21	that	that	SCONJ
ma-182	147	22	fω(f	fω(f	PUNCT
ma-182	147	23	)	)	PUNCT
ma-182	147	24	is	be	AUX
ma-182	147	25	constant	constant	ADJ
ma-182	147	26	on	on	ADP
ma-182	147	27	u	u	PROPN
ma-182	147	28	[	[	X
ma-182	147	29	9	9	NUM
ma-182	147	30	,	,	PUNCT
ma-182	147	31	f.7e	f.7e	NOUN
ma-182	147	32	]	]	PUNCT
ma-182	147	33	.	.	PUNCT
ma-182	148	1	thus	thus	ADV
ma-182	148	2	,	,	PUNCT
ma-182	148	3	b	b	PROPN
ma-182	148	4	is	be	AUX
ma-182	148	5	continuous	continuous	ADJ
ma-182	148	6	on	on	ADP
ma-182	148	7	ĝ.let	ĝ.let	NOUN
ma-182	148	8	ε	ε	PROPN
ma-182	148	9	>	>	X
ma-182	148	10	0	0	PUNCT
ma-182	148	11	and	and	CCONJ
ma-182	148	12	let	let	VERB
ma-182	148	13	γ1	γ1	NOUN
ma-182	148	14	,	,	PUNCT
ma-182	148	15	γ2	γ2	PROPN
ma-182	148	16	,	,	PUNCT
ma-182	148	17	....	....	PUNCT
ma-182	148	18	,	,	PUNCT
ma-182	148	19	γn	γn	PROPN
ma-182	148	20	∈	∈	PROPN
ma-182	148	21	ĝ.	ĝ.	NOUN
ma-182	148	22	via	via	ADP
ma-182	148	23	theorem	theorem	NOUN
ma-182	148	24	2.2	2.2	NUM
ma-182	148	25	,	,	PUNCT
ma-182	148	26	we	we	PRON
ma-182	148	27	can	can	AUX
ma-182	148	28	choose	choose	VERB
ma-182	148	29	g	g	PROPN
ma-182	148	30	∈	∈	PROPN
ma-182	148	31	l1(g	l1(g	PROPN
ma-182	148	32	)	)	PUNCT
ma-182	148	33	suchthat	suchthat	VERB
ma-182	148	34	‖f(g)‖	‖f(g)‖	PUNCT
ma-182	148	35	=	=	SYM
ma-182	148	36	‖g‖1	‖g‖1	NOUN
ma-182	148	37	<	<	X
ma-182	148	38	1	1	NUM
ma-182	148	39	+	+	CCONJ
ma-182	148	40	ε	ε	PROPN
ma-182	148	41	and	and	CCONJ
ma-182	148	42	f(g)(γi	f(g)(γi	NOUN
ma-182	148	43	)	)	PUNCT
ma-182	148	44	=	=	NOUN
ma-182	148	45	1	1	NUM
ma-182	148	46	,	,	PUNCT
ma-182	148	47	i	i	PRON
ma-182	148	48	=	=	NOUN
ma-182	148	49	1	1	NUM
ma-182	148	50	,	,	PUNCT
ma-182	148	51	2	2	NUM
ma-182	148	52	,	,	PUNCT
ma-182	148	53	3	3	NUM
ma-182	148	54	,	,	PUNCT
ma-182	148	55	.....	.....	PUNCT
ma-182	148	56	,	,	PUNCT
ma-182	148	57	n.	n.	PROPN
ma-182	148	58	now	now	ADV
ma-182	148	59	,	,	PUNCT
ma-182	148	60	set	set	VERB
ma-182	148	61	f	f	PROPN
ma-182	148	62	=	=	SYM
ma-182	148	63	g	g	PROPN
ma-182	148	64	ω	ω	PROPN
ma-182	148	65	.	.	PUNCT
ma-182	149	1	then	then	ADV
ma-182	149	2	,	,	PUNCT
ma-182	149	3	f	f	PROPN
ma-182	149	4	∈	∈	PROPN
ma-182	149	5	l1ω(g	l1ω(g	PROPN
ma-182	149	6	)	)	PUNCT
ma-182	149	7	,	,	PUNCT
ma-182	149	8	‖fω(f	‖fω(f	PROPN
ma-182	149	9	)	)	PUNCT
ma-182	149	10	‖	‖	PROPN
ma-182	150	1	=	=	NOUN
ma-182	150	2	‖f	‖f	ADP
ma-182	150	3	‖1,ω	‖1,ω	NOUN
ma-182	150	4	<	<	X
ma-182	150	5	1	1	NUM
ma-182	150	6	+	+	CCONJ
ma-182	150	7	ε	ε	PROPN
ma-182	150	8	and	and	CCONJ
ma-182	150	9	fω(f	fω(f	NUM
ma-182	150	10	)	)	PUNCT
ma-182	150	11	(	(	PUNCT
ma-182	150	12	γi	γi	NOUN
ma-182	150	13	)	)	PUNCT
ma-182	150	14	=	=	SYM
ma-182	150	15	1	1	NUM
ma-182	150	16	,	,	PUNCT
ma-182	150	17	i	i	PRON
ma-182	150	18	=	=	NOUN
ma-182	150	19	1	1	NUM
ma-182	150	20	,	,	PUNCT
ma-182	150	21	2	2	NUM
ma-182	150	22	,	,	PUNCT
ma-182	150	23	3	3	NUM
ma-182	150	24	,	,	PUNCT
ma-182	150	25	.....	.....	PUNCT
ma-182	150	26	,	,	PUNCT
ma-182	150	27	n.for	n.for	PROPN
ma-182	150	28	zi	zi	PROPN
ma-182	150	29	∈	∈	PROPN
ma-182	150	30	c	c	X
ma-182	150	31	,	,	PUNCT
ma-182	150	32	i	i	PRON
ma-182	150	33	=	=	NOUN
ma-182	150	34	1	1	NUM
ma-182	150	35	,	,	PUNCT
ma-182	150	36	2	2	NUM
ma-182	150	37	,	,	PUNCT
ma-182	150	38	.....	.....	PUNCT
ma-182	150	39	,	,	PUNCT
ma-182	150	40	n	n	CCONJ
ma-182	150	41	,	,	PUNCT
ma-182	150	42	one	one	NUM
ma-182	150	43	has∣∣∣∣∣	has∣∣∣∣∣	NOUN
ma-182	150	44	n∑	n∑	X
ma-182	150	45	i=1	i=1	PROPN
ma-182	150	46	zib(γi	zib(γi	NOUN
ma-182	150	47	)	)	PUNCT
ma-182	150	48	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ma-182	150	49	=	=	SYM
ma-182	151	1	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-182	151	2	n∑	n∑	PROPN
ma-182	152	1	i=1	i=1	PROPN
ma-182	152	2	zifω(f	zifω(f	PROPN
ma-182	152	3	)	)	PUNCT
ma-182	152	4	(	(	PUNCT
ma-182	152	5	γi	γi	NOUN
ma-182	152	6	)	)	PUNCT
ma-182	152	7	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ma-182	153	1	=	=	SYM
ma-182	153	2	∣∣∣∣∣∫g	∣∣∣∣∣∫g	NOUN
ma-182	153	3	[	[	PUNCT
ma-182	153	4	n∑	n∑	NOUN
ma-182	153	5	i=1	i=1	PROPN
ma-182	153	6	ziγi(x	ziγi(x	PROPN
ma-182	153	7	)	)	PUNCT
ma-182	153	8	]	]	PUNCT
ma-182	154	1	f	f	X
ma-182	154	2	(	(	PUNCT
ma-182	154	3	x)ω(x)dx	x)ω(x)dx	ADJ
ma-182	154	4	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-182	154	5	6	6	NUM
ma-182	154	6	‖f	‖f	ADP
ma-182	154	7	‖1,ω	‖1,ω	NOUN
ma-182	154	8	∥∥∥∥∥	∥∥∥∥∥	X
ma-182	154	9	n∑	n∑	NOUN
ma-182	154	10	i=1	i=1	PROPN
ma-182	154	11	ziγi	ziγi	PROPN
ma-182	154	12	∥∥∥∥∥	∥∥∥∥∥	VERB
ma-182	154	13	∞	∞	PROPN
ma-182	154	14	<	<	X
ma-182	154	15	(	(	PUNCT
ma-182	154	16	1	1	NUM
ma-182	154	17	+	+	CCONJ
ma-182	154	18	ε	ε	PROPN
ma-182	154	19	)	)	PUNCT
ma-182	154	20	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	155	1	n∑	n∑	PROPN
ma-182	155	2	i=1	i=1	PROPN
ma-182	155	3	ziγi	ziγi	PROPN
ma-182	155	4	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	155	5	∞	∞	PROPN
ma-182	155	6	.	.	PUNCT
ma-182	156	1	since	since	SCONJ
ma-182	156	2	ε	ε	PROPN
ma-182	156	3	is	be	AUX
ma-182	156	4	chosen	choose	VERB
ma-182	156	5	arbitrarily	arbitrarily	ADV
ma-182	156	6	,	,	PUNCT
ma-182	156	7	it	it	PRON
ma-182	156	8	follows	follow	VERB
ma-182	156	9	that∣∣∣∣∣	that∣∣∣∣∣	PROPN
ma-182	156	10	n∑	n∑	PROPN
ma-182	156	11	i=1	i=1	PROPN
ma-182	156	12	zib(γi	zib(γi	NUM
ma-182	156	13	)	)	PUNCT
ma-182	157	1	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-182	157	2	<	<	X
ma-182	157	3	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	157	4	n∑	n∑	X
ma-182	157	5	i=1	i=1	PROPN
ma-182	157	6	ziγi	ziγi	PROPN
ma-182	157	7	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	157	8	∞	∞	NUM
ma-182	157	9	.	.	PUNCT
ma-182	158	1	we	we	PRON
ma-182	158	2	conclude	conclude	VERB
ma-182	158	3	via	via	ADP
ma-182	158	4	theorem	theorem	ADJ
ma-182	158	5	2.1	2.1	NUM
ma-182	158	6	(	(	PUNCT
ma-182	158	7	with	with	ADP
ma-182	158	8	c	c	NOUN
ma-182	158	9	=	=	SYM
ma-182	158	10	1	1	NUM
ma-182	158	11	)	)	PUNCT
ma-182	158	12	that	that	SCONJ
ma-182	158	13	there	there	PRON
ma-182	158	14	exists	exist	VERB
ma-182	158	15	a	a	DET
ma-182	158	16	unique	unique	ADJ
ma-182	158	17	bounded	bound	VERB
ma-182	158	18	measure	measure	NOUN
ma-182	158	19	µ	µ	PRON
ma-182	158	20	such	such	ADJ
ma-182	158	21	that	that	DET
ma-182	158	22	b	b	NOUN
ma-182	158	23	=	=	SYM
ma-182	158	24	f(µ	f(µ	PROPN
ma-182	158	25	)	)	PUNCT
ma-182	158	26	.	.	PUNCT
ma-182	159	1	now	now	ADV
ma-182	159	2	,	,	PUNCT
ma-182	159	3	if	if	SCONJ
ma-182	159	4	we	we	PRON
ma-182	159	5	set	set	VERB
ma-182	159	6	ϕ	ϕ	NOUN
ma-182	159	7	=	=	SYM
ma-182	159	8	ω−1µ	ω−1µ	PROPN
ma-182	159	9	,	,	PUNCT
ma-182	159	10	then	then	ADV
ma-182	159	11	ϕ	ϕ	PROPN
ma-182	159	12	∈	∈	PROPN
ma-182	159	13	m1ω(g	m1ω(g	PROPN
ma-182	159	14	)	)	PUNCT
ma-182	159	15	and	and	CCONJ
ma-182	159	16	b	b	X
ma-182	159	17	=	=	SYM
ma-182	159	18	fω(ϕ).therefore	fω(ϕ).therefore	NOUN
ma-182	159	19	,	,	PUNCT
ma-182	159	20	fω(t	fω(t	NOUN
ma-182	159	21	f	f	NOUN
ma-182	159	22	)	)	PUNCT
ma-182	159	23	=	=	SYM
ma-182	159	24	fω(ϕ)fω(f	fω(ϕ)fω(f	NOUN
ma-182	159	25	)	)	PUNCT
ma-182	159	26	=	=	SYM
ma-182	159	27	fω(ϕ	fω(ϕ	NUM
ma-182	159	28	∗ω	∗ω	PROPN
ma-182	159	29	f	f	PROPN
ma-182	159	30	)	)	PUNCT
ma-182	159	31	.since	.since	NOUN
ma-182	160	1	the	the	DET
ma-182	160	2	fourier	fourier	NOUN
ma-182	160	3	transform	transform	NOUN
ma-182	160	4	is	be	AUX
ma-182	160	5	injective	injective	ADJ
ma-182	160	6	,	,	PUNCT
ma-182	160	7	it	it	PRON
ma-182	160	8	follows	follow	VERB
ma-182	160	9	that	that	SCONJ
ma-182	160	10	t	t	PROPN
ma-182	160	11	f	f	PROPN
ma-182	160	12	=	=	SYM
ma-182	160	13	ϕ	ϕ	PROPN
ma-182	160	14	∗ω	∗ω	PROPN
ma-182	160	15	f	f	PROPN
ma-182	160	16	.(2	.(2	PUNCT
ma-182	160	17	)	)	PUNCT
ma-182	160	18	assume	assume	VERB
ma-182	160	19	that	that	SCONJ
ma-182	160	20	1	1	X
ma-182	160	21	<	<	X
ma-182	160	22	p	p	X
ma-182	160	23	<	<	X
ma-182	160	24	∞.	∞.	PROPN
ma-182	160	25	let	let	VERB
ma-182	160	26	t	t	PROPN
ma-182	160	27	∈	∈	PROPN
ma-182	160	28	m1,p	m1,p	PROPN
ma-182	160	29	ω	ω	NOUN
ma-182	160	30	(	(	PUNCT
ma-182	160	31	g	g	NOUN
ma-182	160	32	)	)	PUNCT
ma-182	160	33	.	.	PUNCT
ma-182	161	1	the	the	DET
ma-182	161	2	weighted	weight	VERB
ma-182	161	3	group	group	NOUN
ma-182	161	4	algebra	algebra	NOUN
ma-182	161	5	(	(	PUNCT
ma-182	161	6	l1ω(g	l1ω(g	PROPN
ma-182	161	7	)	)	PUNCT
ma-182	161	8	,	,	PUNCT
ma-182	161	9	‖	‖	PROPN
ma-182	161	10	·	·	SYM
ma-182	161	11	‖1,ω	‖1,ω	NOUN
ma-182	161	12	,	,	PUNCT
ma-182	161	13	∗ω	∗ω	PROPN
ma-182	161	14	)	)	PUNCT
ma-182	161	15	has	have	VERB
ma-182	161	16	a	a	DET
ma-182	161	17	bounded	bounded	ADJ
ma-182	161	18	approximate	approximate	ADJ
ma-182	161	19	identity	identity	NOUN
ma-182	161	20	[	[	X
ma-182	161	21	10	10	NUM
ma-182	161	22	,	,	PUNCT
ma-182	161	23	theorem	theorem	VERB
ma-182	161	24	2.2	2.2	NUM
ma-182	161	25	]	]	PUNCT
ma-182	161	26	.	.	PUNCT
ma-182	162	1	let	let	AUX
ma-182	162	2	{	{	PUNCT
ma-182	162	3	υn	υn	AUX
ma-182	162	4	}	}	PUNCT
ma-182	162	5	be	be	AUX
ma-182	162	6	a	a	DET
ma-182	162	7	boundedapproximate	boundedapproximate	ADJ
ma-182	162	8	identity	identity	NOUN
ma-182	162	9	for	for	ADP
ma-182	162	10	(	(	PUNCT
ma-182	162	11	l1ω(g	l1ω(g	PROPN
ma-182	162	12	)	)	PUNCT
ma-182	162	13	,	,	PUNCT
ma-182	162	14	‖	‖	PROPN
ma-182	162	15	·	·	SYM
ma-182	162	16	‖1,ω	‖1,ω	NOUN
ma-182	162	17	,	,	PUNCT
ma-182	162	18	∗ω	∗ω	PROPN
ma-182	162	19	)	)	PUNCT
ma-182	162	20	.	.	PUNCT
ma-182	163	1	let	let	VERB
ma-182	163	2	g	g	PROPN
ma-182	163	3	∈	∈	PROPN
ma-182	163	4	l1ω(g	l1ω(g	NOUN
ma-182	163	5	)	)	PUNCT
ma-182	163	6	.	.	PUNCT
ma-182	164	1	then	then	ADV
ma-182	164	2	,	,	PUNCT
ma-182	164	3	‖tg	‖tg	NUM
ma-182	164	4	−	−	NUM
ma-182	164	5	tυn	tυn	NOUN
ma-182	164	6	∗ω	∗ω	PROPN
ma-182	164	7	g‖p	g‖p	NOUN
ma-182	164	8	,	,	PUNCT
ma-182	164	9	ω	ω	X
ma-182	164	10	=	=	PUNCT
ma-182	165	1	‖tg	‖tg	NUM
ma-182	165	2	−	−	PROPN
ma-182	165	3	t	t	NOUN
ma-182	165	4	(	(	PUNCT
ma-182	165	5	υn	υn	NOUN
ma-182	165	6	∗ω	∗ω	PROPN
ma-182	165	7	g)‖p	g)‖p	PROPN
ma-182	165	8	,	,	PUNCT
ma-182	165	9	ω	ω	NUM
ma-182	165	10	6	6	NUM
ma-182	165	11	‖t‖‖g	‖t‖‖g	NUM
ma-182	165	12	−	−	ADP
ma-182	165	13	υn	υn	PROPN
ma-182	165	14	∗ω	∗ω	PROPN
ma-182	165	15	g‖1,ω	g‖1,ω	PROPN
ma-182	165	16	.	.	PUNCT
ma-182	166	1	since	since	SCONJ
ma-182	166	2	‖g	‖g	PROPN
ma-182	166	3	−	−	PROPN
ma-182	166	4	υn	υn	NOUN
ma-182	166	5	∗ω	∗ω	PROPN
ma-182	166	6	g‖1,ω	g‖1,ω	PROPN
ma-182	166	7	tends	tend	VERB
ma-182	166	8	to	to	ADP
ma-182	166	9	0	0	NUM
ma-182	166	10	whenever	whenever	SCONJ
ma-182	166	11	n	n	PRON
ma-182	166	12	goes	go	VERB
ma-182	166	13	to	to	ADP
ma-182	166	14	∞	∞	PROPN
ma-182	166	15	,	,	PUNCT
ma-182	166	16	then	then	ADV
ma-182	166	17	(	(	PUNCT
ma-182	166	18	tυn	tυn	PROPN
ma-182	166	19	∗ω	∗ω	PROPN
ma-182	166	20	g)n	g)n	NOUN
ma-182	166	21	converges	converge	VERB
ma-182	166	22	to	to	ADP
ma-182	166	23	tg	tg	PROPN
ma-182	166	24	in	in	ADP
ma-182	166	25	lpω(g	lpω(g	NOUN
ma-182	166	26	)	)	PUNCT
ma-182	166	27	.	.	PUNCT
ma-182	167	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	167	2	eur	eur	PROPN
ma-182	167	3	.	.	PUNCT
ma-182	168	1	j.	j.	PROPN
ma-182	168	2	math	math	PROPN
ma-182	168	3	.	.	PUNCT
ma-182	169	1	anal	anal	PROPN
ma-182	169	2	.	.	PUNCT
ma-182	170	1	10.28924	10.28924	NUM
ma-182	170	2	/	/	SYM
ma-182	170	3	ada	ada	PROPN
ma-182	170	4	/	/	SYM
ma-182	170	5	ma.3.27	ma.3.27	PROPN
ma-182	171	1	7moreover	7moreover	NUM
ma-182	171	2	,	,	PUNCT
ma-182	171	3	‖tυn‖p	‖tυn‖p	PROPN
ma-182	171	4	,	,	PUNCT
ma-182	171	5	ω	ω	PROPN
ma-182	171	6	6	6	NUM
ma-182	171	7	‖t‖‖υn‖1,ω	‖t‖‖υn‖1,ω	NOUN
ma-182	171	8	=	=	PUNCT
ma-182	171	9	‖t‖.	‖t‖.	PUNCT
ma-182	171	10	therefore	therefore	ADV
ma-182	171	11	,	,	PUNCT
ma-182	171	12	{	{	PUNCT
ma-182	171	13	tυn	tυn	NOUN
ma-182	171	14	}	}	PUNCT
ma-182	171	15	lies	lie	VERB
ma-182	171	16	in	in	ADP
ma-182	171	17	a	a	DET
ma-182	171	18	norm	norm	NOUN
ma-182	171	19	bounded	bound	VERB
ma-182	171	20	subsetof	subsetof	PROPN
ma-182	171	21	lpω(g	lpω(g	PROPN
ma-182	171	22	)	)	PUNCT
ma-182	171	23	=	=	PUNCT
ma-182	171	24	(	(	PUNCT
ma-182	171	25	lqw	lqw	ADJ
ma-182	171	26	(	(	PUNCT
ma-182	171	27	g	g	NOUN
ma-182	171	28	)	)	PUNCT
ma-182	171	29	)	)	PUNCT
ma-182	171	30	′.	′.	NOUN
ma-182	171	31	so	so	ADV
ma-182	171	32	,	,	PUNCT
ma-182	171	33	by	by	ADP
ma-182	171	34	alaoglu	alaoglu	NOUN
ma-182	171	35	’s	’s	PART
ma-182	171	36	theorem	theorem	NOUN
ma-182	171	37	(	(	PUNCT
ma-182	171	38	[	[	X
ma-182	171	39	16	16	NUM
ma-182	171	40	,	,	PUNCT
ma-182	171	41	page	page	NOUN
ma-182	171	42	299	299	NUM
ma-182	171	43	]	]	PUNCT
ma-182	171	44	or	or	CCONJ
ma-182	171	45	[	[	X
ma-182	171	46	9	9	NUM
ma-182	171	47	,	,	PUNCT
ma-182	171	48	theorem	theorem	ADJ
ma-182	171	49	d.4.3	d.4.3	NOUN
ma-182	171	50	.	.	PROPN
ma-182	171	51	]	]	PUNCT
ma-182	171	52	)	)	PUNCT
ma-182	172	1	andthe	andthe	ADJ
ma-182	172	2	reflexivity	reflexivity	NOUN
ma-182	172	3	of	of	ADP
ma-182	172	4	lpω(g	lpω(g	NOUN
ma-182	172	5	)	)	PUNCT
ma-182	172	6	,	,	PUNCT
ma-182	172	7	we	we	PRON
ma-182	172	8	see	see	VERB
ma-182	172	9	that	that	SCONJ
ma-182	172	10	there	there	PRON
ma-182	172	11	exists	exist	VERB
ma-182	172	12	a	a	DET
ma-182	172	13	subnet	subnet	NOUN
ma-182	172	14	{	{	PUNCT
ma-182	172	15	tυm	tυm	NOUN
ma-182	172	16	}	}	PUNCT
ma-182	172	17	of	of	ADP
ma-182	172	18	{	{	PUNCT
ma-182	172	19	tυn	tυn	NOUN
ma-182	172	20	}	}	PUNCT
ma-182	172	21	and	and	CCONJ
ma-182	172	22	ϕ	ϕ	PROPN
ma-182	172	23	∈	∈	PROPN
ma-182	172	24	lpω(g)such	lpω(g)such	INTJ
ma-182	172	25	that	that	SCONJ
ma-182	172	26	{	{	PUNCT
ma-182	172	27	tυm	tυm	NOUN
ma-182	172	28	}	}	PUNCT
ma-182	172	29	converges	converge	VERB
ma-182	172	30	to	to	ADP
ma-182	172	31	ϕ	ϕ	NOUN
ma-182	172	32	in	in	ADP
ma-182	172	33	the	the	DET
ma-182	172	34	weak∗-topology	weak∗-topology	PROPN
ma-182	172	35	.	.	PUNCT
ma-182	173	1	that	that	PRON
ma-182	173	2	is	be	AUX
ma-182	173	3	,	,	PUNCT
ma-182	173	4	lim	lim	PROPN
ma-182	173	5	m	m	PROPN
ma-182	173	6	〈	〈	PROPN
ma-182	173	7	tυm	tυm	NOUN
ma-182	173	8	,	,	PUNCT
ma-182	173	9	u〉ω	u〉ω	PROPN
ma-182	173	10	=	=	SYM
ma-182	173	11	〈	〈	PROPN
ma-182	173	12	ϕ	ϕ	NOUN
ma-182	173	13	,	,	PUNCT
ma-182	173	14	u〉ωfor	u〉ωfor	ADP
ma-182	173	15	all	all	PRON
ma-182	173	16	u	u	PROPN
ma-182	173	17	∈	∈	PROPN
ma-182	173	18	lqw	lqw	ADJ
ma-182	173	19	(	(	PUNCT
ma-182	173	20	g	g	NOUN
ma-182	173	21	)	)	PUNCT
ma-182	173	22	.	.	PUNCT
ma-182	174	1	then	then	ADV
ma-182	174	2	,	,	PUNCT
ma-182	174	3	for	for	ADP
ma-182	174	4	h	h	NOUN
ma-182	174	5	,	,	PUNCT
ma-182	174	6	g	g	PROPN
ma-182	174	7	∈	∈	PROPN
ma-182	174	8	cc(g	cc(g	NOUN
ma-182	174	9	)	)	PUNCT
ma-182	174	10	,	,	PUNCT
ma-182	174	11	we	we	PRON
ma-182	174	12	have	have	VERB
ma-182	174	13	〈	〈	PROPN
ma-182	174	14	th	th	X
ma-182	174	15	,	,	PUNCT
ma-182	174	16	g〉ω	g〉ω	PROPN
ma-182	174	17	=	=	SYM
ma-182	174	18	lim	lim	PROPN
ma-182	174	19	m	m	PROPN
ma-182	174	20	〈	〈	PROPN
ma-182	174	21	tυm	tυm	NOUN
ma-182	174	22	∗ω	∗ω	PROPN
ma-182	174	23	h	h	NOUN
ma-182	174	24	,	,	PUNCT
ma-182	174	25	g〉ω	g〉ω	PROPN
ma-182	174	26	=	=	SYM
ma-182	174	27	lim	lim	PROPN
ma-182	174	28	m	m	PROPN
ma-182	174	29	〈	〈	PROPN
ma-182	174	30	tυm	tυm	NOUN
ma-182	174	31	,	,	PUNCT
ma-182	174	32	(	(	PUNCT
ma-182	174	33	h	h	NOUN
ma-182	174	34	∗ω	∗ω	PROPN
ma-182	174	35	g	g	PROPN
ma-182	174	36	ω	ω	PROPN
ma-182	174	37	)	)	PUNCT
ma-182	174	38	ω〉ω	ω〉ω	ADV
ma-182	174	39	=	=	PUNCT
ma-182	174	40	〈	〈	PROPN
ma-182	174	41	ϕ	ϕ	NOUN
ma-182	174	42	,	,	PUNCT
ma-182	174	43	(	(	PUNCT
ma-182	174	44	h	h	NOUN
ma-182	174	45	∗ω	∗ω	PROPN
ma-182	174	46	g	g	PROPN
ma-182	174	47	ω	ω	PROPN
ma-182	174	48	)	)	PUNCT
ma-182	174	49	ω〉ω	ω〉ω	ADV
ma-182	174	50	=	=	PUNCT
ma-182	174	51	〈	〈	PROPN
ma-182	174	52	ϕ	ϕ	PROPN
ma-182	174	53	∗ω	∗ω	PROPN
ma-182	174	54	h	h	PROPN
ma-182	174	55	,	,	PUNCT
ma-182	174	56	g〉ω	g〉ω	PROPN
ma-182	174	57	.	.	PUNCT
ma-182	175	1	however	however	ADV
ma-182	175	2	,	,	PUNCT
ma-182	175	3	cc(g	cc(g	PUNCT
ma-182	175	4	)	)	PUNCT
ma-182	175	5	is	be	AUX
ma-182	175	6	norm	norm	VERB
ma-182	175	7	dense	dense	ADJ
ma-182	175	8	in	in	ADP
ma-182	175	9	lqw	lqw	ADJ
ma-182	175	10	(	(	PUNCT
ma-182	175	11	g	g	NOUN
ma-182	175	12	)	)	PUNCT
ma-182	175	13	.	.	PUNCT
ma-182	176	1	therefore	therefore	ADV
ma-182	176	2	,	,	PUNCT
ma-182	176	3	th	th	X
ma-182	176	4	=	=	SYM
ma-182	176	5	ϕ	ϕ	PROPN
ma-182	176	6	∗ω	∗ω	PROPN
ma-182	176	7	h	h	NOUN
ma-182	176	8	for	for	ADP
ma-182	176	9	each	each	DET
ma-182	176	10	h	h	NOUN
ma-182	176	11	∈	∈	PROPN
ma-182	176	12	cc(g).moreover	cc(g).moreover	NOUN
ma-182	176	13	,	,	PUNCT
ma-182	176	14	cc(g	cc(g	PUNCT
ma-182	176	15	)	)	PUNCT
ma-182	176	16	is	be	AUX
ma-182	176	17	norm	norm	NOUN
ma-182	176	18	dense	dense	ADJ
ma-182	176	19	in	in	ADP
ma-182	176	20	l1ω(g	l1ω(g	NOUN
ma-182	176	21	)	)	PUNCT
ma-182	176	22	.	.	PUNCT
ma-182	177	1	thus	thus	ADV
ma-182	177	2	,	,	PUNCT
ma-182	177	3	th	th	X
ma-182	177	4	=	=	SYM
ma-182	177	5	ϕ	ϕ	PROPN
ma-182	177	6	∗ω	∗ω	PROPN
ma-182	177	7	h	h	NOUN
ma-182	177	8	for	for	ADP
ma-182	177	9	all	all	DET
ma-182	177	10	h	h	NOUN
ma-182	177	11	∈	∈	NOUN
ma-182	177	12	l1ω(g).(3	l1ω(g).(3	NOUN
ma-182	177	13	)	)	PUNCT
ma-182	177	14	conversely	conversely	ADV
ma-182	177	15	,	,	PUNCT
ma-182	177	16	let	let	VERB
ma-182	177	17	1	1	NUM
ma-182	177	18	≤	≤	NOUN
ma-182	178	1	p	p	NOUN
ma-182	178	2	<	<	X
ma-182	178	3	∞.	∞.	PROPN
ma-182	178	4	assume	assume	VERB
ma-182	178	5	that	that	SCONJ
ma-182	178	6	there	there	PRON
ma-182	178	7	exists	exist	VERB
ma-182	178	8	a	a	DET
ma-182	178	9	measure	measure	NOUN
ma-182	178	10	µ	µ	PRON
ma-182	178	11	∈	∈	PROPN
ma-182	178	12	m1ω(g	m1ω(g	PROPN
ma-182	178	13	)	)	PUNCT
ma-182	178	14	or	or	CCONJ
ma-182	178	15	a	a	DET
ma-182	178	16	function	function	NOUN
ma-182	178	17	ϕ	ϕ	PROPN
ma-182	178	18	∈	∈	PROPN
ma-182	178	19	lpω(g	lpω(g	X
ma-182	178	20	)	)	PUNCT
ma-182	178	21	such	such	ADJ
ma-182	178	22	that	that	SCONJ
ma-182	178	23	th	th	X
ma-182	178	24	=	=	SYM
ma-182	178	25	ϕ	ϕ	PROPN
ma-182	178	26	∗ω	∗ω	PROPN
ma-182	178	27	h	h	NOUN
ma-182	178	28	for	for	ADP
ma-182	178	29	all	all	DET
ma-182	178	30	h	h	NOUN
ma-182	178	31	∈	∈	PROPN
ma-182	178	32	l1ω(g	l1ω(g	NOUN
ma-182	178	33	)	)	PUNCT
ma-182	178	34	.	.	PUNCT
ma-182	179	1	then	then	ADV
ma-182	179	2	,	,	PUNCT
ma-182	179	3	(	(	PUNCT
ma-182	179	4	tγsω)h	tγsω)h	PROPN
ma-182	179	5	=	=	SYM
ma-182	179	6	t	t	PROPN
ma-182	179	7	(	(	PUNCT
ma-182	179	8	γsωh	γsωh	ADV
ma-182	179	9	)	)	PUNCT
ma-182	179	10	=	=	PUNCT
ma-182	180	1	ϕ	ϕ	PROPN
ma-182	180	2	∗ω	∗ω	PROPN
ma-182	180	3	γsωh	γsωh	NOUN
ma-182	180	4	=	=	PUNCT
ma-182	180	5	γsω(ϕ	γsω(ϕ	PROPN
ma-182	180	6	∗ω	∗ω	PROPN
ma-182	180	7	h	h	NOUN
ma-182	180	8	)	)	PUNCT
ma-182	180	9	=	=	SYM
ma-182	180	10	γsω(th	γsω(th	ADJ
ma-182	180	11	)	)	PUNCT
ma-182	180	12	=	=	SYM
ma-182	180	13	(	(	PUNCT
ma-182	180	14	γsωt	γsωt	PROPN
ma-182	180	15	)	)	PUNCT
ma-182	181	1	h.	h.	PROPN
ma-182	181	2	thus	thus	ADV
ma-182	181	3	,	,	PUNCT
ma-182	181	4	t	t	PROPN
ma-182	181	5	∈m1,p	∈m1,p	PROPN
ma-182	181	6	ω	ω	PROPN
ma-182	181	7	(	(	PUNCT
ma-182	181	8	g).(4	g).(4	ADP
ma-182	181	9	)	)	PUNCT
ma-182	181	10	concerning	concern	VERB
ma-182	181	11	the	the	DET
ma-182	181	12	uniqueness	uniqueness	ADJ
ma-182	181	13	statement	statement	NOUN
ma-182	181	14	,	,	PUNCT
ma-182	181	15	let	let	VERB
ma-182	181	16	us	we	PRON
ma-182	181	17	consider	consider	VERB
ma-182	181	18	ϕ	ϕ	NOUN
ma-182	181	19	and	and	CCONJ
ma-182	181	20	ψ	ψ	X
ma-182	181	21	be	be	AUX
ma-182	181	22	such	such	ADJ
ma-182	181	23	that	that	SCONJ
ma-182	181	24	th	th	X
ma-182	181	25	=	=	PUNCT
ma-182	181	26	ϕ∗ω	ϕ∗ω	NUM
ma-182	181	27	h	h	NOUN
ma-182	181	28	=	=	SYM
ma-182	181	29	ψ	ψ	X
ma-182	181	30	∗ω	∗ω	PROPN
ma-182	181	31	h	h	NOUN
ma-182	181	32	for	for	ADP
ma-182	181	33	all	all	DET
ma-182	181	34	h	h	NOUN
ma-182	181	35	∈	∈	PROPN
ma-182	181	36	l1ω(g	l1ω(g	NOUN
ma-182	181	37	)	)	PUNCT
ma-182	181	38	.	.	PUNCT
ma-182	182	1	then	then	ADV
ma-182	182	2	,	,	PUNCT
ma-182	182	3	using	use	VERB
ma-182	182	4	the	the	DET
ma-182	182	5	fourier	fourier	NOUN
ma-182	182	6	transform	transform	NOUN
ma-182	182	7	,	,	PUNCT
ma-182	182	8	we	we	PRON
ma-182	182	9	obtain	obtain	VERB
ma-182	182	10	fω(ϕ)fω(h	fω(ϕ)fω(h	ADP
ma-182	182	11	)	)	PUNCT
ma-182	182	12	=	=	SYM
ma-182	183	1	fω(ψ)fω(h	fω(ψ)fω(h	PROPN
ma-182	183	2	)	)	PUNCT
ma-182	183	3	.	.	PUNCT
ma-182	184	1	so	so	ADV
ma-182	184	2	,	,	PUNCT
ma-182	184	3	fω(ϕ	fω(ϕ	NUM
ma-182	184	4	)	)	PUNCT
ma-182	184	5	=	=	SYM
ma-182	185	1	fω(ψ	fω(ψ	NOUN
ma-182	185	2	)	)	PUNCT
ma-182	185	3	.	.	PUNCT
ma-182	186	1	finally	finally	ADV
ma-182	186	2	,	,	PUNCT
ma-182	186	3	ϕ	ϕ	PROPN
ma-182	186	4	=	=	X
ma-182	186	5	ψ	ψ	NOUN
ma-182	186	6	by	by	ADP
ma-182	186	7	the	the	DET
ma-182	186	8	injectivity	injectivity	NOUN
ma-182	186	9	of	of	ADP
ma-182	186	10	the	the	DET
ma-182	186	11	fouriertransform	fouriertransform	NOUN
ma-182	186	12	.	.	PUNCT
ma-182	187	1	�	�	PROPN
ma-182	187	2	theorem	theorem	VERB
ma-182	187	3	3.4	3.4	NUM
ma-182	187	4	.	.	PUNCT
ma-182	188	1	let	let	VERB
ma-182	188	2	g	g	PRON
ma-182	188	3	be	be	AUX
ma-182	188	4	a	a	DET
ma-182	188	5	locally	locally	ADV
ma-182	188	6	compact	compact	ADJ
ma-182	188	7	abelian	abelian	ADJ
ma-182	188	8	group	group	NOUN
ma-182	188	9	.	.	PUNCT
ma-182	189	1	then	then	ADV
ma-182	189	2	,	,	PUNCT
ma-182	189	3	m1,1	m1,1	PROPN
ma-182	189	4	ω	ω	X
ma-182	189	5	(	(	PUNCT
ma-182	189	6	g	g	NOUN
ma-182	189	7	)	)	PUNCT
ma-182	189	8	is	be	AUX
ma-182	189	9	isometrically	isometrically	PROPN
ma-182	189	10	isomorphic	isomorphic	ADJ
ma-182	189	11	to	to	ADP
ma-182	189	12	m1ω(g	m1ω(g	PROPN
ma-182	189	13	)	)	PUNCT
ma-182	189	14	.	.	PUNCT
ma-182	190	1	proof	proof	NOUN
ma-182	190	2	.	.	PUNCT
ma-182	191	1	we	we	PRON
ma-182	191	2	have	have	AUX
ma-182	191	3	seen	see	VERB
ma-182	191	4	in	in	ADP
ma-182	191	5	theorem	theorem	ADJ
ma-182	191	6	3.3	3.3	NUM
ma-182	191	7	that	that	PRON
ma-182	191	8	t	t	PROPN
ma-182	191	9	∈m1,1	∈m1,1	NOUN
ma-182	191	10	ω	ω	PROPN
ma-182	191	11	(	(	PUNCT
ma-182	191	12	g	g	NOUN
ma-182	191	13	)	)	PUNCT
ma-182	192	1	if	if	SCONJ
ma-182	193	1	and	and	CCONJ
ma-182	193	2	only	only	ADV
ma-182	193	3	if	if	SCONJ
ma-182	193	4	there	there	PRON
ma-182	193	5	exists	exist	VERB
ma-182	193	6	a	a	DET
ma-182	193	7	unique	unique	ADJ
ma-182	193	8	measure	measure	NOUN
ma-182	193	9	µ	µ	PRON
ma-182	193	10	∈	∈	PROPN
ma-182	193	11	m1ω(g	m1ω(g	NOUN
ma-182	193	12	)	)	PUNCT
ma-182	193	13	such	such	ADJ
ma-182	193	14	that	that	SCONJ
ma-182	193	15	t	t	PROPN
ma-182	193	16	f	f	PROPN
ma-182	193	17	=	=	NOUN
ma-182	193	18	µ∗ω	µ∗ω	DET
ma-182	193	19	f	f	PROPN
ma-182	193	20	for	for	ADP
ma-182	193	21	all	all	DET
ma-182	193	22	f	f	PROPN
ma-182	193	23	∈	∈	PROPN
ma-182	193	24	l1ω(g	l1ω(g	NOUN
ma-182	193	25	)	)	PUNCT
ma-182	193	26	.	.	PUNCT
ma-182	194	1	then	then	ADV
ma-182	194	2	,	,	PUNCT
ma-182	194	3	the	the	DET
ma-182	194	4	mapping	mapping	NOUN
ma-182	194	5	t	t	PROPN
ma-182	194	6	7−→	7−→	PROPN
ma-182	194	7	µ	µ	NOUN
ma-182	194	8	defines	define	VERB
ma-182	194	9	a	a	DET
ma-182	194	10	bijectionfrom	bijectionfrom	NOUN
ma-182	194	11	m1,1	m1,1	NOUN
ma-182	194	12	ω	ω	NOUN
ma-182	194	13	(	(	PUNCT
ma-182	194	14	g	g	NOUN
ma-182	194	15	)	)	PUNCT
ma-182	194	16	)	)	PUNCT
ma-182	194	17	onto	onto	ADP
ma-182	194	18	m1ω(g	m1ω(g	PROPN
ma-182	194	19	)	)	PUNCT
ma-182	194	20	.	.	PUNCT
ma-182	195	1	moreover	moreover	ADV
ma-182	195	2	,	,	PUNCT
ma-182	195	3	‖t	‖t	PROPN
ma-182	195	4	f	f	PROPN
ma-182	195	5	‖1,ω	‖1,ω	NOUN
ma-182	195	6	=	=	SYM
ma-182	195	7	‖µ	‖µ	NOUN
ma-182	195	8	∗ω	∗ω	PROPN
ma-182	195	9	f	f	PROPN
ma-182	195	10	‖1,ω	‖1,ω	PROPN
ma-182	195	11	=	=	SYM
ma-182	195	12	∫	∫	PROPN
ma-182	196	1	g	g	PROPN
ma-182	196	2	∣∣∣∣∫	∣∣∣∣∫	PROPN
ma-182	196	3	g	g	PROPN
ma-182	197	1	f	f	PROPN
ma-182	197	2	(	(	PUNCT
ma-182	197	3	y−1x	y−1x	NOUN
ma-182	197	4	)	)	PUNCT
ma-182	197	5	ω(y−1x)ω(y	ω(y−1x)ω(y	NUM
ma-182	197	6	)	)	PUNCT
ma-182	197	7	ω(x	ω(x	NOUN
ma-182	197	8	)	)	PUNCT
ma-182	197	9	dµ(y	dµ(y	PUNCT
ma-182	197	10	)	)	PUNCT
ma-182	198	1	∣∣∣∣ω(x)dx	∣∣∣∣ω(x)dx	NUM
ma-182	198	2	6	6	NUM
ma-182	198	3	∫	∫	NOUN
ma-182	198	4	g	g	PROPN
ma-182	198	5	∫	∫	PROPN
ma-182	198	6	g	g	PROPN
ma-182	199	1	∣∣f	∣∣f	PROPN
ma-182	199	2	(	(	PUNCT
ma-182	199	3	y−1x	y−1x	NOUN
ma-182	199	4	)	)	PUNCT
ma-182	199	5	∣∣	∣∣	NUM
ma-182	199	6	ω(y−1x)ω(y	ω(y−1x)ω(y	NOUN
ma-182	199	7	)	)	PUNCT
ma-182	199	8	ω(x	ω(x	NOUN
ma-182	199	9	)	)	PUNCT
ma-182	199	10	ω(x)d	ω(x)d	PROPN
ma-182	199	11	|µ|(y)dx	|µ|(y)dx	NOUN
ma-182	199	12	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	199	13	eur	eur	PROPN
ma-182	199	14	.	.	PUNCT
ma-182	200	1	j.	j.	PROPN
ma-182	200	2	math	math	PROPN
ma-182	200	3	.	.	PUNCT
ma-182	201	1	anal	anal	PROPN
ma-182	201	2	.	.	PUNCT
ma-182	202	1	10.28924	10.28924	NUM
ma-182	202	2	/	/	SYM
ma-182	202	3	ada	ada	PROPN
ma-182	202	4	/	/	SYM
ma-182	202	5	ma.3.27	ma.3.27	PROPN
ma-182	202	6	8	8	NUM
ma-182	202	7	6	6	NUM
ma-182	202	8	(	(	PUNCT
ma-182	202	9	∫	∫	PROPN
ma-182	202	10	g	g	PROPN
ma-182	202	11	|f	|f	PROPN
ma-182	202	12	(	(	PUNCT
ma-182	202	13	x)|ω(x)dx	x)|ω(x)dx	PROPN
ma-182	202	14	)	)	PUNCT
ma-182	202	15	(	(	PUNCT
ma-182	202	16	∫	∫	PROPN
ma-182	202	17	g	g	PROPN
ma-182	202	18	ω(y)d	ω(y)d	PROPN
ma-182	202	19	|µ|(y	|µ|(y	NUM
ma-182	202	20	)	)	PUNCT
ma-182	202	21	)	)	PUNCT
ma-182	203	1	(	(	PUNCT
ma-182	203	2	invariance	invariance	NOUN
ma-182	203	3	of	of	ADP
ma-182	203	4	the	the	DET
ma-182	203	5	haar	haar	NOUN
ma-182	203	6	measure	measure	NOUN
ma-182	203	7	)	)	PUNCT
ma-182	203	8	6	6	NUM
ma-182	203	9	‖f	‖f	PRON
ma-182	203	10	‖1,ω‖µ‖ω	‖1,ω‖µ‖ω	NOUN
ma-182	203	11	.	.	PUNCT
ma-182	204	1	then	then	ADV
ma-182	204	2	,	,	PUNCT
ma-182	204	3	‖t‖1,ω	‖t‖1,ω	NOUN
ma-182	204	4	6	6	NUM
ma-182	204	5	‖µ‖ω	‖µ‖ω	ADJ
ma-182	204	6	.in	.in	PUNCT
ma-182	204	7	the	the	DET
ma-182	204	8	converse	converse	NOUN
ma-182	204	9	,	,	PUNCT
ma-182	204	10	for	for	ADP
ma-182	204	11	γ1	γ1	NOUN
ma-182	204	12	,	,	PUNCT
ma-182	204	13	·	·	PUNCT
ma-182	204	14	·	·	PUNCT
ma-182	204	15	·	·	PUNCT
ma-182	204	16	,	,	PUNCT
ma-182	204	17	γn	γn	NUM
ma-182	204	18	∈	∈	PROPN
ma-182	204	19	ĝ	ĝ	NOUN
ma-182	204	20	,	,	PUNCT
ma-182	204	21	z1	z1	NOUN
ma-182	204	22	,	,	PUNCT
ma-182	204	23	·	·	PUNCT
ma-182	204	24	·	·	PUNCT
ma-182	204	25	·	·	PUNCT
ma-182	204	26	,	,	PUNCT
ma-182	204	27	zn	zn	PROPN
ma-182	204	28	∈	∈	PROPN
ma-182	204	29	c	c	NOUN
ma-182	204	30	,	,	PUNCT
ma-182	204	31	and	and	CCONJ
ma-182	204	32	ε	ε	PROPN
ma-182	204	33	>	>	X
ma-182	204	34	0	0	PROPN
ma-182	204	35	,	,	PUNCT
ma-182	204	36	let	let	VERB
ma-182	204	37	us	we	PRON
ma-182	204	38	choose	choose	VERB
ma-182	204	39	f	f	PROPN
ma-182	204	40	∈	∈	PROPN
ma-182	204	41	l1ω(g	l1ω(g	PROPN
ma-182	204	42	)	)	PUNCT
ma-182	204	43	suchthat	suchthat	VERB
ma-182	204	44	‖fω(f	‖fω(f	PROPN
ma-182	204	45	)	)	PUNCT
ma-182	204	46	‖	‖	PROPN
ma-182	204	47	=	=	NOUN
ma-182	205	1	‖f	‖f	ADP
ma-182	205	2	‖1,ω	‖1,ω	NOUN
ma-182	205	3	<	<	X
ma-182	205	4	1	1	NUM
ma-182	205	5	+	+	CCONJ
ma-182	205	6	ε	ε	PROPN
ma-182	205	7	and	and	CCONJ
ma-182	205	8	fω(f	fω(f	NUM
ma-182	205	9	)	)	PUNCT
ma-182	205	10	(	(	PUNCT
ma-182	205	11	γi	γi	NOUN
ma-182	205	12	)	)	PUNCT
ma-182	205	13	=	=	SYM
ma-182	205	14	1	1	NUM
ma-182	205	15	,	,	PUNCT
ma-182	205	16	i	i	PRON
ma-182	205	17	=	=	NOUN
ma-182	205	18	1	1	NUM
ma-182	205	19	,	,	PUNCT
ma-182	205	20	2	2	NUM
ma-182	205	21	,	,	PUNCT
ma-182	205	22	3	3	NUM
ma-182	205	23	,	,	PUNCT
ma-182	205	24	.....	.....	PUNCT
ma-182	205	25	,	,	PUNCT
ma-182	205	26	n.	n.	PROPN
ma-182	205	27	then,∣∣∣∣∣	then,∣∣∣∣∣	PROPN
ma-182	205	28	n∑	n∑	PROPN
ma-182	205	29	i=1	i=1	PROPN
ma-182	205	30	zifω(µ)(γi	zifω(µ)(γi	NOUN
ma-182	205	31	)	)	PUNCT
ma-182	205	32	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ma-182	205	33	=	=	SYM
ma-182	206	1	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-182	206	2	n∑	n∑	PROPN
ma-182	206	3	i=1	i=1	PROPN
ma-182	206	4	zifω(µ)(γi)fω(f	zifω(µ)(γi)fω(f	NOUN
ma-182	206	5	)	)	PUNCT
ma-182	206	6	(	(	PUNCT
ma-182	206	7	γi	γi	NOUN
ma-182	206	8	)	)	PUNCT
ma-182	206	9	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-182	206	10	=	=	SYM
ma-182	207	1	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-182	207	2	n∑	n∑	PROPN
ma-182	208	1	i=1	i=1	PROPN
ma-182	208	2	zifω(µ	zifω(µ	PROPN
ma-182	208	3	∗ω	∗ω	PROPN
ma-182	208	4	f	f	PROPN
ma-182	208	5	)	)	PUNCT
ma-182	208	6	(	(	PUNCT
ma-182	208	7	γi	γi	NOUN
ma-182	208	8	)	)	PUNCT
ma-182	208	9	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-182	208	10	=	=	SYM
ma-182	209	1	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-182	209	2	n∑	n∑	PROPN
ma-182	210	1	i=1	i=1	PROPN
ma-182	210	2	zifω(t	zifω(t	NUM
ma-182	210	3	f	f	NOUN
ma-182	210	4	)	)	PUNCT
ma-182	210	5	(	(	PUNCT
ma-182	210	6	γi	γi	NOUN
ma-182	210	7	)	)	PUNCT
ma-182	210	8	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-182	210	9	6	6	NUM
ma-182	210	10	‖t‖(1	‖t‖(1	NOUN
ma-182	210	11	+	+	CCONJ
ma-182	210	12	ε	ε	PROPN
ma-182	210	13	)	)	PUNCT
ma-182	210	14	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	211	1	n∑	n∑	PROPN
ma-182	211	2	i=1	i=1	PROPN
ma-182	211	3	ziγi	ziγi	PROPN
ma-182	211	4	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-182	211	5	∞	∞	PROPN
ma-182	211	6	.	.	PUNCT
ma-182	212	1	since	since	SCONJ
ma-182	212	2	ε	ε	PROPN
ma-182	212	3	is	be	AUX
ma-182	212	4	arbitrary	arbitrary	ADJ
ma-182	212	5	,	,	PUNCT
ma-182	212	6	then	then	ADV
ma-182	212	7	‖t‖	‖t‖	PROPN
ma-182	212	8	>	>	X
ma-182	212	9	‖µ‖ω	‖µ‖ω	X
ma-182	212	10	by	by	ADP
ma-182	212	11	the	the	DET
ma-182	212	12	use	use	NOUN
ma-182	212	13	of	of	ADP
ma-182	212	14	theorem	theorem	ADJ
ma-182	212	15	2.1	2.1	NUM
ma-182	212	16	applied	apply	VERB
ma-182	212	17	with	with	ADP
ma-182	212	18	fω	fω	PRON
ma-182	212	19	instead	instead	ADV
ma-182	212	20	of	of	ADP
ma-182	212	21	f	f	PROPN
ma-182	212	22	.	.	PUNCT
ma-182	213	1	�	�	PROPN
ma-182	213	2	theorem	theorem	VERB
ma-182	213	3	3.5	3.5	NUM
ma-182	213	4	.	.	PUNCT
ma-182	214	1	let	let	VERB
ma-182	214	2	g	g	PRON
ma-182	214	3	be	be	AUX
ma-182	214	4	a	a	DET
ma-182	214	5	locally	locally	ADV
ma-182	214	6	compact	compact	ADJ
ma-182	214	7	abelian	abelian	NOUN
ma-182	214	8	group	group	NOUN
ma-182	214	9	.	.	PUNCT
ma-182	215	1	let	let	VERB
ma-182	215	2	1	1	NUM
ma-182	215	3	6	6	NUM
ma-182	215	4	p	p	NOUN
ma-182	215	5	<	<	X
ma-182	215	6	∞.	∞.	PROPN
ma-182	215	7	if	if	SCONJ
ma-182	215	8	f	f	PROPN
ma-182	215	9	∈	∈	PROPN
ma-182	215	10	lpω(g	lpω(g	PROPN
ma-182	215	11	)	)	PUNCT
ma-182	215	12	,	,	PUNCT
ma-182	215	13	then	then	ADV
ma-182	215	14	the	the	DET
ma-182	215	15	mapping	mapping	NOUN
ma-182	215	16	s	s	PART
ma-182	215	17	7−→	7−→	NOUN
ma-182	215	18	γsωf	γsωf	NOUN
ma-182	215	19	is	be	AUX
ma-182	215	20	continuous	continuous	ADJ
ma-182	215	21	from	from	ADP
ma-182	215	22	g	g	NOUN
ma-182	215	23	into	into	ADP
ma-182	215	24	lpω(g	lpω(g	NOUN
ma-182	215	25	)	)	PUNCT
ma-182	215	26	.	.	PUNCT
ma-182	216	1	proof	proof	NOUN
ma-182	216	2	.	.	PUNCT
ma-182	217	1	the	the	DET
ma-182	217	2	set	set	NOUN
ma-182	217	3	of	of	ADP
ma-182	217	4	complex	complex	ADJ
ma-182	217	5	continuous	continuous	ADJ
ma-182	217	6	functions	function	NOUN
ma-182	217	7	on	on	ADP
ma-182	217	8	g	g	NOUN
ma-182	217	9	with	with	ADP
ma-182	217	10	compact	compact	ADJ
ma-182	217	11	support	support	NOUN
ma-182	217	12	cc(g	cc(g	PUNCT
ma-182	217	13	)	)	PUNCT
ma-182	217	14	is	be	AUX
ma-182	217	15	dense	dense	ADJ
ma-182	217	16	in	in	ADP
ma-182	217	17	lpω(g)under	lpω(g)under	NOUN
ma-182	217	18	the	the	DET
ma-182	217	19	norm	norm	NOUN
ma-182	217	20	‖	‖	PROPN
ma-182	217	21	·	·	PUNCT
ma-182	217	22	‖p	‖p	PROPN
ma-182	217	23	,	,	PUNCT
ma-182	217	24	ω	ω	PROPN
ma-182	217	25	.	.	PUNCT
ma-182	218	1	let	let	VERB
ma-182	218	2	ε	ε	PROPN
ma-182	218	3	>	>	X
ma-182	218	4	0	0	X
ma-182	218	5	.	.	PUNCT
ma-182	219	1	consider	consider	VERB
ma-182	219	2	g	g	PROPN
ma-182	219	3	∈	∈	PROPN
ma-182	219	4	cc(g	cc(g	NOUN
ma-182	219	5	)	)	PUNCT
ma-182	219	6	and	and	CCONJ
ma-182	219	7	set	set	VERB
ma-182	219	8	c1	c1	NOUN
ma-182	219	9	=	=	PROPN
ma-182	219	10	supp(g	supp(g	PROPN
ma-182	219	11	)	)	PUNCT
ma-182	219	12	.	.	PUNCT
ma-182	220	1	let	let	VERB
ma-182	220	2	us	we	PRON
ma-182	220	3	choose	choose	VERB
ma-182	220	4	acompact	acompact	ADJ
ma-182	220	5	neighborhood	neighborhood	NOUN
ma-182	220	6	c2	c2	PROPN
ma-182	220	7	of	of	ADP
ma-182	220	8	the	the	DET
ma-182	220	9	neutral	neutral	ADJ
ma-182	220	10	element	element	NOUN
ma-182	220	11	e	e	PROPN
ma-182	220	12	.	.	PUNCT
ma-182	221	1	set	set	VERB
ma-182	221	2	c	c	PROPN
ma-182	221	3	=	=	SYM
ma-182	221	4	c1∪c2∪	c1∪c2∪	PROPN
ma-182	221	5	(	(	PUNCT
ma-182	221	6	c1c2	c1c2	NOUN
ma-182	221	7	)	)	PUNCT
ma-182	221	8	.	.	PUNCT
ma-182	222	1	we	we	PRON
ma-182	222	2	have	have	VERB
ma-182	222	3	for	for	ADP
ma-182	222	4	s	s	PROPN
ma-182	222	5	∈	∈	PROPN
ma-182	222	6	c2	c2	PROPN
ma-182	222	7	,	,	PUNCT
ma-182	222	8	‖γsωg	‖γsωg	NOUN
ma-182	222	9	−	−	PROPN
ma-182	222	10	g‖pp	g‖pp	PROPN
ma-182	222	11	,	,	PUNCT
ma-182	222	12	ω	ω	PROPN
ma-182	222	13	=	=	SYM
ma-182	222	14	∫	∫	PROPN
ma-182	223	1	c	c	PROPN
ma-182	223	2	|γsωg(x)−	|γsωg(x)−	PROPN
ma-182	224	1	g(x)|pω(x)dx	g(x)|pω(x)dx	PROPN
ma-182	224	2	6	6	NUM
ma-182	224	3	∫	∫	NOUN
ma-182	224	4	c	c	PROPN
ma-182	224	5	|g(s−1x)ω(s−1x)−	|g(s−1x)ω(s−1x)−	PROPN
ma-182	224	6	g(x)ω(x)|pdx	g(x)ω(x)|pdx	PROPN
ma-182	224	7	.	.	PUNCT
ma-182	225	1	the	the	DET
ma-182	225	2	mapping	mapping	NOUN
ma-182	225	3	x	x	SYM
ma-182	225	4	7−→	7−→	NOUN
ma-182	225	5	(	(	PUNCT
ma-182	225	6	gω)(x	gω)(x	PROPN
ma-182	225	7	)	)	PUNCT
ma-182	225	8	is	be	AUX
ma-182	225	9	uniformly	uniformly	ADV
ma-182	225	10	continuous	continuous	ADJ
ma-182	225	11	on	on	ADP
ma-182	225	12	g.	g.	PROPN
ma-182	225	13	thus	thus	ADV
ma-182	225	14	,	,	PUNCT
ma-182	225	15	there	there	PRON
ma-182	225	16	exists	exist	VERB
ma-182	225	17	a	a	DET
ma-182	225	18	neighborhood	neighborhood	NOUN
ma-182	225	19	u	u	NOUN
ma-182	225	20	of	of	ADP
ma-182	225	21	e	e	PRON
ma-182	225	22	which	which	PRON
ma-182	225	23	we	we	PRON
ma-182	225	24	may	may	AUX
ma-182	225	25	assume	assume	VERB
ma-182	225	26	to	to	PART
ma-182	225	27	be	be	AUX
ma-182	225	28	contained	contain	VERB
ma-182	225	29	in	in	ADP
ma-182	225	30	c2	c2	PROPN
ma-182	225	31	,	,	PUNCT
ma-182	225	32	such	such	ADJ
ma-182	225	33	that	that	SCONJ
ma-182	225	34	∀s	∀s	PROPN
ma-182	225	35	∈	∈	PROPN
ma-182	225	36	u	u	PROPN
ma-182	225	37	,	,	PUNCT
ma-182	225	38	|(gω)(s−1x)−	|(gω)(s−1x)−	PROPN
ma-182	225	39	(	(	PUNCT
ma-182	225	40	gω)(x)|p	gω)(x)|p	PROPN
ma-182	225	41	<	<	X
ma-182	225	42	εp	εp	ADP
ma-182	225	43	|c|where	|c|where	ADJ
ma-182	225	44	|c|	|c|	PROPN
ma-182	225	45	is	be	AUX
ma-182	225	46	the	the	DET
ma-182	225	47	measure	measure	NOUN
ma-182	225	48	of	of	ADP
ma-182	225	49	the	the	DET
ma-182	225	50	compact	compact	ADJ
ma-182	225	51	set	set	NOUN
ma-182	225	52	c.	c.	NOUN
ma-182	225	53	then	then	ADV
ma-182	225	54	,	,	PUNCT
ma-182	225	55	for	for	ADP
ma-182	225	56	s	s	PROPN
ma-182	225	57	∈	∈	PROPN
ma-182	225	58	u	u	NOUN
ma-182	225	59	,	,	PUNCT
ma-182	225	60	we	we	PRON
ma-182	225	61	have	have	VERB
ma-182	225	62	‖γsωg	‖γsωg	NOUN
ma-182	225	63	−	−	PROPN
ma-182	225	64	g‖pp	g‖pp	PROPN
ma-182	225	65	,	,	PUNCT
ma-182	225	66	ω	ω	PROPN
ma-182	225	67	6	6	NUM
ma-182	225	68	∫	∫	PROPN
ma-182	225	69	c	c	PROPN
ma-182	225	70	|(gω)(s−1x)−	|(gω)(s−1x)−	PROPN
ma-182	225	71	(	(	PUNCT
ma-182	225	72	gω)(x)|pdx	gω)(x)|pdx	PROPN
ma-182	225	73	<	<	X
ma-182	225	74	ε|c|	ε|c|	NOUN
ma-182	225	75	|c|	|c|	PROPN
ma-182	225	76	=	=	SYM
ma-182	225	77	ε	ε	PROPN
ma-182	225	78	.	.	PUNCT
ma-182	226	1	we	we	PRON
ma-182	226	2	will	will	AUX
ma-182	226	3	show	show	VERB
ma-182	226	4	the	the	DET
ma-182	226	5	claim	claim	NOUN
ma-182	226	6	for	for	ADP
ma-182	226	7	f	f	PROPN
ma-182	226	8	∈	∈	PROPN
ma-182	226	9	lpω(g	lpω(g	PROPN
ma-182	226	10	)	)	PUNCT
ma-182	226	11	.	.	PUNCT
ma-182	227	1	let	let	VERB
ma-182	227	2	k	k	PRON
ma-182	227	3	be	be	AUX
ma-182	227	4	a	a	DET
ma-182	227	5	compact	compact	ADJ
ma-182	227	6	neighborhood	neighborhood	NOUN
ma-182	227	7	of	of	ADP
ma-182	227	8	e	e	PROPN
ma-182	227	9	.	.	PUNCT
ma-182	228	1	since	since	SCONJ
ma-182	228	2	cc(g	cc(g	NOUN
ma-182	228	3	)	)	PUNCT
ma-182	228	4	isdense	isdense	NOUN
ma-182	228	5	in	in	ADP
ma-182	228	6	lpω(g	lpω(g	NOUN
ma-182	228	7	)	)	PUNCT
ma-182	228	8	,	,	PUNCT
ma-182	228	9	then	then	ADV
ma-182	228	10	there	there	PRON
ma-182	228	11	exists	exist	VERB
ma-182	228	12	g	g	PROPN
ma-182	228	13	∈	∈	PROPN
ma-182	228	14	cc(g	cc(g	NOUN
ma-182	228	15	)	)	PUNCT
ma-182	228	16	such	such	ADJ
ma-182	228	17	that	that	SCONJ
ma-182	228	18	‖f	‖f	ADP
ma-182	228	19	−	−	NOUN
ma-182	228	20	g‖p	g‖p	NOUN
ma-182	228	21	,	,	PUNCT
ma-182	228	22	ω	ω	X
ma-182	228	23	<	<	X
ma-182	228	24	ε	ε	PROPN
ma-182	228	25	3	3	NUM
ma-182	228	26	.	.	PUNCT
ma-182	229	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	229	2	eur	eur	PROPN
ma-182	229	3	.	.	PUNCT
ma-182	230	1	j.	j.	PROPN
ma-182	230	2	math	math	PROPN
ma-182	230	3	.	.	PUNCT
ma-182	231	1	anal	anal	PROPN
ma-182	231	2	.	.	PUNCT
ma-182	232	1	10.28924	10.28924	NUM
ma-182	232	2	/	/	SYM
ma-182	232	3	ada	ada	PROPN
ma-182	232	4	/	/	SYM
ma-182	232	5	ma.3.27	ma.3.27	PROPN
ma-182	232	6	9there	9there	NUM
ma-182	232	7	exists	exist	VERB
ma-182	232	8	a	a	DET
ma-182	232	9	compact	compact	ADJ
ma-182	232	10	neighborhood	neighborhood	NOUN
ma-182	232	11	v	v	NOUN
ma-182	232	12	of	of	ADP
ma-182	232	13	e	e	X
ma-182	232	14	which	which	PRON
ma-182	232	15	we	we	PRON
ma-182	232	16	may	may	AUX
ma-182	232	17	assume	assume	VERB
ma-182	232	18	to	to	PART
ma-182	232	19	be	be	AUX
ma-182	232	20	contained	contain	VERB
ma-182	232	21	in	in	ADP
ma-182	232	22	k	k	PROPN
ma-182	232	23	,	,	PUNCT
ma-182	232	24	such	such	ADJ
ma-182	232	25	that	that	SCONJ
ma-182	232	26	‖γsωg	‖γsωg	NOUN
ma-182	232	27	−	−	PROPN
ma-182	232	28	g‖p	g‖p	NOUN
ma-182	232	29	,	,	PUNCT
ma-182	232	30	ω	ω	X
ma-182	232	31	<	<	X
ma-182	232	32	ε	ε	PROPN
ma-182	232	33	3	3	NUM
ma-182	232	34	for	for	ADP
ma-182	232	35	all	all	PRON
ma-182	232	36	s	s	PART
ma-182	232	37	∈	∈	NOUN
ma-182	232	38	v	v	NOUN
ma-182	232	39	.then	.then	NOUN
ma-182	232	40	,	,	PUNCT
ma-182	232	41	for	for	ADP
ma-182	232	42	s	s	PROPN
ma-182	232	43	∈	∈	PROPN
ma-182	232	44	v	v	NOUN
ma-182	232	45	,	,	PUNCT
ma-182	232	46	we	we	PRON
ma-182	232	47	have	have	AUX
ma-182	232	48	‖γsωf	‖γsωf	ADJ
ma-182	232	49	−	−	PROPN
ma-182	232	50	f	f	PROPN
ma-182	232	51	‖p	‖p	PROPN
ma-182	232	52	,	,	PUNCT
ma-182	232	53	ω	ω	PROPN
ma-182	232	54	6	6	NUM
ma-182	232	55	‖γsωf	‖γsωf	VERB
ma-182	232	56	−	−	PRON
ma-182	232	57	γsωg‖p	γsωg‖p	NUM
ma-182	232	58	,	,	PUNCT
ma-182	232	59	ω	ω	PROPN
ma-182	232	60	+	+	CCONJ
ma-182	232	61	‖γsωg	‖γsωg	NOUN
ma-182	232	62	−	−	PRON
ma-182	232	63	g‖p	g‖p	NOUN
ma-182	232	64	,	,	PUNCT
ma-182	232	65	ω	ω	PROPN
ma-182	232	66	+	+	CCONJ
ma-182	232	67	‖f	‖f	ADP
ma-182	232	68	−	−	NOUN
ma-182	232	69	g‖p	g‖p	NOUN
ma-182	232	70	,	,	PUNCT
ma-182	232	71	ω	ω	X
ma-182	232	72	<	<	X
ma-182	232	73	1	1	NUM
ma-182	232	74	ω(s	ω(s	PROPN
ma-182	232	75	)	)	PUNCT
ma-182	232	76	∫	∫	PROPN
ma-182	232	77	g	g	PROPN
ma-182	232	78	|(f	|(f	PROPN
ma-182	233	1	−	−	PROPN
ma-182	233	2	g)(t)|pω(st)dt	g)(t)|pω(st)dt	PROPN
ma-182	233	3	+	+	CCONJ
ma-182	233	4	ε	ε	PROPN
ma-182	233	5	3	3	NUM
ma-182	233	6	+	+	CCONJ
ma-182	233	7	ε	ε	PROPN
ma-182	233	8	3	3	NUM
ma-182	233	9	6	6	NUM
ma-182	233	10	1	1	NUM
ma-182	233	11	ω(s	ω(s	NUM
ma-182	233	12	)	)	PUNCT
ma-182	233	13	∫	∫	PROPN
ma-182	233	14	g	g	PROPN
ma-182	233	15	|(f	|(f	PROPN
ma-182	233	16	−	−	PROPN
ma-182	233	17	g)(t)|pω(s)ω(t)dt	g)(t)|pω(s)ω(t)dt	NOUN
ma-182	233	18	+	+	CCONJ
ma-182	233	19	ε	ε	PROPN
ma-182	233	20	3	3	NUM
ma-182	233	21	+	+	CCONJ
ma-182	233	22	ε	ε	PROPN
ma-182	233	23	3	3	NUM
ma-182	233	24	<	<	X
ma-182	233	25	∫	∫	PROPN
ma-182	233	26	g	g	PROPN
ma-182	233	27	|(f	|(f	PROPN
ma-182	233	28	−	−	PROPN
ma-182	233	29	g)(t)|pω(t)dt	g)(t)|pω(t)dt	PROPN
ma-182	233	30	+	+	CCONJ
ma-182	233	31	ε	ε	PROPN
ma-182	233	32	3	3	NUM
ma-182	233	33	+	+	CCONJ
ma-182	233	34	ε	ε	PROPN
ma-182	233	35	3	3	NUM
ma-182	233	36	‖f	‖f	PRON
ma-182	233	37	−	−	NOUN
ma-182	233	38	g‖p	g‖p	NOUN
ma-182	233	39	,	,	PUNCT
ma-182	233	40	ω	ω	PROPN
ma-182	233	41	+	+	CCONJ
ma-182	233	42	ε	ε	PROPN
ma-182	233	43	3	3	NUM
ma-182	233	44	+	+	CCONJ
ma-182	233	45	ε	ε	PROPN
ma-182	233	46	3	3	NUM
ma-182	233	47	=	=	SYM
ma-182	233	48	ε	ε	PROPN
ma-182	233	49	3	3	NUM
ma-182	233	50	+	+	CCONJ
ma-182	233	51	ε	ε	PROPN
ma-182	233	52	3	3	NUM
ma-182	233	53	+	+	CCONJ
ma-182	233	54	ε	ε	PROPN
ma-182	233	55	3	3	NUM
ma-182	233	56	=	=	SYM
ma-182	233	57	ε	ε	PROPN
ma-182	233	58	.	.	PUNCT
ma-182	233	59	�	�	PROPN
ma-182	233	60	theorem	theorem	VERB
ma-182	233	61	3.6	3.6	NUM
ma-182	233	62	.	.	PUNCT
ma-182	234	1	let	let	VERB
ma-182	234	2	g	g	PRON
ma-182	234	3	be	be	AUX
ma-182	234	4	a	a	DET
ma-182	234	5	locally	locally	ADV
ma-182	234	6	compact	compact	ADJ
ma-182	234	7	abelian	abelian	NOUN
ma-182	234	8	group	group	NOUN
ma-182	234	9	.	.	PUNCT
ma-182	235	1	let	let	VERB
ma-182	235	2	f	f	PRON
ma-182	235	3	∈	∈	PROPN
ma-182	235	4	lpω(g	lpω(g	PROPN
ma-182	235	5	)	)	PUNCT
ma-182	235	6	,	,	PUNCT
ma-182	235	7	1	1	NUM
ma-182	235	8	6	6	NUM
ma-182	235	9	p	p	NOUN
ma-182	235	10	<	<	X
ma-182	235	11	∞.	∞.	PROPN
ma-182	235	12	let	let	VERB
ma-182	235	13	ε	ε	PROPN
ma-182	235	14	>	>	X
ma-182	235	15	0	0	PROPN
ma-182	235	16	.	.	PUNCT
ma-182	236	1	then	then	ADV
ma-182	236	2	,	,	PUNCT
ma-182	236	3	there	there	PRON
ma-182	236	4	exists	exist	VERB
ma-182	236	5	a	a	DET
ma-182	236	6	positive	positive	ADJ
ma-182	236	7	function	function	NOUN
ma-182	236	8	g	g	PROPN
ma-182	236	9	∈	∈	PROPN
ma-182	236	10	cc(g	cc(g	NOUN
ma-182	236	11	)	)	PUNCT
ma-182	236	12	such	such	ADJ
ma-182	236	13	that	that	DET
ma-182	236	14	‖g‖1,ω	‖g‖1,ω	PROPN
ma-182	236	15	=	=	SYM
ma-182	236	16	1	1	NUM
ma-182	236	17	and	and	CCONJ
ma-182	236	18	‖f	‖f	ADP
ma-182	236	19	∗ω	∗ω	NOUN
ma-182	236	20	g	g	NOUN
ma-182	236	21	−	−	PROPN
ma-182	236	22	f	f	PROPN
ma-182	236	23	‖p	‖p	PROPN
ma-182	236	24	,	,	PUNCT
ma-182	236	25	ω	ω	PROPN
ma-182	236	26	6	6	NUM
ma-182	236	27	ε	ε	PROPN
ma-182	236	28	.	.	PUNCT
ma-182	237	1	proof	proof	NOUN
ma-182	237	2	.	.	PUNCT
ma-182	238	1	let	let	VERB
ma-182	238	2	f	f	PRON
ma-182	238	3	∈	∈	PROPN
ma-182	238	4	lpω(g	lpω(g	X
ma-182	238	5	)	)	PUNCT
ma-182	238	6	and	and	CCONJ
ma-182	238	7	ε	ε	X
ma-182	238	8	>	>	X
ma-182	238	9	0	0	PROPN
ma-182	238	10	.	.	PUNCT
ma-182	239	1	according	accord	VERB
ma-182	239	2	to	to	ADP
ma-182	239	3	theorem	theorem	ADJ
ma-182	239	4	3.5	3.5	NUM
ma-182	239	5	,	,	PUNCT
ma-182	239	6	the	the	DET
ma-182	239	7	mapping	mapping	NOUN
ma-182	239	8	s	s	PART
ma-182	239	9	7−→	7−→	NOUN
ma-182	239	10	γsωf	γsωf	NOUN
ma-182	239	11	is	be	AUX
ma-182	239	12	continuousat	continuousat	VERB
ma-182	239	13	the	the	DET
ma-182	239	14	neutral	neutral	ADJ
ma-182	239	15	element	element	NOUN
ma-182	239	16	e	e	PROPN
ma-182	239	17	of	of	ADP
ma-182	239	18	g.	g.	PROPN
ma-182	239	19	then	then	ADV
ma-182	239	20	,	,	PUNCT
ma-182	239	21	there	there	PRON
ma-182	239	22	exists	exist	VERB
ma-182	239	23	a	a	DET
ma-182	239	24	compact	compact	ADJ
ma-182	239	25	neighborhood	neighborhood	NOUN
ma-182	239	26	k	k	NOUN
ma-182	239	27	of	of	ADP
ma-182	239	28	e	e	PRON
ma-182	239	29	such	such	ADJ
ma-182	239	30	that	that	SCONJ
ma-182	239	31	‖γsωf	‖γsωf	ADJ
ma-182	239	32	−	−	PROPN
ma-182	239	33	f	f	PROPN
ma-182	239	34	‖p	‖p	PROPN
ma-182	239	35	,	,	PUNCT
ma-182	239	36	ω	ω	PROPN
ma-182	239	37	6	6	NUM
ma-182	239	38	ε	ε	PROPN
ma-182	239	39	,	,	PUNCT
ma-182	239	40	∀s	∀s	PROPN
ma-182	239	41	∈	∈	PROPN
ma-182	239	42	k.	k.	PROPN
ma-182	239	43	consider	consider	VERB
ma-182	239	44	a	a	DET
ma-182	239	45	positive	positive	ADJ
ma-182	239	46	function	function	NOUN
ma-182	239	47	g	g	ADP
ma-182	239	48	such	such	DET
ma-182	239	49	that	that	DET
ma-182	239	50	supp(g	supp(g	NOUN
ma-182	239	51	)	)	PUNCT
ma-182	240	1	⊂	⊂	PROPN
ma-182	240	2	k	k	PROPN
ma-182	240	3	and	and	CCONJ
ma-182	240	4	∫	∫	PROPN
ma-182	241	1	g	g	PROPN
ma-182	241	2	g(y)ω(y)dy	g(y)ω(y)dy	NOUN
ma-182	241	3	=	=	SYM
ma-182	241	4	1	1	NUM
ma-182	241	5	(	(	PUNCT
ma-182	241	6	that	that	PRON
ma-182	241	7	is	be	AUX
ma-182	241	8	‖g‖1,ω	‖g‖1,ω	PROPN
ma-182	241	9	=	=	SYM
ma-182	241	10	1	1	NUM
ma-182	241	11	)	)	PUNCT
ma-182	241	12	.	.	PUNCT
ma-182	242	1	then	then	ADV
ma-182	242	2	,	,	PUNCT
ma-182	242	3	|(f	|(f	PROPN
ma-182	242	4	∗ω	∗ω	PROPN
ma-182	242	5	g)(x	g)(x	PROPN
ma-182	242	6	)	)	PUNCT
ma-182	243	1	−	−	PROPN
ma-182	243	2	f	f	X
ma-182	243	3	(	(	PUNCT
ma-182	243	4	x)|	x)|	PROPN
ma-182	243	5	6	6	NUM
ma-182	243	6	∫	∫	NOUN
ma-182	243	7	g	g	PROPN
ma-182	243	8	|γsωf	|γsωf	ADJ
ma-182	243	9	(	(	PUNCT
ma-182	243	10	x	x	NOUN
ma-182	243	11	)	)	PUNCT
ma-182	243	12	−	−	PROPN
ma-182	244	1	f	f	PROPN
ma-182	244	2	(	(	PUNCT
ma-182	244	3	x)|g(s)ω(s)ds	x)|g(s)ω(s)ds	PROPN
ma-182	244	4	.	.	PUNCT
ma-182	245	1	using	use	VERB
ma-182	245	2	the	the	DET
ma-182	245	3	hölder	hölder	NOUN
ma-182	245	4	’s	’s	PART
ma-182	245	5	inequality	inequality	NOUN
ma-182	245	6	withrespect	withrespect	ADJ
ma-182	245	7	to	to	ADP
ma-182	245	8	the	the	DET
ma-182	245	9	measure	measure	NOUN
ma-182	245	10	g(s)ω(s)ds	g(s)ω(s)d	VERB
ma-182	245	11	,	,	PUNCT
ma-182	245	12	one	one	PRON
ma-182	245	13	has	have	VERB
ma-182	245	14	|(f	|(f	PROPN
ma-182	245	15	∗ω	∗ω	PROPN
ma-182	245	16	g)(x)−	g)(x)−	PROPN
ma-182	245	17	f	f	PROPN
ma-182	246	1	(	(	PUNCT
ma-182	246	2	x)|	x)|	PROPN
ma-182	246	3	6	6	NUM
ma-182	246	4	(	(	PUNCT
ma-182	246	5	∫	∫	PROPN
ma-182	246	6	g	g	PROPN
ma-182	246	7	|γsωf	|γsωf	PROPN
ma-182	246	8	(	(	PUNCT
ma-182	246	9	x)−	x)−	PROPN
ma-182	246	10	f	f	PROPN
ma-182	246	11	(	(	PUNCT
ma-182	246	12	x)|pg(s)ω(s)ds	x)|pg(s)ω(s)ds	PROPN
ma-182	246	13	)	)	PUNCT
ma-182	246	14	1	1	NUM
ma-182	246	15	p	p	NOUN
ma-182	246	16	(	(	PUNCT
ma-182	246	17	∫	∫	PROPN
ma-182	246	18	g	g	PROPN
ma-182	246	19	g(s)ω(s)ds	g(s)ω(s)ds	PROPN
ma-182	246	20	)	)	PUNCT
ma-182	246	21	1	1	NUM
ma-182	246	22	q	q	NOUN
ma-182	246	23	6	6	NUM
ma-182	246	24	(	(	PUNCT
ma-182	246	25	∫	∫	PROPN
ma-182	246	26	g	g	PROPN
ma-182	246	27	|γsωf	|γsωf	PROPN
ma-182	246	28	(	(	PUNCT
ma-182	246	29	x)−	x)−	PROPN
ma-182	246	30	f	f	PROPN
ma-182	246	31	(	(	PUNCT
ma-182	246	32	x)|pg(s)ω(s)ds	x)|pg(s)ω(s)ds	PROPN
ma-182	246	33	)	)	PUNCT
ma-182	246	34	1	1	NUM
ma-182	246	35	p	p	NOUN
ma-182	246	36	,	,	PUNCT
ma-182	246	37	where	where	SCONJ
ma-182	246	38	q	q	NOUN
ma-182	246	39	is	be	AUX
ma-182	246	40	such	such	ADJ
ma-182	246	41	that	that	SCONJ
ma-182	246	42	1	1	NUM
ma-182	246	43	p	p	NOUN
ma-182	246	44	+	+	NOUN
ma-182	246	45	1	1	NUM
ma-182	246	46	q	q	NOUN
ma-182	246	47	=	=	ADJ
ma-182	246	48	1	1	X
ma-182	246	49	.	.	PUNCT
ma-182	247	1	then	then	ADV
ma-182	247	2	,	,	PUNCT
ma-182	247	3	‖f	‖f	ADP
ma-182	247	4	∗ω	∗ω	NOUN
ma-182	247	5	g	g	NOUN
ma-182	247	6	−	−	PROPN
ma-182	247	7	f	f	PROPN
ma-182	247	8	‖pp	‖pp	PROPN
ma-182	247	9	,	,	PUNCT
ma-182	247	10	ω	ω	PROPN
ma-182	247	11	=	=	SYM
ma-182	247	12	∫	∫	PROPN
ma-182	247	13	g	g	PROPN
ma-182	247	14	|(f	|(f	PROPN
ma-182	247	15	∗ω	∗ω	PROPN
ma-182	247	16	g)(x)−	g)(x)−	PROPN
ma-182	247	17	f	f	PROPN
ma-182	247	18	(	(	PUNCT
ma-182	247	19	x)|pω(x)dx	x)|pω(x)dx	PROPN
ma-182	247	20	6	6	NUM
ma-182	247	21	∫∫	∫∫	PROPN
ma-182	247	22	g×g	g×g	PROPN
ma-182	247	23	|γsωf	|γsωf	NOUN
ma-182	247	24	(	(	PUNCT
ma-182	247	25	x)−	x)−	PROPN
ma-182	247	26	f	f	PROPN
ma-182	247	27	(	(	PUNCT
ma-182	247	28	x)|pg(s)ω(s)dsω(x)dx	x)|pg(s)ω(s)dsω(x)dx	PROPN
ma-182	247	29	6	6	NUM
ma-182	247	30	∫	∫	NOUN
ma-182	247	31	g	g	PROPN
ma-182	247	32	‖γsωf	‖γsωf	VERB
ma-182	247	33	−	−	PROPN
ma-182	247	34	f	f	PROPN
ma-182	247	35	‖pp	‖pp	NUM
ma-182	247	36	,	,	PUNCT
ma-182	247	37	ωg(s)ω(s)ds	ωg(s)ω(s)ds	NUM
ma-182	247	38	=	=	SYM
ma-182	247	39	‖γsωf	‖γsωf	ADP
ma-182	247	40	−	−	PROPN
ma-182	247	41	f	f	PROPN
ma-182	247	42	‖pp	‖pp	PROPN
ma-182	247	43	,	,	PUNCT
ma-182	247	44	ω	ω	NUM
ma-182	247	45	∫	∫	PROPN
ma-182	247	46	g	g	PROPN
ma-182	247	47	g(s)ω(s)ds	g(s)ω(s)ds	PROPN
ma-182	247	48	=	=	SYM
ma-182	247	49	‖γsωf	‖γsωf	VERB
ma-182	247	50	−	−	PROPN
ma-182	247	51	f	f	PROPN
ma-182	247	52	‖pp	‖pp	PROPN
ma-182	247	53	,	,	PUNCT
ma-182	247	54	ω	ω	NUM
ma-182	247	55	6	6	NUM
ma-182	247	56	εp	εp	NOUN
ma-182	247	57	.	.	PUNCT
ma-182	248	1	thus	thus	ADV
ma-182	248	2	,	,	PUNCT
ma-182	248	3	‖f	‖f	ADP
ma-182	248	4	∗ω	∗ω	NOUN
ma-182	248	5	g	g	NOUN
ma-182	248	6	−	−	PROPN
ma-182	248	7	f	f	PROPN
ma-182	248	8	‖p	‖p	PROPN
ma-182	248	9	,	,	PUNCT
ma-182	248	10	ω	ω	PROPN
ma-182	248	11	6	6	NUM
ma-182	248	12	ε	ε	PROPN
ma-182	248	13	.	.	PUNCT
ma-182	248	14	�	�	PROPN
ma-182	248	15	theorem	theorem	VERB
ma-182	248	16	3.7	3.7	NUM
ma-182	248	17	.	.	PUNCT
ma-182	249	1	let	let	VERB
ma-182	249	2	g	g	PRON
ma-182	249	3	be	be	AUX
ma-182	249	4	a	a	DET
ma-182	249	5	locally	locally	ADV
ma-182	249	6	compact	compact	ADJ
ma-182	249	7	abelian	abelian	ADJ
ma-182	249	8	group	group	NOUN
ma-182	249	9	.	.	PUNCT
ma-182	250	1	if	if	SCONJ
ma-182	250	2	t	t	PROPN
ma-182	250	3	∈m1,p	∈m1,p	PROPN
ma-182	250	4	ω	ω	PROPN
ma-182	250	5	(	(	PUNCT
ma-182	250	6	g	g	NOUN
ma-182	250	7	)	)	PUNCT
ma-182	250	8	,	,	PUNCT
ma-182	250	9	then	then	ADV
ma-182	250	10	‖t	‖t	PUNCT
ma-182	250	11	f	f	PROPN
ma-182	250	12	‖p	‖p	PROPN
ma-182	250	13	6	6	NUM
ma-182	250	14	‖t‖‖f	‖t‖‖f	PROPN
ma-182	250	15	‖1	‖1	NOUN
ma-182	250	16	.	.	PUNCT
ma-182	251	1	in	in	ADP
ma-182	251	2	other	other	ADJ
ma-182	251	3	words	word	NOUN
ma-182	251	4	,	,	PUNCT
ma-182	251	5	t	t	NOUN
ma-182	251	6	:	:	PUNCT
ma-182	251	7	l1ω(g	l1ω(g	PROPN
ma-182	251	8	)	)	PUNCT
ma-182	251	9	−→	−→	NOUN
ma-182	251	10	lpω(g	lpω(g	NOUN
ma-182	251	11	)	)	PUNCT
ma-182	251	12	is	be	AUX
ma-182	251	13	a	a	DET
ma-182	251	14	bounded	bounded	ADJ
ma-182	251	15	operator	operator	NOUN
ma-182	251	16	.	.	PUNCT
ma-182	252	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	252	2	eur	eur	PROPN
ma-182	252	3	.	.	PUNCT
ma-182	253	1	j.	j.	PROPN
ma-182	253	2	math	math	PROPN
ma-182	253	3	.	.	PUNCT
ma-182	254	1	anal	anal	PROPN
ma-182	254	2	.	.	PUNCT
ma-182	255	1	10.28924	10.28924	NUM
ma-182	255	2	/	/	SYM
ma-182	255	3	ada	ada	PROPN
ma-182	255	4	/	/	SYM
ma-182	255	5	ma.3.27	ma.3.27	PROPN
ma-182	255	6	10	10	NUM
ma-182	255	7	proof	proof	NOUN
ma-182	255	8	.	.	PUNCT
ma-182	256	1	let	let	VERB
ma-182	256	2	ε	ε	PROPN
ma-182	256	3	>	>	X
ma-182	256	4	0	0	PROPN
ma-182	256	5	.	.	PUNCT
ma-182	256	6	via	via	ADP
ma-182	256	7	theorem	theorem	NOUN
ma-182	256	8	3.6	3.6	NUM
ma-182	256	9	,	,	PUNCT
ma-182	256	10	there	there	PRON
ma-182	256	11	exists	exist	VERB
ma-182	256	12	a	a	DET
ma-182	256	13	positive	positive	ADJ
ma-182	256	14	function	function	NOUN
ma-182	256	15	g	g	NOUN
ma-182	256	16	in	in	ADP
ma-182	256	17	cc(g	cc(g	NOUN
ma-182	256	18	)	)	PUNCT
ma-182	257	1	such	such	ADJ
ma-182	257	2	that∫	that∫	NOUN
ma-182	257	3	g	g	PROPN
ma-182	257	4	g(t)ω(t)dt	g(t)ω(t)dt	NOUN
ma-182	257	5	=	=	SYM
ma-182	257	6	1	1	NUM
ma-182	257	7	and	and	CCONJ
ma-182	257	8	‖g	‖g	PROPN
ma-182	257	9	∗ω	∗ω	PROPN
ma-182	257	10	t	t	PROPN
ma-182	257	11	f	f	PROPN
ma-182	258	1	−	−	PROPN
ma-182	259	1	t	t	PROPN
ma-182	259	2	f	f	PROPN
ma-182	259	3	‖p	‖p	PROPN
ma-182	259	4	6	6	NUM
ma-182	259	5	ε	ε	PROPN
ma-182	259	6	because	because	SCONJ
ma-182	259	7	‖·‖p	‖·‖p	PROPN
ma-182	259	8	6	6	NUM
ma-182	259	9	‖·‖p	‖·‖p	ADJ
ma-182	259	10	,	,	PUNCT
ma-182	259	11	ω	ω	NOUN
ma-182	259	12	.	.	PUNCT
ma-182	260	1	we	we	PRON
ma-182	260	2	have	have	VERB
ma-182	260	3	,	,	PUNCT
ma-182	260	4	‖g	‖g	PROPN
ma-182	260	5	∗ω	∗ω	PROPN
ma-182	261	1	t	t	X
ma-182	261	2	f	f	PROPN
ma-182	262	1	−	−	PROPN
ma-182	262	2	t	t	PROPN
ma-182	262	3	f	f	PROPN
ma-182	262	4	‖p	‖p	PROPN
ma-182	262	5	>	>	X
ma-182	263	1	‖t	‖t	PROPN
ma-182	264	1	f	f	PROPN
ma-182	264	2	‖p	‖p	PROPN
ma-182	265	1	−	−	PROPN
ma-182	265	2	‖g	‖g	PROPN
ma-182	266	1	∗ω	∗ω	PROPN
ma-182	266	2	t	t	PROPN
ma-182	266	3	f	f	PROPN
ma-182	266	4	‖p	‖p	PROPN
ma-182	266	5	.	.	PUNCT
ma-182	267	1	therefore	therefore	ADV
ma-182	267	2	,	,	PUNCT
ma-182	267	3	‖t	‖t	PROPN
ma-182	267	4	f	f	PROPN
ma-182	267	5	‖p	‖p	PROPN
ma-182	267	6	6	6	NUM
ma-182	267	7	‖g	‖g	PROPN
ma-182	267	8	∗ω	∗ω	PROPN
ma-182	268	1	t	t	PROPN
ma-182	268	2	f	f	PROPN
ma-182	268	3	‖p	‖p	PROPN
ma-182	268	4	+	+	CCONJ
ma-182	268	5	ε	ε	PROPN
ma-182	268	6	=	=	SYM
ma-182	268	7	‖tg	‖tg	NUM
ma-182	268	8	∗ω	∗ω	PROPN
ma-182	268	9	f	f	X
ma-182	268	10	‖p	‖p	PROPN
ma-182	269	1	+	+	CCONJ
ma-182	269	2	ε	ε	PROPN
ma-182	269	3	6	6	NUM
ma-182	269	4	‖tg‖p‖f	‖tg‖p‖f	PROPN
ma-182	269	5	‖1	‖1	NOUN
ma-182	269	6	+	+	CCONJ
ma-182	269	7	ε	ε	PROPN
ma-182	269	8	6	6	NUM
ma-182	269	9	‖tg‖p	‖tg‖p	PROPN
ma-182	269	10	,	,	PUNCT
ma-182	269	11	ω‖f	ω‖f	PRON
ma-182	269	12	‖1	‖1	NOUN
ma-182	269	13	+	+	CCONJ
ma-182	269	14	ε	ε	PROPN
ma-182	269	15	6	6	NUM
ma-182	269	16	‖t‖‖‖1,ω‖f	‖t‖‖‖1,ω‖f	PROPN
ma-182	269	17	‖1	‖1	NOUN
ma-182	270	1	+	+	NUM
ma-182	270	2	ε	ε	PROPN
ma-182	270	3	=	=	SYM
ma-182	270	4	‖t‖‖f	‖t‖‖f	PROPN
ma-182	270	5	‖1	‖1	NOUN
ma-182	270	6	+	+	CCONJ
ma-182	270	7	ε	ε	PROPN
ma-182	270	8	.	.	PROPN
ma-182	271	1	since	since	SCONJ
ma-182	271	2	the	the	DET
ma-182	271	3	latter	latter	ADJ
ma-182	271	4	inequality	inequality	NOUN
ma-182	271	5	is	be	AUX
ma-182	271	6	true	true	ADJ
ma-182	271	7	for	for	ADP
ma-182	271	8	arbitrary	arbitrary	ADJ
ma-182	271	9	ε	ε	PROPN
ma-182	271	10	>	>	X
ma-182	271	11	0	0	PROPN
ma-182	271	12	,	,	PUNCT
ma-182	271	13	then	then	ADV
ma-182	271	14	we	we	PRON
ma-182	271	15	obtain	obtain	VERB
ma-182	271	16	‖t	‖t	PROPN
ma-182	271	17	f	f	PROPN
ma-182	271	18	‖p	‖p	PROPN
ma-182	271	19	6	6	NUM
ma-182	271	20	‖t‖‖f	‖t‖‖f	PROPN
ma-182	271	21	‖1	‖1	PROPN
ma-182	271	22	.	.	PUNCT
ma-182	272	1	�	�	PROPN
ma-182	272	2	for	for	ADP
ma-182	272	3	a	a	DET
ma-182	272	4	function	function	NOUN
ma-182	272	5	f	f	PROPN
ma-182	272	6	in	in	ADP
ma-182	272	7	lpω(g	lpω(g	NOUN
ma-182	272	8	)	)	PUNCT
ma-182	273	1	,	,	PUNCT
ma-182	273	2	we	we	PRON
ma-182	273	3	define	define	VERB
ma-182	273	4	the	the	DET
ma-182	273	5	convolution	convolution	NOUN
ma-182	273	6	operator	operator	NOUN
ma-182	273	7	tf	tf	X
ma-182	273	8	by	by	ADP
ma-182	273	9	tf	tf	NUM
ma-182	273	10	g	g	PROPN
ma-182	273	11	=	=	SYM
ma-182	273	12	f	f	PROPN
ma-182	273	13	∗ω	∗ω	PROPN
ma-182	273	14	g.	g.	PROPN
ma-182	273	15	theorem	theorem	VERB
ma-182	273	16	3.8	3.8	NUM
ma-182	273	17	.	.	PUNCT
ma-182	274	1	let	let	VERB
ma-182	274	2	g	g	PRON
ma-182	274	3	be	be	AUX
ma-182	274	4	a	a	DET
ma-182	274	5	locally	locally	ADV
ma-182	274	6	compact	compact	ADJ
ma-182	274	7	abelian	abelian	NOUN
ma-182	274	8	group	group	NOUN
ma-182	274	9	.	.	PUNCT
ma-182	275	1	let	let	VERB
ma-182	275	2	1	1	NUM
ma-182	275	3	<	<	X
ma-182	275	4	p	p	X
ma-182	275	5	<	<	X
ma-182	275	6	∞.	∞.	PROPN
ma-182	275	7	let	let	VERB
ma-182	275	8	f	f	PRON
ma-182	275	9	be	be	AUX
ma-182	275	10	a	a	DET
ma-182	275	11	function	function	NOUN
ma-182	275	12	in	in	ADP
ma-182	275	13	lpω(g	lpω(g	NOUN
ma-182	275	14	)	)	PUNCT
ma-182	275	15	.	.	PUNCT
ma-182	276	1	then	then	ADV
ma-182	276	2	,	,	PUNCT
ma-182	276	3	‖tf	‖tf	PROPN
ma-182	276	4	‖	‖	PROPN
ma-182	276	5	=	=	NOUN
ma-182	276	6	‖f	‖f	ADP
ma-182	276	7	‖p	‖p	PROPN
ma-182	276	8	,	,	PUNCT
ma-182	276	9	ω	ω	NOUN
ma-182	276	10	.	.	PUNCT
ma-182	276	11	proof	proof	NOUN
ma-182	276	12	.	.	PUNCT
ma-182	277	1	let	let	VERB
ma-182	277	2	f	f	PRON
ma-182	277	3	∈	∈	PROPN
ma-182	277	4	lpω(g	lpω(g	X
ma-182	277	5	)	)	PUNCT
ma-182	277	6	and	and	CCONJ
ma-182	277	7	let	let	VERB
ma-182	277	8	ε	ε	PROPN
ma-182	277	9	>	>	X
ma-182	277	10	0	0	PROPN
ma-182	277	11	.	.	PUNCT
ma-182	277	12	from	from	ADP
ma-182	277	13	theorem	theorem	ADJ
ma-182	277	14	3.6	3.6	NUM
ma-182	277	15	,	,	PUNCT
ma-182	277	16	there	there	PRON
ma-182	277	17	exits	exit	VERB
ma-182	277	18	a	a	DET
ma-182	277	19	positive	positive	ADJ
ma-182	277	20	function	function	NOUN
ma-182	277	21	g	g	ADP
ma-182	277	22	such	such	ADJ
ma-182	277	23	that∫	that∫	NOUN
ma-182	277	24	g	g	PROPN
ma-182	277	25	g(t)ω(t)dt	g(t)ω(t)dt	NOUN
ma-182	277	26	=	=	SYM
ma-182	277	27	1	1	NUM
ma-182	277	28	and	and	CCONJ
ma-182	277	29	‖f	‖f	ADP
ma-182	277	30	∗ω	∗ω	NOUN
ma-182	277	31	g	g	NOUN
ma-182	277	32	−	−	PROPN
ma-182	277	33	f	f	PROPN
ma-182	277	34	‖p	‖p	PROPN
ma-182	277	35	,	,	PUNCT
ma-182	277	36	ω	ω	PROPN
ma-182	277	37	6	6	NUM
ma-182	277	38	ε	ε	PROPN
ma-182	277	39	.	.	PUNCT
ma-182	278	1	then	then	ADV
ma-182	278	2	,	,	PUNCT
ma-182	278	3	‖f	‖f	ADJ
ma-182	278	4	‖p	‖p	PROPN
ma-182	278	5	,	,	PUNCT
ma-182	278	6	ω	ω	PROPN
ma-182	278	7	6	6	NUM
ma-182	278	8	ε+	ε+	NOUN
ma-182	278	9	‖f	‖f	ADP
ma-182	278	10	∗ω	∗ω	NOUN
ma-182	278	11	g‖p	g‖p	NOUN
ma-182	278	12	,	,	PUNCT
ma-182	278	13	ω	ω	NUM
ma-182	278	14	=	=	SYM
ma-182	278	15	ε+	ε+	X
ma-182	278	16	‖tf	‖tf	X
ma-182	278	17	g‖p	g‖p	NOUN
ma-182	278	18	,	,	PUNCT
ma-182	278	19	ω	ω	PROPN
ma-182	278	20	6	6	NUM
ma-182	278	21	ε+	ε+	X
ma-182	278	22	‖tf	‖tf	NUM
ma-182	278	23	‖‖g‖1,ω	‖‖g‖1,ω	NOUN
ma-182	278	24	=	=	PUNCT
ma-182	278	25	ε+	ε+	X
ma-182	278	26	‖tf	‖tf	X
ma-182	278	27	‖.	‖.	X
ma-182	278	28	thus	thus	ADV
ma-182	278	29	‖f	‖f	ADP
ma-182	278	30	‖p	‖p	PROPN
ma-182	278	31	,	,	PUNCT
ma-182	278	32	ω	ω	PROPN
ma-182	278	33	6	6	NUM
ma-182	278	34	‖tf	‖tf	NUM
ma-182	278	35	‖.let	‖.let	ADP
ma-182	278	36	us	we	PRON
ma-182	278	37	prove	prove	VERB
ma-182	278	38	the	the	DET
ma-182	278	39	inverse	inverse	NOUN
ma-182	278	40	inequality	inequality	NOUN
ma-182	278	41	.	.	PUNCT
ma-182	279	1	let	let	VERB
ma-182	279	2	g	g	PROPN
ma-182	279	3	∈	∈	PROPN
ma-182	279	4	l1ω(g	l1ω(g	NOUN
ma-182	279	5	)	)	PUNCT
ma-182	279	6	.	.	PUNCT
ma-182	280	1	applying	apply	VERB
ma-182	280	2	the	the	DET
ma-182	280	3	hölder	hölder	NOUN
ma-182	280	4	’s	’s	PART
ma-182	280	5	inequality	inequality	NOUN
ma-182	280	6	withrespect	withrespect	ADJ
ma-182	280	7	to	to	ADP
ma-182	280	8	the	the	DET
ma-182	280	9	measure	measure	NOUN
ma-182	280	10	g(y)ω(y)dy	g(y)ω(y)dy	NOUN
ma-182	280	11	,	,	PUNCT
ma-182	280	12	one	one	PRON
ma-182	280	13	has	have	VERB
ma-182	280	14	‖f	‖f	ADP
ma-182	280	15	∗	∗	NOUN
ma-182	280	16	g‖pp	g‖pp	PROPN
ma-182	280	17	,	,	PUNCT
ma-182	280	18	ω	ω	PROPN
ma-182	281	1	=	=	SYM
ma-182	281	2	∫	∫	PROPN
ma-182	281	3	g	g	PROPN
ma-182	281	4	|g	|g	PROPN
ma-182	281	5	∗ω	∗ω	PROPN
ma-182	281	6	f	f	NOUN
ma-182	281	7	|pω(x)dx	|pω(x)dx	ADV
ma-182	281	8	=	=	PUNCT
ma-182	281	9	∫	∫	PROPN
ma-182	281	10	g	g	PROPN
ma-182	281	11	∣∣∣∣∫	∣∣∣∣∫	PROPN
ma-182	281	12	g	g	PROPN
ma-182	281	13	g(y)γyωf	g(y)γyωf	PRON
ma-182	281	14	(	(	PUNCT
ma-182	281	15	x)ω(y	x)ω(y	PROPN
ma-182	281	16	)	)	PUNCT
ma-182	281	17	∣∣∣∣p	∣∣∣∣p	ADP
ma-182	282	1	ω(x)dx	ω(x)dx	NUM
ma-182	282	2	6	6	NUM
ma-182	282	3	∫	∫	NOUN
ma-182	282	4	g	g	PROPN
ma-182	283	1	[	[	X
ma-182	283	2	∫	∫	X
ma-182	283	3	g	g	PROPN
ma-182	283	4	|γyωf	|γyωf	NOUN
ma-182	283	5	(	(	PUNCT
ma-182	283	6	x)|p|g(y)|ω(y)dy	x)|p|g(y)|ω(y)dy	X
ma-182	283	7	]	]	PUNCT
ma-182	284	1	[	[	X
ma-182	284	2	∫	∫	X
ma-182	284	3	g	g	PROPN
ma-182	284	4	|g(y)|ω(y)dy	|g(y)|ω(y)dy	PROPN
ma-182	284	5	]	]	PUNCT
ma-182	284	6	p	p	X
ma-182	284	7	q	q	PROPN
ma-182	284	8	ω(x)dx	ω(x)dx	NUM
ma-182	284	9	6	6	NUM
ma-182	284	10	∫	∫	NOUN
ma-182	284	11	g	g	PROPN
ma-182	284	12	(	(	PUNCT
ma-182	284	13	|f	|f	PROPN
ma-182	284	14	|p	|p	PROPN
ma-182	284	15	∗ω	∗ω	NOUN
ma-182	284	16	|g|)ω(x)dx	|g|)ω(x)dx	PROPN
ma-182	285	1	[	[	X
ma-182	285	2	∫	∫	X
ma-182	285	3	g	g	PROPN
ma-182	285	4	|g(y)|ω(y)dy	|g(y)|ω(y)dy	PROPN
ma-182	285	5	]	]	PUNCT
ma-182	285	6	p	p	X
ma-182	285	7	q	q	PROPN
ma-182	285	8	6	6	NUM
ma-182	285	9	‖|f	‖|f	NOUN
ma-182	285	10	|p	|p	NOUN
ma-182	285	11	∗ω	∗ω	NOUN
ma-182	285	12	|g|‖1,ω‖g‖	|g|‖1,ω‖g‖	X
ma-182	285	13	p	p	X
ma-182	285	14	q	q	PROPN
ma-182	285	15	1,ω	1,ω	PROPN
ma-182	285	16	6	6	NUM
ma-182	285	17	‖f	‖f	PUNCT
ma-182	285	18	‖	‖	PROPN
ma-182	285	19	p	p	X
ma-182	285	20	p	p	NOUN
ma-182	285	21	,	,	PUNCT
ma-182	285	22	ω‖g‖1,ω‖g‖	ω‖g‖1,ω‖g‖	NOUN
ma-182	285	23	p	p	NOUN
ma-182	285	24	q	q	NOUN
ma-182	285	25	1,ω	1,ω	PROPN
ma-182	285	26	=	=	SYM
ma-182	285	27	‖f	‖f	ADP
ma-182	285	28	‖pp	‖pp	NOUN
ma-182	285	29	,	,	PUNCT
ma-182	285	30	ω‖g‖	ω‖g‖	NOUN
ma-182	285	31	p	p	NOUN
ma-182	285	32	1,ω	1,ω	NUM
ma-182	285	33	.	.	PUNCT
ma-182	286	1	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	286	2	eur	eur	PROPN
ma-182	286	3	.	.	PUNCT
ma-182	287	1	j.	j.	PROPN
ma-182	287	2	math	math	PROPN
ma-182	287	3	.	.	PUNCT
ma-182	288	1	anal	anal	PROPN
ma-182	288	2	.	.	PUNCT
ma-182	289	1	10.28924	10.28924	NUM
ma-182	289	2	/	/	SYM
ma-182	289	3	ada	ada	PROPN
ma-182	289	4	/	/	SYM
ma-182	289	5	ma.3.27	ma.3.27	PROPN
ma-182	290	1	11then	11then	ADV
ma-182	290	2	,	,	PUNCT
ma-182	290	3	‖tf	‖tf	PROPN
ma-182	290	4	g‖pp	g‖pp	PROPN
ma-182	290	5	,	,	PUNCT
ma-182	290	6	ω	ω	NOUN
ma-182	290	7	6	6	NUM
ma-182	290	8	‖f	‖f	ADP
ma-182	290	9	‖pp	‖pp	NUM
ma-182	290	10	,	,	PUNCT
ma-182	290	11	ω‖g‖p1,ω	ω‖g‖p1,ω	PROPN
ma-182	290	12	.	.	PUNCT
ma-182	291	1	thus	thus	ADV
ma-182	291	2	,	,	PUNCT
ma-182	292	1	‖tf	‖tf	PROPN
ma-182	292	2	‖	‖	PROPN
ma-182	292	3	6	6	NUM
ma-182	292	4	‖f	‖f	PUNCT
ma-182	292	5	‖p	‖p	PROPN
ma-182	292	6	,	,	PUNCT
ma-182	292	7	ω	ω	PROPN
ma-182	292	8	.	.	PUNCT
ma-182	292	9	�	�	PROPN
ma-182	292	10	as	as	ADP
ma-182	292	11	a	a	DET
ma-182	292	12	consequence	consequence	NOUN
ma-182	292	13	of	of	ADP
ma-182	292	14	theorem	theorem	ADJ
ma-182	292	15	3.3	3.3	NUM
ma-182	292	16	and	and	CCONJ
ma-182	292	17	theorem	theorem	VERB
ma-182	292	18	3.8	3.8	NUM
ma-182	292	19	,	,	PUNCT
ma-182	292	20	we	we	PRON
ma-182	292	21	have	have	VERB
ma-182	292	22	the	the	DET
ma-182	292	23	following	follow	VERB
ma-182	292	24	result	result	NOUN
ma-182	292	25	.	.	PUNCT
ma-182	293	1	corollary	corollary	ADJ
ma-182	293	2	3.9	3.9	NUM
ma-182	293	3	.	.	PUNCT
ma-182	294	1	let	let	VERB
ma-182	294	2	g	g	PRON
ma-182	294	3	be	be	AUX
ma-182	294	4	a	a	DET
ma-182	294	5	locally	locally	ADV
ma-182	294	6	compact	compact	ADJ
ma-182	294	7	abelian	abelian	NOUN
ma-182	294	8	group	group	NOUN
ma-182	294	9	.	.	PUNCT
ma-182	295	1	let	let	VERB
ma-182	295	2	1	1	NUM
ma-182	295	3	<	<	X
ma-182	295	4	p	p	X
ma-182	295	5	<	<	X
ma-182	295	6	∞.	∞.	PROPN
ma-182	295	7	then	then	ADV
ma-182	295	8	,	,	PUNCT
ma-182	295	9	the	the	DET
ma-182	295	10	multipliers	multiplier	NOUN
ma-182	295	11	space	space	NOUN
ma-182	295	12	m1,p	m1,p	PROPN
ma-182	295	13	ω	ω	PROPN
ma-182	295	14	(	(	PUNCT
ma-182	295	15	g	g	NOUN
ma-182	295	16	)	)	PUNCT
ma-182	295	17	and	and	CCONJ
ma-182	295	18	the	the	DET
ma-182	295	19	beurling	beurling	ADJ
ma-182	295	20	space	space	NOUN
ma-182	295	21	lpω(g	lpω(g	NOUN
ma-182	295	22	)	)	PUNCT
ma-182	295	23	are	be	AUX
ma-182	295	24	isometricaly	isometricaly	ADJ
ma-182	295	25	identified	identify	VERB
ma-182	295	26	by	by	ADP
ma-182	295	27	the	the	DET
ma-182	295	28	mapping	mapping	NOUN
ma-182	295	29	t	t	NOUN
ma-182	295	30	:	:	PUNCT
ma-182	296	1	f	f	X
ma-182	296	2	7−→	7−→	NOUN
ma-182	296	3	tf	tf	INTJ
ma-182	296	4	.	.	PUNCT
ma-182	297	1	conclusion	conclusion	NOUN
ma-182	297	2	in	in	ADP
ma-182	297	3	this	this	DET
ma-182	297	4	paper	paper	NOUN
ma-182	297	5	,	,	PUNCT
ma-182	297	6	we	we	PRON
ma-182	297	7	obtain	obtain	VERB
ma-182	297	8	a	a	DET
ma-182	297	9	characterization	characterization	NOUN
ma-182	297	10	of	of	ADP
ma-182	297	11	multipliers	multiplier	NOUN
ma-182	297	12	for	for	ADP
ma-182	297	13	the	the	DET
ma-182	297	14	pair	pair	NOUN
ma-182	297	15	(	(	PUNCT
ma-182	297	16	l1ω	l1ω	ADJ
ma-182	297	17	,	,	PUNCT
ma-182	297	18	l	l	PROPN
ma-182	297	19	p	p	PROPN
ma-182	297	20	ω	ω	NOUN
ma-182	297	21	)	)	PUNCT
ma-182	297	22	using	use	VERB
ma-182	297	23	the	the	DET
ma-182	297	24	fouriertransform	fouriertransform	NOUN
ma-182	297	25	related	relate	VERB
ma-182	297	26	to	to	ADP
ma-182	297	27	a	a	DET
ma-182	297	28	beurling	beurling	ADJ
ma-182	297	29	weight	weight	NOUN
ma-182	297	30	.	.	PUNCT
ma-182	298	1	we	we	PRON
ma-182	298	2	also	also	ADV
ma-182	298	3	obtain	obtain	VERB
ma-182	298	4	the	the	DET
ma-182	298	5	identification	identification	NOUN
ma-182	298	6	of	of	ADP
ma-182	298	7	the	the	DET
ma-182	298	8	space	space	NOUN
ma-182	298	9	of	of	ADP
ma-182	298	10	suchmultipliers	suchmultiplier	NOUN
ma-182	298	11	with	with	ADP
ma-182	298	12	the	the	DET
ma-182	298	13	beurling	beurling	ADJ
ma-182	298	14	space	space	NOUN
ma-182	298	15	lpω	lpω	VERB
ma-182	298	16	when	when	SCONJ
ma-182	298	17	1	1	NUM
ma-182	298	18	<	<	X
ma-182	298	19	p	p	X
ma-182	298	20	<	<	X
ma-182	298	21	∞.	∞.	PROPN
ma-182	298	22	it	it	PRON
ma-182	298	23	would	would	AUX
ma-182	298	24	be	be	AUX
ma-182	298	25	interesting	interesting	ADJ
ma-182	298	26	in	in	ADP
ma-182	298	27	the	the	DET
ma-182	298	28	future	future	ADJ
ma-182	298	29	toconsider	toconsider	NOUN
ma-182	298	30	the	the	DET
ma-182	298	31	case	case	NOUN
ma-182	298	32	of	of	ADP
ma-182	298	33	the	the	DET
ma-182	298	34	pair	pair	NOUN
ma-182	298	35	(	(	PUNCT
ma-182	298	36	lpω	lpω	PROPN
ma-182	298	37	,	,	PUNCT
ma-182	298	38	l	l	PROPN
ma-182	298	39	q	q	PROPN
ma-182	298	40	ω	ω	PROPN
ma-182	298	41	)	)	PUNCT
ma-182	298	42	in	in	ADP
ma-182	298	43	this	this	DET
ma-182	298	44	framework	framework	NOUN
ma-182	298	45	of	of	ADP
ma-182	298	46	the	the	DET
ma-182	298	47	weight	weight	NOUN
ma-182	298	48	dependent	dependent	ADJ
ma-182	298	49	convolution	convolution	NOUN
ma-182	298	50	.	.	PUNCT
ma-182	299	1	competing	compete	VERB
ma-182	299	2	interests	interest	NOUN
ma-182	299	3	the	the	DET
ma-182	299	4	authors	author	NOUN
ma-182	299	5	declare	declare	VERB
ma-182	299	6	that	that	SCONJ
ma-182	299	7	no	no	DET
ma-182	299	8	competing	compete	VERB
ma-182	299	9	interests	interest	NOUN
ma-182	299	10	exist	exist	VERB
ma-182	299	11	.	.	PUNCT
ma-182	300	1	references	reference	NOUN
ma-182	300	2	[	[	X
ma-182	300	3	1	1	NUM
ma-182	300	4	]	]	PUNCT
ma-182	300	5	r.	r.	PROPN
ma-182	300	6	akylzhanov	akylzhanov	PROPN
ma-182	300	7	,	,	PUNCT
ma-182	300	8	m.	m.	NOUN
ma-182	300	9	ruzhansky	ruzhansky	PROPN
ma-182	300	10	,	,	PUNCT
ma-182	300	11	lp	lp	ADJ
ma-182	300	12	-	-	PUNCT
ma-182	300	13	lq	lq	ADJ
ma-182	300	14	multipliers	multiplier	NOUN
ma-182	300	15	on	on	ADP
ma-182	300	16	locally	locally	ADV
ma-182	300	17	compact	compact	ADJ
ma-182	300	18	groups	group	NOUN
ma-182	300	19	,	,	PUNCT
ma-182	300	20	j.	j.	PROPN
ma-182	300	21	funct	funct	PROPN
ma-182	300	22	.	.	PUNCT
ma-182	301	1	anal	anal	PROPN
ma-182	301	2	.	.	PUNCT
ma-182	302	1	278	278	NUM
ma-182	302	2	(	(	PUNCT
ma-182	302	3	3	3	NUM
ma-182	302	4	)	)	PUNCT
ma-182	302	5	(	(	PUNCT
ma-182	302	6	2020	2020	NUM
ma-182	302	7	)	)	PUNCT
ma-182	302	8	108324	108324	NUM
ma-182	302	9	.	.	PUNCT
ma-182	303	1	https://doi.org/10.1016/j.jfa.2019.108324[2	https://doi.org/10.1016/j.jfa.2019.108324[2	PROPN
ma-182	303	2	]	]	X
ma-182	303	3	a.	a.	NOUN
ma-182	303	4	bourouihiya	bourouihiya	PROPN
ma-182	303	5	,	,	PUNCT
ma-182	303	6	beurling	beurle	VERB
ma-182	303	7	weighted	weight	VERB
ma-182	303	8	spaces	space	NOUN
ma-182	303	9	,	,	PUNCT
ma-182	303	10	product	product	NOUN
ma-182	303	11	-	-	PUNCT
ma-182	303	12	convolution	convolution	NOUN
ma-182	303	13	operators	operator	NOUN
ma-182	303	14	and	and	CCONJ
ma-182	303	15	the	the	DET
ma-182	303	16	tensor	tensor	NOUN
ma-182	303	17	product	product	NOUN
ma-182	303	18	of	of	ADP
ma-182	303	19	frames	frame	NOUN
ma-182	303	20	,	,	PUNCT
ma-182	303	21	phdthesis	phdthesis	NOUN
ma-182	303	22	,	,	PUNCT
ma-182	303	23	university	university	NOUN
ma-182	303	24	of	of	ADP
ma-182	303	25	maryland	maryland	PROPN
ma-182	303	26	,	,	PUNCT
ma-182	303	27	usa	usa	PROPN
ma-182	303	28	,	,	PUNCT
ma-182	303	29	2006	2006	NUM
ma-182	303	30	.	.	PUNCT
ma-182	304	1	http://www.norbertwiener.umd.edu/research/lectures/2006/	http://www.norbertwiener.umd.edu/research/lectures/2006/	PROPN
ma-182	304	2	abourouihiya_thesis.pdf[3	abourouihiya_thesis.pdf[3	PROPN
ma-182	304	3	]	]	PUNCT
ma-182	304	4	g.i	g.i	PROPN
ma-182	304	5	.	.	PROPN
ma-182	304	6	gaudry	gaudry	PROPN
ma-182	304	7	,	,	PUNCT
ma-182	304	8	multipliers	multiplier	NOUN
ma-182	304	9	of	of	ADP
ma-182	304	10	weighted	weight	VERB
ma-182	304	11	lebesgue	lebesgue	NOUN
ma-182	304	12	and	and	CCONJ
ma-182	304	13	measure	measure	NOUN
ma-182	304	14	spaces	space	NOUN
ma-182	304	15	,	,	PUNCT
ma-182	304	16	proc	proc	NOUN
ma-182	304	17	.	.	PUNCT
ma-182	305	1	london	london	PROPN
ma-182	305	2	math	math	PROPN
ma-182	305	3	.	.	PUNCT
ma-182	306	1	soc	soc	PROPN
ma-182	306	2	.	.	PUNCT
ma-182	307	1	19	19	NUM
ma-182	307	2	(	(	PUNCT
ma-182	307	3	3	3	NUM
ma-182	307	4	)	)	PUNCT
ma-182	307	5	(	(	PUNCT
ma-182	307	6	1969	1969	NUM
ma-182	307	7	)	)	PUNCT
ma-182	307	8	327	327	NUM
ma-182	307	9	-	-	SYM
ma-182	307	10	340	340	NUM
ma-182	307	11	.	.	PUNCT
ma-182	308	1	https://doi.org/10.1112/plms/s3-19.2.327[4	https://doi.org/10.1112/plms/s3-19.2.327[4	NOUN
ma-182	308	2	]	]	X
ma-182	308	3	a.	a.	NOUN
ma-182	308	4	issa	issa	PROPN
ma-182	308	5	,	,	PUNCT
ma-182	308	6	y.	y.	PROPN
ma-182	308	7	mensah	mensah	PROPN
ma-182	308	8	,	,	PUNCT
ma-182	308	9	multipliers	multiplier	NOUN
ma-182	308	10	on	on	ADP
ma-182	308	11	weighted	weighted	ADJ
ma-182	308	12	group	group	NOUN
ma-182	308	13	algebras	algebra	NOUN
ma-182	308	14	,	,	PUNCT
ma-182	308	15	gulf	gulf	PROPN
ma-182	308	16	j.	j.	PROPN
ma-182	308	17	math	math	PROPN
ma-182	308	18	.	.	PUNCT
ma-182	309	1	8(2	8(2	NUM
ma-182	309	2	)	)	PUNCT
ma-182	309	3	(	(	PUNCT
ma-182	309	4	2020	2020	NUM
ma-182	309	5	)	)	PUNCT
ma-182	309	6	35	35	NUM
ma-182	309	7	-	-	SYM
ma-182	309	8	45	45	NUM
ma-182	309	9	.	.	PUNCT
ma-182	310	1	https://doi.org/	https://doi.org/	VERB
ma-182	310	2	10.56947	10.56947	NUM
ma-182	310	3	/	/	SYM
ma-182	310	4	gjom.v8i2.283[5	gjom.v8i2.283[5	NOUN
ma-182	310	5	]	]	PUNCT
ma-182	310	6	a.	a.	NOUN
ma-182	310	7	issa	issa	PROPN
ma-182	310	8	,	,	PUNCT
ma-182	310	9	y.	y.	PROPN
ma-182	310	10	mensah	mensah	PROPN
ma-182	310	11	,	,	PUNCT
ma-182	310	12	on	on	ADP
ma-182	310	13	beurling	beurle	VERB
ma-182	310	14	spaces	space	NOUN
ma-182	310	15	provided	provide	VERB
ma-182	310	16	with	with	ADP
ma-182	310	17	a	a	DET
ma-182	310	18	weight	weight	NOUN
ma-182	310	19	dependent	dependent	ADJ
ma-182	310	20	convolution	convolution	NOUN
ma-182	310	21	,	,	PUNCT
ma-182	310	22	funct	funct	NOUN
ma-182	310	23	.	.	PUNCT
ma-182	311	1	anal	anal	PROPN
ma-182	311	2	.	.	PUNCT
ma-182	312	1	1	1	NUM
ma-182	312	2	(	(	PUNCT
ma-182	312	3	2022	2022	NUM
ma-182	312	4	)	)	PUNCT
ma-182	313	1	i	i	PROPN
ma-182	313	2	d	d	PROPN
ma-182	313	3	6	6	X
ma-182	313	4	.	.	PUNCT
ma-182	314	1	https://fac-seams.org/journal/index.php/fa/issue/view/1[6	https://fac-seams.org/journal/index.php/fa/issue/view/1[6	NOUN
ma-182	314	2	]	]	PUNCT
ma-182	314	3	a.	a.	NOUN
ma-182	314	4	issa	issa	PROPN
ma-182	314	5	,	,	PUNCT
ma-182	314	6	y.	y.	PROPN
ma-182	314	7	mensah	mensah	PROPN
ma-182	314	8	,	,	PUNCT
ma-182	314	9	multipliers	multiplier	NOUN
ma-182	314	10	for	for	ADP
ma-182	314	11	gelfand	gelfand	ADJ
ma-182	314	12	pairs	pair	NOUN
ma-182	314	13	,	,	PUNCT
ma-182	314	14	palest	pale	ADJ
ma-182	314	15	.	.	PUNCT
ma-182	315	1	j.	j.	PROPN
ma-182	315	2	math	math	PROPN
ma-182	315	3	.	.	PUNCT
ma-182	316	1	10(1	10(1	NUM
ma-182	316	2	)	)	PUNCT
ma-182	316	3	(	(	PUNCT
ma-182	316	4	2021	2021	NUM
ma-182	316	5	)	)	PUNCT
ma-182	316	6	151	151	NUM
ma-182	316	7	-	-	SYM
ma-182	316	8	159	159	NUM
ma-182	316	9	.	.	PUNCT
ma-182	317	1	https://pjm.ppu.edu/	https://pjm.ppu.edu/	PROPN
ma-182	317	2	paper/801	paper/801	NOUN
ma-182	317	3	-	-	PUNCT
ma-182	317	4	multipliers	multiplier	NOUN
ma-182	317	5	-	-	PUNCT
ma-182	317	6	gelfand	gelfand	NOUN
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ma-182	319	4	)	)	PUNCT
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ma-182	319	6	-	-	SYM
ma-182	319	7	193	193	NUM
ma-182	319	8	.	.	PUNCT
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ma-182	320	22	,	,	PUNCT
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ma-182	322	2	(	(	PUNCT
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ma-182	322	4	)	)	PUNCT
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ma-182	322	6	-	-	SYM
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ma-182	322	8	.	.	PUNCT
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ma-182	323	19	]	]	PUNCT
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ma-182	324	4	(	(	PUNCT
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ma-182	324	6	)	)	PUNCT
ma-182	324	7	(	(	PUNCT
ma-182	324	8	2009	2009	NUM
ma-182	324	9	)	)	PUNCT
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ma-182	324	11	-	-	SYM
ma-182	324	12	82	82	NUM
ma-182	324	13	.	.	PUNCT
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ma-182	325	3	/	/	SYM
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ma-182	325	5	]	]	X
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ma-182	325	10	rajbangshi	rajbangshi	PROPN
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ma-182	325	13	and	and	CCONJ
ma-182	325	14	multipliers	multiplier	NOUN
ma-182	325	15	on	on	ADP
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ma-182	326	4	(	(	PUNCT
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ma-182	326	6	)	)	PUNCT
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ma-182	326	8	-	-	SYM
ma-182	326	9	277	277	NUM
ma-182	326	10	.	.	PUNCT
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ma-182	327	11	-	-	PUNCT
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ma-182	327	13	,	,	PUNCT
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ma-182	329	2	math	math	NOUN
ma-182	329	3	.	.	PUNCT
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ma-182	331	2	(	(	PUNCT
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ma-182	331	4	)	)	PUNCT
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ma-182	331	7	)	)	PUNCT
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ma-182	331	9	-	-	SYM
ma-182	331	10	656	656	NUM
ma-182	331	11	.	.	PUNCT
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ma-182	332	4	/	/	SYM
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ma-182	332	7	https://doi.org/10.1016/j.jfa.2019.108324	https://doi.org/10.1016/j.jfa.2019.108324	PROPN
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ma-182	332	10	https://doi.org/10.1112/plms/s3-19.2.327	https://doi.org/10.1112/plms/s3-19.2.327	ADJ
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ma-182	333	3	https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/253	https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/253	PROPN
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ma-182	334	2	https://www.sid.ir/paper/322483/en	https://www.sid.ir/paper/322483/en	PROPN
ma-182	334	3	https://www.sid.ir/paper/322483/en	https://www.sid.ir/paper/322483/en	PROPN
ma-182	334	4	https://doi.org/10.1007/s13370-018-0645-6	https://doi.org/10.1007/s13370-018-0645-6	PROPN
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ma-182	334	6	https://doi.org/10.1155/s016117120000096x	https://doi.org/10.1155/s016117120000096x	PROPN
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ma-182	334	8	.	.	PUNCT
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ma-182	336	2	.	.	PUNCT
ma-182	337	1	10.28924	10.28924	NUM
ma-182	337	2	/	/	SYM
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ma-182	337	5	ma.3.27	ma.3.27	PROPN
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ma-182	338	1	[	[	X
ma-182	338	2	13	13	NUM
ma-182	338	3	]	]	PUNCT
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ma-182	338	19	,	,	PUNCT
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ma-182	338	21	and	and	CCONJ
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ma-182	338	24	,	,	PUNCT
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ma-182	338	30	]	]	X
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ma-182	338	51	-	-	PUNCT
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ma-182	338	53	.	.	PUNCT
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ma-182	340	2	(	(	PUNCT
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ma-182	340	4	)	)	PUNCT
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ma-182	340	6	-	-	SYM
ma-182	340	7	245	245	NUM
ma-182	340	8	.	.	PUNCT
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ma-182	341	5	,	,	PUNCT
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ma-182	341	11	harmonic	harmonic	ADJ
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ma-182	341	13	and	and	CCONJ
ma-182	341	14	locally	locally	ADV
ma-182	341	15	compact	compact	ADJ
ma-182	341	16	groups	group	NOUN
ma-182	341	17	,	,	PUNCT
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ma-182	342	3	,	,	PUNCT
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ma-182	342	5	,	,	PUNCT
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ma-182	343	7	,	,	PUNCT
ma-182	343	8	(	(	PUNCT
ma-182	343	9	fourth	fourth	PROPN
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ma-182	343	11	)	)	PUNCT
ma-182	343	12	,	,	PUNCT
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ma-182	343	15	press	press	PROPN
ma-182	343	16	,	,	PUNCT
ma-182	343	17	2010.[17	2010.[17	PROPN
ma-182	343	18	]	]	X
ma-182	343	19	w.	w.	PROPN
ma-182	343	20	rudin	rudin	PROPN
ma-182	343	21	,	,	PUNCT
ma-182	343	22	fourier	fourier	ADJ
ma-182	343	23	analysis	analysis	NOUN
ma-182	343	24	on	on	ADP
ma-182	343	25	groups	group	NOUN
ma-182	343	26	,	,	PUNCT
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ma-182	343	29	,	,	PUNCT
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ma-182	343	32	,	,	PUNCT
ma-182	343	33	inc	inc	PROPN
ma-182	343	34	.	.	PROPN
ma-182	343	35	,	,	PUNCT
ma-182	344	1	1962.[18	1962.[18	NUM
ma-182	344	2	]	]	PUNCT
ma-182	344	3	v.	v.	PROPN
ma-182	344	4	s.	s.	PROPN
ma-182	344	5	shulman	shulman	PROPN
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ma-182	344	7	i.	i.	PROPN
ma-182	344	8	g.	g.	PROPN
ma-182	344	9	todorov	todorov	PROPN
ma-182	344	10	,	,	PUNCT
ma-182	344	11	l.	l.	PROPN
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ma-182	344	13	,	,	PUNCT
ma-182	344	14	sets	set	NOUN
ma-182	344	15	of	of	ADP
ma-182	344	16	multiplicity	multiplicity	NOUN
ma-182	344	17	and	and	CCONJ
ma-182	344	18	closable	closable	ADJ
ma-182	344	19	multipliers	multiplier	NOUN
ma-182	344	20	on	on	ADP
ma-182	344	21	group	group	NOUN
ma-182	344	22	algebras	algebras	PROPN
ma-182	344	23	,	,	PUNCT
ma-182	344	24	j.	j.	PROPN
ma-182	344	25	funct.anal	funct.anal	PROPN
ma-182	344	26	.	.	PROPN
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ma-182	344	28	(	(	PUNCT
ma-182	344	29	6	6	NUM
ma-182	344	30	)	)	PUNCT
ma-182	344	31	(	(	PUNCT
ma-182	344	32	2015	2015	NUM
ma-182	344	33	)	)	PUNCT
ma-182	344	34	1454	1454	NUM
ma-182	344	35	-	-	SYM
ma-182	344	36	1508	1508	NUM
ma-182	344	37	.	.	PUNCT
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ma-182	345	7	operators	operator	NOUN
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ma-182	345	9	banach	banach	NOUN
ma-182	345	10	algebras	algebra	VERB
ma-182	345	11	in	in	ADP
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ma-182	345	13	communications	communication	NOUN
ma-182	345	14	,	,	PUNCT
ma-182	345	15	appl	appl	PROPN
ma-182	345	16	.	.	PUNCT
ma-182	346	1	comput	comput	NOUN
ma-182	346	2	.	.	PUNCT
ma-182	347	1	harmon.anal	harmon.anal	X
ma-182	347	2	.	.	NOUN
ma-182	347	3	20	20	NUM
ma-182	347	4	(	(	PUNCT
ma-182	347	5	2006	2006	NUM
ma-182	347	6	)	)	PUNCT
ma-182	347	7	237	237	NUM
ma-182	347	8	-	-	SYM
ma-182	347	9	249	249	NUM
ma-182	347	10	.	.	PUNCT
ma-182	348	1	https://doi.org/10.1016/j.acha.2005.06.003	https://doi.org/10.1016/j.acha.2005.06.003	NUM
ma-182	348	2	https://doi.org/10.28924/ada/ma.3.27	https://doi.org/10.28924/ada/ma.3.27	PROPN
ma-182	348	3	https://doi.org/10.1007/s11868-017-0213-0	https://doi.org/10.1007/s11868-017-0213-0	X
ma-182	348	4	https://doi.org/10.1016/j.jfa.2014.11.019	https://doi.org/10.1016/j.jfa.2014.11.019	X
ma-182	348	5	https://doi.org/10.1016/j.acha.2005.06.003	https://doi.org/10.1016/j.acha.2005.06.003	NOUN
ma-182	348	6	1	1	NUM
ma-182	348	7	.	.	PUNCT
ma-182	349	1	introduction	introduction	NOUN
ma-182	349	2	2	2	NUM
ma-182	349	3	.	.	PUNCT
ma-182	349	4	preliminaries	preliminary	NOUN
ma-182	349	5	2.1	2.1	NUM
ma-182	349	6	.	.	PUNCT
ma-182	350	1	the	the	DET
ma-182	350	2	beurling	beurling	NOUN
ma-182	350	3	spaces	space	VERB
ma-182	350	4	2.2	2.2	NUM
ma-182	350	5	.	.	PUNCT
ma-182	351	1	a	a	DET
ma-182	351	2	generalized	generalized	ADJ
ma-182	351	3	convolution	convolution	NOUN
ma-182	351	4	product	product	NOUN
ma-182	351	5	2.3	2.3	NUM
ma-182	351	6	.	.	PUNCT
ma-182	352	1	some	some	DET
ma-182	352	2	useful	useful	ADJ
ma-182	352	3	facts	fact	NOUN
ma-182	352	4	3	3	NUM
ma-182	352	5	.	.	PUNCT
ma-182	352	6	multipliers	multiplier	NOUN
ma-182	352	7	for	for	ADP
ma-182	352	8	the	the	DET
ma-182	352	9	pair	pair	NOUN
ma-182	352	10	(	(	PUNCT
ma-182	352	11	l1(g),lp(g	l1(g),lp(g	NOUN
ma-182	352	12	)	)	PUNCT
ma-182	352	13	)	)	PUNCT
ma-182	352	14	conclusion	conclusion	NOUN
ma-182	352	15	competing	compete	VERB
ma-182	352	16	interests	interest	NOUN
ma-182	352	17	references	reference	NOUN