id	sid	tid	token	lemma	pos
ma-220	1	1	2024	2024	NUM
ma-220	1	2	ada	ada	PROPN
ma-220	1	3	academica	academica	PROPN
ma-220	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-220	1	5	.	.	PUNCT
ma-220	2	1	j.	j.	PROPN
ma-220	2	2	math	math	PROPN
ma-220	2	3	.	.	PUNCT
ma-220	3	1	anal	anal	ADJ
ma-220	3	2	.	.	PUNCT
ma-220	4	1	4	4	NUM
ma-220	4	2	(	(	PUNCT
ma-220	4	3	2024	2024	NUM
ma-220	4	4	)	)	PUNCT
ma-220	5	1	4doi	4doi	NOUN
ma-220	5	2	:	:	PUNCT
ma-220	5	3	10.28924	10.28924	NUM
ma-220	5	4	/	/	SYM
ma-220	5	5	ada	ada	PROPN
ma-220	5	6	/	/	SYM
ma-220	5	7	ma.4.4	ma.4.4	NOUN
ma-220	5	8	duals	dual	NOUN
ma-220	5	9	of	of	ADP
ma-220	5	10	continuous	continuous	ADJ
ma-220	5	11	frames	frame	NOUN
ma-220	5	12	in	in	ADP
ma-220	5	13	hilbert	hilbert	PROPN
ma-220	5	14	c∗-modules	c∗-modules	PROPN
ma-220	5	15	mohamed	mohamed	PROPN
ma-220	5	16	rossafi1,∗	rossafi1,∗	PROPN
ma-220	5	17	,	,	PUNCT
ma-220	5	18	khadija	khadija	PROPN
ma-220	5	19	mabrouk2	mabrouk2	PROPN
ma-220	5	20	,	,	PUNCT
ma-220	5	21	m’hamed	m’hamed	PROPN
ma-220	5	22	ghiati2	ghiati2	PROPN
ma-220	5	23	,	,	PUNCT
ma-220	5	24	mohammed	mohammed	PROPN
ma-220	5	25	mouniane2	mouniane2	PROPN
ma-220	6	1	1department	1department	NUM
ma-220	6	2	of	of	ADP
ma-220	6	3	mathematics	mathematics	PROPN
ma-220	6	4	faculty	faculty	NOUN
ma-220	6	5	of	of	ADP
ma-220	6	6	sciences	science	NOUN
ma-220	6	7	,	,	PUNCT
ma-220	6	8	dhar	dhar	PROPN
ma-220	6	9	el	el	PROPN
ma-220	6	10	mahraz	mahraz	PROPN
ma-220	6	11	university	university	PROPN
ma-220	6	12	sidi	sidi	PROPN
ma-220	6	13	mohamed	mohamed	PROPN
ma-220	6	14	ben	ben	PROPN
ma-220	6	15	abdellah	abdellah	PROPN
ma-220	6	16	,	,	PUNCT
ma-220	6	17	fez	fez	PROPN
ma-220	6	18	,	,	PUNCT
ma-220	6	19	morocco	morocco	PROPN
ma-220	6	20	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-220	7	1	2department	2department	NUM
ma-220	7	2	of	of	ADP
ma-220	7	3	mathematics	mathematic	NOUN
ma-220	7	4	,	,	PUNCT
ma-220	7	5	faculty	faculty	NOUN
ma-220	7	6	of	of	ADP
ma-220	7	7	sciences	science	NOUN
ma-220	7	8	,	,	PUNCT
ma-220	7	9	university	university	NOUN
ma-220	7	10	of	of	ADP
ma-220	7	11	ibn	ibn	PROPN
ma-220	7	12	tofail	tofail	NOUN
ma-220	7	13	,	,	PUNCT
ma-220	7	14	kenitra	kenitra	PROPN
ma-220	7	15	,	,	PUNCT
ma-220	7	16	morocco	morocco	PROPN
ma-220	7	17	khadija.mabrouk@uit.ac.ma	khadija.mabrouk@uit.ac.ma	PROPN
ma-220	7	18	,	,	PUNCT
ma-220	7	19	mhamed.ghiati@uit.ac.ma	mhamed.ghiati@uit.ac.ma	PROPN
ma-220	7	20	,	,	PUNCT
ma-220	7	21	mouniane.mohammed@uit.ac.ma	mouniane.mohammed@uit.ac.ma	VERB
ma-220	7	22	∗correspondence	∗correspondence	NOUN
ma-220	7	23	:	:	PUNCT
ma-220	7	24	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-220	8	1	abstract	abstract	ADJ
ma-220	8	2	.	.	PUNCT
ma-220	9	1	the	the	DET
ma-220	9	2	concept	concept	NOUN
ma-220	9	3	of	of	ADP
ma-220	9	4	frame	frame	NOUN
ma-220	9	5	is	be	AUX
ma-220	9	6	an	an	DET
ma-220	9	7	exciting	exciting	ADJ
ma-220	9	8	,	,	PUNCT
ma-220	9	9	dynamic	dynamic	ADJ
ma-220	9	10	,	,	PUNCT
ma-220	9	11	and	and	CCONJ
ma-220	9	12	fast	fast	ADV
ma-220	9	13	-	-	PUNCT
ma-220	9	14	paced	pace	VERB
ma-220	9	15	subject	subject	NOUN
ma-220	9	16	with	with	ADP
ma-220	9	17	applications	application	NOUN
ma-220	9	18	innumerous	innumerous	ADJ
ma-220	9	19	fields	field	NOUN
ma-220	9	20	of	of	ADP
ma-220	9	21	mathematics	mathematic	NOUN
ma-220	9	22	and	and	CCONJ
ma-220	9	23	engineering	engineering	NOUN
ma-220	9	24	.	.	PUNCT
ma-220	10	1	the	the	DET
ma-220	10	2	purpose	purpose	NOUN
ma-220	10	3	of	of	ADP
ma-220	10	4	this	this	DET
ma-220	10	5	paper	paper	NOUN
ma-220	10	6	is	be	AUX
ma-220	10	7	to	to	PART
ma-220	10	8	introduce	introduce	VERB
ma-220	10	9	equiv	equiv	NOUN
ma-220	10	10	-	-	PUNCT
ma-220	10	11	alent	alent	NOUN
ma-220	10	12	∗-continuous	∗-continuous	ADJ
ma-220	10	13	frames	frame	NOUN
ma-220	10	14	and	and	CCONJ
ma-220	10	15	to	to	PART
ma-220	10	16	present	present	VERB
ma-220	10	17	ordinary	ordinary	ADJ
ma-220	10	18	duals	dual	NOUN
ma-220	10	19	of	of	ADP
ma-220	10	20	constructed	construct	VERB
ma-220	10	21	∗-continuous	∗-continuous	ADJ
ma-220	10	22	frames	frame	NOUN
ma-220	10	23	by	by	ADP
ma-220	10	24	anadjointable	anadjointable	ADJ
ma-220	10	25	and	and	CCONJ
ma-220	10	26	invertible	invertible	ADJ
ma-220	10	27	operator	operator	NOUN
ma-220	10	28	.	.	PUNCT
ma-220	11	1	also	also	ADV
ma-220	11	2	,	,	PUNCT
ma-220	11	3	we	we	PRON
ma-220	11	4	establish	establish	VERB
ma-220	11	5	some	some	DET
ma-220	11	6	properties	property	NOUN
ma-220	11	7	.	.	PUNCT
ma-220	12	1	1	1	X
ma-220	12	2	.	.	X
ma-220	12	3	introduction	introduction	NOUN
ma-220	12	4	frames	frame	NOUN
ma-220	12	5	in	in	ADP
ma-220	12	6	hilbert	hilbert	PROPN
ma-220	12	7	spaces	space	NOUN
ma-220	12	8	have	have	AUX
ma-220	12	9	been	be	AUX
ma-220	12	10	introduced	introduce	VERB
ma-220	12	11	by	by	ADP
ma-220	12	12	duffin	duffin	PROPN
ma-220	12	13	and	and	CCONJ
ma-220	12	14	schaeffer	schaeffer	VERB
ma-220	12	15	[	[	X
ma-220	12	16	3	3	NUM
ma-220	12	17	]	]	PUNCT
ma-220	12	18	in	in	ADP
ma-220	12	19	1952	1952	NUM
ma-220	12	20	to	to	PART
ma-220	12	21	studysome	studysome	VERB
ma-220	12	22	deep	deep	ADJ
ma-220	12	23	problems	problem	NOUN
ma-220	12	24	in	in	ADP
ma-220	12	25	nonharmonic	nonharmonic	ADJ
ma-220	12	26	fourier	fourier	NOUN
ma-220	12	27	series	series	NOUN
ma-220	12	28	.	.	PUNCT
ma-220	13	1	after	after	ADP
ma-220	13	2	the	the	DET
ma-220	13	3	fundamental	fundamental	ADJ
ma-220	13	4	paper	paper	NOUN
ma-220	13	5	,	,	PUNCT
ma-220	13	6	by	by	ADP
ma-220	13	7	daubechies	daubechie	NOUN
ma-220	13	8	,	,	PUNCT
ma-220	13	9	grossman	grossman	PROPN
ma-220	13	10	and	and	CCONJ
ma-220	13	11	meyer	meyer	PROPN
ma-220	14	1	[	[	X
ma-220	14	2	2	2	NUM
ma-220	14	3	]	]	PUNCT
ma-220	14	4	,	,	PUNCT
ma-220	14	5	frame	frame	NOUN
ma-220	14	6	theory	theory	NOUN
ma-220	14	7	began	begin	VERB
ma-220	14	8	to	to	PART
ma-220	14	9	be	be	AUX
ma-220	14	10	widely	widely	ADV
ma-220	14	11	used	use	VERB
ma-220	14	12	,	,	PUNCT
ma-220	14	13	particularly	particularly	ADV
ma-220	14	14	in	in	ADP
ma-220	14	15	the	the	DET
ma-220	14	16	more	more	ADV
ma-220	14	17	specializedcontext	specializedcontext	NOUN
ma-220	14	18	of	of	ADP
ma-220	14	19	wavelet	wavelet	NOUN
ma-220	14	20	frame	frame	NOUN
ma-220	14	21	and	and	CCONJ
ma-220	14	22	gabor	gabor	PROPN
ma-220	14	23	frame	frame	NOUN
ma-220	15	1	[	[	X
ma-220	15	2	4	4	NUM
ma-220	15	3	]	]	PUNCT
ma-220	15	4	.	.	PUNCT
ma-220	16	1	frames	frame	NOUN
ma-220	16	2	have	have	AUX
ma-220	16	3	been	be	AUX
ma-220	16	4	used	use	VERB
ma-220	16	5	in	in	ADP
ma-220	16	6	signal	signal	ADJ
ma-220	16	7	processing	processing	NOUN
ma-220	16	8	,	,	PUNCT
ma-220	16	9	imageprocessing	imageprocesse	VERB
ma-220	16	10	,	,	PUNCT
ma-220	16	11	data	datum	NOUN
ma-220	16	12	compression	compression	NOUN
ma-220	16	13	and	and	CCONJ
ma-220	16	14	sampling	sample	VERB
ma-220	16	15	theory	theory	NOUN
ma-220	16	16	.	.	PUNCT
ma-220	17	1	for	for	ADP
ma-220	17	2	more	more	ADJ
ma-220	17	3	about	about	ADP
ma-220	17	4	frames	frame	NOUN
ma-220	17	5	,	,	PUNCT
ma-220	17	6	see	see	VERB
ma-220	17	7	[	[	X
ma-220	17	8	5	5	NUM
ma-220	17	9	,	,	PUNCT
ma-220	17	10	7–11].in	7–11].in	NUM
ma-220	17	11	this	this	DET
ma-220	17	12	paper	paper	NOUN
ma-220	17	13	,	,	PUNCT
ma-220	17	14	we	we	PRON
ma-220	17	15	introduce	introduce	VERB
ma-220	17	16	the	the	DET
ma-220	17	17	notions	notion	NOUN
ma-220	17	18	of	of	ADP
ma-220	17	19	continuous	continuous	ADJ
ma-220	17	20	frame	frame	NOUN
ma-220	17	21	on	on	ADP
ma-220	17	22	a	a	DET
ma-220	17	23	hilbert	hilbert	NOUN
ma-220	17	24	c∗-module	c∗-module	NOUN
ma-220	17	25	over	over	ADP
ma-220	17	26	aunital	aunital	ADJ
ma-220	17	27	c∗-algebra	c∗-algebra	NOUN
ma-220	17	28	which	which	PRON
ma-220	17	29	is	be	AUX
ma-220	17	30	a	a	DET
ma-220	17	31	generalization	generalization	NOUN
ma-220	17	32	of	of	ADP
ma-220	17	33	discrete	discrete	ADJ
ma-220	17	34	frame	frame	NOUN
ma-220	17	35	,	,	PUNCT
ma-220	17	36	the	the	DET
ma-220	17	37	∗-continuous	∗-continuous	ADJ
ma-220	17	38	frame	frame	NOUN
ma-220	17	39	,	,	PUNCT
ma-220	17	40	which	which	PRON
ma-220	17	41	are	be	AUX
ma-220	17	42	ageneralization	ageneralization	NOUN
ma-220	17	43	of	of	ADP
ma-220	17	44	∗-frame	∗-frame	PROPN
ma-220	17	45	in	in	ADP
ma-220	17	46	hilbert	hilbert	PROPN
ma-220	17	47	c∗-modules	c∗-modules	PROPN
ma-220	17	48	and	and	CCONJ
ma-220	17	49	we	we	PRON
ma-220	17	50	establish	establish	VERB
ma-220	17	51	some	some	DET
ma-220	17	52	new	new	ADJ
ma-220	17	53	results.the	results.the	DET
ma-220	17	54	paper	paper	NOUN
ma-220	17	55	is	be	AUX
ma-220	17	56	organized	organize	VERB
ma-220	17	57	as	as	SCONJ
ma-220	17	58	follows	follow	VERB
ma-220	17	59	.	.	PUNCT
ma-220	18	1	we	we	PRON
ma-220	18	2	continue	continue	VERB
ma-220	18	3	this	this	DET
ma-220	18	4	introductory	introductory	ADJ
ma-220	18	5	section	section	NOUN
ma-220	18	6	and	and	CCONJ
ma-220	18	7	briefly	briefly	NOUN
ma-220	18	8	recall	recall	VERB
ma-220	18	9	thedefinitions	thedefinition	NOUN
ma-220	18	10	and	and	CCONJ
ma-220	18	11	basic	basic	ADJ
ma-220	18	12	properties	property	NOUN
ma-220	18	13	of	of	ADP
ma-220	18	14	hilbert	hilbert	PROPN
ma-220	18	15	c∗-modules	c∗-modules	PROPN
ma-220	18	16	.	.	PUNCT
ma-220	19	1	in	in	ADP
ma-220	19	2	section	section	NOUN
ma-220	19	3	2	2	NUM
ma-220	19	4	,	,	PUNCT
ma-220	19	5	the	the	DET
ma-220	19	6	generalized	generalized	ADJ
ma-220	19	7	duals	dual	NOUN
ma-220	19	8	fora	forum	NOUN
ma-220	19	9	given	give	VERB
ma-220	19	10	∗-continuous	∗-continuous	ADJ
ma-220	19	11	frame	frame	NOUN
ma-220	19	12	will	will	AUX
ma-220	19	13	be	be	AUX
ma-220	19	14	considered	consider	VERB
ma-220	19	15	.	.	PUNCT
ma-220	20	1	also	also	ADV
ma-220	20	2	,	,	PUNCT
ma-220	20	3	we	we	PRON
ma-220	20	4	study	study	VERB
ma-220	20	5	their	their	PRON
ma-220	20	6	properties	property	NOUN
ma-220	20	7	and	and	CCONJ
ma-220	20	8	characterizeall	characterizeall	ADJ
ma-220	20	9	operator	operator	NOUN
ma-220	20	10	dual	dual	ADJ
ma-220	20	11	∗-continuous	∗-continuous	ADJ
ma-220	20	12	frames	frame	NOUN
ma-220	20	13	associated	associate	VERB
ma-220	20	14	with	with	ADP
ma-220	20	15	the	the	DET
ma-220	20	16	given	give	VERB
ma-220	20	17	∗-continuous	∗-continuous	ADJ
ma-220	20	18	frame	frame	NOUN
ma-220	20	19	in	in	ADP
ma-220	20	20	hilbert	hilbert	PROPN
ma-220	20	21	c∗-modules	c∗-modules	PROPN
ma-220	20	22	.	.	PUNCT
ma-220	21	1	in	in	ADP
ma-220	21	2	section	section	NOUN
ma-220	21	3	3	3	NUM
ma-220	21	4	,	,	PUNCT
ma-220	21	5	we	we	PRON
ma-220	21	6	extend	extend	VERB
ma-220	21	7	this	this	DET
ma-220	21	8	notion	notion	NOUN
ma-220	21	9	for	for	ADP
ma-220	21	10	sequences	sequence	NOUN
ma-220	21	11	(	(	PUNCT
ma-220	21	12	continuous	continuous	ADJ
ma-220	21	13	frames	frame	NOUN
ma-220	21	14	)	)	PUNCT
ma-220	21	15	in	in	ADP
ma-220	21	16	hilbert	hilbert	PROPN
ma-220	21	17	c∗-modules	c∗-modules	PROPN
ma-220	21	18	.	.	PUNCT
ma-220	22	1	also	also	ADV
ma-220	22	2	,	,	PUNCT
ma-220	22	3	some	some	DET
ma-220	22	4	properties	property	NOUN
ma-220	22	5	of	of	ADP
ma-220	22	6	them	they	PRON
ma-220	22	7	will	will	AUX
ma-220	22	8	be	be	AUX
ma-220	22	9	studied	study	VERB
ma-220	22	10	.	.	PUNCT
ma-220	23	1	in	in	ADP
ma-220	23	2	section	section	NOUN
ma-220	23	3	4	4	NUM
ma-220	23	4	,	,	PUNCT
ma-220	23	5	a	a	DET
ma-220	23	6	∗-continuous	∗-continuous	ADJ
ma-220	23	7	frame	frame	NOUN
ma-220	23	8	isconstructed	isconstructe	VERB
ma-220	23	9	by	by	ADP
ma-220	23	10	an	an	DET
ma-220	23	11	orthogonal	orthogonal	ADJ
ma-220	23	12	projection	projection	NOUN
ma-220	23	13	.	.	PUNCT
ma-220	24	1	received	receive	VERB
ma-220	24	2	:	:	PUNCT
ma-220	24	3	21	21	NUM
ma-220	24	4	jan	jan	PROPN
ma-220	24	5	2024.2020	2024.2020	PROPN
ma-220	24	6	mathematics	mathematic	NOUN
ma-220	24	7	subject	subject	ADJ
ma-220	24	8	classification	classification	NOUN
ma-220	24	9	.	.	PUNCT
ma-220	25	1	42c15	42c15	NUM
ma-220	25	2	,	,	PUNCT
ma-220	25	3	41a58	41a58	NOUN
ma-220	25	4	.	.	PUNCT
ma-220	26	1	key	key	ADJ
ma-220	26	2	words	word	NOUN
ma-220	26	3	and	and	CCONJ
ma-220	26	4	phrases	phrase	NOUN
ma-220	26	5	.	.	PUNCT
ma-220	27	1	continuous	continuous	ADJ
ma-220	27	2	frame	frame	NOUN
ma-220	27	3	;	;	PUNCT
ma-220	27	4	∗-continuous	∗-continuous	ADJ
ma-220	27	5	frame	frame	NOUN
ma-220	27	6	;	;	PUNCT
ma-220	27	7	c∗-algebra	c∗-algebra	NOUN
ma-220	27	8	;	;	PUNCT
ma-220	27	9	hilbert	hilbert	NOUN
ma-220	27	10	c∗-module.1	c∗-module.1	NUM
ma-220	27	11	https://adac.ee	https://adac.ee	PROPN
ma-220	27	12	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	27	13	eur	eur	PROPN
ma-220	27	14	.	.	PUNCT
ma-220	28	1	j.	j.	PROPN
ma-220	28	2	math	math	PROPN
ma-220	28	3	.	.	PUNCT
ma-220	29	1	anal	anal	PROPN
ma-220	29	2	.	.	PUNCT
ma-220	30	1	10.28924	10.28924	NUM
ma-220	30	2	/	/	SYM
ma-220	30	3	ada	ada	PROPN
ma-220	30	4	/	/	SYM
ma-220	30	5	ma.4.4	ma.4.4	PROPN
ma-220	30	6	2	2	NUM
ma-220	30	7	definition	definition	NOUN
ma-220	30	8	1.1	1.1	NUM
ma-220	30	9	.	.	PUNCT
ma-220	31	1	[	[	X
ma-220	31	2	6	6	X
ma-220	31	3	]	]	PUNCT
ma-220	31	4	suppose	suppose	VERB
ma-220	31	5	that	that	SCONJ
ma-220	31	6	a	a	PRON
ma-220	31	7	is	be	AUX
ma-220	31	8	a	a	DET
ma-220	31	9	c∗-algebra	c∗-algebra	PROPN
ma-220	31	10	.	.	PUNCT
ma-220	31	11	a	a	DET
ma-220	31	12	linear	linear	ADJ
ma-220	31	13	space	space	NOUN
ma-220	31	14	h	h	NOUN
ma-220	31	15	which	which	PRON
ma-220	31	16	is	be	AUX
ma-220	31	17	also	also	ADV
ma-220	31	18	an	an	DET
ma-220	31	19	algebraicleft	algebraicleft	NOUN
ma-220	31	20	a	a	DET
ma-220	31	21	-	-	PUNCT
ma-220	31	22	module	module	NOUN
ma-220	31	23	together	together	ADV
ma-220	31	24	with	with	ADP
ma-220	31	25	an	an	DET
ma-220	31	26	a	a	PRON
ma-220	31	27	-	-	PUNCT
ma-220	31	28	inner	inner	ADJ
ma-220	31	29	product	product	NOUN
ma-220	31	30	〈	〈	PROPN
ma-220	31	31	·	·	SYM
ma-220	31	32	,	,	PUNCT
ma-220	31	33	·	·	PUNCT
ma-220	31	34	〉	〉	NOUN
ma-220	31	35	:	:	PUNCT
ma-220	31	36	h×h	h×h	PROPN
ma-220	31	37	−→	−→	NOUN
ma-220	31	38	a	a	PRON
ma-220	31	39	and	and	CCONJ
ma-220	31	40	possesses	possess	VERB
ma-220	31	41	the	the	DET
ma-220	31	42	followingproperties	followingpropertie	NOUN
ma-220	31	43	is	be	AUX
ma-220	31	44	called	call	VERB
ma-220	31	45	a	a	DET
ma-220	31	46	pre	pre	ADJ
ma-220	31	47	-	-	ADJ
ma-220	31	48	hilbert	hilbert	ADJ
ma-220	31	49	c∗-module:(1	c∗-module:(1	NOUN
ma-220	31	50	)	)	PUNCT
ma-220	31	51	〈	〈	PROPN
ma-220	31	52	f	f	PROPN
ma-220	31	53	,	,	PUNCT
ma-220	31	54	f	f	PROPN
ma-220	31	55	〉	〉	PROPN
ma-220	31	56	≥	≥	NUM
ma-220	31	57	0	0	NUM
ma-220	31	58	,	,	PUNCT
ma-220	31	59	for	for	ADP
ma-220	31	60	any	any	DET
ma-220	31	61	f	f	PROPN
ma-220	31	62	∈	∈	PROPN
ma-220	31	63	h;(2	h;(2	PROPN
ma-220	31	64	)	)	PUNCT
ma-220	32	1	〈	〈	PROPN
ma-220	32	2	f	f	PROPN
ma-220	32	3	,	,	PUNCT
ma-220	32	4	f	f	PROPN
ma-220	32	5	〉	〉	PROPN
ma-220	32	6	=	=	NOUN
ma-220	32	7	0	0	PUNCT
ma-220	33	1	if	if	SCONJ
ma-220	33	2	and	and	CCONJ
ma-220	33	3	only	only	ADV
ma-220	33	4	if	if	SCONJ
ma-220	33	5	f	f	PROPN
ma-220	33	6	=	=	SYM
ma-220	33	7	0;(3	0;(3	PROPN
ma-220	33	8	)	)	PUNCT
ma-220	34	1	〈	〈	PROPN
ma-220	34	2	f	f	PROPN
ma-220	34	3	,	,	PUNCT
ma-220	34	4	g	g	PROPN
ma-220	34	5	〉	〉	NOUN
ma-220	34	6	=	=	SYM
ma-220	34	7	〈	〈	PROPN
ma-220	34	8	g	g	NOUN
ma-220	34	9	,	,	PUNCT
ma-220	34	10	f	f	PROPN
ma-220	34	11	〉	〉	PROPN
ma-220	34	12	∗	∗	NOUN
ma-220	34	13	,	,	PUNCT
ma-220	34	14	for	for	ADP
ma-220	34	15	any	any	DET
ma-220	34	16	f	f	NOUN
ma-220	34	17	,	,	PUNCT
ma-220	34	18	g	g	PROPN
ma-220	34	19	∈	∈	PROPN
ma-220	34	20	h;(4	h;(4	PROPN
ma-220	34	21	)	)	PUNCT
ma-220	34	22	〈	〈	PROPN
ma-220	34	23	λf	λf	NOUN
ma-220	34	24	,	,	PUNCT
ma-220	34	25	h	h	NOUN
ma-220	34	26	〉	〉	NOUN
ma-220	34	27	=	=	SYM
ma-220	34	28	λ〈f	λ〈f	PROPN
ma-220	34	29	,	,	PUNCT
ma-220	34	30	h	h	NOUN
ma-220	34	31	〉	〉	PROPN
ma-220	34	32	,	,	PUNCT
ma-220	34	33	for	for	ADP
ma-220	34	34	any	any	DET
ma-220	34	35	λ	λ	PROPN
ma-220	34	36	∈	∈	PROPN
ma-220	34	37	c	c	PROPN
ma-220	34	38	and	and	CCONJ
ma-220	34	39	f	f	PROPN
ma-220	34	40	,	,	PUNCT
ma-220	34	41	h	h	PROPN
ma-220	34	42	∈	∈	PROPN
ma-220	34	43	h;(5	h;(5	PROPN
ma-220	34	44	)	)	PUNCT
ma-220	34	45	〈	〈	PROPN
ma-220	34	46	af	af	PROPN
ma-220	34	47	+	+	X
ma-220	34	48	bg	bg	PROPN
ma-220	34	49	,	,	PUNCT
ma-220	34	50	h	h	NOUN
ma-220	34	51	〉	〉	NUM
ma-220	34	52	=	=	SYM
ma-220	34	53	a〈f	a〈f	X
ma-220	34	54	,	,	PUNCT
ma-220	34	55	h〉+	h〉+	PROPN
ma-220	34	56	b〈g	b〈g	X
ma-220	34	57	,	,	PUNCT
ma-220	34	58	h	h	PROPN
ma-220	34	59	〉	〉	PROPN
ma-220	34	60	,	,	PUNCT
ma-220	34	61	for	for	ADP
ma-220	34	62	any	any	DET
ma-220	34	63	a	a	NOUN
ma-220	34	64	,	,	PUNCT
ma-220	34	65	b	b	X
ma-220	34	66	∈	∈	PROPN
ma-220	34	67	a	a	PRON
ma-220	34	68	and	and	CCONJ
ma-220	34	69	f	f	PROPN
ma-220	34	70	,	,	PUNCT
ma-220	34	71	g	g	PROPN
ma-220	34	72	,	,	PUNCT
ma-220	34	73	h	h	PROPN
ma-220	34	74	∈	∈	PROPN
ma-220	34	75	h.	h.	PROPN
ma-220	34	76	for	for	ADP
ma-220	34	77	x	x	PROPN
ma-220	34	78	∈	∈	PROPN
ma-220	34	79	h	h	NOUN
ma-220	34	80	,	,	PUNCT
ma-220	34	81	we	we	PRON
ma-220	34	82	define	define	VERB
ma-220	34	83	‖x‖	‖x‖	PROPN
ma-220	34	84	=	=	SYM
ma-220	34	85	||〈x	||〈x	PROPN
ma-220	34	86	,	,	PUNCT
ma-220	34	87	x〉||	x〉||	NUM
ma-220	34	88	1	1	NUM
ma-220	34	89	2	2	NUM
ma-220	34	90	.	.	PUNCT
ma-220	35	1	if	if	SCONJ
ma-220	35	2	h	h	NOUN
ma-220	35	3	is	be	AUX
ma-220	35	4	complete	complete	ADJ
ma-220	35	5	with	with	ADP
ma-220	35	6	||.||	||.||	NOUN
ma-220	35	7	,	,	PUNCT
ma-220	35	8	it	it	PRON
ma-220	35	9	is	be	AUX
ma-220	35	10	called	call	VERB
ma-220	35	11	a	a	DET
ma-220	35	12	hilbert	hilbert	NOUN
ma-220	35	13	a	a	DET
ma-220	35	14	-	-	PUNCT
ma-220	35	15	module	module	NOUN
ma-220	35	16	or	or	CCONJ
ma-220	35	17	a	a	DET
ma-220	35	18	hilbert	hilbert	NOUN
ma-220	35	19	c∗-module	c∗-module	NOUN
ma-220	35	20	over	over	ADP
ma-220	35	21	a.	a.	NOUN
ma-220	35	22	for	for	ADP
ma-220	35	23	every	every	DET
ma-220	35	24	a	a	PRON
ma-220	35	25	in	in	ADP
ma-220	35	26	c∗-algebra	c∗-algebra	PROPN
ma-220	35	27	a	a	PRON
ma-220	35	28	,	,	PUNCT
ma-220	35	29	we	we	PRON
ma-220	35	30	have	have	VERB
ma-220	35	31	|a|	|a|	NOUN
ma-220	35	32	=	=	SYM
ma-220	35	33	(	(	PUNCT
ma-220	35	34	a∗a	a∗a	X
ma-220	35	35	)	)	PUNCT
ma-220	35	36	1	1	NUM
ma-220	35	37	2	2	NUM
ma-220	35	38	andthe	andthe	NOUN
ma-220	35	39	a	a	ADV
ma-220	35	40	-	-	PUNCT
ma-220	35	41	valued	value	VERB
ma-220	35	42	norm	norm	NOUN
ma-220	35	43	on	on	ADP
ma-220	35	44	h	h	NOUN
ma-220	35	45	is	be	AUX
ma-220	35	46	defined	define	VERB
ma-220	35	47	by	by	ADP
ma-220	35	48	|x	|x	NOUN
ma-220	36	1	|	|	ADV
ma-220	36	2	=	=	SYM
ma-220	36	3	〈	〈	PROPN
ma-220	36	4	x	x	X
ma-220	36	5	,	,	PUNCT
ma-220	36	6	x	x	PROPN
ma-220	36	7	〉	〉	NUM
ma-220	36	8	1	1	NUM
ma-220	36	9	2	2	NUM
ma-220	36	10	for	for	ADP
ma-220	36	11	x	x	PROPN
ma-220	36	12	∈	∈	PROPN
ma-220	36	13	h.	h.	NOUN
ma-220	36	14	let	let	VERB
ma-220	36	15	h	h	NOUN
ma-220	36	16	and	and	CCONJ
ma-220	36	17	k	k	PROPN
ma-220	36	18	be	be	AUX
ma-220	36	19	two	two	NUM
ma-220	36	20	hilbert	hilbert	NOUN
ma-220	36	21	a	a	NOUN
ma-220	36	22	-	-	PUNCT
ma-220	36	23	modules	module	NOUN
ma-220	36	24	.	.	PUNCT
ma-220	37	1	a	a	DET
ma-220	37	2	map	map	NOUN
ma-220	37	3	t	t	NOUN
ma-220	37	4	:	:	PUNCT
ma-220	37	5	h	h	PROPN
ma-220	37	6	→	→	PUNCT
ma-220	37	7	k	k	PROPN
ma-220	37	8	is	be	AUX
ma-220	37	9	said	say	VERB
ma-220	37	10	to	to	PART
ma-220	37	11	be	be	AUX
ma-220	37	12	adjointable	adjointable	ADJ
ma-220	37	13	if	if	SCONJ
ma-220	37	14	there	there	PRON
ma-220	37	15	exists	exist	VERB
ma-220	37	16	a	a	DET
ma-220	37	17	map	map	NOUN
ma-220	37	18	t	t	NOUN
ma-220	37	19	∗	∗	NOUN
ma-220	37	20	:	:	PUNCT
ma-220	38	1	k	k	X
ma-220	38	2	→	→	PUNCT
ma-220	38	3	h	h	NOUN
ma-220	38	4	suchthat	suchthat	VERB
ma-220	38	5	〈	〈	PROPN
ma-220	38	6	tx	tx	PROPN
ma-220	38	7	,	,	PUNCT
ma-220	38	8	y〉a	y〉a	PROPN
ma-220	38	9	=	=	SYM
ma-220	38	10	〈	〈	PROPN
ma-220	38	11	x	x	PRON
ma-220	38	12	,	,	PUNCT
ma-220	38	13	t	t	PROPN
ma-220	38	14	∗y〉a	∗y〉a	PROPN
ma-220	38	15	for	for	ADP
ma-220	38	16	all	all	DET
ma-220	38	17	x	x	SYM
ma-220	38	18	∈	∈	PROPN
ma-220	38	19	h	h	NOUN
ma-220	38	20	and	and	CCONJ
ma-220	38	21	y	y	PROPN
ma-220	38	22	∈	∈	PROPN
ma-220	38	23	k.	k.	PROPN
ma-220	38	24	lemma	lemma	PROPN
ma-220	38	25	1.2	1.2	NUM
ma-220	38	26	.	.	PUNCT
ma-220	39	1	[	[	X
ma-220	39	2	12	12	NUM
ma-220	39	3	]	]	X
ma-220	39	4	let	let	ADJ
ma-220	39	5	(	(	PUNCT
ma-220	39	6	ω,µ	ω,µ	NOUN
ma-220	39	7	)	)	PUNCT
ma-220	39	8	be	be	VERB
ma-220	39	9	a	a	DET
ma-220	39	10	measure	measure	NOUN
ma-220	39	11	space	space	NOUN
ma-220	39	12	,	,	PUNCT
ma-220	39	13	x	x	PUNCT
ma-220	39	14	and	and	CCONJ
ma-220	39	15	y	y	PROPN
ma-220	39	16	be	be	AUX
ma-220	39	17	two	two	NUM
ma-220	39	18	banach	banach	NOUN
ma-220	39	19	spaces	space	NOUN
ma-220	39	20	,	,	PUNCT
ma-220	39	21	λ	λ	X
ma-220	39	22	:	:	PUNCT
ma-220	39	23	x	x	X
ma-220	39	24	→	→	SYM
ma-220	39	25	y	y	X
ma-220	39	26	be	be	AUX
ma-220	39	27	a	a	DET
ma-220	39	28	bounded	bounded	ADJ
ma-220	39	29	linear	linear	ADJ
ma-220	39	30	operator	operator	NOUN
ma-220	39	31	and	and	CCONJ
ma-220	39	32	f	f	NOUN
ma-220	39	33	:	:	PUNCT
ma-220	39	34	ω	ω	PROPN
ma-220	39	35	→	→	SYM
ma-220	39	36	y	y	PROPN
ma-220	39	37	be	be	AUX
ma-220	39	38	a	a	DET
ma-220	39	39	measurable	measurable	ADJ
ma-220	39	40	function	function	NOUN
ma-220	39	41	.	.	PUNCT
ma-220	40	1	then	then	ADV
ma-220	40	2	λ	λ	PROPN
ma-220	40	3	(	(	PUNCT
ma-220	40	4	∫	∫	PROPN
ma-220	40	5	ω	ω	PROPN
ma-220	40	6	f	f	PROPN
ma-220	40	7	dµ	dµ	PROPN
ma-220	40	8	)	)	PUNCT
ma-220	40	9	=	=	SYM
ma-220	40	10	∫	∫	PROPN
ma-220	40	11	ω	ω	PROPN
ma-220	40	12	(	(	PUNCT
ma-220	40	13	λf	λf	PROPN
ma-220	40	14	)	)	PUNCT
ma-220	40	15	dµ.	dµ.	PROPN
ma-220	40	16	lemma	lemma	PROPN
ma-220	40	17	1.3	1.3	NUM
ma-220	40	18	.	.	PUNCT
ma-220	41	1	[	[	X
ma-220	41	2	1	1	X
ma-220	41	3	]	]	PUNCT
ma-220	41	4	let	let	VERB
ma-220	41	5	h	h	NOUN
ma-220	41	6	and	and	CCONJ
ma-220	41	7	k	k	PROPN
ma-220	41	8	be	be	AUX
ma-220	41	9	two	two	NUM
ma-220	41	10	hilbert	hilbert	NOUN
ma-220	41	11	a	a	NOUN
ma-220	41	12	-	-	PUNCT
ma-220	41	13	modules	module	NOUN
ma-220	41	14	and	and	CCONJ
ma-220	41	15	t	t	NOUN
ma-220	41	16	∈	∈	PROPN
ma-220	41	17	end∗(h	end∗(h	PROPN
ma-220	41	18	,	,	PUNCT
ma-220	41	19	k).(i	k).(i	PROPN
ma-220	41	20	)	)	PUNCT
ma-220	41	21	if	if	SCONJ
ma-220	41	22	t	t	PROPN
ma-220	41	23	is	be	AUX
ma-220	41	24	injective	injective	ADJ
ma-220	41	25	and	and	CCONJ
ma-220	41	26	t	t	PROPN
ma-220	41	27	has	have	VERB
ma-220	41	28	a	a	DET
ma-220	41	29	closed	closed	ADJ
ma-220	41	30	range	range	NOUN
ma-220	41	31	,	,	PUNCT
ma-220	41	32	then	then	ADV
ma-220	41	33	the	the	DET
ma-220	41	34	adjointable	adjointable	NOUN
ma-220	41	35	map	map	NOUN
ma-220	41	36	t	t	PROPN
ma-220	41	37	∗t	∗t	PROPN
ma-220	41	38	is	be	AUX
ma-220	41	39	invertible	invertible	ADJ
ma-220	41	40	and	and	CCONJ
ma-220	41	41	‖(t	‖(t	PUNCT
ma-220	41	42	∗t	∗t	ADJ
ma-220	41	43	)	)	PUNCT
ma-220	41	44	−1‖−1	−1‖−1	NOUN
ma-220	41	45	≤	≤	NUM
ma-220	41	46	t	t	X
ma-220	41	47	∗t	∗t	PROPN
ma-220	41	48	≤	≤	PROPN
ma-220	41	49	‖t‖2	‖t‖2	NOUN
ma-220	41	50	.	.	PUNCT
ma-220	42	1	(	(	PUNCT
ma-220	42	2	ii	ii	NOUN
ma-220	42	3	)	)	PUNCT
ma-220	42	4	if	if	SCONJ
ma-220	42	5	t	t	PROPN
ma-220	42	6	is	be	AUX
ma-220	42	7	surjective	surjective	ADJ
ma-220	42	8	,	,	PUNCT
ma-220	42	9	then	then	ADV
ma-220	42	10	the	the	DET
ma-220	42	11	adjointable	adjointable	NOUN
ma-220	42	12	map	map	NOUN
ma-220	42	13	tt	tt	PROPN
ma-220	42	14	∗	∗	NOUN
ma-220	42	15	is	be	AUX
ma-220	42	16	invertible	invertible	ADJ
ma-220	42	17	and	and	CCONJ
ma-220	42	18	‖(tt	‖(tt	PROPN
ma-220	42	19	∗)−1‖−1	∗)−1‖−1	PROPN
ma-220	42	20	≤	≤	ADV
ma-220	42	21	tt	tt	PROPN
ma-220	42	22	∗	∗	VERB
ma-220	42	23	≤	≤	NUM
ma-220	42	24	‖t‖2	‖t‖2	NOUN
ma-220	42	25	.	.	PUNCT
ma-220	43	1	the	the	DET
ma-220	43	2	following	follow	VERB
ma-220	43	3	definition	definition	NOUN
ma-220	43	4	was	be	AUX
ma-220	43	5	introduced	introduce	VERB
ma-220	43	6	in	in	ADP
ma-220	43	7	[	[	X
ma-220	43	8	11	11	NUM
ma-220	43	9	]	]	PUNCT
ma-220	43	10	.	.	PUNCT
ma-220	44	1	definition	definition	NOUN
ma-220	44	2	1.4	1.4	NUM
ma-220	44	3	.	.	PUNCT
ma-220	45	1	let	let	VERB
ma-220	45	2	h	h	PRON
ma-220	45	3	be	be	AUX
ma-220	45	4	a	a	DET
ma-220	45	5	hilbert	hilbert	NOUN
ma-220	45	6	a	a	DET
ma-220	45	7	-	-	PUNCT
ma-220	45	8	module	module	NOUN
ma-220	45	9	and	and	CCONJ
ma-220	45	10	(	(	PUNCT
ma-220	45	11	ω	ω	PROPN
ma-220	45	12	,	,	PUNCT
ma-220	45	13	µ	µ	NOUN
ma-220	45	14	)	)	PUNCT
ma-220	45	15	be	be	AUX
ma-220	45	16	a	a	DET
ma-220	45	17	measure	measure	NOUN
ma-220	45	18	space	space	NOUN
ma-220	45	19	.	.	PUNCT
ma-220	46	1	a	a	DET
ma-220	46	2	map	map	NOUN
ma-220	46	3	f	f	X
ma-220	46	4	:	:	PUNCT
ma-220	46	5	ω→	ω→	PUNCT
ma-220	47	1	his	his	PRON
ma-220	47	2	called	call	VERB
ma-220	47	3	a	a	DET
ma-220	47	4	∗-continuous	∗-continuous	ADJ
ma-220	47	5	frame	frame	NOUN
ma-220	47	6	with	with	ADP
ma-220	47	7	respect	respect	NOUN
ma-220	47	8	to	to	ADP
ma-220	47	9	(	(	PUNCT
ma-220	47	10	ω	ω	PROPN
ma-220	47	11	,	,	PUNCT
ma-220	47	12	µ	µ	NOUN
ma-220	47	13	)	)	PUNCT
ma-220	47	14	if1	if1	NOUN
ma-220	47	15	.	.	PUNCT
ma-220	48	1	for	for	ADP
ma-220	48	2	all	all	DET
ma-220	48	3	f	f	PROPN
ma-220	48	4	∈	∈	PROPN
ma-220	48	5	h	h	NOUN
ma-220	48	6	,	,	PUNCT
ma-220	48	7	w	w	PROPN
ma-220	48	8	→	→	SYM
ma-220	48	9	〈	〈	PROPN
ma-220	48	10	f	f	PROPN
ma-220	48	11	,	,	PUNCT
ma-220	48	12	fw	fw	PROPN
ma-220	48	13	〉	〉	PROPN
ma-220	48	14	is	be	AUX
ma-220	48	15	a	a	DET
ma-220	48	16	measurable	measurable	ADJ
ma-220	48	17	function	function	NOUN
ma-220	48	18	on	on	ADP
ma-220	48	19	ω,2	ω,2	NUM
ma-220	48	20	.	.	PUNCT
ma-220	49	1	there	there	PRON
ma-220	49	2	exist	exist	VERB
ma-220	49	3	two	two	NUM
ma-220	49	4	strictly	strictly	ADV
ma-220	49	5	nonzero	nonzero	ADJ
ma-220	49	6	elements	element	NOUN
ma-220	49	7	a	a	DET
ma-220	49	8	,	,	PUNCT
ma-220	49	9	b	b	X
ma-220	49	10	>	>	X
ma-220	49	11	0	0	NUM
ma-220	49	12	in	in	ADP
ma-220	49	13	a	a	DET
ma-220	49	14	such	such	ADJ
ma-220	49	15	that	that	PRON
ma-220	49	16	a〈f	a〈f	PROPN
ma-220	49	17	,	,	PUNCT
ma-220	49	18	f	f	PROPN
ma-220	49	19	〉	〉	PROPN
ma-220	49	20	a∗	a∗	PROPN
ma-220	49	21	≤	≤	NUM
ma-220	49	22	∫	∫	PROPN
ma-220	49	23	ω	ω	PROPN
ma-220	50	1	〈	〈	PROPN
ma-220	50	2	f	f	PROPN
ma-220	50	3	,	,	PUNCT
ma-220	50	4	fw	fw	PROPN
ma-220	50	5	〉	〉	PROPN
ma-220	50	6	〈	〈	PROPN
ma-220	50	7	fw	fw	PROPN
ma-220	50	8	,	,	PUNCT
ma-220	50	9	f	f	PROPN
ma-220	50	10	〉	〉	PROPN
ma-220	50	11	dµ(w	dµ(w	PUNCT
ma-220	50	12	)	)	PUNCT
ma-220	50	13	≤	≤	NUM
ma-220	50	14	b〈f	b〈f	PUNCT
ma-220	50	15	,	,	PUNCT
ma-220	50	16	f	f	PROPN
ma-220	50	17	〉	〉	PROPN
ma-220	50	18	b∗,∀f	b∗,∀f	PROPN
ma-220	50	19	∈	∈	PROPN
ma-220	50	20	h.	h.	PROPN
ma-220	50	21	(	(	PUNCT
ma-220	50	22	1.1	1.1	NUM
ma-220	50	23	)	)	PUNCT
ma-220	50	24	the	the	DET
ma-220	50	25	elements	element	NOUN
ma-220	50	26	a	a	PRON
ma-220	50	27	and	and	CCONJ
ma-220	50	28	b	b	NOUN
ma-220	50	29	are	be	AUX
ma-220	50	30	called	call	VERB
ma-220	50	31	∗-continuous	∗-continuous	ADJ
ma-220	50	32	frame	frame	NOUN
ma-220	50	33	bounds	bound	NOUN
ma-220	50	34	.	.	PUNCT
ma-220	51	1	if	if	SCONJ
ma-220	51	2	a	a	DET
ma-220	51	3	=	=	SYM
ma-220	51	4	b	b	NOUN
ma-220	51	5	,	,	PUNCT
ma-220	51	6	we	we	PRON
ma-220	51	7	call	call	VERB
ma-220	51	8	this	this	DET
ma-220	51	9	∗-continuousframe	∗-continuousframe	NOUN
ma-220	51	10	a	a	DET
ma-220	51	11	tight	tight	ADJ
ma-220	51	12	∗-continuous	∗-continuous	ADJ
ma-220	51	13	frame	frame	NOUN
ma-220	51	14	,	,	PUNCT
ma-220	51	15	and	and	CCONJ
ma-220	51	16	if	if	SCONJ
ma-220	51	17	a	a	DET
ma-220	51	18	=	=	SYM
ma-220	51	19	b	b	NOUN
ma-220	51	20	=	=	SYM
ma-220	51	21	1	1	NUM
ma-220	51	22	,	,	PUNCT
ma-220	51	23	it	it	PRON
ma-220	51	24	is	be	AUX
ma-220	51	25	called	call	VERB
ma-220	51	26	a	a	DET
ma-220	51	27	parseval	parseval	NOUN
ma-220	51	28	∗-continuous	∗-continuous	ADJ
ma-220	51	29	frame	frame	NOUN
ma-220	51	30	.	.	PUNCT
ma-220	52	1	ifonly	ifonly	ADV
ma-220	52	2	the	the	DET
ma-220	52	3	right	right	ADJ
ma-220	52	4	-	-	PUNCT
ma-220	52	5	hand	hand	NOUN
ma-220	52	6	inequality	inequality	NOUN
ma-220	52	7	of	of	ADP
ma-220	52	8	(	(	PUNCT
ma-220	52	9	1.1	1.1	NUM
ma-220	52	10	)	)	PUNCT
ma-220	52	11	is	be	AUX
ma-220	52	12	satisfied	satisfied	ADJ
ma-220	52	13	,	,	PUNCT
ma-220	52	14	we	we	PRON
ma-220	52	15	call	call	VERB
ma-220	52	16	f	f	X
ma-220	52	17	:	:	PUNCT
ma-220	52	18	ω→	ω→	PUNCT
ma-220	52	19	h	h	PROPN
ma-220	52	20	a	a	DET
ma-220	52	21	∗-continuous	∗-continuous	ADJ
ma-220	52	22	bessel	bessel	ADJ
ma-220	52	23	mapwith	mapwith	ADJ
ma-220	52	24	bessel	bessel	NOUN
ma-220	52	25	bound	bind	VERB
ma-220	52	26	b.	b.	PROPN
ma-220	52	27	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	52	28	eur	eur	PROPN
ma-220	52	29	.	.	PUNCT
ma-220	53	1	j.	j.	PROPN
ma-220	53	2	math	math	PROPN
ma-220	53	3	.	.	PUNCT
ma-220	54	1	anal	anal	PROPN
ma-220	54	2	.	.	PUNCT
ma-220	55	1	10.28924	10.28924	NUM
ma-220	55	2	/	/	SYM
ma-220	55	3	ada	ada	PROPN
ma-220	55	4	/	/	SYM
ma-220	55	5	ma.4.4	ma.4.4	PROPN
ma-220	55	6	3let	3let	NUM
ma-220	55	7	x	x	AUX
ma-220	55	8	be	be	AUX
ma-220	55	9	a	a	DET
ma-220	55	10	banach	banach	NOUN
ma-220	55	11	space	space	NOUN
ma-220	55	12	,	,	PUNCT
ma-220	55	13	(	(	PUNCT
ma-220	55	14	ω	ω	PROPN
ma-220	55	15	,	,	PUNCT
ma-220	55	16	µ	µ	NOUN
ma-220	55	17	)	)	PUNCT
ma-220	55	18	be	be	AUX
ma-220	55	19	a	a	DET
ma-220	55	20	measure	measure	NOUN
ma-220	55	21	space	space	NOUN
ma-220	55	22	and	and	CCONJ
ma-220	55	23	f	f	NOUN
ma-220	55	24	:	:	PUNCT
ma-220	55	25	ω→	ω→	PUNCT
ma-220	55	26	x	x	PUNCT
ma-220	55	27	be	be	AUX
ma-220	55	28	a	a	DET
ma-220	55	29	measurable	measurable	ADJ
ma-220	55	30	function.integral	function.integral	NOUN
ma-220	55	31	of	of	ADP
ma-220	55	32	the	the	DET
ma-220	55	33	banach	banach	ADV
ma-220	55	34	-	-	PUNCT
ma-220	55	35	valued	value	VERB
ma-220	55	36	function	function	NOUN
ma-220	55	37	f	f	PROPN
ma-220	55	38	has	have	AUX
ma-220	55	39	been	be	AUX
ma-220	55	40	defined	define	VERB
ma-220	55	41	by	by	ADP
ma-220	55	42	bochner	bochner	NOUN
ma-220	55	43	and	and	CCONJ
ma-220	55	44	others	other	NOUN
ma-220	55	45	.	.	PUNCT
ma-220	56	1	most	most	ADJ
ma-220	56	2	propertiesof	propertiesof	NOUN
ma-220	56	3	this	this	DET
ma-220	56	4	integral	integral	ADJ
ma-220	56	5	are	be	AUX
ma-220	56	6	similar	similar	ADJ
ma-220	56	7	to	to	ADP
ma-220	56	8	those	those	PRON
ma-220	56	9	of	of	ADP
ma-220	56	10	the	the	DET
ma-220	56	11	integral	integral	ADJ
ma-220	56	12	of	of	ADP
ma-220	56	13	real	real	ADV
ma-220	56	14	-	-	PUNCT
ma-220	56	15	valued	value	VERB
ma-220	56	16	functions	function	NOUN
ma-220	56	17	.	.	PUNCT
ma-220	57	1	since	since	SCONJ
ma-220	57	2	every	every	DET
ma-220	57	3	c∗-algebraand	c∗-algebraand	NOUN
ma-220	57	4	hilbert	hilbert	NOUN
ma-220	57	5	c∗-module	c∗-module	NOUN
ma-220	57	6	is	be	AUX
ma-220	57	7	a	a	DET
ma-220	57	8	banach	banach	NOUN
ma-220	57	9	space	space	NOUN
ma-220	57	10	,	,	PUNCT
ma-220	57	11	we	we	PRON
ma-220	57	12	can	can	AUX
ma-220	57	13	use	use	VERB
ma-220	57	14	this	this	DET
ma-220	57	15	integral	integral	ADJ
ma-220	57	16	and	and	CCONJ
ma-220	57	17	its	its	PRON
ma-220	57	18	properties.let	properties.let	NOUN
ma-220	57	19	(	(	PUNCT
ma-220	57	20	ω	ω	PROPN
ma-220	57	21	,	,	PUNCT
ma-220	57	22	µ	µ	NOUN
ma-220	57	23	)	)	PUNCT
ma-220	57	24	be	be	AUX
ma-220	57	25	a	a	DET
ma-220	57	26	measure	measure	NOUN
ma-220	57	27	space	space	NOUN
ma-220	57	28	.	.	PUNCT
ma-220	58	1	we	we	PRON
ma-220	58	2	define	define	VERB
ma-220	58	3	l2(ω	l2(ω	ADP
ma-220	58	4	,	,	PUNCT
ma-220	58	5	a	a	PRON
ma-220	58	6	)	)	PUNCT
ma-220	58	7	=	=	PRON
ma-220	58	8	{	{	PUNCT
ma-220	58	9	ϕ	ϕ	NOUN
ma-220	58	10	:	:	PUNCT
ma-220	58	11	ω→	ω→	X
ma-220	58	12	a	a	DET
ma-220	58	13	:	:	PUNCT
ma-220	58	14	∥∥∥∥∫	∥∥∥∥∫	PROPN
ma-220	58	15	ω	ω	PROPN
ma-220	58	16	ϕ(ω)ϕ(ω)∗dµ(ω	ϕ(ω)ϕ(ω)∗dµ(ω	PRON
ma-220	58	17	)	)	PUNCT
ma-220	58	18	∥∥∥∥	∥∥∥∥	PUNCT
ma-220	59	1	<	<	X
ma-220	59	2	∞	∞	NUM
ma-220	59	3	}	}	PUNCT
ma-220	59	4	.	.	PUNCT
ma-220	60	1	for	for	ADP
ma-220	60	2	any	any	DET
ma-220	60	3	ϕ,ψ	ϕ,ψ	PROPN
ma-220	60	4	∈	∈	PROPN
ma-220	60	5	l2(ω	l2(ω	PROPN
ma-220	60	6	,	,	PUNCT
ma-220	60	7	a	a	PRON
ma-220	60	8	)	)	PUNCT
ma-220	60	9	,	,	PUNCT
ma-220	60	10	if	if	SCONJ
ma-220	60	11	the	the	DET
ma-220	60	12	a	a	ADV
ma-220	60	13	-	-	PUNCT
ma-220	60	14	valued	value	VERB
ma-220	60	15	inner	inner	ADJ
ma-220	60	16	product	product	NOUN
ma-220	60	17	is	be	AUX
ma-220	60	18	defined	define	VERB
ma-220	60	19	by	by	ADP
ma-220	60	20	〈	〈	PROPN
ma-220	60	21	ϕ,ψ	ϕ,ψ	PROPN
ma-220	60	22	〉	〉	PROPN
ma-220	60	23	=	=	SYM
ma-220	60	24	∫	∫	PROPN
ma-220	60	25	ω	ω	NUM
ma-220	60	26	ϕ(ω)ψ(ω)∗dµ(w	ϕ(ω)ψ(ω)∗dµ(w	PROPN
ma-220	60	27	)	)	PUNCT
ma-220	60	28	,	,	PUNCT
ma-220	60	29	the	the	DET
ma-220	60	30	norm	norm	NOUN
ma-220	60	31	is	be	AUX
ma-220	60	32	defined	define	VERB
ma-220	60	33	by	by	ADP
ma-220	60	34	‖ϕ‖	‖ϕ‖	PROPN
ma-220	60	35	=	=	PUNCT
ma-220	60	36	‖〈ϕ,ϕ〉‖	‖〈ϕ,ϕ〉‖	NOUN
ma-220	60	37	1	1	NUM
ma-220	60	38	2	2	NUM
ma-220	60	39	,	,	PUNCT
ma-220	60	40	then	then	ADV
ma-220	60	41	l2(ω	l2(ω	PROPN
ma-220	60	42	,	,	PUNCT
ma-220	60	43	a	a	PRON
ma-220	60	44	)	)	PUNCT
ma-220	60	45	is	be	AUX
ma-220	60	46	a	a	DET
ma-220	60	47	hilbert	hilbert	NOUN
ma-220	60	48	c∗-module.the	c∗-module.the	DET
ma-220	60	49	frame	frame	NOUN
ma-220	60	50	transform	transform	VERB
ma-220	60	51	or	or	CCONJ
ma-220	60	52	pre	pre	ADJ
ma-220	60	53	-	-	ADJ
ma-220	60	54	frame	frame	ADJ
ma-220	60	55	operator	operator	NOUN
ma-220	60	56	t	t	NOUN
ma-220	60	57	:	:	PUNCT
ma-220	60	58	h	h	PROPN
ma-220	60	59	−→	−→	NOUN
ma-220	60	60	l2(ω	l2(ω	PROPN
ma-220	60	61	,	,	PUNCT
ma-220	60	62	a	a	PRON
ma-220	60	63	)	)	PUNCT
ma-220	60	64	is	be	AUX
ma-220	60	65	defined	define	VERB
ma-220	60	66	by	by	ADP
ma-220	60	67	t	t	PROPN
ma-220	60	68	(	(	PUNCT
ma-220	60	69	f	f	PROPN
ma-220	60	70	)	)	PUNCT
ma-220	61	1	=	=	PRON
ma-220	61	2	{	{	PUNCT
ma-220	61	3	〈	〈	PROPN
ma-220	61	4	f	f	PROPN
ma-220	61	5	,	,	PUNCT
ma-220	61	6	fw	fw	PROPN
ma-220	61	7	〉	〉	PROPN
ma-220	61	8	}	}	PUNCT
ma-220	61	9	w∈ωand	w∈ωand	NOUN
ma-220	61	10	it	it	PRON
ma-220	61	11	is	be	AUX
ma-220	61	12	an	an	DET
ma-220	61	13	injective	injective	ADJ
ma-220	61	14	and	and	CCONJ
ma-220	61	15	closed	closed	ADJ
ma-220	61	16	range	range	NOUN
ma-220	61	17	adjointable	adjointable	NOUN
ma-220	61	18	a	a	DET
ma-220	61	19	-	-	PUNCT
ma-220	61	20	module	module	NOUN
ma-220	61	21	map	map	NOUN
ma-220	61	22	and	and	CCONJ
ma-220	61	23	‖t‖	‖t‖	PROPN
ma-220	61	24	≤	≤	NOUN
ma-220	61	25	‖b‖.	‖b‖.	PUNCT
ma-220	61	26	the	the	DET
ma-220	61	27	adjointoperator	adjointoperator	NOUN
ma-220	61	28	t	t	PROPN
ma-220	61	29	∗	∗	NOUN
ma-220	61	30	is	be	AUX
ma-220	61	31	surjective	surjective	ADJ
ma-220	61	32	and	and	CCONJ
ma-220	61	33	it	it	PRON
ma-220	61	34	is	be	AUX
ma-220	61	35	given	give	VERB
ma-220	61	36	by	by	ADP
ma-220	61	37	t	t	PROPN
ma-220	61	38	∗	∗	NOUN
ma-220	61	39	(	(	PUNCT
ma-220	61	40	ew	ew	NOUN
ma-220	61	41	)	)	PUNCT
ma-220	62	1	=	=	PUNCT
ma-220	62	2	fw	fw	PROPN
ma-220	62	3	for	for	ADP
ma-220	62	4	w	w	PROPN
ma-220	62	5	∈	∈	PROPN
ma-220	62	6	ω	ω	PROPN
ma-220	62	7	,	,	PUNCT
ma-220	62	8	where	where	SCONJ
ma-220	62	9	{	{	PUNCT
ma-220	62	10	ew}w∈ω	ew}w∈ω	PROPN
ma-220	62	11	is	be	AUX
ma-220	62	12	thestandard	thestandard	NOUN
ma-220	62	13	basis	basis	NOUN
ma-220	62	14	for	for	ADP
ma-220	62	15	l2(ω	l2(ω	PROPN
ma-220	62	16	,	,	PUNCT
ma-220	62	17	a	a	PRON
ma-220	62	18	)	)	PUNCT
ma-220	62	19	.	.	PUNCT
ma-220	63	1	definition	definition	NOUN
ma-220	63	2	1.5	1.5	NUM
ma-220	63	3	.	.	PUNCT
ma-220	64	1	let	let	VERB
ma-220	64	2	f	f	PRON
ma-220	64	3	be	be	AUX
ma-220	64	4	a	a	DET
ma-220	64	5	continuous	continuous	ADJ
ma-220	64	6	frame	frame	NOUN
ma-220	64	7	for	for	ADP
ma-220	64	8	h	h	NOUN
ma-220	64	9	with	with	ADP
ma-220	64	10	respect	respect	NOUN
ma-220	64	11	to	to	ADP
ma-220	64	12	(	(	PUNCT
ma-220	64	13	ω	ω	PROPN
ma-220	64	14	,	,	PUNCT
ma-220	64	15	µ	µ	NOUN
ma-220	64	16	)	)	PUNCT
ma-220	64	17	.	.	PUNCT
ma-220	65	1	we	we	PRON
ma-220	65	2	define	define	VERB
ma-220	65	3	the	the	DET
ma-220	65	4	frameoperator	frameoperator	NOUN
ma-220	65	5	s	s	PART
ma-220	65	6	:	:	PUNCT
ma-220	65	7	h	h	NOUN
ma-220	65	8	→	→	SYM
ma-220	65	9	h	h	NOUN
ma-220	65	10	by	by	ADP
ma-220	65	11	sx	sx	PROPN
ma-220	65	12	=	=	SYM
ma-220	65	13	t	t	PROPN
ma-220	65	14	∗ftf	∗ftf	PROPN
ma-220	65	15	x	x	X
ma-220	65	16	=	=	SYM
ma-220	65	17	∫	∫	PROPN
ma-220	65	18	ω〈x	ω〈x	PROPN
ma-220	65	19	,	,	PUNCT
ma-220	65	20	fw	fw	PROPN
ma-220	65	21	〉	〉	PROPN
ma-220	65	22	fwdµ(w),∀x	fwdµ(w),∀x	NOUN
ma-220	65	23	∈	∈	PROPN
ma-220	65	24	h	h	NOUN
ma-220	65	25	,	,	PUNCT
ma-220	65	26	that	that	PRON
ma-220	65	27	is	be	AUX
ma-220	65	28	positive	positive	ADJ
ma-220	65	29	,	,	PUNCT
ma-220	65	30	invertibleand	invertibleand	ADJ
ma-220	65	31	adjointable	adjointable	NOUN
ma-220	65	32	and	and	CCONJ
ma-220	65	33	the	the	DET
ma-220	65	34	inequality	inequality	NOUN
ma-220	65	35	∥∥a−1	∥∥a−1	NOUN
ma-220	65	36	∥∥−2	∥∥−2	NUM
ma-220	65	37	≤	≤	PROPN
ma-220	65	38	‖s‖	‖s‖	PROPN
ma-220	65	39	≤	≤	PROPN
ma-220	65	40	‖b‖2	‖b‖2	ADJ
ma-220	65	41	holds	hold	NOUN
ma-220	65	42	,	,	PUNCT
ma-220	65	43	and	and	CCONJ
ma-220	65	44	the	the	DET
ma-220	65	45	reconstruction	reconstruction	NOUN
ma-220	65	46	formula	formula	NOUN
ma-220	66	1	f	f	NOUN
ma-220	66	2	=	=	SYM
ma-220	66	3	∫	∫	PROPN
ma-220	66	4	ω	ω	NUM
ma-220	66	5	〈	〈	PROPN
ma-220	66	6	f	f	PROPN
ma-220	66	7	,	,	PUNCT
ma-220	66	8	s−1fw	s−1fw	ADJ
ma-220	66	9	〉	〉	NOUN
ma-220	66	10	fwdµ(w	fwdµ(w	NOUN
ma-220	66	11	)	)	PUNCT
ma-220	66	12	holds	hold	VERB
ma-220	66	13	for	for	ADP
ma-220	66	14	all	all	DET
ma-220	66	15	f	f	PROPN
ma-220	66	16	∈	∈	PROPN
ma-220	66	17	h.	h.	NOUN
ma-220	66	18	2	2	X
ma-220	66	19	.	.	X
ma-220	66	20	∗-continuous	∗-continuous	ADJ
ma-220	66	21	operator	operator	NOUN
ma-220	66	22	duals	dual	NOUN
ma-220	66	23	definition	definition	NOUN
ma-220	66	24	2.1	2.1	NUM
ma-220	66	25	.	.	PUNCT
ma-220	67	1	let	let	VERB
ma-220	67	2	{	{	PUNCT
ma-220	67	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	67	4	and	and	CCONJ
ma-220	67	5	{	{	PUNCT
ma-220	67	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	67	7	be	be	AUX
ma-220	67	8	two	two	NUM
ma-220	67	9	∗-continuous	∗-continuous	ADJ
ma-220	67	10	frames	frame	NOUN
ma-220	67	11	for	for	ADP
ma-220	67	12	h.	h.	PROPN
ma-220	67	13	if	if	SCONJ
ma-220	67	14	there	there	PRON
ma-220	67	15	exists	exist	VERB
ma-220	67	16	aninvertible	aninvertible	ADJ
ma-220	67	17	adjointable	adjointable	NOUN
ma-220	67	18	a	a	DET
ma-220	67	19	-	-	PUNCT
ma-220	67	20	module	module	NOUN
ma-220	67	21	map	map	NOUN
ma-220	67	22	on	on	ADP
ma-220	67	23	h	h	NOUN
ma-220	67	24	such	such	ADJ
ma-220	67	25	that	that	SCONJ
ma-220	67	26	x	x	X
ma-220	68	1	=	=	SYM
ma-220	68	2	∫	∫	PROPN
ma-220	68	3	ω	ω	X
ma-220	68	4	〈	〈	PROPN
ma-220	68	5	γx	γx	NOUN
ma-220	68	6	,	,	PUNCT
ma-220	68	7	gw	gw	PROPN
ma-220	68	8	〉	〉	PROPN
ma-220	68	9	fwdµ(ω	fwdµ(ω	NOUN
ma-220	68	10	)	)	PUNCT
ma-220	68	11	,	,	PUNCT
ma-220	68	12	∀x	∀x	VERB
ma-220	68	13	∈	∈	PROPN
ma-220	68	14	h	h	NOUN
ma-220	68	15	,	,	PUNCT
ma-220	68	16	(	(	PUNCT
ma-220	68	17	2.1	2.1	NUM
ma-220	68	18	)	)	PUNCT
ma-220	68	19	then	then	ADV
ma-220	68	20	{	{	PUNCT
ma-220	68	21	gw}w∈ω	gw}w∈ω	NOUN
ma-220	68	22	is	be	AUX
ma-220	68	23	called	call	VERB
ma-220	68	24	a	a	DET
ma-220	68	25	∗-continuous	∗-continuous	ADJ
ma-220	68	26	operator	operator	NOUN
ma-220	68	27	dual	dual	ADJ
ma-220	68	28	of	of	ADP
ma-220	68	29	{	{	PUNCT
ma-220	68	30	fw}w∈ω	fw}w∈ω	PROPN
ma-220	68	31	.	.	PROPN
ma-220	68	32	remark	remark	PROPN
ma-220	68	33	2.2	2.2	NUM
ma-220	68	34	.	.	PUNCT
ma-220	69	1	every	every	DET
ma-220	69	2	∗-continuous	∗-continuous	ADJ
ma-220	69	3	frame	frame	NOUN
ma-220	69	4	{	{	PUNCT
ma-220	69	5	fw}w∈ω	fw}w∈ω	NOUN
ma-220	69	6	with	with	ADP
ma-220	69	7	continuous	continuous	ADJ
ma-220	69	8	frame	frame	NOUN
ma-220	69	9	operator	operator	NOUN
ma-220	69	10	s	s	PART
ma-220	69	11	is	be	AUX
ma-220	69	12	a	a	DET
ma-220	69	13	∗-continuousoperator	∗-continuousoperator	NOUN
ma-220	69	14	dual	dual	ADJ
ma-220	69	15	for	for	ADP
ma-220	69	16	itself	itself	PRON
ma-220	69	17	.	.	PUNCT
ma-220	70	1	to	to	PART
ma-220	70	2	see	see	VERB
ma-220	70	3	this	this	PRON
ma-220	70	4	,	,	PUNCT
ma-220	70	5	set	set	VERB
ma-220	70	6	γ	γ	X
ma-220	70	7	:	:	PUNCT
ma-220	70	8	=	=	SYM
ma-220	70	9	s−1	s−1	NOUN
ma-220	70	10	and	and	CCONJ
ma-220	70	11	the	the	DET
ma-220	70	12	reconstruction	reconstruction	NOUN
ma-220	70	13	formula	formula	NOUN
ma-220	70	14	concludes	conclude	VERB
ma-220	70	15	it	it	PRON
ma-220	70	16	.	.	PUNCT
ma-220	71	1	remark	remark	VERB
ma-220	71	2	2.3	2.3	NUM
ma-220	71	3	.	.	PUNCT
ma-220	72	1	every	every	DET
ma-220	72	2	dual	dual	ADJ
ma-220	72	3	∗-continuous	∗-continuous	ADJ
ma-220	72	4	frame	frame	NOUN
ma-220	72	5	{	{	PUNCT
ma-220	72	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	72	7	of	of	ADP
ma-220	72	8	∗-continuous	∗-continuous	ADJ
ma-220	72	9	frame	frame	NOUN
ma-220	72	10	{	{	PUNCT
ma-220	72	11	fw}w∈ω	fw}w∈ω	NOUN
ma-220	72	12	is	be	AUX
ma-220	72	13	a	a	DET
ma-220	72	14	∗-continuous	∗-continuous	ADJ
ma-220	72	15	operator	operator	NOUN
ma-220	72	16	dual	dual	ADJ
ma-220	72	17	when	when	SCONJ
ma-220	72	18	γ	γ	PROPN
ma-220	72	19	=	=	SYM
ma-220	72	20	i	i	PROPN
ma-220	72	21	,	,	PUNCT
ma-220	72	22	i	i	PRON
ma-220	72	23	is	be	AUX
ma-220	72	24	the	the	DET
ma-220	72	25	identity	identity	NOUN
ma-220	72	26	operator	operator	NOUN
ma-220	72	27	on	on	ADP
ma-220	72	28	h.	h.	PROPN
ma-220	72	29	remark	remark	PROPN
ma-220	72	30	2.4	2.4	NUM
ma-220	72	31	.	.	PUNCT
ma-220	73	1	let	let	VERB
ma-220	73	2	g	g	NOUN
ma-220	73	3	=	=	PRON
ma-220	73	4	{	{	PUNCT
ma-220	73	5	gw}w∈ω	gw}w∈ω	NOUN
ma-220	73	6	be	be	VERB
ma-220	73	7	an	an	DET
ma-220	73	8	operator	operator	NOUN
ma-220	73	9	dual	dual	ADJ
ma-220	73	10	of	of	ADP
ma-220	73	11	a	a	DET
ma-220	73	12	∗-continuous	∗-continuous	ADJ
ma-220	73	13	frame	frame	NOUN
ma-220	73	14	f	f	X
ma-220	73	15	=	=	PRON
ma-220	73	16	{	{	PUNCT
ma-220	73	17	fw}w∈ω	fw}w∈ω	NOUN
ma-220	73	18	in	in	ADP
ma-220	73	19	h.then	h.then	NOUN
ma-220	73	20	for	for	ADP
ma-220	73	21	some	some	DET
ma-220	73	22	invertible	invertible	ADJ
ma-220	73	23	adjointable	adjointable	NOUN
ma-220	73	24	map	map	NOUN
ma-220	73	25	γ	γ	PROPN
ma-220	73	26	∈	∈	PROPN
ma-220	73	27	b∗(h	b∗(h	PROPN
ma-220	73	28	)	)	PUNCT
ma-220	73	29	x	x	X
ma-220	74	1	=	=	SYM
ma-220	74	2	∫	∫	PROPN
ma-220	74	3	ω	ω	X
ma-220	74	4	〈	〈	PROPN
ma-220	74	5	γx	γx	NOUN
ma-220	74	6	,	,	PUNCT
ma-220	74	7	gw	gw	PROPN
ma-220	74	8	〉	〉	PROPN
ma-220	74	9	fwdµ(ω	fwdµ(ω	NOUN
ma-220	74	10	)	)	PUNCT
ma-220	74	11	,	,	PUNCT
ma-220	74	12	∀x	∀x	PROPN
ma-220	74	13	∈	∈	PROPN
ma-220	74	14	h.	h.	PROPN
ma-220	74	15	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	74	16	eur	eur	PROPN
ma-220	74	17	.	.	PUNCT
ma-220	75	1	j.	j.	PROPN
ma-220	75	2	math	math	PROPN
ma-220	75	3	.	.	PUNCT
ma-220	76	1	anal	anal	PROPN
ma-220	76	2	.	.	PUNCT
ma-220	77	1	10.28924	10.28924	NUM
ma-220	77	2	/	/	SYM
ma-220	77	3	ada	ada	PROPN
ma-220	77	4	/	/	SYM
ma-220	77	5	ma.4.4	ma.4.4	PROPN
ma-220	77	6	4the	4the	NUM
ma-220	77	7	equality	equality	NOUN
ma-220	77	8	shows	show	VERB
ma-220	77	9	that	that	SCONJ
ma-220	77	10	i	i	PRON
ma-220	77	11	=	=	PRON
ma-220	77	12	(	(	PUNCT
ma-220	77	13	t	t	NOUN
ma-220	77	14	∗ftg	∗ftg	NUM
ma-220	77	15	)	)	PUNCT
ma-220	77	16	γ	γ	X
ma-220	77	17	,	,	PUNCT
ma-220	77	18	where	where	SCONJ
ma-220	77	19	i	i	PRON
ma-220	77	20	is	be	AUX
ma-220	77	21	the	the	DET
ma-220	77	22	identity	identity	NOUN
ma-220	77	23	map	map	NOUN
ma-220	77	24	on	on	ADP
ma-220	77	25	h	h	NOUN
ma-220	77	26	,	,	PUNCT
ma-220	77	27	and	and	CCONJ
ma-220	77	28	tf	tf	INTJ
ma-220	77	29	and	and	CCONJ
ma-220	77	30	tg	tg	PROPN
ma-220	77	31	arepre	arepre	ADJ
ma-220	77	32	-	-	PUNCT
ma-220	77	33	frame	frame	NOUN
ma-220	77	34	operators	operator	NOUN
ma-220	77	35	of	of	ADP
ma-220	77	36	f	f	PROPN
ma-220	77	37	and	and	CCONJ
ma-220	77	38	g	g	NOUN
ma-220	77	39	,	,	PUNCT
ma-220	77	40	respectively	respectively	ADV
ma-220	77	41	.	.	PUNCT
ma-220	78	1	therefore	therefore	ADV
ma-220	78	2	,	,	PUNCT
ma-220	78	3	the	the	DET
ma-220	78	4	operator	operator	NOUN
ma-220	78	5	γ	γ	NOUN
ma-220	78	6	is	be	AUX
ma-220	78	7	unique	unique	ADJ
ma-220	78	8	and	and	CCONJ
ma-220	78	9	γ−1	γ−1	PROPN
ma-220	78	10	=	=	SYM
ma-220	78	11	t	t	PROPN
ma-220	78	12	∗ftg	∗ftg	PROPN
ma-220	78	13	.	.	PUNCT
ma-220	79	1	by	by	ADP
ma-220	79	2	remark	remark	NOUN
ma-220	79	3	2.4	2.4	NUM
ma-220	79	4	,	,	PUNCT
ma-220	79	5	we	we	PRON
ma-220	79	6	say	say	VERB
ma-220	79	7	that	that	SCONJ
ma-220	79	8	{	{	PUNCT
ma-220	79	9	gw}w∈ω	gw}w∈ω	NOUN
ma-220	79	10	is	be	AUX
ma-220	79	11	an	an	DET
ma-220	79	12	operator	operator	NOUN
ma-220	79	13	dual	dual	ADJ
ma-220	79	14	for	for	ADP
ma-220	79	15	{	{	PUNCT
ma-220	79	16	fw}w∈ω	fw}w∈ω	NOUN
ma-220	79	17	with	with	ADP
ma-220	79	18	the	the	DET
ma-220	79	19	correspondingoperator	correspondingoperator	NOUN
ma-220	79	20	γ	γ	PROPN
ma-220	79	21	.	.	PROPN
ma-220	80	1	moreover	moreover	ADV
ma-220	80	2	,	,	PUNCT
ma-220	80	3	we	we	PRON
ma-220	80	4	mention	mention	VERB
ma-220	80	5	that	that	SCONJ
ma-220	80	6	the	the	DET
ma-220	80	7	operator	operator	NOUN
ma-220	80	8	duality	duality	NOUN
ma-220	80	9	relation	relation	NOUN
ma-220	80	10	of	of	ADP
ma-220	80	11	x	x	NOUN
ma-220	80	12	-	-	NOUN
ma-220	80	13	frames	frame	NOUN
ma-220	80	14	is	be	AUX
ma-220	80	15	symmetric	symmetric	ADJ
ma-220	80	16	.	.	PUNCT
ma-220	81	1	it	it	PRON
ma-220	81	2	isconsidered	isconsidere	VERB
ma-220	81	3	in	in	ADP
ma-220	81	4	the	the	DET
ma-220	81	5	next	next	ADJ
ma-220	81	6	remark	remark	NOUN
ma-220	81	7	.	.	PUNCT
ma-220	82	1	remark	remark	VERB
ma-220	82	2	2.5	2.5	NUM
ma-220	82	3	.	.	PUNCT
ma-220	83	1	if	if	SCONJ
ma-220	83	2	g	g	PROPN
ma-220	83	3	=	=	PRON
ma-220	83	4	{	{	PUNCT
ma-220	83	5	gw}w∈ω	gw}w∈ω	NOUN
ma-220	83	6	is	be	AUX
ma-220	83	7	an	an	DET
ma-220	83	8	operator	operator	NOUN
ma-220	83	9	dual	dual	ADV
ma-220	83	10	of	of	ADP
ma-220	83	11	a	a	DET
ma-220	83	12	given	give	VERB
ma-220	83	13	∗-continuous	∗-continuous	ADJ
ma-220	83	14	frame	frame	NOUN
ma-220	83	15	f	f	NOUN
ma-220	84	1	=	=	PRON
ma-220	84	2	{	{	PUNCT
ma-220	84	3	fw}w∈ωwith	fw}w∈ωwith	VERB
ma-220	84	4	the	the	DET
ma-220	84	5	corresponding	correspond	VERB
ma-220	84	6	operator	operator	NOUN
ma-220	84	7	γ	γ	NOUN
ma-220	84	8	,	,	PUNCT
ma-220	84	9	then	then	ADV
ma-220	84	10	{	{	PUNCT
ma-220	84	11	fw}w∈ω	fw}w∈ω	NOUN
ma-220	84	12	is	be	AUX
ma-220	84	13	an	an	DET
ma-220	84	14	operator	operator	NOUN
ma-220	84	15	dual	dual	ADJ
ma-220	84	16	for	for	ADP
ma-220	84	17	{	{	PUNCT
ma-220	84	18	gw}w∈ω	gw}w∈ω	NOUN
ma-220	84	19	with	with	ADP
ma-220	84	20	the	the	DET
ma-220	84	21	cor	cor	NOUN
ma-220	84	22	-	-	ADJ
ma-220	84	23	responding	respond	VERB
ma-220	84	24	operator	operator	NOUN
ma-220	84	25	γ∗.	γ∗.	ADV
ma-220	84	26	in	in	ADP
ma-220	84	27	order	order	NOUN
ma-220	84	28	to	to	PART
ma-220	84	29	see	see	VERB
ma-220	84	30	this	this	PRON
ma-220	84	31	,	,	PUNCT
ma-220	84	32	assume	assume	VERB
ma-220	84	33	that	that	SCONJ
ma-220	84	34	tf	tf	PROPN
ma-220	84	35	and	and	CCONJ
ma-220	84	36	tg	tg	PROPN
ma-220	84	37	are	be	AUX
ma-220	84	38	pre	pre	ADJ
ma-220	84	39	-	-	ADJ
ma-220	84	40	frame	frame	ADJ
ma-220	84	41	operators	operator	NOUN
ma-220	84	42	of	of	ADP
ma-220	84	43	fand	fand	PROPN
ma-220	84	44	g	g	PROPN
ma-220	84	45	,	,	PUNCT
ma-220	84	46	respectively	respectively	ADV
ma-220	84	47	.	.	PUNCT
ma-220	85	1	by	by	ADP
ma-220	85	2	the	the	DET
ma-220	85	3	definition	definition	NOUN
ma-220	85	4	of	of	ADP
ma-220	85	5	operator	operator	NOUN
ma-220	85	6	duals	dual	NOUN
ma-220	85	7	,	,	PUNCT
ma-220	85	8	we	we	PRON
ma-220	85	9	have	have	VERB
ma-220	85	10	i	i	PRON
ma-220	85	11	=	=	SYM
ma-220	85	12	∫	∫	PROPN
ma-220	85	13	ω	ω	X
ma-220	86	1	〈	〈	PROPN
ma-220	86	2	γx	γx	NOUN
ma-220	86	3	,	,	PUNCT
ma-220	86	4	gw	gw	PROPN
ma-220	86	5	〉	〉	PROPN
ma-220	86	6	fwdµ(ω	fwdµ(ω	NOUN
ma-220	86	7	)	)	PUNCT
ma-220	87	1	=	=	PUNCT
ma-220	87	2	(	(	PUNCT
ma-220	87	3	t	t	PROPN
ma-220	87	4	∗ftg	∗ftg	PROPN
ma-220	87	5	)	)	PUNCT
ma-220	87	6	γ	γ	PROPN
ma-220	87	7	.	.	PROPN
ma-220	87	8	since	since	SCONJ
ma-220	87	9	γ	γ	PROPN
ma-220	87	10	is	be	AUX
ma-220	87	11	invertible	invertible	ADJ
ma-220	87	12	,	,	PUNCT
ma-220	87	13	γ−1	γ−1	PROPN
ma-220	87	14	=	=	SYM
ma-220	87	15	t	t	PROPN
ma-220	87	16	∗ftg	∗ftg	NUM
ma-220	88	1	and	and	CCONJ
ma-220	88	2	i	i	PRON
ma-220	88	3	=	=	SYM
ma-220	88	4	γ	γ	X
ma-220	88	5	(	(	PUNCT
ma-220	88	6	t	t	PROPN
ma-220	88	7	∗ftg	∗ftg	PROPN
ma-220	88	8	)	)	PUNCT
ma-220	88	9	=	=	PUNCT
ma-220	89	1	(	(	PUNCT
ma-220	89	2	t	t	NOUN
ma-220	89	3	∗gtf	∗gtf	PROPN
ma-220	89	4	)	)	PUNCT
ma-220	89	5	γ∗	γ∗	NOUN
ma-220	89	6	=	=	SYM
ma-220	90	1	∫	∫	PROPN
ma-220	90	2	ω	ω	PROPN
ma-220	90	3	〈	〈	PROPN
ma-220	90	4	γ∗f	γ∗f	PROPN
ma-220	90	5	,	,	PUNCT
ma-220	90	6	fw	fw	PROPN
ma-220	90	7	〉	〉	PROPN
ma-220	90	8	gwdµ(ω	gwdµ(ω	PROPN
ma-220	90	9	)	)	PUNCT
ma-220	90	10	.	.	PUNCT
ma-220	91	1	the	the	DET
ma-220	91	2	following	follow	VERB
ma-220	91	3	lemma	lemma	PROPN
ma-220	91	4	is	be	AUX
ma-220	91	5	obtained	obtain	VERB
ma-220	91	6	by	by	ADP
ma-220	91	7	using	use	VERB
ma-220	91	8	the	the	DET
ma-220	91	9	last	last	ADJ
ma-220	91	10	remark	remark	NOUN
ma-220	91	11	and	and	CCONJ
ma-220	91	12	some	some	DET
ma-220	91	13	properties	property	NOUN
ma-220	91	14	of	of	ADP
ma-220	91	15	pre	pre	NOUN
ma-220	91	16	-	-	NOUN
ma-220	91	17	frameoperators	frameoperator	NOUN
ma-220	91	18	.	.	PUNCT
ma-220	92	1	lemma	lemma	PROPN
ma-220	92	2	2.6	2.6	NUM
ma-220	92	3	.	.	PUNCT
ma-220	93	1	let	let	VERB
ma-220	93	2	f	f	NOUN
ma-220	93	3	=	=	PRON
ma-220	93	4	{	{	PUNCT
ma-220	93	5	fw}w∈ω	fw}w∈ω	NOUN
ma-220	93	6	and	and	CCONJ
ma-220	93	7	g	g	NOUN
ma-220	93	8	=	=	PUNCT
ma-220	93	9	{	{	PUNCT
ma-220	93	10	gw}w∈ω	gw}w∈ω	NOUN
ma-220	93	11	be	be	VERB
ma-220	93	12	∗-bessel	∗-bessel	ADJ
ma-220	93	13	sequences	sequence	NOUN
ma-220	93	14	for	for	ADP
ma-220	93	15	h	h	NOUN
ma-220	93	16	with	with	ADP
ma-220	93	17	the	the	DET
ma-220	93	18	pre	pre	ADJ
ma-220	93	19	-	-	ADJ
ma-220	93	20	frame	frame	ADJ
ma-220	93	21	operators	operator	NOUN
ma-220	93	22	tf	tf	X
ma-220	93	23	and	and	CCONJ
ma-220	93	24	tg	tg	PROPN
ma-220	93	25	,	,	PUNCT
ma-220	93	26	respectively	respectively	ADV
ma-220	93	27	.	.	PUNCT
ma-220	94	1	assume	assume	VERB
ma-220	94	2	that	that	SCONJ
ma-220	94	3	γ	γ	PROPN
ma-220	94	4	is	be	AUX
ma-220	94	5	an	an	DET
ma-220	94	6	invertible	invertible	ADJ
ma-220	94	7	and	and	CCONJ
ma-220	94	8	adjointable	adjointable	ADJ
ma-220	94	9	a	a	DET
ma-220	94	10	-	-	PUNCT
ma-220	94	11	module	module	NOUN
ma-220	94	12	map	map	NOUN
ma-220	94	13	on	on	ADP
ma-220	94	14	h.	h.	PROPN
ma-220	94	15	then	then	ADV
ma-220	94	16	for	for	ADP
ma-220	94	17	x	x	PROPN
ma-220	94	18	∈	∈	PROPN
ma-220	94	19	h	h	NOUN
ma-220	94	20	,	,	PUNCT
ma-220	94	21	the	the	DET
ma-220	94	22	following	following	ADJ
ma-220	94	23	statements	statement	NOUN
ma-220	94	24	are	be	AUX
ma-220	94	25	equivalent	equivalent	ADJ
ma-220	94	26	:	:	PUNCT
ma-220	94	27	(	(	PUNCT
ma-220	94	28	i	i	NOUN
ma-220	94	29	)	)	PUNCT
ma-220	94	30	x	x	PUNCT
ma-220	95	1	=	=	SYM
ma-220	95	2	∫	∫	PROPN
ma-220	95	3	ω	ω	X
ma-220	95	4	〈	〈	PROPN
ma-220	95	5	γx	γx	NOUN
ma-220	95	6	,	,	PUNCT
ma-220	95	7	gw	gw	PROPN
ma-220	95	8	〉	〉	PROPN
ma-220	95	9	fwdµ(ω	fwdµ(ω	NOUN
ma-220	95	10	)	)	PUNCT
ma-220	95	11	.	.	PUNCT
ma-220	96	1	(	(	PUNCT
ma-220	96	2	i	i	PRON
ma-220	96	3	i	i	PROPN
ma-220	96	4	)	)	PUNCT
ma-220	96	5	x	x	PUNCT
ma-220	97	1	=	=	SYM
ma-220	97	2	∫	∫	PROPN
ma-220	97	3	ω	ω	NUM
ma-220	97	4	〈	〈	PROPN
ma-220	97	5	γ	γ	NOUN
ma-220	97	6	∗x	∗x	NOUN
ma-220	97	7	,	,	PUNCT
ma-220	97	8	fw	fw	PROPN
ma-220	97	9	〉	〉	PROPN
ma-220	97	10	gwdµ(ω	gwdµ(ω	PROPN
ma-220	97	11	)	)	PUNCT
ma-220	97	12	.	.	PUNCT
ma-220	98	1	in	in	ADP
ma-220	98	2	case	case	NOUN
ma-220	98	3	that	that	SCONJ
ma-220	98	4	one	one	NUM
ma-220	98	5	of	of	ADP
ma-220	98	6	the	the	DET
ma-220	98	7	above	above	ADJ
ma-220	98	8	equalities	equality	NOUN
ma-220	98	9	is	be	AUX
ma-220	98	10	satisfied	satisfied	ADJ
ma-220	98	11	,	,	PUNCT
ma-220	98	12	{	{	PUNCT
ma-220	98	13	fw}w∈ω	fw}w∈ω	NOUN
ma-220	98	14	and	and	CCONJ
ma-220	98	15	{	{	PUNCT
ma-220	98	16	gw}w∈ω	gw}w∈ω	NOUN
ma-220	98	17	are	be	AUX
ma-220	98	18	operator	operator	NOUN
ma-220	98	19	dual	dual	ADJ
ma-220	98	20	∗-frames	∗-frame	NOUN
ma-220	98	21	.	.	PUNCT
ma-220	99	1	moreover	moreover	ADV
ma-220	99	2	,	,	PUNCT
ma-220	99	3	if	if	SCONJ
ma-220	99	4	b	b	PROPN
ma-220	99	5	is	be	AUX
ma-220	99	6	an	an	DET
ma-220	99	7	upper	upper	ADJ
ma-220	99	8	bound	bind	VERB
ma-220	99	9	for	for	ADP
ma-220	99	10	{	{	PUNCT
ma-220	99	11	fw}w∈ω	fw}w∈ω	NOUN
ma-220	99	12	and	and	CCONJ
ma-220	99	13	s	s	NOUN
ma-220	99	14	is	be	AUX
ma-220	99	15	frame	frame	NOUN
ma-220	99	16	operator	operator	NOUN
ma-220	99	17	of	of	ADP
ma-220	99	18	{	{	PUNCT
ma-220	99	19	fw}w∈ω	fw}w∈ω	PROPN
ma-220	99	20	,	,	PUNCT
ma-220	99	21	then	then	ADV
ma-220	99	22	b	b	PROPN
ma-220	99	23	∥∥s−1	∥∥s−1	PROPN
ma-220	99	24	∥∥−	∥∥−	PROPN
ma-220	99	25	1	1	NUM
ma-220	99	26	2	2	NUM
ma-220	99	27	‖tf‖−1	‖tf‖−1	NOUN
ma-220	99	28	‖γ‖−1	‖γ‖−1	NOUN
ma-220	99	29	is	be	AUX
ma-220	99	30	a	a	DET
ma-220	99	31	lower	lower	ADV
ma-220	99	32	bound	bind	VERB
ma-220	99	33	for	for	ADP
ma-220	99	34	{	{	PUNCT
ma-220	99	35	gw}w∈ω	gw}w∈ω	NOUN
ma-220	99	36	.	.	NOUN
ma-220	99	37	proof	proof	NOUN
ma-220	99	38	.	.	PUNCT
ma-220	100	1	the	the	DET
ma-220	100	2	equivalency	equivalency	NOUN
ma-220	100	3	of	of	ADP
ma-220	100	4	the	the	DET
ma-220	100	5	two	two	NUM
ma-220	100	6	conditions	condition	NOUN
ma-220	100	7	is	be	AUX
ma-220	100	8	given	give	VERB
ma-220	100	9	from	from	ADP
ma-220	100	10	remark	remark	NOUN
ma-220	100	11	2.5.now	2.5.now	NUM
ma-220	100	12	,	,	PUNCT
ma-220	100	13	let	let	VERB
ma-220	100	14	b	b	X
ma-220	100	15	be	be	AUX
ma-220	100	16	a	a	DET
ma-220	100	17	∗-bessel	∗-bessel	NOUN
ma-220	100	18	bound	bind	VERB
ma-220	100	19	for	for	ADP
ma-220	100	20	{	{	PUNCT
ma-220	100	21	fw}w∈ω	fw}w∈ω	NOUN
ma-220	100	22	and	and	CCONJ
ma-220	100	23	(	(	PUNCT
ma-220	100	24	i	i	NOUN
ma-220	100	25	)	)	PUNCT
ma-220	100	26	holds	hold	VERB
ma-220	100	27	.	.	PUNCT
ma-220	101	1	by	by	ADP
ma-220	101	2	the	the	DET
ma-220	101	3	definition	definition	NOUN
ma-220	101	4	of	of	ADP
ma-220	101	5	∗-besselsequence	∗-besselsequence	X
ma-220	101	6	{	{	PUNCT
ma-220	101	7	fw}w∈ω	fw}w∈ω	NOUN
ma-220	101	8	and	and	CCONJ
ma-220	101	9	t	t	X
ma-220	101	10	∗ftgγ	∗ftgγ	NOUN
ma-220	101	11	=	=	SYM
ma-220	101	12	idh	idh	PROPN
ma-220	101	13	,	,	PUNCT
ma-220	101	14	we	we	PRON
ma-220	101	15	can	can	AUX
ma-220	101	16	write	write	VERB
ma-220	101	17	,	,	PUNCT
ma-220	101	18	for	for	ADP
ma-220	101	19	x	x	PROPN
ma-220	101	20	∈	∈	PROPN
ma-220	101	21	h	h	NOUN
ma-220	101	22	,	,	PUNCT
ma-220	101	23	〈	〈	PROPN
ma-220	101	24	tfx	tfx	PROPN
ma-220	101	25	,	,	PUNCT
ma-220	101	26	tfx	tfx	PROPN
ma-220	101	27	〉	〉	PROPN
ma-220	101	28	≤	≤	NUM
ma-220	101	29	b〈x	b〈x	PUNCT
ma-220	101	30	,	,	PUNCT
ma-220	101	31	x〉b∗	x〉b∗	PROPN
ma-220	101	32	(	(	PUNCT
ma-220	101	33	2.2	2.2	NUM
ma-220	101	34	)	)	PUNCT
ma-220	101	35	=	=	SYM
ma-220	102	1	b	b	X
ma-220	102	2	〈	〈	PROPN
ma-220	102	3	t	t	PROPN
ma-220	102	4	∗ftgγx	∗ftgγx	PROPN
ma-220	102	5	,	,	PUNCT
ma-220	102	6	t	t	PROPN
ma-220	102	7	∗ftgγx〉b∗	∗ftgγx〉b∗	PROPN
ma-220	102	8	≤	≤	PROPN
ma-220	102	9	b	b	X
ma-220	102	10	‖tf‖2	‖tf‖2	PUNCT
ma-220	102	11	〈	〈	PROPN
ma-220	102	12	tgγx	tgγx	NOUN
ma-220	102	13	,	,	PUNCT
ma-220	102	14	tgγx〉b∗.	tgγx〉b∗.	NUM
ma-220	102	15	using	use	VERB
ma-220	102	16	lemma	lemma	PROPN
ma-220	102	17	1.3	1.3	NUM
ma-220	102	18	,	,	PUNCT
ma-220	102	19	we	we	PRON
ma-220	102	20	have∥∥∥(t	have∥∥∥(t	VERB
ma-220	102	21	∗ftf)−1	∗ftf)−1	VERB
ma-220	102	22	∥∥∥−1	∥∥∥−1	PROPN
ma-220	102	23	〈	〈	PROPN
ma-220	102	24	x	x	X
ma-220	102	25	,	,	PUNCT
ma-220	102	26	x	x	PROPN
ma-220	102	27	〉	〉	NOUN
ma-220	102	28	≤	≤	NUM
ma-220	102	29	〈	〈	PROPN
ma-220	102	30	tfx	tfx	PROPN
ma-220	102	31	,	,	PUNCT
ma-220	102	32	tfx	tfx	PROPN
ma-220	102	33	〉	〉	PROPN
ma-220	102	34	,	,	PUNCT
ma-220	102	35	∀f	∀f	PROPN
ma-220	102	36	∈	∈	PROPN
ma-220	102	37	h.	h.	NOUN
ma-220	102	38	(	(	PUNCT
ma-220	102	39	2.3	2.3	NUM
ma-220	102	40	)	)	PUNCT
ma-220	102	41	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	102	42	eur	eur	NOUN
ma-220	102	43	.	.	PUNCT
ma-220	103	1	j.	j.	PROPN
ma-220	103	2	math	math	PROPN
ma-220	103	3	.	.	PUNCT
ma-220	104	1	anal	anal	PROPN
ma-220	104	2	.	.	PUNCT
ma-220	105	1	10.28924	10.28924	NUM
ma-220	105	2	/	/	SYM
ma-220	105	3	ada	ada	PROPN
ma-220	105	4	/	/	SYM
ma-220	105	5	ma.4.4	ma.4.4	PROPN
ma-220	105	6	5it	5it	PROPN
ma-220	105	7	follows	follow	VERB
ma-220	105	8	from	from	ADP
ma-220	105	9	lemma	lemma	PROPN
ma-220	105	10	1.3	1.3	NUM
ma-220	105	11	,	,	PUNCT
ma-220	105	12	(	(	PUNCT
ma-220	105	13	2.2	2.2	NUM
ma-220	105	14	)	)	PUNCT
ma-220	105	15	,	,	PUNCT
ma-220	105	16	and	and	CCONJ
ma-220	105	17	(	(	PUNCT
ma-220	105	18	2.3	2.3	NUM
ma-220	105	19	)	)	PUNCT
ma-220	106	1	that	that	SCONJ
ma-220	106	2	for	for	ADP
ma-220	106	3	x	x	SYM
ma-220	106	4	∈	∈	PROPN
ma-220	106	5	h,∥∥s−1	h,∥∥s−1	PROPN
ma-220	106	6	∥∥−1	∥∥−1	PROPN
ma-220	106	7	‖γγ∗‖−1	‖γγ∗‖−1	PROPN
ma-220	106	8	〈	〈	PROPN
ma-220	106	9	f	f	PROPN
ma-220	106	10	,	,	PUNCT
ma-220	106	11	f	f	PROPN
ma-220	106	12	〉	〉	PROPN
ma-220	106	13	≤	≤	PROPN
ma-220	106	14	∥∥s−1	∥∥s−1	NOUN
ma-220	106	15	∥∥−1	∥∥−1	PROPN
ma-220	106	16	〈	〈	PROPN
ma-220	106	17	γ−1f	γ−1f	NOUN
ma-220	106	18	,	,	PUNCT
ma-220	106	19	γ−1f	γ−1f	X
ma-220	106	20	〉	〉	NOUN
ma-220	106	21	≤	≤	PROPN
ma-220	106	22	b	b	X
ma-220	106	23	‖tf‖2	‖tf‖2	PUNCT
ma-220	106	24	〈	〈	PROPN
ma-220	106	25	tgf	tgf	PROPN
ma-220	106	26	tgf	tgf	PROPN
ma-220	106	27	〉	〉	PROPN
ma-220	106	28	b∗	b∗	PROPN
ma-220	106	29	,	,	PUNCT
ma-220	106	30	(	(	PUNCT
ma-220	106	31	b−1	b−1	PROPN
ma-220	106	32	∥∥s−1	∥∥s−1	NOUN
ma-220	106	33	∥∥−	∥∥−	PROPN
ma-220	106	34	1	1	NUM
ma-220	106	35	2	2	NUM
ma-220	106	36	‖tf‖−1	‖tf‖−1	NOUN
ma-220	106	37	‖γ‖−1	‖γ‖−1	NOUN
ma-220	106	38	)	)	PUNCT
ma-220	107	1	〈	〈	PROPN
ma-220	107	2	f	f	PROPN
ma-220	107	3	,	,	PUNCT
ma-220	107	4	f	f	PROPN
ma-220	107	5	〉	〉	PROPN
ma-220	107	6	(	(	PUNCT
ma-220	107	7	b−1	b−1	PROPN
ma-220	107	8	∥∥s−1	∥∥s−1	NOUN
ma-220	107	9	∥∥−	∥∥−	PROPN
ma-220	107	10	1	1	NUM
ma-220	107	11	2	2	NUM
ma-220	107	12	‖tf‖−1	‖tf‖−1	NOUN
ma-220	107	13	‖γ‖−1	‖γ‖−1	NOUN
ma-220	107	14	)	)	PUNCT
ma-220	107	15	∗	∗	NOUN
ma-220	107	16	≤	≤	NUM
ma-220	108	1	〈	〈	PROPN
ma-220	108	2	tgf	tgf	PROPN
ma-220	108	3	,	,	PUNCT
ma-220	108	4	tgf	tgf	PROPN
ma-220	108	5	〉	〉	PROPN
ma-220	108	6	.	.	PUNCT
ma-220	109	1	therefore	therefore	ADV
ma-220	109	2	,	,	PUNCT
ma-220	109	3	b	b	PROPN
ma-220	109	4	∥∥s−1	∥∥s−1	ADJ
ma-220	109	5	∥∥∣∣−	∥∥∣∣−	PROPN
ma-220	109	6	1	1	NUM
ma-220	109	7	2	2	NUM
ma-220	109	8	‖γ‖−1	‖γ‖−1	NOUN
ma-220	109	9	‖tf‖−1	‖tf‖−1	NOUN
ma-220	109	10	is	be	AUX
ma-220	109	11	a	a	DET
ma-220	109	12	lower	low	ADJ
ma-220	109	13	∗-frame	∗-frame	NOUN
ma-220	109	14	bound	bind	VERB
ma-220	109	15	for	for	ADP
ma-220	109	16	{	{	PUNCT
ma-220	109	17	gj}j∈j	gj}j∈j	X
ma-220	109	18	and	and	CCONJ
ma-220	109	19	{	{	PUNCT
ma-220	109	20	gj}j∈j	gj}j∈j	NOUN
ma-220	109	21	is	be	AUX
ma-220	109	22	a	a	DET
ma-220	109	23	∗-frame.similarly	∗-frame.similarly	ADJ
ma-220	109	24	,	,	PUNCT
ma-220	109	25	{	{	PUNCT
ma-220	109	26	fj}j∈j	fj}j∈j	PROPN
ma-220	109	27	is	be	AUX
ma-220	109	28	also	also	ADV
ma-220	109	29	a	a	DET
ma-220	109	30	∗-frame	∗-frame	NOUN
ma-220	109	31	.	.	PUNCT
ma-220	110	1	�	�	PROPN
ma-220	110	2	proposition	proposition	PROPN
ma-220	110	3	2.7	2.7	NUM
ma-220	110	4	.	.	PUNCT
ma-220	111	1	let	let	AUX
ma-220	111	2	(	(	PUNCT
ma-220	111	3	{	{	PUNCT
ma-220	111	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	111	5	,	,	PUNCT
ma-220	111	6	γ	γ	PROPN
ma-220	111	7	)	)	PUNCT
ma-220	111	8	be	be	AUX
ma-220	111	9	an	an	DET
ma-220	111	10	operator	operator	NOUN
ma-220	111	11	dual	dual	ADJ
ma-220	111	12	of	of	ADP
ma-220	111	13	a	a	DET
ma-220	111	14	∗-continuous	∗-continuous	ADJ
ma-220	111	15	frame	frame	NOUN
ma-220	111	16	{	{	PUNCT
ma-220	111	17	fw}w∈ω.1	fw}w∈ω.1	ADJ
ma-220	111	18	.	.	PUNCT
ma-220	112	1	for	for	ADP
ma-220	112	2	a	a	DET
ma-220	112	3	strictly	strictly	ADV
ma-220	112	4	nonzero	nonzero	ADJ
ma-220	112	5	element	element	NOUN
ma-220	112	6	α	α	NOUN
ma-220	112	7	in	in	ADP
ma-220	112	8	the	the	DET
ma-220	112	9	center	center	NOUN
ma-220	112	10	of	of	ADP
ma-220	112	11	a	a	PRON
ma-220	112	12	,	,	PUNCT
ma-220	112	13	the	the	DET
ma-220	112	14	pair	pair	NOUN
ma-220	112	15	(	(	PUNCT
ma-220	112	16	{	{	PUNCT
ma-220	112	17	αgw}w∈ω	αgw}w∈ω	INTJ
ma-220	112	18	,	,	PUNCT
ma-220	112	19	α	α	NOUN
ma-220	112	20	−1γ	−1γ	X
ma-220	112	21	)	)	PUNCT
ma-220	112	22	is	be	AUX
ma-220	112	23	an	an	DET
ma-220	112	24	operator	operator	NOUN
ma-220	112	25	dual	dual	ADJ
ma-220	112	26	for	for	ADP
ma-220	112	27	{	{	PUNCT
ma-220	112	28	fw}w∈ω.2	fw}w∈ω.2	NOUN
ma-220	112	29	.	.	PUNCT
ma-220	113	1	if	if	SCONJ
ma-220	113	2	υ	υ	PROPN
ma-220	113	3	is	be	AUX
ma-220	113	4	an	an	DET
ma-220	113	5	invertible	invertible	ADJ
ma-220	113	6	and	and	CCONJ
ma-220	113	7	adjointable	adjointable	ADJ
ma-220	113	8	operator	operator	NOUN
ma-220	113	9	on	on	ADP
ma-220	113	10	h	h	NOUN
ma-220	113	11	,	,	PUNCT
ma-220	113	12	then	then	ADV
ma-220	113	13	(	(	PUNCT
ma-220	113	14	{	{	PUNCT
ma-220	113	15	υgw}w∈ω	υgw}w∈ω	PROPN
ma-220	113	16	,	,	PUNCT
ma-220	113	17	(	(	PUNCT
ma-220	113	18	υ)−1γ	υ)−1γ	NOUN
ma-220	113	19	)	)	PUNCT
ma-220	113	20	is	be	AUX
ma-220	113	21	an	an	DET
ma-220	113	22	operator	operator	NOUN
ma-220	113	23	dual	dual	ADJ
ma-220	113	24	for	for	ADP
ma-220	113	25	{	{	PUNCT
ma-220	113	26	fw}w∈ω.3	fw}w∈ω.3	NOUN
ma-220	113	27	.	.	PUNCT
ma-220	114	1	the	the	DET
ma-220	114	2	sequence	sequence	NOUN
ma-220	114	3	{	{	PUNCT
ma-220	114	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	114	5	is	be	AUX
ma-220	114	6	a	a	DET
ma-220	114	7	dual	dual	ADJ
ma-220	114	8	of	of	ADP
ma-220	114	9	{	{	PUNCT
ma-220	114	10	γ∗fw}w∈ω.4	γ∗fw}w∈ω.4	NUM
ma-220	114	11	.	.	PUNCT
ma-220	115	1	assume	assume	VERB
ma-220	115	2	that	that	SCONJ
ma-220	115	3	(	(	PUNCT
ma-220	115	4	{	{	PUNCT
ma-220	115	5	hw}w∈ω	hw}w∈ω	NOUN
ma-220	115	6	,	,	PUNCT
ma-220	115	7	λ	λ	PROPN
ma-220	115	8	)	)	PUNCT
ma-220	115	9	is	be	AUX
ma-220	115	10	another	another	DET
ma-220	115	11	operator	operator	NOUN
ma-220	115	12	dual	dual	ADV
ma-220	115	13	of	of	ADP
ma-220	115	14	{	{	PUNCT
ma-220	115	15	fw}w∈ω	fw}w∈ω	NOUN
ma-220	115	16	.	.	PUNCT
ma-220	116	1	then	then	ADV
ma-220	116	2	(	(	PUNCT
ma-220	116	3	{	{	PUNCT
ma-220	116	4	gw	gw	PROPN
ma-220	116	5	+	+	X
ma-220	116	6	hw}w∈j	hw}w∈j	NOUN
ma-220	116	7	,	,	PUNCT
ma-220	116	8	(	(	PUNCT
ma-220	116	9	γ−1	γ−1	PROPN
ma-220	116	10	+	+	PROPN
ma-220	116	11	λ−1	λ−1	PROPN
ma-220	116	12	)	)	PUNCT
ma-220	116	13	−1	−1	NOUN
ma-220	116	14	)	)	PUNCT
ma-220	116	15	is	be	AUX
ma-220	116	16	an	an	DET
ma-220	116	17	operator	operator	NOUN
ma-220	116	18	dual	dual	ADJ
ma-220	116	19	for	for	ADP
ma-220	116	20	{	{	PUNCT
ma-220	116	21	fw}w∈ω	fw}w∈ω	PROPN
ma-220	116	22	.	.	NOUN
ma-220	116	23	proposition	proposition	NOUN
ma-220	116	24	2.8	2.8	NUM
ma-220	116	25	.	.	PUNCT
ma-220	117	1	let	let	AUX
ma-220	117	2	(	(	PUNCT
ma-220	117	3	{	{	PUNCT
ma-220	117	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	117	5	,	,	PUNCT
ma-220	117	6	γ	γ	PROPN
ma-220	117	7	)	)	PUNCT
ma-220	117	8	be	be	AUX
ma-220	117	9	an	an	DET
ma-220	117	10	operator	operator	NOUN
ma-220	117	11	dual	dual	ADJ
ma-220	117	12	of	of	ADP
ma-220	117	13	{	{	PUNCT
ma-220	117	14	fw}w∈ω	fw}w∈ω	PROPN
ma-220	117	15	for	for	ADP
ma-220	117	16	h.	h.	PROPN
ma-220	117	17	if	if	SCONJ
ma-220	117	18	f	f	PROPN
ma-220	117	19	is	be	AUX
ma-220	117	20	an	an	DET
ma-220	117	21	element	element	NOUN
ma-220	117	22	of	of	ADP
ma-220	117	23	h	h	NOUN
ma-220	117	24	such	such	ADJ
ma-220	117	25	that	that	SCONJ
ma-220	117	26	〈	〈	PROPN
ma-220	117	27	x	x	X
ma-220	117	28	,	,	PUNCT
ma-220	117	29	x	x	PRON
ma-220	117	30	〉	〉	NOUN
ma-220	117	31	is	be	AUX
ma-220	117	32	a	a	DET
ma-220	117	33	strictly	strictly	ADV
ma-220	117	34	nonzero	nonzero	ADJ
ma-220	117	35	element	element	NOUN
ma-220	117	36	in	in	ADP
ma-220	117	37	the	the	DET
ma-220	117	38	center	center	NOUN
ma-220	117	39	of	of	ADP
ma-220	117	40	a	a	PRON
ma-220	117	41	,	,	PUNCT
ma-220	117	42	then	then	ADV
ma-220	117	43	{	{	PUNCT
ma-220	117	44	〈	〈	PROPN
ma-220	117	45	gw	gw	PROPN
ma-220	117	46	,	,	PUNCT
ma-220	117	47	(	(	PUNCT
ma-220	117	48	〈	〈	PROPN
ma-220	117	49	x	x	X
ma-220	117	50	,	,	PUNCT
ma-220	117	51	x〉)−1γx	x〉)−1γx	NOUN
ma-220	117	52	〉	〉	NOUN
ma-220	117	53	}	}	PUNCT
ma-220	117	54	w∈ω	w∈ω	NOUN
ma-220	117	55	is	be	AUX
ma-220	117	56	a	a	DET
ma-220	117	57	dual	dual	ADJ
ma-220	117	58	of	of	ADP
ma-220	117	59	{	{	PUNCT
ma-220	117	60	〈	〈	PROPN
ma-220	117	61	fw	fw	PROPN
ma-220	117	62	,	,	PUNCT
ma-220	117	63	x〉}w∈ω	x〉}w∈ω	PROPN
ma-220	117	64	.	.	PUNCT
ma-220	118	1	proof	proof	NOUN
ma-220	118	2	.	.	PUNCT
ma-220	119	1	suppose	suppose	VERB
ma-220	119	2	that	that	SCONJ
ma-220	119	3	a	a	DET
ma-220	119	4	∈	∈	PROPN
ma-220	119	5	a.	a.	NOUN
ma-220	119	6	then∫	then∫	NOUN
ma-220	119	7	ω	ω	NUM
ma-220	119	8	〈	〈	PROPN
ma-220	119	9	a	a	X
ma-220	119	10	,	,	PUNCT
ma-220	119	11	〈	〈	PROPN
ma-220	119	12	gw	gw	NOUN
ma-220	119	13	,	,	PUNCT
ma-220	119	14	〈	〈	PROPN
ma-220	119	15	x	x	X
ma-220	119	16	,	,	PUNCT
ma-220	119	17	x〉−1γx	x〉−1γx	ADJ
ma-220	119	18	〉	〉	NOUN
ma-220	119	19	〉	〉	NOUN
ma-220	119	20	〈	〈	NOUN
ma-220	119	21	fw	fw	NOUN
ma-220	119	22	,	,	PUNCT
ma-220	119	23	x	x	PROPN
ma-220	119	24	〉	〉	NOUN
ma-220	119	25	dµ(ω	dµ(ω	PUNCT
ma-220	119	26	)	)	PUNCT
ma-220	120	1	=	=	SYM
ma-220	120	2	∫	∫	PROPN
ma-220	120	3	ω	ω	PROPN
ma-220	120	4	a	a	DET
ma-220	120	5	〈	〈	PROPN
ma-220	120	6	〈	〈	PROPN
ma-220	120	7	x	x	X
ma-220	120	8	,	,	PUNCT
ma-220	120	9	x〉−1γx	x〉−1γx	PROPN
ma-220	120	10	,	,	PUNCT
ma-220	120	11	gw	gw	PROPN
ma-220	120	12	〉	〉	NOUN
ma-220	120	13	〈	〈	NOUN
ma-220	120	14	fw	fw	NOUN
ma-220	120	15	,	,	PUNCT
ma-220	120	16	x	x	PROPN
ma-220	120	17	〉	〉	NOUN
ma-220	120	18	dµ(ω	dµ(ω	PUNCT
ma-220	120	19	)	)	PUNCT
ma-220	120	20	=	=	SYM
ma-220	120	21	a〈x	a〈x	PROPN
ma-220	120	22	,	,	PUNCT
ma-220	120	23	x〉−1	x〉−1	PROPN
ma-220	120	24	〈	〈	PROPN
ma-220	120	25	∫	∫	PROPN
ma-220	120	26	ω	ω	NUM
ma-220	120	27	〈	〈	PROPN
ma-220	120	28	γx	γx	NOUN
ma-220	120	29	,	,	PUNCT
ma-220	120	30	gw	gw	PROPN
ma-220	120	31	〉	〉	PROPN
ma-220	120	32	fwdµ(ω	fwdµ(ω	NOUN
ma-220	120	33	)	)	PUNCT
ma-220	120	34	,	,	PUNCT
ma-220	120	35	x	x	SYM
ma-220	120	36	〉	〉	NOUN
ma-220	120	37	=	=	SYM
ma-220	120	38	a〈x	a〈x	PROPN
ma-220	120	39	,	,	PUNCT
ma-220	120	40	x〉−1〈x	x〉−1〈x	PROPN
ma-220	120	41	,	,	PUNCT
ma-220	120	42	x	x	NOUN
ma-220	120	43	〉	〉	NUM
ma-220	120	44	=	=	SYM
ma-220	120	45	a.this	a.this	PROPN
ma-220	120	46	completes	complete	VERB
ma-220	120	47	the	the	DET
ma-220	120	48	proof	proof	NOUN
ma-220	120	49	.	.	PUNCT
ma-220	121	1	�	�	PROPN
ma-220	121	2	proposition	proposition	NOUN
ma-220	121	3	2.9	2.9	NUM
ma-220	121	4	.	.	PUNCT
ma-220	122	1	let	let	VERB
ma-220	122	2	{	{	PUNCT
ma-220	122	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	122	4	be	be	AUX
ma-220	122	5	a	a	DET
ma-220	122	6	∗-continuous	∗-continuous	ADJ
ma-220	122	7	frame	frame	NOUN
ma-220	122	8	for	for	ADP
ma-220	122	9	h	h	NOUN
ma-220	122	10	with	with	ADP
ma-220	122	11	frame	frame	NOUN
ma-220	122	12	operator	operator	NOUN
ma-220	122	13	s.	s.	PROPN
ma-220	122	14	if	if	SCONJ
ma-220	122	15	θ	θ	PROPN
ma-220	122	16	is	be	AUX
ma-220	122	17	an	an	DET
ma-220	122	18	adjointable	adjointable	ADJ
ma-220	122	19	and	and	CCONJ
ma-220	122	20	invertible	invertible	ADJ
ma-220	122	21	operator	operator	NOUN
ma-220	122	22	on	on	ADP
ma-220	122	23	h	h	NOUN
ma-220	122	24	,	,	PUNCT
ma-220	122	25	then	then	ADV
ma-220	122	26	(	(	PUNCT
ma-220	122	27	{	{	PUNCT
ma-220	122	28	θfw}w∈ω	θfw}w∈ω	NOUN
ma-220	122	29	,	,	PUNCT
ma-220	122	30	(	(	PUNCT
ma-220	122	31	θ−1	θ−1	PROPN
ma-220	122	32	)	)	PUNCT
ma-220	122	33	∗	∗	NOUN
ma-220	122	34	s−1	s−1	PROPN
ma-220	122	35	)	)	PUNCT
ma-220	122	36	is	be	AUX
ma-220	122	37	an	an	DET
ma-220	122	38	operator	operator	NOUN
ma-220	122	39	dual	dual	ADJ
ma-220	122	40	for	for	ADP
ma-220	122	41	{	{	PUNCT
ma-220	122	42	fw}w∈ω	fw}w∈ω	NOUN
ma-220	122	43	.	.	NOUN
ma-220	122	44	proof	proof	NOUN
ma-220	122	45	.	.	PUNCT
ma-220	123	1	let	let	VERB
ma-220	123	2	x	x	SYM
ma-220	123	3	∈	∈	PROPN
ma-220	123	4	h.	h.	PROPN
ma-220	123	5	then∫	then∫	PROPN
ma-220	123	6	ω	ω	PROPN
ma-220	124	1	〈	〈	PROPN
ma-220	124	2	(	(	PUNCT
ma-220	124	3	s−1θ−1	s−1θ−1	PROPN
ma-220	124	4	)	)	PUNCT
ma-220	124	5	x	x	NOUN
ma-220	124	6	,	,	PUNCT
ma-220	124	7	fw	fw	ADJ
ma-220	124	8	〉	〉	NOUN
ma-220	124	9	θfwdµ(ω	θfwdµ(ω	NOUN
ma-220	124	10	)	)	PUNCT
ma-220	125	1	=	=	SYM
ma-220	125	2	θ	θ	PROPN
ma-220	125	3	(	(	PUNCT
ma-220	125	4	∫	∫	PROPN
ma-220	125	5	ω	ω	PROPN
ma-220	125	6	〈	〈	PROPN
ma-220	125	7	(	(	PUNCT
ma-220	125	8	s−1θ−1	s−1θ−1	PROPN
ma-220	125	9	)	)	PUNCT
ma-220	125	10	xdµ(ω	xdµ(ω	PROPN
ma-220	125	11	)	)	PUNCT
ma-220	125	12	,	,	PUNCT
ma-220	125	13	fw	fw	ADJ
ma-220	125	14	〉	〉	NOUN
ma-220	125	15	fw	fw	NOUN
ma-220	125	16	)	)	PUNCT
ma-220	125	17	=	=	SYM
ma-220	125	18	θ	θ	NOUN
ma-220	125	19	(	(	PUNCT
ma-220	125	20	θ−1x	θ−1x	NOUN
ma-220	125	21	)	)	PUNCT
ma-220	125	22	=	=	PUNCT
ma-220	126	1	x.	x.	NOUN
ma-220	126	2	this	this	PRON
ma-220	126	3	completes	complete	VERB
ma-220	126	4	the	the	DET
ma-220	126	5	proof	proof	NOUN
ma-220	126	6	.	.	PUNCT
ma-220	127	1	�	�	PROPN
ma-220	127	2	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	127	3	eur	eur	PROPN
ma-220	127	4	.	.	PUNCT
ma-220	128	1	j.	j.	PROPN
ma-220	128	2	math	math	PROPN
ma-220	128	3	.	.	PUNCT
ma-220	129	1	anal	anal	PROPN
ma-220	129	2	.	.	PUNCT
ma-220	130	1	10.28924	10.28924	NUM
ma-220	130	2	/	/	SYM
ma-220	130	3	ada	ada	PROPN
ma-220	130	4	/	/	SYM
ma-220	130	5	ma.4.4	ma.4.4	PROPN
ma-220	130	6	6	6	NUM
ma-220	130	7	proposition	proposition	NOUN
ma-220	130	8	2.10	2.10	NUM
ma-220	130	9	.	.	PUNCT
ma-220	131	1	let	let	VERB
ma-220	131	2	{	{	PUNCT
ma-220	131	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	131	4	be	be	AUX
ma-220	131	5	a	a	DET
ma-220	131	6	∗-continuous	∗-continuous	ADJ
ma-220	131	7	frame	frame	NOUN
ma-220	131	8	and	and	CCONJ
ma-220	131	9	θ	θ	PROPN
ma-220	131	10	be	be	AUX
ma-220	131	11	an	an	DET
ma-220	131	12	adjointable	adjointable	NOUN
ma-220	131	13	and	and	CCONJ
ma-220	131	14	invertible	invertible	ADJ
ma-220	131	15	operator	operator	NOUN
ma-220	131	16	on	on	ADP
ma-220	131	17	h.	h.	PROPN
ma-220	131	18	then	then	ADV
ma-220	131	19	the	the	DET
ma-220	131	20	sets	set	NOUN
ma-220	131	21	of	of	ADP
ma-220	131	22	operator	operator	NOUN
ma-220	131	23	duals	dual	NOUN
ma-220	131	24	of	of	ADP
ma-220	131	25	{	{	PUNCT
ma-220	131	26	fw}w∈ω	fw}w∈ω	NOUN
ma-220	131	27	and	and	CCONJ
ma-220	131	28	{	{	PUNCT
ma-220	131	29	θfw}w∈ω	θfw}w∈ω	NOUN
ma-220	131	30	are	be	AUX
ma-220	131	31	in	in	ADP
ma-220	131	32	one	one	NUM
ma-220	131	33	to	to	ADP
ma-220	131	34	one	one	NUM
ma-220	131	35	correspondence	correspondence	NOUN
ma-220	131	36	.	.	PUNCT
ma-220	132	1	proof	proof	NOUN
ma-220	132	2	.	.	PUNCT
ma-220	133	1	first	first	ADV
ma-220	133	2	,	,	PUNCT
ma-220	133	3	suppose	suppose	VERB
ma-220	133	4	that	that	SCONJ
ma-220	133	5	(	(	PUNCT
ma-220	133	6	{	{	PUNCT
ma-220	133	7	gw}w∈ω	gw}w∈ω	NOUN
ma-220	133	8	,	,	PUNCT
ma-220	133	9	γ	γ	PROPN
ma-220	133	10	)	)	PUNCT
ma-220	133	11	is	be	AUX
ma-220	133	12	an	an	DET
ma-220	133	13	operator	operator	NOUN
ma-220	133	14	dual	dual	ADJ
ma-220	133	15	for	for	ADP
ma-220	133	16	{	{	PUNCT
ma-220	133	17	fw}w∈ω	fw}w∈ω	NOUN
ma-220	133	18	.	.	PUNCT
ma-220	133	19	for	for	ADP
ma-220	133	20	x	x	PROPN
ma-220	133	21	∈	∈	PROPN
ma-220	133	22	h	h	NOUN
ma-220	133	23	,	,	PUNCT
ma-220	133	24	we	we	PRON
ma-220	133	25	obtain	obtain	VERB
ma-220	133	26	x	x	PUNCT
ma-220	133	27	=	=	SYM
ma-220	133	28	∫	∫	PROPN
ma-220	133	29	ω	ω	NUM
ma-220	133	30	〈	〈	PROPN
ma-220	133	31	γ∗x	γ∗x	PROPN
ma-220	133	32	,	,	PUNCT
ma-220	133	33	fw	fw	PROPN
ma-220	133	34	〉	〉	PROPN
ma-220	133	35	gwdµ(ω	gwdµ(ω	PROPN
ma-220	133	36	)	)	PUNCT
ma-220	134	1	=	=	SYM
ma-220	134	2	∫	∫	PROPN
ma-220	134	3	ω	ω	NUM
ma-220	134	4	〈	〈	PROPN
ma-220	134	5	θ∗	θ∗	NOUN
ma-220	134	6	(	(	PUNCT
ma-220	134	7	θ−1	θ−1	PROPN
ma-220	134	8	)	)	PUNCT
ma-220	134	9	∗	∗	PROPN
ma-220	134	10	γ∗x	γ∗x	NOUN
ma-220	134	11	,	,	PUNCT
ma-220	134	12	fw	fw	PRON
ma-220	134	13	〉	〉	NOUN
ma-220	134	14	gwdµ(ω	gwdµ(ω	NOUN
ma-220	134	15	)	)	PUNCT
ma-220	135	1	=	=	SYM
ma-220	135	2	∫	∫	PROPN
ma-220	135	3	ω	ω	X
ma-220	135	4	〈	〈	PROPN
ma-220	135	5	(	(	PUNCT
ma-220	135	6	θ−1	θ−1	PROPN
ma-220	135	7	)	)	PUNCT
ma-220	135	8	∗	∗	PROPN
ma-220	135	9	γ∗x	γ∗x	NOUN
ma-220	135	10	,	,	PUNCT
ma-220	135	11	θfw	θfw	X
ma-220	135	12	〉	〉	NOUN
ma-220	135	13	gwdµ(ω	gwdµ(ω	PROPN
ma-220	135	14	)	)	PUNCT
ma-220	135	15	.	.	PUNCT
ma-220	136	1	so	so	ADV
ma-220	136	2	(	(	PUNCT
ma-220	136	3	{	{	PUNCT
ma-220	136	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	136	5	,	,	PUNCT
ma-220	136	6	γθ−1	γθ−1	PROPN
ma-220	136	7	)	)	PUNCT
ma-220	136	8	is	be	AUX
ma-220	136	9	an	an	DET
ma-220	136	10	operator	operator	NOUN
ma-220	136	11	dual	dual	ADJ
ma-220	136	12	for	for	ADP
ma-220	136	13	{	{	PUNCT
ma-220	136	14	θfw}w∈ω.now	θfw}w∈ω.now	PROPN
ma-220	136	15	,	,	PUNCT
ma-220	136	16	if	if	SCONJ
ma-220	136	17	(	(	PUNCT
ma-220	136	18	{	{	PUNCT
ma-220	136	19	gw}w∈ω	gw}w∈ω	NOUN
ma-220	136	20	,	,	PUNCT
ma-220	136	21	γ	γ	PROPN
ma-220	136	22	)	)	PUNCT
ma-220	136	23	is	be	AUX
ma-220	136	24	an	an	DET
ma-220	136	25	operator	operator	NOUN
ma-220	136	26	dual	dual	ADJ
ma-220	136	27	of	of	ADP
ma-220	136	28	{	{	PUNCT
ma-220	136	29	θfw}w∈ω	θfw}w∈ω	PROPN
ma-220	136	30	,	,	PUNCT
ma-220	136	31	then	then	ADV
ma-220	136	32	(	(	PUNCT
ma-220	136	33	{	{	PUNCT
ma-220	136	34	gw}w∈ω	gw}w∈ω	NOUN
ma-220	136	35	,	,	PUNCT
ma-220	136	36	γ∗θ	γ∗θ	NUM
ma-220	136	37	)	)	PUNCT
ma-220	136	38	is	be	AUX
ma-220	136	39	an	an	DET
ma-220	136	40	operatordual	operatordual	NOUN
ma-220	136	41	of	of	ADP
ma-220	136	42	{	{	PUNCT
ma-220	136	43	fw}w∈ω	fw}w∈ω	NOUN
ma-220	136	44	,	,	PUNCT
ma-220	136	45	since	since	SCONJ
ma-220	136	46	x	x	PROPN
ma-220	136	47	=	=	SYM
ma-220	136	48	∫	∫	PROPN
ma-220	136	49	ω	ω	NUM
ma-220	136	50	〈	〈	PROPN
ma-220	136	51	γx	γx	NOUN
ma-220	136	52	,	,	PUNCT
ma-220	136	53	θfw	θfw	NOUN
ma-220	136	54	〉	〉	PROPN
ma-220	136	55	gwdµ(ω	gwdµ(ω	PROPN
ma-220	136	56	)	)	PUNCT
ma-220	137	1	=	=	SYM
ma-220	137	2	∫	∫	PROPN
ma-220	137	3	ω	ω	X
ma-220	138	1	〈	〈	PROPN
ma-220	138	2	θ∗γx	θ∗γx	PROPN
ma-220	138	3	,	,	PUNCT
ma-220	138	4	fw	fw	PROPN
ma-220	138	5	〉	〉	PROPN
ma-220	138	6	gwdµ(ω	gwdµ(ω	PROPN
ma-220	138	7	)	)	PUNCT
ma-220	138	8	,	,	PUNCT
ma-220	138	9	∀x	∀x	PROPN
ma-220	138	10	∈	∈	PROPN
ma-220	138	11	h.	h.	NOUN
ma-220	138	12	this	this	PRON
ma-220	138	13	completes	complete	VERB
ma-220	138	14	the	the	DET
ma-220	138	15	proof	proof	NOUN
ma-220	138	16	.	.	PUNCT
ma-220	139	1	�	�	PROPN
ma-220	139	2	proposition	proposition	NOUN
ma-220	139	3	2.11	2.11	NUM
ma-220	139	4	.	.	PUNCT
ma-220	140	1	let	let	VERB
ma-220	140	2	f	f	NOUN
ma-220	140	3	=	=	PRON
ma-220	140	4	{	{	PUNCT
ma-220	140	5	fw}w∈ω	fw}w∈ω	NOUN
ma-220	140	6	be	be	AUX
ma-220	140	7	a	a	DET
ma-220	140	8	∗-continuous	∗-continuous	ADJ
ma-220	140	9	frame	frame	NOUN
ma-220	140	10	for	for	ADP
ma-220	140	11	h	h	NOUN
ma-220	140	12	with	with	ADP
ma-220	140	13	pre	pre	ADJ
ma-220	140	14	-	-	ADJ
ma-220	140	15	frame	frame	ADJ
ma-220	140	16	operator	operator	NOUN
ma-220	140	17	tf	tf	NOUN
ma-220	140	18	and	and	CCONJ
ma-220	140	19	frame	frame	NOUN
ma-220	140	20	operator	operator	NOUN
ma-220	140	21	s.	s.	PROPN
ma-220	140	22	then	then	ADV
ma-220	140	23	the	the	DET
ma-220	140	24	set	set	NOUN
ma-220	140	25	of	of	ADP
ma-220	140	26	all	all	DET
ma-220	140	27	the	the	DET
ma-220	140	28	operator	operator	NOUN
ma-220	140	29	duals	dual	NOUN
ma-220	140	30	of	of	ADP
ma-220	140	31	{	{	PUNCT
ma-220	140	32	fw}w∈ω	fw}w∈ω	NOUN
ma-220	140	33	is	be	AUX
ma-220	140	34	precisely	precisely	ADV
ma-220	140	35	the	the	DET
ma-220	140	36	following	follow	VERB
ma-220	140	37	{	{	PUNCT
ma-220	140	38	gw}w∈ω	gw}w∈ω	NOUN
ma-220	140	39	=	=	NOUN
ma-220	140	40	{	{	PUNCT
ma-220	140	41	γfw	γfw	NOUN
ma-220	140	42	+	+	NUM
ma-220	140	43	ϕew	ϕew	PROPN
ma-220	141	1	−	−	PROPN
ma-220	141	2	∫	∫	PROPN
ma-220	141	3	ω	ω	NUM
ma-220	141	4	〈	〈	PROPN
ma-220	141	5	s−1fw	s−1fw	NOUN
ma-220	141	6	,	,	PUNCT
ma-220	141	7	fi	fi	NOUN
ma-220	141	8	〉	〉	NOUN
ma-220	141	9	ϕewdµ(ω	ϕewdµ(ω	NOUN
ma-220	141	10	)	)	PUNCT
ma-220	141	11	}	}	PUNCT
ma-220	141	12	w∈ω	w∈ω	NOUN
ma-220	141	13	,	,	PUNCT
ma-220	141	14	where	where	SCONJ
ma-220	141	15	{	{	PUNCT
ma-220	141	16	ew}w∈ω	ew}w∈ω	NOUN
ma-220	141	17	is	be	AUX
ma-220	141	18	the	the	DET
ma-220	141	19	standard	standard	ADJ
ma-220	141	20	orthonormal	orthonormal	ADJ
ma-220	141	21	basis	basis	NOUN
ma-220	141	22	for	for	ADP
ma-220	141	23	l2(ω	l2(ω	PROPN
ma-220	141	24	,	,	PUNCT
ma-220	141	25	a	a	PRON
ma-220	141	26	)	)	PUNCT
ma-220	141	27	,	,	PUNCT
ma-220	141	28	ϕ	ϕ	PROPN
ma-220	141	29	∈	∈	PROPN
ma-220	141	30	b∗	b∗	ADJ
ma-220	141	31	(	(	PUNCT
ma-220	141	32	h	h	NOUN
ma-220	141	33	,	,	PUNCT
ma-220	141	34	l2(ω	l2(ω	PROPN
ma-220	141	35	,	,	PUNCT
ma-220	141	36	a	a	PRON
ma-220	141	37	)	)	PUNCT
ma-220	141	38	)	)	PUNCT
ma-220	141	39	,	,	PUNCT
ma-220	141	40	and	and	CCONJ
ma-220	141	41	γ	γ	X
ma-220	141	42	is	be	AUX
ma-220	141	43	an	an	DET
ma-220	141	44	invertible	invertible	ADJ
ma-220	141	45	adjointable	adjointable	NOUN
ma-220	141	46	operator	operator	NOUN
ma-220	141	47	on	on	ADP
ma-220	141	48	h.	h.	PROPN
ma-220	141	49	proof	proof	PROPN
ma-220	141	50	.	.	PUNCT
ma-220	142	1	assume	assume	VERB
ma-220	142	2	that	that	SCONJ
ma-220	142	3	{	{	PUNCT
ma-220	142	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	142	5	is	be	AUX
ma-220	142	6	a	a	DET
ma-220	142	7	sequence	sequence	NOUN
ma-220	142	8	as	as	ADP
ma-220	142	9	above	above	ADV
ma-220	142	10	.	.	PUNCT
ma-220	143	1	then	then	ADV
ma-220	143	2	its	its	PRON
ma-220	143	3	pre	pre	ADJ
ma-220	143	4	-	-	ADJ
ma-220	143	5	frame	frame	ADJ
ma-220	143	6	operator	operator	NOUN
ma-220	143	7	is	be	AUX
ma-220	143	8	tg	tg	NOUN
ma-220	143	9	=	=	PUNCT
ma-220	143	10	tfγ	tfγ	NOUN
ma-220	143	11	+	+	CCONJ
ma-220	143	12	ϕ−	ϕ−	PROPN
ma-220	143	13	tfs−1	tfs−1	PROPN
ma-220	143	14	t	t	PROPN
ma-220	143	15	∗fϕ	∗fϕ	NOUN
ma-220	143	16	and	and	CCONJ
ma-220	143	17	so	so	ADV
ma-220	143	18	(	(	PUNCT
ma-220	143	19	sγ)−1	sγ)−1	NOUN
ma-220	143	20	(	(	PUNCT
ma-220	143	21	t	t	NOUN
ma-220	143	22	∗ftg	∗ftg	PROPN
ma-220	143	23	)	)	PUNCT
ma-220	143	24	=	=	SYM
ma-220	143	25	(	(	PUNCT
ma-220	143	26	sγ)−1	sγ)−1	PROPN
ma-220	143	27	(	(	PUNCT
ma-220	143	28	t	t	NOUN
ma-220	143	29	∗ftfγ	∗ftfγ	NOUN
ma-220	144	1	+	+	X
ma-220	145	1	t	t	X
ma-220	146	1	∗fϕ−	∗fϕ−	X
ma-220	146	2	t	t	PROPN
ma-220	146	3	∗ftfs−1	∗ftfs−1	PROPN
ma-220	146	4	t	t	PROPN
ma-220	146	5	∗fϕ	∗fϕ	PUNCT
ma-220	146	6	)	)	PUNCT
ma-220	147	1	=	=	PRON
ma-220	147	2	(	(	PUNCT
ma-220	147	3	sγ)−1	sγ)−1	PROPN
ma-220	147	4	(	(	PUNCT
ma-220	147	5	t	t	PROPN
ma-220	147	6	∗ftfs	∗ftfs	NUM
ma-220	148	1	−1sγ	−1sγ	NOUN
ma-220	149	1	+	+	NUM
ma-220	149	2	t	t	X
ma-220	149	3	∗fϕ−	∗fϕ−	X
ma-220	149	4	t	t	PROPN
ma-220	149	5	∗ftfs−1	∗ftfs−1	PROPN
ma-220	149	6	t	t	PROPN
ma-220	149	7	∗fϕ	∗fϕ	PUNCT
ma-220	149	8	)	)	PUNCT
ma-220	150	1	=	=	PRON
ma-220	150	2	(	(	PUNCT
ma-220	150	3	sγ)−1(sγ	sγ)−1(sγ	NOUN
ma-220	150	4	)	)	PUNCT
ma-220	151	1	=	=	SYM
ma-220	151	2	i.	i.	NOUN
ma-220	151	3	by	by	ADP
ma-220	151	4	a	a	DET
ma-220	151	5	similar	similar	ADJ
ma-220	151	6	relation	relation	NOUN
ma-220	151	7	with	with	ADP
ma-220	151	8	the	the	DET
ma-220	151	9	given	give	VERB
ma-220	151	10	equality	equality	NOUN
ma-220	151	11	in	in	ADP
ma-220	151	12	remark	remark	NOUN
ma-220	151	13	2.5	2.5	NUM
ma-220	151	14	,	,	PUNCT
ma-220	151	15	we	we	PRON
ma-220	151	16	can	can	AUX
ma-220	151	17	conclude	conclude	VERB
ma-220	151	18	that	that	PRON
ma-220	151	19	{	{	PUNCT
ma-220	151	20	gw}w∈ω	gw}w∈ω	NOUN
ma-220	151	21	isan	isan	ADJ
ma-220	151	22	operator	operator	NOUN
ma-220	151	23	dual	dual	ADJ
ma-220	151	24	for	for	ADP
ma-220	151	25	{	{	PUNCT
ma-220	151	26	fw}w∈ω	fw}w∈ω	NOUN
ma-220	151	27	with	with	ADP
ma-220	151	28	the	the	DET
ma-220	151	29	corresponding	correspond	VERB
ma-220	151	30	operator	operator	NOUN
ma-220	151	31	(	(	PUNCT
ma-220	151	32	sγ)−1	sγ)−1	PROPN
ma-220	151	33	.	.	PUNCT
ma-220	151	34	�	�	PROPN
ma-220	151	35	theorem	theorem	VERB
ma-220	151	36	2.12	2.12	NUM
ma-220	151	37	.	.	PUNCT
ma-220	152	1	let	let	AUX
ma-220	152	2	(	(	PUNCT
ma-220	152	3	{	{	PUNCT
ma-220	152	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	152	5	,	,	PUNCT
ma-220	152	6	γ	γ	PROPN
ma-220	152	7	)	)	PUNCT
ma-220	152	8	be	be	AUX
ma-220	152	9	an	an	DET
ma-220	152	10	operator	operator	NOUN
ma-220	152	11	dual	dual	ADJ
ma-220	152	12	of	of	ADP
ma-220	152	13	∗-continuous	∗-continuous	ADJ
ma-220	152	14	frame	frame	NOUN
ma-220	152	15	{	{	PUNCT
ma-220	152	16	fw}w∈ω	fw}w∈ω	NOUN
ma-220	152	17	for	for	ADP
ma-220	152	18	h.	h.	PROPN
ma-220	152	19	then	then	ADV
ma-220	152	20	there	there	PRON
ma-220	152	21	exist	exist	VERB
ma-220	152	22	a	a	DET
ma-220	152	23	hilbert	hilbert	NOUN
ma-220	152	24	a	a	DET
ma-220	152	25	-	-	PUNCT
ma-220	152	26	module	module	NOUN
ma-220	152	27	k	k	PROPN
ma-220	152	28	⊇	⊇	PROPN
ma-220	152	29	h	h	PROPN
ma-220	152	30	and	and	CCONJ
ma-220	152	31	a	a	DET
ma-220	152	32	riesz	riesz	PROPN
ma-220	152	33	basis	basis	NOUN
ma-220	152	34	{	{	PUNCT
ma-220	152	35	uw}w∈ω	uw}w∈ω	NOUN
ma-220	152	36	of	of	ADP
ma-220	152	37	k	k	PROPN
ma-220	152	38	which	which	PRON
ma-220	152	39	has	have	VERB
ma-220	152	40	a	a	DET
ma-220	152	41	unique	unique	ADJ
ma-220	152	42	dual	dual	ADJ
ma-220	152	43	{	{	PUNCT
ma-220	152	44	vw}w∈ω	vw}w∈ω	NOUN
ma-220	152	45	and	and	CCONJ
ma-220	152	46	satisfies	satisfie	NOUN
ma-220	152	47	(	(	PUNCT
ma-220	152	48	pu)uw	pu)uw	PUNCT
ma-220	152	49	=	=	SYM
ma-220	152	50	fw	fw	PROPN
ma-220	152	51	and	and	CCONJ
ma-220	152	52	(	(	PUNCT
ma-220	152	53	pv0)vw	pv0)vw	PROPN
ma-220	152	54	=	=	SYM
ma-220	152	55	gw	gw	PROPN
ma-220	152	56	for	for	ADP
ma-220	152	57	all	all	DET
ma-220	152	58	w	w	PROPN
ma-220	152	59	∈	∈	PROPN
ma-220	152	60	ω	ω	NOUN
ma-220	152	61	,	,	PUNCT
ma-220	152	62	where	where	SCONJ
ma-220	152	63	p	p	NOUN
ma-220	152	64	is	be	AUX
ma-220	152	65	the	the	DET
ma-220	152	66	projection	projection	NOUN
ma-220	152	67	from	from	ADP
ma-220	152	68	k	k	PROPN
ma-220	152	69	onto	onto	ADP
ma-220	152	70	h.	h.	PROPN
ma-220	152	71	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	152	72	eur	eur	PROPN
ma-220	152	73	.	.	PUNCT
ma-220	153	1	j.	j.	PROPN
ma-220	153	2	math	math	PROPN
ma-220	153	3	.	.	PUNCT
ma-220	154	1	anal	anal	PROPN
ma-220	154	2	.	.	PUNCT
ma-220	155	1	10.28924	10.28924	NUM
ma-220	155	2	/	/	SYM
ma-220	155	3	ada	ada	PROPN
ma-220	155	4	/	/	SYM
ma-220	155	5	ma.4.4	ma.4.4	PROPN
ma-220	155	6	7	7	NUM
ma-220	155	7	proof	proof	NOUN
ma-220	155	8	.	.	PUNCT
ma-220	156	1	assume	assume	VERB
ma-220	156	2	that	that	SCONJ
ma-220	156	3	tf	tf	INTJ
ma-220	156	4	,	,	PUNCT
ma-220	156	5	tg	tg	PROPN
ma-220	156	6	and	and	CCONJ
ma-220	156	7	sf	sf	PROPN
ma-220	156	8	,	,	PUNCT
ma-220	156	9	sg	sg	PROPN
ma-220	156	10	are	be	AUX
ma-220	156	11	pre	pre	ADJ
ma-220	156	12	-	-	ADJ
ma-220	156	13	frame	frame	ADJ
ma-220	156	14	operators	operator	NOUN
ma-220	156	15	and	and	CCONJ
ma-220	156	16	frame	frame	NOUN
ma-220	156	17	operators	operator	NOUN
ma-220	156	18	of	of	ADP
ma-220	156	19	{	{	PUNCT
ma-220	156	20	fw}w∈ωand	fw}w∈ωand	PROPN
ma-220	156	21	{	{	PUNCT
ma-220	156	22	gw}w∈ω	gw}w∈ω	NOUN
ma-220	156	23	,	,	PUNCT
ma-220	156	24	respectively	respectively	ADV
ma-220	156	25	.	.	PUNCT
ma-220	157	1	also	also	ADV
ma-220	157	2	,	,	PUNCT
ma-220	157	3	the	the	DET
ma-220	157	4	orthogonal	orthogonal	ADJ
ma-220	157	5	projections	projection	NOUN
ma-220	157	6	onto	onto	ADP
ma-220	157	7	the	the	DET
ma-220	157	8	range	range	NOUN
ma-220	157	9	of	of	ADP
ma-220	157	10	tf	tf	INTJ
ma-220	157	11	,	,	PUNCT
ma-220	157	12	r	r	NOUN
ma-220	157	13	(	(	PUNCT
ma-220	157	14	tf	tf	NOUN
ma-220	157	15	)	)	PUNCT
ma-220	157	16	,	,	PUNCT
ma-220	157	17	and	and	CCONJ
ma-220	157	18	therange	therange	NOUN
ma-220	157	19	of	of	ADP
ma-220	157	20	tg	tg	PROPN
ma-220	157	21	,	,	PUNCT
ma-220	157	22	r	r	NOUN
ma-220	157	23	(	(	PUNCT
ma-220	157	24	tg	tg	PROPN
ma-220	157	25	)	)	PUNCT
ma-220	157	26	,	,	PUNCT
ma-220	157	27	are	be	AUX
ma-220	157	28	pf	pf	NOUN
ma-220	157	29	and	and	CCONJ
ma-220	157	30	pg	pg	INTJ
ma-220	157	31	,	,	PUNCT
ma-220	157	32	respectively	respectively	ADV
ma-220	157	33	.	.	PUNCT
ma-220	158	1	now	now	ADV
ma-220	158	2	,	,	PUNCT
ma-220	158	3	for	for	ADP
ma-220	158	4	x	x	PROPN
ma-220	158	5	∈	∈	PROPN
ma-220	158	6	h	h	NOUN
ma-220	158	7	,	,	PUNCT
ma-220	158	8	〈	〈	PROPN
ma-220	158	9	tgx	tgx	PROPN
ma-220	158	10	,	,	PUNCT
ma-220	158	11	tgs	tgs	PROPN
ma-220	158	12	−1	−1	NOUN
ma-220	158	13	g	g	NOUN
ma-220	158	14	gw	gw	PROPN
ma-220	158	15	〉	〉	NOUN
ma-220	158	16	=	=	PUNCT
ma-220	158	17	〈	〈	PROPN
ma-220	158	18	t	t	NOUN
ma-220	158	19	∗gtgx	∗gtgx	PROPN
ma-220	158	20	,	,	PUNCT
ma-220	158	21	s	s	VERB
ma-220	158	22	−1	−1	NOUN
ma-220	158	23	g	g	NOUN
ma-220	158	24	gw	gw	PROPN
ma-220	158	25	〉	〉	NOUN
ma-220	158	26	=	=	PUNCT
ma-220	158	27	〈	〈	PROPN
ma-220	158	28	s−1	s−1	PROPN
ma-220	158	29	g	g	PROPN
ma-220	158	30	sgx	sgx	PROPN
ma-220	158	31	,	,	PUNCT
ma-220	158	32	gw	gw	PROPN
ma-220	158	33	〉	〉	NOUN
ma-220	158	34	=	=	SYM
ma-220	158	35	〈	〈	PROPN
ma-220	158	36	x	x	X
ma-220	158	37	,	,	PUNCT
ma-220	158	38	gw	gw	PROPN
ma-220	158	39	〉	〉	PROPN
ma-220	158	40	=	=	SYM
ma-220	159	1	〈	〈	PROPN
ma-220	159	2	tgx	tgx	PROPN
ma-220	159	3	,	,	PUNCT
ma-220	159	4	ew	ew	INTJ
ma-220	159	5	〉	〉	NOUN
ma-220	159	6	=	=	SYM
ma-220	159	7	〈	〈	PROPN
ma-220	159	8	tgx	tgx	PROPN
ma-220	159	9	,	,	PUNCT
ma-220	159	10	pgew	pgew	PROPN
ma-220	159	11	〉	〉	PROPN
ma-220	159	12	′and	′and	PROPN
ma-220	159	13	so	so	ADV
ma-220	159	14	pgew	pgew	NOUN
ma-220	159	15	=	=	PROPN
ma-220	160	1	tgs	tgs	PROPN
ma-220	160	2	−1	−1	NOUN
ma-220	160	3	g	g	PROPN
ma-220	160	4	gw	gw	PROPN
ma-220	160	5	,	,	PUNCT
ma-220	160	6	∀w	∀w	PROPN
ma-220	160	7	∈	∈	PROPN
ma-220	160	8	ω	ω	PROPN
ma-220	160	9	,	,	PUNCT
ma-220	160	10	(	(	PUNCT
ma-220	160	11	2.4	2.4	NUM
ma-220	160	12	)	)	PUNCT
ma-220	160	13	where	where	SCONJ
ma-220	160	14	{	{	PUNCT
ma-220	160	15	ew}w∈ω	ew}w∈ω	NOUN
ma-220	160	16	is	be	AUX
ma-220	160	17	the	the	DET
ma-220	160	18	standard	standard	ADJ
ma-220	160	19	orthonormal	orthonormal	ADJ
ma-220	160	20	basis	basis	NOUN
ma-220	160	21	of	of	ADP
ma-220	160	22	l2(ω	l2(ω	PROPN
ma-220	160	23	,	,	PUNCT
ma-220	160	24	a	a	PRON
ma-220	160	25	)	)	PUNCT
ma-220	160	26	.	.	PUNCT
ma-220	161	1	by	by	ADP
ma-220	161	2	(	(	PUNCT
ma-220	161	3	2.4	2.4	NUM
ma-220	161	4	)	)	PUNCT
ma-220	161	5	,	,	PUNCT
ma-220	161	6	for	for	ADP
ma-220	161	7	x	x	PROPN
ma-220	161	8	∈	∈	PROPN
ma-220	161	9	h	h	NOUN
ma-220	161	10	,	,	PUNCT
ma-220	161	11	we	we	PRON
ma-220	161	12	give	give	VERB
ma-220	161	13	pgtf	pgtf	NOUN
ma-220	161	14	(	(	PUNCT
ma-220	161	15	γ∗x	γ∗x	NOUN
ma-220	161	16	)	)	PUNCT
ma-220	161	17	=	=	SYM
ma-220	162	1	pg	pg	PROPN
ma-220	162	2	(	(	PUNCT
ma-220	162	3	∫	∫	PROPN
ma-220	162	4	ω	ω	PROPN
ma-220	162	5	〈	〈	PROPN
ma-220	162	6	γ∗x	γ∗x	PROPN
ma-220	162	7	,	,	PUNCT
ma-220	162	8	fw	fw	PROPN
ma-220	162	9	〉	〉	PROPN
ma-220	162	10	ew	ew	NOUN
ma-220	162	11	)	)	PUNCT
ma-220	163	1	=	=	SYM
ma-220	163	2	∫	∫	PROPN
ma-220	163	3	ω	ω	X
ma-220	163	4	〈	〈	PROPN
ma-220	163	5	γ∗x	γ∗x	PROPN
ma-220	163	6	,	,	PUNCT
ma-220	163	7	fw	fw	ADJ
ma-220	163	8	〉	〉	PROPN
ma-220	163	9	pgew	pgew	NOUN
ma-220	163	10	=	=	SYM
ma-220	163	11	∫	∫	PROPN
ma-220	163	12	ω	ω	PROPN
ma-220	163	13	〈	〈	PROPN
ma-220	163	14	γ∗f	γ∗f	PROPN
ma-220	163	15	,	,	PUNCT
ma-220	163	16	fw	fw	PROPN
ma-220	163	17	〉	〉	PROPN
ma-220	163	18	tgs−1	tgs−1	PROPN
ma-220	163	19	g	g	NOUN
ma-220	163	20	gw	gw	PROPN
ma-220	163	21	=	=	PROPN
ma-220	163	22	tgs	tgs	PROPN
ma-220	163	23	−1	−1	NOUN
ma-220	163	24	g	g	PROPN
ma-220	163	25	(	(	PUNCT
ma-220	163	26	∫	∫	PROPN
ma-220	163	27	ω	ω	PROPN
ma-220	163	28	〈	〈	PROPN
ma-220	163	29	γ∗x	γ∗x	PROPN
ma-220	163	30	,	,	PUNCT
ma-220	163	31	fw	fw	ADJ
ma-220	163	32	〉	〉	PROPN
ma-220	163	33	gw	gw	PROPN
ma-220	163	34	)	)	PUNCT
ma-220	164	1	=	=	PUNCT
ma-220	164	2	tgs	tgs	AUX
ma-220	164	3	−1	−1	NOUN
ma-220	164	4	g	g	PROPN
ma-220	164	5	x.	x.	PROPN
ma-220	164	6	set	set	VERB
ma-220	164	7	k	k	PROPN
ma-220	164	8	=	=	SYM
ma-220	164	9	h⊕	h⊕	PROPN
ma-220	164	10	p⊥g	p⊥g	PROPN
ma-220	164	11	l2(a	l2(a	PROPN
ma-220	164	12	)	)	PUNCT
ma-220	164	13	,	,	PUNCT
ma-220	164	14	uw	uw	PROPN
ma-220	164	15	=	=	PROPN
ma-220	164	16	fw	fw	PROPN
ma-220	164	17	⊕	⊕	PROPN
ma-220	164	18	p⊥g	p⊥g	PROPN
ma-220	164	19	ew	ew	INTJ
ma-220	164	20	,	,	PUNCT
ma-220	164	21	∀w	∀w	PROPN
ma-220	164	22	∈	∈	PROPN
ma-220	164	23	ω	ω	NOUN
ma-220	164	24	.	.	PUNCT
ma-220	165	1	if	if	SCONJ
ma-220	165	2	tu	tu	PROPN
ma-220	165	3	is	be	AUX
ma-220	165	4	a	a	DET
ma-220	165	5	pre	pre	ADJ
ma-220	165	6	-	-	ADJ
ma-220	165	7	frame	frame	ADJ
ma-220	165	8	operator	operator	NOUN
ma-220	165	9	of	of	ADP
ma-220	165	10	the	the	DET
ma-220	165	11	sequence	sequence	NOUN
ma-220	165	12	{	{	PUNCT
ma-220	165	13	uw}w∈ω	uw}w∈ω	INTJ
ma-220	165	14	,	,	PUNCT
ma-220	165	15	then	then	ADV
ma-220	165	16	tu(x	tu(x	PUNCT
ma-220	165	17	⊕	⊕	PROPN
ma-220	165	18	v	v	NOUN
ma-220	165	19	)	)	PUNCT
ma-220	165	20	=	=	VERB
ma-220	166	1	tfx	tfx	PROPN
ma-220	166	2	+	+	CCONJ
ma-220	166	3	v	v	NOUN
ma-220	166	4	and	and	CCONJ
ma-220	166	5	‖tu(x	‖tu(x	PROPN
ma-220	166	6	⊕	⊕	PROPN
ma-220	166	7	v)‖	v)‖	NOUN
ma-220	167	1	=	=	PUNCT
ma-220	167	2	‖tf	‖tf	NUM
ma-220	167	3	f	f	PROPN
ma-220	167	4	+	+	CCONJ
ma-220	167	5	w‖	w‖	PROPN
ma-220	167	6	≤	≤	X
ma-220	167	7	b(‖x‖+	b(‖x‖+	VERB
ma-220	167	8	‖v‖	‖v‖	PROPN
ma-220	167	9	)	)	PUNCT
ma-220	167	10	=	=	SYM
ma-220	167	11	b‖x	b‖x	NOUN
ma-220	167	12	⊕	⊕	PROPN
ma-220	167	13	v‖	v‖	PROPN
ma-220	167	14	,	,	PUNCT
ma-220	167	15	∀x	∀x	X
ma-220	167	16	⊕	⊕	PROPN
ma-220	167	17	v	v	ADP
ma-220	167	18	∈	∈	PROPN
ma-220	167	19	k	k	NOUN
ma-220	167	20	for	for	ADP
ma-220	167	21	some	some	DET
ma-220	167	22	b	b	NOUN
ma-220	167	23	>	>	X
ma-220	167	24	0	0	PUNCT
ma-220	168	1	and	and	CCONJ
ma-220	168	2	so	so	ADV
ma-220	168	3	{	{	PUNCT
ma-220	168	4	uw}w∈ω	uw}w∈ω	INTJ
ma-220	168	5	is	be	AUX
ma-220	168	6	a	a	DET
ma-220	168	7	bessel	bessel	ADJ
ma-220	168	8	sequence	sequence	NOUN
ma-220	168	9	.	.	PUNCT
ma-220	169	1	we	we	PRON
ma-220	169	2	show	show	VERB
ma-220	169	3	that	that	SCONJ
ma-220	169	4	tu	tu	PROPN
ma-220	169	5	has	have	VERB
ma-220	169	6	a	a	DET
ma-220	169	7	closed	closed	ADJ
ma-220	169	8	range.suppose	range.suppose	X
ma-220	169	9	{	{	PUNCT
ma-220	169	10	ηn}n∈n	ηn}n∈n	ADP
ma-220	169	11	⊆	⊆	NUM
ma-220	169	12	r	r	NOUN
ma-220	169	13	(	(	PUNCT
ma-220	169	14	tu	tu	PROPN
ma-220	169	15	)	)	PUNCT
ma-220	169	16	such	such	ADJ
ma-220	169	17	that	that	SCONJ
ma-220	169	18	ηn	ηn	PROPN
ma-220	169	19	n→∞−→	n→∞−→	PROPN
ma-220	169	20	η	η	PROPN
ma-220	169	21	.	.	PROPN
ma-220	169	22	since	since	SCONJ
ma-220	169	23	γ	γ	PROPN
ma-220	169	24	is	be	AUX
ma-220	169	25	invertible	invertible	ADJ
ma-220	169	26	and	and	CCONJ
ma-220	169	27	adjointable	adjointable	ADJ
ma-220	169	28	,	,	PUNCT
ma-220	169	29	there	there	PRON
ma-220	169	30	exists	exist	VERB
ma-220	169	31	γ∗fn	γ∗fn	PROPN
ma-220	169	32	⊕	⊕	PROPN
ma-220	169	33	vn	vn	PROPN
ma-220	169	34	∈	∈	PROPN
ma-220	169	35	h	h	PROPN
ma-220	169	36	⊕	⊕	PROPN
ma-220	169	37	p⊥g	p⊥g	PROPN
ma-220	169	38	(	(	PUNCT
ma-220	169	39	l2(ω	l2(ω	PROPN
ma-220	169	40	,	,	PUNCT
ma-220	169	41	a);tu	a);tu	VERB
ma-220	169	42	(	(	PUNCT
ma-220	169	43	γ∗fn	γ∗fn	PROPN
ma-220	169	44	⊕	⊕	PROPN
ma-220	169	45	vn	vn	PROPN
ma-220	169	46	)	)	PUNCT
ma-220	169	47	=	=	VERB
ma-220	170	1	ηn	ηn	ADJ
ma-220	170	2	.	.	PUNCT
ma-220	171	1	on	on	ADP
ma-220	171	2	the	the	DET
ma-220	171	3	other	other	ADJ
ma-220	171	4	hand	hand	NOUN
ma-220	171	5	,	,	PUNCT
ma-220	171	6	tu	tu	PROPN
ma-220	171	7	(	(	PUNCT
ma-220	171	8	γ∗fn	γ∗fn	PROPN
ma-220	171	9	⊕	⊕	PROPN
ma-220	171	10	vn	vn	PROPN
ma-220	171	11	)	)	PUNCT
ma-220	172	1	=	=	SYM
ma-220	172	2	tf	tf	INTJ
ma-220	172	3	(	(	PUNCT
ma-220	172	4	γ∗fn	γ∗fn	PROPN
ma-220	172	5	)	)	PUNCT
ma-220	173	1	+	+	NUM
ma-220	173	2	vn	vn	X
ma-220	173	3	=	=	PUNCT
ma-220	173	4	ηn	ηn	PROPN
ma-220	173	5	n→∞−→	n→∞−→	PROPN
ma-220	173	6	η	η	PROPN
ma-220	173	7	and	and	CCONJ
ma-220	173	8	remark	remark	VERB
ma-220	173	9	2.4	2.4	NUM
ma-220	173	10	gives	give	VERB
ma-220	173	11	fn	fn	NOUN
ma-220	173	12	=	=	SYM
ma-220	173	13	t	t	NOUN
ma-220	173	14	∗gtfγ∗fn	∗gtfγ∗fn	PUNCT
ma-220	173	15	=	=	SYM
ma-220	173	16	t	t	PROPN
ma-220	173	17	∗g	∗g	PROPN
ma-220	173	18	(	(	PUNCT
ma-220	173	19	tfγ∗fn	tfγ∗fn	PROPN
ma-220	173	20	+	+	CCONJ
ma-220	173	21	vn	vn	X
ma-220	173	22	)	)	PUNCT
ma-220	173	23	n→∞−→	n→∞−→	NOUN
ma-220	173	24	t	t	NOUN
ma-220	173	25	∗gη	∗gη	NOUN
ma-220	173	26	.	.	PUNCT
ma-220	174	1	r	r	NOUN
ma-220	174	2	(	(	PUNCT
ma-220	174	3	tu	tu	PROPN
ma-220	174	4	)	)	PUNCT
ma-220	174	5	is	be	AUX
ma-220	174	6	closed	close	VERB
ma-220	174	7	,	,	PUNCT
ma-220	174	8	since	since	SCONJ
ma-220	174	9	r	r	NOUN
ma-220	174	10	(	(	PUNCT
ma-220	174	11	tf	tf	INTJ
ma-220	174	12	)	)	PUNCT
ma-220	174	13	is	be	AUX
ma-220	174	14	closed	close	VERB
ma-220	174	15	.	.	PUNCT
ma-220	175	1	also	also	ADV
ma-220	175	2	,	,	PUNCT
ma-220	175	3	t	t	PROPN
ma-220	175	4	∗u	∗u	NOUN
ma-220	175	5	has	have	VERB
ma-220	175	6	a	a	DET
ma-220	175	7	closed	closed	ADJ
ma-220	175	8	range	range	NOUN
ma-220	175	9	.	.	PUNCT
ma-220	176	1	this	this	DET
ma-220	176	2	step	step	NOUN
ma-220	176	3	will	will	AUX
ma-220	176	4	obtain	obtain	VERB
ma-220	176	5	theinjectivity	theinjectivity	NOUN
ma-220	176	6	of	of	ADP
ma-220	176	7	t	t	PROPN
ma-220	176	8	∗u	∗u	PROPN
ma-220	176	9	.	.	PUNCT
ma-220	177	1	if	if	SCONJ
ma-220	177	2	t	t	PROPN
ma-220	177	3	∗u	∗u	PROPN
ma-220	177	4	(	(	PUNCT
ma-220	177	5	∫ω	∫ω	NOUN
ma-220	177	6	awew	awew	PROPN
ma-220	177	7	)	)	PUNCT
ma-220	178	1	=	=	SYM
ma-220	178	2	0	0	NUM
ma-220	178	3	,	,	PUNCT
ma-220	178	4	then	then	ADV
ma-220	178	5	0	0	NUM
ma-220	178	6	=	=	SYM
ma-220	179	1	∫	∫	PROPN
ma-220	179	2	ω	ω	INTJ
ma-220	179	3	aw	aw	INTJ
ma-220	179	4	(	(	PUNCT
ma-220	179	5	fw	fw	PROPN
ma-220	179	6	⊕	⊕	PROPN
ma-220	179	7	p⊥g	p⊥g	PROPN
ma-220	179	8	ew	ew	INTJ
ma-220	179	9	)	)	PUNCT
ma-220	179	10	=	=	SYM
ma-220	180	1	∫	∫	PROPN
ma-220	180	2	ω	ω	PROPN
ma-220	180	3	awfw	awfw	PROPN
ma-220	180	4	⊕	⊕	PROPN
ma-220	180	5	p⊥g	p⊥g	PROPN
ma-220	180	6	(	(	PUNCT
ma-220	180	7	∫	∫	PROPN
ma-220	180	8	ω	ω	PROPN
ma-220	180	9	awew	awew	PROPN
ma-220	180	10	)	)	PUNCT
ma-220	180	11	.	.	PUNCT
ma-220	181	1	it	it	PRON
ma-220	181	2	concludes	conclude	VERB
ma-220	181	3	that	that	SCONJ
ma-220	181	4	(	(	PUNCT
ma-220	181	5	i	i	NOUN
ma-220	181	6	)	)	PUNCT
ma-220	181	7	∫	∫	PROPN
ma-220	181	8	ω	ω	NUM
ma-220	181	9	awfw	awfw	NOUN
ma-220	181	10	=	=	SYM
ma-220	181	11	0	0	NUM
ma-220	182	1	and	and	CCONJ
ma-220	182	2	(	(	PUNCT
ma-220	182	3	i	i	PRON
ma-220	182	4	i	i	PROPN
ma-220	182	5	)	)	PUNCT
ma-220	182	6	p⊥g	p⊥g	PROPN
ma-220	182	7	(	(	PUNCT
ma-220	182	8	∫	∫	PROPN
ma-220	182	9	ω	ω	PROPN
ma-220	182	10	awew	awew	PROPN
ma-220	182	11	)	)	PUNCT
ma-220	182	12	=	=	PUNCT
ma-220	183	1	0	0	X
ma-220	183	2	.	.	PUNCT
ma-220	184	1	from	from	ADP
ma-220	184	2	(	(	PUNCT
ma-220	184	3	i	i	PRON
ma-220	184	4	i	i	PROPN
ma-220	184	5	)	)	PUNCT
ma-220	184	6	,	,	PUNCT
ma-220	184	7	we	we	PRON
ma-220	184	8	have∫	have∫	VERB
ma-220	184	9	ω	ω	NUM
ma-220	184	10	awew	awew	PROPN
ma-220	184	11	∈	∈	PROPN
ma-220	184	12	r	r	NOUN
ma-220	184	13	(	(	PUNCT
ma-220	184	14	tg	tg	INTJ
ma-220	184	15	)	)	PUNCT
ma-220	185	1	=	=	NOUN
ma-220	185	2	⇒	⇒	NOUN
ma-220	185	3	∃h	∃h	NOUN
ma-220	185	4	∈	∈	PROPN
ma-220	185	5	h;tgh	h;tgh	PROPN
ma-220	185	6	=	=	SYM
ma-220	185	7	∫	∫	PROPN
ma-220	185	8	ω	ω	PROPN
ma-220	185	9	awew	awew	PROPN
ma-220	185	10	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	185	11	eur	eur	PROPN
ma-220	185	12	.	.	PUNCT
ma-220	186	1	j.	j.	PROPN
ma-220	186	2	math	math	PROPN
ma-220	186	3	.	.	PUNCT
ma-220	187	1	anal	anal	PROPN
ma-220	187	2	.	.	PUNCT
ma-220	188	1	10.28924	10.28924	NUM
ma-220	188	2	/	/	SYM
ma-220	188	3	ada	ada	PROPN
ma-220	188	4	/	/	SYM
ma-220	188	5	ma.4.4	ma.4.4	PROPN
ma-220	188	6	8and	8and	NUM
ma-220	188	7	on	on	ADP
ma-220	188	8	the	the	DET
ma-220	188	9	other	other	ADJ
ma-220	188	10	hand	hand	NOUN
ma-220	188	11	,	,	PUNCT
ma-220	188	12	tgh	tgh	PROPN
ma-220	188	13	=	=	SYM
ma-220	188	14	∫	∫	PROPN
ma-220	188	15	ω	ω	NUM
ma-220	189	1	〈	〈	PROPN
ma-220	189	2	h	h	NOUN
ma-220	189	3	,	,	PUNCT
ma-220	189	4	gw	gw	PROPN
ma-220	189	5	〉	〉	PROPN
ma-220	189	6	ew	ew	NOUN
ma-220	189	7	=	=	NOUN
ma-220	189	8	⇒	⇒	VERB
ma-220	190	1	aw	aw	INTJ
ma-220	190	2	=	=	SYM
ma-220	190	3	〈	〈	PROPN
ma-220	190	4	h	h	NOUN
ma-220	190	5	,	,	PUNCT
ma-220	190	6	gw	gw	PROPN
ma-220	190	7	〉	〉	NUM
ma-220	190	8	,	,	PUNCT
ma-220	190	9	∀w	∀w	X
ma-220	190	10	∈	∈	PROPN
ma-220	190	11	ω	ω	PROPN
ma-220	190	12	.	.	PUNCT
ma-220	191	1	from	from	ADP
ma-220	191	2	(	(	PUNCT
ma-220	191	3	i	i	NOUN
ma-220	191	4	)	)	PUNCT
ma-220	191	5	,	,	PUNCT
ma-220	191	6	we	we	PRON
ma-220	191	7	have	have	VERB
ma-220	191	8	0	0	NUM
ma-220	191	9	=	=	SYM
ma-220	191	10	∫	∫	PROPN
ma-220	191	11	ω	ω	NUM
ma-220	191	12	awfw	awfw	NOUN
ma-220	191	13	=	=	SYM
ma-220	191	14	∫	∫	PROPN
ma-220	191	15	ω	ω	NUM
ma-220	192	1	〈	〈	PROPN
ma-220	192	2	h	h	NOUN
ma-220	192	3	,	,	PUNCT
ma-220	192	4	gw	gw	PROPN
ma-220	192	5	〉	〉	PROPN
ma-220	192	6	fw	fw	PROPN
ma-220	192	7	=	=	SYM
ma-220	192	8	∫	∫	PROPN
ma-220	192	9	ω	ω	NUM
ma-220	192	10	〈	〈	PROPN
ma-220	192	11	γγ−1h	γγ−1h	PROPN
ma-220	192	12	,	,	PUNCT
ma-220	192	13	gw	gw	PROPN
ma-220	192	14	〉	〉	NOUN
ma-220	192	15	fw	fw	PROPN
ma-220	192	16	=	=	SYM
ma-220	192	17	γ−1h	γ−1h	NOUN
ma-220	192	18	.	.	PUNCT
ma-220	193	1	since	since	SCONJ
ma-220	193	2	γ	γ	PROPN
ma-220	193	3	is	be	AUX
ma-220	193	4	injective	injective	ADJ
ma-220	193	5	,	,	PUNCT
ma-220	193	6	h	h	NOUN
ma-220	193	7	=	=	NOUN
ma-220	193	8	0	0	NUM
ma-220	194	1	and	and	CCONJ
ma-220	194	2	aw	aw	INTJ
ma-220	194	3	=	=	NOUN
ma-220	194	4	0	0	NUM
ma-220	194	5	for	for	ADP
ma-220	194	6	w	w	PROPN
ma-220	194	7	∈	∈	PROPN
ma-220	194	8	ω	ω	PROPN
ma-220	194	9	.	.	PUNCT
ma-220	195	1	so	so	ADV
ma-220	195	2	∫ω	∫ω	VERB
ma-220	195	3	awew	awew	PROPN
ma-220	195	4	=	=	SYM
ma-220	195	5	0	0	NUM
ma-220	195	6	,	,	PUNCT
ma-220	195	7	and	and	CCONJ
ma-220	195	8	t	t	PROPN
ma-220	195	9	∗u	∗u	NOUN
ma-220	195	10	is	be	AUX
ma-220	195	11	injective	injective	ADJ
ma-220	195	12	.	.	PUNCT
ma-220	196	1	theoperator	theoperator	PROPN
ma-220	196	2	t	t	PROPN
ma-220	196	3	∗utu	∗utu	PROPN
ma-220	196	4	is	be	AUX
ma-220	196	5	an	an	DET
ma-220	196	6	invertible	invertible	ADJ
ma-220	196	7	selfadjoint	selfadjoint	NOUN
ma-220	196	8	operator	operator	NOUN
ma-220	196	9	such	such	ADJ
ma-220	196	10	that	that	SCONJ
ma-220	196	11	it	it	PRON
ma-220	196	12	has	have	AUX
ma-220	196	13	an	an	DET
ma-220	196	14	upper	upper	ADJ
ma-220	196	15	bound	bind	VERB
ma-220	196	16	and	and	CCONJ
ma-220	196	17	a	a	DET
ma-220	196	18	lowerbound	lowerbound	NOUN
ma-220	196	19	by	by	ADP
ma-220	196	20	lemma	lemma	PROPN
ma-220	196	21	1.3	1.3	NUM
ma-220	196	22	,	,	PUNCT
ma-220	196	23	and	and	CCONJ
ma-220	196	24	also	also	ADV
ma-220	196	25	{	{	PUNCT
ma-220	196	26	uw}w∈ω	uw}w∈ω	INTJ
ma-220	196	27	is	be	AUX
ma-220	196	28	a	a	DET
ma-220	196	29	frame	frame	NOUN
ma-220	196	30	for	for	ADP
ma-220	196	31	k	k	PROPN
ma-220	196	32	with	with	ADP
ma-220	196	33	frame	frame	NOUN
ma-220	196	34	operator	operator	NOUN
ma-220	196	35	su	su	PROPN
ma-220	196	36	=	=	PROPN
ma-220	196	37	t	t	PROPN
ma-220	196	38	∗utu	∗utu	PROPN
ma-220	196	39	.	.	PUNCT
ma-220	196	40	�	�	PROPN
ma-220	196	41	3	3	NUM
ma-220	196	42	.	.	PUNCT
ma-220	197	1	equivalent	equivalent	ADJ
ma-220	197	2	∗-continuous	∗-continuous	ADJ
ma-220	197	3	frames	frame	NOUN
ma-220	197	4	definition	definition	NOUN
ma-220	197	5	3.1	3.1	NUM
ma-220	197	6	.	.	PUNCT
ma-220	198	1	two	two	NUM
ma-220	198	2	sequences	sequence	NOUN
ma-220	198	3	{	{	PUNCT
ma-220	198	4	fw}w∈ω	fw}w∈ω	NOUN
ma-220	198	5	and	and	CCONJ
ma-220	198	6	{	{	PUNCT
ma-220	198	7	gw}w∈ω	gw}w∈ω	NOUN
ma-220	198	8	in	in	ADP
ma-220	198	9	h	h	NOUN
ma-220	198	10	are	be	AUX
ma-220	198	11	said	say	VERB
ma-220	198	12	to	to	PART
ma-220	198	13	be	be	AUX
ma-220	198	14	equivalent	equivalent	ADJ
ma-220	198	15	sequencesif	sequencesif	NOUN
ma-220	198	16	there	there	PRON
ma-220	198	17	exists	exist	VERB
ma-220	198	18	an	an	DET
ma-220	198	19	adjointable	adjointable	NOUN
ma-220	198	20	and	and	CCONJ
ma-220	198	21	invertible	invertible	ADJ
ma-220	198	22	operator	operator	NOUN
ma-220	198	23	λ	λ	NOUN
ma-220	198	24	on	on	ADP
ma-220	198	25	h	h	PRON
ma-220	199	1	such	such	ADJ
ma-220	199	2	that	that	SCONJ
ma-220	199	3	λfw	λfw	X
ma-220	199	4	=	=	X
ma-220	199	5	gw	gw	PROPN
ma-220	199	6	,	,	PUNCT
ma-220	199	7	for	for	ADP
ma-220	199	8	w	w	PROPN
ma-220	199	9	∈	∈	PROPN
ma-220	199	10	ω	ω	PROPN
ma-220	199	11	.	.	PUNCT
ma-220	199	12	theorem	theorem	PROPN
ma-220	199	13	3.2	3.2	NUM
ma-220	199	14	.	.	PUNCT
ma-220	200	1	let	let	VERB
ma-220	200	2	{	{	PUNCT
ma-220	200	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	200	4	be	be	AUX
ma-220	200	5	a	a	DET
ma-220	200	6	∗-continuous	∗-continuous	ADJ
ma-220	200	7	frame	frame	NOUN
ma-220	200	8	for	for	ADP
ma-220	200	9	h	h	NOUN
ma-220	200	10	and	and	CCONJ
ma-220	200	11	ξ	ξ	PROPN
ma-220	200	12	be	be	AUX
ma-220	200	13	an	an	DET
ma-220	200	14	adjointable	adjointable	NOUN
ma-220	200	15	and	and	CCONJ
ma-220	200	16	invertible	invertible	ADJ
ma-220	200	17	operator	operator	NOUN
ma-220	200	18	on	on	ADP
ma-220	200	19	h.	h.	PROPN
ma-220	200	20	then	then	ADV
ma-220	200	21	every	every	DET
ma-220	200	22	dual	dual	ADJ
ma-220	200	23	of	of	ADP
ma-220	200	24	the	the	DET
ma-220	200	25	∗-continuous	∗-continuous	ADJ
ma-220	200	26	frame	frame	NOUN
ma-220	200	27	{	{	PUNCT
ma-220	200	28	ξfw}w∈ω	ξfw}w∈ω	PROPN
ma-220	200	29	is	be	AUX
ma-220	200	30	equivalent	equivalent	ADJ
ma-220	200	31	to	to	ADP
ma-220	200	32	a	a	DET
ma-220	200	33	dual	dual	ADJ
ma-220	200	34	of	of	ADP
ma-220	200	35	{	{	PUNCT
ma-220	200	36	fw}w∈ω	fw}w∈ω	NOUN
ma-220	200	37	,	,	PUNCT
ma-220	200	38	and	and	CCONJ
ma-220	200	39	the	the	DET
ma-220	200	40	converse	converse	NOUN
ma-220	200	41	of	of	ADP
ma-220	200	42	the	the	DET
ma-220	200	43	relation	relation	NOUN
ma-220	200	44	is	be	AUX
ma-220	200	45	valid	valid	ADJ
ma-220	200	46	.	.	PUNCT
ma-220	201	1	proof	proof	NOUN
ma-220	201	2	.	.	PUNCT
ma-220	202	1	first	first	ADV
ma-220	202	2	,	,	PUNCT
ma-220	202	3	suppose	suppose	VERB
ma-220	202	4	that	that	SCONJ
ma-220	202	5	{	{	PUNCT
ma-220	202	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	202	7	is	be	AUX
ma-220	202	8	a	a	DET
ma-220	202	9	dual	dual	ADJ
ma-220	202	10	of	of	ADP
ma-220	202	11	{	{	PUNCT
ma-220	202	12	fw}w∈ω	fw}w∈ω	NOUN
ma-220	202	13	.	.	PUNCT
ma-220	202	14	then	then	ADV
ma-220	202	15	for	for	ADP
ma-220	202	16	x	x	PROPN
ma-220	202	17	∈	∈	PROPN
ma-220	202	18	h	h	NOUN
ma-220	202	19	,	,	PUNCT
ma-220	202	20	we	we	PRON
ma-220	202	21	obtain	obtain	VERB
ma-220	202	22	x	x	X
ma-220	202	23	=	=	SYM
ma-220	202	24	ξ	ξ	PROPN
ma-220	202	25	(	(	PUNCT
ma-220	202	26	ξ−1	ξ−1	PROPN
ma-220	202	27	)	)	PUNCT
ma-220	202	28	x	x	X
ma-220	202	29	=	=	SYM
ma-220	202	30	ξ	ξ	PROPN
ma-220	202	31	(	(	PUNCT
ma-220	202	32	∫	∫	PROPN
ma-220	202	33	ω	ω	NUM
ma-220	202	34	〈	〈	PROPN
ma-220	202	35	ξ−1x	ξ−1x	NOUN
ma-220	202	36	,	,	PUNCT
ma-220	202	37	gw	gw	PROPN
ma-220	202	38	〉	〉	NOUN
ma-220	202	39	fwdµ(ω	fwdµ(ω	NOUN
ma-220	202	40	)	)	PUNCT
ma-220	202	41	)	)	PUNCT
ma-220	203	1	=	=	SYM
ma-220	203	2	∫	∫	PROPN
ma-220	203	3	ω	ω	NUM
ma-220	203	4	〈	〈	PROPN
ma-220	203	5	x	x	PRON
ma-220	203	6	,	,	PUNCT
ma-220	203	7	(	(	PUNCT
ma-220	203	8	ξ−1	ξ−1	PROPN
ma-220	203	9	)	)	PUNCT
ma-220	203	10	∗	∗	NOUN
ma-220	203	11	gw	gw	PROPN
ma-220	203	12	〉	〉	NOUN
ma-220	203	13	ξfwdµ(ω	ξfwdµ(ω	NUM
ma-220	203	14	)	)	PUNCT
ma-220	203	15	.	.	PUNCT
ma-220	204	1	so	so	ADV
ma-220	204	2	{	{	PUNCT
ma-220	204	3	(	(	PUNCT
ma-220	204	4	ξ−1	ξ−1	PROPN
ma-220	204	5	)	)	PUNCT
ma-220	204	6	∗	∗	NOUN
ma-220	204	7	gw	gw	PROPN
ma-220	204	8	}	}	PUNCT
ma-220	204	9	is	be	AUX
ma-220	204	10	a	a	DET
ma-220	204	11	dual	dual	ADJ
ma-220	204	12	for	for	ADP
ma-220	204	13	{	{	PUNCT
ma-220	204	14	ξfw}w∈ω	ξfw}w∈ω	PROPN
ma-220	204	15	,	,	PUNCT
ma-220	204	16	and	and	CCONJ
ma-220	204	17	it	it	PRON
ma-220	204	18	is	be	AUX
ma-220	204	19	also	also	ADV
ma-220	204	20	equivalent	equivalent	ADJ
ma-220	204	21	to	to	ADP
ma-220	204	22	{	{	PUNCT
ma-220	204	23	gw}w∈ω.now	gw}w∈ω.now	PROPN
ma-220	204	24	,	,	PUNCT
ma-220	204	25	suppose	suppose	VERB
ma-220	204	26	that	that	SCONJ
ma-220	204	27	{	{	PUNCT
ma-220	204	28	hw}w∈ω	hw}w∈ω	NOUN
ma-220	204	29	is	be	AUX
ma-220	204	30	a	a	DET
ma-220	204	31	dual	dual	ADJ
ma-220	204	32	frame	frame	NOUN
ma-220	204	33	for	for	ADP
ma-220	204	34	{	{	PUNCT
ma-220	204	35	ξfw}w∈ω	ξfw}w∈ω	PROPN
ma-220	204	36	.	.	PUNCT
ma-220	205	1	set	set	VERB
ma-220	205	2	gw	gw	PROPN
ma-220	205	3	=	=	SYM
ma-220	205	4	ξ∗hw	ξ∗hw	PROPN
ma-220	205	5	,	,	PUNCT
ma-220	205	6	for	for	ADP
ma-220	205	7	w	w	PROPN
ma-220	205	8	∈	∈	PROPN
ma-220	205	9	ω	ω	PROPN
ma-220	205	10	.	.	PUNCT
ma-220	206	1	thenfor	thenfor	PROPN
ma-220	206	2	x	x	PUNCT
ma-220	206	3	∈	∈	PROPN
ma-220	206	4	h	h	NOUN
ma-220	206	5	,	,	PUNCT
ma-220	206	6	∫	∫	PROPN
ma-220	206	7	ω	ω	NUM
ma-220	206	8	〈	〈	PROPN
ma-220	206	9	x	x	X
ma-220	206	10	,	,	PUNCT
ma-220	206	11	gw	gw	PROPN
ma-220	206	12	〉	〉	PROPN
ma-220	206	13	fwdµ(ω	fwdµ(ω	NOUN
ma-220	206	14	)	)	PUNCT
ma-220	207	1	=	=	SYM
ma-220	207	2	∫	∫	PROPN
ma-220	208	1	ω	ω	NUM
ma-220	208	2	〈	〈	PROPN
ma-220	208	3	x	x	X
ma-220	208	4	,	,	PUNCT
ma-220	208	5	ξ∗hw	ξ∗hw	NOUN
ma-220	208	6	〉	〉	PROPN
ma-220	208	7	ξ−1ξfwdµ(ω	ξ−1ξfwdµ(ω	NOUN
ma-220	208	8	)	)	PUNCT
ma-220	209	1	=	=	SYM
ma-220	209	2	ξ−1	ξ−1	PROPN
ma-220	209	3	(	(	PUNCT
ma-220	209	4	∫	∫	PROPN
ma-220	209	5	ω	ω	PROPN
ma-220	210	1	〈	〈	PROPN
ma-220	210	2	ξf	ξf	PROPN
ma-220	210	3	,	,	PUNCT
ma-220	210	4	hw	hw	PROPN
ma-220	210	5	〉	〉	PROPN
ma-220	210	6	ξfwdµ(ω	ξfwdµ(ω	NUM
ma-220	210	7	)	)	PUNCT
ma-220	210	8	)	)	PUNCT
ma-220	211	1	=	=	SYM
ma-220	211	2	ξ−1ξx	ξ−1ξx	NOUN
ma-220	211	3	=	=	PUNCT
ma-220	211	4	x.thus	x.thu	NOUN
ma-220	211	5	{	{	PUNCT
ma-220	211	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	211	7	is	be	AUX
ma-220	211	8	a	a	DET
ma-220	211	9	dual	dual	ADJ
ma-220	211	10	for	for	ADP
ma-220	211	11	{	{	PUNCT
ma-220	211	12	fw}w∈ω	fw}w∈ω	NOUN
ma-220	211	13	and	and	CCONJ
ma-220	211	14	hw	hw	VERB
ma-220	211	15	=	=	PUNCT
ma-220	211	16	(	(	PUNCT
ma-220	211	17	ξ−1	ξ−1	PROPN
ma-220	211	18	)	)	PUNCT
ma-220	211	19	∗	∗	NOUN
ma-220	211	20	gw	gw	PROPN
ma-220	211	21	.	.	PUNCT
ma-220	212	1	�	�	PROPN
ma-220	212	2	theorem	theorem	VERB
ma-220	212	3	3.3	3.3	NUM
ma-220	212	4	.	.	PUNCT
ma-220	213	1	if	if	SCONJ
ma-220	213	2	{	{	PUNCT
ma-220	213	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	213	4	and	and	CCONJ
ma-220	213	5	{	{	PUNCT
ma-220	213	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	213	7	are	be	AUX
ma-220	213	8	∗-continuous	∗-continuous	ADJ
ma-220	213	9	frames	frame	NOUN
ma-220	213	10	with	with	ADP
ma-220	213	11	the	the	DET
ma-220	213	12	continuous	continuous	ADJ
ma-220	213	13	frame	frame	NOUN
ma-220	213	14	operators	operator	NOUN
ma-220	213	15	sf	sf	INTJ
ma-220	213	16	and	and	CCONJ
ma-220	213	17	sg	sg	INTJ
ma-220	213	18	,	,	PUNCT
ma-220	213	19	respectively	respectively	ADV
ma-220	213	20	,	,	PUNCT
ma-220	213	21	then	then	ADV
ma-220	213	22	there	there	PRON
ma-220	213	23	exists	exist	VERB
ma-220	213	24	a	a	DET
ma-220	213	25	∗-continuous	∗-continuous	ADJ
ma-220	213	26	frame	frame	NOUN
ma-220	213	27	that	that	PRON
ma-220	213	28	is	be	AUX
ma-220	213	29	equivalent	equivalent	ADJ
ma-220	213	30	to	to	ADP
ma-220	213	31	{	{	PUNCT
ma-220	213	32	gw}w∈ω	gw}w∈ω	NOUN
ma-220	213	33	and	and	CCONJ
ma-220	213	34	its	its	PRON
ma-220	213	35	frame	frame	NOUN
ma-220	213	36	operator	operator	NOUN
ma-220	213	37	is	be	AUX
ma-220	213	38	sf	sf	NOUN
ma-220	213	39	.	.	PUNCT
ma-220	214	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	214	2	eur	eur	PROPN
ma-220	214	3	.	.	PUNCT
ma-220	215	1	j.	j.	PROPN
ma-220	215	2	math	math	PROPN
ma-220	215	3	.	.	PUNCT
ma-220	216	1	anal	anal	PROPN
ma-220	216	2	.	.	PUNCT
ma-220	217	1	10.28924	10.28924	NUM
ma-220	217	2	/	/	SYM
ma-220	217	3	ada	ada	PROPN
ma-220	217	4	/	/	SYM
ma-220	217	5	ma.4.4	ma.4.4	NOUN
ma-220	217	6	9	9	NUM
ma-220	217	7	proof	proof	NOUN
ma-220	217	8	.	.	PUNCT
ma-220	218	1	for	for	ADP
ma-220	218	2	the	the	DET
ma-220	218	3	adjointable	adjointable	ADJ
ma-220	218	4	and	and	CCONJ
ma-220	218	5	invertible	invertible	ADJ
ma-220	218	6	operator	operator	NOUN
ma-220	218	7	ξ	ξ	NOUN
ma-220	218	8	=	=	SYM
ma-220	218	9	s	s	PART
ma-220	218	10	1	1	NUM
ma-220	218	11	2	2	NUM
ma-220	218	12	fs	fs	ADP
ma-220	218	13	−	−	NUM
ma-220	218	14	1	1	NUM
ma-220	218	15	2	2	NUM
ma-220	218	16	g	g	NOUN
ma-220	218	17	,	,	PUNCT
ma-220	218	18	the	the	DET
ma-220	218	19	sequence	sequence	NOUN
ma-220	218	20	{	{	PUNCT
ma-220	218	21	ξgw}w∈ω	ξgw}w∈ω	NOUN
ma-220	218	22	is	be	AUX
ma-220	218	23	a	a	DET
ma-220	218	24	∗-continuous	∗-continuous	ADJ
ma-220	218	25	frame	frame	NOUN
ma-220	218	26	with	with	ADP
ma-220	218	27	the	the	DET
ma-220	218	28	frame	frame	NOUN
ma-220	218	29	operator	operator	NOUN
ma-220	218	30	sξ	sξ	ADP
ma-220	218	31	=	=	NOUN
ma-220	218	32	ξsgξ	ξsgξ	NOUN
ma-220	218	33	∗.	∗.	VERB
ma-220	218	34	so	so	CCONJ
ma-220	218	35	sξ	sξ	ADP
ma-220	218	36	=	=	PUNCT
ma-220	218	37	ξsgξ	ξsgξ	NOUN
ma-220	218	38	∗	∗	NOUN
ma-220	218	39	=	=	PUNCT
ma-220	218	40	(	(	PUNCT
ma-220	218	41	s	s	NOUN
ma-220	218	42	1	1	NUM
ma-220	218	43	2	2	NUM
ma-220	218	44	fs	fs	ADP
ma-220	218	45	−	−	NUM
ma-220	218	46	1	1	NUM
ma-220	218	47	2	2	NUM
ma-220	218	48	g	g	NOUN
ma-220	218	49	)	)	PUNCT
ma-220	218	50	sg	sg	PROPN
ma-220	218	51	(	(	PUNCT
ma-220	218	52	s	s	NOUN
ma-220	218	53	1	1	NUM
ma-220	218	54	2	2	NUM
ma-220	218	55	fs	fs	ADP
ma-220	218	56	−	−	NUM
ma-220	218	57	1	1	NUM
ma-220	218	58	2	2	NUM
ma-220	218	59	g	g	NOUN
ma-220	218	60	)	)	PUNCT
ma-220	218	61	∗	∗	NOUN
ma-220	218	62	=	=	PUNCT
ma-220	218	63	sf	sf	NOUN
ma-220	218	64	.this	.this	PROPN
ma-220	218	65	completes	complete	VERB
ma-220	218	66	the	the	DET
ma-220	218	67	proof	proof	NOUN
ma-220	218	68	.	.	PUNCT
ma-220	219	1	�	�	PROPN
ma-220	219	2	theorem	theorem	VERB
ma-220	219	3	3.4	3.4	NUM
ma-220	219	4	.	.	PUNCT
ma-220	220	1	let	let	VERB
ma-220	220	2	{	{	PUNCT
ma-220	220	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	220	4	and	and	CCONJ
ma-220	220	5	{	{	PUNCT
ma-220	220	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	220	7	be	be	AUX
ma-220	220	8	∗-continuous	∗-continuous	ADJ
ma-220	220	9	frames	frame	NOUN
ma-220	220	10	for	for	ADP
ma-220	220	11	h.	h.	PROPN
ma-220	220	12	then	then	ADV
ma-220	220	13	the	the	DET
ma-220	220	14	following	follow	VERB
ma-220	220	15	statements	statement	NOUN
ma-220	220	16	are	be	AUX
ma-220	220	17	valid.(1	valid.(1	NUM
ma-220	220	18	)	)	PUNCT
ma-220	220	19	{	{	PUNCT
ma-220	220	20	gw}w∈ω	gw}w∈ω	NOUN
ma-220	220	21	is	be	AUX
ma-220	220	22	equivalent	equivalent	ADJ
ma-220	220	23	to	to	ADP
ma-220	220	24	a	a	DET
ma-220	220	25	dual	dual	ADJ
ma-220	220	26	frame	frame	NOUN
ma-220	220	27	of	of	ADP
ma-220	220	28	{	{	PUNCT
ma-220	220	29	fw}w∈ω	fw}w∈ω	NOUN
ma-220	220	30	if	if	SCONJ
ma-220	221	1	and	and	CCONJ
ma-220	221	2	only	only	ADV
ma-220	221	3	if	if	SCONJ
ma-220	221	4	there	there	PRON
ma-220	221	5	exists	exist	VERB
ma-220	221	6	an	an	DET
ma-220	221	7	adjointable	adjointable	NOUN
ma-220	221	8	and	and	CCONJ
ma-220	221	9	invertible	invertible	ADJ
ma-220	221	10	operator	operator	NOUN
ma-220	221	11	ξ	ξ	PROPN
ma-220	221	12	on	on	ADP
ma-220	221	13	h	h	PRON
ma-220	221	14	such	such	ADJ
ma-220	221	15	that	that	SCONJ
ma-220	221	16	ξf	ξf	PRON
ma-220	221	17	x	x	X
ma-220	221	18	=	=	SYM
ma-220	221	19	∫	∫	PROPN
ma-220	221	20	ω	ω	NUM
ma-220	222	1	〈	〈	PROPN
ma-220	222	2	f	f	PROPN
ma-220	222	3	,	,	PUNCT
ma-220	222	4	fw	fw	PROPN
ma-220	222	5	〉	〉	PROPN
ma-220	222	6	gwdµ(ω	gwdµ(ω	PROPN
ma-220	222	7	)	)	PUNCT
ma-220	222	8	,	,	PUNCT
ma-220	222	9	∀x	∀x	PROPN
ma-220	222	10	∈	∈	PROPN
ma-220	222	11	h.	h.	NOUN
ma-220	222	12	(	(	PUNCT
ma-220	222	13	2	2	NUM
ma-220	222	14	)	)	PUNCT
ma-220	222	15	{	{	PUNCT
ma-220	222	16	gw}w∈ω	gw}w∈ω	NOUN
ma-220	222	17	is	be	AUX
ma-220	222	18	equivalent	equivalent	ADJ
ma-220	222	19	to	to	ADP
ma-220	222	20	a	a	DET
ma-220	222	21	dual	dual	ADJ
ma-220	222	22	frame	frame	NOUN
ma-220	222	23	of	of	ADP
ma-220	222	24	{	{	PUNCT
ma-220	222	25	fw}w∈ω	fw}w∈ω	NOUN
ma-220	222	26	if	if	SCONJ
ma-220	222	27	and	and	CCONJ
ma-220	222	28	only	only	ADV
ma-220	222	29	if	if	SCONJ
ma-220	222	30	there	there	PRON
ma-220	222	31	exists	exist	VERB
ma-220	222	32	an	an	DET
ma-220	222	33	adjointable	adjointable	NOUN
ma-220	222	34	and	and	CCONJ
ma-220	222	35	invertible	invertible	ADJ
ma-220	222	36	operator	operator	NOUN
ma-220	222	37	γ	γ	NOUN
ma-220	222	38	such	such	ADJ
ma-220	222	39	that	that	SCONJ
ma-220	222	40	(	(	PUNCT
ma-220	222	41	{	{	PUNCT
ma-220	222	42	gw}w∈ω	gw}w∈ω	NOUN
ma-220	222	43	,	,	PUNCT
ma-220	222	44	γ	γ	PROPN
ma-220	222	45	)	)	PUNCT
ma-220	222	46	is	be	AUX
ma-220	222	47	an	an	DET
ma-220	222	48	operator	operator	NOUN
ma-220	222	49	dual	dual	ADJ
ma-220	222	50	for	for	ADP
ma-220	222	51	{	{	PUNCT
ma-220	222	52	fw}w∈ω	fw}w∈ω	NOUN
ma-220	222	53	.	.	NOUN
ma-220	222	54	proof	proof	NOUN
ma-220	222	55	.	.	PUNCT
ma-220	223	1	first	first	ADV
ma-220	223	2	,	,	PUNCT
ma-220	223	3	assume	assume	VERB
ma-220	223	4	that	that	SCONJ
ma-220	223	5	{	{	PUNCT
ma-220	223	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	223	7	is	be	AUX
ma-220	223	8	equivalent	equivalent	ADJ
ma-220	223	9	to	to	ADP
ma-220	223	10	a	a	DET
ma-220	223	11	dual	dual	ADJ
ma-220	223	12	frame	frame	NOUN
ma-220	223	13	of	of	ADP
ma-220	223	14	{	{	PUNCT
ma-220	223	15	fw}w∈ω	fw}w∈ω	NOUN
ma-220	223	16	.	.	PUNCT
ma-220	224	1	then	then	ADV
ma-220	224	2	there	there	ADV
ma-220	224	3	existsan	existsan	VERB
ma-220	224	4	adjointable	adjointable	NOUN
ma-220	224	5	and	and	CCONJ
ma-220	224	6	invertible	invertible	ADJ
ma-220	224	7	operator	operator	NOUN
ma-220	224	8	γ	γ	NOUN
ma-220	224	9	on	on	ADP
ma-220	224	10	h	h	PRON
ma-220	224	11	such	such	ADJ
ma-220	224	12	that	that	SCONJ
ma-220	224	13	{	{	PUNCT
ma-220	224	14	γgw}w∈ω	γgw}w∈ω	NOUN
ma-220	224	15	is	be	AUX
ma-220	224	16	a	a	DET
ma-220	224	17	dual	dual	ADJ
ma-220	224	18	for	for	ADP
ma-220	224	19	{	{	PUNCT
ma-220	224	20	fw}w∈ω	fw}w∈ω	NOUN
ma-220	224	21	.	.	PUNCT
ma-220	225	1	now	now	ADV
ma-220	225	2	,	,	PUNCT
ma-220	225	3	for	for	ADP
ma-220	225	4	x	x	PROPN
ma-220	225	5	∈	∈	PROPN
ma-220	225	6	h	h	NOUN
ma-220	225	7	,	,	PUNCT
ma-220	225	8	=	=	SYM
ma-220	225	9	∫	∫	PROPN
ma-220	225	10	ω	ω	NUM
ma-220	225	11	〈	〈	PROPN
ma-220	225	12	x	x	PRON
ma-220	225	13	,	,	PUNCT
ma-220	225	14	fw	fw	PROPN
ma-220	225	15	〉	〉	PROPN
ma-220	225	16	γgwdµ(ω	γgwdµ(ω	PROPN
ma-220	225	17	)	)	PUNCT
ma-220	225	18	.	.	PUNCT
ma-220	226	1	set	set	VERB
ma-220	226	2	ξ	ξ	PROPN
ma-220	226	3	=	=	SYM
ma-220	226	4	γ−1	γ−1	PROPN
ma-220	226	5	.	.	PUNCT
ma-220	227	1	then	then	ADV
ma-220	227	2	it	it	PRON
ma-220	227	3	concludes	conclude	VERB
ma-220	227	4	ξx	ξx	NOUN
ma-220	227	5	=	=	SYM
ma-220	227	6	γ−1x	γ−1x	PROPN
ma-220	227	7	=	=	SYM
ma-220	227	8	γ−1	γ−1	PROPN
ma-220	227	9	(	(	PUNCT
ma-220	227	10	∫	∫	PROPN
ma-220	227	11	ω	ω	NUM
ma-220	227	12	〈	〈	PROPN
ma-220	227	13	x	x	PRON
ma-220	227	14	,	,	PUNCT
ma-220	227	15	fw	fw	PROPN
ma-220	227	16	〉	〉	PROPN
ma-220	227	17	γgwdµ(ω	γgwdµ(ω	PROPN
ma-220	227	18	)	)	PUNCT
ma-220	227	19	)	)	PUNCT
ma-220	228	1	=	=	SYM
ma-220	229	1	∫	∫	PROPN
ma-220	229	2	ω	ω	NUM
ma-220	229	3	〈	〈	PROPN
ma-220	229	4	x	x	PRON
ma-220	229	5	,	,	PUNCT
ma-220	229	6	fw	fw	PROPN
ma-220	229	7	〉	〉	PROPN
ma-220	229	8	gwdµ(ω	gwdµ(ω	PROPN
ma-220	229	9	)	)	PUNCT
ma-220	229	10	.	.	PUNCT
ma-220	230	1	in	in	ADP
ma-220	230	2	the	the	DET
ma-220	230	3	second	second	ADJ
ma-220	230	4	step	step	NOUN
ma-220	230	5	,	,	PUNCT
ma-220	230	6	the	the	DET
ma-220	230	7	adjointable	adjointable	ADJ
ma-220	230	8	and	and	CCONJ
ma-220	230	9	invertible	invertible	ADJ
ma-220	230	10	operator	operator	NOUN
ma-220	230	11	ξ	ξ	PROPN
ma-220	230	12	on	on	ADP
ma-220	230	13	h	h	NOUN
ma-220	230	14	satisfies	satisfie	NOUN
ma-220	230	15	the	the	DET
ma-220	230	16	following	follow	VERB
ma-220	230	17	property	property	NOUN
ma-220	230	18	ξx	ξx	NOUN
ma-220	231	1	=	=	SYM
ma-220	231	2	∫	∫	PROPN
ma-220	231	3	ω	ω	NUM
ma-220	231	4	〈	〈	PROPN
ma-220	231	5	x	x	PRON
ma-220	231	6	,	,	PUNCT
ma-220	231	7	fw	fw	PROPN
ma-220	231	8	〉	〉	PROPN
ma-220	231	9	gwdµ(ω	gwdµ(ω	PROPN
ma-220	231	10	)	)	PUNCT
ma-220	231	11	,	,	PUNCT
ma-220	231	12	∀x	∀x	X
ma-220	231	13	∈	∈	PROPN
ma-220	231	14	h.	h.	NOUN
ma-220	231	15	since	since	SCONJ
ma-220	231	16	ξ	ξ	PROPN
ma-220	231	17	is	be	AUX
ma-220	231	18	invertible	invertible	ADJ
ma-220	231	19	,	,	PUNCT
ma-220	231	20	x	x	SYM
ma-220	232	1	=	=	SYM
ma-220	232	2	∫	∫	PROPN
ma-220	232	3	ω	ω	NUM
ma-220	232	4	〈	〈	PROPN
ma-220	232	5	x	x	PRON
ma-220	232	6	,	,	PUNCT
ma-220	232	7	fw	fw	PROPN
ma-220	232	8	〉	〉	PROPN
ma-220	232	9	ξ−1gwdµ(ω	ξ−1gwdµ(ω	NOUN
ma-220	232	10	)	)	PUNCT
ma-220	232	11	,	,	PUNCT
ma-220	232	12	∀x	∀x	PROPN
ma-220	232	13	∈	∈	PROPN
ma-220	232	14	h.	h.	NOUN
ma-220	232	15	it	it	PRON
ma-220	232	16	shows	show	VERB
ma-220	232	17	that	that	SCONJ
ma-220	232	18	{	{	PUNCT
ma-220	232	19	ξ−1gw	ξ−1gw	NOUN
ma-220	232	20	}	}	PUNCT
ma-220	232	21	w∈ω	w∈ω	PROPN
ma-220	232	22	is	be	AUX
ma-220	232	23	a	a	DET
ma-220	232	24	dual	dual	ADJ
ma-220	232	25	for	for	ADP
ma-220	232	26	{	{	PUNCT
ma-220	232	27	fw}w∈ω	fw}w∈ω	NOUN
ma-220	232	28	,	,	PUNCT
ma-220	232	29	and	and	CCONJ
ma-220	232	30	is	be	AUX
ma-220	232	31	equivalent	equivalent	ADJ
ma-220	232	32	to	to	PART
ma-220	232	33	{	{	PUNCT
ma-220	232	34	gw}w∈ω.for	gw}w∈ω.for	ADP
ma-220	232	35	the	the	DET
ma-220	232	36	proof	proof	NOUN
ma-220	232	37	of	of	ADP
ma-220	232	38	“	"	PUNCT
ma-220	232	39	if	if	SCONJ
ma-220	232	40	"	"	PUNCT
ma-220	232	41	part	part	NOUN
ma-220	232	42	,	,	PUNCT
ma-220	232	43	assume	assume	VERB
ma-220	232	44	that	that	SCONJ
ma-220	232	45	there	there	PRON
ma-220	232	46	exists	exist	VERB
ma-220	232	47	a	a	DET
ma-220	232	48	dual	dual	ADJ
ma-220	232	49	frame	frame	NOUN
ma-220	232	50	{	{	PUNCT
ma-220	232	51	hw}w∈ω	hw}w∈ω	NOUN
ma-220	232	52	for	for	ADP
ma-220	232	53	{	{	PUNCT
ma-220	232	54	fw}w∈ω	fw}w∈ω	NOUN
ma-220	232	55	suchthat	suchthat	PROPN
ma-220	232	56	{	{	PUNCT
ma-220	232	57	hw}w∈ω	hw}w∈ω	NOUN
ma-220	232	58	and	and	CCONJ
ma-220	232	59	{	{	PUNCT
ma-220	232	60	gw}w∈ω	gw}w∈ω	NOUN
ma-220	232	61	are	be	AUX
ma-220	232	62	equivalent	equivalent	ADJ
ma-220	232	63	.	.	PUNCT
ma-220	233	1	then	then	ADV
ma-220	233	2	there	there	PRON
ma-220	233	3	is	be	VERB
ma-220	233	4	an	an	DET
ma-220	233	5	adjointable	adjointable	ADJ
ma-220	233	6	and	and	CCONJ
ma-220	233	7	invertible	invertible	ADJ
ma-220	233	8	operator	operator	NOUN
ma-220	233	9	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	233	10	eur	eur	NOUN
ma-220	233	11	.	.	PUNCT
ma-220	234	1	j.	j.	PROPN
ma-220	234	2	math	math	PROPN
ma-220	234	3	.	.	PUNCT
ma-220	235	1	anal	anal	PROPN
ma-220	235	2	.	.	PUNCT
ma-220	236	1	10.28924	10.28924	NUM
ma-220	236	2	/	/	SYM
ma-220	236	3	ada	ada	PROPN
ma-220	236	4	/	/	SYM
ma-220	236	5	ma.4.4	ma.4.4	PROPN
ma-220	236	6	10	10	NUM
ma-220	236	7	λ	λ	NOUN
ma-220	236	8	:	:	PUNCT
ma-220	236	9	h	h	PROPN
ma-220	236	10	→	→	PUNCT
ma-220	236	11	h	h	NOUN
ma-220	236	12	such	such	ADJ
ma-220	236	13	that	that	SCONJ
ma-220	236	14	λgw	λgw	X
ma-220	236	15	=	=	SYM
ma-220	236	16	hw	hw	PROPN
ma-220	236	17	for	for	ADP
ma-220	236	18	all	all	DET
ma-220	236	19	w	w	PROPN
ma-220	236	20	∈	∈	PROPN
ma-220	236	21	ω	ω	NOUN
ma-220	236	22	.	.	PUNCT
ma-220	237	1	by	by	ADP
ma-220	237	2	theorem	theorem	NOUN
ma-220	237	3	3.2	3.2	NUM
ma-220	237	4	,	,	PUNCT
ma-220	237	5	the	the	DET
ma-220	237	6	sequence	sequence	NOUN
ma-220	237	7	{	{	PUNCT
ma-220	237	8	gw}w∈ω	gw}w∈ω	NOUN
ma-220	237	9	is	be	AUX
ma-220	237	10	a	a	DET
ma-220	237	11	∗-continuous	∗-continuous	ADJ
ma-220	237	12	frame.on	frame.on	NOUN
ma-220	237	13	the	the	DET
ma-220	237	14	other	other	ADJ
ma-220	237	15	hand	hand	NOUN
ma-220	237	16	,	,	PUNCT
ma-220	237	17	for	for	ADP
ma-220	237	18	x	x	PROPN
ma-220	237	19	∈	∈	PROPN
ma-220	237	20	h	h	NOUN
ma-220	237	21	,	,	PUNCT
ma-220	237	22	x	x	PUNCT
ma-220	238	1	=	=	SYM
ma-220	238	2	∫	∫	PROPN
ma-220	238	3	ω	ω	NUM
ma-220	238	4	〈	〈	PROPN
ma-220	238	5	x	x	PRON
ma-220	238	6	,	,	PUNCT
ma-220	238	7	hw	hw	PROPN
ma-220	238	8	〉	〉	NOUN
ma-220	238	9	fwdµ(ω	fwdµ(ω	NOUN
ma-220	238	10	)	)	PUNCT
ma-220	239	1	=	=	SYM
ma-220	239	2	∫	∫	PROPN
ma-220	240	1	ω	ω	NUM
ma-220	240	2	〈	〈	PROPN
ma-220	240	3	x	x	PRON
ma-220	240	4	,	,	PUNCT
ma-220	240	5	λgw	λgw	PRON
ma-220	240	6	〉	〉	NOUN
ma-220	240	7	fwdµ(ω	fwdµ(ω	NOUN
ma-220	240	8	)	)	PUNCT
ma-220	241	1	=	=	SYM
ma-220	241	2	∫	∫	PROPN
ma-220	241	3	ω	ω	X
ma-220	241	4	〈	〈	PROPN
ma-220	241	5	λ∗x	λ∗x	PROPN
ma-220	241	6	,	,	PUNCT
ma-220	241	7	gw	gw	PROPN
ma-220	241	8	〉	〉	PROPN
ma-220	241	9	fwdµ(ω	fwdµ(ω	PROPN
ma-220	241	10	)	)	PUNCT
ma-220	241	11	and	and	CCONJ
ma-220	241	12	so	so	ADV
ma-220	241	13	x	x	X
ma-220	241	14	=	=	SYM
ma-220	241	15	∫	∫	PROPN
ma-220	241	16	ω	ω	PROPN
ma-220	241	17	〈	〈	PROPN
ma-220	241	18	λ∗x	λ∗x	PROPN
ma-220	241	19	,	,	PUNCT
ma-220	241	20	gw	gw	PROPN
ma-220	241	21	〉	〉	PROPN
ma-220	241	22	fwdµ(ω).this	fwdµ(ω).this	PRON
ma-220	241	23	shows	show	VERB
ma-220	241	24	that	that	SCONJ
ma-220	241	25	{	{	PUNCT
ma-220	241	26	(	(	PUNCT
ma-220	241	27	gw	gw	PROPN
ma-220	241	28	,	,	PUNCT
ma-220	241	29	λ∗	λ∗	PROPN
ma-220	241	30	)	)	PUNCT
ma-220	241	31	}	}	PUNCT
ma-220	241	32	is	be	AUX
ma-220	241	33	an	an	DET
ma-220	241	34	operator	operator	NOUN
ma-220	241	35	dual	dual	ADJ
ma-220	241	36	for	for	ADP
ma-220	241	37	{	{	PUNCT
ma-220	241	38	fw}w∈ω	fw}w∈ω	NOUN
ma-220	241	39	.	.	PUNCT
ma-220	242	1	the	the	DET
ma-220	242	2	converse	converse	NOUN
ma-220	242	3	part	part	NOUN
ma-220	242	4	is	be	AUX
ma-220	242	5	clear	clear	ADJ
ma-220	242	6	by	by	ADP
ma-220	242	7	thelast	thelast	ADJ
ma-220	242	8	equalities	equality	NOUN
ma-220	242	9	.	.	PUNCT
ma-220	243	1	�	�	PROPN
ma-220	243	2	theorem	theorem	VERB
ma-220	243	3	3.5	3.5	NUM
ma-220	243	4	.	.	PUNCT
ma-220	244	1	let	let	VERB
ma-220	244	2	{	{	PUNCT
ma-220	244	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	244	4	and	and	CCONJ
ma-220	244	5	{	{	PUNCT
ma-220	244	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	244	7	be	be	AUX
ma-220	244	8	∗-frames	∗-frame	NOUN
ma-220	244	9	for	for	ADP
ma-220	244	10	h.	h.	PROPN
ma-220	244	11	then	then	ADV
ma-220	244	12	{	{	PUNCT
ma-220	244	13	gw}w∈ω	gw}w∈ω	NOUN
ma-220	244	14	is	be	AUX
ma-220	244	15	equivalent	equivalent	ADJ
ma-220	244	16	to	to	ADP
ma-220	244	17	an	an	DET
ma-220	244	18	operator	operator	NOUN
ma-220	244	19	dual	dual	ADJ
ma-220	244	20	frame	frame	NOUN
ma-220	244	21	of	of	ADP
ma-220	244	22	{	{	PUNCT
ma-220	244	23	fw}w∈ω	fw}w∈ω	NOUN
ma-220	244	24	if	if	SCONJ
ma-220	245	1	and	and	CCONJ
ma-220	245	2	only	only	ADV
ma-220	245	3	if	if	SCONJ
ma-220	245	4	there	there	PRON
ma-220	245	5	exists	exist	VERB
ma-220	245	6	an	an	DET
ma-220	245	7	adjointable	adjointable	NOUN
ma-220	245	8	and	and	CCONJ
ma-220	245	9	invertible	invertible	ADJ
ma-220	245	10	operator	operator	NOUN
ma-220	245	11	ξ	ξ	PROPN
ma-220	245	12	on	on	ADP
ma-220	245	13	h	h	PRON
ma-220	246	1	such	such	ADJ
ma-220	246	2	that	that	PRON
ma-220	246	3	ξx	ξx	PROPN
ma-220	246	4	=	=	SYM
ma-220	246	5	∫	∫	PROPN
ma-220	246	6	ω	ω	NUM
ma-220	246	7	〈	〈	PROPN
ma-220	246	8	x	x	X
ma-220	246	9	,	,	PUNCT
ma-220	246	10	gw	gw	PROPN
ma-220	246	11	〉	〉	PROPN
ma-220	246	12	fwdµ(ω	fwdµ(ω	NOUN
ma-220	246	13	)	)	PUNCT
ma-220	246	14	,	,	PUNCT
ma-220	246	15	∀x	∀x	VERB
ma-220	246	16	∈	∈	PROPN
ma-220	246	17	h.	h.	NOUN
ma-220	246	18	proof	proof	NOUN
ma-220	246	19	.	.	PUNCT
ma-220	247	1	suppose	suppose	VERB
ma-220	247	2	that	that	SCONJ
ma-220	247	3	{	{	PUNCT
ma-220	247	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	247	5	is	be	AUX
ma-220	247	6	equivalent	equivalent	ADJ
ma-220	247	7	to	to	ADP
ma-220	247	8	{	{	PUNCT
ma-220	247	9	hw}w∈ω	hw}w∈ω	PROPN
ma-220	247	10	,	,	PUNCT
ma-220	247	11	where	where	SCONJ
ma-220	247	12	(	(	PUNCT
ma-220	247	13	{	{	PUNCT
ma-220	247	14	hw}w∈ω	hw}w∈ω	PROPN
ma-220	247	15	,	,	PUNCT
ma-220	247	16	γ	γ	PROPN
ma-220	247	17	)	)	PUNCT
ma-220	247	18	is	be	AUX
ma-220	247	19	an	an	DET
ma-220	247	20	operator	operator	NOUN
ma-220	247	21	dualfor	dualfor	NOUN
ma-220	247	22	{	{	PUNCT
ma-220	247	23	fw}w∈ω	fw}w∈ω	PROPN
ma-220	247	24	.	.	PUNCT
ma-220	248	1	then	then	ADV
ma-220	248	2	there	there	PRON
ma-220	248	3	exists	exist	VERB
ma-220	248	4	an	an	DET
ma-220	248	5	adjointable	adjointable	NOUN
ma-220	248	6	and	and	CCONJ
ma-220	248	7	invertible	invertible	ADJ
ma-220	248	8	operator	operator	NOUN
ma-220	248	9	θ	θ	PROPN
ma-220	248	10	on	on	ADP
ma-220	248	11	h	h	PRON
ma-220	248	12	such	such	ADJ
ma-220	248	13	that	that	DET
ma-220	248	14	θgw	θgw	NOUN
ma-220	248	15	=	=	PUNCT
ma-220	248	16	hwfor	hwfor	ADP
ma-220	248	17	all	all	PRON
ma-220	248	18	w	w	PROPN
ma-220	248	19	∈	∈	PROPN
ma-220	248	20	ω	ω	NOUN
ma-220	248	21	and	and	CCONJ
ma-220	248	22	for	for	ADP
ma-220	248	23	x	x	PROPN
ma-220	248	24	∈	∈	PROPN
ma-220	248	25	h	h	NOUN
ma-220	248	26	,	,	PUNCT
ma-220	248	27	x	x	PUNCT
ma-220	248	28	=	=	SYM
ma-220	248	29	∫	∫	PROPN
ma-220	248	30	ω	ω	X
ma-220	248	31	〈	〈	PROPN
ma-220	248	32	γx	γx	NOUN
ma-220	248	33	,	,	PUNCT
ma-220	248	34	hw	hw	PROPN
ma-220	248	35	〉	〉	PROPN
ma-220	248	36	fw	fw	NOUN
ma-220	248	37	=	=	SYM
ma-220	248	38	∫	∫	PROPN
ma-220	248	39	ω	ω	NUM
ma-220	249	1	〈	〈	PROPN
ma-220	249	2	γx	γx	NOUN
ma-220	249	3	,	,	PUNCT
ma-220	249	4	θgw	θgw	ADJ
ma-220	249	5	〉	〉	NUM
ma-220	249	6	fwdµ(ω	fwdµ(ω	NOUN
ma-220	249	7	)	)	PUNCT
ma-220	249	8	=	=	SYM
ma-220	250	1	∫	∫	PROPN
ma-220	250	2	ω	ω	X
ma-220	250	3	〈	〈	PROPN
ma-220	250	4	θ∗γx	θ∗γx	PROPN
ma-220	250	5	,	,	PUNCT
ma-220	250	6	gw	gw	PROPN
ma-220	250	7	〉	〉	PROPN
ma-220	250	8	fwdµ(ω	fwdµ(ω	NOUN
ma-220	250	9	)	)	PUNCT
ma-220	250	10	.	.	PUNCT
ma-220	251	1	set	set	VERB
ma-220	251	2	ξ	ξ	PROPN
ma-220	251	3	=	=	PUNCT
ma-220	251	4	(	(	PUNCT
ma-220	251	5	θ∗γ)−1	θ∗γ)−1	PROPN
ma-220	251	6	.	.	PUNCT
ma-220	252	1	then	then	ADV
ma-220	252	2	the	the	DET
ma-220	252	3	result	result	NOUN
ma-220	252	4	is	be	AUX
ma-220	252	5	obtained.for	obtained.for	ADP
ma-220	252	6	the	the	DET
ma-220	252	7	converse	converse	NOUN
ma-220	252	8	,	,	PUNCT
ma-220	252	9	let	let	VERB
ma-220	252	10	ξ	ξ	X
ma-220	252	11	be	be	AUX
ma-220	252	12	an	an	DET
ma-220	252	13	adjointable	adjointable	NOUN
ma-220	252	14	and	and	CCONJ
ma-220	252	15	invertible	invertible	ADJ
ma-220	252	16	operator	operator	NOUN
ma-220	252	17	on	on	ADP
ma-220	252	18	h	h	NOUN
ma-220	253	1	that	that	PRON
ma-220	253	2	ξx	ξx	NOUN
ma-220	254	1	=	=	SYM
ma-220	254	2	∫	∫	PROPN
ma-220	254	3	ω	ω	NUM
ma-220	254	4	〈	〈	PROPN
ma-220	254	5	x	x	X
ma-220	254	6	,	,	PUNCT
ma-220	254	7	gw	gw	PROPN
ma-220	254	8	〉	〉	PROPN
ma-220	254	9	fwdµ(ω	fwdµ(ω	NOUN
ma-220	254	10	)	)	PUNCT
ma-220	254	11	.	.	PUNCT
ma-220	255	1	then	then	ADV
ma-220	255	2	x	x	X
ma-220	255	3	=	=	SYM
ma-220	255	4	∫	∫	PROPN
ma-220	255	5	ω	ω	NUM
ma-220	255	6	〈	〈	PROPN
ma-220	255	7	ξ−1x	ξ−1x	NOUN
ma-220	255	8	,	,	PUNCT
ma-220	255	9	gw	gw	PROPN
ma-220	255	10	〉	〉	NOUN
ma-220	255	11	fwdµ(ω	fwdµ(ω	NOUN
ma-220	255	12	)	)	PUNCT
ma-220	255	13	,	,	PUNCT
ma-220	255	14	∀x	∀x	VERB
ma-220	255	15	∈	∈	PROPN
ma-220	255	16	h	h	NOUN
ma-220	255	17	and	and	CCONJ
ma-220	255	18	(	(	PUNCT
ma-220	255	19	{	{	PUNCT
ma-220	255	20	gw}w∈ω	gw}w∈ω	NOUN
ma-220	255	21	,	,	PUNCT
ma-220	255	22	ξ	ξ	PROPN
ma-220	255	23	−1	−1	NOUN
ma-220	255	24	)	)	PUNCT
ma-220	255	25	is	be	AUX
ma-220	255	26	an	an	DET
ma-220	255	27	operator	operator	NOUN
ma-220	255	28	dual	dual	ADV
ma-220	255	29	of	of	ADP
ma-220	255	30	{	{	PUNCT
ma-220	255	31	fw}w∈ω	fw}w∈ω	NOUN
ma-220	255	32	and	and	CCONJ
ma-220	255	33	{	{	PUNCT
ma-220	255	34	gw}w∈ω	gw}w∈ω	NOUN
ma-220	255	35	is	be	AUX
ma-220	255	36	equivalent	equivalent	ADJ
ma-220	255	37	to	to	ADP
ma-220	255	38	itself	itself	PRON
ma-220	255	39	.	.	PUNCT
ma-220	256	1	�	�	PROPN
ma-220	256	2	moreover	moreover	ADV
ma-220	256	3	,	,	PUNCT
ma-220	256	4	some	some	DET
ma-220	256	5	equivalence	equivalence	NOUN
ma-220	256	6	frames	frame	NOUN
ma-220	256	7	have	have	VERB
ma-220	256	8	the	the	DET
ma-220	256	9	same	same	ADJ
ma-220	256	10	grammian	grammian	ADJ
ma-220	256	11	matrices	matrix	NOUN
ma-220	256	12	.	.	PUNCT
ma-220	257	1	these	these	DET
ma-220	257	2	frames	frame	NOUN
ma-220	257	3	are	be	AUX
ma-220	257	4	intro	intro	VERB
ma-220	257	5	-	-	PUNCT
ma-220	257	6	duced	duce	VERB
ma-220	257	7	in	in	ADP
ma-220	257	8	the	the	DET
ma-220	257	9	following	follow	VERB
ma-220	257	10	proposition	proposition	NOUN
ma-220	257	11	.	.	PUNCT
ma-220	258	1	proposition	proposition	NOUN
ma-220	258	2	3.6	3.6	NUM
ma-220	258	3	.	.	PUNCT
ma-220	259	1	let	let	VERB
ma-220	259	2	{	{	PUNCT
ma-220	259	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	259	4	and	and	CCONJ
ma-220	259	5	{	{	PUNCT
ma-220	259	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	259	7	be	be	VERB
ma-220	259	8	equivalent	equivalent	ADJ
ma-220	259	9	parseval	parseval	NOUN
ma-220	259	10	frames	frame	NOUN
ma-220	259	11	for	for	ADP
ma-220	259	12	h	h	NOUN
ma-220	259	13	and	and	CCONJ
ma-220	259	14	let	let	VERB
ma-220	259	15	gf	gf	PRON
ma-220	259	16	and	and	CCONJ
ma-220	259	17	gg	gg	PROPN
ma-220	259	18	be	be	AUX
ma-220	259	19	grammian	grammian	ADJ
ma-220	259	20	matrices	matrix	NOUN
ma-220	259	21	of	of	ADP
ma-220	259	22	{	{	PUNCT
ma-220	259	23	fw}w∈ω	fw}w∈ω	NOUN
ma-220	259	24	and	and	CCONJ
ma-220	259	25	{	{	PUNCT
ma-220	259	26	gw}w∈ω	gw}w∈ω	NOUN
ma-220	259	27	,	,	PUNCT
ma-220	259	28	respectively	respectively	ADV
ma-220	259	29	.	.	PUNCT
ma-220	260	1	then	then	ADV
ma-220	260	2	gf	gf	PROPN
ma-220	260	3	=	=	PUNCT
ma-220	260	4	gg	gg	PROPN
ma-220	260	5	.	.	PUNCT
ma-220	261	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	261	2	eur	eur	PROPN
ma-220	261	3	.	.	PUNCT
ma-220	262	1	j.	j.	PROPN
ma-220	262	2	math	math	PROPN
ma-220	262	3	.	.	PUNCT
ma-220	263	1	anal	anal	PROPN
ma-220	263	2	.	.	PUNCT
ma-220	264	1	10.28924	10.28924	NUM
ma-220	264	2	/	/	SYM
ma-220	264	3	ada	ada	PROPN
ma-220	264	4	/	/	SYM
ma-220	264	5	ma.4.4	ma.4.4	PROPN
ma-220	264	6	11	11	NUM
ma-220	264	7	proof	proof	NOUN
ma-220	264	8	.	.	PUNCT
ma-220	265	1	since	since	SCONJ
ma-220	265	2	two	two	NUM
ma-220	265	3	frames	frame	NOUN
ma-220	265	4	{	{	PUNCT
ma-220	265	5	fw}w∈ω	fw}w∈ω	NOUN
ma-220	265	6	and	and	CCONJ
ma-220	265	7	{	{	PUNCT
ma-220	265	8	gw}w∈ω	gw}w∈ω	NOUN
ma-220	265	9	are	be	AUX
ma-220	265	10	equivalent	equivalent	ADJ
ma-220	265	11	,	,	PUNCT
ma-220	265	12	there	there	PRON
ma-220	265	13	exists	exist	VERB
ma-220	265	14	an	an	DET
ma-220	265	15	adjointable	adjointable	ADJ
ma-220	265	16	andinvertible	andinvertible	ADJ
ma-220	265	17	operator	operator	NOUN
ma-220	265	18	ξ	ξ	NOUN
ma-220	265	19	:	:	PUNCT
ma-220	265	20	h	h	PROPN
ma-220	265	21	−→	−→	ADJ
ma-220	265	22	h	h	NOUN
ma-220	265	23	by	by	ADP
ma-220	265	24	ξfw	ξfw	NOUN
ma-220	265	25	=	=	SYM
ma-220	265	26	gw	gw	PROPN
ma-220	265	27	for	for	ADP
ma-220	265	28	w	w	PROPN
ma-220	265	29	∈	∈	PROPN
ma-220	265	30	ω	ω	PROPN
ma-220	265	31	.	.	PUNCT
ma-220	266	1	since	since	SCONJ
ma-220	266	2	their	their	PRON
ma-220	266	3	frame	frame	NOUN
ma-220	266	4	operators	operator	NOUN
ma-220	266	5	are	be	AUX
ma-220	266	6	theidentity	theidentity	NOUN
ma-220	266	7	operator	operator	NOUN
ma-220	266	8	on	on	ADP
ma-220	266	9	h	h	NOUN
ma-220	266	10	,	,	PUNCT
ma-220	266	11	by	by	ADP
ma-220	266	12	theorem	theorem	NOUN
ma-220	266	13	3.2	3.2	NUM
ma-220	266	14	,	,	PUNCT
ma-220	266	15	ξξ∗	ξξ∗	ADJ
ma-220	266	16	=	=	SYM
ma-220	266	17	ξidξ∗	ξidξ∗	NOUN
ma-220	266	18	=	=	SYM
ma-220	266	19	i	i	PROPN
ma-220	266	20	d.	d.	PROPN
ma-220	266	21	so	so	PROPN
ma-220	266	22	ξ	ξ	PROPN
ma-220	266	23	is	be	AUX
ma-220	266	24	a	a	DET
ma-220	266	25	unitary	unitary	ADJ
ma-220	266	26	operator	operator	NOUN
ma-220	266	27	and	and	CCONJ
ma-220	266	28	then	then	ADV
ma-220	266	29	for	for	ADP
ma-220	266	30	i	i	PRON
ma-220	266	31	,	,	PUNCT
ma-220	266	32	w	w	PROPN
ma-220	266	33	∈	∈	PROPN
ma-220	266	34	ω	ω	PROPN
ma-220	266	35	,	,	PUNCT
ma-220	266	36	〈	〈	PROPN
ma-220	266	37	gi	gi	X
ma-220	266	38	,	,	PUNCT
ma-220	266	39	gw	gw	PROPN
ma-220	266	40	〉	〉	PROPN
ma-220	266	41	=	=	PUNCT
ma-220	267	1	〈	〈	PROPN
ma-220	267	2	ξfi	ξfi	NOUN
ma-220	267	3	,	,	PUNCT
ma-220	267	4	ξfw	ξfw	PROPN
ma-220	267	5	〉	〉	PROPN
ma-220	267	6	=	=	SYM
ma-220	267	7	〈	〈	PROPN
ma-220	267	8	fi	fi	NOUN
ma-220	267	9	,	,	PUNCT
ma-220	267	10	fw	fw	PROPN
ma-220	267	11	〉	〉	PROPN
ma-220	267	12	,	,	PUNCT
ma-220	267	13	which	which	PRON
ma-220	267	14	shows	show	VERB
ma-220	267	15	that	that	SCONJ
ma-220	267	16	gf	gf	NOUN
ma-220	267	17	=	=	PUNCT
ma-220	268	1	[	[	X
ma-220	268	2	〈	〈	PROPN
ma-220	268	3	fi	fi	NOUN
ma-220	268	4	,	,	PUNCT
ma-220	268	5	fw	fw	PROPN
ma-220	268	6	〉	〉	PROPN
ma-220	268	7	]	]	PUNCT
ma-220	268	8	w∈ω	w∈ω	PROPN
ma-220	268	9	=	=	PUNCT
ma-220	269	1	[	[	X
ma-220	269	2	〈	〈	X
ma-220	269	3	gi	gi	X
ma-220	269	4	,	,	PUNCT
ma-220	269	5	gw	gw	PROPN
ma-220	269	6	〉	〉	PROPN
ma-220	269	7	]	]	PUNCT
ma-220	269	8	w∈ω	w∈ω	PROPN
ma-220	269	9	=	=	SYM
ma-220	269	10	gg	gg	PROPN
ma-220	269	11	.	.	PUNCT
ma-220	270	1	�	�	PROPN
ma-220	270	2	4	4	NUM
ma-220	270	3	.	.	NUM
ma-220	270	4	constructed	construct	VERB
ma-220	270	5	∗-continuous	∗-continuous	ADJ
ma-220	270	6	frames	frame	NOUN
ma-220	270	7	and	and	CCONJ
ma-220	270	8	some	some	DET
ma-220	270	9	properties	property	NOUN
ma-220	270	10	theorem	theorem	VERB
ma-220	270	11	4.1	4.1	NUM
ma-220	270	12	.	.	PUNCT
ma-220	271	1	let	let	VERB
ma-220	271	2	{	{	PUNCT
ma-220	271	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	271	4	be	be	AUX
ma-220	271	5	a	a	DET
ma-220	271	6	∗-continuous	∗-continuous	ADJ
ma-220	271	7	frame	frame	NOUN
ma-220	271	8	for	for	ADP
ma-220	271	9	h	h	NOUN
ma-220	271	10	and	and	CCONJ
ma-220	271	11	ξ	ξ	PROPN
ma-220	271	12	be	be	AUX
ma-220	271	13	an	an	DET
ma-220	271	14	adjointable	adjointable	NOUN
ma-220	271	15	and	and	CCONJ
ma-220	271	16	invertible	invertible	ADJ
ma-220	271	17	operator	operator	NOUN
ma-220	271	18	on	on	ADP
ma-220	271	19	h.	h.	PROPN
ma-220	271	20	then	then	ADV
ma-220	271	21	the	the	DET
ma-220	271	22	set	set	NOUN
ma-220	271	23	(	(	PUNCT
ma-220	271	24	{	{	PUNCT
ma-220	271	25	gw}w∈ω	gw}w∈ω	NOUN
ma-220	271	26	,	,	PUNCT
ma-220	271	27	γξ−1	γξ−1	PROPN
ma-220	271	28	)	)	PUNCT
ma-220	271	29	is	be	AUX
ma-220	271	30	all	all	PRON
ma-220	271	31	of	of	ADP
ma-220	271	32	operator	operator	NOUN
ma-220	271	33	duals	dual	NOUN
ma-220	271	34	of	of	ADP
ma-220	271	35	{	{	PUNCT
ma-220	271	36	ξfw}w∈ω	ξfw}w∈ω	PROPN
ma-220	271	37	,	,	PUNCT
ma-220	271	38	where	where	SCONJ
ma-220	271	39	(	(	PUNCT
ma-220	271	40	{	{	PUNCT
ma-220	271	41	gw}w∈ω	gw}w∈ω	NOUN
ma-220	271	42	,	,	PUNCT
ma-220	271	43	γ	γ	PROPN
ma-220	271	44	)	)	PUNCT
ma-220	271	45	is	be	AUX
ma-220	271	46	an	an	DET
ma-220	271	47	operator	operator	NOUN
ma-220	271	48	dual	dual	ADJ
ma-220	271	49	for	for	ADP
ma-220	271	50	{	{	PUNCT
ma-220	271	51	fw}w∈ω	fw}w∈ω	NOUN
ma-220	271	52	.	.	NOUN
ma-220	271	53	proof	proof	NOUN
ma-220	271	54	.	.	PUNCT
ma-220	272	1	let	let	AUX
ma-220	272	2	(	(	PUNCT
ma-220	272	3	{	{	PUNCT
ma-220	272	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	272	5	,	,	PUNCT
ma-220	272	6	γ	γ	PROPN
ma-220	272	7	)	)	PUNCT
ma-220	272	8	be	be	AUX
ma-220	272	9	an	an	DET
ma-220	272	10	operator	operator	NOUN
ma-220	272	11	dual	dual	ADJ
ma-220	272	12	of	of	ADP
ma-220	272	13	{	{	PUNCT
ma-220	272	14	fw}w∈ω	fw}w∈ω	NOUN
ma-220	272	15	.	.	PUNCT
ma-220	272	16	then	then	ADV
ma-220	272	17	for	for	SCONJ
ma-220	272	18	x	x	PROPN
ma-220	272	19	∈	∈	PROPN
ma-220	272	20	h,∫	h,∫	NOUN
ma-220	272	21	ω	ω	NUM
ma-220	272	22	〈	〈	PROPN
ma-220	272	23	γξ−1x	γξ−1x	PROPN
ma-220	272	24	,	,	PUNCT
ma-220	272	25	gw	gw	PROPN
ma-220	272	26	〉	〉	NOUN
ma-220	272	27	ξxwdµ(ω	ξxwdµ(ω	NOUN
ma-220	272	28	)	)	PUNCT
ma-220	273	1	=	=	SYM
ma-220	273	2	ξ	ξ	PROPN
ma-220	273	3	(	(	PUNCT
ma-220	273	4	∫	∫	PROPN
ma-220	273	5	ω	ω	NUM
ma-220	273	6	〈	〈	PROPN
ma-220	273	7	γξ−1f	γξ−1f	PROPN
ma-220	273	8	,	,	PUNCT
ma-220	273	9	gw	gw	PROPN
ma-220	273	10	〉	〉	NOUN
ma-220	273	11	fwdµ(ω	fwdµ(ω	NOUN
ma-220	273	12	)	)	PUNCT
ma-220	273	13	)	)	PUNCT
ma-220	274	1	=	=	PUNCT
ma-220	274	2	ξ	ξ	X
ma-220	274	3	(	(	PUNCT
ma-220	274	4	ξ−1	ξ−1	PROPN
ma-220	274	5	)	)	PUNCT
ma-220	274	6	x	x	X
ma-220	274	7	=	=	PUNCT
ma-220	275	1	x.	x.	NOUN
ma-220	275	2	this	this	PRON
ma-220	275	3	shows	show	VERB
ma-220	275	4	that	that	SCONJ
ma-220	275	5	(	(	PUNCT
ma-220	275	6	{	{	PUNCT
ma-220	275	7	gj}j∈j	gj}j∈j	INTJ
ma-220	275	8	,	,	PUNCT
ma-220	275	9	γξ−1	γξ−1	PROPN
ma-220	275	10	)	)	PUNCT
ma-220	275	11	is	be	AUX
ma-220	275	12	an	an	DET
ma-220	275	13	operator	operator	NOUN
ma-220	275	14	dual	dual	ADV
ma-220	275	15	of	of	ADP
ma-220	275	16	{	{	PUNCT
ma-220	275	17	ξfj}j∈j	ξfj}j∈j	ADP
ma-220	275	18	.now	.now	NOUN
ma-220	275	19	,	,	PUNCT
ma-220	275	20	if	if	SCONJ
ma-220	275	21	(	(	PUNCT
ma-220	275	22	{	{	PUNCT
ma-220	275	23	gj}j∈j	gj}j∈j	INTJ
ma-220	275	24	,	,	PUNCT
ma-220	275	25	γ	γ	PROPN
ma-220	275	26	)	)	PUNCT
ma-220	275	27	is	be	AUX
ma-220	275	28	an	an	DET
ma-220	275	29	operator	operator	NOUN
ma-220	275	30	dual	dual	ADJ
ma-220	275	31	for	for	ADP
ma-220	275	32	{	{	PUNCT
ma-220	275	33	ξfj}j∈j	ξfj}j∈j	NOUN
ma-220	275	34	,	,	PUNCT
ma-220	275	35	then	then	ADV
ma-220	275	36	it	it	PRON
ma-220	275	37	is	be	AUX
ma-220	275	38	enough	enough	ADJ
ma-220	275	39	to	to	PART
ma-220	275	40	set	set	VERB
ma-220	275	41	γ	γ	NOUN
ma-220	275	42	:	:	PUNCT
ma-220	275	43	=	=	SYM
ma-220	275	44	γξ	γξ	ADP
ma-220	275	45	in	in	ADP
ma-220	275	46	thelast	thelast	ADJ
ma-220	275	47	equalities	equality	NOUN
ma-220	275	48	which	which	PRON
ma-220	275	49	follows	follow	VERB
ma-220	275	50	that	that	SCONJ
ma-220	275	51	(	(	PUNCT
ma-220	275	52	{	{	PUNCT
ma-220	275	53	gj}j∈j	gj}j∈j	INTJ
ma-220	275	54	,	,	PUNCT
ma-220	275	55	γ	γ	PROPN
ma-220	275	56	)	)	PUNCT
ma-220	275	57	is	be	AUX
ma-220	275	58	an	an	DET
ma-220	275	59	operator	operator	NOUN
ma-220	275	60	dual	dual	ADJ
ma-220	275	61	for	for	ADP
ma-220	275	62	{	{	PUNCT
ma-220	275	63	fj}j∈j	fj}j∈j	PROPN
ma-220	275	64	.	.	PUNCT
ma-220	276	1	�	�	PROPN
ma-220	276	2	an	an	DET
ma-220	276	3	orthogonal	orthogonal	ADJ
ma-220	276	4	projection	projection	NOUN
ma-220	276	5	will	will	AUX
ma-220	276	6	obtain	obtain	VERB
ma-220	276	7	a	a	DET
ma-220	276	8	∗-frame	∗-frame	NOUN
ma-220	276	9	,	,	PUNCT
ma-220	276	10	and	and	CCONJ
ma-220	276	11	relation	relation	NOUN
ma-220	276	12	will	will	AUX
ma-220	276	13	also	also	ADV
ma-220	276	14	be	be	AUX
ma-220	276	15	given	give	VERB
ma-220	276	16	for	for	ADP
ma-220	276	17	this	this	PRON
ma-220	276	18	projection.to	projection.to	PRON
ma-220	276	19	see	see	VERB
ma-220	276	20	this	this	PRON
ma-220	276	21	,	,	PUNCT
ma-220	276	22	we	we	PRON
ma-220	276	23	must	must	AUX
ma-220	276	24	show	show	VERB
ma-220	276	25	that	that	SCONJ
ma-220	276	26	the	the	DET
ma-220	276	27	inverse	inverse	NOUN
ma-220	276	28	of	of	ADP
ma-220	276	29	the	the	DET
ma-220	276	30	frame	frame	NOUN
ma-220	276	31	operator	operator	NOUN
ma-220	276	32	is	be	AUX
ma-220	276	33	unique	unique	ADJ
ma-220	276	34	in	in	ADP
ma-220	276	35	the	the	DET
ma-220	276	36	reconstructionformula	reconstructionformula	NOUN
ma-220	276	37	.	.	PUNCT
ma-220	277	1	so	so	ADV
ma-220	277	2	,	,	PUNCT
ma-220	277	3	firstly	firstly	ADV
ma-220	277	4	this	this	DET
ma-220	277	5	fact	fact	NOUN
ma-220	277	6	will	will	AUX
ma-220	277	7	be	be	AUX
ma-220	277	8	considered	consider	VERB
ma-220	277	9	.	.	PUNCT
ma-220	278	1	theorem	theorem	VERB
ma-220	278	2	4.2	4.2	NUM
ma-220	278	3	.	.	PUNCT
ma-220	279	1	if	if	SCONJ
ma-220	279	2	{	{	PUNCT
ma-220	279	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	279	4	is	be	AUX
ma-220	279	5	a	a	DET
ma-220	279	6	∗-continuous	∗-continuous	ADJ
ma-220	279	7	frame	frame	NOUN
ma-220	279	8	for	for	ADP
ma-220	279	9	h	h	NOUN
ma-220	279	10	,	,	PUNCT
ma-220	279	11	then	then	ADV
ma-220	279	12	there	there	PRON
ma-220	279	13	exists	exist	VERB
ma-220	279	14	a	a	DET
ma-220	279	15	unique	unique	ADJ
ma-220	279	16	adjointable	adjointable	NOUN
ma-220	279	17	operator	operator	NOUN
ma-220	279	18	λ	λ	PROPN
ma-220	279	19	on	on	ADP
ma-220	279	20	h	h	PRON
ma-220	279	21	such	such	ADJ
ma-220	279	22	that	that	SCONJ
ma-220	279	23	x	x	X
ma-220	279	24	=	=	SYM
ma-220	279	25	∫	∫	PROPN
ma-220	279	26	ω	ω	NUM
ma-220	279	27	〈	〈	PROPN
ma-220	279	28	x	x	X
ma-220	279	29	,	,	PUNCT
ma-220	279	30	λfw	λfw	ADJ
ma-220	279	31	〉	〉	NUM
ma-220	279	32	fwdµ(ω	fwdµ(ω	NOUN
ma-220	279	33	)	)	PUNCT
ma-220	279	34	,	,	PUNCT
ma-220	279	35	∀x	∀x	VERB
ma-220	279	36	∈	∈	PROPN
ma-220	279	37	h.	h.	NOUN
ma-220	279	38	proof	proof	NOUN
ma-220	279	39	.	.	PUNCT
ma-220	280	1	by	by	ADP
ma-220	280	2	the	the	DET
ma-220	280	3	reconstruction	reconstruction	NOUN
ma-220	280	4	formula	formula	NOUN
ma-220	280	5	,	,	PUNCT
ma-220	280	6	there	there	PRON
ma-220	280	7	exists	exist	VERB
ma-220	280	8	λ	λ	X
ma-220	280	9	=	=	SYM
ma-220	280	10	s−1	s−1	PROPN
ma-220	280	11	.	.	PUNCT
ma-220	281	1	for	for	ADP
ma-220	281	2	the	the	DET
ma-220	281	3	uniqueness	uniqueness	NOUN
ma-220	281	4	of	of	ADP
ma-220	281	5	s−1	s−1	PROPN
ma-220	281	6	with	with	ADP
ma-220	281	7	thisproperty	thisproperty	NOUN
ma-220	281	8	,	,	PUNCT
ma-220	281	9	we	we	PRON
ma-220	281	10	know	know	VERB
ma-220	281	11	that	that	SCONJ
ma-220	281	12	{	{	PUNCT
ma-220	281	13	s−	s−	NOUN
ma-220	281	14	1	1	NUM
ma-220	281	15	2fw	2fw	ADJ
ma-220	281	16	}	}	PUNCT
ma-220	281	17	w∈ω	w∈ω	NOUN
ma-220	281	18	is	be	AUX
ma-220	281	19	a	a	DET
ma-220	281	20	continuous	continuous	ADJ
ma-220	281	21	parseval	parseval	NOUN
ma-220	281	22	frame	frame	NOUN
ma-220	281	23	for	for	ADP
ma-220	281	24	h.	h.	PROPN
ma-220	281	25	set	set	PROPN
ma-220	281	26	gw	gw	PROPN
ma-220	281	27	=	=	PUNCT
ma-220	281	28	s−	s−	PROPN
ma-220	281	29	1	1	NUM
ma-220	281	30	2fw	2fw	NOUN
ma-220	281	31	.then	.then	VERB
ma-220	282	1	fw	fw	PROPN
ma-220	282	2	=	=	SYM
ma-220	282	3	s	s	PART
ma-220	282	4	1	1	NUM
ma-220	282	5	2	2	NUM
ma-220	282	6	gw	gw	NOUN
ma-220	282	7	.	.	PUNCT
ma-220	283	1	now	now	ADV
ma-220	283	2	,	,	PUNCT
ma-220	283	3	suppose	suppose	VERB
ma-220	283	4	that	that	SCONJ
ma-220	283	5	λ	λ	PROPN
ma-220	283	6	is	be	AUX
ma-220	283	7	an	an	DET
ma-220	283	8	adjointable	adjointable	ADJ
ma-220	283	9	operator	operator	NOUN
ma-220	283	10	such	such	ADJ
ma-220	283	11	that	that	PRON
ma-220	283	12	x	x	PUNCT
ma-220	283	13	=	=	SYM
ma-220	283	14	∫	∫	PROPN
ma-220	283	15	ω	ω	NUM
ma-220	283	16	〈	〈	PROPN
ma-220	283	17	x	x	X
ma-220	283	18	,	,	PUNCT
ma-220	283	19	λfw	λfw	ADJ
ma-220	283	20	〉	〉	NUM
ma-220	283	21	fwdµ(ω	fwdµ(ω	NOUN
ma-220	283	22	)	)	PUNCT
ma-220	283	23	,	,	PUNCT
ma-220	283	24	∀x	∀x	PROPN
ma-220	283	25	∈	∈	PROPN
ma-220	283	26	h.	h.	PROPN
ma-220	283	27	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	283	28	eur	eur	PROPN
ma-220	283	29	.	.	PUNCT
ma-220	284	1	j.	j.	PROPN
ma-220	284	2	math	math	PROPN
ma-220	284	3	.	.	PUNCT
ma-220	285	1	anal	anal	PROPN
ma-220	285	2	.	.	PUNCT
ma-220	286	1	10.28924	10.28924	NUM
ma-220	286	2	/	/	SYM
ma-220	286	3	ada	ada	PROPN
ma-220	286	4	/	/	SYM
ma-220	286	5	ma.4.4	ma.4.4	PROPN
ma-220	287	1	12then	12then	INTJ
ma-220	287	2	we	we	PRON
ma-220	287	3	have	have	VERB
ma-220	287	4	x	x	NOUN
ma-220	287	5	=	=	SYM
ma-220	287	6	∫	∫	PROPN
ma-220	287	7	ω	ω	NUM
ma-220	287	8	〈	〈	PROPN
ma-220	287	9	x	x	X
ma-220	287	10	,	,	PUNCT
ma-220	287	11	λfw	λfw	ADJ
ma-220	287	12	〉	〉	NUM
ma-220	287	13	fwdµ(ω	fwdµ(ω	NOUN
ma-220	287	14	)	)	PUNCT
ma-220	288	1	=	=	SYM
ma-220	288	2	∫	∫	PROPN
ma-220	289	1	ω	ω	NUM
ma-220	289	2	〈	〈	PROPN
ma-220	289	3	x	x	NOUN
ma-220	289	4	,	,	PUNCT
ma-220	289	5	λs	λs	ADP
ma-220	289	6	1	1	NUM
ma-220	289	7	2	2	NUM
ma-220	289	8	gw	gw	NOUN
ma-220	289	9	〉	〉	NOUN
ma-220	289	10	s	s	PART
ma-220	289	11	1	1	NUM
ma-220	289	12	2	2	NUM
ma-220	289	13	gwdµ(ω	gwdµ(ω	NOUN
ma-220	289	14	)	)	PUNCT
ma-220	290	1	=	=	SYM
ma-220	290	2	s	s	PART
ma-220	290	3	1	1	NUM
ma-220	290	4	2	2	NUM
ma-220	290	5	(	(	PUNCT
ma-220	290	6	∫	∫	PROPN
ma-220	290	7	ω	ω	NUM
ma-220	290	8	〈	〈	PROPN
ma-220	290	9	x	x	NOUN
ma-220	290	10	,	,	PUNCT
ma-220	290	11	λs	λs	ADP
ma-220	290	12	1	1	NUM
ma-220	290	13	2	2	NUM
ma-220	290	14	gw	gw	PROPN
ma-220	290	15	〉	〉	NOUN
ma-220	290	16	gwdµ(ω	gwdµ(ω	PROPN
ma-220	290	17	)	)	PUNCT
ma-220	290	18	)	)	PUNCT
ma-220	291	1	=	=	PUNCT
ma-220	291	2	s	s	VERB
ma-220	291	3	1	1	NUM
ma-220	291	4	2	2	NUM
ma-220	291	5	(	(	PUNCT
ma-220	291	6	∫	∫	PROPN
ma-220	291	7	ω	ω	NUM
ma-220	291	8	〈	〈	PROPN
ma-220	291	9	s	s	NOUN
ma-220	291	10	1	1	NUM
ma-220	291	11	2	2	NUM
ma-220	291	12	λ∗x	λ∗x	NUM
ma-220	291	13	,	,	PUNCT
ma-220	291	14	gw	gw	PROPN
ma-220	291	15	〉	〉	NOUN
ma-220	291	16	gw	gw	NOUN
ma-220	291	17	)	)	PUNCT
ma-220	292	1	=	=	SYM
ma-220	292	2	s	s	PART
ma-220	292	3	1	1	NUM
ma-220	292	4	2	2	NUM
ma-220	292	5	(	(	PUNCT
ma-220	292	6	s	s	NOUN
ma-220	292	7	1	1	NUM
ma-220	292	8	2	2	NUM
ma-220	292	9	λ∗x	λ∗x	NUM
ma-220	292	10	)	)	PUNCT
ma-220	292	11	=	=	SYM
ma-220	292	12	sλ∗x	sλ∗x	NOUN
ma-220	292	13	,	,	PUNCT
ma-220	292	14	∀x	∀x	X
ma-220	292	15	∈	∈	PROPN
ma-220	292	16	h.this	h.this	PRON
ma-220	292	17	concludes	conclude	VERB
ma-220	292	18	that	that	SCONJ
ma-220	292	19	sλ∗	sλ∗	NOUN
ma-220	292	20	=	=	PUNCT
ma-220	292	21	i	i	PROPN
ma-220	292	22	d	d	PROPN
ma-220	292	23	and	and	CCONJ
ma-220	292	24	then	then	ADV
ma-220	292	25	λ∗	λ∗	NOUN
ma-220	293	1	=	=	SYM
ma-220	293	2	s−1	s−1	PROPN
ma-220	293	3	.	.	PUNCT
ma-220	294	1	more	more	ADV
ma-220	294	2	precisely	precisely	ADV
ma-220	294	3	,	,	PUNCT
ma-220	294	4	λ	λ	PROPN
ma-220	294	5	is	be	AUX
ma-220	294	6	self	self	NOUN
ma-220	294	7	-	-	PUNCT
ma-220	294	8	adjoint	adjoint	NOUN
ma-220	294	9	,	,	PUNCT
ma-220	294	10	positive	positive	ADJ
ma-220	294	11	andinvertible	andinvertible	ADJ
ma-220	294	12	.	.	PUNCT
ma-220	295	1	�	�	PROPN
ma-220	295	2	now	now	ADV
ma-220	295	3	,	,	PUNCT
ma-220	295	4	a	a	DET
ma-220	295	5	∗-continuous	∗-continuous	ADJ
ma-220	295	6	frame	frame	NOUN
ma-220	295	7	is	be	AUX
ma-220	295	8	constructed	construct	VERB
ma-220	295	9	by	by	ADP
ma-220	295	10	an	an	DET
ma-220	295	11	orthogonal	orthogonal	ADJ
ma-220	295	12	projection	projection	NOUN
ma-220	295	13	.	.	PUNCT
ma-220	296	1	proposition	proposition	NOUN
ma-220	296	2	4.3	4.3	NUM
ma-220	296	3	.	.	PUNCT
ma-220	297	1	let	let	VERB
ma-220	297	2	{	{	PUNCT
ma-220	297	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	297	4	be	be	AUX
ma-220	297	5	a	a	DET
ma-220	297	6	∗-continuous	∗-continuous	ADJ
ma-220	297	7	frame	frame	NOUN
ma-220	297	8	for	for	ADP
ma-220	297	9	h	h	NOUN
ma-220	297	10	with	with	ADP
ma-220	297	11	the	the	DET
ma-220	297	12	frame	frame	NOUN
ma-220	297	13	operator	operator	NOUN
ma-220	297	14	s	s	PART
ma-220	297	15	and	and	CCONJ
ma-220	297	16	∗continuous	∗continuous	ADJ
ma-220	297	17	frame	frame	NOUN
ma-220	297	18	bounds	bound	VERB
ma-220	297	19	a	a	PRON
ma-220	297	20	and	and	CCONJ
ma-220	297	21	b.	b.	PROPN
ma-220	297	22	also	also	ADV
ma-220	297	23	,	,	PUNCT
ma-220	297	24	suppose	suppose	VERB
ma-220	297	25	that	that	SCONJ
ma-220	297	26	p	p	PROPN
ma-220	297	27	is	be	AUX
ma-220	297	28	an	an	DET
ma-220	297	29	orthogonal	orthogonal	ADJ
ma-220	297	30	projection	projection	NOUN
ma-220	297	31	on	on	ADP
ma-220	297	32	h.	h.	PROPN
ma-220	297	33	then	then	ADV
ma-220	297	34	{	{	PUNCT
ma-220	297	35	pfw}w∈ω	pfw}w∈ω	NOUN
ma-220	297	36	is	be	AUX
ma-220	297	37	a	a	DET
ma-220	297	38	∗-continuous	∗-continuous	ADJ
ma-220	297	39	frame	frame	NOUN
ma-220	297	40	for	for	ADP
ma-220	297	41	rp	rp	NOUN
ma-220	297	42	with	with	SCONJ
ma-220	297	43	∗-continuous	∗-continuous	ADJ
ma-220	297	44	frame	frame	NOUN
ma-220	297	45	bounds	bound	VERB
ma-220	297	46	a	a	PRON
ma-220	297	47	and	and	CCONJ
ma-220	297	48	b.	b.	PROPN
ma-220	297	49	moreover	moreover	ADV
ma-220	297	50	,	,	PUNCT
ma-220	297	51	if	if	SCONJ
ma-220	297	52	(	(	PUNCT
ma-220	297	53	{	{	PUNCT
ma-220	297	54	gw}w∈ω	gw}w∈ω	NOUN
ma-220	297	55	,	,	PUNCT
ma-220	297	56	γ	γ	PROPN
ma-220	297	57	)	)	PUNCT
ma-220	297	58	is	be	AUX
ma-220	297	59	an	an	DET
ma-220	297	60	operator	operator	NOUN
ma-220	297	61	dual	dual	ADJ
ma-220	297	62	of	of	ADP
ma-220	297	63	{	{	PUNCT
ma-220	297	64	fw}w∈ω	fw}w∈ω	PROPN
ma-220	297	65	,	,	PUNCT
ma-220	297	66	then	then	ADV
ma-220	297	67	{	{	PUNCT
ma-220	297	68	pγ∗gw}w∈ω	pγ∗gw}w∈ω	NOUN
ma-220	297	69	is	be	AUX
ma-220	297	70	a	a	DET
ma-220	297	71	dual	dual	ADJ
ma-220	297	72	∗-continuous	∗-continuous	ADJ
ma-220	297	73	frame	frame	NOUN
ma-220	297	74	for	for	ADP
ma-220	297	75	{	{	PUNCT
ma-220	297	76	p	p	NOUN
ma-220	297	77	fw}w∈ω	fw}w∈ω	PROPN
ma-220	297	78	.	.	PUNCT
ma-220	297	79	proof	proof	NOUN
ma-220	297	80	.	.	PUNCT
ma-220	298	1	for	for	ADP
ma-220	298	2	x	x	PROPN
ma-220	298	3	∈	∈	PROPN
ma-220	298	4	rp	rp	NOUN
ma-220	298	5	,	,	PUNCT
ma-220	298	6	∫	∫	PROPN
ma-220	298	7	ω	ω	NUM
ma-220	298	8	〈	〈	PROPN
ma-220	298	9	x	x	PROPN
ma-220	298	10	,	,	PUNCT
ma-220	298	11	pfw	pfw	PROPN
ma-220	298	12	〉	〉	PROPN
ma-220	298	13	〈	〈	PROPN
ma-220	298	14	pfw	pfw	PROPN
ma-220	298	15	,	,	PUNCT
ma-220	298	16	x	x	PROPN
ma-220	298	17	〉	〉	NOUN
ma-220	298	18	dµ(ω	dµ(ω	PUNCT
ma-220	298	19	)	)	PUNCT
ma-220	298	20	=	=	SYM
ma-220	298	21	∫	∫	PROPN
ma-220	298	22	ω	ω	NUM
ma-220	298	23	〈	〈	PROPN
ma-220	298	24	px	px	PROPN
ma-220	298	25	,	,	PUNCT
ma-220	298	26	fw	fw	ADJ
ma-220	298	27	〉	〉	PROPN
ma-220	298	28	〈	〈	PROPN
ma-220	298	29	fw	fw	PROPN
ma-220	298	30	,	,	PUNCT
ma-220	298	31	p	p	X
ma-220	298	32	x	x	X
ma-220	298	33	〉	〉	NOUN
ma-220	298	34	dµ(ω	dµ(ω	PUNCT
ma-220	298	35	)	)	PUNCT
ma-220	298	36	=	=	SYM
ma-220	298	37	∫	∫	PROPN
ma-220	298	38	ω	ω	NUM
ma-220	298	39	〈	〈	PROPN
ma-220	298	40	x	x	PRON
ma-220	298	41	,	,	PUNCT
ma-220	298	42	fw	fw	PROPN
ma-220	298	43	〉	〉	PROPN
ma-220	298	44	〈	〈	PROPN
ma-220	298	45	fw	fw	PROPN
ma-220	298	46	,	,	PUNCT
ma-220	298	47	x	x	PROPN
ma-220	298	48	〉	〉	NOUN
ma-220	298	49	dµ(ω	dµ(ω	PUNCT
ma-220	298	50	)	)	PUNCT
ma-220	298	51	and	and	CCONJ
ma-220	298	52	by	by	ADP
ma-220	298	53	the	the	DET
ma-220	298	54	definition	definition	NOUN
ma-220	298	55	of	of	ADP
ma-220	298	56	∗-continuous	∗-continuous	ADJ
ma-220	298	57	frame	frame	NOUN
ma-220	298	58	{	{	PUNCT
ma-220	298	59	fw}w∈ω	fw}w∈ω	NOUN
ma-220	298	60	,	,	PUNCT
ma-220	298	61	we	we	PRON
ma-220	298	62	have	have	VERB
ma-220	298	63	a〈x	a〈x	NOUN
ma-220	298	64	,	,	PUNCT
ma-220	298	65	x〉a∗	x〉a∗	PROPN
ma-220	298	66	≤	≤	NUM
ma-220	298	67	∫	∫	PROPN
ma-220	298	68	ω	ω	NUM
ma-220	298	69	〈	〈	PROPN
ma-220	298	70	x	x	PROPN
ma-220	298	71	,	,	PUNCT
ma-220	298	72	pfw	pfw	PROPN
ma-220	298	73	〉	〉	PROPN
ma-220	298	74	〈	〈	PROPN
ma-220	298	75	pfw	pfw	PROPN
ma-220	298	76	,	,	PUNCT
ma-220	298	77	x	x	PROPN
ma-220	298	78	〉	〉	NOUN
ma-220	298	79	dµ(ω	dµ(ω	PUNCT
ma-220	298	80	)	)	PUNCT
ma-220	298	81	≤	≤	NUM
ma-220	298	82	b〈x	b〈x	PUNCT
ma-220	298	83	,	,	PUNCT
ma-220	298	84	x〉b∗.	x〉b∗.	PROPN
ma-220	298	85	now	now	ADV
ma-220	298	86	,	,	PUNCT
ma-220	298	87	if	if	SCONJ
ma-220	298	88	(	(	PUNCT
ma-220	298	89	{	{	PUNCT
ma-220	298	90	gw}w∈ω	gw}w∈ω	NOUN
ma-220	298	91	,	,	PUNCT
ma-220	298	92	γ	γ	PROPN
ma-220	298	93	)	)	PUNCT
ma-220	298	94	is	be	AUX
ma-220	298	95	an	an	DET
ma-220	298	96	operator	operator	NOUN
ma-220	298	97	dual	dual	ADJ
ma-220	298	98	of	of	ADP
ma-220	298	99	{	{	PUNCT
ma-220	298	100	fw}w∈ω	fw}w∈ω	PROPN
ma-220	298	101	,	,	PUNCT
ma-220	298	102	then	then	ADV
ma-220	298	103	for	for	ADP
ma-220	298	104	x	x	PROPN
ma-220	298	105	∈	∈	PROPN
ma-220	298	106	rp	rp	NOUN
ma-220	298	107	,	,	PUNCT
ma-220	298	108	x	x	PROPN
ma-220	298	109	=	=	SYM
ma-220	298	110	px	px	X
ma-220	298	111	=	=	SYM
ma-220	298	112	p	p	X
ma-220	298	113	(	(	PUNCT
ma-220	298	114	∫	∫	PROPN
ma-220	298	115	ω	ω	PROPN
ma-220	298	116	〈	〈	PROPN
ma-220	298	117	γpx	γpx	PROPN
ma-220	298	118	,	,	PUNCT
ma-220	298	119	gw	gw	PROPN
ma-220	298	120	〉	〉	PROPN
ma-220	298	121	fwdµ(ω	fwdµ(ω	NOUN
ma-220	298	122	)	)	PUNCT
ma-220	298	123	)	)	PUNCT
ma-220	299	1	=	=	SYM
ma-220	300	1	∫	∫	PROPN
ma-220	300	2	ω	ω	NUM
ma-220	300	3	〈	〈	PROPN
ma-220	300	4	x	x	PRON
ma-220	300	5	,	,	PUNCT
ma-220	300	6	pγ∗gw	pγ∗gw	PROPN
ma-220	300	7	〉	〉	NOUN
ma-220	300	8	pfwdµ(ω	pfwdµ(ω	NOUN
ma-220	300	9	)	)	PUNCT
ma-220	300	10	.	.	PUNCT
ma-220	301	1	if	if	SCONJ
ma-220	301	2	{	{	PUNCT
ma-220	301	3	gw}w∈ω	gw}w∈ω	NOUN
ma-220	301	4	is	be	AUX
ma-220	301	5	also	also	ADV
ma-220	301	6	a	a	DET
ma-220	301	7	dual	dual	ADJ
ma-220	301	8	of	of	ADP
ma-220	301	9	{	{	PUNCT
ma-220	301	10	fw}w∈ω	fw}w∈ω	PROPN
ma-220	301	11	,	,	PUNCT
ma-220	301	12	then	then	ADV
ma-220	301	13	{	{	PUNCT
ma-220	301	14	pgw}w∈ω	pgw}w∈ω	PROPN
ma-220	301	15	is	be	AUX
ma-220	301	16	a	a	DET
ma-220	301	17	dual	dual	ADJ
ma-220	301	18	of	of	ADP
ma-220	301	19	{	{	PUNCT
ma-220	301	20	pfw}w∈ω	pfw}w∈ω	PROPN
ma-220	301	21	.	.	PUNCT
ma-220	302	1	so	so	ADV
ma-220	302	2	the	the	DET
ma-220	302	3	result	result	NOUN
ma-220	302	4	isclear	isclear	NOUN
ma-220	302	5	by	by	ADP
ma-220	302	6	γ	γ	PROPN
ma-220	302	7	=	=	SYM
ma-220	302	8	idh	idh	PROPN
ma-220	302	9	.	.	PUNCT
ma-220	303	1	�	�	PROPN
ma-220	303	2	by	by	ADP
ma-220	303	3	the	the	DET
ma-220	303	4	last	last	ADJ
ma-220	303	5	theorem	theorem	NOUN
ma-220	303	6	,	,	PUNCT
ma-220	303	7	a	a	DET
ma-220	303	8	necessary	necessary	ADJ
ma-220	303	9	and	and	CCONJ
ma-220	303	10	sufficient	sufficient	ADJ
ma-220	303	11	condition	condition	NOUN
ma-220	303	12	is	be	AUX
ma-220	303	13	found	find	VERB
ma-220	303	14	for	for	ADP
ma-220	303	15	commutating	commutate	VERB
ma-220	303	16	a	a	DET
ma-220	303	17	projectionwith	projectionwith	ADP
ma-220	303	18	the	the	DET
ma-220	303	19	inverse	inverse	NOUN
ma-220	303	20	of	of	ADP
ma-220	303	21	the	the	DET
ma-220	303	22	frame	frame	NOUN
ma-220	303	23	operator	operator	NOUN
ma-220	303	24	of	of	ADP
ma-220	303	25	a	a	DET
ma-220	303	26	given	give	VERB
ma-220	303	27	∗-frame	∗-frame	NOUN
ma-220	303	28	.	.	PUNCT
ma-220	304	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	304	2	eur	eur	PROPN
ma-220	304	3	.	.	PUNCT
ma-220	305	1	j.	j.	PROPN
ma-220	305	2	math	math	PROPN
ma-220	305	3	.	.	PUNCT
ma-220	306	1	anal	anal	PROPN
ma-220	306	2	.	.	PUNCT
ma-220	307	1	10.28924	10.28924	NUM
ma-220	307	2	/	/	SYM
ma-220	307	3	ada	ada	PROPN
ma-220	307	4	/	/	SYM
ma-220	307	5	ma.4.4	ma.4.4	PROPN
ma-220	307	6	13	13	NUM
ma-220	307	7	theorem	theorem	NOUN
ma-220	307	8	4.4	4.4	NUM
ma-220	307	9	.	.	PUNCT
ma-220	308	1	let	let	AUX
ma-220	308	2	{	{	PUNCT
ma-220	308	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	308	4	be	be	AUX
ma-220	308	5	a	a	DET
ma-220	308	6	∗-continuous	∗-continuous	ADJ
ma-220	308	7	frame	frame	NOUN
ma-220	308	8	for	for	ADP
ma-220	308	9	h	h	NOUN
ma-220	308	10	with	with	ADP
ma-220	308	11	the	the	DET
ma-220	308	12	frame	frame	NOUN
ma-220	308	13	operator	operator	NOUN
ma-220	308	14	s.	s.	PROPN
ma-220	308	15	suppose	suppose	VERB
ma-220	308	16	that	that	SCONJ
ma-220	308	17	p	p	PROPN
ma-220	308	18	is	be	AUX
ma-220	308	19	an	an	DET
ma-220	308	20	orthogonal	orthogonal	ADJ
ma-220	308	21	projection	projection	NOUN
ma-220	308	22	on	on	ADP
ma-220	308	23	h.	h.	PROPN
ma-220	308	24	then	then	ADV
ma-220	308	25	ps−1fw	ps−1fw	ADJ
ma-220	308	26	=	=	SYM
ma-220	308	27	s−1	s−1	PROPN
ma-220	308	28	p	p	PROPN
ma-220	308	29	pfw	pfw	PROPN
ma-220	308	30	,	,	PUNCT
ma-220	308	31	for	for	ADP
ma-220	308	32	all	all	DET
ma-220	308	33	w	w	PROPN
ma-220	308	34	∈	∈	PROPN
ma-220	308	35	ω	ω	NOUN
ma-220	308	36	if	if	SCONJ
ma-220	308	37	and	and	CCONJ
ma-220	308	38	only	only	ADV
ma-220	308	39	if	if	SCONJ
ma-220	308	40	ps−1	ps−1	PROPN
ma-220	308	41	=	=	SYM
ma-220	308	42	s−1p	s−1p	PROPN
ma-220	308	43	,	,	PUNCT
ma-220	308	44	where	where	SCONJ
ma-220	308	45	sp	sp	NOUN
ma-220	308	46	is	be	AUX
ma-220	308	47	the	the	DET
ma-220	308	48	continuous	continuous	ADJ
ma-220	308	49	frame	frame	NOUN
ma-220	308	50	operator	operator	NOUN
ma-220	308	51	of	of	ADP
ma-220	308	52	the	the	DET
ma-220	308	53	∗-continuous	∗-continuous	ADJ
ma-220	308	54	frame	frame	NOUN
ma-220	308	55	{	{	PUNCT
ma-220	308	56	pfw}w∈ω	pfw}w∈ω	NOUN
ma-220	308	57	.	.	PUNCT
ma-220	309	1	proof	proof	NOUN
ma-220	309	2	.	.	PUNCT
ma-220	310	1	first	first	ADV
ma-220	310	2	,	,	PUNCT
ma-220	310	3	assume	assume	VERB
ma-220	310	4	that	that	SCONJ
ma-220	310	5	ps−1fw	ps−1fw	NUM
ma-220	310	6	=	=	SYM
ma-220	310	7	s−1	s−1	PROPN
ma-220	310	8	p	p	PROPN
ma-220	310	9	pfw	pfw	PROPN
ma-220	310	10	,	,	PUNCT
ma-220	310	11	for	for	ADP
ma-220	310	12	all	all	DET
ma-220	310	13	w	w	PROPN
ma-220	310	14	∈	∈	PROPN
ma-220	310	15	ω	ω	NOUN
ma-220	310	16	.	.	PUNCT
ma-220	311	1	now	now	ADV
ma-220	311	2	,	,	PUNCT
ma-220	311	3	let	let	VERB
ma-220	311	4	x	x	X
ma-220	311	5	∈	∈	PROPN
ma-220	311	6	h.	h.	NOUN
ma-220	311	7	then	then	ADV
ma-220	311	8	we	we	PRON
ma-220	311	9	have	have	VERB
ma-220	311	10	s−1	s−1	PROPN
ma-220	311	11	p	p	NOUN
ma-220	312	1	px	px	NOUN
ma-220	312	2	=	=	PUNCT
ma-220	312	3	s−1	s−1	PROPN
ma-220	312	4	p	p	PROPN
ma-220	312	5	p	p	PROPN
ma-220	312	6	(	(	PUNCT
ma-220	312	7	∫	∫	PROPN
ma-220	312	8	ω	ω	NUM
ma-220	312	9	〈	〈	PROPN
ma-220	312	10	x	x	PROPN
ma-220	312	11	,	,	PUNCT
ma-220	312	12	s−1fw	s−1fw	ADJ
ma-220	312	13	〉	〉	NOUN
ma-220	312	14	fwdµ(ω	fwdµ(ω	NOUN
ma-220	312	15	)	)	PUNCT
ma-220	312	16	)	)	PUNCT
ma-220	313	1	=	=	SYM
ma-220	313	2	∫	∫	PROPN
ma-220	313	3	ω	ω	NUM
ma-220	313	4	〈	〈	PROPN
ma-220	313	5	x	x	PROPN
ma-220	313	6	,	,	PUNCT
ma-220	313	7	s−1fw	s−1fw	ADJ
ma-220	313	8	〉	〉	NOUN
ma-220	313	9	s−1	s−1	PROPN
ma-220	313	10	p	p	NOUN
ma-220	313	11	pfwdµ(ω	pfwdµ(ω	NOUN
ma-220	313	12	)	)	PUNCT
ma-220	313	13	=	=	SYM
ma-220	314	1	∫	∫	PROPN
ma-220	314	2	ω	ω	NUM
ma-220	314	3	〈	〈	PROPN
ma-220	314	4	x	x	PROPN
ma-220	314	5	,	,	PUNCT
ma-220	314	6	s−1fw	s−1fw	ADJ
ma-220	314	7	〉	〉	NOUN
ma-220	314	8	ps−1fwdµ(ω	ps−1fwdµ(ω	NOUN
ma-220	314	9	)	)	PUNCT
ma-220	315	1	=	=	PUNCT
ma-220	315	2	ps−1	ps−1	PROPN
ma-220	315	3	(	(	PUNCT
ma-220	315	4	∫	∫	PROPN
ma-220	315	5	ω	ω	NUM
ma-220	315	6	〈	〈	PROPN
ma-220	315	7	x	x	PROPN
ma-220	315	8	,	,	PUNCT
ma-220	315	9	s−1fw	s−1fw	ADJ
ma-220	315	10	〉	〉	NOUN
ma-220	315	11	fwdµ(ω	fwdµ(ω	NOUN
ma-220	315	12	)	)	PUNCT
ma-220	315	13	)	)	PUNCT
ma-220	316	1	=	=	SYM
ma-220	316	2	ps−1x	ps−1x	NOUN
ma-220	316	3	.	.	PUNCT
ma-220	317	1	therefore	therefore	ADV
ma-220	317	2	,	,	PUNCT
ma-220	317	3	ps−1x	ps−1x	NOUN
ma-220	317	4	=	=	PUNCT
ma-220	317	5	s−1	s−1	PROPN
ma-220	317	6	p	p	NOUN
ma-220	317	7	px	px	PROPN
ma-220	317	8	,	,	PUNCT
ma-220	317	9	for	for	ADP
ma-220	317	10	all	all	DET
ma-220	317	11	x	x	SYM
ma-220	317	12	∈	∈	PROPN
ma-220	317	13	h	h	NOUN
ma-220	317	14	,	,	PUNCT
ma-220	317	15	and	and	CCONJ
ma-220	317	16	so	so	ADV
ma-220	317	17	s−1	s−1	PROPN
ma-220	317	18	p	p	NOUN
ma-220	317	19	p	p	NOUN
ma-220	317	20	=	=	X
ma-220	317	21	ps−1p	ps−1p	PROPN
ma-220	317	22	⇒	⇒	VERB
ma-220	317	23	ps−1p	ps−1p	PROPN
ma-220	317	24	=	=	SYM
ma-220	317	25	ps−1	ps−1	PROPN
ma-220	317	26	.	.	PUNCT
ma-220	318	1	thus	thus	ADV
ma-220	318	2	ps−1p	ps−1p	NUM
ma-220	318	3	=	=	SYM
ma-220	318	4	(	(	PUNCT
ma-220	318	5	ps−1p	ps−1p	NUM
ma-220	318	6	)	)	PUNCT
ma-220	318	7	∗	∗	NOUN
ma-220	318	8	=	=	SYM
ma-220	318	9	(	(	PUNCT
ma-220	318	10	ps−1	ps−1	PROPN
ma-220	318	11	)	)	PUNCT
ma-220	318	12	∗	∗	NOUN
ma-220	318	13	=	=	PUNCT
ma-220	318	14	s−1p.for	s−1p.for	ADP
ma-220	318	15	the	the	DET
ma-220	318	16	proof	proof	NOUN
ma-220	318	17	of	of	ADP
ma-220	318	18	converse	converse	NOUN
ma-220	318	19	,	,	PUNCT
ma-220	318	20	suppose	suppose	VERB
ma-220	318	21	that	that	SCONJ
ma-220	318	22	ps−1	ps−1	PROPN
ma-220	318	23	=	=	SYM
ma-220	318	24	s−1p	s−1p	PROPN
ma-220	318	25	.	.	PUNCT
ma-220	319	1	let	let	VERB
ma-220	319	2	x	x	PUNCT
ma-220	319	3	∈	∈	PROPN
ma-220	319	4	rp	rp	NOUN
ma-220	319	5	.	.	PUNCT
ma-220	320	1	then	then	ADV
ma-220	320	2	we	we	PRON
ma-220	320	3	have	have	VERB
ma-220	320	4	x	x	NOUN
ma-220	320	5	=	=	SYM
ma-220	320	6	px	px	X
ma-220	320	7	=	=	SYM
ma-220	320	8	p	p	X
ma-220	320	9	(	(	PUNCT
ma-220	320	10	∫	∫	PROPN
ma-220	320	11	ω	ω	NUM
ma-220	320	12	〈	〈	PROPN
ma-220	320	13	x	x	PROPN
ma-220	320	14	,	,	PUNCT
ma-220	320	15	s−1fw	s−1fw	ADJ
ma-220	320	16	〉	〉	NOUN
ma-220	320	17	fwdµ(ω	fwdµ(ω	NOUN
ma-220	320	18	)	)	PUNCT
ma-220	320	19	)	)	PUNCT
ma-220	321	1	=	=	SYM
ma-220	321	2	∫	∫	PROPN
ma-220	321	3	ω	ω	NUM
ma-220	321	4	〈	〈	PROPN
ma-220	321	5	x	x	PROPN
ma-220	321	6	,	,	PUNCT
ma-220	321	7	s−1fw	s−1fw	ADJ
ma-220	321	8	〉	〉	NOUN
ma-220	321	9	pfwdµ(ω	pfwdµ(ω	NOUN
ma-220	321	10	)	)	PUNCT
ma-220	321	11	=	=	SYM
ma-220	322	1	∫	∫	PROPN
ma-220	322	2	ω	ω	NUM
ma-220	322	3	〈	〈	PROPN
ma-220	322	4	px	px	PROPN
ma-220	322	5	,	,	PUNCT
ma-220	322	6	s−1fw	s−1fw	ADJ
ma-220	322	7	〉	〉	NOUN
ma-220	322	8	pfwdµ(ω	pfwdµ(ω	NOUN
ma-220	322	9	)	)	PUNCT
ma-220	322	10	=	=	SYM
ma-220	323	1	∫	∫	PROPN
ma-220	323	2	ω	ω	NUM
ma-220	323	3	〈	〈	PROPN
ma-220	323	4	x	x	SYM
ma-220	323	5	,	,	PUNCT
ma-220	323	6	ps−1fw	ps−1fw	NUM
ma-220	323	7	〉	〉	NOUN
ma-220	323	8	pfwdµ(ω	pfwdµ(ω	NOUN
ma-220	323	9	)	)	PUNCT
ma-220	323	10	=	=	SYM
ma-220	324	1	∫	∫	PROPN
ma-220	324	2	ω	ω	NUM
ma-220	324	3	〈	〈	PROPN
ma-220	324	4	x	x	PROPN
ma-220	324	5	,	,	PUNCT
ma-220	324	6	s−1pfw	s−1pfw	PROPN
ma-220	324	7	〉	〉	NOUN
ma-220	324	8	pfwdµ(ω).by	pfwdµ(ω).by	PROPN
ma-220	324	9	theorem	theorem	VERB
ma-220	324	10	4.2	4.2	NUM
ma-220	324	11	and	and	CCONJ
ma-220	324	12	the	the	DET
ma-220	324	13	assumption	assumption	NOUN
ma-220	324	14	,	,	PUNCT
ma-220	324	15	for	for	ADP
ma-220	324	16	w	w	PROPN
ma-220	324	17	∈	∈	PROPN
ma-220	324	18	ω	ω	PROPN
ma-220	324	19	,	,	PUNCT
ma-220	324	20	s−1pfw	s−1pfw	PROPN
ma-220	324	21	=	=	PUNCT
ma-220	324	22	s−1	s−1	PROPN
ma-220	324	23	p	p	NOUN
ma-220	324	24	fw	fw	PROPN
ma-220	324	25	=	=	SYM
ma-220	324	26	s−1	s−1	PROPN
ma-220	324	27	p	p	NOUN
ma-220	324	28	pfw	pfw	NOUN
ma-220	324	29	=	=	SYM
ma-220	324	30	ps−1fw	ps−1fw	NUM
ma-220	324	31	and	and	CCONJ
ma-220	324	32	the	the	DET
ma-220	324	33	proof	proof	NOUN
ma-220	324	34	is	be	AUX
ma-220	324	35	complete	complete	ADJ
ma-220	324	36	.	.	PUNCT
ma-220	325	1	�	�	PROPN
ma-220	325	2	theorem	theorem	VERB
ma-220	325	3	4.5	4.5	NUM
ma-220	325	4	.	.	PUNCT
ma-220	326	1	let	let	VERB
ma-220	326	2	{	{	PUNCT
ma-220	326	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	326	4	be	be	AUX
ma-220	326	5	a	a	DET
ma-220	326	6	continuous	continuous	ADJ
ma-220	326	7	parseval	parseval	NOUN
ma-220	326	8	frame	frame	NOUN
ma-220	326	9	of	of	ADP
ma-220	326	10	h	h	NOUN
ma-220	326	11	with	with	ADP
ma-220	326	12	pre	pre	ADJ
ma-220	326	13	-	-	ADJ
ma-220	326	14	frame	frame	ADJ
ma-220	326	15	operator	operator	NOUN
ma-220	326	16	θf	θf	NOUN
ma-220	326	17	and	and	CCONJ
ma-220	326	18	let	let	VERB
ma-220	326	19	{	{	PUNCT
ma-220	326	20	gw}w∈ω	gw}w∈ω	NOUN
ma-220	326	21	be	be	AUX
ma-220	326	22	a	a	DET
ma-220	326	23	∗-continuous	∗-continuous	ADJ
ma-220	326	24	frame	frame	NOUN
ma-220	326	25	with	with	ADP
ma-220	326	26	the	the	DET
ma-220	326	27	pre	pre	ADJ
ma-220	326	28	-	-	ADJ
ma-220	326	29	frame	frame	ADJ
ma-220	326	30	operator	operator	NOUN
ma-220	326	31	θg	θg	PRON
ma-220	326	32	.	.	PUNCT
ma-220	327	1	then	then	ADV
ma-220	327	2	(	(	PUNCT
ma-220	327	3	{	{	PUNCT
ma-220	327	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	327	5	,	,	PUNCT
ma-220	327	6	γ	γ	PROPN
ma-220	327	7	)	)	PUNCT
ma-220	327	8	is	be	AUX
ma-220	327	9	an	an	DET
ma-220	327	10	operator	operator	NOUN
ma-220	327	11	dual	dual	ADJ
ma-220	327	12	for	for	ADP
ma-220	327	13	{	{	PUNCT
ma-220	327	14	fw}w∈ω	fw}w∈ω	NOUN
ma-220	327	15	if	if	SCONJ
ma-220	328	1	and	and	CCONJ
ma-220	328	2	only	only	ADV
ma-220	328	3	if	if	SCONJ
ma-220	328	4	pθf	pθf	PROPN
ma-220	328	5	θgγ	θgγ	NOUN
ma-220	328	6	=	=	SYM
ma-220	328	7	θf	θf	NOUN
ma-220	328	8	,	,	PUNCT
ma-220	328	9	where	where	SCONJ
ma-220	328	10	pθf	pθf	PROPN
ma-220	328	11	is	be	AUX
ma-220	328	12	the	the	DET
ma-220	328	13	orthogonal	orthogonal	ADJ
ma-220	328	14	projection	projection	NOUN
ma-220	328	15	on	on	ADP
ma-220	328	16	the	the	DET
ma-220	328	17	range	range	NOUN
ma-220	328	18	of	of	ADP
ma-220	328	19	θf	θf	INTJ
ma-220	328	20	.	.	PUNCT
ma-220	329	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	329	2	eur	eur	PROPN
ma-220	329	3	.	.	PUNCT
ma-220	330	1	j.	j.	PROPN
ma-220	330	2	math	math	PROPN
ma-220	330	3	.	.	PUNCT
ma-220	331	1	anal	anal	PROPN
ma-220	331	2	.	.	PUNCT
ma-220	332	1	10.28924	10.28924	NUM
ma-220	332	2	/	/	SYM
ma-220	332	3	ada	ada	PROPN
ma-220	332	4	/	/	SYM
ma-220	332	5	ma.4.4	ma.4.4	PROPN
ma-220	332	6	14	14	NUM
ma-220	332	7	proof	proof	NOUN
ma-220	332	8	.	.	PUNCT
ma-220	332	9	suppose	suppose	VERB
ma-220	332	10	that	that	SCONJ
ma-220	332	11	(	(	PUNCT
ma-220	332	12	{	{	PUNCT
ma-220	332	13	gw}w∈ω	gw}w∈ω	NOUN
ma-220	332	14	,	,	PUNCT
ma-220	332	15	γ	γ	PROPN
ma-220	332	16	)	)	PUNCT
ma-220	332	17	is	be	AUX
ma-220	332	18	an	an	DET
ma-220	332	19	operator	operator	NOUN
ma-220	332	20	dual	dual	ADJ
ma-220	332	21	for	for	ADP
ma-220	332	22	{	{	PUNCT
ma-220	332	23	fw}w∈ω	fw}w∈ω	NOUN
ma-220	332	24	.	.	PUNCT
ma-220	333	1	since	since	SCONJ
ma-220	333	2	{	{	PUNCT
ma-220	333	3	fw}w∈ω	fw}w∈ω	PROPN
ma-220	333	4	is	be	AUX
ma-220	333	5	a	a	DET
ma-220	333	6	contin	contin	NOUN
ma-220	333	7	-	-	PUNCT
ma-220	333	8	uous	uous	ADJ
ma-220	333	9	parseval	parseval	NOUN
ma-220	333	10	frame	frame	NOUN
ma-220	333	11	,	,	PUNCT
ma-220	333	12	the	the	DET
ma-220	333	13	pre	pre	ADJ
ma-220	333	14	-	-	ADJ
ma-220	333	15	frame	frame	ADJ
ma-220	333	16	operator	operator	NOUN
ma-220	333	17	θf	θf	VERB
ma-220	333	18	is	be	AUX
ma-220	333	19	an	an	DET
ma-220	333	20	isometry	isometry	NOUN
ma-220	333	21	〈	〈	PROPN
ma-220	333	22	θfx	θfx	NOUN
ma-220	333	23	,	,	PUNCT
ma-220	333	24	θfx	θfx	ADJ
ma-220	333	25	〉	〉	NUM
ma-220	333	26	=	=	SYM
ma-220	333	27	∫	∫	PROPN
ma-220	333	28	ω	ω	NUM
ma-220	333	29	〈	〈	PROPN
ma-220	333	30	x	x	PRON
ma-220	333	31	,	,	PUNCT
ma-220	333	32	fw	fw	PROPN
ma-220	333	33	〉	〉	PROPN
ma-220	333	34	〈	〈	PROPN
ma-220	333	35	fw	fw	PROPN
ma-220	333	36	,	,	PUNCT
ma-220	333	37	x	x	PROPN
ma-220	333	38	〉	〉	NOUN
ma-220	333	39	dµ(ω	dµ(ω	PUNCT
ma-220	333	40	)	)	PUNCT
ma-220	334	1	=	=	SYM
ma-220	334	2	〈	〈	PROPN
ma-220	334	3	x	x	X
ma-220	334	4	,	,	PUNCT
ma-220	334	5	x	x	PROPN
ma-220	334	6	〉	〉	NOUN
ma-220	334	7	,	,	PUNCT
ma-220	334	8	∀x	∀x	X
ma-220	334	9	∈	∈	PROPN
ma-220	334	10	h.	h.	NOUN
ma-220	334	11	then	then	ADV
ma-220	334	12	for	for	ADP
ma-220	334	13	x	x	PROPN
ma-220	334	14	∈	∈	PROPN
ma-220	334	15	h	h	NOUN
ma-220	334	16	,	,	PUNCT
ma-220	334	17	〈	〈	PROPN
ma-220	334	18	pθf	pθf	NOUN
ma-220	334	19	θgγx	θgγx	NOUN
ma-220	334	20	,	,	PUNCT
ma-220	334	21	θfx	θfx	ADJ
ma-220	334	22	〉	〉	NOUN
ma-220	334	23	=	=	SYM
ma-220	334	24	〈	〈	PROPN
ma-220	334	25	θgγx	θgγx	NOUN
ma-220	334	26	,	,	PUNCT
ma-220	334	27	pθf	pθf	X
ma-220	334	28	θfx	θfx	ADJ
ma-220	334	29	〉	〉	NOUN
ma-220	334	30	=	=	SYM
ma-220	334	31	〈	〈	PROPN
ma-220	334	32	θgγf	θgγf	NOUN
ma-220	334	33	,	,	PUNCT
ma-220	334	34	θfx	θfx	ADJ
ma-220	334	35	〉	〉	NOUN
ma-220	334	36	=	=	SYM
ma-220	334	37	〈	〈	PROPN
ma-220	334	38	∫	∫	PROPN
ma-220	334	39	ω	ω	NUM
ma-220	334	40	〈	〈	PROPN
ma-220	334	41	γx	γx	NOUN
ma-220	334	42	,	,	PUNCT
ma-220	334	43	gw	gw	PROPN
ma-220	334	44	〉	〉	PROPN
ma-220	334	45	fwdµ(ω	fwdµ(ω	NOUN
ma-220	334	46	)	)	PUNCT
ma-220	334	47	,	,	PUNCT
ma-220	334	48	x	x	SYM
ma-220	334	49	〉	〉	NOUN
ma-220	334	50	=	=	SYM
ma-220	334	51	〈	〈	PROPN
ma-220	334	52	x	x	X
ma-220	334	53	,	,	PUNCT
ma-220	334	54	x	x	NOUN
ma-220	334	55	〉	〉	NOUN
ma-220	334	56	=	=	SYM
ma-220	334	57	〈	〈	PROPN
ma-220	334	58	θfx	θfx	NOUN
ma-220	334	59	,	,	PUNCT
ma-220	334	60	θfx	θfx	NOUN
ma-220	334	61	〉	〉	NOUN
ma-220	334	62	.thus	.thus	PUNCT
ma-220	334	63	pθf	pθf	PROPN
ma-220	334	64	θgγ	θgγ	NOUN
ma-220	334	65	=	=	PUNCT
ma-220	334	66	θf	θf	NOUN
ma-220	334	67	.conversely	.conversely	ADV
ma-220	334	68	,	,	PUNCT
ma-220	334	69	since	since	SCONJ
ma-220	334	70	θf	θf	ADV
ma-220	334	71	is	be	VERB
ma-220	334	72	an	an	DET
ma-220	334	73	isometry	isometry	NOUN
ma-220	334	74	,	,	PUNCT
ma-220	334	75	by	by	ADP
ma-220	334	76	a	a	DET
ma-220	334	77	similar	similar	ADJ
ma-220	334	78	method	method	NOUN
ma-220	334	79	,	,	PUNCT
ma-220	334	80	we	we	PRON
ma-220	334	81	have	have	VERB
ma-220	334	82	〈	〈	PROPN
ma-220	334	83	x	x	X
ma-220	334	84	,	,	PUNCT
ma-220	334	85	g	g	PROPN
ma-220	334	86	〉	〉	NOUN
ma-220	334	87	=	=	SYM
ma-220	334	88	〈	〈	PROPN
ma-220	334	89	θfx	θfx	NOUN
ma-220	334	90	,	,	PUNCT
ma-220	334	91	θfg	θfg	PROPN
ma-220	334	92	〉	〉	PROPN
ma-220	334	93	=	=	SYM
ma-220	334	94	〈	〈	PROPN
ma-220	334	95	pθf	pθf	NOUN
ma-220	334	96	θgγx	θgγx	NOUN
ma-220	334	97	,	,	PUNCT
ma-220	334	98	θfg	θfg	PROPN
ma-220	334	99	〉	〉	NOUN
ma-220	334	100	=	=	SYM
ma-220	334	101	〈	〈	PROPN
ma-220	334	102	∫	∫	PROPN
ma-220	334	103	ω	ω	NUM
ma-220	334	104	〈	〈	PROPN
ma-220	334	105	γx	γx	NOUN
ma-220	334	106	,	,	PUNCT
ma-220	334	107	gw	gw	PROPN
ma-220	334	108	〉	〉	PROPN
ma-220	334	109	fwdµ(ω	fwdµ(ω	NOUN
ma-220	334	110	)	)	PUNCT
ma-220	334	111	,	,	PUNCT
ma-220	334	112	g	g	PROPN
ma-220	334	113	〉	〉	NOUN
ma-220	334	114	,	,	PUNCT
ma-220	334	115	∀x	∀x	X
ma-220	334	116	∈	∈	PROPN
ma-220	334	117	h	h	NOUN
ma-220	334	118	,	,	PUNCT
ma-220	334	119	and	and	CCONJ
ma-220	334	120	so	so	ADV
ma-220	334	121	x	x	X
ma-220	334	122	=	=	SYM
ma-220	334	123	∫	∫	PROPN
ma-220	334	124	ω	ω	X
ma-220	335	1	〈	〈	PROPN
ma-220	335	2	γx	γx	NOUN
ma-220	335	3	,	,	PUNCT
ma-220	335	4	gw	gw	PROPN
ma-220	335	5	〉	〉	PROPN
ma-220	335	6	fwdµ(ω	fwdµ(ω	NOUN
ma-220	335	7	)	)	PUNCT
ma-220	335	8	,	,	PUNCT
ma-220	335	9	∀x	∀x	PROPN
ma-220	335	10	∈	∈	PROPN
ma-220	335	11	h.	h.	NOUN
ma-220	335	12	thus	thus	ADV
ma-220	335	13	(	(	PUNCT
ma-220	335	14	{	{	PUNCT
ma-220	335	15	gw}w∈ω	gw}w∈ω	NOUN
ma-220	335	16	,	,	PUNCT
ma-220	335	17	γ	γ	PROPN
ma-220	335	18	)	)	PUNCT
ma-220	335	19	is	be	AUX
ma-220	335	20	a	a	DET
ma-220	335	21	dual	dual	ADJ
ma-220	335	22	for	for	ADP
ma-220	335	23	{	{	PUNCT
ma-220	335	24	fw}w∈ω	fw}w∈ω	PROPN
ma-220	335	25	.	.	PROPN
ma-220	335	26	�	�	PROPN
ma-220	335	27	corollary	corollary	NOUN
ma-220	335	28	4.6	4.6	NUM
ma-220	335	29	.	.	PUNCT
ma-220	336	1	let	let	VERB
ma-220	336	2	{	{	PUNCT
ma-220	336	3	fw}w∈ω	fw}w∈ω	NOUN
ma-220	336	4	and	and	CCONJ
ma-220	336	5	{	{	PUNCT
ma-220	336	6	gw}w∈ω	gw}w∈ω	NOUN
ma-220	336	7	be	be	AUX
ma-220	336	8	two	two	NUM
ma-220	336	9	∗-continuous	∗-continuous	ADJ
ma-220	336	10	frames	frame	NOUN
ma-220	336	11	for	for	ADP
ma-220	336	12	h	h	NOUN
ma-220	336	13	with	with	ADP
ma-220	336	14	pre	pre	ADJ
ma-220	336	15	-	-	ADJ
ma-220	336	16	frame	frame	ADJ
ma-220	336	17	operators	operator	NOUN
ma-220	336	18	θf	θf	VERB
ma-220	336	19	and	and	CCONJ
ma-220	336	20	θg	θg	PRON
ma-220	336	21	,	,	PUNCT
ma-220	336	22	respectively	respectively	ADV
ma-220	336	23	.	.	PUNCT
ma-220	337	1	then	then	ADV
ma-220	337	2	(	(	PUNCT
ma-220	337	3	{	{	PUNCT
ma-220	337	4	gw}w∈ω	gw}w∈ω	NOUN
ma-220	337	5	,	,	PUNCT
ma-220	337	6	γ	γ	PROPN
ma-220	337	7	)	)	PUNCT
ma-220	337	8	is	be	AUX
ma-220	337	9	an	an	DET
ma-220	337	10	operator	operator	NOUN
ma-220	337	11	dual	dual	ADJ
ma-220	337	12	for	for	ADP
ma-220	337	13	{	{	PUNCT
ma-220	337	14	fw}w∈ω	fw}w∈ω	NOUN
ma-220	337	15	if	if	SCONJ
ma-220	338	1	and	and	CCONJ
ma-220	338	2	only	only	ADV
ma-220	338	3	if	if	SCONJ
ma-220	338	4	θ∗fpθf	θ∗fpθf	ADV
ma-220	338	5	θgγ	θgγ	NOUN
ma-220	339	1	=	=	PUNCT
ma-220	339	2	i	i	PROPN
ma-220	339	3	d	d	PROPN
ma-220	339	4	.	.	PUNCT
ma-220	340	1	authors	author	NOUN
ma-220	340	2	’	'	PUNCT
ma-220	340	3	contributionsthe	contributionsthe	DET
ma-220	340	4	authors	author	NOUN
ma-220	340	5	equally	equally	ADV
ma-220	340	6	conceived	conceive	VERB
ma-220	340	7	of	of	ADP
ma-220	340	8	the	the	DET
ma-220	340	9	study	study	NOUN
ma-220	340	10	,	,	PUNCT
ma-220	340	11	participated	participate	VERB
ma-220	340	12	in	in	ADP
ma-220	340	13	its	its	PRON
ma-220	340	14	design	design	NOUN
ma-220	340	15	and	and	CCONJ
ma-220	340	16	coordination	coordination	NOUN
ma-220	340	17	,	,	PUNCT
ma-220	340	18	drafted	draft	VERB
ma-220	340	19	themanuscript	themanuscript	NOUN
ma-220	340	20	,	,	PUNCT
ma-220	340	21	participated	participate	VERB
ma-220	340	22	in	in	ADP
ma-220	340	23	the	the	DET
ma-220	340	24	sequence	sequence	NOUN
ma-220	340	25	alignment	alignment	NOUN
ma-220	340	26	,	,	PUNCT
ma-220	340	27	and	and	CCONJ
ma-220	340	28	read	read	VERB
ma-220	340	29	and	and	CCONJ
ma-220	340	30	approved	approve	VERB
ma-220	340	31	the	the	DET
ma-220	340	32	final	final	ADJ
ma-220	340	33	manuscript	manuscript	NOUN
ma-220	340	34	.	.	PUNCT
ma-220	341	1	references	reference	NOUN
ma-220	341	2	[	[	X
ma-220	341	3	1	1	NUM
ma-220	341	4	]	]	PUNCT
ma-220	341	5	a.	a.	NOUN
ma-220	341	6	alijani	alijani	PROPN
ma-220	341	7	,	,	PUNCT
ma-220	341	8	m.a	m.a	PROPN
ma-220	341	9	.	.	PROPN
ma-220	341	10	dehghan	dehghan	PROPN
ma-220	341	11	,	,	PUNCT
ma-220	341	12	g	g	NOUN
ma-220	341	13	-	-	PUNCT
ma-220	341	14	frames	frame	NOUN
ma-220	341	15	and	and	CCONJ
ma-220	341	16	their	their	PRON
ma-220	341	17	duals	dual	NOUN
ma-220	341	18	in	in	ADP
ma-220	341	19	hilbert	hilbert	PROPN
ma-220	341	20	c∗-modules	c∗-modules	PROPN
ma-220	341	21	,	,	PUNCT
ma-220	341	22	bull	bull	NOUN
ma-220	341	23	.	.	PUNCT
ma-220	342	1	iran	iran	PROPN
ma-220	342	2	.	.	PUNCT
ma-220	343	1	math	math	NOUN
ma-220	343	2	.	.	PUNCT
ma-220	344	1	soc	soc	PROPN
ma-220	344	2	.	.	PUNCT
ma-220	345	1	38	38	NUM
ma-220	345	2	(	(	PUNCT
ma-220	345	3	2012),567–580.[2	2012),567–580.[2	NUM
ma-220	345	4	]	]	X
ma-220	345	5	i.	i.	PROPN
ma-220	345	6	daubechies	daubechies	PROPN
ma-220	345	7	,	,	PUNCT
ma-220	345	8	a.	a.	NOUN
ma-220	345	9	grossmann	grossmann	PROPN
ma-220	345	10	,	,	PUNCT
ma-220	345	11	y.	y.	PROPN
ma-220	345	12	meyer	meyer	PROPN
ma-220	345	13	,	,	PUNCT
ma-220	345	14	painless	painless	ADJ
ma-220	345	15	nonorthogonal	nonorthogonal	ADJ
ma-220	345	16	expansions	expansion	NOUN
ma-220	345	17	,	,	PUNCT
ma-220	345	18	j.	j.	PROPN
ma-220	345	19	math	math	PROPN
ma-220	345	20	.	.	PUNCT
ma-220	346	1	phys	phy	NOUN
ma-220	346	2	.	.	PUNCT
ma-220	347	1	27	27	NUM
ma-220	347	2	(	(	PUNCT
ma-220	347	3	1986	1986	NUM
ma-220	347	4	)	)	PUNCT
ma-220	347	5	,	,	PUNCT
ma-220	347	6	1271–1283.[3	1271–1283.[3	NUM
ma-220	347	7	]	]	X
ma-220	347	8	r.	r.	PROPN
ma-220	347	9	j.	j.	PROPN
ma-220	347	10	duffin	duffin	PROPN
ma-220	347	11	,	,	PUNCT
ma-220	347	12	a.	a.	PROPN
ma-220	347	13	c.	c.	PROPN
ma-220	347	14	schaeffer	schaeffer	PROPN
ma-220	347	15	,	,	PUNCT
ma-220	347	16	a	a	DET
ma-220	347	17	class	class	NOUN
ma-220	347	18	of	of	ADP
ma-220	347	19	nonharmonic	nonharmonic	ADJ
ma-220	347	20	fourier	fourier	NOUN
ma-220	347	21	series	series	NOUN
ma-220	347	22	,	,	PUNCT
ma-220	347	23	trans	trans	PROPN
ma-220	347	24	.	.	PROPN
ma-220	347	25	amer	amer	PROPN
ma-220	347	26	.	.	PUNCT
ma-220	347	27	math	math	PROPN
ma-220	347	28	.	.	PUNCT
ma-220	348	1	soc	soc	PROPN
ma-220	348	2	.	.	PUNCT
ma-220	349	1	72	72	NUM
ma-220	349	2	(	(	PUNCT
ma-220	349	3	1952	1952	NUM
ma-220	349	4	)	)	PUNCT
ma-220	349	5	,	,	PUNCT
ma-220	349	6	341–366.[4	341–366.[4	NUM
ma-220	349	7	]	]	X
ma-220	349	8	d.	d.	PROPN
ma-220	349	9	gabor	gabor	PROPN
ma-220	349	10	,	,	PUNCT
ma-220	349	11	theory	theory	NOUN
ma-220	349	12	of	of	ADP
ma-220	349	13	communications	communication	NOUN
ma-220	349	14	,	,	PUNCT
ma-220	349	15	j.	j.	PROPN
ma-220	349	16	inst	inst	PROPN
ma-220	349	17	.	.	PUNCT
ma-220	350	1	electr	electr	PROPN
ma-220	350	2	.	.	PUNCT
ma-220	351	1	eng	eng	PROPN
ma-220	351	2	.	.	PROPN
ma-220	352	1	93	93	NUM
ma-220	352	2	(	(	PUNCT
ma-220	352	3	1946	1946	NUM
ma-220	352	4	)	)	PUNCT
ma-220	352	5	,	,	PUNCT
ma-220	352	6	429–457.[5	429–457.[5	NUM
ma-220	352	7	]	]	X
ma-220	352	8	f.	f.	PROPN
ma-220	352	9	d.	d.	PROPN
ma-220	352	10	nhari	nhari	PROPN
ma-220	352	11	,	,	PUNCT
ma-220	352	12	r.	r.	PROPN
ma-220	352	13	echarghaoui	echarghaoui	PROPN
ma-220	352	14	,	,	PUNCT
ma-220	352	15	m.	m.	NOUN
ma-220	352	16	rossafi	rossafi	PROPN
ma-220	352	17	,	,	PUNCT
ma-220	352	18	k	k	NOUN
ma-220	352	19	-	-	PUNCT
ma-220	352	20	g	g	NOUN
ma-220	352	21	-	-	PUNCT
ma-220	352	22	fusion	fusion	NOUN
ma-220	352	23	frames	frame	NOUN
ma-220	352	24	in	in	ADP
ma-220	352	25	hilbert	hilbert	PROPN
ma-220	352	26	c∗-modules	c∗-modules	PROPN
ma-220	352	27	,	,	PUNCT
ma-220	352	28	int	int	NOUN
ma-220	352	29	.	.	PUNCT
ma-220	353	1	j.	j.	PROPN
ma-220	353	2	anal	anal	PROPN
ma-220	353	3	.	.	PUNCT
ma-220	354	1	appl	appl	PROPN
ma-220	354	2	.	.	PROPN
ma-220	355	1	19	19	NUM
ma-220	355	2	(	(	PUNCT
ma-220	355	3	2021),836–857.[6	2021),836–857.[6	NUM
ma-220	355	4	]	]	X
ma-220	355	5	w.	w.	PROPN
ma-220	355	6	paschke	paschke	PROPN
ma-220	355	7	,	,	PUNCT
ma-220	355	8	inner	inner	ADJ
ma-220	355	9	product	product	NOUN
ma-220	355	10	modules	module	NOUN
ma-220	355	11	over	over	ADP
ma-220	355	12	b∗-algebras	b∗-algebra	NOUN
ma-220	355	13	,	,	PUNCT
ma-220	355	14	trans	tran	NOUN
ma-220	355	15	.	.	PUNCT
ma-220	355	16	am	be	AUX
ma-220	355	17	.	.	PUNCT
ma-220	356	1	math	math	NOUN
ma-220	356	2	.	.	PUNCT
ma-220	357	1	soc	soc	PROPN
ma-220	357	2	.	.	PUNCT
ma-220	358	1	182	182	NUM
ma-220	358	2	(	(	PUNCT
ma-220	358	3	1973	1973	NUM
ma-220	358	4	)	)	PUNCT
ma-220	358	5	,	,	PUNCT
ma-220	358	6	443–468.[7	443–468.[7	NUM
ma-220	358	7	]	]	X
ma-220	358	8	m.	m.	NOUN
ma-220	358	9	rossafi	rossafi	PROPN
ma-220	358	10	,	,	PUNCT
ma-220	358	11	s.	s.	PROPN
ma-220	358	12	kabbaj	kabbaj	PROPN
ma-220	358	13	,	,	PUNCT
ma-220	358	14	∗-k	∗-k	ADJ
ma-220	358	15	-	-	PUNCT
ma-220	358	16	g	g	NOUN
ma-220	358	17	-	-	PUNCT
ma-220	358	18	frames	frame	NOUN
ma-220	358	19	in	in	ADP
ma-220	358	20	hilbert	hilbert	PROPN
ma-220	358	21	a	a	PROPN
ma-220	358	22	-	-	PUNCT
ma-220	358	23	modules	module	NOUN
ma-220	358	24	,	,	PUNCT
ma-220	358	25	j.	j.	PROPN
ma-220	358	26	linear	linear	PROPN
ma-220	358	27	topol	topol	PROPN
ma-220	358	28	.	.	PUNCT
ma-220	359	1	algebra	algebra	NOUN
ma-220	359	2	7	7	NUM
ma-220	359	3	(	(	PUNCT
ma-220	359	4	2018	2018	NUM
ma-220	359	5	)	)	PUNCT
ma-220	359	6	,	,	PUNCT
ma-220	359	7	63–71	63–71	NOUN
ma-220	359	8	.	.	PUNCT
ma-220	360	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	360	2	eur	eur	PROPN
ma-220	360	3	.	.	PUNCT
ma-220	361	1	j.	j.	PROPN
ma-220	361	2	math	math	PROPN
ma-220	361	3	.	.	PUNCT
ma-220	362	1	anal	anal	PROPN
ma-220	362	2	.	.	PUNCT
ma-220	363	1	10.28924	10.28924	NUM
ma-220	363	2	/	/	SYM
ma-220	363	3	ada	ada	PROPN
ma-220	363	4	/	/	SYM
ma-220	363	5	ma.4.4	ma.4.4	PROPN
ma-220	363	6	15	15	NUM
ma-220	363	7	[	[	SYM
ma-220	363	8	8	8	NUM
ma-220	363	9	]	]	PUNCT
ma-220	363	10	m.	m.	NOUN
ma-220	363	11	rossafi	rossafi	PROPN
ma-220	363	12	,	,	PUNCT
ma-220	363	13	s.	s.	PROPN
ma-220	363	14	kabbaj	kabbaj	PROPN
ma-220	363	15	,	,	PUNCT
ma-220	363	16	∗-g	∗-g	NOUN
ma-220	363	17	-	-	PUNCT
ma-220	363	18	frames	frame	NOUN
ma-220	363	19	in	in	ADP
ma-220	363	20	tensor	tensor	NOUN
ma-220	363	21	products	product	NOUN
ma-220	363	22	of	of	ADP
ma-220	363	23	hilbert	hilbert	PROPN
ma-220	363	24	c∗-modules	c∗-modules	PROPN
ma-220	363	25	,	,	PUNCT
ma-220	363	26	ann	ann	PROPN
ma-220	363	27	.	.	PROPN
ma-220	363	28	univ	univ	PROPN
ma-220	363	29	.	.	PUNCT
ma-220	364	1	paedagog	paedagog	PROPN
ma-220	364	2	.	.	PUNCT
ma-220	365	1	crac	crac	PROPN
ma-220	365	2	.	.	PROPN
ma-220	365	3	stud	stud	PROPN
ma-220	365	4	.	.	PUNCT
ma-220	366	1	math.17	math.17	PROPN
ma-220	366	2	(	(	PUNCT
ma-220	366	3	2018	2018	NUM
ma-220	366	4	)	)	PUNCT
ma-220	366	5	,	,	PUNCT
ma-220	366	6	17–25.[9	17–25.[9	NUM
ma-220	366	7	]	]	X
ma-220	366	8	m.	m.	NOUN
ma-220	366	9	rossafi	rossafi	PROPN
ma-220	366	10	,	,	PUNCT
ma-220	366	11	s.	s.	PROPN
ma-220	366	12	kabbaj	kabbaj	PROPN
ma-220	366	13	,	,	PUNCT
ma-220	366	14	operator	operator	NOUN
ma-220	366	15	frame	frame	NOUN
ma-220	366	16	for	for	ADP
ma-220	366	17	end∗a(h	end∗a(h	NOUN
ma-220	366	18	)	)	PUNCT
ma-220	366	19	,	,	PUNCT
ma-220	366	20	j.	j.	PROPN
ma-220	366	21	linear	linear	PROPN
ma-220	366	22	topol	topol	PROPN
ma-220	366	23	.	.	PUNCT
ma-220	367	1	algebra	algebra	NOUN
ma-220	367	2	8	8	NUM
ma-220	367	3	(	(	PUNCT
ma-220	367	4	2019	2019	NUM
ma-220	367	5	)	)	PUNCT
ma-220	367	6	,	,	PUNCT
ma-220	367	7	85–95.[10	85–95.[10	NUM
ma-220	367	8	]	]	PUNCT
ma-220	367	9	m.	m.	NOUN
ma-220	367	10	rossafi	rossafi	PROPN
ma-220	367	11	,	,	PUNCT
ma-220	367	12	s.	s.	PROPN
ma-220	367	13	kabbaj	kabbaj	PROPN
ma-220	367	14	,	,	PUNCT
ma-220	367	15	∗-k	∗-k	NOUN
ma-220	367	16	-	-	PUNCT
ma-220	367	17	operator	operator	NOUN
ma-220	367	18	frame	frame	NOUN
ma-220	367	19	for	for	ADP
ma-220	367	20	end∗a(h	end∗a(h	NOUN
ma-220	367	21	)	)	PUNCT
ma-220	367	22	,	,	PUNCT
ma-220	367	23	asian	asian	ADJ
ma-220	367	24	-	-	PUNCT
ma-220	367	25	eur	eur	NOUN
ma-220	367	26	.	.	PUNCT
ma-220	368	1	j.	j.	PROPN
ma-220	368	2	math	math	PROPN
ma-220	368	3	.	.	PUNCT
ma-220	369	1	13	13	NUM
ma-220	369	2	(	(	PUNCT
ma-220	369	3	2020	2020	NUM
ma-220	369	4	)	)	PUNCT
ma-220	369	5	,	,	PUNCT
ma-220	369	6	2050060.[11	2050060.[11	NUM
ma-220	369	7	]	]	PUNCT
ma-220	369	8	m.	m.	NOUN
ma-220	369	9	rossafi	rossafi	PROPN
ma-220	369	10	,	,	PUNCT
ma-220	369	11	f.	f.	PROPN
ma-220	369	12	d.	d.	PROPN
ma-220	369	13	nhari	nhari	PROPN
ma-220	369	14	,	,	PUNCT
ma-220	369	15	c.	c.	PROPN
ma-220	369	16	park	park	PROPN
ma-220	369	17	,	,	PUNCT
ma-220	369	18	s.	s.	PROPN
ma-220	369	19	kabbaj	kabbaj	PROPN
ma-220	369	20	,	,	PUNCT
ma-220	369	21	continuous	continuous	ADJ
ma-220	369	22	g	g	NOUN
ma-220	369	23	-	-	PUNCT
ma-220	369	24	frames	frame	NOUN
ma-220	369	25	with	with	ADP
ma-220	369	26	c∗-valued	c∗-value	VERB
ma-220	369	27	bounds	bound	NOUN
ma-220	369	28	and	and	CCONJ
ma-220	369	29	their	their	PRON
ma-220	369	30	properties	property	NOUN
ma-220	369	31	,	,	PUNCT
ma-220	369	32	complex	complex	ADJ
ma-220	369	33	anal	anal	NOUN
ma-220	369	34	.	.	PUNCT
ma-220	370	1	oper	oper	PROPN
ma-220	370	2	.	.	PROPN
ma-220	370	3	theory	theory	NOUN
ma-220	370	4	16	16	NUM
ma-220	370	5	(	(	PUNCT
ma-220	370	6	2022	2022	NUM
ma-220	370	7	)	)	PUNCT
ma-220	370	8	,	,	PUNCT
ma-220	370	9	44.[12	44.[12	NUM
ma-220	370	10	]	]	X
ma-220	370	11	k.	k.	PROPN
ma-220	370	12	yosida	yosida	PROPN
ma-220	370	13	,	,	PUNCT
ma-220	370	14	functional	functional	ADJ
ma-220	370	15	analysis	analysis	NOUN
ma-220	370	16	,	,	PUNCT
ma-220	370	17	vol	vol	NOUN
ma-220	370	18	.	.	NOUN
ma-220	370	19	123	123	NUM
ma-220	370	20	,	,	PUNCT
ma-220	370	21	grundlehren	grundlehren	PROPN
ma-220	370	22	der	der	PROPN
ma-220	370	23	mathematischen	mathematischen	PROPN
ma-220	370	24	wissenschaften	wissenschaften	NOUN
ma-220	370	25	,	,	PUNCT
ma-220	370	26	springer	springer	NOUN
ma-220	370	27	,	,	PUNCT
ma-220	370	28	berlin	berlin	PROPN
ma-220	370	29	andnew	andnew	PROPN
ma-220	370	30	york	york	PROPN
ma-220	370	31	,	,	PUNCT
ma-220	370	32	1980	1980	NUM
ma-220	370	33	.	.	PUNCT
ma-220	371	1	https://doi.org/10.28924/ada/ma.4.4	https://doi.org/10.28924/ada/ma.4.4	PROPN
ma-220	371	2	1	1	NUM
ma-220	371	3	.	.	PUNCT
ma-220	371	4	introduction	introduction	NOUN
ma-220	371	5	2	2	NUM
ma-220	371	6	.	.	PUNCT
ma-220	371	7	-continuous	-continuous	ADJ
ma-220	371	8	operator	operator	NOUN
ma-220	371	9	duals	dual	NOUN
ma-220	371	10	3	3	NUM
ma-220	371	11	.	.	PUNCT
ma-220	372	1	equivalent	equivalent	ADJ
ma-220	372	2	*	*	PUNCT
ma-220	372	3	-continuous	-continuous	ADJ
ma-220	372	4	frames	frame	NOUN
ma-220	372	5	4	4	NUM
ma-220	372	6	.	.	PUNCT
ma-220	372	7	constructed	construct	VERB
ma-220	372	8	-continuous	-continuous	ADJ
ma-220	372	9	frames	frame	NOUN
ma-220	372	10	and	and	CCONJ
ma-220	372	11	some	some	DET
ma-220	372	12	properties	property	NOUN
ma-220	372	13	references	reference	NOUN