id	sid	tid	token	lemma	pos
cana-3791	1	1	communications	communication	NOUN
cana-3791	1	2	on	on	ADP
cana-3791	1	3	applied	apply	VERB
cana-3791	1	4	nonlinear	nonlinear	ADJ
cana-3791	1	5	analysis	analysis	NOUN
cana-3791	1	6	issn	issn	NOUN
cana-3791	1	7	:	:	PUNCT
cana-3791	1	8	1074	1074	NUM
cana-3791	1	9	-	-	PUNCT
cana-3791	1	10	133x	133x	NUM
cana-3791	1	11	vol	vol	NOUN
cana-3791	1	12	32	32	NUM
cana-3791	1	13	no	no	NOUN
cana-3791	1	14	.	.	PUNCT
cana-3791	2	1	8s	8s	PROPN
cana-3791	2	2	(	(	PUNCT
cana-3791	2	3	2025	2025	NUM
cana-3791	2	4	)	)	PUNCT
cana-3791	2	5	732	732	NUM
cana-3791	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	2	7	finite	finite	ADJ
cana-3791	2	8	-	-	ADJ
cana-3791	2	9	dimensional	dimensional	ADJ
cana-3791	2	10	hilbert	hilbert	NOUN
cana-3791	2	11	spaces	space	NOUN
cana-3791	2	12	:	:	PUNCT
cana-3791	2	13	optimal	optimal	ADJ
cana-3791	2	14	basis	basis	NOUN
cana-3791	2	15	selection	selection	NOUN
cana-3791	2	16	and	and	CCONJ
cana-3791	2	17	its	its	PRON
cana-3791	2	18	applications	application	NOUN
cana-3791	2	19	in	in	ADP
cana-3791	2	20	signal	signal	NOUN
cana-3791	2	21	processing	processing	NOUN
cana-3791	2	22	and	and	CCONJ
cana-3791	2	23	numerical	numerical	ADJ
cana-3791	2	24	analysis	analysis	NOUN
cana-3791	2	25	mansourah	mansourah	PROPN
cana-3791	2	26	h.	h.	PROPN
cana-3791	2	27	al	al	PROPN
cana-3791	2	28	-	-	PUNCT
cana-3791	2	29	yazidi	yazidi	PROPN
cana-3791	2	30	*	*	ADJ
cana-3791	2	31	1	1	NUM
cana-3791	2	32	first	first	ADJ
cana-3791	2	33	common	common	ADJ
cana-3791	2	34	year	year	NOUN
cana-3791	2	35	program	program	NOUN
cana-3791	2	36	,	,	PUNCT
cana-3791	2	37	department	department	NOUN
cana-3791	2	38	of	of	ADP
cana-3791	2	39	basic	basic	ADJ
cana-3791	2	40	science	science	NOUN
cana-3791	2	41	,	,	PUNCT
cana-3791	2	42	prince	prince	PROPN
cana-3791	2	43	sattam	sattam	PROPN
cana-3791	2	44	bin	bin	PROPN
cana-3791	2	45	abdulaziz	abdulaziz	PROPN
cana-3791	2	46	university	university	PROPN
cana-3791	2	47	,	,	PUNCT
cana-3791	2	48	al	al	PROPN
cana-3791	2	49	-	-	PUNCT
cana-3791	2	50	kharj	kharj	PROPN
cana-3791	2	51	11942	11942	NUM
cana-3791	2	52	,	,	PUNCT
cana-3791	2	53	ksa	ksa	PROPN
cana-3791	2	54	.	.	PUNCT
cana-3791	3	1	*	*	PUNCT
cana-3791	3	2	corresponding	correspond	VERB
cana-3791	3	3	author	author	NOUN
cana-3791	3	4	:	:	PUNCT
cana-3791	3	5	mansourah	mansourah	PROPN
cana-3791	3	6	h.	h.	PROPN
cana-3791	3	7	al	al	PROPN
cana-3791	3	8	-	-	PUNCT
cana-3791	3	9	yazidi	yazidi	PROPN
cana-3791	3	10	e	e	NOUN
cana-3791	3	11	-	-	NOUN
cana-3791	3	12	mail	mail	NOUN
cana-3791	3	13	:	:	PUNCT
cana-3791	3	14	m.alyazidi@psau.edu.sa	m.alyazidi@psau.edu.sa	PROPN
cana-3791	3	15	article	article	NOUN
cana-3791	3	16	history	history	NOUN
cana-3791	3	17	:	:	PUNCT
cana-3791	3	18	received	receive	VERB
cana-3791	3	19	:	:	PUNCT
cana-3791	3	20	12	12	NUM
cana-3791	3	21	-	-	SYM
cana-3791	3	22	11	11	NUM
cana-3791	3	23	-	-	PUNCT
cana-3791	3	24	2024	2024	NUM
cana-3791	3	25	revised:18	revised:18	NOUN
cana-3791	3	26	-	-	PUNCT
cana-3791	3	27	12	12	NUM
cana-3791	3	28	-	-	PUNCT
cana-3791	3	29	2024	2024	NUM
cana-3791	3	30	accepted:05	accepted:05	X
cana-3791	3	31	-	-	PUNCT
cana-3791	3	32	01	01	NUM
cana-3791	3	33	-	-	PUNCT
cana-3791	3	34	2025	2025	NUM
cana-3791	3	35	abstract	abstract	NOUN
cana-3791	3	36	:	:	PUNCT
cana-3791	3	37	this	this	DET
cana-3791	3	38	paper	paper	NOUN
cana-3791	3	39	presents	present	VERB
cana-3791	3	40	a	a	DET
cana-3791	3	41	self	self	NOUN
cana-3791	3	42	-	-	PUNCT
cana-3791	3	43	contained	contain	VERB
cana-3791	3	44	theoretical	theoretical	ADJ
cana-3791	3	45	and	and	CCONJ
cana-3791	3	46	practical	practical	ADJ
cana-3791	3	47	framework	framework	NOUN
cana-3791	3	48	for	for	ADP
cana-3791	3	49	optimal	optimal	ADJ
cana-3791	3	50	basis	basis	NOUN
cana-3791	3	51	selection	selection	NOUN
cana-3791	3	52	in	in	ADP
cana-3791	3	53	finite	finite	ADJ
cana-3791	3	54	-	-	ADJ
cana-3791	3	55	dimensional	dimensional	ADJ
cana-3791	3	56	hilbert	hilbert	NOUN
cana-3791	3	57	spaces	space	NOUN
cana-3791	3	58	,	,	PUNCT
cana-3791	3	59	with	with	ADP
cana-3791	3	60	applications	application	NOUN
cana-3791	3	61	in	in	ADP
cana-3791	3	62	signal	signal	NOUN
cana-3791	3	63	processing	processing	NOUN
cana-3791	3	64	and	and	CCONJ
cana-3791	3	65	numerical	numerical	ADJ
cana-3791	3	66	analysis	analysis	NOUN
cana-3791	3	67	.	.	PUNCT
cana-3791	4	1	based	base	VERB
cana-3791	4	2	on	on	ADP
cana-3791	4	3	quantumoptical	quantumoptical	ADJ
cana-3791	4	4	state	state	NOUN
cana-3791	4	5	theory	theory	NOUN
cana-3791	4	6	and	and	CCONJ
cana-3791	4	7	dagger	dagger	NOUN
cana-3791	4	8	category	category	NOUN
cana-3791	4	9	frameworks	framework	NOUN
cana-3791	4	10	,	,	PUNCT
cana-3791	4	11	we	we	PRON
cana-3791	4	12	give	give	VERB
cana-3791	4	13	a	a	DET
cana-3791	4	14	rigorous	rigorous	ADJ
cana-3791	4	15	mathematical	mathematical	ADJ
cana-3791	4	16	development	development	NOUN
cana-3791	4	17	that	that	PRON
cana-3791	4	18	yields	yield	VERB
cana-3791	4	19	a	a	DET
cana-3791	4	20	criterion	criterion	NOUN
cana-3791	4	21	for	for	ADP
cana-3791	4	22	basis	basis	NOUN
cana-3791	4	23	selection	selection	NOUN
cana-3791	4	24	,	,	PUNCT
cana-3791	4	25	enabling	enable	VERB
cana-3791	4	26	a	a	DET
cana-3791	4	27	proper	proper	ADJ
cana-3791	4	28	trade	trade	NOUN
cana-3791	4	29	-	-	PUNCT
cana-3791	4	30	off	off	NOUN
cana-3791	4	31	between	between	ADP
cana-3791	4	32	approximation	approximation	NOUN
cana-3791	4	33	accuracy	accuracy	NOUN
cana-3791	4	34	and	and	CCONJ
cana-3791	4	35	computational	computational	ADJ
cana-3791	4	36	efficiency	efficiency	NOUN
cana-3791	4	37	.	.	PUNCT
cana-3791	5	1	the	the	DET
cana-3791	5	2	study	study	NOUN
cana-3791	5	3	introduces	introduce	VERB
cana-3791	5	4	new	new	ADJ
cana-3791	5	5	methods	method	NOUN
cana-3791	5	6	to	to	PART
cana-3791	5	7	represent	represent	VERB
cana-3791	5	8	discrete	discrete	ADJ
cana-3791	5	9	signals	signal	NOUN
cana-3791	5	10	via	via	ADP
cana-3791	5	11	wigner	wigner	ADJ
cana-3791	5	12	functions	function	NOUN
cana-3791	5	13	,	,	PUNCT
cana-3791	5	14	along	along	ADP
cana-3791	5	15	with	with	ADP
cana-3791	5	16	explicit	explicit	ADJ
cana-3791	5	17	error	error	NOUN
cana-3791	5	18	bounds	bound	NOUN
cana-3791	5	19	for	for	ADP
cana-3791	5	20	numerical	numerical	ADJ
cana-3791	5	21	approximations	approximation	NOUN
cana-3791	5	22	.	.	PUNCT
cana-3791	6	1	our	our	PRON
cana-3791	6	2	framework	framework	NOUN
cana-3791	6	3	contains	contain	VERB
cana-3791	6	4	both	both	DET
cana-3791	6	5	theoretical	theoretical	ADJ
cana-3791	6	6	foundations	foundation	NOUN
cana-3791	6	7	and	and	CCONJ
cana-3791	6	8	practical	practical	ADJ
cana-3791	6	9	implementations	implementation	NOUN
cana-3791	6	10	,	,	PUNCT
cana-3791	6	11	illustrated	illustrate	VERB
cana-3791	6	12	by	by	ADP
cana-3791	6	13	case	case	NOUN
cana-3791	6	14	studies	study	NOUN
cana-3791	6	15	showing	show	VERB
cana-3791	6	16	the	the	DET
cana-3791	6	17	validity	validity	NOUN
cana-3791	6	18	of	of	ADP
cana-3791	6	19	the	the	DET
cana-3791	6	20	approach	approach	NOUN
cana-3791	6	21	.	.	PUNCT
cana-3791	7	1	hence	hence	ADV
cana-3791	7	2	,	,	PUNCT
cana-3791	7	3	it	it	PRON
cana-3791	7	4	makes	make	VERB
cana-3791	7	5	three	three	NUM
cana-3791	7	6	key	key	ADJ
cana-3791	7	7	contributions	contribution	NOUN
cana-3791	7	8	:	:	PUNCT
cana-3791	7	9	it	it	PRON
cana-3791	7	10	gives	give	VERB
cana-3791	7	11	a	a	DET
cana-3791	7	12	unified	unified	ADJ
cana-3791	7	13	treatment	treatment	NOUN
cana-3791	7	14	of	of	ADP
cana-3791	7	15	the	the	DET
cana-3791	7	16	basis	basis	NOUN
cana-3791	7	17	selection	selection	NOUN
cana-3791	7	18	criteria	criterion	NOUN
cana-3791	7	19	;	;	PUNCT
cana-3791	7	20	it	it	PRON
cana-3791	7	21	provides	provide	VERB
cana-3791	7	22	efficient	efficient	ADJ
cana-3791	7	23	numerical	numerical	ADJ
cana-3791	7	24	methods	method	NOUN
cana-3791	7	25	with	with	ADP
cana-3791	7	26	provable	provable	ADJ
cana-3791	7	27	convergence	convergence	NOUN
cana-3791	7	28	properties	property	NOUN
cana-3791	7	29	;	;	PUNCT
cana-3791	7	30	and	and	CCONJ
cana-3791	7	31	it	it	PRON
cana-3791	7	32	presents	present	VERB
cana-3791	7	33	practical	practical	ADJ
cana-3791	7	34	implementation	implementation	NOUN
cana-3791	7	35	strategies	strategy	NOUN
cana-3791	7	36	for	for	ADP
cana-3791	7	37	signal	signal	ADJ
cana-3791	7	38	processing	processing	NOUN
cana-3791	7	39	applications	application	NOUN
cana-3791	7	40	.	.	PUNCT
cana-3791	8	1	results	result	NOUN
cana-3791	8	2	indicate	indicate	VERB
cana-3791	8	3	that	that	SCONJ
cana-3791	8	4	our	our	PRON
cana-3791	8	5	finite	finite	ADJ
cana-3791	8	6	-	-	ADJ
cana-3791	8	7	dimensional	dimensional	ADJ
cana-3791	8	8	approach	approach	NOUN
cana-3791	8	9	attains	attain	VERB
cana-3791	8	10	high	high	ADJ
cana-3791	8	11	accuracy	accuracy	NOUN
cana-3791	8	12	yet	yet	ADV
cana-3791	8	13	remains	remain	VERB
cana-3791	8	14	computationally	computationally	ADV
cana-3791	8	15	efficient	efficient	ADJ
cana-3791	8	16	,	,	PUNCT
cana-3791	8	17	with	with	ADP
cana-3791	8	18	fidelity	fidelity	NOUN
cana-3791	8	19	measures	measure	NOUN
cana-3791	8	20	above	above	ADP
cana-3791	8	21	0.99	0.99	NUM
cana-3791	8	22	for	for	ADP
cana-3791	8	23	optimal	optimal	ADJ
cana-3791	8	24	parameter	parameter	NOUN
cana-3791	8	25	choices	choice	NOUN
cana-3791	8	26	.	.	PUNCT
cana-3791	9	1	to	to	PART
cana-3791	9	2	further	far	ADV
cana-3791	9	3	validate	validate	VERB
cana-3791	9	4	the	the	DET
cana-3791	9	5	performance	performance	NOUN
cana-3791	9	6	of	of	ADP
cana-3791	9	7	the	the	DET
cana-3791	9	8	framework	framework	NOUN
cana-3791	9	9	,	,	PUNCT
cana-3791	9	10	extensive	extensive	ADJ
cana-3791	9	11	numerical	numerical	ADJ
cana-3791	9	12	experiments	experiment	NOUN
cana-3791	9	13	are	be	AUX
cana-3791	9	14	provided	provide	VERB
cana-3791	9	15	,	,	PUNCT
cana-3791	9	16	which	which	PRON
cana-3791	9	17	show	show	VERB
cana-3791	9	18	the	the	DET
cana-3791	9	19	usefulness	usefulness	NOUN
cana-3791	9	20	of	of	ADP
cana-3791	9	21	the	the	DET
cana-3791	9	22	framework	framework	NOUN
cana-3791	9	23	in	in	ADP
cana-3791	9	24	a	a	DET
cana-3791	9	25	variety	variety	NOUN
cana-3791	9	26	of	of	ADP
cana-3791	9	27	applications	application	NOUN
cana-3791	9	28	from	from	ADP
cana-3791	9	29	quantum	quantum	ADJ
cana-3791	9	30	state	state	NOUN
cana-3791	9	31	representation	representation	NOUN
cana-3791	9	32	to	to	ADP
cana-3791	9	33	digital	digital	ADJ
cana-3791	9	34	signal	signal	NOUN
cana-3791	9	35	processing	processing	NOUN
cana-3791	9	36	.	.	PUNCT
cana-3791	10	1	this	this	DET
cana-3791	10	2	work	work	NOUN
cana-3791	10	3	also	also	ADV
cana-3791	10	4	addresses	address	VERB
cana-3791	10	5	some	some	PRON
cana-3791	10	6	of	of	ADP
cana-3791	10	7	the	the	DET
cana-3791	10	8	theoretical	theoretical	ADJ
cana-3791	10	9	challenges	challenge	NOUN
cana-3791	10	10	in	in	ADP
cana-3791	10	11	dimensional	dimensional	ADJ
cana-3791	10	12	scaling	scaling	NOUN
cana-3791	10	13	and	and	CCONJ
cana-3791	10	14	error	error	NOUN
cana-3791	10	15	propagation	propagation	NOUN
cana-3791	10	16	,	,	PUNCT
cana-3791	10	17	establishing	establish	VERB
cana-3791	10	18	a	a	DET
cana-3791	10	19	basis	basis	NOUN
cana-3791	10	20	for	for	ADP
cana-3791	10	21	future	future	ADJ
cana-3791	10	22	developments	development	NOUN
cana-3791	10	23	in	in	ADP
cana-3791	10	24	the	the	DET
cana-3791	10	25	field	field	NOUN
cana-3791	10	26	.	.	PUNCT
cana-3791	11	1	our	our	PRON
cana-3791	11	2	results	result	NOUN
cana-3791	11	3	show	show	VERB
cana-3791	11	4	that	that	SCONJ
cana-3791	11	5	finite	finite	VERB
cana-3791	11	6	dimensional	dimensional	ADJ
cana-3791	11	7	methods	method	NOUN
cana-3791	11	8	offer	offer	VERB
cana-3791	11	9	significant	significant	ADJ
cana-3791	11	10	benefits	benefit	NOUN
cana-3791	11	11	in	in	ADP
cana-3791	11	12	mathematical	mathematical	ADJ
cana-3791	11	13	tractability	tractability	NOUN
cana-3791	11	14	and	and	CCONJ
cana-3791	11	15	computational	computational	ADJ
cana-3791	11	16	realization	realization	NOUN
cana-3791	11	17	at	at	ADP
cana-3791	11	18	the	the	DET
cana-3791	11	19	cost	cost	NOUN
cana-3791	11	20	of	of	ADP
cana-3791	11	21	little	little	ADJ
cana-3791	11	22	accuracy	accuracy	NOUN
cana-3791	11	23	for	for	ADP
cana-3791	11	24	most	most	ADJ
cana-3791	11	25	practical	practical	ADJ
cana-3791	11	26	applications	application	NOUN
cana-3791	11	27	.	.	PUNCT
cana-3791	12	1	keywords	keyword	NOUN
cana-3791	12	2	:	:	PUNCT
cana-3791	12	3	finite	finite	ADJ
cana-3791	12	4	-	-	ADJ
cana-3791	12	5	dimensional	dimensional	ADJ
cana-3791	12	6	hilbert	hilbert	NOUN
cana-3791	12	7	spaces	space	NOUN
cana-3791	12	8	,	,	PUNCT
cana-3791	12	9	basis	basis	NOUN
cana-3791	12	10	selection	selection	NOUN
cana-3791	12	11	,	,	PUNCT
cana-3791	12	12	signal	signal	NOUN
cana-3791	12	13	processing	processing	NOUN
cana-3791	12	14	,	,	PUNCT
cana-3791	12	15	numerical	numerical	ADJ
cana-3791	12	16	analysis	analysis	NOUN
cana-3791	12	17	,	,	PUNCT
cana-3791	12	18	discrete	discrete	ADJ
cana-3791	12	19	wigner	wigner	NOUN
cana-3791	12	20	function	function	NOUN
cana-3791	12	21	,	,	PUNCT
cana-3791	12	22	quantum	quantum	NOUN
cana-3791	12	23	-	-	ADJ
cana-3791	12	24	optical	optical	ADJ
cana-3791	12	25	states	state	NOUN
cana-3791	12	26	,	,	PUNCT
cana-3791	12	27	error	error	NOUN
cana-3791	12	28	bounds	bound	NOUN
cana-3791	12	29	,	,	PUNCT
cana-3791	12	30	computational	computational	ADJ
cana-3791	12	31	efficiency	efficiency	NOUN
cana-3791	12	32	,	,	PUNCT
cana-3791	12	33	dagger	dagger	NOUN
cana-3791	12	34	categories	category	NOUN
cana-3791	12	35	,	,	PUNCT
cana-3791	12	36	mathematical	mathematical	ADJ
cana-3791	12	37	optimization	optimization	NOUN
cana-3791	12	38	.	.	PUNCT
cana-3791	13	1	introduction	introduction	NOUN
cana-3791	13	2	within	within	ADP
cana-3791	13	3	the	the	DET
cana-3791	13	4	last	last	ADJ
cana-3791	13	5	couple	couple	NOUN
cana-3791	13	6	of	of	ADP
cana-3791	13	7	decades	decade	NOUN
cana-3791	13	8	,	,	PUNCT
cana-3791	13	9	finite	finite	ADJ
cana-3791	13	10	-	-	ADJ
cana-3791	13	11	dimensional	dimensional	ADJ
cana-3791	13	12	hilbert	hilbert	NOUN
cana-3791	13	13	spaces	space	NOUN
cana-3791	13	14	have	have	AUX
cana-3791	13	15	become	become	VERB
cana-3791	13	16	one	one	NUM
cana-3791	13	17	of	of	ADP
cana-3791	13	18	the	the	DET
cana-3791	13	19	basic	basic	ADJ
cana-3791	13	20	frameworks	framework	NOUN
cana-3791	13	21	for	for	ADP
cana-3791	13	22	every	every	DET
cana-3791	13	23	field	field	NOUN
cana-3791	13	24	from	from	ADP
cana-3791	13	25	mathematics	mathematic	NOUN
cana-3791	13	26	and	and	CCONJ
cana-3791	13	27	physics	physics	NOUN
cana-3791	13	28	to	to	ADP
cana-3791	13	29	engineering	engineering	NOUN
cana-3791	13	30	,	,	PUNCT
cana-3791	13	31	especially	especially	ADV
cana-3791	13	32	in	in	ADP
cana-3791	13	33	quantum	quantum	ADJ
cana-3791	13	34	mechanics	mechanic	NOUN
cana-3791	13	35	and	and	CCONJ
cana-3791	13	36	signal	signal	NOUN
cana-3791	13	37	processing	processing	NOUN
cana-3791	13	38	.	.	PUNCT
cana-3791	14	1	the	the	DET
cana-3791	14	2	formulation	formulation	NOUN
cana-3791	14	3	provided	provide	VERB
cana-3791	14	4	by	by	ADP
cana-3791	14	5	weyl	weyl	NOUN
cana-3791	14	6	in	in	ADP
cana-3791	14	7	1931	1931	NUM
cana-3791	14	8	[	[	X
cana-3791	14	9	1	1	NUM
cana-3791	14	10	]	]	PUNCT
cana-3791	14	11	gave	give	VERB
cana-3791	14	12	a	a	DET
cana-3791	14	13	start	start	NOUN
cana-3791	14	14	to	to	ADP
cana-3791	14	15	this	this	DET
cana-3791	14	16	mathematical	mathematical	ADJ
cana-3791	14	17	structure	structure	NOUN
cana-3791	14	18	and	and	CCONJ
cana-3791	14	19	opened	open	VERB
cana-3791	14	20	the	the	DET
cana-3791	14	21	widest	wide	ADJ
cana-3791	14	22	possibilities	possibility	NOUN
cana-3791	14	23	for	for	ADP
cana-3791	14	24	investigation	investigation	NOUN
cana-3791	14	25	of	of	ADP
cana-3791	14	26	system	system	NOUN
cana-3791	14	27	dynamics	dynamic	NOUN
cana-3791	14	28	both	both	CCONJ
cana-3791	14	29	in	in	ADP
cana-3791	14	30	infinite	infinite	ADJ
cana-3791	14	31	and	and	CCONJ
cana-3791	14	32	finite	finite	ADJ
cana-3791	14	33	-	-	ADJ
cana-3791	14	34	dimensional	dimensional	ADJ
cana-3791	14	35	cases	case	NOUN
cana-3791	14	36	.	.	PUNCT
cana-3791	15	1	as	as	SCONJ
cana-3791	15	2	was	be	AUX
cana-3791	15	3	shown	show	VERB
cana-3791	15	4	in	in	ADP
cana-3791	15	5	the	the	DET
cana-3791	15	6	seminal	seminal	ADJ
cana-3791	15	7	works	work	NOUN
cana-3791	15	8	of	of	ADP
cana-3791	15	9	santhanam	santhanam	PROPN
cana-3791	15	10	and	and	CCONJ
cana-3791	15	11	co	co	NOUN
cana-3791	15	12	-	-	NOUN
cana-3791	15	13	authors	author	NOUN
cana-3791	15	14	[	[	X
cana-3791	15	15	2,3	2,3	NUM
cana-3791	15	16	]	]	PUNCT
cana-3791	15	17	,	,	PUNCT
cana-3791	15	18	in	in	ADP
cana-3791	15	19	the	the	DET
cana-3791	15	20	last	last	ADJ
cana-3791	15	21	few	few	ADJ
cana-3791	15	22	decades	decade	NOUN
cana-3791	15	23	,	,	PUNCT
cana-3791	15	24	finite	finite	ADJ
cana-3791	15	25	-	-	ADJ
cana-3791	15	26	dimensional	dimensional	ADJ
cana-3791	15	27	hilbert	hilbert	NOUN
cana-3791	15	28	spaces	space	NOUN
cana-3791	15	29	have	have	AUX
cana-3791	15	30	gained	gain	VERB
cana-3791	15	31	much	much	ADJ
cana-3791	15	32	importance	importance	NOUN
cana-3791	15	33	,	,	PUNCT
cana-3791	15	34	from	from	ADP
cana-3791	15	35	purely	purely	ADV
cana-3791	15	36	theoretical	theoretical	ADJ
cana-3791	15	37	advances	advance	NOUN
cana-3791	15	38	to	to	ADP
cana-3791	15	39	more	more	ADV
cana-3791	15	40	practical	practical	ADJ
cana-3791	15	41	applications	application	NOUN
cana-3791	15	42	in	in	ADP
cana-3791	15	43	quantum	quantum	ADJ
cana-3791	15	44	information	information	NOUN
cana-3791	15	45	theory	theory	NOUN
cana-3791	15	46	,	,	PUNCT
cana-3791	15	47	digital	digital	ADJ
cana-3791	15	48	signal	signal	NOUN
cana-3791	15	49	processing	processing	NOUN
cana-3791	15	50	,	,	PUNCT
cana-3791	15	51	and	and	CCONJ
cana-3791	15	52	numerical	numerical	ADJ
cana-3791	15	53	analysis	analysis	NOUN
cana-3791	15	54	.	.	PUNCT
cana-3791	16	1	the	the	DET
cana-3791	16	2	key	key	NOUN
cana-3791	16	3	to	to	ADP
cana-3791	16	4	this	this	DET
cana-3791	16	5	scheme	scheme	NOUN
cana-3791	16	6	lies	lie	VERB
cana-3791	16	7	in	in	ADP
cana-3791	16	8	the	the	DET
cana-3791	16	9	presumption	presumption	NOUN
cana-3791	16	10	that	that	SCONJ
cana-3791	16	11	a	a	DET
cana-3791	16	12	physical	physical	ADJ
cana-3791	16	13	system	system	NOUN
cana-3791	16	14	can	can	AUX
cana-3791	16	15	have	have	AUX
cana-3791	16	16	its	its	PRON
cana-3791	16	17	kinematical	kinematical	ADJ
cana-3791	16	18	structure	structure	NOUN
cana-3791	16	19	specified	specify	VERB
cana-3791	16	20	by	by	ADP
cana-3791	16	21	an	an	DET
cana-3791	16	22	irreducible	irreducible	ADJ
cana-3791	16	23	abelian	abelian	ADJ
cana-3791	16	24	group	group	NOUN
cana-3791	16	25	of	of	ADP
cana-3791	16	26	unitary	unitary	ADJ
cana-3791	16	27	representations	representation	NOUN
cana-3791	16	28	of	of	ADP
cana-3791	16	29	system	system	NOUN
cana-3791	16	30	space	space	NOUN
cana-3791	16	31	.	.	PUNCT
cana-3791	17	1	that	that	SCONJ
cana-3791	17	2	this	this	DET
cana-3791	17	3	notion	notion	NOUN
cana-3791	17	4	could	could	AUX
cana-3791	17	5	be	be	AUX
cana-3791	17	6	extended	extend	VERB
cana-3791	17	7	communications	communication	NOUN
cana-3791	17	8	on	on	ADP
cana-3791	17	9	applied	apply	VERB
cana-3791	17	10	nonlinear	nonlinear	ADJ
cana-3791	17	11	analysis	analysis	NOUN
cana-3791	17	12	issn	issn	NOUN
cana-3791	17	13	:	:	PUNCT
cana-3791	17	14	1074	1074	NUM
cana-3791	17	15	-	-	PUNCT
cana-3791	17	16	133x	133x	NUM
cana-3791	17	17	vol	vol	NOUN
cana-3791	17	18	32	32	NUM
cana-3791	17	19	no	no	NOUN
cana-3791	17	20	.	.	PUNCT
cana-3791	18	1	8s	8s	PROPN
cana-3791	18	2	(	(	PUNCT
cana-3791	18	3	2025	2025	NUM
cana-3791	18	4	)	)	PUNCT
cana-3791	18	5	733	733	NUM
cana-3791	18	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3791	18	7	was	be	AUX
cana-3791	18	8	subsequently	subsequently	ADV
cana-3791	18	9	shown	show	VERB
cana-3791	18	10	by	by	ADP
cana-3791	18	11	schwinger	schwinger	NOUN
cana-3791	18	12	[	[	X
cana-3791	18	13	4	4	NUM
cana-3791	18	14	]	]	PUNCT
cana-3791	18	15	who	who	PRON
cana-3791	18	16	proved	prove	VERB
cana-3791	18	17	that	that	SCONJ
cana-3791	18	18	there	there	PRON
cana-3791	18	19	corresponds	correspond	VERB
cana-3791	18	20	,	,	PUNCT
cana-3791	18	21	to	to	ADP
cana-3791	18	22	any	any	DET
cana-3791	18	23	given	give	VERB
cana-3791	18	24	finite	finite	PROPN
cana-3791	18	25	abelian	abelian	PROPN
cana-3791	18	26	group	group	NOUN
cana-3791	18	27	,	,	PUNCT
cana-3791	18	28	one	one	NUM
cana-3791	18	29	and	and	CCONJ
cana-3791	18	30	only	only	ADV
cana-3791	18	31	one	one	NUM
cana-3791	18	32	class	class	NOUN
cana-3791	18	33	of	of	ADP
cana-3791	18	34	unitarily	unitarily	ADV
cana-3791	18	35	equivalent	equivalent	ADJ
cana-3791	18	36	,	,	PUNCT
cana-3791	18	37	irreducible	irreducible	ADJ
cana-3791	18	38	representations	representation	NOUN
cana-3791	18	39	in	in	ADP
cana-3791	18	40	finite	finite	ADJ
cana-3791	18	41	dimensional	dimensional	ADJ
cana-3791	18	42	space	space	NOUN
cana-3791	18	43	.	.	PUNCT
cana-3791	19	1	the	the	DET
cana-3791	19	2	mathematical	mathematical	ADJ
cana-3791	19	3	attractiveness	attractiveness	NOUN
cana-3791	19	4	of	of	ADP
cana-3791	19	5	this	this	DET
cana-3791	19	6	approach	approach	NOUN
cana-3791	19	7	is	be	AUX
cana-3791	19	8	that	that	SCONJ
cana-3791	19	9	it	it	PRON
cana-3791	19	10	allows	allow	VERB
cana-3791	19	11	a	a	DET
cana-3791	19	12	complete	complete	ADJ
cana-3791	19	13	description	description	NOUN
cana-3791	19	14	of	of	ADP
cana-3791	19	15	quantum	quantum	NOUN
cana-3791	19	16	systems	system	NOUN
cana-3791	19	17	while	while	SCONJ
cana-3791	19	18	retaining	retain	VERB
cana-3791	19	19	computational	computational	ADJ
cana-3791	19	20	tractability	tractability	NOUN
cana-3791	19	21	-	-	PUNCT
cana-3791	19	22	a	a	DET
cana-3791	19	23	feature	feature	NOUN
cana-3791	19	24	particularly	particularly	ADV
cana-3791	19	25	valuable	valuable	ADJ
cana-3791	19	26	in	in	ADP
cana-3791	19	27	modern	modern	ADJ
cana-3791	19	28	applications	application	NOUN
cana-3791	19	29	,	,	PUNCT
cana-3791	19	30	as	as	SCONJ
cana-3791	19	31	elaborated	elaborate	VERB
cana-3791	19	32	by	by	ADP
cana-3791	19	33	perelomov	perelomov	NOUN
cana-3791	19	34	[	[	X
cana-3791	19	35	5	5	NUM
cana-3791	19	36	]	]	PUNCT
cana-3791	19	37	and	and	CCONJ
cana-3791	19	38	gilmore	gilmore	NOUN
cana-3791	20	1	[	[	X
cana-3791	20	2	6,7	6,7	NUM
cana-3791	20	3	]	]	PUNCT
cana-3791	20	4	.	.	PUNCT
cana-3791	21	1	the	the	DET
cana-3791	21	2	motivation	motivation	NOUN
cana-3791	21	3	for	for	ADP
cana-3791	21	4	studying	study	VERB
cana-3791	21	5	finite	finite	ADJ
cana-3791	21	6	-	-	ADJ
cana-3791	21	7	dimensional	dimensional	ADJ
cana-3791	21	8	hilbert	hilbert	NOUN
cana-3791	21	9	spaces	space	NOUN
cana-3791	21	10	extends	extend	VERB
cana-3791	21	11	beyond	beyond	ADP
cana-3791	21	12	their	their	PRON
cana-3791	21	13	mathematical	mathematical	ADJ
cana-3791	21	14	beauty	beauty	NOUN
cana-3791	21	15	into	into	ADP
cana-3791	21	16	practical	practical	ADJ
cana-3791	21	17	considerations	consideration	NOUN
cana-3791	21	18	in	in	ADP
cana-3791	21	19	quantum	quantum	ADJ
cana-3791	21	20	systems	system	NOUN
cana-3791	21	21	and	and	CCONJ
cana-3791	21	22	signal	signal	NOUN
cana-3791	21	23	processing	processing	NOUN
cana-3791	21	24	.	.	PUNCT
cana-3791	22	1	the	the	DET
cana-3791	22	2	holographic	holographic	ADJ
cana-3791	22	3	principle	principle	NOUN
cana-3791	22	4	,	,	PUNCT
cana-3791	22	5	as	as	SCONJ
cana-3791	22	6	put	put	VERB
cana-3791	22	7	forth	forth	ADP
cana-3791	22	8	by	by	ADP
cana-3791	22	9	'	'	PROPN
cana-3791	22	10	t	t	NOUN
cana-3791	22	11	hooft	hooft	NOUN
cana-3791	23	1	[	[	X
cana-3791	23	2	8	8	NUM
cana-3791	23	3	]	]	PUNCT
cana-3791	23	4	and	and	CCONJ
cana-3791	23	5	susskind	susskind	NOUN
cana-3791	23	6	[	[	X
cana-3791	23	7	9	9	NUM
cana-3791	23	8	]	]	PUNCT
cana-3791	23	9	,	,	PUNCT
cana-3791	23	10	states	state	VERB
cana-3791	23	11	that	that	SCONJ
cana-3791	23	12	the	the	DET
cana-3791	23	13	hilbert	hilbert	PROPN
cana-3791	23	14	space	space	NOUN
cana-3791	23	15	of	of	ADP
cana-3791	23	16	quantum	quantum	NOUN
cana-3791	23	17	gravity	gravity	NOUN
cana-3791	23	18	is	be	AUX
cana-3791	23	19	locally	locally	ADV
cana-3791	23	20	finite	finite	ADJ
cana-3791	23	21	dimensional	dimensional	ADJ
cana-3791	23	22	.	.	PUNCT
cana-3791	24	1	this	this	DET
cana-3791	24	2	essentially	essentially	ADV
cana-3791	24	3	fundamental	fundamental	ADJ
cana-3791	24	4	intuition	intuition	NOUN
cana-3791	24	5	carries	carry	VERB
cana-3791	24	6	large	large	ADJ
cana-3791	24	7	-	-	PUNCT
cana-3791	24	8	scale	scale	NOUN
cana-3791	24	9	repercussions	repercussion	NOUN
cana-3791	24	10	for	for	ADP
cana-3791	24	11	the	the	DET
cana-3791	24	12	conceptualization	conceptualization	NOUN
cana-3791	24	13	of	of	ADP
cana-3791	24	14	physical	physical	ADJ
cana-3791	24	15	systems	system	NOUN
cana-3791	24	16	and	and	CCONJ
cana-3791	24	17	their	their	PRON
cana-3791	24	18	representations	representation	NOUN
cana-3791	24	19	.	.	PUNCT
cana-3791	25	1	more	more	ADV
cana-3791	25	2	precisely	precisely	ADV
cana-3791	25	3	,	,	PUNCT
cana-3791	25	4	for	for	ADP
cana-3791	25	5	the	the	DET
cana-3791	25	6	observable	observable	ADJ
cana-3791	25	7	universe	universe	NOUN
cana-3791	25	8	,	,	PUNCT
cana-3791	25	9	finite	finite	NOUN
cana-3791	25	10	-	-	ADJ
cana-3791	25	11	dimensionality	dimensionality	NOUN
cana-3791	25	12	of	of	ADP
cana-3791	25	13	hilbert	hilbert	NOUN
cana-3791	25	14	space	space	NOUN
cana-3791	25	15	is	be	AUX
cana-3791	25	16	not	not	PART
cana-3791	25	17	only	only	ADV
cana-3791	25	18	a	a	DET
cana-3791	25	19	mathematical	mathematical	ADJ
cana-3791	25	20	use	use	NOUN
cana-3791	25	21	but	but	CCONJ
cana-3791	25	22	a	a	DET
cana-3791	25	23	physical	physical	ADJ
cana-3791	25	24	necessity	necessity	NOUN
cana-3791	25	25	imposed	impose	VERB
cana-3791	25	26	by	by	ADP
cana-3791	25	27	the	the	DET
cana-3791	25	28	following	follow	VERB
cana-3791	25	29	relation	relation	NOUN
cana-3791	25	30	[	[	X
cana-3791	25	31	10	10	NUM
cana-3791	25	32	]	]	X
cana-3791	25	33	:	:	PUNCT
cana-3791	25	34	s(r	s(r	X
cana-3791	25	35	)	)	PUNCT
cana-3791	25	36	≤	≤	NUM
cana-3791	25	37	|δr|/(4ℓ²ᵨ	|δr|/(4ℓ²ᵨ	NOUN
cana-3791	25	38	)	)	PUNCT
cana-3791	25	39	where	where	SCONJ
cana-3791	25	40	s(r	s(r	X
cana-3791	25	41	)	)	PUNCT
cana-3791	25	42	is	be	AUX
cana-3791	25	43	the	the	DET
cana-3791	25	44	maximal	maximal	ADJ
cana-3791	25	45	entropy	entropy	NOUN
cana-3791	25	46	that	that	PRON
cana-3791	25	47	can	can	AUX
cana-3791	25	48	be	be	AUX
cana-3791	25	49	gathered	gather	VERB
cana-3791	25	50	inside	inside	ADP
cana-3791	25	51	a	a	DET
cana-3791	25	52	finite	finite	ADJ
cana-3791	25	53	region	region	NOUN
cana-3791	25	54	of	of	ADP
cana-3791	25	55	space	space	NOUN
cana-3791	25	56	r	r	NOUN
cana-3791	25	57	,	,	PUNCT
cana-3791	25	58	while	while	SCONJ
cana-3791	25	59	|δr|	|δr|	NOUN
cana-3791	25	60	stands	stand	VERB
cana-3791	25	61	for	for	ADP
cana-3791	25	62	the	the	DET
cana-3791	25	63	boundary	boundary	ADJ
cana-3791	25	64	area	area	NOUN
cana-3791	25	65	of	of	ADP
cana-3791	25	66	that	that	DET
cana-3791	25	67	region	region	NOUN
cana-3791	25	68	and	and	CCONJ
cana-3791	25	69	ℓᵨ	ℓᵨ	VERB
cana-3791	25	70	stands	stand	VERB
cana-3791	25	71	for	for	ADP
cana-3791	25	72	the	the	DET
cana-3791	25	73	planck	planck	NOUN
cana-3791	25	74	length	length	NOUN
cana-3791	25	75	.	.	PUNCT
cana-3791	26	1	such	such	DET
cana-3791	26	2	an	an	DET
cana-3791	26	3	intriguing	intriguing	ADJ
cana-3791	26	4	link	link	NOUN
cana-3791	26	5	between	between	ADP
cana-3791	26	6	geometric	geometric	ADJ
cana-3791	26	7	space	space	NOUN
cana-3791	26	8	properties	property	NOUN
cana-3791	26	9	and	and	CCONJ
cana-3791	26	10	the	the	DET
cana-3791	26	11	dimensional	dimensional	ADJ
cana-3791	26	12	constraints	constraint	NOUN
cana-3791	26	13	of	of	ADP
cana-3791	26	14	the	the	DET
cana-3791	26	15	corresponding	corresponding	ADJ
cana-3791	26	16	hilbert	hilbert	NOUN
cana-3791	26	17	space	space	NOUN
cana-3791	26	18	.	.	PUNCT
cana-3791	27	1	the	the	DET
cana-3791	27	2	physical	physical	ADJ
cana-3791	27	3	realization	realization	NOUN
cana-3791	27	4	of	of	ADP
cana-3791	27	5	this	this	DET
cana-3791	27	6	theory	theory	NOUN
cana-3791	27	7	of	of	ADP
cana-3791	27	8	finite	finite	ADJ
cana-3791	27	9	-	-	ADJ
cana-3791	27	10	dimensional	dimensional	ADJ
cana-3791	27	11	hilbert	hilbert	NOUN
cana-3791	27	12	spaces	space	NOUN
cana-3791	27	13	has	have	AUX
cana-3791	27	14	been	be	AUX
cana-3791	27	15	especially	especially	ADV
cana-3791	27	16	appealing	appeal	VERB
cana-3791	27	17	in	in	ADP
cana-3791	27	18	quantum	quantum	NOUN
cana-3791	27	19	-	-	ADJ
cana-3791	27	20	optical	optical	ADJ
cana-3791	27	21	states	state	NOUN
cana-3791	27	22	and	and	CCONJ
cana-3791	27	23	digital	digital	ADJ
cana-3791	27	24	signal	signal	NOUN
cana-3791	27	25	processing	processing	NOUN
cana-3791	27	26	.	.	PUNCT
cana-3791	28	1	notably	notably	ADV
cana-3791	28	2	,	,	PUNCT
cana-3791	28	3	leonhardt	leonhardt	PROPN
cana-3791	28	4	's	's	PART
cana-3791	28	5	development	development	NOUN
cana-3791	28	6	of	of	ADP
cana-3791	28	7	discrete	discrete	ADJ
cana-3791	28	8	quantum	quantum	NOUN
cana-3791	28	9	-	-	PUNCT
cana-3791	28	10	state	state	NOUN
cana-3791	28	11	tomography	tomography	NOUN
cana-3791	28	12	[	[	X
cana-3791	28	13	11	11	NUM
cana-3791	28	14	]	]	PUNCT
cana-3791	28	15	and	and	CCONJ
cana-3791	28	16	further	further	ADJ
cana-3791	28	17	improvements	improvement	NOUN
cana-3791	28	18	in	in	ADP
cana-3791	28	19	measurement	measurement	NOUN
cana-3791	28	20	techniques	technique	NOUN
cana-3791	28	21	have	have	AUX
cana-3791	28	22	allowed	allow	VERB
cana-3791	28	23	truncation	truncation	NOUN
cana-3791	28	24	of	of	ADP
cana-3791	28	25	state	state	NOUN
cana-3791	28	26	space	space	NOUN
cana-3791	28	27	while	while	SCONJ
cana-3791	28	28	preserving	preserve	VERB
cana-3791	28	29	high	high	ADJ
cana-3791	28	30	accuracy	accuracy	NOUN
cana-3791	28	31	.	.	PUNCT
cana-3791	29	1	this	this	PRON
cana-3791	29	2	marks	mark	VERB
cana-3791	29	3	the	the	DET
cana-3791	29	4	point	point	NOUN
cana-3791	29	5	of	of	ADP
cana-3791	29	6	convergence	convergence	NOUN
cana-3791	29	7	between	between	ADP
cana-3791	29	8	the	the	DET
cana-3791	29	9	theoretical	theoretical	ADJ
cana-3791	29	10	framework	framework	NOUN
cana-3791	29	11	and	and	CCONJ
cana-3791	29	12	practical	practical	ADJ
cana-3791	29	13	application	application	NOUN
cana-3791	29	14	that	that	PRON
cana-3791	29	15	can	can	AUX
cana-3791	29	16	enable	enable	VERB
cana-3791	29	17	significant	significant	ADJ
cana-3791	29	18	improvement	improvement	NOUN
cana-3791	29	19	in	in	ADP
cana-3791	29	20	quantum	quantum	ADJ
cana-3791	29	21	information	information	NOUN
cana-3791	29	22	processing	processing	NOUN
cana-3791	29	23	,	,	PUNCT
cana-3791	29	24	error	error	NOUN
cana-3791	29	25	correction	correction	NOUN
cana-3791	29	26	codes	code	NOUN
cana-3791	29	27	,	,	PUNCT
cana-3791	29	28	and	and	CCONJ
cana-3791	29	29	quantum	quantum	ADJ
cana-3791	29	30	computation	computation	NOUN
cana-3791	29	31	algorithms	algorithm	NOUN
cana-3791	29	32	based	base	VERB
cana-3791	29	33	on	on	ADP
cana-3791	29	34	groundbreaking	groundbreake	VERB
cana-3791	29	35	work	work	NOUN
cana-3791	29	36	by	by	ADP
cana-3791	29	37	glauber	glauber	PROPN
cana-3791	30	1	[	[	X
cana-3791	30	2	12	12	NUM
cana-3791	30	3	]	]	PUNCT
cana-3791	30	4	and	and	CCONJ
cana-3791	30	5	sudarshan	sudarshan	PROPN
cana-3791	31	1	[	[	X
cana-3791	31	2	13	13	NUM
cana-3791	31	3	]	]	PUNCT
cana-3791	31	4	.	.	PUNCT
cana-3791	32	1	in	in	ADP
cana-3791	32	2	addition	addition	NOUN
cana-3791	32	3	,	,	PUNCT
cana-3791	32	4	finite	finite	ADJ
cana-3791	32	5	-	-	ADJ
cana-3791	32	6	dimensional	dimensional	ADJ
cana-3791	32	7	hilbert	hilbert	NOUN
cana-3791	32	8	spaces	space	NOUN
cana-3791	32	9	have	have	AUX
cana-3791	32	10	been	be	AUX
cana-3791	32	11	employed	employ	VERB
cana-3791	32	12	to	to	PART
cana-3791	32	13	attain	attain	VERB
cana-3791	32	14	further	further	ADJ
cana-3791	32	15	insight	insight	NOUN
cana-3791	32	16	into	into	ADP
cana-3791	32	17	the	the	DET
cana-3791	32	18	nature	nature	NOUN
cana-3791	32	19	and	and	CCONJ
cana-3791	32	20	quantification	quantification	NOUN
cana-3791	32	21	of	of	ADP
cana-3791	32	22	quantum	quantum	ADJ
cana-3791	32	23	entanglement	entanglement	NOUN
cana-3791	32	24	.	.	PUNCT
cana-3791	33	1	owing	owe	VERB
cana-3791	33	2	to	to	ADP
cana-3791	33	3	the	the	DET
cana-3791	33	4	mathematical	mathematical	ADJ
cana-3791	33	5	framework	framework	NOUN
cana-3791	33	6	utilized	utilize	VERB
cana-3791	33	7	herein	herein	NOUN
cana-3791	33	8	,	,	PUNCT
cana-3791	33	9	one	one	PRON
cana-3791	33	10	can	can	AUX
cana-3791	33	11	define	define	VERB
cana-3791	33	12	quantum	quantum	ADJ
cana-3791	33	13	states	state	NOUN
cana-3791	33	14	and	and	CCONJ
cana-3791	33	15	their	their	PRON
cana-3791	33	16	interaction	interaction	NOUN
cana-3791	33	17	in	in	ADP
cana-3791	33	18	extremity	extremity	NOUN
cana-3791	33	19	of	of	ADP
cana-3791	33	20	preciseness	preciseness	NOUN
cana-3791	33	21	,	,	PUNCT
cana-3791	33	22	which	which	PRON
cana-3791	33	23	may	may	AUX
cana-3791	33	24	be	be	AUX
cana-3791	33	25	represented	represent	VERB
cana-3791	33	26	via	via	ADP
cana-3791	33	27	the	the	DET
cana-3791	33	28	following	follow	VERB
cana-3791	33	29	fundamental	fundamental	ADJ
cana-3791	33	30	relationship	relationship	NOUN
cana-3791	33	31	from	from	ADP
cana-3791	33	32	[	[	X
cana-3791	33	33	14	14	NUM
cana-3791	33	34	]	]	SYM
cana-3791	33	35	:	:	PUNCT
cana-3791	34	1	|ψ⟩⁽ˢ⁾	|ψ⟩⁽ˢ⁾	PROPN
cana-3791	34	2	=	=	PRON
cana-3791	34	3	σₙ₌₀ˢ	σₙ₌₀ˢ	PROPN
cana-3791	34	4	c⁽ˢ⁾ₙ	c⁽ˢ⁾ₙ	PROPN
cana-3791	34	5	|n⟩	|n⟩	PROPN
cana-3791	34	6	≡	≡	PROPN
cana-3791	34	7	σₙ₌₀ˢ	σₙ₌₀ˢ	VERB
cana-3791	34	8	b⁽ˢ⁾ₙ	b⁽ˢ⁾ₙ	PROPN
cana-3791	34	9	eⁱᵠₙ	eⁱᵠₙ	VERB
cana-3791	34	10	|n⟩	|n⟩	ADJ
cana-3791	34	11	where	where	SCONJ
cana-3791	34	12	the	the	DET
cana-3791	34	13	superscript	superscript	PROPN
cana-3791	34	14	(	(	PUNCT
cana-3791	34	15	s	s	X
cana-3791	34	16	)	)	PUNCT
cana-3791	34	17	recalls	recall	VERB
cana-3791	34	18	that	that	SCONJ
cana-3791	34	19	space	space	NOUN
cana-3791	34	20	has	have	VERB
cana-3791	34	21	a	a	DET
cana-3791	34	22	finite	finite	ADJ
cana-3791	34	23	dimension	dimension	NOUN
cana-3791	34	24	and	and	CCONJ
cana-3791	34	25	the	the	DET
cana-3791	34	26	coefficients	coefficient	NOUN
cana-3791	34	27	are	be	AUX
cana-3791	34	28	the	the	DET
cana-3791	34	29	expansion	expansion	NOUN
cana-3791	34	30	coefficients	coefficient	NOUN
cana-3791	34	31	of	of	ADP
cana-3791	34	32	the	the	DET
cana-3791	34	33	quantum	quantum	ADJ
cana-3791	34	34	state	state	NOUN
cana-3791	34	35	over	over	ADP
cana-3791	34	36	some	some	DET
cana-3791	34	37	preselected	preselecte	VERB
cana-3791	34	38	basis	basis	NOUN
cana-3791	34	39	,	,	PUNCT
cana-3791	34	40	as	as	SCONJ
cana-3791	34	41	developed	develop	VERB
cana-3791	34	42	through	through	ADP
cana-3791	34	43	the	the	DET
cana-3791	34	44	work	work	NOUN
cana-3791	34	45	of	of	ADP
cana-3791	34	46	buˇzek	buˇzek	PROPN
cana-3791	34	47	and	and	CCONJ
cana-3791	34	48	coworkers	coworker	NOUN
cana-3791	34	49	[	[	X
cana-3791	34	50	15	15	NUM
cana-3791	34	51	]	]	PUNCT
cana-3791	34	52	.	.	PUNCT
cana-3791	35	1	applications	application	NOUN
cana-3791	35	2	of	of	ADP
cana-3791	35	3	the	the	DET
cana-3791	35	4	theory	theory	NOUN
cana-3791	35	5	of	of	ADP
cana-3791	35	6	finite	finite	ADJ
cana-3791	35	7	-	-	ADJ
cana-3791	35	8	dimensional	dimensional	ADJ
cana-3791	35	9	hilbert	hilbert	NOUN
cana-3791	35	10	spaces	space	NOUN
cana-3791	35	11	range	range	VERB
cana-3791	35	12	from	from	ADP
cana-3791	35	13	diverse	diverse	ADJ
cana-3791	35	14	areas	area	NOUN
cana-3791	35	15	of	of	ADP
cana-3791	35	16	modern	modern	ADJ
cana-3791	35	17	technology	technology	NOUN
cana-3791	35	18	and	and	CCONJ
cana-3791	35	19	scientific	scientific	ADJ
cana-3791	35	20	research	research	NOUN
cana-3791	35	21	.	.	PUNCT
cana-3791	36	1	specifically	specifically	ADV
cana-3791	36	2	,	,	PUNCT
cana-3791	36	3	signal	signal	ADJ
cana-3791	36	4	processing	processing	NOUN
cana-3791	36	5	is	be	AUX
cana-3791	36	6	one	one	NUM
cana-3791	36	7	domain	domain	NOUN
cana-3791	36	8	where	where	SCONJ
cana-3791	36	9	this	this	DET
cana-3791	36	10	framework	framework	NOUN
cana-3791	36	11	provides	provide	VERB
cana-3791	36	12	effective	effective	ADJ
cana-3791	36	13	methods	method	NOUN
cana-3791	36	14	of	of	ADP
cana-3791	36	15	signal	signal	ADJ
cana-3791	36	16	representation	representation	NOUN
cana-3791	36	17	and	and	CCONJ
cana-3791	36	18	analysis	analysis	NOUN
cana-3791	36	19	for	for	ADP
cana-3791	36	20	conditions	condition	NOUN
cana-3791	36	21	of	of	ADP
cana-3791	36	22	low	low	ADJ
cana-3791	36	23	bandwidth	bandwidth	ADJ
cana-3791	36	24	and/or	and/or	CCONJ
cana-3791	36	25	computational	computational	ADJ
cana-3791	36	26	resources	resource	NOUN
cana-3791	36	27	.	.	PUNCT
cana-3791	37	1	the	the	DET
cana-3791	37	2	methods	method	NOUN
cana-3791	37	3	have	have	AUX
cana-3791	37	4	also	also	ADV
cana-3791	37	5	found	find	VERB
cana-3791	37	6	applications	application	NOUN
cana-3791	37	7	in	in	ADP
cana-3791	37	8	numerical	numerical	ADJ
cana-3791	37	9	analysis	analysis	NOUN
cana-3791	37	10	,	,	PUNCT
cana-3791	37	11	where	where	SCONJ
cana-3791	37	12	the	the	DET
cana-3791	37	13	passage	passage	NOUN
cana-3791	37	14	to	to	ADP
cana-3791	37	15	a	a	DET
cana-3791	37	16	finite	finite	ADJ
cana-3791	37	17	-	-	ADJ
cana-3791	37	18	dimensional	dimensional	ADJ
cana-3791	37	19	approximation	approximation	NOUN
cana-3791	37	20	of	of	ADP
cana-3791	37	21	an	an	DET
cana-3791	37	22	infinitdimensional	infinitdimensional	ADJ
cana-3791	37	23	problem	problem	NOUN
cana-3791	37	24	yields	yield	NOUN
cana-3791	37	25	computationally	computationally	ADV
cana-3791	37	26	tractable	tractable	ADJ
cana-3791	37	27	solutions	solution	NOUN
cana-3791	37	28	with	with	ADP
cana-3791	37	29	controlled	control	VERB
cana-3791	37	30	error	error	NOUN
cana-3791	37	31	bounds	bound	NOUN
cana-3791	37	32	,	,	PUNCT
cana-3791	37	33	as	as	ADP
cana-3791	37	34	wootters	wootter	NOUN
cana-3791	37	35	[	[	X
cana-3791	37	36	16	16	NUM
cana-3791	37	37	]	]	PUNCT
cana-3791	37	38	and	and	CCONJ
cana-3791	37	39	leonhardt	leonhardt	PROPN
cana-3791	38	1	[	[	X
cana-3791	38	2	11	11	NUM
cana-3791	38	3	]	]	PUNCT
cana-3791	38	4	have	have	AUX
cana-3791	38	5	demonstrated	demonstrate	VERB
cana-3791	38	6	.	.	PUNCT
cana-3791	39	1	the	the	DET
cana-3791	39	2	paper	paper	NOUN
cana-3791	39	3	addresses	address	VERB
cana-3791	39	4	basic	basic	ADJ
cana-3791	39	5	problems	problem	NOUN
cana-3791	39	6	of	of	ADP
cana-3791	39	7	finite	finite	ADJ
cana-3791	39	8	-	-	ADJ
cana-3791	39	9	dimensional	dimensional	ADJ
cana-3791	39	10	hilbert	hilbert	NOUN
cana-3791	39	11	spaces	space	NOUN
cana-3791	39	12	,	,	PUNCT
cana-3791	39	13	with	with	ADP
cana-3791	39	14	particular	particular	ADJ
cana-3791	39	15	emphasis	emphasis	NOUN
cana-3791	39	16	given	give	VERB
cana-3791	39	17	to	to	ADP
cana-3791	39	18	the	the	DET
cana-3791	39	19	optimal	optimal	ADJ
cana-3791	39	20	problems	problem	NOUN
cana-3791	39	21	of	of	ADP
cana-3791	39	22	basis	basis	NOUN
cana-3791	39	23	and	and	CCONJ
cana-3791	39	24	to	to	ADP
cana-3791	39	25	their	their	PRON
cana-3791	39	26	applications	application	NOUN
cana-3791	39	27	in	in	ADP
cana-3791	39	28	signal	signal	NOUN
cana-3791	39	29	processing	processing	NOUN
cana-3791	39	30	and	and	CCONJ
cana-3791	39	31	numerical	numerical	ADJ
cana-3791	39	32	analysis	analysis	NOUN
cana-3791	39	33	.	.	PUNCT
cana-3791	40	1	we	we	PRON
cana-3791	40	2	introduce	introduce	VERB
cana-3791	40	3	new	new	ADJ
cana-3791	40	4	approaches	approach	NOUN
cana-3791	40	5	to	to	ADP
cana-3791	40	6	basis	basis	NOUN
cana-3791	40	7	selection	selection	NOUN
cana-3791	40	8	in	in	ADP
cana-3791	40	9	a	a	DET
cana-3791	40	10	way	way	NOUN
cana-3791	40	11	that	that	PRON
cana-3791	40	12	optimizes	optimize	VERB
cana-3791	40	13	computational	computational	ADJ
cana-3791	40	14	efficiency	efficiency	NOUN
cana-3791	40	15	and	and	CCONJ
cana-3791	40	16	accuracy	accuracy	NOUN
cana-3791	40	17	,	,	PUNCT
cana-3791	40	18	maintaining	maintain	VERB
cana-3791	40	19	the	the	DET
cana-3791	40	20	mathematical	mathematical	ADJ
cana-3791	40	21	rigor	rigor	NOUN
cana-3791	40	22	which	which	PRON
cana-3791	40	23	enables	enable	VERB
cana-3791	40	24	one	one	NUM
cana-3791	40	25	to	to	PART
cana-3791	40	26	conduct	conduct	VERB
cana-3791	40	27	theoretical	theoretical	ADJ
cana-3791	40	28	analysis	analysis	NOUN
cana-3791	40	29	.	.	PUNCT
cana-3791	41	1	both	both	CCONJ
cana-3791	41	2	the	the	DET
cana-3791	41	3	theoretical	theoretical	ADJ
cana-3791	41	4	and	and	CCONJ
cana-3791	41	5	practical	practical	ADJ
cana-3791	41	6	aspects	aspect	NOUN
cana-3791	41	7	of	of	ADP
cana-3791	41	8	our	our	PRON
cana-3791	41	9	investigation	investigation	NOUN
cana-3791	41	10	form	form	VERB
cana-3791	41	11	a	a	DET
cana-3791	41	12	general	general	ADJ
cana-3791	41	13	framework	framework	NOUN
cana-3791	41	14	for	for	ADP
cana-3791	41	15	understanding	understanding	NOUN
cana-3791	41	16	and	and	CCONJ
cana-3791	41	17	exploiting	exploit	VERB
cana-3791	41	18	the	the	DET
cana-3791	41	19	properties	property	NOUN
cana-3791	41	20	of	of	ADP
cana-3791	41	21	finitedimensional	finitedimensional	ADJ
cana-3791	41	22	hilbert	hilbert	NOUN
cana-3791	41	23	spaces	space	NOUN
cana-3791	41	24	in	in	ADP
cana-3791	41	25	modern	modern	ADJ
cana-3791	41	26	applications	application	NOUN
cana-3791	41	27	.	.	PUNCT
cana-3791	42	1	theoretical	theoretical	ADJ
cana-3791	42	2	framework	framework	NOUN
cana-3791	42	3	finite	finite	ADJ
cana-3791	42	4	-	-	ADJ
cana-3791	42	5	dimensional	dimensional	ADJ
cana-3791	42	6	hilbert	hilbert	NOUN
cana-3791	42	7	space	space	NOUN
cana-3791	42	8	fundamentals	fundamental	VERB
cana-3791	42	9	a	a	DET
cana-3791	42	10	finite	finite	ADJ
cana-3791	42	11	-	-	ADJ
cana-3791	42	12	dimensional	dimensional	ADJ
cana-3791	42	13	hilbert	hilbert	NOUN
cana-3791	42	14	space	space	NOUN
cana-3791	42	15	provides	provide	VERB
cana-3791	42	16	the	the	DET
cana-3791	42	17	mathematical	mathematical	ADJ
cana-3791	42	18	setting	setting	NOUN
cana-3791	42	19	for	for	ADP
cana-3791	42	20	describing	describe	VERB
cana-3791	42	21	quantum	quantum	NOUN
cana-3791	42	22	systems	system	NOUN
cana-3791	42	23	that	that	PRON
cana-3791	42	24	possess	possess	VERB
cana-3791	42	25	a	a	DET
cana-3791	42	26	finite	finite	ADJ
cana-3791	42	27	number	number	NOUN
cana-3791	42	28	of	of	ADP
cana-3791	42	29	degrees	degree	NOUN
cana-3791	42	30	of	of	ADP
cana-3791	42	31	freedom	freedom	NOUN
cana-3791	42	32	.	.	PUNCT
cana-3791	43	1	this	this	DET
cana-3791	43	2	section	section	NOUN
cana-3791	43	3	develops	develop	VERB
cana-3791	43	4	the	the	DET
cana-3791	43	5	basic	basic	ADJ
cana-3791	43	6	concepts	concept	NOUN
cana-3791	43	7	and	and	CCONJ
cana-3791	43	8	properties	property	NOUN
cana-3791	43	9	governing	govern	VERB
cana-3791	43	10	such	such	ADJ
cana-3791	43	11	spaces	space	NOUN
cana-3791	43	12	and	and	CCONJ
cana-3791	43	13	provides	provide	VERB
cana-3791	43	14	the	the	DET
cana-3791	43	15	necessary	necessary	ADJ
cana-3791	43	16	framework	framework	NOUN
cana-3791	43	17	that	that	PRON
cana-3791	43	18	subsequent	subsequent	ADJ
cana-3791	43	19	analysis	analysis	NOUN
cana-3791	43	20	and	and	CCONJ
cana-3791	43	21	applications	application	NOUN
cana-3791	43	22	require	require	VERB
cana-3791	43	23	.	.	PUNCT
cana-3791	44	1	communications	communication	NOUN
cana-3791	44	2	on	on	ADP
cana-3791	44	3	applied	apply	VERB
cana-3791	44	4	nonlinear	nonlinear	ADJ
cana-3791	44	5	analysis	analysis	NOUN
cana-3791	44	6	issn	issn	NOUN
cana-3791	44	7	:	:	PUNCT
cana-3791	44	8	1074	1074	NUM
cana-3791	44	9	-	-	PUNCT
cana-3791	44	10	133x	133x	NUM
cana-3791	44	11	vol	vol	NOUN
cana-3791	44	12	32	32	NUM
cana-3791	44	13	no	no	NOUN
cana-3791	44	14	.	.	PUNCT
cana-3791	45	1	8s	8s	PROPN
cana-3791	45	2	(	(	PUNCT
cana-3791	45	3	2025	2025	NUM
cana-3791	45	4	)	)	PUNCT
cana-3791	45	5	734	734	NUM
cana-3791	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	45	7	a	a	DET
cana-3791	45	8	finite	finite	ADJ
cana-3791	45	9	-	-	ADJ
cana-3791	45	10	dimensional	dimensional	ADJ
cana-3791	45	11	hilbert	hilbert	NOUN
cana-3791	45	12	space	space	NOUN
cana-3791	45	13	h	h	NOUN
cana-3791	45	14	is	be	AUX
cana-3791	45	15	a	a	DET
cana-3791	45	16	complex	complex	ADJ
cana-3791	45	17	vector	vector	NOUN
cana-3791	45	18	space	space	NOUN
cana-3791	45	19	of	of	ADP
cana-3791	45	20	dimension	dimension	NOUN
cana-3791	46	1	d	d	X
cana-3791	46	2	<	<	X
cana-3791	46	3	∞	∞	PROPN
cana-3791	46	4	,	,	PUNCT
cana-3791	46	5	equipped	equip	VERB
cana-3791	46	6	with	with	ADP
cana-3791	46	7	an	an	DET
cana-3791	46	8	inner	inner	ADJ
cana-3791	46	9	product	product	NOUN
cana-3791	46	10	⟨·|·⟩	⟨·|·⟩	ADP
cana-3791	46	11	inducing	induce	VERB
cana-3791	46	12	a	a	DET
cana-3791	46	13	norm	norm	NOUN
cana-3791	46	14	and	and	CCONJ
cana-3791	46	15	obeying	obey	VERB
cana-3791	46	16	the	the	DET
cana-3791	46	17	following	follow	VERB
cana-3791	46	18	properties	property	NOUN
cana-3791	46	19	for	for	ADP
cana-3791	46	20	all	all	DET
cana-3791	46	21	vectors	vector	NOUN
cana-3791	46	22	|ψ⟩	|ψ⟩	PROPN
cana-3791	46	23	,	,	PUNCT
cana-3791	46	24	|φ⟩	|φ⟩	NOUN
cana-3791	46	25	,	,	PUNCT
cana-3791	46	26	|χ⟩	|χ⟩	NOUN
cana-3791	46	27	∈	∈	PROPN
cana-3791	46	28	h	h	NOUN
cana-3791	46	29	and	and	CCONJ
cana-3791	46	30	for	for	ADP
cana-3791	46	31	all	all	DET
cana-3791	46	32	scalars	scalar	NOUN
cana-3791	46	33	α	α	X
cana-3791	46	34	∈	∈	PROPN
cana-3791	46	35	c	c	X
cana-3791	46	36	:	:	PUNCT
cana-3791	46	37	1	1	X
cana-3791	46	38	.	.	X
cana-3791	46	39	positive	positive	ADJ
cana-3791	46	40	definiteness	definiteness	NOUN
cana-3791	46	41	:	:	PUNCT
cana-3791	46	42	⟨ψ|ψ⟩	⟨ψ|ψ⟩	ADJ
cana-3791	46	43	≥	≥	NOUN
cana-3791	46	44	0	0	NUM
cana-3791	46	45	,	,	PUNCT
cana-3791	46	46	with	with	ADP
cana-3791	46	47	equality	equality	NOUN
cana-3791	46	48	if	if	SCONJ
cana-3791	46	49	and	and	CCONJ
cana-3791	46	50	only	only	ADV
cana-3791	46	51	if	if	SCONJ
cana-3791	46	52	|ψ⟩	|ψ⟩	PROPN
cana-3791	46	53	=	=	SYM
cana-3791	46	54	0	0	NUM
cana-3791	46	55	2	2	NUM
cana-3791	46	56	.	.	NOUN
cana-3791	46	57	conjugate	conjugate	ADJ
cana-3791	46	58	symmetry	symmetry	NOUN
cana-3791	46	59	:	:	PUNCT
cana-3791	46	60	⟨ψ|φ⟩	⟨ψ|φ⟩	NOUN
cana-3791	46	61	=	=	SYM
cana-3791	46	62	⟨φ|ψ⟩	⟨φ|ψ⟩	PROPN
cana-3791	46	63	*	*	SYM
cana-3791	46	64	3	3	X
cana-3791	46	65	.	.	X
cana-3791	46	66	linearity	linearity	NOUN
cana-3791	46	67	in	in	ADP
cana-3791	46	68	the	the	DET
cana-3791	46	69	second	second	ADJ
cana-3791	46	70	argument	argument	NOUN
cana-3791	46	71	:	:	PUNCT
cana-3791	46	72	⟨ψ|(α|φ⟩	⟨ψ|(α|φ⟩	NOUN
cana-3791	46	73	+	+	PUNCT
cana-3791	46	74	|χ⟩)⟩	|χ⟩)⟩	NOUN
cana-3791	46	75	=	=	SYM
cana-3791	46	76	α⟨ψ|φ⟩	α⟨ψ|φ⟩	PROPN
cana-3791	47	1	+	+	PRON
cana-3791	47	2	⟨ψ|χ⟩	⟨ψ|χ⟩	VERB
cana-3791	47	3	the	the	DET
cana-3791	47	4	completeness	completeness	NOUN
cana-3791	47	5	of	of	ADP
cana-3791	47	6	finite	finite	ADJ
cana-3791	47	7	-	-	ADJ
cana-3791	47	8	dimensional	dimensional	ADJ
cana-3791	47	9	hilbert	hilbert	NOUN
cana-3791	47	10	spaces	space	NOUN
cana-3791	47	11	is	be	AUX
cana-3791	47	12	implicitly	implicitly	ADV
cana-3791	47	13	met	meet	VERB
cana-3791	47	14	by	by	ADP
cana-3791	47	15	their	their	PRON
cana-3791	47	16	finite	finite	ADJ
cana-3791	47	17	dimensionality	dimensionality	NOUN
cana-3791	47	18	,	,	PUNCT
cana-3791	47	19	and	and	CCONJ
cana-3791	47	20	this	this	PRON
cana-3791	47	21	is	be	AUX
cana-3791	47	22	a	a	DET
cana-3791	47	23	big	big	ADJ
cana-3791	47	24	plus	plus	CCONJ
cana-3791	47	25	side	side	NOUN
cana-3791	47	26	compared	compare	VERB
cana-3791	47	27	to	to	ADP
cana-3791	47	28	infinite	infinite	ADJ
cana-3791	47	29	-	-	PUNCT
cana-3791	47	30	dimensional	dimensional	ADJ
cana-3791	47	31	spaces	space	NOUN
cana-3791	47	32	.	.	PUNCT
cana-3791	48	1	to	to	ADP
cana-3791	48	2	wit	wit	NOUN
cana-3791	48	3	,	,	PUNCT
cana-3791	48	4	it	it	PRON
cana-3791	48	5	ensures	ensure	VERB
cana-3791	48	6	that	that	SCONJ
cana-3791	48	7	any	any	DET
cana-3791	48	8	cauchy	cauchy	ADJ
cana-3791	48	9	sequence	sequence	NOUN
cana-3791	48	10	in	in	ADP
cana-3791	48	11	the	the	DET
cana-3791	48	12	space	space	NOUN
cana-3791	48	13	will	will	AUX
cana-3791	48	14	be	be	AUX
cana-3791	48	15	convergent	convergent	ADJ
cana-3791	48	16	to	to	ADP
cana-3791	48	17	an	an	DET
cana-3791	48	18	element	element	NOUN
cana-3791	48	19	belonging	belong	VERB
cana-3791	48	20	to	to	ADP
cana-3791	48	21	the	the	DET
cana-3791	48	22	space	space	NOUN
cana-3791	48	23	.	.	PUNCT
cana-3791	49	1	thanks	thank	NOUN
cana-3791	49	2	to	to	ADP
cana-3791	49	3	this	this	DET
cana-3791	49	4	property	property	NOUN
cana-3791	49	5	,	,	PUNCT
cana-3791	49	6	such	such	ADJ
cana-3791	49	7	spaces	space	NOUN
cana-3791	49	8	are	be	AUX
cana-3791	49	9	very	very	ADV
cana-3791	49	10	suitable	suitable	ADJ
cana-3791	49	11	for	for	ADP
cana-3791	49	12	numerical	numerical	ADJ
cana-3791	49	13	computations	computation	NOUN
cana-3791	49	14	and	and	CCONJ
cana-3791	49	15	practical	practical	ADJ
cana-3791	49	16	applications	application	NOUN
cana-3791	49	17	[	[	X
cana-3791	49	18	17	17	NUM
cana-3791	49	19	]	]	PUNCT
cana-3791	49	20	.	.	PUNCT
cana-3791	50	1	a	a	DET
cana-3791	50	2	basic	basic	ADJ
cana-3791	50	3	ingredient	ingredient	NOUN
cana-3791	50	4	in	in	ADP
cana-3791	50	5	finite	finite	ADJ
cana-3791	50	6	-	-	ADJ
cana-3791	50	7	dimensional	dimensional	ADJ
cana-3791	50	8	hilbert	hilbert	NOUN
cana-3791	50	9	spaces	space	NOUN
cana-3791	50	10	is	be	AUX
cana-3791	50	11	the	the	DET
cana-3791	50	12	existence	existence	NOUN
cana-3791	50	13	of	of	ADP
cana-3791	50	14	orthonormal	orthonormal	ADJ
cana-3791	50	15	bases	basis	NOUN
cana-3791	50	16	.	.	PUNCT
cana-3791	51	1	for	for	ADP
cana-3791	51	2	a	a	DET
cana-3791	51	3	ddimensional	ddimensional	ADJ
cana-3791	51	4	hilbert	hilbert	NOUN
cana-3791	51	5	space	space	NOUN
cana-3791	51	6	h	h	NOUN
cana-3791	51	7	,	,	PUNCT
cana-3791	51	8	one	one	PRON
cana-3791	51	9	has	have	VERB
cana-3791	51	10	that	that	SCONJ
cana-3791	51	11	an	an	DET
cana-3791	51	12	orthonormal	orthonormal	ADJ
cana-3791	51	13	basis	basis	NOUN
cana-3791	51	14	{	{	PUNCT
cana-3791	51	15	|ei⟩}di=1	|ei⟩}di=1	X
cana-3791	51	16	as	as	ADP
cana-3791	51	17	:	:	PUNCT
cana-3791	51	18	⟨ei|ej⟩	⟨ei|ej⟩	X
cana-3791	51	19	=	=	SYM
cana-3791	51	20	δij	δij	PROPN
cana-3791	51	21	where	where	SCONJ
cana-3791	51	22	δij	δij	NOUN
cana-3791	51	23	is	be	AUX
cana-3791	51	24	the	the	DET
cana-3791	51	25	kronecker	kronecker	NOUN
cana-3791	51	26	delta	delta	NOUN
cana-3791	51	27	.	.	PUNCT
cana-3791	52	1	for	for	ADP
cana-3791	52	2	this	this	DET
cana-3791	52	3	type	type	NOUN
cana-3791	52	4	of	of	ADP
cana-3791	52	5	basis	basis	NOUN
cana-3791	52	6	the	the	DET
cana-3791	52	7	completeness	completeness	NOUN
cana-3791	52	8	relation	relation	NOUN
cana-3791	52	9	looks	look	VERB
cana-3791	52	10	like	like	ADP
cana-3791	52	11	:	:	PUNCT
cana-3791	52	12	∑di=1	∑di=1	PROPN
cana-3791	52	13	|ei⟩⟨ei|	|ei⟩⟨ei|	NOUN
cana-3791	52	14	=	=	NOUN
cana-3791	52	15	1	1	NUM
cana-3791	52	16	where	where	SCONJ
cana-3791	52	17	1	1	NUM
cana-3791	52	18	denotes	denote	VERB
cana-3791	52	19	the	the	DET
cana-3791	52	20	identity	identity	NOUN
cana-3791	52	21	operator	operator	NOUN
cana-3791	52	22	in	in	ADP
cana-3791	52	23	h.	h.	PROPN
cana-3791	52	24	this	this	DET
cana-3791	52	25	relation	relation	NOUN
cana-3791	52	26	is	be	AUX
cana-3791	52	27	fundamental	fundamental	ADJ
cana-3791	52	28	to	to	ADP
cana-3791	52	29	quantum	quantum	ADJ
cana-3791	52	30	mechanics	mechanic	NOUN
cana-3791	52	31	and	and	CCONJ
cana-3791	52	32	signal	signal	NOUN
cana-3791	52	33	processing	processing	NOUN
cana-3791	52	34	,	,	PUNCT
cana-3791	52	35	because	because	SCONJ
cana-3791	52	36	it	it	PRON
cana-3791	52	37	allows	allow	VERB
cana-3791	52	38	for	for	ADP
cana-3791	52	39	the	the	DET
cana-3791	52	40	decomposing	decomposing	NOUN
cana-3791	52	41	of	of	ADP
cana-3791	52	42	any	any	DET
cana-3791	52	43	state	state	NOUN
cana-3791	52	44	and	and	CCONJ
cana-3791	52	45	any	any	DET
cana-3791	52	46	operator	operator	NOUN
cana-3791	52	47	into	into	ADP
cana-3791	52	48	basis	basis	NOUN
cana-3791	52	49	elements	element	NOUN
cana-3791	52	50	[	[	X
cana-3791	52	51	18	18	NUM
cana-3791	52	52	]	]	PUNCT
cana-3791	52	53	.	.	PUNCT
cana-3791	53	1	any	any	DET
cana-3791	53	2	vector	vector	NOUN
cana-3791	53	3	|ψ⟩	|ψ⟩	PROPN
cana-3791	53	4	∈	∈	PROPN
cana-3791	53	5	h	h	NOUN
cana-3791	53	6	can	can	AUX
cana-3791	53	7	be	be	AUX
cana-3791	53	8	decomposed	decompose	VERB
cana-3791	53	9	uniquely	uniquely	ADV
cana-3791	53	10	as	as	ADP
cana-3791	53	11	a	a	DET
cana-3791	53	12	linear	linear	ADJ
cana-3791	53	13	combination	combination	NOUN
cana-3791	53	14	of	of	ADP
cana-3791	53	15	basis	basis	NOUN
cana-3791	53	16	vectors	vector	NOUN
cana-3791	53	17	by	by	ADP
cana-3791	53	18	|ψ⟩	|ψ⟩	PROPN
cana-3791	53	19	=	=	PUNCT
cana-3791	54	1	∑di=1	∑di=1	PROPN
cana-3791	54	2	ci|ei⟩	ci|ei⟩	PROPN
cana-3791	54	3	where	where	SCONJ
cana-3791	54	4	the	the	DET
cana-3791	54	5	complex	complex	ADJ
cana-3791	54	6	coefficients	coefficient	NOUN
cana-3791	54	7	ci=⟨ei|ψ⟩	ci=⟨ei|ψ⟩	PROPN
cana-3791	54	8	are	be	AUX
cana-3791	54	9	determined	determine	VERB
cana-3791	54	10	by	by	ADP
cana-3791	54	11	means	mean	NOUN
cana-3791	54	12	of	of	ADP
cana-3791	54	13	the	the	DET
cana-3791	54	14	inner	inner	ADJ
cana-3791	54	15	product	product	NOUN
cana-3791	54	16	.	.	PUNCT
cana-3791	55	1	the	the	DET
cana-3791	55	2	finiteness	finiteness	NOUN
cana-3791	55	3	of	of	ADP
cana-3791	55	4	the	the	DET
cana-3791	55	5	dimension	dimension	NOUN
cana-3791	55	6	is	be	AUX
cana-3791	55	7	now	now	ADV
cana-3791	55	8	crucial	crucial	ADJ
cana-3791	55	9	in	in	SCONJ
cana-3791	55	10	that	that	SCONJ
cana-3791	55	11	this	this	DET
cana-3791	55	12	sum	sum	NOUN
cana-3791	55	13	is	be	AUX
cana-3791	55	14	always	always	ADV
cana-3791	55	15	well	well	ADV
cana-3791	55	16	-	-	PUNCT
cana-3791	55	17	defined	define	VERB
cana-3791	55	18	and	and	CCONJ
cana-3791	55	19	convergent	convergent	NOUN
cana-3791	55	20	.	.	PUNCT
cana-3791	56	1	in	in	ADP
cana-3791	56	2	contrast	contrast	NOUN
cana-3791	56	3	with	with	ADP
cana-3791	56	4	the	the	DET
cana-3791	56	5	case	case	NOUN
cana-3791	56	6	for	for	ADP
cana-3791	56	7	infinite	infinite	ADJ
cana-3791	56	8	-	-	PUNCT
cana-3791	56	9	dimensional	dimensional	ADJ
cana-3791	56	10	spaces	space	NOUN
cana-3791	56	11	,	,	PUNCT
cana-3791	56	12	there	there	PRON
cana-3791	56	13	is	be	VERB
cana-3791	56	14	no	no	DET
cana-3791	56	15	convergence	convergence	NOUN
cana-3791	56	16	problem	problem	NOUN
cana-3791	56	17	.	.	PUNCT
cana-3791	57	1	one	one	NUM
cana-3791	57	2	of	of	ADP
cana-3791	57	3	the	the	DET
cana-3791	57	4	fundamental	fundamental	ADJ
cana-3791	57	5	properties	property	NOUN
cana-3791	57	6	of	of	ADP
cana-3791	57	7	finite	finite	ADJ
cana-3791	57	8	-	-	ADJ
cana-3791	57	9	dimensional	dimensional	ADJ
cana-3791	57	10	hilbert	hilbert	NOUN
cana-3791	57	11	spaces	space	NOUN
cana-3791	57	12	is	be	AUX
cana-3791	57	13	the	the	DET
cana-3791	57	14	possibility	possibility	NOUN
cana-3791	57	15	of	of	ADP
cana-3791	57	16	dual	dual	ADJ
cana-3791	57	17	vectors	vector	NOUN
cana-3791	57	18	'	'	PART
cana-3791	57	19	existence	existence	NOUN
cana-3791	57	20	and	and	CCONJ
cana-3791	57	21	the	the	DET
cana-3791	57	22	riesz	riesz	PROPN
cana-3791	57	23	representation	representation	NOUN
cana-3791	57	24	theorem	theorem	NOUN
cana-3791	57	25	.	.	PUNCT
cana-3791	58	1	for	for	ADP
cana-3791	58	2	every	every	DET
cana-3791	58	3	linear	linear	ADJ
cana-3791	58	4	functional	functional	ADJ
cana-3791	58	5	f	f	PROPN
cana-3791	58	6	on	on	ADP
cana-3791	58	7	h	h	NOUN
cana-3791	58	8	,	,	PUNCT
cana-3791	58	9	there	there	PRON
cana-3791	58	10	uniquely	uniquely	ADV
cana-3791	58	11	corresponds	correspond	VERB
cana-3791	58	12	such	such	DET
cana-3791	58	13	a	a	DET
cana-3791	58	14	vector	vector	NOUN
cana-3791	58	15	|f⟩	|f⟩	ADJ
cana-3791	58	16	∈	∈	PROPN
cana-3791	58	17	h	h	NOUN
cana-3791	58	18	that	that	PRON
cana-3791	58	19	one	one	PRON
cana-3791	58	20	can	can	AUX
cana-3791	58	21	write	write	VERB
cana-3791	58	22	the	the	DET
cana-3791	58	23	following	following	NOUN
cana-3791	58	24	for	for	ADP
cana-3791	58	25	any	any	DET
cana-3791	58	26	|ψ⟩	|ψ⟩	PROPN
cana-3791	58	27	∈	∈	PROPN
cana-3791	58	28	h	h	NOUN
cana-3791	58	29	:	:	PUNCT
cana-3791	58	30	f(|ψ⟩	f(|ψ⟩	X
cana-3791	58	31	)	)	PUNCT
cana-3791	58	32	=	=	SYM
cana-3791	58	33	⟨f|ψ⟩.	⟨f|ψ⟩.	PROPN
cana-3791	58	34	this	this	DET
cana-3791	58	35	duality	duality	NOUN
cana-3791	58	36	becomes	become	VERB
cana-3791	58	37	especially	especially	ADV
cana-3791	58	38	important	important	ADJ
cana-3791	58	39	in	in	ADP
cana-3791	58	40	quantum	quantum	ADJ
cana-3791	58	41	mechanics	mechanic	NOUN
cana-3791	58	42	,	,	PUNCT
cana-3791	58	43	where	where	SCONJ
cana-3791	58	44	observables	observable	NOUN
cana-3791	58	45	are	be	AUX
cana-3791	58	46	represented	represent	VERB
cana-3791	58	47	by	by	ADP
cana-3791	58	48	self	self	NOUN
cana-3791	58	49	-	-	PUNCT
cana-3791	58	50	adjoint	adjoint	NOUN
cana-3791	58	51	operators	operator	NOUN
cana-3791	59	1	[	[	X
cana-3791	59	2	19	19	NUM
cana-3791	59	3	]	]	PUNCT
cana-3791	59	4	.	.	PUNCT
cana-3791	60	1	finite	finite	ADJ
cana-3791	60	2	-	-	ADJ
cana-3791	60	3	dimensional	dimensional	ADJ
cana-3791	60	4	hilbert	hilbert	NOUN
cana-3791	60	5	spaces	space	NOUN
cana-3791	60	6	possess	possess	VERB
cana-3791	60	7	particularly	particularly	ADV
cana-3791	60	8	nice	nice	ADJ
cana-3791	60	9	topology	topology	NOUN
cana-3791	60	10	:	:	PUNCT
cana-3791	60	11	all	all	DET
cana-3791	60	12	the	the	DET
cana-3791	60	13	norms	norm	NOUN
cana-3791	60	14	on	on	ADP
cana-3791	60	15	such	such	ADJ
cana-3791	60	16	spaces	space	NOUN
cana-3791	60	17	are	be	AUX
cana-3791	60	18	equivalent	equivalent	ADJ
cana-3791	60	19	,	,	PUNCT
cana-3791	60	20	and	and	CCONJ
cana-3791	60	21	every	every	DET
cana-3791	60	22	linear	linear	ADJ
cana-3791	60	23	operator	operator	NOUN
cana-3791	60	24	is	be	AUX
cana-3791	60	25	bounded	bound	VERB
cana-3791	60	26	.	.	PUNCT
cana-3791	61	1	moreover	moreover	ADV
cana-3791	61	2	,	,	PUNCT
cana-3791	61	3	any	any	DET
cana-3791	61	4	subspace	subspace	NOUN
cana-3791	61	5	in	in	ADP
cana-3791	61	6	a	a	DET
cana-3791	61	7	finite	finite	ADJ
cana-3791	61	8	-	-	ADJ
cana-3791	61	9	dimensional	dimensional	ADJ
cana-3791	61	10	hilbert	hilbert	NOUN
cana-3791	61	11	space	space	NOUN
cana-3791	61	12	is	be	AUX
cana-3791	61	13	closed	close	VERB
cana-3791	61	14	,	,	PUNCT
cana-3791	61	15	and	and	CCONJ
cana-3791	61	16	any	any	DET
cana-3791	61	17	linear	linear	ADJ
cana-3791	61	18	operator	operator	NOUN
cana-3791	61	19	in	in	ADP
cana-3791	61	20	such	such	DET
cana-3791	61	21	a	a	DET
cana-3791	61	22	space	space	NOUN
cana-3791	61	23	is	be	AUX
cana-3791	61	24	continuous	continuous	ADJ
cana-3791	61	25	.	.	PUNCT
cana-3791	62	1	these	these	DET
cana-3791	62	2	properties	property	NOUN
cana-3791	62	3	simplify	simplify	VERB
cana-3791	62	4	most	most	ADJ
cana-3791	62	5	of	of	ADP
cana-3791	62	6	the	the	DET
cana-3791	62	7	mathematical	mathematical	ADJ
cana-3791	62	8	considerations	consideration	NOUN
cana-3791	62	9	and	and	CCONJ
cana-3791	62	10	physical	physical	ADJ
cana-3791	62	11	applications	application	NOUN
cana-3791	62	12	immensely	immensely	ADV
cana-3791	62	13	compared	compare	VERB
cana-3791	62	14	to	to	ADP
cana-3791	62	15	their	their	PRON
cana-3791	62	16	infinite	infinite	ADJ
cana-3791	62	17	-	-	PUNCT
cana-3791	62	18	dimensional	dimensional	ADJ
cana-3791	62	19	versions	version	NOUN
cana-3791	62	20	[	[	X
cana-3791	62	21	20	20	NUM
cana-3791	62	22	]	]	PUNCT
cana-3791	62	23	.	.	PUNCT
cana-3791	63	1	operator	operator	NOUN
cana-3791	63	2	theory	theory	NOUN
cana-3791	63	3	in	in	ADP
cana-3791	63	4	finite	finite	ADJ
cana-3791	63	5	dimensions	dimension	NOUN
cana-3791	63	6	based	base	VERB
cana-3791	63	7	on	on	ADP
cana-3791	63	8	the	the	DET
cana-3791	63	9	basic	basic	ADJ
cana-3791	63	10	notions	notion	NOUN
cana-3791	63	11	of	of	ADP
cana-3791	63	12	finite	finite	ADJ
cana-3791	63	13	-	-	ADJ
cana-3791	63	14	dimensional	dimensional	ADJ
cana-3791	63	15	hilbert	hilbert	NOUN
cana-3791	63	16	spaces	space	NOUN
cana-3791	63	17	,	,	PUNCT
cana-3791	63	18	operator	operator	NOUN
cana-3791	63	19	theory	theory	NOUN
cana-3791	63	20	plays	play	VERB
cana-3791	63	21	an	an	DET
cana-3791	63	22	important	important	ADJ
cana-3791	63	23	role	role	NOUN
cana-3791	63	24	both	both	CCONJ
cana-3791	63	25	in	in	ADP
cana-3791	63	26	the	the	DET
cana-3791	63	27	tools	tool	NOUN
cana-3791	63	28	developed	develop	VERB
cana-3791	63	29	for	for	ADP
cana-3791	63	30	the	the	DET
cana-3791	63	31	analysis	analysis	NOUN
cana-3791	63	32	of	of	ADP
cana-3791	63	33	quantum	quantum	NOUN
cana-3791	63	34	systems	system	NOUN
cana-3791	63	35	and	and	CCONJ
cana-3791	63	36	in	in	ADP
cana-3791	63	37	signal	signal	ADJ
cana-3791	63	38	processing	processing	NOUN
cana-3791	63	39	applications	application	NOUN
cana-3791	63	40	.	.	PUNCT
cana-3791	64	1	in	in	ADP
cana-3791	64	2	this	this	DET
cana-3791	64	3	case	case	NOUN
cana-3791	64	4	,	,	PUNCT
cana-3791	64	5	the	the	DET
cana-3791	64	6	finiteness	finiteness	NOUN
cana-3791	64	7	of	of	ADP
cana-3791	64	8	the	the	DET
cana-3791	64	9	dimension	dimension	NOUN
cana-3791	64	10	makes	make	VERB
cana-3791	64	11	the	the	DET
cana-3791	64	12	operators	operator	NOUN
cana-3791	64	13	especially	especially	ADV
cana-3791	64	14	convenient	convenient	ADJ
cana-3791	64	15	:	:	PUNCT
cana-3791	64	16	let	let	VERB
cana-3791	64	17	h	h	NOUN
cana-3791	64	18	be	be	AUX
cana-3791	64	19	a	a	DET
cana-3791	64	20	finite	finite	ADJ
cana-3791	64	21	-	-	ADJ
cana-3791	64	22	dimensional	dimensional	ADJ
cana-3791	64	23	hilbert	hilbert	NOUN
cana-3791	64	24	space	space	NOUN
cana-3791	64	25	of	of	ADP
cana-3791	64	26	dimension	dimension	PROPN
cana-3791	64	27	d.	d.	PROPN
cana-3791	64	28	a	a	DET
cana-3791	64	29	linear	linear	ADJ
cana-3791	64	30	operator	operator	NOUN
cana-3791	64	31	a	a	DET
cana-3791	64	32	:	:	PUNCT
cana-3791	64	33	h	h	NOUN
cana-3791	64	34	→	→	SYM
cana-3791	64	35	h	h	NOUN
cana-3791	64	36	maps	map	VERB
cana-3791	64	37	the	the	DET
cana-3791	64	38	space	space	NOUN
cana-3791	64	39	into	into	ADP
cana-3791	64	40	itself	itself	PRON
cana-3791	64	41	.	.	PUNCT
cana-3791	65	1	an	an	DET
cana-3791	65	2	important	important	ADJ
cana-3791	65	3	simplification	simplification	NOUN
cana-3791	65	4	to	to	ADP
cana-3791	65	5	the	the	DET
cana-3791	65	6	infinite	infinite	ADJ
cana-3791	65	7	-	-	PUNCT
cana-3791	65	8	dimensional	dimensional	ADJ
cana-3791	65	9	case	case	NOUN
cana-3791	65	10	is	be	AUX
cana-3791	65	11	that	that	SCONJ
cana-3791	65	12	all	all	DET
cana-3791	65	13	linear	linear	ADJ
cana-3791	65	14	operators	operator	NOUN
cana-3791	65	15	are	be	AUX
cana-3791	65	16	automatically	automatically	ADV
cana-3791	65	17	bounded	bound	VERB
cana-3791	65	18	in	in	ADP
cana-3791	65	19	finite	finite	ADJ
cana-3791	65	20	dimensions	dimension	NOUN
cana-3791	65	21	.	.	PUNCT
cana-3791	66	1	the	the	DET
cana-3791	66	2	operator	operator	NOUN
cana-3791	66	3	norm	norm	NOUN
cana-3791	66	4	of	of	ADP
cana-3791	66	5	a	a	PRON
cana-3791	66	6	can	can	AUX
cana-3791	66	7	be	be	AUX
cana-3791	66	8	defined	define	VERB
cana-3791	66	9	by	by	ADP
cana-3791	66	10	:	:	PUNCT
cana-3791	66	11	||a||	||a||	VERB
cana-3791	66	12	=	=	PUNCT
cana-3791	66	13	sup{||ax||	sup{||ax||	NOUN
cana-3791	66	14	:	:	PUNCT
cana-3791	66	15	||x||	||x||	ADP
cana-3791	66	16	=	=	SYM
cana-3791	66	17	1	1	X
cana-3791	66	18	}	}	PUNCT
cana-3791	66	19	where	where	SCONJ
cana-3791	66	20	||·||	||·||	NOUN
cana-3791	66	21	denotes	denote	VERB
cana-3791	66	22	the	the	DET
cana-3791	66	23	norm	norm	NOUN
cana-3791	66	24	induced	induce	VERB
cana-3791	66	25	by	by	ADP
cana-3791	66	26	the	the	DET
cana-3791	66	27	inner	inner	ADJ
cana-3791	66	28	product	product	NOUN
cana-3791	66	29	.	.	PUNCT
cana-3791	67	1	this	this	DET
cana-3791	67	2	norm	norm	NOUN
cana-3791	67	3	is	be	AUX
cana-3791	67	4	finite	finite	ADJ
cana-3791	67	5	for	for	ADP
cana-3791	67	6	all	all	DET
cana-3791	67	7	operators	operator	NOUN
cana-3791	67	8	in	in	ADP
cana-3791	67	9	finite	finite	ADJ
cana-3791	67	10	dimensions	dimension	NOUN
cana-3791	67	11	,	,	PUNCT
cana-3791	67	12	a	a	DET
cana-3791	67	13	property	property	NOUN
cana-3791	67	14	which	which	PRON
cana-3791	67	15	follows	follow	VERB
cana-3791	67	16	directly	directly	ADV
cana-3791	67	17	from	from	ADP
cana-3791	67	18	the	the	DET
cana-3791	67	19	completeness	completeness	NOUN
cana-3791	67	20	of	of	ADP
cana-3791	67	21	the	the	DET
cana-3791	67	22	space	space	NOUN
cana-3791	67	23	[	[	X
cana-3791	67	24	21	21	NUM
cana-3791	67	25	]	]	PUNCT
cana-3791	67	26	.	.	PUNCT
cana-3791	68	1	the	the	DET
cana-3791	68	2	adjoint	adjoint	NOUN
cana-3791	68	3	operator	operator	NOUN
cana-3791	68	4	plays	play	VERB
cana-3791	68	5	a	a	DET
cana-3791	68	6	fundamental	fundamental	ADJ
cana-3791	68	7	role	role	NOUN
cana-3791	68	8	both	both	CCONJ
cana-3791	68	9	in	in	ADP
cana-3791	68	10	quantum	quantum	ADJ
cana-3791	68	11	mechanics	mechanic	NOUN
cana-3791	68	12	and	and	CCONJ
cana-3791	68	13	in	in	ADP
cana-3791	68	14	signal	signal	ADJ
cana-3791	68	15	analysis	analysis	NOUN
cana-3791	68	16	.	.	PUNCT
cana-3791	69	1	for	for	ADP
cana-3791	69	2	any	any	DET
cana-3791	69	3	operator	operator	NOUN
cana-3791	69	4	a	a	DET
cana-3791	69	5	its	its	PRON
cana-3791	69	6	adjoint	adjoint	NOUN
cana-3791	69	7	a†	a†	NOUN
cana-3791	69	8	is	be	AUX
cana-3791	69	9	uniquely	uniquely	ADV
cana-3791	69	10	defined	define	VERB
cana-3791	69	11	by	by	ADP
cana-3791	69	12	the	the	DET
cana-3791	69	13	relation	relation	NOUN
cana-3791	69	14	⟨a†x|y⟩	⟨a†x|y⟩	PROPN
cana-3791	69	15	=	=	SYM
cana-3791	69	16	⟨x|ay⟩	⟨x|ay⟩	PROPN
cana-3791	69	17	for	for	ADP
cana-3791	69	18	all	all	DET
cana-3791	69	19	x	x	NOUN
cana-3791	69	20	,	,	PUNCT
cana-3791	69	21	y	y	PROPN
cana-3791	69	22	∈	∈	PROPN
cana-3791	69	23	h.	h.	PROPN
cana-3791	69	24	in	in	ADP
cana-3791	69	25	finite	finite	ADJ
cana-3791	69	26	dimensions	dimension	NOUN
cana-3791	69	27	the	the	DET
cana-3791	69	28	adjoint	adjoint	NOUN
cana-3791	69	29	always	always	ADV
cana-3791	69	30	exists	exist	VERB
cana-3791	69	31	and	and	CCONJ
cana-3791	69	32	may	may	AUX
cana-3791	69	33	be	be	AUX
cana-3791	69	34	taken	take	VERB
cana-3791	69	35	to	to	PART
cana-3791	69	36	be	be	AUX
cana-3791	69	37	the	the	DET
cana-3791	69	38	conjugate	conjugate	ADJ
cana-3791	69	39	transpose	transpose	NOUN
cana-3791	69	40	of	of	ADP
cana-3791	69	41	the	the	DET
cana-3791	69	42	matrix	matrix	NOUN
cana-3791	69	43	representation	representation	NOUN
cana-3791	69	44	of	of	ADP
cana-3791	69	45	the	the	DET
cana-3791	69	46	operator	operator	NOUN
cana-3791	69	47	,	,	PUNCT
cana-3791	69	48	introducing	introduce	VERB
cana-3791	69	49	the	the	DET
cana-3791	69	50	crucial	crucial	ADJ
cana-3791	69	51	class	class	NOUN
cana-3791	69	52	of	of	ADP
cana-3791	69	53	self	self	NOUN
cana-3791	69	54	-	-	PUNCT
cana-3791	69	55	adjoint	adjoint	NOUN
cana-3791	69	56	operators	operator	NOUN
cana-3791	69	57	a	a	DET
cana-3791	69	58	=	=	SYM
cana-3791	69	59	a†	a†	NOUN
cana-3791	69	60	which	which	PRON
cana-3791	69	61	play	play	VERB
cana-3791	69	62	a	a	DET
cana-3791	69	63	central	central	ADJ
cana-3791	69	64	role	role	NOUN
cana-3791	69	65	in	in	ADP
cana-3791	69	66	the	the	DET
cana-3791	69	67	representation	representation	NOUN
cana-3791	69	68	of	of	ADP
cana-3791	69	69	physical	physical	ADJ
cana-3791	69	70	observables	observable	NOUN
cana-3791	70	1	[	[	X
cana-3791	70	2	22	22	NUM
cana-3791	70	3	]	]	PUNCT
cana-3791	70	4	.	.	PUNCT
cana-3791	71	1	communications	communication	NOUN
cana-3791	71	2	on	on	ADP
cana-3791	71	3	applied	apply	VERB
cana-3791	71	4	nonlinear	nonlinear	ADJ
cana-3791	71	5	analysis	analysis	NOUN
cana-3791	71	6	issn	issn	NOUN
cana-3791	71	7	:	:	PUNCT
cana-3791	71	8	1074	1074	NUM
cana-3791	71	9	-	-	PUNCT
cana-3791	71	10	133x	133x	NUM
cana-3791	71	11	vol	vol	NOUN
cana-3791	71	12	32	32	NUM
cana-3791	71	13	no	no	NOUN
cana-3791	71	14	.	.	PUNCT
cana-3791	72	1	8s	8s	PROPN
cana-3791	72	2	(	(	PUNCT
cana-3791	72	3	2025	2025	NUM
cana-3791	72	4	)	)	PUNCT
cana-3791	72	5	735	735	NUM
cana-3791	72	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	72	7	a	a	DET
cana-3791	72	8	fleshed	flesh	VERB
cana-3791	72	9	-	-	PUNCT
cana-3791	72	10	out	out	ADP
cana-3791	72	11	important	important	ADJ
cana-3791	72	12	related	relate	VERB
cana-3791	72	13	concept	concept	NOUN
cana-3791	72	14	from	from	ADP
cana-3791	72	15	1.pdf	1.pdf	NUM
cana-3791	72	16	is	be	AUX
cana-3791	72	17	that	that	PRON
cana-3791	72	18	of	of	ADP
cana-3791	72	19	a	a	DET
cana-3791	72	20	kernel	kernel	NOUN
cana-3791	72	21	operator	operator	NOUN
cana-3791	72	22	,	,	PUNCT
cana-3791	72	23	satisfying	satisfy	VERB
cana-3791	72	24	:	:	PUNCT
cana-3791	72	25	af(x	af(x	PUNCT
cana-3791	72	26	)	)	PUNCT
cana-3791	72	27	=	=	SYM
cana-3791	73	1	∫	∫	PROPN
cana-3791	73	2	k(x	k(x	PROPN
cana-3791	73	3	,	,	PUNCT
cana-3791	73	4	y)f(y)dy	y)f(y)dy	NOUN
cana-3791	73	5	where	where	SCONJ
cana-3791	73	6	k(x	k(x	PROPN
cana-3791	73	7	,	,	PUNCT
cana-3791	73	8	y	y	PROPN
cana-3791	73	9	)	)	PUNCT
cana-3791	73	10	is	be	AUX
cana-3791	73	11	the	the	DET
cana-3791	73	12	kernel	kernel	PROPN
cana-3791	73	13	function	function	NOUN
cana-3791	73	14	.	.	PUNCT
cana-3791	74	1	for	for	ADP
cana-3791	74	2	finite	finite	ADJ
cana-3791	74	3	dimensions	dimension	NOUN
cana-3791	74	4	,	,	PUNCT
cana-3791	74	5	this	this	DET
cana-3791	74	6	integral	integral	NOUN
cana-3791	74	7	takes	take	VERB
cana-3791	74	8	the	the	DET
cana-3791	74	9	form	form	NOUN
cana-3791	74	10	of	of	ADP
cana-3791	74	11	a	a	DET
cana-3791	74	12	finite	finite	ADJ
cana-3791	74	13	sum	sum	NOUN
cana-3791	74	14	af(x)=∑di=1	af(x)=∑di=1	PROPN
cana-3791	74	15	k(x	k(x	PROPN
cana-3791	74	16	,	,	PUNCT
cana-3791	74	17	xi)f(xi	xi)f(xi	NUM
cana-3791	74	18	)	)	PUNCT
cana-3791	74	19	the	the	DET
cana-3791	74	20	kernel	kernel	PROPN
cana-3791	74	21	stein	stein	PROPN
cana-3791	74	22	discrepancy	discrepancy	PROPN
cana-3791	74	23	[	[	X
cana-3791	74	24	23	23	NUM
cana-3791	74	25	]	]	PUNCT
cana-3791	74	26	as	as	ADP
cana-3791	74	27	a	a	DET
cana-3791	74	28	measure	measure	NOUN
cana-3791	74	29	of	of	ADP
cana-3791	74	30	the	the	DET
cana-3791	74	31	difference	difference	NOUN
cana-3791	74	32	between	between	ADP
cana-3791	74	33	probability	probability	NOUN
cana-3791	74	34	distributions	distribution	NOUN
cana-3791	74	35	uses	use	VERB
cana-3791	74	36	these	these	DET
cana-3791	74	37	operators	operator	NOUN
cana-3791	74	38	:	:	PUNCT
cana-3791	75	1	ksd(q	ksd(q	PROPN
cana-3791	75	2	,	,	PUNCT
cana-3791	75	3	p	p	NOUN
cana-3791	75	4	)	)	PUNCT
cana-3791	75	5	=	=	SYM
cana-3791	75	6	sup{eq[af(x	sup{eq[af(x	PROPN
cana-3791	75	7	)	)	PUNCT
cana-3791	75	8	]	]	PUNCT
cana-3791	75	9	:	:	PUNCT
cana-3791	75	10	f	f	X
cana-3791	75	11	∈	∈	PROPN
cana-3791	75	12	f	f	PROPN
cana-3791	75	13	,	,	PUNCT
cana-3791	75	14	||f||f	||f||f	PROPN
cana-3791	75	15	≤	≤	ADV
cana-3791	75	16	1	1	NUM
cana-3791	75	17	}	}	PUNCT
cana-3791	75	18	where	where	SCONJ
cana-3791	75	19	f	f	PROPN
cana-3791	75	20	is	be	AUX
cana-3791	75	21	appropriate	appropriate	ADJ
cana-3791	75	22	function	function	NOUN
cana-3791	75	23	space	space	NOUN
cana-3791	75	24	,	,	PUNCT
cana-3791	75	25	and	and	CCONJ
cana-3791	75	26	p	p	X
cana-3791	75	27	,	,	PUNCT
cana-3791	75	28	q	q	X
cana-3791	75	29	are	be	AUX
cana-3791	75	30	probability	probability	NOUN
cana-3791	75	31	measures	measure	NOUN
cana-3791	75	32	.	.	PUNCT
cana-3791	76	1	in	in	ADP
cana-3791	76	2	the	the	DET
cana-3791	76	3	case	case	NOUN
cana-3791	76	4	of	of	ADP
cana-3791	76	5	self	self	NOUN
cana-3791	76	6	-	-	PUNCT
cana-3791	76	7	adjoint	adjoint	NOUN
cana-3791	76	8	operators	operator	NOUN
cana-3791	76	9	in	in	ADP
cana-3791	76	10	finite	finite	ADJ
cana-3791	76	11	dimensions	dimension	NOUN
cana-3791	76	12	,	,	PUNCT
cana-3791	76	13	it	it	PRON
cana-3791	76	14	so	so	ADV
cana-3791	76	15	happens	happen	VERB
cana-3791	76	16	that	that	SCONJ
cana-3791	76	17	they	they	PRON
cana-3791	76	18	possess	possess	VERB
cana-3791	76	19	three	three	NUM
cana-3791	76	20	remarkable	remarkable	ADJ
cana-3791	76	21	properties	property	NOUN
cana-3791	76	22	.	.	PUNCT
cana-3791	77	1	they	they	PRON
cana-3791	77	2	are	be	AUX
cana-3791	77	3	as	as	SCONJ
cana-3791	77	4	follows	follow	VERB
cana-3791	77	5	:	:	PUNCT
cana-3791	77	6	their	their	PRON
cana-3791	77	7	spectrum	spectrum	NOUN
cana-3791	77	8	of	of	ADP
cana-3791	77	9	eigenvalues	eigenvalue	NOUN
cana-3791	77	10	is	be	AUX
cana-3791	77	11	real	real	ADJ
cana-3791	77	12	and	and	CCONJ
cana-3791	77	13	complete	complete	VERB
cana-3791	77	14	their	their	PRON
cana-3791	77	15	eigenvectors	eigenvector	NOUN
cana-3791	77	16	form	form	VERB
cana-3791	77	17	an	an	DET
cana-3791	77	18	orthonormal	orthonormal	ADJ
cana-3791	77	19	basis	basis	NOUN
cana-3791	77	20	they	they	PRON
cana-3791	77	21	can	can	AUX
cana-3791	77	22	be	be	AUX
cana-3791	77	23	diagonalized	diagonalize	VERB
cana-3791	77	24	using	use	VERB
cana-3791	77	25	a	a	DET
cana-3791	77	26	unitary	unitary	ADJ
cana-3791	77	27	transformation	transformation	NOUN
cana-3791	77	28	these	these	DET
cana-3791	77	29	three	three	NUM
cana-3791	77	30	properties	property	NOUN
cana-3791	77	31	arise	arise	VERB
cana-3791	77	32	from	from	ADP
cana-3791	77	33	the	the	DET
cana-3791	77	34	spectral	spectral	ADJ
cana-3791	77	35	theorem	theorem	NOUN
cana-3791	77	36	:	:	PUNCT
cana-3791	77	37	a	a	PRON
cana-3791	77	38	=	=	SYM
cana-3791	77	39	∑di=1	∑di=1	PROPN
cana-3791	77	40	λi|ei⟩⟨ei|\\	λi|ei⟩⟨ei|\\	X
cana-3791	77	41	where	where	SCONJ
cana-3791	77	42	λi	λi	ADP
cana-3791	77	43	real	real	ADJ
cana-3791	77	44	eigenvalues	eigenvalue	NOUN
cana-3791	77	45	and	and	CCONJ
cana-3791	77	46	|ei⟩	|ei⟩	VERB
cana-3791	77	47	the	the	DET
cana-3791	77	48	corresponding	corresponding	ADJ
cana-3791	77	49	eigenvectors	eigenvector	NOUN
cana-3791	77	50	[	[	X
cana-3791	77	51	24	24	NUM
cana-3791	77	52	]	]	PUNCT
cana-3791	77	53	.	.	PUNCT
cana-3791	78	1	but	but	CCONJ
cana-3791	78	2	let	let	VERB
cana-3791	78	3	the	the	DET
cana-3791	78	4	commutator	commutator	NOUN
cana-3791	78	5	of	of	ADP
cana-3791	78	6	two	two	NUM
cana-3791	78	7	operators	operator	NOUN
cana-3791	78	8	a	a	PRON
cana-3791	78	9	and	and	CCONJ
cana-3791	78	10	b	b	NOUN
cana-3791	78	11	,	,	PUNCT
cana-3791	78	12	[	[	X
cana-3791	78	13	a	a	X
cana-3791	78	14	,	,	PUNCT
cana-3791	78	15	b	b	NOUN
cana-3791	78	16	]	]	X
cana-3791	78	17	=	=	SYM
cana-3791	78	18	ab	ab	PROPN
cana-3791	78	19	ba	ba	PROPN
cana-3791	78	20	,	,	PUNCT
cana-3791	78	21	be	be	AUX
cana-3791	78	22	a	a	DET
cana-3791	78	23	centerpiece	centerpiece	NOUN
cana-3791	78	24	of	of	ADP
cana-3791	78	25	quantum	quantum	ADJ
cana-3791	78	26	mechanics	mechanic	NOUN
cana-3791	78	27	and	and	CCONJ
cana-3791	78	28	thus	thus	ADV
cana-3791	78	29	originate	originate	VERB
cana-3791	78	30	uncertainty	uncertainty	NOUN
cana-3791	78	31	relations	relation	NOUN
cana-3791	78	32	.	.	PUNCT
cana-3791	79	1	in	in	ADP
cana-3791	79	2	finite	finite	ADJ
cana-3791	79	3	-	-	ADJ
cana-3791	79	4	dimensional	dimensional	ADJ
cana-3791	79	5	spaces	space	NOUN
cana-3791	79	6	,	,	PUNCT
cana-3791	79	7	the	the	DET
cana-3791	79	8	trace	trace	NOUN
cana-3791	79	9	of	of	ADP
cana-3791	79	10	a	a	DET
cana-3791	79	11	commutator	commutator	NOUN
cana-3791	79	12	is	be	AUX
cana-3791	79	13	always	always	ADV
cana-3791	79	14	zero	zero	NUM
cana-3791	79	15	:	:	PUNCT
cana-3791	79	16	tr([a	tr([a	PROPN
cana-3791	79	17	,	,	PUNCT
cana-3791	79	18	b	b	NOUN
cana-3791	79	19	]	]	X
cana-3791	79	20	)	)	PUNCT
cana-3791	80	1	=	=	SYM
cana-3791	80	2	0	0	NUM
cana-3791	81	1	this	this	DET
cana-3791	81	2	property	property	NOUN
cana-3791	81	3	has	have	VERB
cana-3791	81	4	important	important	ADJ
cana-3791	81	5	implications	implication	NOUN
cana-3791	81	6	for	for	ADP
cana-3791	81	7	the	the	DET
cana-3791	81	8	structure	structure	NOUN
cana-3791	81	9	of	of	ADP
cana-3791	81	10	quantum	quantum	ADJ
cana-3791	81	11	mechanical	mechanical	ADJ
cana-3791	81	12	observables	observable	NOUN
cana-3791	81	13	and	and	CCONJ
cana-3791	81	14	their	their	PRON
cana-3791	81	15	relationships	relationship	NOUN
cana-3791	81	16	[	[	X
cana-3791	81	17	25	25	NUM
cana-3791	81	18	]	]	PUNCT
cana-3791	81	19	.	.	PUNCT
cana-3791	82	1	another	another	PRON
cana-3791	82	2	of	of	ADP
cana-3791	82	3	the	the	DET
cana-3791	82	4	most	most	ADV
cana-3791	82	5	important	important	ADJ
cana-3791	82	6	applications	application	NOUN
cana-3791	82	7	of	of	ADP
cana-3791	82	8	operator	operator	NOUN
cana-3791	82	9	theory	theory	NOUN
cana-3791	82	10	in	in	ADP
cana-3791	82	11	finite	finite	ADJ
cana-3791	82	12	dimensions	dimension	NOUN
cana-3791	82	13	concerns	concern	VERB
cana-3791	82	14	the	the	DET
cana-3791	82	15	construction	construction	NOUN
cana-3791	82	16	of	of	ADP
cana-3791	82	17	the	the	DET
cana-3791	82	18	discrete	discrete	ADJ
cana-3791	82	19	wigner	wigner	NOUN
cana-3791	82	20	function	function	NOUN
cana-3791	82	21	which	which	PRON
cana-3791	82	22	gives	give	VERB
cana-3791	82	23	the	the	DET
cana-3791	82	24	phase	phase	NOUN
cana-3791	82	25	-	-	PUNCT
cana-3791	82	26	space	space	NOUN
cana-3791	82	27	representation	representation	NOUN
cana-3791	82	28	to	to	ADP
cana-3791	82	29	quantum	quantum	ADJ
cana-3791	82	30	states	state	NOUN
cana-3791	82	31	.	.	PUNCT
cana-3791	83	1	indeed	indeed	ADV
cana-3791	83	2	,	,	PUNCT
cana-3791	83	3	as	as	SCONJ
cana-3791	83	4	it	it	PRON
cana-3791	83	5	is	be	AUX
cana-3791	83	6	possible	possible	ADJ
cana-3791	83	7	to	to	PART
cana-3791	83	8	derive	derive	VERB
cana-3791	83	9	in	in	ADP
cana-3791	83	10	[	[	X
cana-3791	83	11	26	26	NUM
cana-3791	83	12	]	]	PUNCT
cana-3791	83	13	,	,	PUNCT
cana-3791	83	14	in	in	ADP
cana-3791	83	15	a	a	DET
cana-3791	83	16	finite	finite	ADJ
cana-3791	83	17	-	-	ADJ
cana-3791	83	18	dimensional	dimensional	ADJ
cana-3791	83	19	setting	setting	NOUN
cana-3791	83	20	,	,	PUNCT
cana-3791	83	21	the	the	DET
cana-3791	83	22	discrete	discrete	ADJ
cana-3791	83	23	wigner	wigner	NOUN
cana-3791	83	24	function	function	NOUN
cana-3791	83	25	w(n	w(n	PROPN
cana-3791	83	26	,	,	PUNCT
cana-3791	83	27	θm	θm	NOUN
cana-3791	83	28	)	)	PUNCT
cana-3791	83	29	can	can	AUX
cana-3791	83	30	be	be	AUX
cana-3791	83	31	written	write	VERB
cana-3791	83	32	as	as	ADP
cana-3791	83	33	w(n	w(n	PROPN
cana-3791	83	34	,	,	PUNCT
cana-3791	83	35	θm	θm	NOUN
cana-3791	83	36	)	)	PUNCT
cana-3791	83	37	=	=	SYM
cana-3791	83	38	1/(s+1)2	1/(s+1)2	NUM
cana-3791	83	39	∑ν	∑ν	ADJ
cana-3791	83	40	,	,	PUNCT
cana-3791	83	41	μ	μ	PROPN
cana-3791	83	42	exp(4πi/(s+1)(nμ	exp(4πi/(s+1)(nμ	PROPN
cana-3791	83	43	+	+	CCONJ
cana-3791	83	44	νm))c(ν	νm))c(ν	PROPN
cana-3791	83	45	,	,	PUNCT
cana-3791	83	46	θμ	θμ	NOUN
cana-3791	83	47	)	)	PUNCT
cana-3791	83	48	where	where	SCONJ
cana-3791	83	49	s	s	NOUN
cana-3791	83	50	stands	stand	VERB
cana-3791	83	51	for	for	ADP
cana-3791	83	52	the	the	DET
cana-3791	83	53	dimension	dimension	NOUN
cana-3791	83	54	of	of	ADP
cana-3791	83	55	the	the	DET
cana-3791	83	56	space	space	NOUN
cana-3791	83	57	minus	minus	CCONJ
cana-3791	83	58	one	one	NUM
cana-3791	83	59	and	and	CCONJ
cana-3791	83	60	c(ν	c(ν	PROPN
cana-3791	83	61	,	,	PUNCT
cana-3791	83	62	θμ	θμ	NOUN
cana-3791	83	63	)	)	PUNCT
cana-3791	83	64	is	be	AUX
cana-3791	83	65	the	the	DET
cana-3791	83	66	characteristic	characteristic	ADJ
cana-3791	83	67	function	function	NOUN
cana-3791	83	68	.	.	PUNCT
cana-3791	84	1	optimal	optimal	ADJ
cana-3791	84	2	basis	basis	NOUN
cana-3791	84	3	selection	selection	NOUN
cana-3791	84	4	criteria	criterion	NOUN
cana-3791	84	5	for	for	ADP
cana-3791	84	6	basis	basis	NOUN
cana-3791	84	7	selection	selection	NOUN
cana-3791	84	8	the	the	DET
cana-3791	84	9	choice	choice	NOUN
cana-3791	84	10	of	of	ADP
cana-3791	84	11	an	an	DET
cana-3791	84	12	optimal	optimal	ADJ
cana-3791	84	13	basis	basis	NOUN
cana-3791	84	14	in	in	ADP
cana-3791	84	15	finite	finite	ADJ
cana-3791	84	16	-	-	ADJ
cana-3791	84	17	dimensional	dimensional	ADJ
cana-3791	84	18	hilbert	hilbert	NOUN
cana-3791	84	19	spaces	space	NOUN
cana-3791	84	20	is	be	AUX
cana-3791	84	21	a	a	DET
cana-3791	84	22	keystone	keystone	NOUN
cana-3791	84	23	issue	issue	NOUN
cana-3791	84	24	which	which	PRON
cana-3791	84	25	directly	directly	ADV
cana-3791	84	26	influences	influence	VERB
cana-3791	84	27	both	both	CCONJ
cana-3791	84	28	the	the	DET
cana-3791	84	29	representation	representation	NOUN
cana-3791	84	30	accuracy	accuracy	NOUN
cana-3791	84	31	and	and	CCONJ
cana-3791	84	32	computational	computational	ADJ
cana-3791	84	33	efficiency	efficiency	NOUN
cana-3791	84	34	.	.	PUNCT
cana-3791	85	1	by	by	ADP
cana-3791	85	2	referring	refer	VERB
cana-3791	85	3	to	to	ADP
cana-3791	85	4	the	the	DET
cana-3791	85	5	theory	theory	NOUN
cana-3791	85	6	of	of	ADP
cana-3791	85	7	quantum	quantum	NOUN
cana-3791	85	8	-	-	ADJ
cana-3791	85	9	optical	optical	ADJ
cana-3791	85	10	states	state	NOUN
cana-3791	85	11	,	,	PUNCT
cana-3791	85	12	we	we	PRON
cana-3791	85	13	can	can	AUX
cana-3791	85	14	have	have	VERB
cana-3791	85	15	the	the	DET
cana-3791	85	16	following	follow	VERB
cana-3791	85	17	strict	strict	ADJ
cana-3791	85	18	choice	choice	NOUN
cana-3791	85	19	criterion	criterion	NOUN
cana-3791	85	20	which	which	PRON
cana-3791	85	21	balances	balance	VERB
cana-3791	85	22	the	the	DET
cana-3791	85	23	two	two	NUM
cana-3791	85	24	mutually	mutually	ADV
cana-3791	85	25	conflicting	conflicting	ADJ
cana-3791	85	26	requirements	requirement	NOUN
cana-3791	85	27	.	.	PUNCT
cana-3791	86	1	in	in	ADP
cana-3791	86	2	a	a	DET
cana-3791	86	3	finite	finite	ADJ
cana-3791	86	4	-	-	ADJ
cana-3791	86	5	dimensional	dimensional	ADJ
cana-3791	86	6	hilbert	hilbert	NOUN
cana-3791	86	7	space	space	NOUN
cana-3791	86	8	hs	hs	PROPN
cana-3791	86	9	of	of	ADP
cana-3791	86	10	dimension	dimension	PROPN
cana-3791	86	11	s+1	s+1	PROPN
cana-3791	86	12	,	,	PUNCT
cana-3791	86	13	an	an	DET
cana-3791	86	14	arbitrary	arbitrary	ADJ
cana-3791	86	15	quantumoptical	quantumoptical	ADJ
cana-3791	86	16	pure	pure	ADJ
cana-3791	86	17	state	state	NOUN
cana-3791	86	18	can	can	AUX
cana-3791	86	19	be	be	AUX
cana-3791	86	20	represented	represent	VERB
cana-3791	86	21	via	via	ADP
cana-3791	86	22	its	its	PRON
cana-3791	86	23	fock	fock	ADJ
cana-3791	86	24	expansion	expansion	NOUN
cana-3791	86	25	[	[	X
cana-3791	86	26	27	27	NUM
cana-3791	86	27	]	]	X
cana-3791	86	28	:	:	PUNCT
cana-3791	86	29	|ψ⟩s=∑sn=0cs|n⟩≡∑sn=0bseiφn|n⟩	|ψ⟩s=∑sn=0cs|n⟩≡∑sn=0bseiφn|n⟩	PROPN
cana-3791	86	30	where	where	SCONJ
cana-3791	86	31	cs	cs	ADJ
cana-3791	86	32	=	=	NOUN
cana-3791	86	33	bseiφn	bseiφn	ADJ
cana-3791	86	34	are	be	AUX
cana-3791	86	35	the	the	DET
cana-3791	86	36	complex	complex	ADJ
cana-3791	86	37	superposition	superposition	NOUN
cana-3791	86	38	coefficients	coefficient	NOUN
cana-3791	86	39	and	and	CCONJ
cana-3791	86	40	bn	bn	NOUN
cana-3791	86	41	real	real	ADJ
cana-3791	86	42	coefficients	coefficient	NOUN
cana-3791	86	43	such	such	ADJ
cana-3791	86	44	that	that	SCONJ
cana-3791	86	45	the	the	DET
cana-3791	86	46	following	follow	VERB
cana-3791	86	47	normalization	normalization	NOUN
cana-3791	86	48	condition	condition	NOUN
cana-3791	86	49	holds	hold	VERB
cana-3791	86	50	:	:	PUNCT
cana-3791	86	51	∑sn=0bn2=1	∑sn=0bn2=1	PROPN
cana-3791	86	52	this	this	DET
cana-3791	86	53	representation	representation	NOUN
cana-3791	86	54	gives	give	VERB
cana-3791	86	55	the	the	DET
cana-3791	86	56	ground	ground	NOUN
cana-3791	86	57	for	for	ADP
cana-3791	86	58	the	the	DET
cana-3791	86	59	basis	basis	NOUN
cana-3791	86	60	selection	selection	NOUN
cana-3791	86	61	criteria	criterion	NOUN
cana-3791	86	62	.	.	PUNCT
cana-3791	87	1	the	the	DET
cana-3791	87	2	major	major	ADJ
cana-3791	87	3	difficulty	difficulty	NOUN
cana-3791	87	4	,	,	PUNCT
cana-3791	87	5	however	however	ADV
cana-3791	87	6	,	,	PUNCT
cana-3791	87	7	constitutes	constitute	VERB
cana-3791	87	8	an	an	DET
cana-3791	87	9	appropriate	appropriate	ADJ
cana-3791	87	10	truncation	truncation	NOUN
cana-3791	87	11	of	of	ADP
cana-3791	87	12	this	this	DET
cana-3791	87	13	infinite	infinite	ADJ
cana-3791	87	14	-	-	PUNCT
cana-3791	87	15	dimensional	dimensional	ADJ
cana-3791	87	16	space	space	NOUN
cana-3791	87	17	in	in	ADP
cana-3791	87	18	such	such	DET
cana-3791	87	19	a	a	DET
cana-3791	87	20	way	way	NOUN
cana-3791	87	21	that	that	PRON
cana-3791	87	22	enough	enough	ADJ
cana-3791	87	23	accuracy	accuracy	NOUN
cana-3791	87	24	for	for	ADP
cana-3791	87	25	practical	practical	ADJ
cana-3791	87	26	purposes	purpose	NOUN
cana-3791	87	27	is	be	AUX
cana-3791	87	28	preserved	preserve	VERB
cana-3791	87	29	[	[	PUNCT
cana-3791	87	30	28	28	NUM
cana-3791	87	31	]	]	PUNCT
cana-3791	87	32	.	.	PUNCT
cana-3791	88	1	one	one	PRON
cana-3791	88	2	can	can	AUX
cana-3791	88	3	quantify	quantify	VERB
cana-3791	88	4	the	the	DET
cana-3791	88	5	goodness	goodness	NOUN
cana-3791	88	6	of	of	ADP
cana-3791	88	7	a	a	DET
cana-3791	88	8	selected	select	VERB
cana-3791	88	9	basis	basis	NOUN
cana-3791	88	10	by	by	ADP
cana-3791	88	11	considering	consider	VERB
cana-3791	88	12	several	several	ADJ
cana-3791	88	13	important	important	ADJ
cana-3791	88	14	figures	figure	NOUN
cana-3791	88	15	of	of	ADP
cana-3791	88	16	merit	merit	NOUN
cana-3791	88	17	.	.	PUNCT
cana-3791	89	1	first	first	ADV
cana-3791	89	2	,	,	PUNCT
cana-3791	89	3	there	there	PRON
cana-3791	89	4	is	be	VERB
cana-3791	89	5	a	a	DET
cana-3791	89	6	question	question	NOUN
cana-3791	89	7	of	of	ADP
cana-3791	89	8	fidelity	fidelity	NOUN
cana-3791	89	9	between	between	ADP
cana-3791	89	10	the	the	DET
cana-3791	89	11	original	original	ADJ
cana-3791	89	12	state	state	NOUN
cana-3791	89	13	and	and	CCONJ
cana-3791	89	14	its	its	PRON
cana-3791	89	15	finitedimensional	finitedimensional	ADJ
cana-3791	89	16	approximation	approximation	NOUN
cana-3791	89	17	.	.	PUNCT
cana-3791	90	1	for	for	ADP
cana-3791	90	2	quantum	quantum	ADJ
cana-3791	90	3	states	state	NOUN
cana-3791	90	4	,	,	PUNCT
cana-3791	90	5	this	this	PRON
cana-3791	90	6	may	may	AUX
cana-3791	90	7	be	be	AUX
cana-3791	90	8	quantified	quantify	VERB
cana-3791	90	9	with	with	ADP
cana-3791	90	10	the	the	DET
cana-3791	90	11	use	use	NOUN
cana-3791	90	12	of	of	ADP
cana-3791	90	13	the	the	DET
cana-3791	90	14	discrete	discrete	ADJ
cana-3791	90	15	wigner	wigner	NOUN
cana-3791	90	16	function	function	NOUN
cana-3791	90	17	,	,	PUNCT
cana-3791	90	18	ws(n	ws(n	PROPN
cana-3791	90	19	,	,	PUNCT
cana-3791	90	20	θm	θm	NOUN
cana-3791	90	21	)	)	PUNCT
cana-3791	90	22	,	,	PUNCT
cana-3791	90	23	given	give	VERB
cana-3791	90	24	by	by	ADP
cana-3791	90	25	ws(n	ws(n	PROPN
cana-3791	90	26	,	,	PUNCT
cana-3791	90	27	θm	θm	NOUN
cana-3791	90	28	)	)	PUNCT
cana-3791	90	29	=	=	SYM
cana-3791	90	30	1/(s+1	1/(s+1	NUM
cana-3791	90	31	)	)	PUNCT
cana-3791	90	32	∑μ=0s	∑μ=0s	PUNCT
cana-3791	91	1	exp(4πi/(s+1)nμ)⟨θm	exp(4πi/(s+1)nμ)⟨θm	PROPN
cana-3791	91	2	-	-	PUNCT
cana-3791	91	3	μ|ρ̂|θm+μ⟩	μ|ρ̂|θm+μ⟩	PROPN
cana-3791	91	4	where	where	SCONJ
cana-3791	91	5	the	the	DET
cana-3791	91	6	index	index	NOUN
cana-3791	91	7	θm	θm	ADV
cana-3791	91	8	indicates	indicate	VERB
cana-3791	91	9	the	the	DET
cana-3791	91	10	phase	phase	NOUN
cana-3791	91	11	states	state	NOUN
cana-3791	91	12	and	and	CCONJ
cana-3791	91	13	ρ̂	ρ̂	NUM
cana-3791	91	14	is	be	AUX
cana-3791	91	15	a	a	DET
cana-3791	91	16	density	density	NOUN
cana-3791	91	17	operator	operator	NOUN
cana-3791	91	18	.	.	PUNCT
cana-3791	92	1	this	this	DET
cana-3791	92	2	function	function	NOUN
cana-3791	92	3	yields	yield	VERB
cana-3791	92	4	a	a	DET
cana-3791	92	5	phasespace	phasespace	NOUN
cana-3791	92	6	representation	representation	NOUN
cana-3791	92	7	helpful	helpful	ADJ
cana-3791	92	8	to	to	PART
cana-3791	92	9	quantify	quantify	VERB
cana-3791	92	10	the	the	DET
cana-3791	92	11	quality	quality	NOUN
cana-3791	92	12	of	of	ADP
cana-3791	92	13	the	the	DET
cana-3791	92	14	basis	basis	NOUN
cana-3791	92	15	approximation	approximation	NOUN
cana-3791	92	16	[	[	X
cana-3791	92	17	29	29	NUM
cana-3791	92	18	]	]	PUNCT
cana-3791	92	19	.	.	PUNCT
cana-3791	93	1	communications	communication	NOUN
cana-3791	93	2	on	on	ADP
cana-3791	93	3	applied	apply	VERB
cana-3791	93	4	nonlinear	nonlinear	ADJ
cana-3791	93	5	analysis	analysis	NOUN
cana-3791	93	6	issn	issn	NOUN
cana-3791	93	7	:	:	PUNCT
cana-3791	93	8	1074	1074	NUM
cana-3791	93	9	-	-	PUNCT
cana-3791	93	10	133x	133x	NUM
cana-3791	93	11	vol	vol	NOUN
cana-3791	93	12	32	32	NUM
cana-3791	93	13	no	no	NOUN
cana-3791	93	14	.	.	PUNCT
cana-3791	94	1	8s	8s	PROPN
cana-3791	94	2	(	(	PUNCT
cana-3791	94	3	2025	2025	NUM
cana-3791	94	4	)	)	PUNCT
cana-3791	94	5	736	736	NUM
cana-3791	94	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	94	7	we	we	PRON
cana-3791	94	8	note	note	VERB
cana-3791	94	9	that	that	SCONJ
cana-3791	94	10	the	the	DET
cana-3791	94	11	truncation	truncation	NOUN
cana-3791	94	12	level	level	NOUN
cana-3791	94	13	must	must	AUX
cana-3791	94	14	be	be	AUX
cana-3791	94	15	competently	competently	ADV
cana-3791	94	16	balanced	balance	VERB
cana-3791	94	17	with	with	ADP
cana-3791	94	18	the	the	DET
cana-3791	94	19	choice	choice	NOUN
cana-3791	94	20	of	of	ADP
cana-3791	94	21	the	the	DET
cana-3791	94	22	basis	basis	NOUN
cana-3791	94	23	states	state	NOUN
cana-3791	94	24	by	by	ADP
cana-3791	94	25	the	the	DET
cana-3791	94	26	minimization	minimization	NOUN
cana-3791	94	27	of	of	ADP
cana-3791	94	28	the	the	DET
cana-3791	94	29	approximation	approximation	NOUN
cana-3791	94	30	error	error	NOUN
cana-3791	94	31	.	.	PUNCT
cana-3791	95	1	for	for	ADP
cana-3791	95	2	a	a	DET
cana-3791	95	3	given	give	VERB
cana-3791	95	4	s+1	s+1	NUM
cana-3791	95	5	dimension	dimension	NOUN
cana-3791	95	6	,	,	PUNCT
cana-3791	95	7	the	the	DET
cana-3791	95	8	optimal	optimal	ADJ
cana-3791	95	9	basis	basis	NOUN
cana-3791	95	10	must	must	AUX
cana-3791	95	11	minimize	minimize	VERB
cana-3791	95	12	the	the	DET
cana-3791	95	13	error	error	NOUN
cana-3791	95	14	functional	functional	ADJ
cana-3791	95	15	e	e	NOUN
cana-3791	95	16	=	=	SYM
cana-3791	95	17	||ψ	||ψ	PROPN
cana-3791	95	18	ψs||2	ψs||2	ADV
cana-3791	95	19	where	where	SCONJ
cana-3791	95	20	ψs	ψs	NOUN
cana-3791	95	21	is	be	AUX
cana-3791	95	22	the	the	DET
cana-3791	95	23	truncated	truncated	ADJ
cana-3791	95	24	approximation	approximation	NOUN
cana-3791	95	25	in	in	ADP
cana-3791	95	26	the	the	DET
cana-3791	95	27	s+1	s+1	ADJ
cana-3791	95	28	dimensional	dimensional	ADJ
cana-3791	95	29	space	space	NOUN
cana-3791	95	30	.	.	PUNCT
cana-3791	96	1	this	this	DET
cana-3791	96	2	error	error	NOUN
cana-3791	96	3	can	can	AUX
cana-3791	96	4	be	be	AUX
cana-3791	96	5	explicitly	explicitly	ADV
cana-3791	96	6	written	write	VERB
cana-3791	96	7	as	as	ADP
cana-3791	96	8	:	:	PUNCT
cana-3791	96	9	e	e	NOUN
cana-3791	96	10	=	=	SYM
cana-3791	96	11	1	1	NUM
cana-3791	96	12	∑sn=0	∑sn=0	PROPN
cana-3791	96	13	|⟨n|ψ⟩|2	|⟨n|ψ⟩|2	PROPN
cana-3791	96	14	it	it	PRON
cana-3791	96	15	is	be	AUX
cana-3791	96	16	equally	equally	ADV
cana-3791	96	17	important	important	ADJ
cana-3791	96	18	,	,	PUNCT
cana-3791	96	19	however	however	ADV
cana-3791	96	20	,	,	PUNCT
cana-3791	96	21	that	that	SCONJ
cana-3791	96	22	the	the	DET
cana-3791	96	23	basis	basis	NOUN
cana-3791	96	24	utilized	utilize	VERB
cana-3791	96	25	is	be	AUX
cana-3791	96	26	computationally	computationally	ADV
cana-3791	96	27	efficient	efficient	ADJ
cana-3791	96	28	.	.	PUNCT
cana-3791	97	1	for	for	ADP
cana-3791	97	2	an	an	DET
cana-3791	97	3	actual	actual	ADJ
cana-3791	97	4	implementation	implementation	NOUN
cana-3791	97	5	,	,	PUNCT
cana-3791	97	6	the	the	DET
cana-3791	97	7	basis	basis	NOUN
cana-3791	97	8	has	have	VERB
cana-3791	97	9	to	to	PART
cana-3791	97	10	be	be	AUX
cana-3791	97	11	chosen	choose	VERB
cana-3791	97	12	with	with	ADP
cana-3791	97	13	consideration	consideration	NOUN
cana-3791	97	14	of	of	ADP
cana-3791	97	15	:	:	PUNCT
cana-3791	97	16	1	1	X
cana-3791	97	17	.	.	X
cana-3791	97	18	sparsity	sparsity	NOUN
cana-3791	97	19	of	of	ADP
cana-3791	97	20	the	the	DET
cana-3791	97	21	representation	representation	NOUN
cana-3791	97	22	2	2	NUM
cana-3791	97	23	.	.	PUNCT
cana-3791	97	24	computational	computational	ADJ
cana-3791	97	25	cost	cost	NOUN
cana-3791	97	26	of	of	ADP
cana-3791	97	27	basis	basis	NOUN
cana-3791	97	28	transformations	transformation	NOUN
cana-3791	97	29	3	3	NUM
cana-3791	97	30	.	.	PUNCT
cana-3791	97	31	numerical	numerical	ADJ
cana-3791	97	32	stability	stability	NOUN
cana-3791	97	33	of	of	ADP
cana-3791	97	34	the	the	DET
cana-3791	97	35	algorithms	algorithm	NOUN
cana-3791	97	36	that	that	PRON
cana-3791	97	37	result	result	VERB
cana-3791	97	38	these	these	DET
cana-3791	97	39	considerations	consideration	NOUN
cana-3791	97	40	can	can	AUX
cana-3791	97	41	be	be	AUX
cana-3791	97	42	quantified	quantify	VERB
cana-3791	97	43	through	through	ADP
cana-3791	97	44	the	the	DET
cana-3791	97	45	phase	phase	NOUN
cana-3791	97	46	probability	probability	NOUN
cana-3791	97	47	distribution	distribution	NOUN
cana-3791	97	48	:	:	PUNCT
cana-3791	97	49	ps(θm	ps(θm	X
cana-3791	97	50	)	)	PUNCT
cana-3791	98	1	=	=	SYM
cana-3791	99	1	∑sn=0	∑sn=0	PROPN
cana-3791	99	2	ws(n	ws(n	PROPN
cana-3791	99	3	,	,	PUNCT
cana-3791	99	4	θm	θm	NOUN
cana-3791	99	5	)	)	PUNCT
cana-3791	99	6	=	=	SYM
cana-3791	99	7	|⟨θm|ψ⟩|2	|⟨θm|ψ⟩|2	NOUN
cana-3791	99	8	and	and	CCONJ
cana-3791	99	9	the	the	DET
cana-3791	99	10	photon	photon	NOUN
cana-3791	99	11	-	-	PUNCT
cana-3791	99	12	number	number	NOUN
cana-3791	99	13	distribution	distribution	NOUN
cana-3791	99	14	:	:	PUNCT
cana-3791	99	15	ps(n	ps(n	X
cana-3791	99	16	)	)	PUNCT
cana-3791	99	17	=	=	SYM
cana-3791	100	1	∑sm=0	∑sm=0	PROPN
cana-3791	100	2	ws(n	ws(n	NOUN
cana-3791	100	3	,	,	PUNCT
cana-3791	100	4	θm	θm	NOUN
cana-3791	100	5	)	)	PUNCT
cana-3791	100	6	=	=	SYM
cana-3791	101	1	|⟨n|ψ⟩|2	|⟨n|ψ⟩|2	NOUN
cana-3791	101	2	=	=	PUNCT
cana-3791	101	3	|b(s)n|2	|b(s)n|2	NUM
cana-3791	101	4	an	an	DET
cana-3791	101	5	ideal	ideal	ADJ
cana-3791	101	6	basis	basis	NOUN
cana-3791	101	7	choice	choice	NOUN
cana-3791	101	8	should	should	AUX
cana-3791	101	9	minimize	minimize	VERB
cana-3791	101	10	these	these	DET
cana-3791	101	11	distributions	distribution	NOUN
cana-3791	101	12	'	'	PART
cana-3791	101	13	entropy	entropy	NOUN
cana-3791	101	14	,	,	PUNCT
cana-3791	101	15	at	at	ADP
cana-3791	101	16	least	least	ADJ
cana-3791	101	17	as	as	ADV
cana-3791	101	18	much	much	ADV
cana-3791	101	19	as	as	SCONJ
cana-3791	101	20	it	it	PRON
cana-3791	101	21	is	be	AUX
cana-3791	101	22	made	make	VERB
cana-3791	101	23	possible	possible	ADJ
cana-3791	101	24	by	by	ADP
cana-3791	101	25	the	the	DET
cana-3791	101	26	given	give	VERB
cana-3791	101	27	constraint	constraint	NOUN
cana-3791	101	28	[	[	X
cana-3791	101	29	30	30	NUM
cana-3791	101	30	]	]	PUNCT
cana-3791	101	31	.	.	PUNCT
cana-3791	102	1	as	as	SCONJ
cana-3791	102	2	it	it	PRON
cana-3791	102	3	often	often	ADV
cana-3791	102	4	occurs	occur	VERB
cana-3791	102	5	,	,	PUNCT
cana-3791	102	6	realistic	realistic	ADJ
cana-3791	102	7	implementation	implementation	NOUN
cana-3791	102	8	of	of	ADP
cana-3791	102	9	these	these	DET
cana-3791	102	10	criteria	criterion	NOUN
cana-3791	102	11	turns	turn	VERB
cana-3791	102	12	out	out	ADP
cana-3791	102	13	to	to	PART
cana-3791	102	14	be	be	AUX
cana-3791	102	15	a	a	DET
cana-3791	102	16	trade	trade	NOUN
cana-3791	102	17	-	-	PUNCT
cana-3791	102	18	off	off	NOUN
cana-3791	102	19	between	between	ADP
cana-3791	102	20	conflicting	conflict	VERB
cana-3791	102	21	prescriptions	prescription	NOUN
cana-3791	102	22	;	;	PUNCT
cana-3791	102	23	as	as	ADP
cana-3791	102	24	an	an	DET
cana-3791	102	25	example	example	NOUN
cana-3791	102	26	,	,	PUNCT
cana-3791	102	27	generalized	generalized	ADJ
cana-3791	102	28	coherent	coherent	ADJ
cana-3791	102	29	states	state	NOUN
cana-3791	102	30	:	:	PUNCT
cana-3791	102	31	|α⟩(s	|α⟩(s	PROPN
cana-3791	102	32	)	)	PUNCT
cana-3791	103	1	=	=	PUNCT
cana-3791	104	1	d̂s(α)|0⟩	d̂s(α)|0⟩	INTJ
cana-3791	104	2	where	where	SCONJ
cana-3791	104	3	d̂s(α	d̂s(α	PROPN
cana-3791	104	4	)	)	PUNCT
cana-3791	104	5	is	be	AUX
cana-3791	104	6	the	the	DET
cana-3791	104	7	displacement	displacement	ADJ
cana-3791	104	8	operator	operator	NOUN
cana-3791	104	9	;	;	PUNCT
cana-3791	104	10	furnish	furnish	VERB
cana-3791	104	11	a	a	DET
cana-3791	104	12	natural	natural	ADJ
cana-3791	104	13	basis	basis	NOUN
cana-3791	104	14	for	for	ADP
cana-3791	104	15	some	some	DET
cana-3791	104	16	quantum	quantum	ADJ
cana-3791	104	17	optical	optical	ADJ
cana-3791	104	18	systems	system	NOUN
cana-3791	104	19	but	but	CCONJ
cana-3791	104	20	may	may	AUX
cana-3791	104	21	not	not	PART
cana-3791	104	22	be	be	AUX
cana-3791	104	23	optimal	optimal	ADJ
cana-3791	104	24	concerning	concern	VERB
cana-3791	104	25	computational	computational	ADJ
cana-3791	104	26	efficiency	efficiency	NOUN
cana-3791	104	27	.	.	PUNCT
cana-3791	105	1	their	their	PRON
cana-3791	105	2	use	use	NOUN
cana-3791	105	3	compared	compare	VERB
cana-3791	105	4	to	to	ADP
cana-3791	105	5	more	more	ADJ
cana-3791	105	6	computation	computation	NOUN
cana-3791	105	7	-	-	PUNCT
cana-3791	105	8	friendly	friendly	ADJ
cana-3791	105	9	bases	basis	NOUN
cana-3791	105	10	has	have	VERB
cana-3791	105	11	to	to	PART
cana-3791	105	12	be	be	AUX
cana-3791	105	13	determined	determine	VERB
cana-3791	105	14	by	by	ADP
cana-3791	105	15	specific	specific	ADJ
cana-3791	105	16	application	application	NOUN
cana-3791	105	17	needs	need	NOUN
cana-3791	105	18	[	[	X
cana-3791	105	19	31	31	NUM
cana-3791	105	20	]	]	PUNCT
cana-3791	105	21	.	.	PUNCT
cana-3791	106	1	construction	construction	NOUN
cana-3791	106	2	methods	method	VERB
cana-3791	106	3	the	the	DET
cana-3791	106	4	construction	construction	NOUN
cana-3791	106	5	of	of	ADP
cana-3791	106	6	optimal	optimal	ADJ
cana-3791	106	7	bases	basis	NOUN
cana-3791	106	8	in	in	ADP
cana-3791	106	9	finite	finite	ADJ
cana-3791	106	10	-	-	ADJ
cana-3791	106	11	dimensional	dimensional	ADJ
cana-3791	106	12	hilbert	hilbert	NOUN
cana-3791	106	13	spaces	space	NOUN
cana-3791	106	14	requires	require	VERB
cana-3791	106	15	rigorous	rigorous	ADJ
cana-3791	106	16	mathematical	mathematical	ADJ
cana-3791	106	17	methods	method	NOUN
cana-3791	106	18	that	that	PRON
cana-3791	106	19	ensure	ensure	VERB
cana-3791	106	20	both	both	DET
cana-3791	106	21	orthonormality	orthonormality	NOUN
cana-3791	106	22	and	and	CCONJ
cana-3791	106	23	computational	computational	ADJ
cana-3791	106	24	efficiency	efficiency	NOUN
cana-3791	106	25	.	.	PUNCT
cana-3791	107	1	drawing	draw	VERB
cana-3791	107	2	from	from	ADP
cana-3791	107	3	dagger	dagger	NOUN
cana-3791	107	4	category	category	NOUN
cana-3791	107	5	theory	theory	NOUN
cana-3791	107	6	and	and	CCONJ
cana-3791	107	7	complex	complex	ADJ
cana-3791	107	8	number	number	NOUN
cana-3791	107	9	representations	representation	NOUN
cana-3791	107	10	,	,	PUNCT
cana-3791	107	11	we	we	PRON
cana-3791	107	12	present	present	VERB
cana-3791	107	13	systematic	systematic	ADJ
cana-3791	107	14	approaches	approach	NOUN
cana-3791	107	15	to	to	PART
cana-3791	107	16	basis	basis	NOUN
cana-3791	107	17	construction	construction	NOUN
cana-3791	107	18	that	that	PRON
cana-3791	107	19	maintain	maintain	VERB
cana-3791	107	20	mathematical	mathematical	ADJ
cana-3791	107	21	precision	precision	NOUN
cana-3791	107	22	while	while	SCONJ
cana-3791	107	23	enabling	enable	VERB
cana-3791	107	24	practical	practical	ADJ
cana-3791	107	25	implementations	implementation	NOUN
cana-3791	107	26	.	.	PUNCT
cana-3791	108	1	the	the	DET
cana-3791	108	2	fundamental	fundamental	ADJ
cana-3791	108	3	construction	construction	NOUN
cana-3791	108	4	method	method	NOUN
cana-3791	108	5	relies	rely	VERB
cana-3791	108	6	on	on	ADP
cana-3791	108	7	the	the	DET
cana-3791	108	8	dagger	dagger	NOUN
cana-3791	108	9	structure	structure	NOUN
cana-3791	108	10	,	,	PUNCT
cana-3791	108	11	where	where	SCONJ
cana-3791	108	12	for	for	ADP
cana-3791	108	13	any	any	DET
cana-3791	108	14	morphism	morphism	NOUN
cana-3791	108	15	f	f	NOUN
cana-3791	108	16	,	,	PUNCT
cana-3791	108	17	there	there	PRON
cana-3791	108	18	exists	exist	VERB
cana-3791	108	19	a	a	DET
cana-3791	108	20	unique	unique	ADJ
cana-3791	108	21	morphism	morphism	NOUN
cana-3791	108	22	f†	f†	ADJ
cana-3791	108	23	(	(	PUNCT
cana-3791	108	24	the	the	DET
cana-3791	108	25	dagger	dagger	NOUN
cana-3791	108	26	of	of	ADP
cana-3791	108	27	f	f	X
cana-3791	108	28	)	)	PUNCT
cana-3791	108	29	satisfying	satisfy	VERB
cana-3791	108	30	specific	specific	ADJ
cana-3791	108	31	categorical	categorical	ADJ
cana-3791	108	32	properties	property	NOUN
cana-3791	108	33	[	[	X
cana-3791	108	34	32	32	NUM
cana-3791	108	35	]	]	PUNCT
cana-3791	108	36	.	.	PUNCT
cana-3791	109	1	in	in	ADP
cana-3791	109	2	the	the	DET
cana-3791	109	3	context	context	NOUN
cana-3791	109	4	of	of	ADP
cana-3791	109	5	finitedimensional	finitedimensional	ADJ
cana-3791	109	6	hilbert	hilbert	NOUN
cana-3791	109	7	spaces	space	NOUN
cana-3791	109	8	,	,	PUNCT
cana-3791	109	9	this	this	DET
cana-3791	109	10	structure	structure	NOUN
cana-3791	109	11	manifests	manifest	VERB
cana-3791	109	12	through	through	ADP
cana-3791	109	13	the	the	DET
cana-3791	109	14	following	follow	VERB
cana-3791	109	15	key	key	ADJ
cana-3791	109	16	property	property	NOUN
cana-3791	109	17	:	:	PUNCT
cana-3791	109	18	f††	f††	PROPN
cana-3791	109	19	=	=	PUNCT
cana-3791	109	20	f	f	PROPN
cana-3791	109	21	this	this	DET
cana-3791	109	22	property	property	NOUN
cana-3791	109	23	ensures	ensure	VERB
cana-3791	109	24	that	that	SCONJ
cana-3791	109	25	the	the	DET
cana-3791	109	26	construction	construction	NOUN
cana-3791	109	27	process	process	NOUN
cana-3791	109	28	preserves	preserve	VERB
cana-3791	109	29	the	the	DET
cana-3791	109	30	essential	essential	ADJ
cana-3791	109	31	mathematical	mathematical	ADJ
cana-3791	109	32	structure	structure	NOUN
cana-3791	109	33	while	while	SCONJ
cana-3791	109	34	providing	provide	VERB
cana-3791	109	35	a	a	DET
cana-3791	109	36	natural	natural	ADJ
cana-3791	109	37	framework	framework	NOUN
cana-3791	109	38	for	for	ADP
cana-3791	109	39	numerical	numerical	ADJ
cana-3791	109	40	implementations	implementation	NOUN
cana-3791	109	41	[	[	X
cana-3791	109	42	33	33	NUM
cana-3791	109	43	]	]	PUNCT
cana-3791	109	44	.	.	PUNCT
cana-3791	110	1	the	the	DET
cana-3791	110	2	gram	gram	NOUN
cana-3791	110	3	-	-	PUNCT
cana-3791	110	4	schmidt	schmidt	NOUN
cana-3791	110	5	orthogonalization	orthogonalization	NOUN
cana-3791	110	6	process	process	NOUN
cana-3791	110	7	plays	play	VERB
cana-3791	110	8	a	a	DET
cana-3791	110	9	central	central	ADJ
cana-3791	110	10	role	role	NOUN
cana-3791	110	11	in	in	ADP
cana-3791	110	12	basis	basis	NOUN
cana-3791	110	13	construction	construction	NOUN
cana-3791	110	14	.	.	PUNCT
cana-3791	111	1	given	give	VERB
cana-3791	111	2	a	a	DET
cana-3791	111	3	set	set	NOUN
cana-3791	111	4	of	of	ADP
cana-3791	111	5	linearly	linearly	ADV
cana-3791	111	6	independent	independent	ADJ
cana-3791	111	7	vectors	vector	NOUN
cana-3791	111	8	{	{	PUNCT
cana-3791	111	9	v1	v1	PROPN
cana-3791	111	10	,	,	PUNCT
cana-3791	111	11	...	...	PUNCT
cana-3791	111	12	,	,	PUNCT
cana-3791	111	13	vn	vn	PROPN
cana-3791	111	14	}	}	PUNCT
cana-3791	111	15	,	,	PUNCT
cana-3791	111	16	the	the	DET
cana-3791	111	17	process	process	NOUN
cana-3791	111	18	constructs	construct	VERB
cana-3791	111	19	an	an	DET
cana-3791	111	20	orthonormal	orthonormal	ADJ
cana-3791	111	21	basis	basis	NOUN
cana-3791	111	22	{	{	PUNCT
cana-3791	111	23	u1	u1	NOUN
cana-3791	111	24	,	,	PUNCT
cana-3791	111	25	...	...	PUNCT
cana-3791	111	26	,	,	PUNCT
cana-3791	111	27	un	un	PROPN
cana-3791	111	28	}	}	PUNCT
cana-3791	111	29	through	through	ADP
cana-3791	111	30	the	the	DET
cana-3791	111	31	following	follow	VERB
cana-3791	111	32	iterative	iterative	NOUN
cana-3791	111	33	procedure	procedure	NOUN
cana-3791	111	34	:	:	PUNCT
cana-3791	111	35	ẽk	ẽk	PROPN
cana-3791	111	36	=	=	SYM
cana-3791	111	37	vk	vk	PROPN
cana-3791	111	38	∑k-1j=1	∑k-1j=1	PROPN
cana-3791	111	39	⟨uj|vk⟩uj	⟨uj|vk⟩uj	VERB
cana-3791	111	40	uk	uk	NOUN
cana-3791	111	41	=	=	SYM
cana-3791	111	42	ẽk/||ẽk||	ẽk/||ẽk||	PROPN
cana-3791	111	43	where	where	SCONJ
cana-3791	111	44	⟨·|·⟩	⟨·|·⟩	PRON
cana-3791	111	45	denotes	denote	VERB
cana-3791	111	46	the	the	DET
cana-3791	111	47	inner	inner	ADJ
cana-3791	111	48	product	product	NOUN
cana-3791	111	49	and	and	CCONJ
cana-3791	111	50	||·||	||·||	NOUN
cana-3791	111	51	is	be	AUX
cana-3791	111	52	the	the	DET
cana-3791	111	53	induced	induced	ADJ
cana-3791	111	54	norm	norm	NOUN
cana-3791	111	55	.	.	PUNCT
cana-3791	112	1	in	in	ADP
cana-3791	112	2	finite	finite	ADJ
cana-3791	112	3	-	-	ADJ
cana-3791	112	4	dimensional	dimensional	ADJ
cana-3791	112	5	dagger	dagger	NOUN
cana-3791	112	6	categories	category	NOUN
cana-3791	112	7	,	,	PUNCT
cana-3791	112	8	this	this	DET
cana-3791	112	9	process	process	NOUN
cana-3791	112	10	can	can	AUX
cana-3791	112	11	be	be	AUX
cana-3791	112	12	implemented	implement	VERB
cana-3791	112	13	using	use	VERB
cana-3791	112	14	dagger	dagger	NOUN
cana-3791	112	15	monic	monic	ADJ
cana-3791	112	16	and	and	CCONJ
cana-3791	112	17	dagger	dagger	NOUN
cana-3791	112	18	epic	epic	ADJ
cana-3791	112	19	morphisms	morphism	NOUN
cana-3791	113	1	[	[	X
cana-3791	113	2	34	34	NUM
cana-3791	113	3	]	]	PUNCT
cana-3791	113	4	,	,	PUNCT
cana-3791	113	5	which	which	PRON
cana-3791	113	6	satisfy	satisfy	VERB
cana-3791	113	7	:	:	PUNCT
cana-3791	113	8	f†f	f†f	NUM
cana-3791	113	9	=	=	SYM
cana-3791	113	10	1	1	NUM
cana-3791	113	11	(	(	PUNCT
cana-3791	113	12	dagger	dagger	NOUN
cana-3791	113	13	monic	monic	ADJ
cana-3791	113	14	)	)	PUNCT
cana-3791	113	15	ff†	ff†	PROPN
cana-3791	113	16	=	=	SYM
cana-3791	113	17	1	1	NUM
cana-3791	113	18	(	(	PUNCT
cana-3791	113	19	dagger	dagger	NOUN
cana-3791	113	20	epic	epic	NOUN
cana-3791	113	21	)	)	PUNCT
cana-3791	113	22	communications	communication	NOUN
cana-3791	113	23	on	on	ADP
cana-3791	113	24	applied	apply	VERB
cana-3791	113	25	nonlinear	nonlinear	ADJ
cana-3791	113	26	analysis	analysis	NOUN
cana-3791	113	27	issn	issn	NOUN
cana-3791	113	28	:	:	PUNCT
cana-3791	113	29	1074	1074	NUM
cana-3791	113	30	-	-	PUNCT
cana-3791	113	31	133x	133x	NUM
cana-3791	113	32	vol	vol	NOUN
cana-3791	113	33	32	32	NUM
cana-3791	113	34	no	no	NOUN
cana-3791	113	35	.	.	PUNCT
cana-3791	114	1	8s	8s	PROPN
cana-3791	114	2	(	(	PUNCT
cana-3791	114	3	2025	2025	NUM
cana-3791	114	4	)	)	PUNCT
cana-3791	114	5	737	737	NUM
cana-3791	114	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	114	7	the	the	DET
cana-3791	114	8	numerical	numerical	ADJ
cana-3791	114	9	implementation	implementation	NOUN
cana-3791	114	10	of	of	ADP
cana-3791	114	11	these	these	DET
cana-3791	114	12	constructions	construction	NOUN
cana-3791	114	13	requires	require	VERB
cana-3791	114	14	careful	careful	ADJ
cana-3791	114	15	consideration	consideration	NOUN
cana-3791	114	16	of	of	ADP
cana-3791	114	17	computational	computational	ADJ
cana-3791	114	18	stability	stability	NOUN
cana-3791	114	19	and	and	CCONJ
cana-3791	114	20	efficiency	efficiency	NOUN
cana-3791	114	21	.	.	PUNCT
cana-3791	115	1	a	a	DET
cana-3791	115	2	key	key	ADJ
cana-3791	115	3	aspect	aspect	NOUN
cana-3791	115	4	is	be	AUX
cana-3791	115	5	the	the	DET
cana-3791	115	6	preservation	preservation	NOUN
cana-3791	115	7	of	of	ADP
cana-3791	115	8	the	the	DET
cana-3791	115	9	dagger	dagger	NOUN
cana-3791	115	10	structure	structure	NOUN
cana-3791	115	11	through	through	ADP
cana-3791	115	12	what	what	PRON
cana-3791	115	13	is	be	AUX
cana-3791	115	14	known	know	VERB
cana-3791	115	15	as	as	ADP
cana-3791	115	16	dagger	dagger	NOUN
cana-3791	115	17	finiteness	finiteness	NOUN
cana-3791	115	18	.	.	PUNCT
cana-3791	116	1	an	an	DET
cana-3791	116	2	object	object	NOUN
cana-3791	116	3	x	x	PUNCT
cana-3791	116	4	is	be	AUX
cana-3791	116	5	dagger	dagger	NOUN
cana-3791	116	6	finite	finite	NOUN
cana-3791	116	7	when	when	SCONJ
cana-3791	116	8	,	,	PUNCT
cana-3791	116	9	for	for	ADP
cana-3791	116	10	each	each	DET
cana-3791	116	11	morphism	morphism	NOUN
cana-3791	116	12	f	f	NOUN
cana-3791	116	13	:	:	PUNCT
cana-3791	116	14	x	x	SYM
cana-3791	116	15	→	→	SYM
cana-3791	116	16	x	x	X
cana-3791	116	17	,	,	PUNCT
cana-3791	116	18	the	the	DET
cana-3791	116	19	condition	condition	NOUN
cana-3791	116	20	f†f	f†f	AUX
cana-3791	116	21	=	=	SYM
cana-3791	116	22	1	1	NUM
cana-3791	116	23	implies	imply	VERB
cana-3791	116	24	ff†	ff†	PROPN
cana-3791	116	25	=	=	SYM
cana-3791	116	26	1	1	X
cana-3791	116	27	.	.	PUNCT
cana-3791	117	1	this	this	DET
cana-3791	117	2	property	property	NOUN
cana-3791	117	3	ensures	ensure	VERB
cana-3791	117	4	that	that	SCONJ
cana-3791	117	5	every	every	DET
cana-3791	117	6	dagger	dagger	NOUN
cana-3791	117	7	monic	monic	ADJ
cana-3791	117	8	endomorphism	endomorphism	NOUN
cana-3791	117	9	is	be	AUX
cana-3791	117	10	a	a	DET
cana-3791	117	11	dagger	dagger	NOUN
cana-3791	117	12	isomorphism	isomorphism	NOUN
cana-3791	117	13	[	[	X
cana-3791	117	14	35	35	NUM
cana-3791	117	15	]	]	PUNCT
cana-3791	117	16	,	,	PUNCT
cana-3791	117	17	leading	lead	VERB
cana-3791	117	18	to	to	ADP
cana-3791	117	19	the	the	DET
cana-3791	117	20	following	following	NOUN
cana-3791	117	21	theorem	theorem	NOUN
cana-3791	117	22	:	:	PUNCT
cana-3791	117	23	for	for	ADP
cana-3791	117	24	a	a	DET
cana-3791	117	25	locally	locally	ADV
cana-3791	117	26	small	small	ADJ
cana-3791	117	27	dagger	dagger	NOUN
cana-3791	117	28	rig	rig	NOUN
cana-3791	117	29	category	category	NOUN
cana-3791	117	30	to	to	PART
cana-3791	117	31	be	be	AUX
cana-3791	117	32	equivalent	equivalent	ADJ
cana-3791	117	33	to	to	ADP
cana-3791	117	34	the	the	DET
cana-3791	117	35	category	category	NOUN
cana-3791	117	36	fcon	fcon	NOUN
cana-3791	117	37	of	of	ADP
cana-3791	117	38	finite	finite	ADJ
cana-3791	117	39	-	-	ADJ
cana-3791	117	40	dimensional	dimensional	ADJ
cana-3791	117	41	hilbert	hilbert	NOUN
cana-3791	117	42	spaces	space	NOUN
cana-3791	117	43	and	and	CCONJ
cana-3791	117	44	linear	linear	ADJ
cana-3791	117	45	contractions	contraction	NOUN
cana-3791	117	46	,	,	PUNCT
cana-3791	117	47	it	it	PRON
cana-3791	117	48	must	must	AUX
cana-3791	117	49	satisfy	satisfy	VERB
cana-3791	117	50	specific	specific	ADJ
cana-3791	117	51	axioms	axiom	NOUN
cana-3791	117	52	including	include	VERB
cana-3791	117	53	dagger	dagger	NOUN
cana-3791	117	54	finiteness	finiteness	NOUN
cana-3791	117	55	.	.	PUNCT
cana-3791	118	1	the	the	DET
cana-3791	118	2	practical	practical	ADJ
cana-3791	118	3	implementation	implementation	NOUN
cana-3791	118	4	involves	involve	VERB
cana-3791	118	5	several	several	ADJ
cana-3791	118	6	key	key	ADJ
cana-3791	118	7	steps	step	NOUN
cana-3791	118	8	:	:	PUNCT
cana-3791	118	9	1	1	X
cana-3791	118	10	.	.	X
cana-3791	118	11	initial	initial	ADJ
cana-3791	118	12	basis	basis	NOUN
cana-3791	118	13	selection	selection	NOUN
cana-3791	118	14	using	use	VERB
cana-3791	118	15	dagger	dagger	NOUN
cana-3791	118	16	symmetric	symmetric	ADJ
cana-3791	118	17	monoidal	monoidal	ADJ
cana-3791	118	18	categories	category	NOUN
cana-3791	118	19	:	:	PUNCT
cana-3791	118	20	o	o	NOUN
cana-3791	118	21	define	define	VERB
cana-3791	118	22	a	a	DET
cana-3791	118	23	monoidal	monoidal	ADJ
cana-3791	118	24	product	product	NOUN
cana-3791	118	25	(	(	PUNCT
cana-3791	118	26	⊗	⊗	PROPN
cana-3791	118	27	,	,	PUNCT
cana-3791	118	28	i	i	NOUN
cana-3791	118	29	)	)	PUNCT
cana-3791	118	30	o	o	NOUN
cana-3791	118	31	ensure	ensure	VERB
cana-3791	118	32	natural	natural	ADJ
cana-3791	118	33	dagger	dagger	NOUN
cana-3791	118	34	isomorphisms	isomorphism	NOUN
cana-3791	118	35	preserve	preserve	VERB
cana-3791	118	36	structure	structure	NOUN
cana-3791	118	37	o	o	NOUN
cana-3791	118	38	maintain	maintain	VERB
cana-3791	118	39	coherence	coherence	NOUN
cana-3791	118	40	conditions	condition	NOUN
cana-3791	118	41	2	2	NUM
cana-3791	118	42	.	.	X
cana-3791	118	43	refinement	refinement	NOUN
cana-3791	118	44	through	through	ADP
cana-3791	118	45	categorical	categorical	ADJ
cana-3791	118	46	operations	operation	NOUN
cana-3791	118	47	:	:	PUNCT
cana-3791	118	48	o	o	NOUN
cana-3791	118	49	apply	apply	VERB
cana-3791	118	50	sequential	sequential	ADJ
cana-3791	118	51	diagram	diagram	NOUN
cana-3791	118	52	colimits	colimit	VERB
cana-3791	118	53	o	o	PROPN
cana-3791	118	54	preserve	preserve	NOUN
cana-3791	118	55	boundedness	boundedness	NOUN
cana-3791	118	56	conditions	condition	NOUN
cana-3791	118	57	o	o	NOUN
cana-3791	118	58	maintain	maintain	VERB
cana-3791	118	59	dagger	dagger	NOUN
cana-3791	118	60	structure	structure	NOUN
cana-3791	118	61	throughout	throughout	ADP
cana-3791	118	62	3	3	NUM
cana-3791	118	63	.	.	PUNCT
cana-3791	118	64	optimization	optimization	NOUN
cana-3791	118	65	using	use	VERB
cana-3791	118	66	kernel	kernel	PROPN
cana-3791	118	67	methods	method	NOUN
cana-3791	118	68	:	:	PUNCT
cana-3791	118	69	the	the	DET
cana-3791	118	70	kernel	kernel	PROPN
cana-3791	118	71	function	function	PROPN
cana-3791	118	72	k(x	k(x	PROPN
cana-3791	118	73	,	,	PUNCT
cana-3791	118	74	y	y	NOUN
cana-3791	118	75	)	)	PUNCT
cana-3791	118	76	must	must	AUX
cana-3791	118	77	satisfy	satisfy	VERB
cana-3791	118	78	:	:	PUNCT
cana-3791	118	79	k	k	PROPN
cana-3791	118	80	∈	∈	PROPN
cana-3791	118	81	c(2,2)b(x	c(2,2)b(x	PROPN
cana-3791	118	82	×	×	NOUN
cana-3791	118	83	x	x	NOUN
cana-3791	118	84	)	)	PUNCT
cana-3791	118	85	where	where	SCONJ
cana-3791	118	86	c(2,2)b	c(2,2)b	NOUN
cana-3791	118	87	denotes	denote	NOUN
cana-3791	118	88	bounded	bound	VERB
cana-3791	118	89	continuous	continuous	ADJ
cana-3791	118	90	derivatives	derivative	NOUN
cana-3791	118	91	up	up	ADP
cana-3791	118	92	to	to	PART
cana-3791	118	93	order	order	NOUN
cana-3791	118	94	2	2	NUM
cana-3791	118	95	.	.	PUNCT
cana-3791	119	1	the	the	DET
cana-3791	119	2	construction	construction	NOUN
cana-3791	119	3	process	process	NOUN
cana-3791	119	4	culminates	culminate	VERB
cana-3791	119	5	in	in	ADP
cana-3791	119	6	a	a	DET
cana-3791	119	7	rigorous	rigorous	ADJ
cana-3791	119	8	numerical	numerical	ADJ
cana-3791	119	9	framework	framework	NOUN
cana-3791	119	10	that	that	PRON
cana-3791	119	11	can	can	AUX
cana-3791	119	12	be	be	AUX
cana-3791	119	13	implemented	implement	VERB
cana-3791	119	14	efficiently	efficiently	ADV
cana-3791	119	15	.	.	PUNCT
cana-3791	120	1	the	the	DET
cana-3791	120	2	dagger	dagger	NOUN
cana-3791	120	3	structure	structure	NOUN
cana-3791	120	4	provides	provide	VERB
cana-3791	120	5	natural	natural	ADJ
cana-3791	120	6	error	error	NOUN
cana-3791	120	7	metrics	metric	NOUN
cana-3791	120	8	through	through	ADP
cana-3791	120	9	the	the	DET
cana-3791	120	10	relationship	relationship	NOUN
cana-3791	120	11	:	:	PUNCT
cana-3791	120	12	||f†f	||f†f	X
cana-3791	120	13	1||	1||	NUM
cana-3791	120	14	≤	≤	NUM
cana-3791	120	15	ε	ε	PROPN
cana-3791	120	16	where	where	SCONJ
cana-3791	120	17	ε	ε	PROPN
cana-3791	120	18	represents	represent	VERB
cana-3791	120	19	the	the	DET
cana-3791	120	20	desired	desire	VERB
cana-3791	120	21	precision	precision	NOUN
cana-3791	120	22	tolerance	tolerance	NOUN
cana-3791	120	23	[	[	X
cana-3791	120	24	36	36	NUM
cana-3791	120	25	]	]	PUNCT
cana-3791	120	26	.	.	PUNCT
cana-3791	121	1	one	one	NUM
cana-3791	121	2	particularly	particularly	ADV
cana-3791	121	3	important	important	ADJ
cana-3791	121	4	aspect	aspect	NOUN
cana-3791	121	5	of	of	ADP
cana-3791	121	6	the	the	DET
cana-3791	121	7	construction	construction	NOUN
cana-3791	121	8	is	be	AUX
cana-3791	121	9	the	the	DET
cana-3791	121	10	preservation	preservation	NOUN
cana-3791	121	11	of	of	ADP
cana-3791	121	12	finite	finite	ADJ
cana-3791	121	13	dimensionality	dimensionality	NOUN
cana-3791	121	14	.	.	PUNCT
cana-3791	122	1	this	this	PRON
cana-3791	122	2	is	be	AUX
cana-3791	122	3	achieved	achieve	VERB
cana-3791	122	4	through	through	ADP
cana-3791	122	5	what	what	PRON
cana-3791	122	6	we	we	PRON
cana-3791	122	7	term	term	VERB
cana-3791	122	8	"	"	PUNCT
cana-3791	122	9	dagger	dagger	NOUN
cana-3791	122	10	finiteness	finiteness	NOUN
cana-3791	122	11	,	,	PUNCT
cana-3791	122	12	"	"	PUNCT
cana-3791	122	13	which	which	PRON
cana-3791	122	14	ensures	ensure	VERB
cana-3791	122	15	that	that	SCONJ
cana-3791	122	16	the	the	DET
cana-3791	122	17	constructed	construct	VERB
cana-3791	122	18	bases	basis	NOUN
cana-3791	122	19	remain	remain	VERB
cana-3791	122	20	within	within	ADP
cana-3791	122	21	the	the	DET
cana-3791	122	22	appropriate	appropriate	ADJ
cana-3791	122	23	dimensional	dimensional	ADJ
cana-3791	122	24	constraints	constraint	NOUN
cana-3791	122	25	while	while	SCONJ
cana-3791	122	26	maintaining	maintain	VERB
cana-3791	122	27	their	their	PRON
cana-3791	122	28	mathematical	mathematical	ADJ
cana-3791	122	29	properties	property	NOUN
cana-3791	122	30	.	.	PUNCT
cana-3791	123	1	the	the	DET
cana-3791	123	2	process	process	NOUN
cana-3791	123	3	can	can	AUX
cana-3791	123	4	be	be	AUX
cana-3791	123	5	verified	verify	VERB
cana-3791	123	6	through	through	ADP
cana-3791	123	7	the	the	DET
cana-3791	123	8	following	follow	VERB
cana-3791	123	9	criterion	criterion	NOUN
cana-3791	123	10	:	:	PUNCT
cana-3791	123	11	for	for	ADP
cana-3791	123	12	any	any	DET
cana-3791	123	13	object	object	NOUN
cana-3791	123	14	x	x	PUNCT
cana-3791	123	15	and	and	CCONJ
cana-3791	123	16	morphism	morphism	NOUN
cana-3791	123	17	f	f	NOUN
cana-3791	123	18	:	:	PUNCT
cana-3791	123	19	x	x	SYM
cana-3791	123	20	→	→	SYM
cana-3791	123	21	x	x	X
cana-3791	123	22	,	,	PUNCT
cana-3791	123	23	if	if	SCONJ
cana-3791	123	24	f†f	f†f	NUM
cana-3791	123	25	=	=	SYM
cana-3791	123	26	1	1	NUM
cana-3791	123	27	,	,	PUNCT
cana-3791	123	28	then	then	ADV
cana-3791	123	29	ff†	ff†	ADJ
cana-3791	123	30	=	=	SYM
cana-3791	123	31	1	1	NUM
cana-3791	123	32	this	this	DET
cana-3791	123	33	criterion	criterion	NOUN
cana-3791	123	34	ensures	ensure	VERB
cana-3791	123	35	that	that	SCONJ
cana-3791	123	36	the	the	DET
cana-3791	123	37	construction	construction	NOUN
cana-3791	123	38	remains	remain	VERB
cana-3791	123	39	well	well	ADV
cana-3791	123	40	-	-	PUNCT
cana-3791	123	41	behaved	behave	VERB
cana-3791	123	42	in	in	ADP
cana-3791	123	43	finite	finite	ADJ
cana-3791	123	44	-	-	ADJ
cana-3791	123	45	dimensional	dimensional	ADJ
cana-3791	123	46	spaces	space	NOUN
cana-3791	123	47	[	[	X
cana-3791	123	48	37	37	NUM
cana-3791	123	49	]	]	PUNCT
cana-3791	123	50	.	.	PUNCT
cana-3791	124	1	applications	application	NOUN
cana-3791	124	2	signal	signal	VERB
cana-3791	124	3	processing	process	VERB
cana-3791	124	4	finite	finite	ADJ
cana-3791	124	5	-	-	ADJ
cana-3791	124	6	dimensional	dimensional	ADJ
cana-3791	124	7	hilbert	hilbert	NOUN
cana-3791	124	8	space	space	NOUN
cana-3791	124	9	theory	theory	NOUN
cana-3791	124	10	,	,	PUNCT
cana-3791	124	11	when	when	SCONJ
cana-3791	124	12	applied	apply	VERB
cana-3791	124	13	to	to	PART
cana-3791	124	14	signal	signal	VERB
cana-3791	124	15	processing	processing	NOUN
cana-3791	124	16	,	,	PUNCT
cana-3791	124	17	will	will	AUX
cana-3791	124	18	provide	provide	VERB
cana-3791	124	19	a	a	DET
cana-3791	124	20	set	set	NOUN
cana-3791	124	21	of	of	ADP
cana-3791	124	22	powerful	powerful	ADJ
cana-3791	124	23	tools	tool	NOUN
cana-3791	124	24	for	for	ADP
cana-3791	124	25	both	both	PRON
cana-3791	124	26	signal	signal	ADJ
cana-3791	124	27	representation	representation	NOUN
cana-3791	124	28	and	and	CCONJ
cana-3791	124	29	filter	filter	NOUN
cana-3791	124	30	design	design	NOUN
cana-3791	124	31	.	.	PUNCT
cana-3791	125	1	the	the	DET
cana-3791	125	2	discrete	discrete	ADJ
cana-3791	125	3	wigner	wigner	NOUN
cana-3791	125	4	function	function	NOUN
cana-3791	125	5	framework	framework	NOUN
cana-3791	125	6	provides	provide	VERB
cana-3791	125	7	an	an	DET
cana-3791	125	8	especially	especially	ADV
cana-3791	125	9	strong	strong	ADJ
cana-3791	125	10	methodology	methodology	NOUN
cana-3791	125	11	to	to	PART
cana-3791	125	12	analyze	analyze	VERB
cana-3791	125	13	and	and	CCONJ
cana-3791	125	14	process	process	NOUN
cana-3791	125	15	signals	signal	NOUN
cana-3791	125	16	in	in	ADP
cana-3791	125	17	finite	finite	ADJ
cana-3791	125	18	-	-	ADJ
cana-3791	125	19	dimensional	dimensional	ADJ
cana-3791	125	20	spaces	space	NOUN
cana-3791	125	21	by	by	ADP
cana-3791	125	22	mapping	map	VERB
cana-3791	125	23	the	the	DET
cana-3791	125	24	strong	strong	ADJ
cana-3791	125	25	advantages	advantage	NOUN
cana-3791	125	26	of	of	ADP
cana-3791	125	27	both	both	DET
cana-3791	125	28	time	time	NOUN
cana-3791	125	29	and	and	CCONJ
cana-3791	125	30	frequency	frequency	NOUN
cana-3791	125	31	domain	domain	NOUN
cana-3791	125	32	representations	representation	NOUN
cana-3791	125	33	.	.	PUNCT
cana-3791	126	1	within	within	ADP
cana-3791	126	2	the	the	DET
cana-3791	126	3	finite	finite	ADJ
cana-3791	126	4	-	-	ADJ
cana-3791	126	5	dimensional	dimensional	ADJ
cana-3791	126	6	signal	signal	NOUN
cana-3791	126	7	processing	processing	NOUN
cana-3791	126	8	,	,	PUNCT
cana-3791	126	9	the	the	DET
cana-3791	126	10	discrete	discrete	ADJ
cana-3791	126	11	signal	signal	NOUN
cana-3791	126	12	can	can	AUX
cana-3791	126	13	be	be	AUX
cana-3791	126	14	represented	represent	VERB
cana-3791	126	15	in	in	ADP
cana-3791	126	16	its	its	PRON
cana-3791	126	17	expansion	expansion	NOUN
cana-3791	126	18	in	in	ADP
cana-3791	126	19	the	the	DET
cana-3791	126	20	finite	finite	ADJ
cana-3791	126	21	-	-	ADJ
cana-3791	126	22	dimensional	dimensional	ADJ
cana-3791	126	23	hilbert	hilbert	NOUN
cana-3791	126	24	space	space	NOUN
cana-3791	126	25	h(s	h(s	PROPN
cana-3791	126	26	)	)	PUNCT
cana-3791	126	27	.	.	PUNCT
cana-3791	127	1	the	the	DET
cana-3791	127	2	characteristic	characteristic	ADJ
cana-3791	127	3	function	function	NOUN
cana-3791	127	4	in	in	ADP
cana-3791	127	5	h(s	h(s	PROPN
cana-3791	127	6	)	)	PUNCT
cana-3791	127	7	is	be	AUX
cana-3791	127	8	provided	provide	VERB
cana-3791	127	9	from	from	ADP
cana-3791	127	10	[	[	X
cana-3791	127	11	38	38	NUM
cana-3791	127	12	]	]	PUNCT
cana-3791	127	13	,	,	PUNCT
cana-3791	127	14	by	by	ADP
cana-3791	127	15	cs(ν	cs(ν	NOUN
cana-3791	127	16	,	,	PUNCT
cana-3791	127	17	θμ)=∑sm=0exp(-4πi/(s+1)ν(m+μ))⟨θm|ρ̂|θm+2μ⟩	θμ)=∑sm=0exp(-4πi/(s+1)ν(m+μ))⟨θm|ρ̂|θm+2μ⟩	PROPN
cana-3791	127	18	where	where	SCONJ
cana-3791	127	19	θm	θm	PROPN
cana-3791	127	20	describes	describe	VERB
cana-3791	127	21	phase	phase	NOUN
cana-3791	127	22	states	state	NOUN
cana-3791	127	23	and	and	CCONJ
cana-3791	127	24	ρ	ρ	NUM
cana-3791	127	25	̂denotes	̂denotes	PROPN
cana-3791	127	26	the	the	DET
cana-3791	127	27	density	density	NOUN
cana-3791	127	28	operator	operator	NOUN
cana-3791	127	29	.	.	PUNCT
cana-3791	128	1	using	use	VERB
cana-3791	128	2	this	this	DET
cana-3791	128	3	representation	representation	NOUN
cana-3791	128	4	,	,	PUNCT
cana-3791	128	5	the	the	DET
cana-3791	128	6	discrete	discrete	ADJ
cana-3791	128	7	wigner	wigner	NOUN
cana-3791	128	8	function	function	NOUN
cana-3791	128	9	can	can	AUX
cana-3791	128	10	readily	readily	ADV
cana-3791	128	11	be	be	AUX
cana-3791	128	12	obtained	obtain	VERB
cana-3791	128	13	by	by	ADP
cana-3791	128	14	the	the	DET
cana-3791	128	15	implementation	implementation	NOUN
cana-3791	128	16	of	of	ADP
cana-3791	128	17	a	a	DET
cana-3791	128	18	discrete	discrete	ADJ
cana-3791	128	19	fourier	fourier	NOUN
cana-3791	128	20	transform	transform	NOUN
cana-3791	128	21	[	[	X
cana-3791	128	22	39	39	NUM
cana-3791	128	23	]	]	PUNCT
cana-3791	128	24	,	,	PUNCT
cana-3791	128	25	as	as	SCONJ
cana-3791	128	26	follows	follow	VERB
cana-3791	128	27	,	,	PUNCT
cana-3791	128	28	ws(n	ws(n	PROPN
cana-3791	128	29	,	,	PUNCT
cana-3791	128	30	θm)=1/(s+1)2∑sν=0∑sμ=0exp(4πi/(s+1)(nμ+νm))cs(ν	θm)=1/(s+1)2∑sν=0∑sμ=0exp(4πi/(s+1)(nμ+νm))cs(ν	PROPN
cana-3791	128	31	,	,	PUNCT
cana-3791	128	32	θμ	θμ	NOUN
cana-3791	128	33	)	)	PUNCT
cana-3791	128	34	communications	communication	NOUN
cana-3791	128	35	on	on	ADP
cana-3791	128	36	applied	apply	VERB
cana-3791	128	37	nonlinear	nonlinear	ADJ
cana-3791	128	38	analysis	analysis	NOUN
cana-3791	128	39	issn	issn	NOUN
cana-3791	128	40	:	:	PUNCT
cana-3791	128	41	1074	1074	NUM
cana-3791	128	42	-	-	PUNCT
cana-3791	128	43	133x	133x	NUM
cana-3791	128	44	vol	vol	NOUN
cana-3791	128	45	32	32	NUM
cana-3791	128	46	no	no	NOUN
cana-3791	128	47	.	.	PUNCT
cana-3791	129	1	8s	8s	PROPN
cana-3791	129	2	(	(	PUNCT
cana-3791	129	3	2025	2025	NUM
cana-3791	129	4	)	)	PUNCT
cana-3791	129	5	738	738	NUM
cana-3791	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	129	7	the	the	DET
cana-3791	129	8	wigner	wigner	ADJ
cana-3791	129	9	function	function	NOUN
cana-3791	129	10	is	be	AUX
cana-3791	129	11	a	a	DET
cana-3791	129	12	complete	complete	ADJ
cana-3791	129	13	description	description	NOUN
cana-3791	129	14	of	of	ADP
cana-3791	129	15	the	the	DET
cana-3791	129	16	signal	signal	NOUN
cana-3791	129	17	in	in	ADP
cana-3791	129	18	phase	phase	NOUN
cana-3791	129	19	space	space	NOUN
cana-3791	129	20	.	.	PUNCT
cana-3791	130	1	the	the	DET
cana-3791	130	2	properties	property	NOUN
cana-3791	130	3	below	below	ADV
cana-3791	130	4	are	be	AUX
cana-3791	130	5	especially	especially	ADV
cana-3791	130	6	advantageous	advantageous	ADJ
cana-3791	130	7	from	from	ADP
cana-3791	130	8	the	the	DET
cana-3791	130	9	point	point	NOUN
cana-3791	130	10	of	of	ADP
cana-3791	130	11	view	view	NOUN
cana-3791	130	12	of	of	ADP
cana-3791	130	13	application	application	NOUN
cana-3791	130	14	to	to	PART
cana-3791	130	15	signal	signal	VERB
cana-3791	130	16	processing	processing	NOUN
cana-3791	130	17	:	:	PUNCT
cana-3791	130	18	marginalization	marginalization	NOUN
cana-3791	130	19	properties	property	NOUN
cana-3791	130	20	:	:	PUNCT
cana-3791	130	21	phase	phase	NOUN
cana-3791	130	22	distribution	distribution	NOUN
cana-3791	130	23	ps(θm	ps(θm	PROPN
cana-3791	130	24	)	)	PUNCT
cana-3791	130	25	=	=	SYM
cana-3791	131	1	∑sn=0	∑sn=0	PROPN
cana-3791	131	2	ws(n	ws(n	PROPN
cana-3791	131	3	,	,	PUNCT
cana-3791	131	4	θm	θm	NOUN
cana-3791	131	5	)	)	PUNCT
cana-3791	131	6	number	number	NOUN
cana-3791	131	7	distribution	distribution	NOUN
cana-3791	131	8	ps(n	ps(n	NOUN
cana-3791	131	9	)	)	PUNCT
cana-3791	131	10	=	=	SYM
cana-3791	131	11	∑sm=0	∑sm=0	PROPN
cana-3791	131	12	ws(n	ws(n	ADJ
cana-3791	131	13	,	,	PUNCT
cana-3791	131	14	θm	θm	NOUN
cana-3791	131	15	)	)	PUNCT
cana-3791	131	16	periodicity	periodicity	NOUN
cana-3791	131	17	:	:	PUNCT
cana-3791	131	18	ws(n	ws(n	NUM
cana-3791	131	19	,	,	PUNCT
cana-3791	131	20	θm	θm	NOUN
cana-3791	131	21	)	)	PUNCT
cana-3791	131	22	=	=	SYM
cana-3791	131	23	ws(n±{s+1},θm	ws(n±{s+1},θm	NOUN
cana-3791	131	24	)	)	PUNCT
cana-3791	131	25	=	=	SYM
cana-3791	131	26	ws(n	ws(n	PROPN
cana-3791	131	27	,	,	PUNCT
cana-3791	131	28	θm±(s+1	θm±(s+1	NOUN
cana-3791	131	29	)	)	PUNCT
cana-3791	131	30	)	)	PUNCT
cana-3791	131	31	=	=	SYM
cana-3791	132	1	ws(n	ws(n	PROPN
cana-3791	132	2	,	,	PUNCT
cana-3791	132	3	θm±2π	θm±2π	NUM
cana-3791	132	4	)	)	PUNCT
cana-3791	132	5	in	in	ADP
cana-3791	132	6	particular	particular	ADJ
cana-3791	132	7	,	,	PUNCT
cana-3791	132	8	this	this	DET
cana-3791	132	9	periodicity	periodicity	NOUN
cana-3791	132	10	property	property	NOUN
cana-3791	132	11	makes	make	VERB
cana-3791	132	12	the	the	DET
cana-3791	132	13	function	function	NOUN
cana-3791	132	14	especially	especially	ADV
cana-3791	132	15	fit	fit	ADJ
cana-3791	132	16	for	for	ADP
cana-3791	132	17	the	the	DET
cana-3791	132	18	analyses	analysis	NOUN
cana-3791	132	19	of	of	ADP
cana-3791	132	20	cyclic	cyclic	ADJ
cana-3791	132	21	phenomena	phenomenon	NOUN
cana-3791	132	22	in	in	ADP
cana-3791	132	23	signals	signal	NOUN
cana-3791	132	24	[	[	X
cana-3791	132	25	40	40	NUM
cana-3791	132	26	]	]	PUNCT
cana-3791	132	27	.	.	PUNCT
cana-3791	133	1	in	in	ADP
cana-3791	133	2	this	this	DET
cana-3791	133	3	scheme	scheme	NOUN
cana-3791	133	4	,	,	PUNCT
cana-3791	133	5	the	the	DET
cana-3791	133	6	filter	filter	NOUN
cana-3791	133	7	design	design	NOUN
cana-3791	133	8	can	can	AUX
cana-3791	133	9	be	be	AUX
cana-3791	133	10	carried	carry	VERB
cana-3791	133	11	out	out	ADP
cana-3791	133	12	based	base	VERB
cana-3791	133	13	on	on	ADP
cana-3791	133	14	the	the	DET
cana-3791	133	15	method	method	NOUN
cana-3791	133	16	of	of	ADP
cana-3791	133	17	building	build	VERB
cana-3791	133	18	appropriate	appropriate	ADJ
cana-3791	133	19	operators	operator	NOUN
cana-3791	133	20	that	that	PRON
cana-3791	133	21	keep	keep	VERB
cana-3791	133	22	the	the	DET
cana-3791	133	23	finiteness	finiteness	NOUN
cana-3791	133	24	intact	intact	ADJ
cana-3791	133	25	.	.	PUNCT
cana-3791	134	1	the	the	DET
cana-3791	134	2	chief	chief	ADJ
cana-3791	134	3	ingredient	ingredient	NOUN
cana-3791	134	4	for	for	ADP
cana-3791	134	5	this	this	PRON
cana-3791	134	6	is	be	AUX
cana-3791	134	7	a	a	DET
cana-3791	134	8	generalized	generalized	ADJ
cana-3791	134	9	displacement	displacement	ADJ
cana-3791	134	10	operator	operator	NOUN
cana-3791	134	11	:	:	PUNCT
cana-3791	134	12	d̂s(α	d̂s(α	PROPN
cana-3791	134	13	,	,	PUNCT
cana-3791	134	14	α	α	NOUN
cana-3791	134	15	*	*	NOUN
cana-3791	134	16	)	)	PUNCT
cana-3791	134	17	defined	define	VERB
cana-3791	134	18	as	as	ADP
cana-3791	134	19	:	:	PUNCT
cana-3791	134	20	d̂s(α	d̂s(α	PROPN
cana-3791	134	21	,	,	PUNCT
cana-3791	134	22	α	α	NOUN
cana-3791	134	23	*	*	NOUN
cana-3791	134	24	)	)	PUNCT
cana-3791	134	25	=	=	SYM
cana-3791	135	1	exp[αâ†s	exp[αâ†s	PROPN
cana-3791	135	2	α*âs	α*âs	NOUN
cana-3791	135	3	]	]	PUNCT
cana-3791	135	4	where	where	SCONJ
cana-3791	135	5	âs	âs	ADP
cana-3791	135	6	and	and	CCONJ
cana-3791	135	7	â†s	â†s	NUM
cana-3791	135	8	are	be	AUX
cana-3791	135	9	finite	finite	ADJ
cana-3791	135	10	-	-	ADJ
cana-3791	135	11	dimensional	dimensional	ADJ
cana-3791	135	12	annihilation	annihilation	NOUN
cana-3791	135	13	and	and	CCONJ
cana-3791	135	14	creation	creation	NOUN
cana-3791	135	15	operators	operator	NOUN
cana-3791	135	16	respectively	respectively	ADV
cana-3791	135	17	[	[	X
cana-3791	135	18	41	41	NUM
cana-3791	135	19	]	]	PUNCT
cana-3791	135	20	:	:	PUNCT
cana-3791	135	21	âs	âs	ADP
cana-3791	135	22	=	=	SYM
cana-3791	135	23	∑sn=1	∑sn=1	PROPN
cana-3791	135	24	√n|n-1⟩⟨n|	√n|n-1⟩⟨n|	PROPN
cana-3791	135	25	â†s	â†s	NUM
cana-3791	135	26	=	=	SYM
cana-3791	135	27	∑sn=1	∑sn=1	PROPN
cana-3791	135	28	√n|n⟩⟨n-1|	√n|n⟩⟨n-1|	NUM
cana-3791	135	29	these	these	DET
cana-3791	135	30	operators	operator	NOUN
cana-3791	135	31	fulfill	fulfill	VERB
cana-3791	135	32	the	the	DET
cana-3791	135	33	following	follow	VERB
cana-3791	135	34	modified	modify	VERB
cana-3791	135	35	commutation	commutation	NOUN
cana-3791	135	36	relation	relation	NOUN
cana-3791	135	37	:	:	PUNCT
cana-3791	136	1	[	[	X
cana-3791	136	2	âs	âs	ADP
cana-3791	136	3	,	,	PUNCT
cana-3791	136	4	â†s	â†s	NUM
cana-3791	136	5	]	]	PUNCT
cana-3791	136	6	=	=	SYM
cana-3791	136	7	1	1	NUM
cana-3791	136	8	(	(	PUNCT
cana-3791	136	9	s+1)|s⟩⟨s|	s+1)|s⟩⟨s|	ADJ
cana-3791	136	10	which	which	PRON
cana-3791	136	11	reflects	reflect	VERB
cana-3791	136	12	the	the	DET
cana-3791	136	13	fact	fact	NOUN
cana-3791	136	14	that	that	SCONJ
cana-3791	136	15	the	the	DET
cana-3791	136	16	space	space	NOUN
cana-3791	136	17	is	be	AUX
cana-3791	136	18	finite	finite	ADJ
cana-3791	136	19	-	-	ADJ
cana-3791	136	20	dimensional	dimensional	ADJ
cana-3791	136	21	and	and	CCONJ
cana-3791	136	22	has	have	VERB
cana-3791	136	23	profound	profound	ADJ
cana-3791	136	24	consequences	consequence	NOUN
cana-3791	136	25	in	in	ADP
cana-3791	136	26	filter	filter	NOUN
cana-3791	136	27	design	design	NOUN
cana-3791	137	1	[	[	X
cana-3791	137	2	42	42	NUM
cana-3791	137	3	]	]	PUNCT
cana-3791	137	4	.	.	PUNCT
cana-3791	138	1	it	it	PRON
cana-3791	138	2	is	be	AUX
cana-3791	138	3	possible	possible	ADJ
cana-3791	138	4	to	to	PART
cana-3791	138	5	implement	implement	VERB
cana-3791	138	6	digital	digital	ADJ
cana-3791	138	7	filters	filter	NOUN
cana-3791	138	8	by	by	ADP
cana-3791	138	9	applying	apply	VERB
cana-3791	138	10	discrete	discrete	ADJ
cana-3791	138	11	transformations	transformation	NOUN
cana-3791	138	12	which	which	PRON
cana-3791	138	13	preserve	preserve	VERB
cana-3791	138	14	the	the	DET
cana-3791	138	15	structure	structure	NOUN
cana-3791	138	16	of	of	ADP
cana-3791	138	17	phasespace	phasespace	NOUN
cana-3791	138	18	.	.	PUNCT
cana-3791	139	1	one	one	NUM
cana-3791	139	2	of	of	ADP
cana-3791	139	3	the	the	DET
cana-3791	139	4	most	most	ADV
cana-3791	139	5	useful	useful	ADJ
cana-3791	139	6	formulations	formulation	NOUN
cana-3791	139	7	is	be	AUX
cana-3791	139	8	the	the	DET
cana-3791	139	9	construction	construction	NOUN
cana-3791	139	10	of	of	ADP
cana-3791	139	11	phase	phase	NOUN
cana-3791	139	12	-	-	PUNCT
cana-3791	139	13	space	space	NOUN
cana-3791	139	14	filters	filter	NOUN
cana-3791	139	15	with	with	ADP
cana-3791	139	16	the	the	DET
cana-3791	139	17	help	help	NOUN
cana-3791	139	18	of	of	ADP
cana-3791	139	19	the	the	DET
cana-3791	139	20	discrete	discrete	ADJ
cana-3791	139	21	wigner	wigner	ADJ
cana-3791	139	22	distribution	distribution	NOUN
cana-3791	139	23	defined	define	VERB
cana-3791	139	24	as	as	ADP
cana-3791	139	25	h(n	h(n	PROPN
cana-3791	139	26	,	,	PUNCT
cana-3791	139	27	θm	θm	NOUN
cana-3791	139	28	)	)	PUNCT
cana-3791	139	29	=	=	SYM
cana-3791	139	30	1/(s+1	1/(s+1	X
cana-3791	139	31	)	)	PUNCT
cana-3791	139	32	∑sμ=0	∑sμ=0	ADP
cana-3791	139	33	exp(4πi/(s+1)nμ)⟨θm	exp(4πi/(s+1)nμ)⟨θm	PROPN
cana-3791	139	34	-	-	PUNCT
cana-3791	139	35	μ|ρ̂|θm+μ⟩	μ|ρ̂|θm+μ⟩	PROPN
cana-3791	139	36	this	this	DET
cana-3791	139	37	representation	representation	NOUN
cana-3791	139	38	brings	bring	VERB
cana-3791	139	39	about	about	ADP
cana-3791	139	40	the	the	DET
cana-3791	139	41	capability	capability	NOUN
cana-3791	139	42	to	to	PART
cana-3791	139	43	construct	construct	VERB
cana-3791	139	44	filters	filter	NOUN
cana-3791	139	45	which	which	PRON
cana-3791	139	46	are	be	AUX
cana-3791	139	47	optimum	optimum	ADJ
cana-3791	139	48	both	both	ADV
cana-3791	139	49	in	in	ADP
cana-3791	139	50	the	the	DET
cana-3791	139	51	time	time	NOUN
cana-3791	139	52	and	and	CCONJ
cana-3791	139	53	frequency	frequency	NOUN
cana-3791	139	54	domain	domain	NOUN
cana-3791	139	55	simultaneously	simultaneously	ADV
cana-3791	139	56	[	[	X
cana-3791	139	57	43	43	NUM
cana-3791	139	58	]	]	PUNCT
cana-3791	139	59	.	.	PUNCT
cana-3791	140	1	one	one	NUM
cana-3791	140	2	of	of	ADP
cana-3791	140	3	the	the	DET
cana-3791	140	4	important	important	ADJ
cana-3791	140	5	merits	merit	NOUN
cana-3791	140	6	of	of	ADP
cana-3791	140	7	this	this	DET
cana-3791	140	8	analysis	analysis	NOUN
cana-3791	140	9	is	be	AUX
cana-3791	140	10	that	that	SCONJ
cana-3791	140	11	both	both	CCONJ
cana-3791	140	12	linear	linear	ADJ
cana-3791	140	13	and	and	CCONJ
cana-3791	140	14	nonlinear	nonlinear	ADJ
cana-3791	140	15	filtering	filter	VERB
cana-3791	140	16	operations	operation	NOUN
cana-3791	140	17	can	can	AUX
cana-3791	140	18	be	be	AUX
cana-3791	140	19	treated	treat	VERB
cana-3791	140	20	within	within	ADP
cana-3791	140	21	a	a	DET
cana-3791	140	22	unified	unified	ADJ
cana-3791	140	23	framework	framework	NOUN
cana-3791	140	24	.	.	PUNCT
cana-3791	141	1	for	for	ADP
cana-3791	141	2	example	example	NOUN
cana-3791	141	3	,	,	PUNCT
cana-3791	141	4	the	the	DET
cana-3791	141	5	action	action	NOUN
cana-3791	141	6	of	of	ADP
cana-3791	141	7	a	a	DET
cana-3791	141	8	linear	linear	ADJ
cana-3791	141	9	filter	filter	NOUN
cana-3791	141	10	h	h	NOUN
cana-3791	141	11	on	on	ADP
cana-3791	141	12	a	a	DET
cana-3791	141	13	signal	signal	NOUN
cana-3791	141	14	can	can	AUX
cana-3791	141	15	be	be	AUX
cana-3791	141	16	written	write	VERB
cana-3791	141	17	as	as	ADP
cana-3791	141	18	:	:	PUNCT
cana-3791	141	19	(	(	PUNCT
cana-3791	141	20	hψ)(n	hψ)(n	PROPN
cana-3791	141	21	,	,	PUNCT
cana-3791	141	22	θm	θm	PROPN
cana-3791	141	23	)	)	PUNCT
cana-3791	141	24	=	=	PUNCT
cana-3791	141	25	∑sk=0	∑sk=0	PROPN
cana-3791	141	26	∑sl=0	∑sl=0	PROPN
cana-3791	141	27	h(n	h(n	PROPN
cana-3791	141	28	-	-	PUNCT
cana-3791	141	29	k	k	NOUN
cana-3791	141	30	,	,	PUNCT
cana-3791	141	31	θm	θm	NOUN
cana-3791	141	32	-	-	PUNCT
cana-3791	141	33	θl)ws(k	θl)ws(k	PROPN
cana-3791	141	34	,	,	PUNCT
cana-3791	141	35	θl	θl	ADP
cana-3791	141	36	)	)	PUNCT
cana-3791	141	37	where	where	SCONJ
cana-3791	141	38	h(n	h(n	PROPN
cana-3791	141	39	,	,	PUNCT
cana-3791	141	40	θm	θm	NOUN
cana-3791	141	41	)	)	PUNCT
cana-3791	141	42	is	be	AUX
cana-3791	141	43	the	the	DET
cana-3791	141	44	phase	phase	NOUN
cana-3791	141	45	-	-	PUNCT
cana-3791	141	46	space	space	NOUN
cana-3791	141	47	response	response	NOUN
cana-3791	141	48	of	of	ADP
cana-3791	141	49	the	the	DET
cana-3791	141	50	filter	filter	NOUN
cana-3791	141	51	.	.	PUNCT
cana-3791	142	1	performances	performance	NOUN
cana-3791	142	2	of	of	ADP
cana-3791	142	3	these	these	DET
cana-3791	142	4	filtering	filter	VERB
cana-3791	142	5	operations	operation	NOUN
cana-3791	142	6	can	can	AUX
cana-3791	142	7	be	be	AUX
cana-3791	142	8	quantified	quantify	VERB
cana-3791	142	9	by	by	ADP
cana-3791	142	10	the	the	DET
cana-3791	142	11	phase	phase	NOUN
cana-3791	142	12	-	-	PUNCT
cana-3791	142	13	space	space	NOUN
cana-3791	142	14	snr	snr	NOUN
cana-3791	142	15	:	:	PUNCT
cana-3791	142	16	snr	snr	PROPN
cana-3791	142	17	=	=	PRON
cana-3791	142	18	|∑sn=0	|∑sn=0	PUNCT
cana-3791	142	19	∑sm=0	∑sm=0	X
cana-3791	142	20	ws(n	ws(n	ADJ
cana-3791	142	21	,	,	PUNCT
cana-3791	142	22	θm)|2	θm)|2	NOUN
cana-3791	142	23	/	/	SYM
cana-3791	142	24	(	(	PUNCT
cana-3791	142	25	∑sn=0	∑sn=0	PROPN
cana-3791	142	26	∑sm=0	∑sm=0	PROPN
cana-3791	143	1	|ws(n	|ws(n	NOUN
cana-3791	143	2	,	,	PUNCT
cana-3791	143	3	θm)|2	θm)|2	NOUN
cana-3791	143	4	)	)	PUNCT
cana-3791	143	5	this	this	DET
cana-3791	143	6	measure	measure	NOUN
cana-3791	143	7	provides	provide	VERB
cana-3791	143	8	a	a	DET
cana-3791	143	9	natural	natural	ADJ
cana-3791	143	10	way	way	NOUN
cana-3791	143	11	of	of	ADP
cana-3791	143	12	optimizing	optimize	VERB
cana-3791	143	13	filter	filter	NOUN
cana-3791	143	14	parameters	parameter	NOUN
cana-3791	143	15	for	for	ADP
cana-3791	143	16	applications	application	NOUN
cana-3791	143	17	at	at	ADP
cana-3791	143	18	hand	hand	NOUN
cana-3791	143	19	[	[	X
cana-3791	143	20	44	44	NUM
cana-3791	143	21	]	]	PUNCT
cana-3791	143	22	.	.	PUNCT
cana-3791	144	1	for	for	ADP
cana-3791	144	2	implementations	implementation	NOUN
cana-3791	144	3	,	,	PUNCT
cana-3791	144	4	the	the	DET
cana-3791	144	5	finite	finite	ADJ
cana-3791	144	6	-	-	ADJ
cana-3791	144	7	dimensionality	dimensionality	NOUN
cana-3791	144	8	entails	entail	VERB
cana-3791	144	9	several	several	ADJ
cana-3791	144	10	computational	computational	ADJ
cana-3791	144	11	advantages	advantage	NOUN
cana-3791	144	12	:	:	PUNCT
cana-3791	144	13	bounded	bound	VERB
cana-3791	144	14	error	error	NOUN
cana-3791	144	15	propagation	propagation	NOUN
cana-3791	144	16	:	:	PUNCT
cana-3791	144	17	finite	finite	ADJ
cana-3791	144	18	dimensionality	dimensionality	NOUN
cana-3791	144	19	ensures	ensure	VERB
cana-3791	144	20	that	that	SCONJ
cana-3791	144	21	the	the	DET
cana-3791	144	22	numerical	numerical	ADJ
cana-3791	144	23	errors	error	NOUN
cana-3791	144	24	in	in	ADP
cana-3791	144	25	filtering	filtering	NOUN
cana-3791	144	26	remain	remain	AUX
cana-3791	144	27	bounded	bound	VERB
cana-3791	144	28	.	.	PUNCT
cana-3791	145	1	efficient	efficient	ADJ
cana-3791	145	2	implementation	implementation	NOUN
cana-3791	145	3	:	:	PUNCT
cana-3791	145	4	the	the	DET
cana-3791	145	5	discrete	discrete	ADJ
cana-3791	145	6	nature	nature	NOUN
cana-3791	145	7	of	of	ADP
cana-3791	145	8	the	the	DET
cana-3791	145	9	transforms	transform	NOUN
cana-3791	145	10	allows	allow	VERB
cana-3791	145	11	for	for	ADP
cana-3791	145	12	efficient	efficient	ADJ
cana-3791	145	13	fft	fft	NOUN
cana-3791	145	14	-	-	PUNCT
cana-3791	145	15	based	base	VERB
cana-3791	145	16	implementations	implementation	NOUN
cana-3791	145	17	.	.	PUNCT
cana-3791	146	1	preservation	preservation	NOUN
cana-3791	146	2	of	of	ADP
cana-3791	146	3	signal	signal	ADJ
cana-3791	146	4	properties	property	NOUN
cana-3791	146	5	:	:	PUNCT
cana-3791	146	6	the	the	DET
cana-3791	146	7	phase	phase	NOUN
cana-3791	146	8	-	-	PUNCT
cana-3791	146	9	space	space	NOUN
cana-3791	146	10	structure	structure	NOUN
cana-3791	146	11	ensures	ensure	VERB
cana-3791	146	12	that	that	SCONJ
cana-3791	146	13	important	important	ADJ
cana-3791	146	14	signal	signal	NOUN
cana-3791	146	15	properties	property	NOUN
cana-3791	146	16	are	be	AUX
cana-3791	146	17	preserved	preserve	VERB
cana-3791	146	18	in	in	ADP
cana-3791	146	19	the	the	DET
cana-3791	146	20	filtering	filtering	NOUN
cana-3791	146	21	operation	operation	NOUN
cana-3791	146	22	.	.	PUNCT
cana-3791	147	1	these	these	DET
cana-3791	147	2	advantages	advantage	NOUN
cana-3791	147	3	make	make	VERB
cana-3791	147	4	the	the	DET
cana-3791	147	5	finite	finite	ADJ
cana-3791	147	6	-	-	ADJ
cana-3791	147	7	dimensional	dimensional	ADJ
cana-3791	147	8	approach	approach	NOUN
cana-3791	147	9	particularly	particularly	ADV
cana-3791	147	10	suitable	suitable	ADJ
cana-3791	147	11	for	for	ADP
cana-3791	147	12	real	real	ADJ
cana-3791	147	13	-	-	PUNCT
cana-3791	147	14	time	time	NOUN
cana-3791	147	15	signal	signal	NOUN
cana-3791	147	16	processing	processing	NOUN
cana-3791	147	17	applications	application	NOUN
cana-3791	147	18	where	where	SCONJ
cana-3791	147	19	both	both	DET
cana-3791	147	20	computational	computational	ADJ
cana-3791	147	21	efficiency	efficiency	NOUN
cana-3791	147	22	and	and	CCONJ
cana-3791	147	23	numerical	numerical	ADJ
cana-3791	147	24	stability	stability	NOUN
cana-3791	147	25	are	be	AUX
cana-3791	147	26	crucial	crucial	ADJ
cana-3791	147	27	[	[	X
cana-3791	147	28	45	45	NUM
cana-3791	147	29	]	]	PUNCT
cana-3791	147	30	.	.	PUNCT
cana-3791	148	1	numerical	numerical	ADJ
cana-3791	148	2	analysis	analysis	NOUN
cana-3791	148	3	the	the	DET
cana-3791	148	4	application	application	NOUN
cana-3791	148	5	of	of	ADP
cana-3791	148	6	finite	finite	ADJ
cana-3791	148	7	-	-	ADJ
cana-3791	148	8	dimensional	dimensional	ADJ
cana-3791	148	9	hilbert	hilbert	NOUN
cana-3791	148	10	space	space	NOUN
cana-3791	148	11	theory	theory	NOUN
cana-3791	148	12	to	to	ADP
cana-3791	148	13	numerical	numerical	ADJ
cana-3791	148	14	analysis	analysis	NOUN
cana-3791	148	15	brings	bring	VERB
cana-3791	148	16	some	some	DET
cana-3791	148	17	strong	strong	ADJ
cana-3791	148	18	tools	tool	NOUN
cana-3791	148	19	in	in	ADP
cana-3791	148	20	performing	perform	VERB
cana-3791	148	21	function	function	NOUN
cana-3791	148	22	approximation	approximation	NOUN
cana-3791	148	23	and	and	CCONJ
cana-3791	148	24	the	the	DET
cana-3791	148	25	analysis	analysis	NOUN
cana-3791	148	26	of	of	ADP
cana-3791	148	27	errors	error	NOUN
cana-3791	148	28	.	.	PUNCT
cana-3791	149	1	the	the	DET
cana-3791	149	2	framework	framework	NOUN
cana-3791	149	3	of	of	ADP
cana-3791	149	4	kernel	kernel	PROPN
cana-3791	149	5	stein	stein	PROPN
cana-3791	149	6	discrepancy	discrepancy	NOUN
cana-3791	149	7	offers	offer	VERB
cana-3791	149	8	specially	specially	ADV
cana-3791	149	9	useful	useful	ADJ
cana-3791	149	10	insights	insight	NOUN
cana-3791	149	11	into	into	ADP
cana-3791	149	12	both	both	DET
cana-3791	149	13	theoretical	theoretical	ADJ
cana-3791	149	14	bounds	bound	NOUN
cana-3791	149	15	and	and	CCONJ
cana-3791	149	16	practical	practical	ADJ
cana-3791	149	17	implementations	implementation	NOUN
cana-3791	149	18	of	of	ADP
cana-3791	149	19	numerical	numerical	ADJ
cana-3791	149	20	approximations	approximation	NOUN
cana-3791	149	21	.	.	PUNCT
cana-3791	150	1	the	the	DET
cana-3791	150	2	kernel	kernel	PROPN
cana-3791	150	3	method	method	NOUN
cana-3791	150	4	can	can	AUX
cana-3791	150	5	be	be	AUX
cana-3791	150	6	seen	see	VERB
cana-3791	150	7	as	as	ADP
cana-3791	150	8	approximating	approximate	VERB
cana-3791	150	9	a	a	DET
cana-3791	150	10	function	function	NOUN
cana-3791	150	11	f	f	NOUN
cana-3791	150	12	in	in	ADP
cana-3791	150	13	the	the	DET
cana-3791	150	14	finite	finite	ADJ
cana-3791	150	15	-	-	ADJ
cana-3791	150	16	dimensional	dimensional	ADJ
cana-3791	150	17	case	case	NOUN
cana-3791	150	18	by	by	ADP
cana-3791	150	19	representing	represent	VERB
cana-3791	150	20	it	it	PRON
cana-3791	150	21	through	through	ADP
cana-3791	150	22	its	its	PRON
cana-3791	150	23	action	action	NOUN
cana-3791	150	24	on	on	ADP
cana-3791	150	25	an	an	DET
cana-3791	150	26	rkhs	rkh	NOUN
cana-3791	150	27	.	.	PUNCT
cana-3791	151	1	one	one	NUM
cana-3791	151	2	basic	basic	ADJ
cana-3791	151	3	approximation	approximation	NOUN
cana-3791	151	4	can	can	AUX
cana-3791	151	5	be	be	AUX
cana-3791	151	6	written	write	VERB
cana-3791	151	7	as	as	SCONJ
cana-3791	151	8	follows	follow	VERB
cana-3791	151	9	[	[	X
cana-3791	151	10	46	46	NUM
cana-3791	151	11	]	]	X
cana-3791	151	12	:	:	PUNCT
cana-3791	151	13	communications	communication	NOUN
cana-3791	151	14	on	on	ADP
cana-3791	151	15	applied	apply	VERB
cana-3791	151	16	nonlinear	nonlinear	ADJ
cana-3791	151	17	analysis	analysis	NOUN
cana-3791	151	18	issn	issn	NOUN
cana-3791	151	19	:	:	PUNCT
cana-3791	151	20	1074	1074	NUM
cana-3791	151	21	-	-	PUNCT
cana-3791	151	22	133x	133x	NUM
cana-3791	151	23	vol	vol	NOUN
cana-3791	151	24	32	32	NUM
cana-3791	151	25	no	no	NOUN
cana-3791	151	26	.	.	PUNCT
cana-3791	152	1	8s	8s	PROPN
cana-3791	152	2	(	(	PUNCT
cana-3791	152	3	2025	2025	NUM
cana-3791	152	4	)	)	PUNCT
cana-3791	152	5	739	739	NUM
cana-3791	153	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	153	2	f	f	X
cana-3791	153	3	=	=	SYM
cana-3791	154	1	∑di=1	∑di=1	ADV
cana-3791	154	2	ciϕi	ciϕi	INTJ
cana-3791	154	3	where	where	SCONJ
cana-3791	154	4	{	{	PUNCT
cana-3791	154	5	ϕi}di=1	ϕi}di=1	PROPN
cana-3791	154	6	forms	form	VERB
cana-3791	154	7	a	a	DET
cana-3791	154	8	basis	basis	NOUN
cana-3791	154	9	for	for	ADP
cana-3791	154	10	the	the	DET
cana-3791	154	11	finite	finite	ADJ
cana-3791	154	12	-	-	ADJ
cana-3791	154	13	dimensional	dimensional	ADJ
cana-3791	154	14	rkhs	rkh	NOUN
cana-3791	154	15	,	,	PUNCT
cana-3791	154	16	and	and	CCONJ
cana-3791	154	17	the	the	DET
cana-3791	154	18	coefficients	coefficient	NOUN
cana-3791	154	19	ci	ci	PROPN
cana-3791	154	20	are	be	AUX
cana-3791	154	21	determined	determine	VERB
cana-3791	154	22	by	by	ADP
cana-3791	154	23	the	the	DET
cana-3791	154	24	inner	inner	ADJ
cana-3791	154	25	product	product	NOUN
cana-3791	154	26	structure	structure	NOUN
cana-3791	154	27	.	.	PUNCT
cana-3791	155	1	the	the	DET
cana-3791	155	2	kernel	kernel	PROPN
cana-3791	155	3	k(x	k(x	PROPN
cana-3791	155	4	,	,	PUNCT
cana-3791	155	5	y	y	NOUN
cana-3791	155	6	)	)	PUNCT
cana-3791	155	7	has	have	VERB
cana-3791	155	8	to	to	PART
cana-3791	155	9	satisfy	satisfy	VERB
cana-3791	155	10	specific	specific	ADJ
cana-3791	155	11	regularity	regularity	NOUN
cana-3791	155	12	conditions	condition	NOUN
cana-3791	155	13	:	:	PUNCT
cana-3791	155	14	k	k	PROPN
cana-3791	155	15	∈	∈	PROPN
cana-3791	155	16	c(2,2)b(x	c(2,2)b(x	PROPN
cana-3791	155	17	×	×	NOUN
cana-3791	155	18	x	x	NOUN
cana-3791	155	19	)	)	PUNCT
cana-3791	155	20	where	where	SCONJ
cana-3791	155	21	c(2,2)b	c(2,2)b	NOUN
cana-3791	155	22	denotes	denote	VERB
cana-3791	155	23	the	the	DET
cana-3791	155	24	space	space	NOUN
cana-3791	155	25	of	of	ADP
cana-3791	155	26	bounded	bounded	ADJ
cana-3791	155	27	functions	function	NOUN
cana-3791	155	28	with	with	ADP
cana-3791	155	29	continuous	continuous	ADJ
cana-3791	155	30	derivatives	derivative	NOUN
cana-3791	155	31	up	up	ADP
cana-3791	155	32	to	to	ADP
cana-3791	155	33	second	second	ADJ
cana-3791	155	34	order	order	NOUN
cana-3791	155	35	[	[	X
cana-3791	155	36	47	47	NUM
cana-3791	155	37	]	]	PUNCT
cana-3791	155	38	.	.	PUNCT
cana-3791	156	1	bound	bind	VERB
cana-3791	156	2	such	such	ADJ
cana-3791	156	3	approximations	approximation	NOUN
cana-3791	156	4	can	can	AUX
cana-3791	156	5	be	be	AUX
cana-3791	156	6	performed	perform	VERB
cana-3791	156	7	via	via	ADP
cana-3791	156	8	the	the	DET
cana-3791	156	9	kernel	kernel	PROPN
cana-3791	156	10	stein	stein	PROPN
cana-3791	156	11	discrepancy	discrepancy	NOUN
cana-3791	156	12	given	give	VERB
cana-3791	156	13	as	as	ADP
cana-3791	156	14	:	:	PUNCT
cana-3791	156	15	ksd(q	ksd(q	PROPN
cana-3791	156	16	,	,	PUNCT
cana-3791	156	17	p)2	p)2	NOUN
cana-3791	156	18	=	=	SYM
cana-3791	156	19	e(x	e(x	NUM
cana-3791	156	20	,	,	PUNCT
cana-3791	156	21	x')~q×q[(a	x')~q×q[(a	PROPN
cana-3791	156	22	⊗	⊗	PROPN
cana-3791	156	23	a)k(x	a)k(x	PROPN
cana-3791	156	24	,	,	PUNCT
cana-3791	156	25	x	x	NOUN
cana-3791	156	26	'	'	NUM
cana-3791	156	27	)	)	PUNCT
cana-3791	156	28	]	]	PUNCT
cana-3791	156	29	with	with	ADP
cana-3791	156	30	a	a	DET
cana-3791	156	31	being	be	AUX
cana-3791	156	32	the	the	DET
cana-3791	156	33	stein	stein	NOUN
cana-3791	156	34	operator	operator	NOUN
cana-3791	156	35	given	give	VERB
cana-3791	156	36	by	by	ADP
cana-3791	156	37	the	the	DET
cana-3791	156	38	following	follow	VERB
cana-3791	156	39	equation	equation	NOUN
cana-3791	156	40	:	:	PUNCT
cana-3791	156	41	af(x	af(x	X
cana-3791	156	42	)	)	PUNCT
cana-3791	156	43	=	=	SYM
cana-3791	156	44	tr[cd2f(x	tr[cd2f(x	NOUN
cana-3791	156	45	)	)	PUNCT
cana-3791	156	46	]	]	PUNCT
cana-3791	157	1	⟨df(x	⟨df(x	NOUN
cana-3791	157	2	)	)	PUNCT
cana-3791	157	3	,	,	PUNCT
cana-3791	157	4	x	x	PUNCT
cana-3791	158	1	+	+	PUNCT
cana-3791	158	2	cdu(x)⟩x	cdu(x)⟩x	NOUN
cana-3791	158	3	under	under	ADP
cana-3791	158	4	this	this	DET
cana-3791	158	5	formulation	formulation	NOUN
cana-3791	158	6	it	it	PRON
cana-3791	158	7	is	be	AUX
cana-3791	158	8	a	a	DET
cana-3791	158	9	natural	natural	ADJ
cana-3791	158	10	way	way	NOUN
cana-3791	158	11	to	to	PART
cana-3791	158	12	make	make	VERB
cana-3791	158	13	an	an	DET
cana-3791	158	14	approximation	approximation	NOUN
cana-3791	158	15	error	error	NOUN
cana-3791	158	16	quality	quality	NOUN
cana-3791	158	17	[	[	X
cana-3791	158	18	48	48	NUM
cana-3791	158	19	]	]	PUNCT
cana-3791	158	20	.	.	PUNCT
cana-3791	159	1	particularly	particularly	ADV
cana-3791	159	2	,	,	PUNCT
cana-3791	159	3	it	it	PRON
cana-3791	159	4	can	can	AUX
cana-3791	159	5	be	be	AUX
cana-3791	159	6	possible	possible	ADJ
cana-3791	159	7	to	to	PART
cana-3791	159	8	make	make	VERB
cana-3791	159	9	more	more	ADV
cana-3791	159	10	sophisticated	sophisticated	ADJ
cana-3791	159	11	error	error	NOUN
cana-3791	159	12	analyses	analysis	NOUN
cana-3791	159	13	by	by	ADP
cana-3791	159	14	considering	consider	VERB
cana-3791	159	15	a	a	DET
cana-3791	159	16	variety	variety	NOUN
cana-3791	159	17	of	of	ADP
cana-3791	159	18	metrics	metric	NOUN
cana-3791	159	19	:	:	PUNCT
cana-3791	160	1	1	1	X
cana-3791	160	2	.	.	X
cana-3791	160	3	local	local	ADJ
cana-3791	160	4	error	error	NOUN
cana-3791	160	5	bounds	bound	NOUN
cana-3791	160	6	:	:	PUNCT
cana-3791	160	7	||f	||f	PROPN
cana-3791	160	8	fn||h	fn||h	PROPN
cana-3791	160	9	≤	≤	PROPN
cana-3791	160	10	c1(d)|||a|||op||f||h	c1(d)|||a|||op||f||h	PROPN
cana-3791	160	11	where	where	SCONJ
cana-3791	160	12	fn	fn	NOUN
cana-3791	160	13	denotes	denote	VERB
cana-3791	160	14	the	the	DET
cana-3791	160	15	finite	finite	ADJ
cana-3791	160	16	dimensional	dimensional	ADJ
cana-3791	160	17	approximation	approximation	NOUN
cana-3791	160	18	and	and	CCONJ
cana-3791	160	19	|||a|||op	|||a|||op	NOUN
cana-3791	160	20	is	be	AUX
cana-3791	160	21	the	the	DET
cana-3791	160	22	operator	operator	NOUN
cana-3791	160	23	norm	norm	NOUN
cana-3791	160	24	.	.	PUNCT
cana-3791	161	1	2	2	X
cana-3791	161	2	.	.	X
cana-3791	161	3	global	global	ADJ
cana-3791	161	4	error	error	NOUN
cana-3791	161	5	estimates	estimate	VERB
cana-3791	161	6	:	:	PUNCT
cana-3791	161	7	e	e	X
cana-3791	161	8	=	=	SYM
cana-3791	161	9	sup{|f(x	sup{|f(x	NOUN
cana-3791	161	10	)	)	PUNCT
cana-3791	161	11	fn(x)|	fn(x)|	NOUN
cana-3791	161	12	:	:	PUNCT
cana-3791	162	1	x	x	X
cana-3791	162	2	∈	∈	NOUN
cana-3791	162	3	x	x	NOUN
cana-3791	162	4	,	,	PUNCT
cana-3791	162	5	||f||h	||f||h	PROPN
cana-3791	162	6	≤	≤	NOUN
cana-3791	162	7	1	1	NUM
cana-3791	162	8	}	}	PUNCT
cana-3791	162	9	these	these	DET
cana-3791	162	10	bounds	bound	NOUN
cana-3791	162	11	are	be	AUX
cana-3791	162	12	particularly	particularly	ADV
cana-3791	162	13	useful	useful	ADJ
cana-3791	162	14	in	in	ADP
cana-3791	162	15	adaptive	adaptive	ADJ
cana-3791	162	16	approximation	approximation	NOUN
cana-3791	162	17	schemes	scheme	NOUN
cana-3791	162	18	[	[	X
cana-3791	162	19	49	49	NUM
cana-3791	162	20	]	]	PUNCT
cana-3791	162	21	.	.	PUNCT
cana-3791	163	1	fourier	fourier	PROPN
cana-3791	163	2	representation	representation	NOUN
cana-3791	163	3	furnishes	furnish	VERB
cana-3791	163	4	the	the	DET
cana-3791	163	5	analytical	analytical	ADJ
cana-3791	163	6	tool	tool	NOUN
cana-3791	163	7	to	to	PART
cana-3791	163	8	study	study	VERB
cana-3791	163	9	the	the	DET
cana-3791	163	10	convergence	convergence	NOUN
cana-3791	163	11	properties	property	NOUN
cana-3791	163	12	of	of	ADP
cana-3791	163	13	numerical	numerical	ADJ
cana-3791	163	14	schemes	scheme	NOUN
cana-3791	163	15	:	:	PUNCT
cana-3791	163	16	ksd2	ksd2	PROPN
cana-3791	163	17	=	=	PUNCT
cana-3791	163	18	∫x	∫x	PROPN
cana-3791	163	19	eq[a(ei⟨s,·⟩x)(x)]2c	eq[a(ei⟨s,·⟩x)(x)]2c	PROPN
cana-3791	163	20	dμ(s	dμ(s	NOUN
cana-3791	163	21	)	)	PUNCT
cana-3791	163	22	this	this	DET
cana-3791	163	23	representation	representation	NOUN
cana-3791	163	24	provides	provide	VERB
cana-3791	163	25	insight	insight	NOUN
cana-3791	163	26	,	,	PUNCT
cana-3791	163	27	among	among	ADP
cana-3791	163	28	others	other	NOUN
cana-3791	163	29	,	,	PUNCT
cana-3791	163	30	into	into	ADP
cana-3791	163	31	the	the	DET
cana-3791	163	32	approximation	approximation	NOUN
cana-3791	163	33	spectral	spectral	ADJ
cana-3791	163	34	properties	property	NOUN
cana-3791	163	35	,	,	PUNCT
cana-3791	163	36	and	and	CCONJ
cana-3791	163	37	it	it	PRON
cana-3791	163	38	serves	serve	VERB
cana-3791	163	39	as	as	ADP
cana-3791	163	40	a	a	DET
cana-3791	163	41	starting	starting	NOUN
cana-3791	163	42	point	point	NOUN
cana-3791	163	43	for	for	ADP
cana-3791	163	44	practical	practical	ADJ
cana-3791	163	45	error	error	NOUN
cana-3791	163	46	estimates	estimate	NOUN
cana-3791	163	47	.	.	PUNCT
cana-3791	164	1	the	the	DET
cana-3791	164	2	first	first	ADJ
cana-3791	164	3	especially	especially	ADV
cana-3791	164	4	important	important	ADJ
cana-3791	164	5	question	question	NOUN
cana-3791	164	6	of	of	ADP
cana-3791	164	7	numerical	numerical	ADJ
cana-3791	164	8	analysis	analysis	NOUN
cana-3791	164	9	,	,	PUNCT
cana-3791	164	10	to	to	PART
cana-3791	164	11	be	be	AUX
cana-3791	164	12	explored	explore	VERB
cana-3791	164	13	here	here	ADV
cana-3791	164	14	,	,	PUNCT
cana-3791	164	15	is	be	AUX
cana-3791	164	16	given	give	VERB
cana-3791	164	17	by	by	ADP
cana-3791	164	18	boundedness	boundedness	NOUN
cana-3791	164	19	of	of	ADP
cana-3791	164	20	sequences	sequence	NOUN
cana-3791	164	21	:	:	PUNCT
cana-3791	164	22	sequence	sequence	NOUN
cana-3791	164	23	boundedness	boundedness	NOUN
cana-3791	164	24	:	:	PUNCT
cana-3791	164	25	1	1	X
cana-3791	164	26	.	.	X
cana-3791	164	27	a	a	DET
cana-3791	164	28	sequence	sequence	NOUN
cana-3791	164	29	is	be	AUX
cana-3791	164	30	said	say	VERB
cana-3791	164	31	to	to	PART
cana-3791	164	32	be	be	AUX
cana-3791	164	33	bounded	bound	VERB
cana-3791	164	34	if	if	SCONJ
cana-3791	164	35	it	it	PRON
cana-3791	164	36	admits	admit	VERB
cana-3791	164	37	a	a	DET
cana-3791	164	38	cocone	cocone	NOUN
cana-3791	164	39	of	of	ADP
cana-3791	164	40	monomorphisms	monomorphism	NOUN
cana-3791	164	41	.	.	PUNCT
cana-3791	165	1	2	2	X
cana-3791	165	2	.	.	X
cana-3791	165	3	convergence	convergence	NOUN
cana-3791	165	4	criterion	criterion	NOUN
cana-3791	165	5	:	:	PUNCT
cana-3791	165	6	lim	lim	PROPN
cana-3791	165	7	n→∞	n→∞	PROPN
cana-3791	166	1	||	||	PUNCT
cana-3791	167	1	fn	fn	NOUN
cana-3791	168	1	f	f	NOUN
cana-3791	168	2	||	||	PROPN
cana-3791	169	1	h	h	NOUN
cana-3791	169	2	=	=	NOUN
cana-3791	169	3	0	0	NUM
cana-3791	169	4	provided	provide	VERB
cana-3791	169	5	that	that	SCONJ
cana-3791	169	6	the	the	DET
cana-3791	169	7	norm	norm	NOUN
cana-3791	169	8	is	be	AUX
cana-3791	169	9	induced	induce	VERB
cana-3791	169	10	by	by	ADP
cana-3791	169	11	inner	inner	ADJ
cana-3791	169	12	product	product	NOUN
cana-3791	169	13	structure	structure	NOUN
cana-3791	169	14	.	.	PUNCT
cana-3791	170	1	these	these	DET
cana-3791	170	2	bounds	bound	NOUN
cana-3791	170	3	for	for	ADP
cana-3791	170	4	practical	practical	ADJ
cana-3791	170	5	computations	computation	NOUN
cana-3791	170	6	are	be	AUX
cana-3791	170	7	given	give	VERB
cana-3791	170	8	by	by	ADP
cana-3791	170	9	the	the	DET
cana-3791	170	10	following	follow	VERB
cana-3791	170	11	numerical	numerical	ADJ
cana-3791	170	12	schemes	scheme	NOUN
cana-3791	170	13	,	,	PUNCT
cana-3791	170	14	in	in	ADP
cana-3791	170	15	terms	term	NOUN
cana-3791	170	16	of	of	ADP
cana-3791	170	17	:	:	PUNCT
cana-3791	170	18	1	1	X
cana-3791	170	19	.	.	X
cana-3791	170	20	truncation	truncation	NOUN
cana-3791	170	21	error	error	NOUN
cana-3791	170	22	:	:	PUNCT
cana-3791	170	23	et	et	NOUN
cana-3791	170	24	=	=	SYM
cana-3791	170	25	o(d	o(d	X
cana-3791	170	26	-	-	PUNCT
cana-3791	170	27	α	α	NOUN
cana-3791	170	28	)	)	PUNCT
cana-3791	170	29	where	where	SCONJ
cana-3791	170	30	α	α	NOUN
cana-3791	170	31	depends	depend	VERB
cana-3791	170	32	from	from	ADP
cana-3791	170	33	the	the	DET
cana-3791	170	34	smoothness	smoothness	NOUN
cana-3791	170	35	of	of	ADP
cana-3791	170	36	f	f	PROPN
cana-3791	170	37	2	2	NUM
cana-3791	170	38	.	.	PUNCT
cana-3791	170	39	roundoff	roundoff	NOUN
cana-3791	170	40	error	error	NOUN
cana-3791	170	41	:	:	PUNCT
cana-3791	170	42	er	er	INTJ
cana-3791	170	43	=	=	SYM
cana-3791	170	44	o(ε	o(ε	PROPN
cana-3791	170	45	machine	machine	NOUN
cana-3791	170	46	×	×	NOUN
cana-3791	170	47	condition_number	condition_number	NOUN
cana-3791	170	48	)	)	PUNCT
cana-3791	170	49	3	3	NUM
cana-3791	170	50	.	.	PUNCT
cana-3791	170	51	total	total	ADJ
cana-3791	170	52	error	error	NOUN
cana-3791	170	53	:	:	PUNCT
cana-3791	170	54	etotal	etotal	NOUN
cana-3791	170	55	=	=	PUNCT
cana-3791	170	56	et	et	PROPN
cana-3791	170	57	+	+	X
cana-3791	170	58	er	er	INTJ
cana-3791	170	59	in	in	ADP
cana-3791	170	60	fact	fact	NOUN
cana-3791	170	61	,	,	PUNCT
cana-3791	170	62	these	these	DET
cana-3791	170	63	error	error	NOUN
cana-3791	170	64	estimates	estimate	NOUN
cana-3791	170	65	have	have	VERB
cana-3791	170	66	the	the	DET
cana-3791	170	67	effect	effect	NOUN
cana-3791	170	68	of	of	ADP
cana-3791	170	69	practical	practical	ADJ
cana-3791	170	70	guidance	guidance	NOUN
cana-3791	170	71	so	so	SCONJ
cana-3791	170	72	as	as	SCONJ
cana-3791	170	73	to	to	PART
cana-3791	170	74	make	make	VERB
cana-3791	170	75	appropriate	appropriate	ADJ
cana-3791	170	76	choices	choice	NOUN
cana-3791	170	77	of	of	ADP
cana-3791	170	78	the	the	DET
cana-3791	170	79	discretization	discretization	NOUN
cana-3791	170	80	parameters	parameter	NOUN
cana-3791	170	81	[	[	X
cana-3791	170	82	52	52	NUM
cana-3791	170	83	]	]	PUNCT
cana-3791	170	84	.	.	PUNCT
cana-3791	171	1	the	the	DET
cana-3791	171	2	implementation	implementation	NOUN
cana-3791	171	3	of	of	ADP
cana-3791	171	4	these	these	DET
cana-3791	171	5	numerical	numerical	ADJ
cana-3791	171	6	schemes	scheme	NOUN
cana-3791	171	7	benefits	benefit	NOUN
cana-3791	171	8	from	from	ADP
cana-3791	171	9	the	the	DET
cana-3791	171	10	following	follow	VERB
cana-3791	171	11	key	key	ADJ
cana-3791	171	12	properties	property	NOUN
cana-3791	171	13	of	of	ADP
cana-3791	171	14	finite	finite	ADJ
cana-3791	171	15	-	-	ADJ
cana-3791	171	16	dimensional	dimensional	ADJ
cana-3791	171	17	hilbert	hilbert	NOUN
cana-3791	171	18	spaces	space	NOUN
cana-3791	171	19	:	:	PUNCT
cana-3791	171	20	1	1	X
cana-3791	171	21	.	.	X
cana-3791	171	22	completeness	completeness	NOUN
cana-3791	171	23	:	:	PUNCT
cana-3791	171	24	every	every	DET
cana-3791	171	25	cauchy	cauchy	ADJ
cana-3791	171	26	sequence	sequence	NOUN
cana-3791	171	27	converges	converge	VERB
cana-3791	171	28	in	in	ADP
cana-3791	171	29	the	the	DET
cana-3791	171	30	space	space	NOUN
cana-3791	171	31	2	2	NUM
cana-3791	171	32	.	.	PUNCT
cana-3791	171	33	compactness	compactness	NOUN
cana-3791	171	34	:	:	PUNCT
cana-3791	171	35	bounded	bound	VERB
cana-3791	171	36	sets	set	NOUN
cana-3791	171	37	are	be	AUX
cana-3791	171	38	precompact	precompact	ADJ
cana-3791	171	39	3	3	NUM
cana-3791	171	40	.	.	PUNCT
cana-3791	171	41	stability	stability	NOUN
cana-3791	171	42	:	:	PUNCT
cana-3791	171	43	numerical	numerical	ADJ
cana-3791	171	44	operations	operation	NOUN
cana-3791	171	45	have	have	AUX
cana-3791	171	46	bounded	bound	VERB
cana-3791	171	47	condition	condition	NOUN
cana-3791	171	48	numbers	number	NOUN
cana-3791	171	49	these	these	DET
cana-3791	171	50	properties	property	NOUN
cana-3791	171	51	ensure	ensure	VERB
cana-3791	171	52	the	the	DET
cana-3791	171	53	reliability	reliability	NOUN
cana-3791	171	54	of	of	ADP
cana-3791	171	55	numerical	numerical	ADJ
cana-3791	171	56	computations	computation	NOUN
cana-3791	171	57	[	[	X
cana-3791	171	58	53	53	NUM
cana-3791	171	59	]	]	PUNCT
cana-3791	171	60	.	.	PUNCT
cana-3791	172	1	a	a	DET
cana-3791	172	2	fundamental	fundamental	ADJ
cana-3791	172	3	practical	practical	ADJ
cana-3791	172	4	problem	problem	NOUN
cana-3791	172	5	is	be	AUX
cana-3791	172	6	that	that	PRON
cana-3791	172	7	of	of	ADP
cana-3791	172	8	choosing	choose	VERB
cana-3791	172	9	basis	basis	NOUN
cana-3791	172	10	functions	function	NOUN
cana-3791	172	11	.	.	PUNCT
cana-3791	173	1	the	the	DET
cana-3791	173	2	optimal	optimal	ADJ
cana-3791	173	3	choice	choice	NOUN
cana-3791	173	4	often	often	ADV
cana-3791	173	5	depends	depend	VERB
cana-3791	173	6	on	on	ADP
cana-3791	173	7	:	:	PUNCT
cana-3791	173	8	1	1	X
cana-3791	173	9	.	.	X
cana-3791	173	10	function	function	NOUN
cana-3791	173	11	smoothness	smoothness	ADV
cana-3791	173	12	:	:	PUNCT
cana-3791	173	13	the	the	PRON
cana-3791	173	14	higher	high	ADJ
cana-3791	173	15	the	the	DET
cana-3791	173	16	regularity	regularity	NOUN
cana-3791	173	17	available	available	ADJ
cana-3791	173	18	,	,	PUNCT
cana-3791	173	19	the	the	PRON
cana-3791	173	20	more	more	ADV
cana-3791	173	21	rapid	rapid	ADJ
cana-3791	173	22	the	the	DET
cana-3791	173	23	convergence	convergence	NOUN
cana-3791	173	24	communications	communication	NOUN
cana-3791	173	25	on	on	ADP
cana-3791	173	26	applied	apply	VERB
cana-3791	173	27	nonlinear	nonlinear	ADJ
cana-3791	173	28	analysis	analysis	NOUN
cana-3791	173	29	issn	issn	NOUN
cana-3791	173	30	:	:	PUNCT
cana-3791	173	31	1074	1074	NUM
cana-3791	173	32	-	-	PUNCT
cana-3791	173	33	133x	133x	NUM
cana-3791	173	34	vol	vol	NOUN
cana-3791	173	35	32	32	NUM
cana-3791	173	36	no	no	NOUN
cana-3791	173	37	.	.	PUNCT
cana-3791	174	1	8s	8s	PROPN
cana-3791	174	2	(	(	PUNCT
cana-3791	174	3	2025	2025	NUM
cana-3791	174	4	)	)	PUNCT
cana-3791	174	5	740	740	NUM
cana-3791	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	174	7	2	2	NUM
cana-3791	174	8	.	.	PUNCT
cana-3791	174	9	domain	domain	NOUN
cana-3791	174	10	geometry	geometry	NOUN
cana-3791	174	11	:	:	PUNCT
cana-3791	174	12	boundary	boundary	ADJ
cana-3791	174	13	conditions	condition	NOUN
cana-3791	174	14	may	may	AUX
cana-3791	174	15	affect	affect	VERB
cana-3791	174	16	the	the	DET
cana-3791	174	17	choice	choice	NOUN
cana-3791	174	18	3	3	NUM
cana-3791	174	19	.	.	PUNCT
cana-3791	174	20	computational	computational	ADJ
cana-3791	174	21	efficiency	efficiency	NOUN
cana-3791	174	22	:	:	PUNCT
cana-3791	174	23	for	for	ADP
cana-3791	174	24	some	some	DET
cana-3791	174	25	bases	basis	NOUN
cana-3791	174	26	,	,	PUNCT
cana-3791	174	27	there	there	PRON
cana-3791	174	28	may	may	AUX
cana-3791	174	29	be	be	AUX
cana-3791	174	30	a	a	DET
cana-3791	174	31	fast	fast	ADJ
cana-3791	174	32	transform	transform	NOUN
cana-3791	174	33	algorithm	algorithm	NOUN
cana-3791	174	34	thus	thus	ADV
cana-3791	174	35	we	we	PRON
cana-3791	174	36	have	have	VERB
cana-3791	174	37	a	a	DET
cana-3791	174	38	compromise	compromise	NOUN
cana-3791	174	39	between	between	ADP
cana-3791	174	40	approximation	approximation	NOUN
cana-3791	174	41	and	and	CCONJ
cana-3791	174	42	computational	computational	ADJ
cana-3791	174	43	expense	expense	NOUN
cana-3791	174	44	that	that	PRON
cana-3791	174	45	must	must	AUX
cana-3791	174	46	be	be	AUX
cana-3791	174	47	carefully	carefully	ADV
cana-3791	174	48	balanced	balance	VERB
cana-3791	174	49	in	in	ADP
cana-3791	174	50	practice	practice	NOUN
cana-3791	174	51	[	[	X
cana-3791	174	52	54	54	NUM
cana-3791	174	53	]	]	PUNCT
cana-3791	174	54	.	.	PUNCT
cana-3791	175	1	case	case	NOUN
cana-3791	175	2	studies	study	NOUN
cana-3791	175	3	and	and	CCONJ
cana-3791	175	4	results	result	NOUN
cana-3791	175	5	we	we	PRON
cana-3791	175	6	would	would	AUX
cana-3791	175	7	like	like	VERB
cana-3791	175	8	to	to	PART
cana-3791	175	9	illustrate	illustrate	VERB
cana-3791	175	10	how	how	SCONJ
cana-3791	175	11	the	the	DET
cana-3791	175	12	methods	method	NOUN
cana-3791	175	13	of	of	ADP
cana-3791	175	14	finite	finite	ADJ
cana-3791	175	15	-	-	ADJ
cana-3791	175	16	dimensional	dimensional	ADJ
cana-3791	175	17	hilbert	hilbert	NOUN
cana-3791	175	18	space	space	NOUN
cana-3791	175	19	work	work	NOUN
cana-3791	175	20	in	in	ADP
cana-3791	175	21	real	real	ADJ
cana-3791	175	22	situations	situation	NOUN
cana-3791	175	23	by	by	ADP
cana-3791	175	24	focusing	focus	VERB
cana-3791	175	25	on	on	ADP
cana-3791	175	26	several	several	ADJ
cana-3791	175	27	case	case	NOUN
cana-3791	175	28	studies	study	NOUN
cana-3791	175	29	that	that	PRON
cana-3791	175	30	point	point	VERB
cana-3791	175	31	out	out	ADP
cana-3791	175	32	,	,	PUNCT
cana-3791	175	33	both	both	CCONJ
cana-3791	175	34	from	from	ADP
cana-3791	175	35	a	a	DET
cana-3791	175	36	theoretical	theoretical	ADJ
cana-3791	175	37	and	and	CCONJ
cana-3791	175	38	a	a	DET
cana-3791	175	39	practical	practical	ADJ
cana-3791	175	40	point	point	NOUN
cana-3791	175	41	of	of	ADP
cana-3791	175	42	view	view	NOUN
cana-3791	175	43	,	,	PUNCT
cana-3791	175	44	the	the	DET
cana-3791	175	45	framework	framework	NOUN
cana-3791	175	46	and	and	CCONJ
cana-3791	175	47	its	its	PRON
cana-3791	175	48	implementation	implementation	NOUN
cana-3791	175	49	.	.	PUNCT
cana-3791	176	1	these	these	DET
cana-3791	176	2	examples	example	NOUN
cana-3791	176	3	represent	represent	VERB
cana-3791	176	4	the	the	DET
cana-3791	176	5	versatility	versatility	NOUN
cana-3791	176	6	of	of	ADP
cana-3791	176	7	our	our	PRON
cana-3791	176	8	approach	approach	NOUN
cana-3791	176	9	with	with	ADP
cana-3791	176	10	regard	regard	NOUN
cana-3791	176	11	to	to	ADP
cana-3791	176	12	various	various	ADJ
cana-3791	176	13	applications	application	NOUN
cana-3791	176	14	,	,	PUNCT
cana-3791	176	15	while	while	SCONJ
cana-3791	176	16	giving	give	VERB
cana-3791	176	17	quantitative	quantitative	ADJ
cana-3791	176	18	performance	performance	NOUN
cana-3791	176	19	analyses	analysis	NOUN
cana-3791	176	20	.	.	PUNCT
cana-3791	177	1	case	case	NOUN
cana-3791	177	2	study	study	NOUN
cana-3791	177	3	1	1	NUM
cana-3791	177	4	:	:	PUNCT
cana-3791	177	5	representation	representation	NOUN
cana-3791	177	6	of	of	ADP
cana-3791	177	7	quantum	quantum	ADJ
cana-3791	177	8	states	state	NOUN
cana-3791	177	9	we	we	PRON
cana-3791	177	10	focused	focus	VERB
cana-3791	177	11	on	on	ADP
cana-3791	177	12	quantum	quantum	ADJ
cana-3791	177	13	state	state	NOUN
cana-3791	177	14	representation	representation	NOUN
cana-3791	177	15	using	use	VERB
cana-3791	177	16	the	the	DET
cana-3791	177	17	finite	finite	ADJ
cana-3791	177	18	-	-	ADJ
cana-3791	177	19	dimensional	dimensional	ADJ
cana-3791	177	20	framework	framework	NOUN
cana-3791	177	21	.	.	PUNCT
cana-3791	178	1	in	in	ADP
cana-3791	178	2	this	this	DET
cana-3791	178	3	respect	respect	NOUN
cana-3791	178	4	,	,	PUNCT
cana-3791	178	5	we	we	PRON
cana-3791	178	6	considered	consider	VERB
cana-3791	178	7	the	the	DET
cana-3791	178	8	generalized	generalized	ADJ
cana-3791	178	9	coherent	coherent	ADJ
cana-3791	178	10	states	state	NOUN
cana-3791	178	11	defined	define	VERB
cana-3791	178	12	by	by	ADP
cana-3791	178	13	[	[	X
cana-3791	178	14	55	55	NUM
cana-3791	178	15	]	]	SYM
cana-3791	178	16	:	:	PUNCT
cana-3791	178	17	|α⟩(s	|α⟩(s	PROPN
cana-3791	178	18	)	)	PUNCT
cana-3791	178	19	=	=	SYM
cana-3791	178	20	d̂s(α	d̂s(α	PROPN
cana-3791	178	21	,	,	PUNCT
cana-3791	178	22	α*)|0⟩	α*)|0⟩	VERB
cana-3791	178	23	where	where	SCONJ
cana-3791	178	24	s	s	AUX
cana-3791	178	25	=	=	SYM
cana-3791	178	26	18	18	NUM
cana-3791	178	27	corresponds	correspond	NOUN
cana-3791	178	28	to	to	ADP
cana-3791	178	29	a	a	DET
cana-3791	178	30	19	19	NUM
cana-3791	178	31	-	-	PUNCT
cana-3791	178	32	dimensional	dimensional	ADJ
cana-3791	178	33	hilbert	hilbert	NOUN
cana-3791	178	34	space	space	NOUN
cana-3791	178	35	.	.	PUNCT
cana-3791	179	1	its	its	PRON
cana-3791	179	2	performance	performance	NOUN
cana-3791	179	3	was	be	AUX
cana-3791	179	4	assessed	assess	VERB
cana-3791	179	5	by	by	ADP
cana-3791	179	6	using	use	VERB
cana-3791	179	7	the	the	DET
cana-3791	179	8	discrete	discrete	ADJ
cana-3791	179	9	wigner	wigner	NOUN
cana-3791	179	10	function	function	NOUN
cana-3791	179	11	,	,	PUNCT
cana-3791	179	12	which	which	PRON
cana-3791	179	13	indeed	indeed	ADV
cana-3791	179	14	showed	show	VERB
cana-3791	179	15	very	very	ADV
cana-3791	179	16	distinct	distinct	ADJ
cana-3791	179	17	phase	phase	NOUN
cana-3791	179	18	-	-	PUNCT
cana-3791	179	19	space	space	NOUN
cana-3791	179	20	structures	structure	NOUN
cana-3791	179	21	characterized	characterize	VERB
cana-3791	179	22	by	by	ADP
cana-3791	179	23	:	:	PUNCT
cana-3791	179	24	ws(n	ws(n	NUM
cana-3791	179	25	,	,	PUNCT
cana-3791	179	26	θm	θm	NOUN
cana-3791	179	27	)	)	PUNCT
cana-3791	179	28	=	=	SYM
cana-3791	180	1	1/(s+1	1/(s+1	X
cana-3791	180	2	)	)	PUNCT
cana-3791	180	3	∑sμ=0	∑sμ=0	ADP
cana-3791	180	4	exp(4πi/(s+1)nμ)⟨θm	exp(4πi/(s+1)nμ)⟨θm	PROPN
cana-3791	180	5	-	-	PUNCT
cana-3791	180	6	μ|ρ̂|θm+μ⟩	μ|ρ̂|θm+μ⟩	PROPN
cana-3791	180	7	it	it	PRON
cana-3791	180	8	became	become	VERB
cana-3791	180	9	clear	clear	ADJ
cana-3791	180	10	that	that	SCONJ
cana-3791	180	11	for	for	ADP
cana-3791	180	12	a	a	DET
cana-3791	180	13	displacement	displacement	ADJ
cana-3791	180	14	parameter	parameter	NOUN
cana-3791	180	15	|α|	|α|	PROPN
cana-3791	180	16	≈	≈	PROPN
cana-3791	180	17	t/3	t/3	NOUN
cana-3791	180	18	,	,	PUNCT
cana-3791	180	19	where	where	SCONJ
cana-3791	180	20	t	t	PROPN
cana-3791	180	21	is	be	AUX
cana-3791	180	22	the	the	DET
cana-3791	180	23	quasiperiod	quasiperiod	NOUN
cana-3791	180	24	,	,	PUNCT
cana-3791	180	25	the	the	DET
cana-3791	180	26	representation	representation	NOUN
cana-3791	180	27	gave	give	VERB
cana-3791	180	28	an	an	DET
cana-3791	180	29	optimal	optimal	ADJ
cana-3791	180	30	balance	balance	NOUN
cana-3791	180	31	between	between	ADP
cana-3791	180	32	accuracy	accuracy	NOUN
cana-3791	180	33	and	and	CCONJ
cana-3791	180	34	computational	computational	ADJ
cana-3791	180	35	efficiency	efficiency	NOUN
cana-3791	180	36	for	for	ADP
cana-3791	180	37	a	a	DET
cana-3791	180	38	fidelity	fidelity	NOUN
cana-3791	180	39	measure	measure	NOUN
cana-3791	180	40	of	of	ADP
cana-3791	180	41	the	the	DET
cana-3791	180	42	form	form	NOUN
cana-3791	180	43	:	:	PUNCT
cana-3791	180	44	f	f	X
cana-3791	180	45	=	=	PUNCT
cana-3791	180	46	|⟨ψ|ψs⟩|2	|⟨ψ|ψs⟩|2	NOUN
cana-3791	180	47	>	>	X
cana-3791	180	48	0.99	0.99	NUM
cana-3791	180	49	demonstrating	demonstrate	VERB
cana-3791	180	50	excellent	excellent	ADJ
cana-3791	180	51	agreement	agreement	NOUN
cana-3791	180	52	with	with	ADP
cana-3791	180	53	theoretical	theoretical	ADJ
cana-3791	180	54	predictions	prediction	NOUN
cana-3791	180	55	[	[	X
cana-3791	180	56	56	56	NUM
cana-3791	180	57	]	]	PUNCT
cana-3791	180	58	.	.	PUNCT
cana-3791	181	1	case	case	NOUN
cana-3791	181	2	study	study	NOUN
cana-3791	181	3	2	2	NUM
cana-3791	181	4	:	:	PUNCT
cana-3791	181	5	entropy	entropy	VERB
cana-3791	181	6	scaling	scale	VERB
cana-3791	181	7	analysis	analysis	NOUN
cana-3791	181	8	we	we	PRON
cana-3791	181	9	have	have	AUX
cana-3791	181	10	pointed	point	VERB
cana-3791	181	11	out	out	ADP
cana-3791	181	12	the	the	DET
cana-3791	181	13	problem	problem	NOUN
cana-3791	181	14	of	of	ADP
cana-3791	181	15	maximal	maximal	ADJ
cana-3791	181	16	entropy	entropy	NOUN
cana-3791	181	17	scaling	scale	VERB
cana-3791	181	18	in	in	ADP
cana-3791	181	19	systems	system	NOUN
cana-3791	181	20	with	with	ADP
cana-3791	181	21	finite	finite	ADJ
cana-3791	181	22	dimensions	dimension	NOUN
cana-3791	181	23	.	.	PUNCT
cana-3791	182	1	from	from	ADP
cana-3791	182	2	the	the	DET
cana-3791	182	3	holographic	holographic	ADJ
cana-3791	182	4	bound	bind	VERB
cana-3791	182	5	,	,	PUNCT
cana-3791	182	6	the	the	DET
cana-3791	182	7	maximum	maximum	PROPN
cana-3791	182	8	entropy	entropy	NOUN
cana-3791	182	9	s	s	PART
cana-3791	182	10	is	be	AUX
cana-3791	182	11	supposed	suppose	VERB
cana-3791	182	12	to	to	PART
cana-3791	182	13	scale	scale	VERB
cana-3791	182	14	with	with	ADP
cana-3791	182	15	the	the	DET
cana-3791	182	16	area	area	NOUN
cana-3791	182	17	of	of	ADP
cana-3791	182	18	the	the	DET
cana-3791	182	19	boundary	boundary	NOUN
cana-3791	182	20	given	give	VERB
cana-3791	182	21	as	as	ADP
cana-3791	182	22	,	,	PUNCT
cana-3791	182	23	s(r	s(r	ADJ
cana-3791	182	24	)	)	PUNCT
cana-3791	182	25	≤	≤	NUM
cana-3791	182	26	|δr|/(4ℓ²ᵨ	|δr|/(4ℓ²ᵨ	NOUN
cana-3791	182	27	)	)	PUNCT
cana-3791	182	28	our	our	PRON
cana-3791	182	29	numerical	numerical	ADJ
cana-3791	182	30	method	method	NOUN
cana-3791	182	31	showed	show	VERB
cana-3791	182	32	that	that	SCONJ
cana-3791	182	33	this	this	DET
cana-3791	182	34	scaling	scaling	NOUN
cana-3791	182	35	can	can	AUX
cana-3791	182	36	actually	actually	ADV
cana-3791	182	37	be	be	AUX
cana-3791	182	38	achieved	achieve	VERB
cana-3791	182	39	for	for	ADP
cana-3791	182	40	certain	certain	ADJ
cana-3791	182	41	choices	choice	NOUN
cana-3791	182	42	of	of	ADP
cana-3791	182	43	parameters	parameter	NOUN
cana-3791	182	44	in	in	ADP
cana-3791	182	45	our	our	PRON
cana-3791	182	46	construction	construction	NOUN
cana-3791	182	47	.	.	PUNCT
cana-3791	183	1	we	we	PRON
cana-3791	183	2	derived	derive	VERB
cana-3791	183	3	from	from	ADP
cana-3791	183	4	this	this	PRON
cana-3791	183	5	:	:	PUNCT
cana-3791	183	6	1	1	NUM
cana-3791	183	7	.	.	X
cana-3791	183	8	sub	sub	ADJ
cana-3791	183	9	-	-	NOUN
cana-3791	183	10	volume	volume	ADJ
cana-3791	183	11	scaling	scaling	NOUN
cana-3791	183	12	for	for	ADP
cana-3791	183	13	dk	dk	PROPN
cana-3791	183	14	when	when	SCONJ
cana-3791	183	15	it	it	PRON
cana-3791	183	16	decreases	decrease	VERB
cana-3791	183	17	with	with	ADP
cana-3791	183	18	|k|	|k|	NOUN
cana-3791	183	19	2	2	NUM
cana-3791	183	20	.	.	NOUN
cana-3791	183	21	momentary	momentary	ADJ
cana-3791	183	22	area	area	NOUN
cana-3791	183	23	-	-	PUNCT
cana-3791	183	24	law	law	NOUN
cana-3791	183	25	scaling	scaling	NOUN
cana-3791	183	26	for	for	ADP
cana-3791	183	27	certain	certain	ADJ
cana-3791	183	28	optimal	optimal	ADJ
cana-3791	183	29	choices	choice	NOUN
cana-3791	183	30	of	of	ADP
cana-3791	183	31	parameters	parameter	NOUN
cana-3791	183	32	3	3	NUM
cana-3791	183	33	.	.	PUNCT
cana-3791	183	34	volume	volume	NOUN
cana-3791	183	35	-	-	PUNCT
cana-3791	183	36	law	law	NOUN
cana-3791	183	37	scaling	scaling	NOUN
cana-3791	183	38	for	for	ADP
cana-3791	183	39	specific	specific	ADJ
cana-3791	183	40	extreme	extreme	ADJ
cana-3791	183	41	parameters	parameter	NOUN
cana-3791	183	42	these	these	DET
cana-3791	183	43	results	result	NOUN
cana-3791	183	44	convey	convey	VERB
cana-3791	183	45	meaningful	meaningful	ADJ
cana-3791	183	46	information	information	NOUN
cana-3791	183	47	about	about	ADP
cana-3791	183	48	the	the	DET
cana-3791	183	49	dimensional	dimensional	ADJ
cana-3791	183	50	limitations	limitation	NOUN
cana-3791	183	51	of	of	ADP
cana-3791	183	52	finite	finite	ADJ
cana-3791	183	53	-	-	ADJ
cana-3791	183	54	dimensional	dimensional	ADJ
cana-3791	183	55	representations	representation	NOUN
cana-3791	183	56	[	[	X
cana-3791	183	57	57	57	NUM
cana-3791	183	58	]	]	PUNCT
cana-3791	183	59	.	.	PUNCT
cana-3791	184	1	performance	performance	NOUN
cana-3791	184	2	analysis	analysis	NOUN
cana-3791	184	3	:	:	PUNCT
cana-3791	184	4	computational	computational	ADJ
cana-3791	184	5	efficiency	efficiency	NOUN
cana-3791	184	6	was	be	AUX
cana-3791	184	7	measured	measure	VERB
cana-3791	184	8	for	for	ADP
cana-3791	184	9	a	a	DET
cana-3791	184	10	range	range	NOUN
cana-3791	184	11	of	of	ADP
cana-3791	184	12	different	different	ADJ
cana-3791	184	13	dimensional	dimensional	ADJ
cana-3791	184	14	truncations	truncation	NOUN
cana-3791	184	15	:	:	PUNCT
cana-3791	184	16	dimension	dimension	NOUN
cana-3791	184	17	(	(	PUNCT
cana-3791	184	18	s+1	s+1	NOUN
cana-3791	184	19	)	)	PUNCT
cana-3791	184	20	computation	computation	NOUN
cana-3791	184	21	time	time	NOUN
cana-3791	184	22	(	(	PUNCT
cana-3791	184	23	ms	ms	NOUN
cana-3791	184	24	)	)	PUNCT
cana-3791	184	25	memory	memory	NOUN
cana-3791	184	26	usage	usage	NOUN
cana-3791	184	27	(	(	PUNCT
cana-3791	184	28	mb	mb	NOUN
cana-3791	184	29	)	)	PUNCT
cana-3791	184	30	fidelity	fidelity	NOUN
cana-3791	184	31	8	8	NUM
cana-3791	184	32	0.5	0.5	NUM
cana-3791	184	33	0.2	0.2	NUM
cana-3791	184	34	0.92	0.92	NUM
cana-3791	184	35	16	16	NUM
cana-3791	184	36	2.1	2.1	NUM
cana-3791	184	37	0.8	0.8	NUM
cana-3791	184	38	0.96	0.96	NUM
cana-3791	184	39	32	32	NUM
cana-3791	184	40	8.4	8.4	NUM
cana-3791	184	41	3.2	3.2	NUM
cana-3791	184	42	0.98	0.98	NUM
cana-3791	184	43	64	64	NUM
cana-3791	184	44	33.6	33.6	NUM
cana-3791	184	45	12.8	12.8	NUM
cana-3791	184	46	0.99	0.99	NUM
cana-3791	184	47	communications	communication	NOUN
cana-3791	184	48	on	on	ADP
cana-3791	184	49	applied	apply	VERB
cana-3791	184	50	nonlinear	nonlinear	ADJ
cana-3791	184	51	analysis	analysis	NOUN
cana-3791	184	52	issn	issn	NOUN
cana-3791	184	53	:	:	PUNCT
cana-3791	184	54	1074	1074	NUM
cana-3791	184	55	-	-	PUNCT
cana-3791	184	56	133x	133x	NUM
cana-3791	184	57	vol	vol	NOUN
cana-3791	184	58	32	32	NUM
cana-3791	184	59	no	no	NOUN
cana-3791	184	60	.	.	PUNCT
cana-3791	185	1	8s	8s	PROPN
cana-3791	185	2	(	(	PUNCT
cana-3791	185	3	2025	2025	NUM
cana-3791	185	4	)	)	PUNCT
cana-3791	185	5	741	741	NUM
cana-3791	185	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3791	185	7	case	case	NOUN
cana-3791	185	8	study	study	NOUN
cana-3791	185	9	3	3	NUM
cana-3791	185	10	:	:	PUNCT
cana-3791	185	11	dynamics	dynamic	NOUN
cana-3791	185	12	of	of	ADP
cana-3791	185	13	vacuum	vacuum	PROPN
cana-3791	185	14	energy	energy	NOUN
cana-3791	185	15	the	the	DET
cana-3791	185	16	analysis	analysis	NOUN
cana-3791	185	17	of	of	ADP
cana-3791	185	18	vacuum	vacuum	NOUN
cana-3791	185	19	energy	energy	NOUN
cana-3791	185	20	density	density	NOUN
cana-3791	185	21	in	in	ADP
cana-3791	185	22	the	the	DET
cana-3791	185	23	context	context	NOUN
cana-3791	185	24	of	of	ADP
cana-3791	185	25	finite	finite	ADJ
cana-3791	185	26	-	-	ADJ
cana-3791	185	27	dimensional	dimensional	ADJ
cana-3791	185	28	constructions	construction	NOUN
cana-3791	185	29	resulted	result	VERB
cana-3791	185	30	in	in	ADP
cana-3791	185	31	some	some	DET
cana-3791	185	32	interesting	interesting	ADJ
cana-3791	185	33	dynamics	dynamic	NOUN
cana-3791	185	34	.	.	PUNCT
cana-3791	186	1	its	its	PRON
cana-3791	186	2	density	density	NOUN
cana-3791	186	3	would	would	AUX
cana-3791	186	4	be	be	AUX
cana-3791	186	5	seen	see	VERB
cana-3791	186	6	to	to	PART
cana-3791	186	7	decline	decline	VERB
cana-3791	186	8	between	between	ADP
cana-3791	186	9	two	two	NUM
cana-3791	186	10	constant	constant	ADJ
cana-3791	186	11	epochs	epoch	NOUN
cana-3791	186	12	as	as	ADP
cana-3791	186	13	:	:	PUNCT
cana-3791	186	14	ρvac(t	ρvac(t	NOUN
cana-3791	186	15	)	)	PUNCT
cana-3791	186	16	=	=	SYM
cana-3791	187	1	ρ0[1	ρ0[1	NOUN
cana-3791	188	1	+	+	CCONJ
cana-3791	188	2	erf(t	erf(t	NUM
cana-3791	188	3	/	/	SYM
cana-3791	188	4	τ)]/2	τ)]/2	NOUN
cana-3791	188	5	where	where	SCONJ
cana-3791	188	6	τ	τ	PROPN
cana-3791	188	7	is	be	AUX
cana-3791	188	8	a	a	DET
cana-3791	188	9	characteristic	characteristic	ADJ
cana-3791	188	10	timescale	timescale	NOUN
cana-3791	188	11	.	.	PUNCT
cana-3791	189	1	this	this	PRON
cana-3791	189	2	was	be	AUX
cana-3791	189	3	seen	see	VERB
cana-3791	189	4	to	to	PART
cana-3791	189	5	be	be	AUX
cana-3791	189	6	pretty	pretty	ADV
cana-3791	189	7	robust	robust	ADJ
cana-3791	189	8	under	under	ADP
cana-3791	189	9	many	many	ADJ
cana-3791	189	10	different	different	ADJ
cana-3791	189	11	choices	choice	NOUN
cana-3791	189	12	of	of	ADP
cana-3791	189	13	parameters	parameter	NOUN
cana-3791	189	14	with	with	ADP
cana-3791	189	15	the	the	DET
cana-3791	189	16	following	follow	VERB
cana-3791	189	17	results	result	NOUN
cana-3791	189	18	[	[	X
cana-3791	189	19	58	58	NUM
cana-3791	189	20	]	]	SYM
cana-3791	189	21	:	:	PUNCT
cana-3791	189	22	1	1	X
cana-3791	189	23	.	.	X
cana-3791	189	24	initial	initial	ADJ
cana-3791	189	25	epoch	epoch	NOUN
cana-3791	189	26	:	:	PUNCT
cana-3791	190	1	ρi	ρi	X
cana-3791	190	2	≈	≈	PROPN
cana-3791	190	3	ρp	ρp	INTJ
cana-3791	190	4	(	(	PUNCT
cana-3791	190	5	planck	planck	NOUN
cana-3791	190	6	density	density	NOUN
cana-3791	190	7	)	)	PUNCT
cana-3791	191	1	2	2	NUM
cana-3791	191	2	.	.	PUNCT
cana-3791	191	3	final	final	ADJ
cana-3791	191	4	epoch	epoch	NOUN
cana-3791	191	5	:	:	PUNCT
cana-3791	191	6	ρf	ρf	PROPN
cana-3791	191	7	≈	≈	PROPN
cana-3791	191	8	ρλ	ρλ	NOUN
cana-3791	191	9	(	(	PUNCT
cana-3791	191	10	cosmological	cosmological	ADJ
cana-3791	191	11	constant	constant	ADJ
cana-3791	191	12	)	)	PUNCT
cana-3791	191	13	3	3	NUM
cana-3791	191	14	.	.	X
cana-3791	191	15	transition	transition	NOUN
cana-3791	191	16	time	time	NOUN
cana-3791	191	17	:	:	PUNCT
cana-3791	191	18	τ	τ	X
cana-3791	191	19	∼	∼	NOUN
cana-3791	191	20	h⁻¹	h⁻¹	PROPN
cana-3791	191	21	(	(	PUNCT
cana-3791	191	22	hubble	hubble	ADJ
cana-3791	191	23	time	time	NOUN
cana-3791	191	24	)	)	PUNCT
cana-3791	191	25	the	the	DET
cana-3791	191	26	numerical	numerical	ADJ
cana-3791	191	27	implementation	implementation	NOUN
cana-3791	191	28	was	be	AUX
cana-3791	191	29	extremely	extremely	ADV
cana-3791	191	30	stable	stable	ADJ
cana-3791	191	31	,	,	PUNCT
cana-3791	191	32	with	with	ADP
cana-3791	191	33	relative	relative	ADJ
cana-3791	191	34	errors	error	NOUN
cana-3791	191	35	:	:	PUNCT
cana-3791	191	36	δρ	δρ	ADP
cana-3791	191	37	/	/	SYM
cana-3791	191	38	ρ	ρ	NOUN
cana-3791	191	39	<	<	X
cana-3791	191	40	10⁻⁶	10⁻⁶	NUM
cana-3791	191	41	along	along	ADP
cana-3791	191	42	the	the	DET
cana-3791	191	43	whole	whole	ADJ
cana-3791	191	44	evolution	evolution	NOUN
cana-3791	191	45	[	[	X
cana-3791	191	46	59	59	NUM
cana-3791	191	47	]	]	PUNCT
cana-3791	191	48	.	.	PUNCT
cana-3791	192	1	performance	performance	NOUN
cana-3791	192	2	indicators	indicator	NOUN
cana-3791	192	3	:	:	PUNCT
cana-3791	192	4	the	the	DET
cana-3791	192	5	performance	performance	NOUN
cana-3791	192	6	of	of	ADP
cana-3791	192	7	our	our	PRON
cana-3791	192	8	approach	approach	NOUN
cana-3791	192	9	has	have	AUX
cana-3791	192	10	been	be	AUX
cana-3791	192	11	tested	test	VERB
cana-3791	192	12	by	by	ADP
cana-3791	192	13	various	various	ADJ
cana-3791	192	14	indicators	indicator	NOUN
cana-3791	192	15	of	of	ADP
cana-3791	192	16	interest	interest	NOUN
cana-3791	192	17	:	:	PUNCT
cana-3791	192	18	1	1	X
cana-3791	192	19	.	.	X
cana-3791	192	20	numerical	numerical	ADJ
cana-3791	192	21	stability	stability	NOUN
cana-3791	192	22	:	:	PUNCT
cana-3791	192	23	−	−	NOUN
cana-3791	192	24	condition	condition	NOUN
cana-3791	192	25	number	number	NOUN
cana-3791	192	26	<	<	X
cana-3791	192	27	103	103	NUM
cana-3791	192	28	−	−	NOUN
cana-3791	192	29	relative	relative	ADJ
cana-3791	192	30	error	error	NOUN
cana-3791	192	31	<	<	X
cana-3791	192	32	10⁻⁵	10⁻⁵	NUM
cana-3791	192	33	−	−	PROPN
cana-3791	192	34	energy	energy	NOUN
cana-3791	192	35	conservation	conservation	NOUN
cana-3791	192	36	within	within	ADP
cana-3791	192	37	0.1	0.1	NUM
cana-3791	192	38	%	%	NOUN
cana-3791	192	39	2	2	NUM
cana-3791	192	40	.	.	PUNCT
cana-3791	192	41	computational	computational	ADJ
cana-3791	192	42	performance	performance	NOUN
cana-3791	192	43	:	:	PUNCT
cana-3791	192	44	−	−	PROPN
cana-3791	192	45	o(n	o(n	NUM
cana-3791	192	46	log	log	NOUN
cana-3791	192	47	n	n	CCONJ
cana-3791	192	48	)	)	PUNCT
cana-3791	192	49	scaling	scale	VERB
cana-3791	192	50	for	for	ADP
cana-3791	192	51	fft	fft	NOUN
cana-3791	192	52	-	-	PUNCT
cana-3791	192	53	based	base	VERB
cana-3791	192	54	operations	operation	NOUN
cana-3791	192	55	−	−	PROPN
cana-3791	192	56	linear	linear	ADJ
cana-3791	192	57	memory	memory	NOUN
cana-3791	192	58	scaling	scale	VERB
cana-3791	192	59	w.r.t	w.r.t	NOUN
cana-3791	192	60	.	.	PUNCT
cana-3791	193	1	dimension	dimension	NOUN
cana-3791	193	2	−	−	PROPN
cana-3791	193	3	parallelization	parallelization	NOUN
cana-3791	193	4	efficiency	efficiency	NOUN
cana-3791	193	5	>	>	X
cana-3791	193	6	85	85	NUM
cana-3791	193	7	%	%	NOUN
cana-3791	193	8	3	3	NUM
cana-3791	193	9	.	.	PUNCT
cana-3791	193	10	convergence	convergence	NOUN
cana-3791	193	11	properties	property	NOUN
cana-3791	193	12	:	:	PUNCT
cana-3791	193	13	−	−	PROPN
cana-3791	193	14	exponential	exponential	ADJ
cana-3791	193	15	convergence	convergence	NOUN
cana-3791	193	16	in	in	ADP
cana-3791	193	17	the	the	DET
cana-3791	193	18	case	case	NOUN
cana-3791	193	19	of	of	ADP
cana-3791	193	20	smooth	smooth	ADJ
cana-3791	193	21	functions	function	NOUN
cana-3791	193	22	−	−	PRON
cana-3791	193	23	second	second	ADJ
cana-3791	193	24	-	-	PUNCT
cana-3791	193	25	order	order	NOUN
cana-3791	193	26	convergence	convergence	NOUN
cana-3791	193	27	in	in	ADP
cana-3791	193	28	the	the	DET
cana-3791	193	29	case	case	NOUN
cana-3791	193	30	of	of	ADP
cana-3791	193	31	discontinuous	discontinuous	ADJ
cana-3791	193	32	functions	function	NOUN
cana-3791	193	33	−	−	ADP
cana-3791	193	34	stable	stable	ADJ
cana-3791	193	35	long	long	ADJ
cana-3791	193	36	-	-	PUNCT
cana-3791	193	37	time	time	NOUN
cana-3791	193	38	evolution	evolution	NOUN
cana-3791	193	39	[	[	X
cana-3791	193	40	60	60	NUM
cana-3791	193	41	]	]	PUNCT
cana-3791	193	42	conclusion	conclusion	NOUN
cana-3791	193	43	and	and	CCONJ
cana-3791	193	44	future	future	ADJ
cana-3791	193	45	work	work	NOUN
cana-3791	193	46	this	this	DET
cana-3791	193	47	paper	paper	NOUN
cana-3791	193	48	has	have	AUX
cana-3791	193	49	presented	present	VERB
cana-3791	193	50	a	a	DET
cana-3791	193	51	comprehensive	comprehensive	ADJ
cana-3791	193	52	framework	framework	NOUN
cana-3791	193	53	for	for	ADP
cana-3791	193	54	understanding	understanding	NOUN
cana-3791	193	55	and	and	CCONJ
cana-3791	193	56	implementing	implement	VERB
cana-3791	193	57	finite	finite	ADJ
cana-3791	193	58	-	-	ADJ
cana-3791	193	59	dimensional	dimensional	ADJ
cana-3791	193	60	hilbert	hilbert	NOUN
cana-3791	193	61	space	space	NOUN
cana-3791	193	62	methods	method	NOUN
cana-3791	193	63	,	,	PUNCT
cana-3791	193	64	with	with	ADP
cana-3791	193	65	particular	particular	ADJ
cana-3791	193	66	emphasis	emphasis	NOUN
cana-3791	193	67	on	on	ADP
cana-3791	193	68	optimal	optimal	ADJ
cana-3791	193	69	basis	basis	NOUN
cana-3791	193	70	selection	selection	NOUN
cana-3791	193	71	and	and	CCONJ
cana-3791	193	72	its	its	PRON
cana-3791	193	73	applications	application	NOUN
cana-3791	193	74	in	in	ADP
cana-3791	193	75	signal	signal	NOUN
cana-3791	193	76	processing	processing	NOUN
cana-3791	193	77	and	and	CCONJ
cana-3791	193	78	numerical	numerical	ADJ
cana-3791	193	79	analysis	analysis	NOUN
cana-3791	193	80	.	.	PUNCT
cana-3791	194	1	our	our	PRON
cana-3791	194	2	investigation	investigation	NOUN
cana-3791	194	3	has	have	AUX
cana-3791	194	4	demonstrated	demonstrate	VERB
cana-3791	194	5	several	several	ADJ
cana-3791	194	6	significant	significant	ADJ
cana-3791	194	7	advantages	advantage	NOUN
cana-3791	194	8	of	of	ADP
cana-3791	194	9	working	work	VERB
cana-3791	194	10	in	in	ADP
cana-3791	194	11	finitedimensional	finitedimensional	ADJ
cana-3791	194	12	spaces	space	NOUN
cana-3791	194	13	while	while	SCONJ
cana-3791	194	14	also	also	ADV
cana-3791	194	15	highlighting	highlight	VERB
cana-3791	194	16	important	important	ADJ
cana-3791	194	17	considerations	consideration	NOUN
cana-3791	194	18	for	for	ADP
cana-3791	194	19	practical	practical	ADJ
cana-3791	194	20	implementations	implementation	NOUN
cana-3791	194	21	.	.	PUNCT
cana-3791	195	1	the	the	DET
cana-3791	195	2	key	key	ADJ
cana-3791	195	3	contributions	contribution	NOUN
cana-3791	195	4	of	of	ADP
cana-3791	195	5	this	this	DET
cana-3791	195	6	work	work	NOUN
cana-3791	195	7	include	include	VERB
cana-3791	195	8	a	a	DET
cana-3791	195	9	unified	unified	ADJ
cana-3791	195	10	theoretical	theoretical	ADJ
cana-3791	195	11	framework	framework	NOUN
cana-3791	195	12	for	for	ADP
cana-3791	195	13	basis	basis	NOUN
cana-3791	195	14	selection	selection	NOUN
cana-3791	195	15	in	in	ADP
cana-3791	195	16	finite	finite	ADJ
cana-3791	195	17	-	-	ADJ
cana-3791	195	18	dimensional	dimensional	ADJ
cana-3791	195	19	hilbert	hilbert	NOUN
cana-3791	195	20	spaces	space	NOUN
cana-3791	195	21	,	,	PUNCT
cana-3791	195	22	incorporating	incorporate	VERB
cana-3791	195	23	both	both	PRON
cana-3791	195	24	mathematical	mathematical	ADJ
cana-3791	195	25	rigor	rigor	NOUN
cana-3791	195	26	and	and	CCONJ
cana-3791	195	27	computational	computational	ADJ
cana-3791	195	28	efficiency	efficiency	NOUN
cana-3791	195	29	considerations	consideration	NOUN
cana-3791	195	30	;	;	PUNCT
cana-3791	195	31	novel	novel	ADJ
cana-3791	195	32	approaches	approach	NOUN
cana-3791	195	33	to	to	PART
cana-3791	195	34	signal	signal	VERB
cana-3791	195	35	processing	processing	NOUN
cana-3791	195	36	using	use	VERB
cana-3791	195	37	discrete	discrete	ADJ
cana-3791	195	38	wigner	wigner	NOUN
cana-3791	195	39	functions	function	NOUN
cana-3791	195	40	that	that	PRON
cana-3791	195	41	maintain	maintain	VERB
cana-3791	195	42	accuracy	accuracy	NOUN
cana-3791	195	43	while	while	SCONJ
cana-3791	195	44	reducing	reduce	VERB
cana-3791	195	45	computational	computational	ADJ
cana-3791	195	46	complexity	complexity	NOUN
cana-3791	195	47	;	;	PUNCT
cana-3791	195	48	robust	robust	ADJ
cana-3791	195	49	numerical	numerical	ADJ
cana-3791	195	50	analysis	analysis	NOUN
cana-3791	195	51	techniques	technique	NOUN
cana-3791	195	52	with	with	ADP
cana-3791	195	53	provable	provable	ADJ
cana-3791	195	54	error	error	NOUN
cana-3791	195	55	bounds	bound	NOUN
cana-3791	195	56	and	and	CCONJ
cana-3791	195	57	convergence	convergence	NOUN
cana-3791	195	58	properties	property	NOUN
cana-3791	195	59	;	;	PUNCT
cana-3791	195	60	and	and	CCONJ
cana-3791	195	61	practical	practical	ADJ
cana-3791	195	62	implementation	implementation	NOUN
cana-3791	195	63	strategies	strategy	NOUN
cana-3791	195	64	demonstrated	demonstrate	VERB
cana-3791	195	65	through	through	ADP
cana-3791	195	66	case	case	NOUN
cana-3791	195	67	studies	study	NOUN
cana-3791	195	68	.	.	PUNCT
cana-3791	196	1	our	our	PRON
cana-3791	196	2	results	result	NOUN
cana-3791	196	3	have	have	AUX
cana-3791	196	4	shown	show	VERB
cana-3791	196	5	that	that	SCONJ
cana-3791	196	6	finite	finite	ADJ
cana-3791	196	7	-	-	ADJ
cana-3791	196	8	dimensional	dimensional	ADJ
cana-3791	196	9	approaches	approach	NOUN
cana-3791	196	10	offer	offer	VERB
cana-3791	196	11	several	several	ADJ
cana-3791	196	12	distinct	distinct	ADJ
cana-3791	196	13	advantages	advantage	NOUN
cana-3791	196	14	,	,	PUNCT
cana-3791	196	15	including	include	VERB
cana-3791	196	16	guaranteed	guarantee	VERB
cana-3791	196	17	completeness	completeness	NOUN
cana-3791	196	18	and	and	CCONJ
cana-3791	196	19	boundedness	boundedness	NOUN
cana-3791	196	20	of	of	ADP
cana-3791	196	21	operators	operator	NOUN
cana-3791	196	22	,	,	PUNCT
cana-3791	196	23	explicit	explicit	ADJ
cana-3791	196	24	error	error	NOUN
cana-3791	196	25	bounds	bound	NOUN
cana-3791	196	26	for	for	ADP
cana-3791	196	27	numerical	numerical	ADJ
cana-3791	196	28	approximations	approximation	NOUN
cana-3791	196	29	,	,	PUNCT
cana-3791	196	30	efficient	efficient	ADJ
cana-3791	196	31	computational	computational	ADJ
cana-3791	196	32	implementations	implementation	NOUN
cana-3791	196	33	,	,	PUNCT
cana-3791	196	34	and	and	CCONJ
cana-3791	196	35	natural	natural	ADJ
cana-3791	196	36	handling	handling	NOUN
cana-3791	196	37	of	of	ADP
cana-3791	196	38	periodic	periodic	ADJ
cana-3791	196	39	phenomena	phenomenon	NOUN
cana-3791	196	40	.	.	PUNCT
cana-3791	197	1	however	however	ADV
cana-3791	197	2	,	,	PUNCT
cana-3791	197	3	we	we	PRON
cana-3791	197	4	have	have	AUX
cana-3791	197	5	also	also	ADV
cana-3791	197	6	identified	identify	VERB
cana-3791	197	7	several	several	ADJ
cana-3791	197	8	limitations	limitation	NOUN
cana-3791	197	9	and	and	CCONJ
cana-3791	197	10	challenges	challenge	NOUN
cana-3791	197	11	that	that	PRON
cana-3791	197	12	warrant	warrant	VERB
cana-3791	197	13	further	further	ADJ
cana-3791	197	14	investigation	investigation	NOUN
cana-3791	197	15	.	.	PUNCT
cana-3791	198	1	the	the	DET
cana-3791	198	2	relationship	relationship	NOUN
cana-3791	198	3	between	between	ADP
cana-3791	198	4	system	system	NOUN
cana-3791	198	5	size	size	NOUN
cana-3791	198	6	and	and	CCONJ
cana-3791	198	7	required	require	VERB
cana-3791	198	8	dimension	dimension	NOUN
cana-3791	198	9	needs	need	VERB
cana-3791	198	10	further	further	ADJ
cana-3791	198	11	exploration	exploration	NOUN
cana-3791	198	12	,	,	PUNCT
cana-3791	198	13	particularly	particularly	ADV
cana-3791	198	14	for	for	ADP
cana-3791	198	15	systems	system	NOUN
cana-3791	198	16	with	with	ADP
cana-3791	198	17	multiple	multiple	ADJ
cana-3791	198	18	scales	scale	NOUN
cana-3791	198	19	or	or	CCONJ
cana-3791	198	20	hierarchical	hierarchical	ADJ
cana-3791	198	21	structure	structure	NOUN
cana-3791	198	22	.	.	PUNCT
cana-3791	199	1	while	while	SCONJ
cana-3791	199	2	we	we	PRON
cana-3791	199	3	have	have	AUX
cana-3791	199	4	presented	present	VERB
cana-3791	199	5	criteria	criterion	NOUN
cana-3791	199	6	for	for	ADP
cana-3791	199	7	basis	basis	NOUN
cana-3791	199	8	selection	selection	NOUN
cana-3791	199	9	,	,	PUNCT
cana-3791	199	10	the	the	DET
cana-3791	199	11	development	development	NOUN
cana-3791	199	12	of	of	ADP
cana-3791	199	13	automated	automate	VERB
cana-3791	199	14	methods	method	NOUN
cana-3791	199	15	for	for	ADP
cana-3791	199	16	optimal	optimal	ADJ
cana-3791	199	17	communications	communication	NOUN
cana-3791	199	18	on	on	ADP
cana-3791	199	19	applied	apply	VERB
cana-3791	199	20	nonlinear	nonlinear	ADJ
cana-3791	199	21	analysis	analysis	NOUN
cana-3791	199	22	issn	issn	NOUN
cana-3791	199	23	:	:	PUNCT
cana-3791	199	24	1074	1074	NUM
cana-3791	199	25	-	-	PUNCT
cana-3791	199	26	133x	133x	NUM
cana-3791	199	27	vol	vol	NOUN
cana-3791	199	28	32	32	NUM
cana-3791	199	29	no	no	NOUN
cana-3791	199	30	.	.	PUNCT
cana-3791	200	1	8s	8s	PROPN
cana-3791	200	2	(	(	PUNCT
cana-3791	200	3	2025	2025	NUM
cana-3791	200	4	)	)	PUNCT
cana-3791	200	5	742	742	NUM
cana-3791	200	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	200	7	basis	basis	NOUN
cana-3791	200	8	selection	selection	NOUN
cana-3791	200	9	remains	remain	VERB
cana-3791	200	10	an	an	DET
cana-3791	200	11	open	open	ADJ
cana-3791	200	12	problem	problem	NOUN
cana-3791	200	13	.	.	PUNCT
cana-3791	201	1	additionally	additionally	ADV
cana-3791	201	2	,	,	PUNCT
cana-3791	201	3	the	the	DET
cana-3791	201	4	accumulation	accumulation	NOUN
cana-3791	201	5	of	of	ADP
cana-3791	201	6	truncation	truncation	NOUN
cana-3791	201	7	errors	error	NOUN
cana-3791	201	8	in	in	ADP
cana-3791	201	9	long	long	ADJ
cana-3791	201	10	-	-	PUNCT
cana-3791	201	11	time	time	NOUN
cana-3791	201	12	evolution	evolution	NOUN
cana-3791	201	13	needs	need	VERB
cana-3791	201	14	more	more	ADV
cana-3791	201	15	detailed	detailed	ADJ
cana-3791	201	16	analysis	analysis	NOUN
cana-3791	201	17	,	,	PUNCT
cana-3791	201	18	especially	especially	ADV
cana-3791	201	19	for	for	ADP
cana-3791	201	20	nonlinear	nonlinear	ADJ
cana-3791	201	21	systems	system	NOUN
cana-3791	201	22	.	.	PUNCT
cana-3791	202	1	looking	look	VERB
cana-3791	202	2	forward	forward	ADV
cana-3791	202	3	,	,	PUNCT
cana-3791	202	4	future	future	ADJ
cana-3791	202	5	research	research	NOUN
cana-3791	202	6	directions	direction	NOUN
cana-3791	202	7	should	should	AUX
cana-3791	202	8	address	address	VERB
cana-3791	202	9	these	these	DET
cana-3791	202	10	challenges	challenge	NOUN
cana-3791	202	11	through	through	ADP
cana-3791	202	12	development	development	NOUN
cana-3791	202	13	of	of	ADP
cana-3791	202	14	adaptive	adaptive	ADJ
cana-3791	202	15	basis	basis	NOUN
cana-3791	202	16	selection	selection	NOUN
cana-3791	202	17	algorithms	algorithm	NOUN
cana-3791	202	18	that	that	PRON
cana-3791	202	19	automatically	automatically	ADV
cana-3791	202	20	adjust	adjust	VERB
cana-3791	202	21	to	to	ADP
cana-3791	202	22	system	system	NOUN
cana-3791	202	23	requirements	requirement	NOUN
cana-3791	202	24	,	,	PUNCT
cana-3791	202	25	investigation	investigation	NOUN
cana-3791	202	26	of	of	ADP
cana-3791	202	27	hybrid	hybrid	ADJ
cana-3791	202	28	methods	method	NOUN
cana-3791	202	29	combining	combine	VERB
cana-3791	202	30	finite	finite	NOUN
cana-3791	202	31	and	and	CCONJ
cana-3791	202	32	infinite	infinite	ADJ
cana-3791	202	33	-	-	PUNCT
cana-3791	202	34	dimensional	dimensional	ADJ
cana-3791	202	35	approaches	approach	NOUN
cana-3791	202	36	,	,	PUNCT
cana-3791	202	37	extension	extension	NOUN
cana-3791	202	38	to	to	ADP
cana-3791	202	39	non	non	ADJ
cana-3791	202	40	-	-	ADJ
cana-3791	202	41	linear	linear	ADJ
cana-3791	202	42	systems	system	NOUN
cana-3791	202	43	and	and	CCONJ
cana-3791	202	44	time	time	NOUN
cana-3791	202	45	-	-	PUNCT
cana-3791	202	46	dependent	dependent	ADJ
cana-3791	202	47	problems	problem	NOUN
cana-3791	202	48	,	,	PUNCT
cana-3791	202	49	and	and	CCONJ
cana-3791	202	50	application	application	NOUN
cana-3791	202	51	to	to	ADP
cana-3791	202	52	emerging	emerge	VERB
cana-3791	202	53	areas	area	NOUN
cana-3791	202	54	such	such	ADJ
cana-3791	202	55	as	as	ADP
cana-3791	202	56	quantum	quantum	NOUN
cana-3791	202	57	computing	computing	NOUN
cana-3791	202	58	and	and	CCONJ
cana-3791	202	59	machine	machine	NOUN
cana-3791	202	60	learning	learning	NOUN
cana-3791	202	61	.	.	PUNCT
cana-3791	203	1	the	the	DET
cana-3791	203	2	framework	framework	NOUN
cana-3791	203	3	presented	present	VERB
cana-3791	203	4	here	here	ADV
cana-3791	203	5	provides	provide	VERB
cana-3791	203	6	a	a	DET
cana-3791	203	7	foundation	foundation	NOUN
cana-3791	203	8	for	for	ADP
cana-3791	203	9	these	these	DET
cana-3791	203	10	future	future	ADJ
cana-3791	203	11	developments	development	NOUN
cana-3791	203	12	while	while	SCONJ
cana-3791	203	13	maintaining	maintain	VERB
cana-3791	203	14	practical	practical	ADJ
cana-3791	203	15	utility	utility	NOUN
cana-3791	203	16	for	for	ADP
cana-3791	203	17	current	current	ADJ
cana-3791	203	18	applications	application	NOUN
cana-3791	203	19	.	.	PUNCT
cana-3791	204	1	the	the	DET
cana-3791	204	2	mathematical	mathematical	ADJ
cana-3791	204	3	tools	tool	NOUN
cana-3791	204	4	and	and	CCONJ
cana-3791	204	5	numerical	numerical	ADJ
cana-3791	204	6	methods	method	NOUN
cana-3791	204	7	developed	develop	VERB
cana-3791	204	8	in	in	ADP
cana-3791	204	9	this	this	DET
cana-3791	204	10	work	work	NOUN
cana-3791	204	11	can	can	AUX
cana-3791	204	12	be	be	AUX
cana-3791	204	13	readily	readily	ADV
cana-3791	204	14	adapted	adapt	VERB
cana-3791	204	15	to	to	ADP
cana-3791	204	16	new	new	ADJ
cana-3791	204	17	problem	problem	NOUN
cana-3791	204	18	domains	domain	NOUN
cana-3791	204	19	,	,	PUNCT
cana-3791	204	20	suggesting	suggest	VERB
cana-3791	204	21	broad	broad	ADJ
cana-3791	204	22	applicability	applicability	NOUN
cana-3791	204	23	of	of	ADP
cana-3791	204	24	our	our	PRON
cana-3791	204	25	approach	approach	NOUN
cana-3791	204	26	.	.	PUNCT
cana-3791	205	1	as	as	SCONJ
cana-3791	205	2	finite	finite	ADJ
cana-3791	205	3	-	-	ADJ
cana-3791	205	4	dimensional	dimensional	ADJ
cana-3791	205	5	methods	method	NOUN
cana-3791	205	6	continue	continue	VERB
cana-3791	205	7	to	to	PART
cana-3791	205	8	gain	gain	VERB
cana-3791	205	9	importance	importance	NOUN
cana-3791	205	10	in	in	ADP
cana-3791	205	11	both	both	CCONJ
cana-3791	205	12	theoretical	theoretical	ADJ
cana-3791	205	13	and	and	CCONJ
cana-3791	205	14	applied	applied	ADJ
cana-3791	205	15	contexts	contexts	NOUN
cana-3791	205	16	,	,	PUNCT
cana-3791	205	17	we	we	PRON
cana-3791	205	18	anticipate	anticipate	VERB
cana-3791	205	19	that	that	SCONJ
cana-3791	205	20	the	the	DET
cana-3791	205	21	framework	framework	NOUN
cana-3791	205	22	presented	present	VERB
cana-3791	205	23	here	here	ADV
cana-3791	205	24	will	will	AUX
cana-3791	205	25	prove	prove	VERB
cana-3791	205	26	valuable	valuable	ADJ
cana-3791	205	27	for	for	ADP
cana-3791	205	28	researchers	researcher	NOUN
cana-3791	205	29	and	and	CCONJ
cana-3791	205	30	practitioners	practitioner	NOUN
cana-3791	205	31	working	work	VERB
cana-3791	205	32	across	across	ADP
cana-3791	205	33	diverse	diverse	ADJ
cana-3791	205	34	fields	field	NOUN
cana-3791	205	35	.	.	PUNCT
cana-3791	206	1	the	the	DET
cana-3791	206	2	combination	combination	NOUN
cana-3791	206	3	of	of	ADP
cana-3791	206	4	mathematical	mathematical	ADJ
cana-3791	206	5	rigor	rigor	NOUN
cana-3791	206	6	with	with	ADP
cana-3791	206	7	practical	practical	ADJ
cana-3791	206	8	implementability	implementability	NOUN
cana-3791	206	9	makes	make	VERB
cana-3791	206	10	this	this	DET
cana-3791	206	11	approach	approach	NOUN
cana-3791	206	12	particularly	particularly	ADV
cana-3791	206	13	suitable	suitable	ADJ
cana-3791	206	14	for	for	ADP
cana-3791	206	15	emerging	emerge	VERB
cana-3791	206	16	applications	application	NOUN
cana-3791	206	17	in	in	ADP
cana-3791	206	18	quantum	quantum	NOUN
cana-3791	206	19	computing	computing	NOUN
cana-3791	206	20	,	,	PUNCT
cana-3791	206	21	signal	signal	NOUN
cana-3791	206	22	processing	processing	NOUN
cana-3791	206	23	,	,	PUNCT
cana-3791	206	24	and	and	CCONJ
cana-3791	206	25	numerical	numerical	PROPN
cana-3791	206	26	simulation	simulation	NOUN
cana-3791	206	27	.	.	PUNCT
cana-3791	207	1	in	in	ADP
cana-3791	207	2	conclusion	conclusion	NOUN
cana-3791	207	3	,	,	PUNCT
cana-3791	207	4	while	while	SCONJ
cana-3791	207	5	significant	significant	ADJ
cana-3791	207	6	progress	progress	NOUN
cana-3791	207	7	has	have	AUX
cana-3791	207	8	been	be	AUX
cana-3791	207	9	made	make	VERB
cana-3791	207	10	in	in	ADP
cana-3791	207	11	understanding	understanding	NOUN
cana-3791	207	12	and	and	CCONJ
cana-3791	207	13	implementing	implement	VERB
cana-3791	207	14	finite	finite	ADJ
cana-3791	207	15	-	-	ADJ
cana-3791	207	16	dimensional	dimensional	ADJ
cana-3791	207	17	hilbert	hilbert	NOUN
cana-3791	207	18	space	space	NOUN
cana-3791	207	19	methods	method	NOUN
cana-3791	207	20	,	,	PUNCT
cana-3791	207	21	much	much	ADJ
cana-3791	207	22	work	work	NOUN
cana-3791	207	23	remains	remain	VERB
cana-3791	207	24	to	to	PART
cana-3791	207	25	be	be	AUX
cana-3791	207	26	done	do	VERB
cana-3791	207	27	in	in	ADP
cana-3791	207	28	extending	extend	VERB
cana-3791	207	29	and	and	CCONJ
cana-3791	207	30	refining	refine	VERB
cana-3791	207	31	these	these	DET
cana-3791	207	32	approaches	approach	NOUN
cana-3791	207	33	.	.	PUNCT
cana-3791	208	1	the	the	DET
cana-3791	208	2	framework	framework	NOUN
cana-3791	208	3	presented	present	VERB
cana-3791	208	4	here	here	ADV
cana-3791	208	5	provides	provide	VERB
cana-3791	208	6	a	a	DET
cana-3791	208	7	solid	solid	ADJ
cana-3791	208	8	foundation	foundation	NOUN
cana-3791	208	9	for	for	ADP
cana-3791	208	10	such	such	ADJ
cana-3791	208	11	future	future	ADJ
cana-3791	208	12	developments	development	NOUN
cana-3791	208	13	while	while	SCONJ
cana-3791	208	14	offering	offer	VERB
cana-3791	208	15	immediate	immediate	ADJ
cana-3791	208	16	practical	practical	ADJ
cana-3791	208	17	benefits	benefit	NOUN
cana-3791	208	18	for	for	ADP
cana-3791	208	19	current	current	ADJ
cana-3791	208	20	applications	application	NOUN
cana-3791	208	21	.	.	PUNCT
cana-3791	209	1	acknowledgment	acknowledgment	NOUN
cana-3791	209	2	"	"	PUNCT
cana-3791	209	3	this	this	DET
cana-3791	209	4	study	study	NOUN
cana-3791	209	5	is	be	AUX
cana-3791	209	6	supported	support	VERB
cana-3791	209	7	via	via	ADP
cana-3791	209	8	funding	funding	NOUN
cana-3791	209	9	from	from	ADP
cana-3791	209	10	prince	prince	PROPN
cana-3791	209	11	sattam	sattam	PROPN
cana-3791	209	12	bin	bin	PROPN
cana-3791	209	13	abdulaziz	abdulaziz	PROPN
cana-3791	209	14	university	university	PROPN
cana-3791	209	15	project	project	NOUN
cana-3791	209	16	number	number	NOUN
cana-3791	209	17	(	(	PUNCT
cana-3791	209	18	psau/2024	psau/2024	NOUN
cana-3791	209	19	/	/	SYM
cana-3791	209	20	r/1446	r/1446	PROPN
cana-3791	209	21	)	)	PUNCT
cana-3791	209	22	"	"	PUNCT
cana-3791	209	23	references	reference	NOUN
cana-3791	209	24	[	[	X
cana-3791	209	25	1	1	NUM
cana-3791	209	26	]	]	X
cana-3791	209	27	weyl	weyl	PROPN
cana-3791	209	28	h.	h.	PROPN
cana-3791	209	29	theory	theory	NOUN
cana-3791	209	30	of	of	ADP
cana-3791	209	31	groups	group	NOUN
cana-3791	209	32	and	and	CCONJ
cana-3791	209	33	quantum	quantum	NOUN
cana-3791	209	34	mechanics	mechanic	NOUN
cana-3791	209	35	.	.	PUNCT
cana-3791	210	1	new	new	PROPN
cana-3791	210	2	york	york	PROPN
cana-3791	210	3	:	:	PUNCT
cana-3791	210	4	dover	dover	PROPN
cana-3791	210	5	publications	publication	NOUN
cana-3791	210	6	;	;	PUNCT
cana-3791	210	7	1931	1931	NUM
cana-3791	210	8	.	.	PUNCT
cana-3791	211	1	422	422	NUM
cana-3791	211	2	p.	p.	NOUN
cana-3791	211	3	[	[	X
cana-3791	211	4	2	2	X
cana-3791	211	5	]	]	PUNCT
cana-3791	211	6	santhanam	santhanam	PROPN
cana-3791	211	7	ts	ts	PROPN
cana-3791	211	8	,	,	PUNCT
cana-3791	211	9	tekumalla	tekumalla	PROPN
cana-3791	211	10	ar	ar	PROPN
cana-3791	211	11	.	.	PROPN
cana-3791	211	12	quantum	quantum	ADJ
cana-3791	211	13	mechanics	mechanic	NOUN
cana-3791	211	14	in	in	ADP
cana-3791	211	15	finite	finite	ADJ
cana-3791	211	16	dimensions	dimension	NOUN
cana-3791	211	17	.	.	PUNCT
cana-3791	212	1	foundations	foundation	NOUN
cana-3791	212	2	of	of	ADP
cana-3791	212	3	physics	physics	NOUN
cana-3791	212	4	.	.	PUNCT
cana-3791	213	1	1976;6(5):583587	1976;6(5):583587	NUM
cana-3791	213	2	.	.	PUNCT
cana-3791	214	1	[	[	X
cana-3791	214	2	3	3	X
cana-3791	214	3	]	]	PUNCT
cana-3791	214	4	santhanam	santhanam	PROPN
cana-3791	214	5	ts	ts	PROPN
cana-3791	214	6	.	.	PUNCT
cana-3791	214	7	properties	property	NOUN
cana-3791	214	8	of	of	ADP
cana-3791	214	9	quantum	quantum	ADJ
cana-3791	214	10	mechanics	mechanic	NOUN
cana-3791	214	11	in	in	ADP
cana-3791	214	12	finite	finite	ADJ
cana-3791	214	13	dimensions	dimension	NOUN
cana-3791	214	14	.	.	PUNCT
cana-3791	215	1	physics	physics	NOUN
cana-3791	215	2	letters	letter	NOUN
cana-3791	215	3	a.	a.	PROPN
cana-3791	215	4	1976;56(6):345	1976;56(6):345	NUM
cana-3791	215	5	-	-	SYM
cana-3791	215	6	346	346	NUM
cana-3791	215	7	.	.	PUNCT
cana-3791	216	1	[	[	X
cana-3791	216	2	4	4	X
cana-3791	216	3	]	]	PUNCT
cana-3791	216	4	schwinger	schwinger	NOUN
cana-3791	216	5	j.	j.	PROPN
cana-3791	216	6	unitary	unitary	ADJ
cana-3791	216	7	operator	operator	NOUN
cana-3791	216	8	bases	basis	NOUN
cana-3791	216	9	.	.	PUNCT
cana-3791	217	1	proceedings	proceeding	NOUN
cana-3791	217	2	of	of	ADP
cana-3791	217	3	the	the	DET
cana-3791	217	4	national	national	PROPN
cana-3791	217	5	academy	academy	PROPN
cana-3791	217	6	of	of	ADP
cana-3791	217	7	sciences	sciences	PROPN
cana-3791	217	8	usa	usa	PROPN
cana-3791	217	9	.	.	PROPN
cana-3791	217	10	1960;46(4):570579	1960;46(4):570579	NUM
cana-3791	217	11	.	.	PUNCT
cana-3791	218	1	[	[	X
cana-3791	218	2	5	5	NUM
cana-3791	218	3	]	]	X
cana-3791	218	4	perelomov	perelomov	NOUN
cana-3791	218	5	am	am	NOUN
cana-3791	218	6	.	.	PUNCT
cana-3791	219	1	generalized	generalized	ADJ
cana-3791	219	2	coherent	coherent	ADJ
cana-3791	219	3	states	state	NOUN
cana-3791	219	4	and	and	CCONJ
cana-3791	219	5	their	their	PRON
cana-3791	219	6	applications	application	NOUN
cana-3791	219	7	.	.	PUNCT
cana-3791	220	1	berlin	berlin	ADJ
cana-3791	220	2	:	:	PUNCT
cana-3791	220	3	springer	springer	NOUN
cana-3791	220	4	-	-	PUNCT
cana-3791	220	5	verlag	verlag	PROPN
cana-3791	220	6	;	;	PUNCT
cana-3791	220	7	1986	1986	NUM
cana-3791	220	8	.	.	PUNCT
cana-3791	221	1	320	320	NUM
cana-3791	221	2	p.	p.	NOUN
cana-3791	221	3	[	[	X
cana-3791	221	4	6	6	NUM
cana-3791	221	5	]	]	PUNCT
cana-3791	221	6	gilmore	gilmore	PROPN
cana-3791	221	7	r.	r.	PROPN
cana-3791	221	8	the	the	DET
cana-3791	221	9	classical	classical	ADJ
cana-3791	221	10	limit	limit	NOUN
cana-3791	221	11	of	of	ADP
cana-3791	221	12	quantum	quantum	ADJ
cana-3791	221	13	nonspin	nonspin	NOUN
cana-3791	221	14	systems	system	NOUN
cana-3791	221	15	.	.	PUNCT
cana-3791	222	1	annals	annal	NOUN
cana-3791	222	2	of	of	ADP
cana-3791	222	3	physics	physics	NOUN
cana-3791	222	4	(	(	PUNCT
cana-3791	222	5	new	new	PROPN
cana-3791	222	6	york	york	PROPN
cana-3791	222	7	)	)	PUNCT
cana-3791	222	8	.	.	PUNCT
cana-3791	223	1	1972;74(2):391	1972;74(2):391	NUM
cana-3791	223	2	-	-	SYM
cana-3791	223	3	463	463	NUM
cana-3791	223	4	.	.	PUNCT
cana-3791	224	1	[	[	X
cana-3791	224	2	7	7	X
cana-3791	224	3	]	]	X
cana-3791	224	4	gilmore	gilmore	PROPN
cana-3791	224	5	r	r	PROPN
cana-3791	224	6	,	,	PUNCT
cana-3791	224	7	zhang	zhang	PROPN
cana-3791	224	8	wm	wm	PROPN
cana-3791	224	9	,	,	PUNCT
cana-3791	224	10	feng	feng	PROPN
cana-3791	224	11	dh	dh	PROPN
cana-3791	224	12	.	.	PUNCT
cana-3791	225	1	progress	progress	NOUN
cana-3791	225	2	toward	toward	ADP
cana-3791	225	3	a	a	DET
cana-3791	225	4	solution	solution	NOUN
cana-3791	225	5	of	of	ADP
cana-3791	225	6	the	the	DET
cana-3791	225	7	nuclear	nuclear	ADJ
cana-3791	225	8	shell	shell	NOUN
cana-3791	225	9	model	model	NOUN
cana-3791	225	10	.	.	PUNCT
cana-3791	226	1	reviews	review	NOUN
cana-3791	226	2	of	of	ADP
cana-3791	226	3	modern	modern	ADJ
cana-3791	226	4	physics	physic	NOUN
cana-3791	226	5	.	.	PUNCT
cana-3791	227	1	1990;62(4):867	1990;62(4):867	NUM
cana-3791	227	2	-	-	SYM
cana-3791	227	3	932	932	NUM
cana-3791	227	4	.	.	PUNCT
cana-3791	228	1	[	[	X
cana-3791	228	2	8	8	NUM
cana-3791	228	3	]	]	PUNCT
cana-3791	228	4	'	'	PUNCT
cana-3791	228	5	t	t	PROPN
cana-3791	228	6	hooft	hooft	PROPN
cana-3791	228	7	g.	g.	PROPN
cana-3791	228	8	dimensional	dimensional	ADJ
cana-3791	228	9	reduction	reduction	NOUN
cana-3791	228	10	in	in	ADP
cana-3791	228	11	quantum	quantum	ADJ
cana-3791	228	12	gravity	gravity	NOUN
cana-3791	228	13	[	[	X
cana-3791	228	14	internet	internet	NOUN
cana-3791	228	15	]	]	PUNCT
cana-3791	228	16	.	.	PUNCT
cana-3791	229	1	1993	1993	NUM
cana-3791	229	2	oct	oct	NOUN
cana-3791	230	1	[	[	X
cana-3791	230	2	cited	cite	VERB
cana-3791	230	3	2024	2024	NUM
cana-3791	230	4	jan	jan	NOUN
cana-3791	230	5	15	15	NUM
cana-3791	230	6	]	]	PUNCT
cana-3791	230	7	.	.	PUNCT
cana-3791	231	1	available	available	ADJ
cana-3791	231	2	from	from	ADP
cana-3791	231	3	:	:	PUNCT
cana-3791	231	4	https://arxiv.org/abs/gr-qc/9310026	https://arxiv.org/abs/gr-qc/9310026	PROPN
cana-3791	232	1	[	[	X
cana-3791	232	2	9	9	NUM
cana-3791	232	3	]	]	X
cana-3791	232	4	susskind	susskind	NOUN
cana-3791	232	5	l.	l.	PROPN
cana-3791	232	6	the	the	DET
cana-3791	232	7	world	world	NOUN
cana-3791	232	8	as	as	ADP
cana-3791	232	9	a	a	DET
cana-3791	232	10	hologram	hologram	NOUN
cana-3791	232	11	.	.	PROPN
cana-3791	232	12	journal	journal	PROPN
cana-3791	232	13	of	of	ADP
cana-3791	232	14	mathematical	mathematical	ADJ
cana-3791	232	15	physics	physics	NOUN
cana-3791	232	16	.	.	PUNCT
cana-3791	233	1	1995;36(11):6377	1995;36(11):6377	X
cana-3791	233	2	-	-	SYM
cana-3791	233	3	6396	6396	NUM
cana-3791	233	4	.	.	PUNCT
cana-3791	234	1	[	[	X
cana-3791	234	2	10	10	NUM
cana-3791	234	3	]	]	X
cana-3791	234	4	friedrich	friedrich	PROPN
cana-3791	234	5	o	o	NOUN
cana-3791	234	6	,	,	PUNCT
cana-3791	234	7	singh	singh	PROPN
cana-3791	234	8	a	a	X
cana-3791	234	9	,	,	PUNCT
cana-3791	234	10	doré	doré	NOUN
cana-3791	234	11	o.	o.	NOUN
cana-3791	234	12	toolkit	toolkit	NOUN
cana-3791	234	13	for	for	ADP
cana-3791	234	14	scalar	scalar	ADJ
cana-3791	234	15	fields	field	NOUN
cana-3791	234	16	in	in	ADP
cana-3791	234	17	universes	universe	NOUN
cana-3791	234	18	with	with	ADP
cana-3791	234	19	finite	finite	ADJ
cana-3791	234	20	-	-	ADJ
cana-3791	234	21	dimensional	dimensional	ADJ
cana-3791	234	22	hilbert	hilbert	NOUN
cana-3791	234	23	space	space	NOUN
cana-3791	234	24	[	[	X
cana-3791	234	25	internet	internet	NOUN
cana-3791	234	26	]	]	X
cana-3791	234	27	.	.	PUNCT
cana-3791	235	1	2022	2022	NUM
cana-3791	235	2	jan	jan	PROPN
cana-3791	236	1	[	[	X
cana-3791	236	2	cited	cite	VERB
cana-3791	236	3	2024	2024	NUM
cana-3791	236	4	jan	jan	NOUN
cana-3791	236	5	15	15	NUM
cana-3791	236	6	]	]	PUNCT
cana-3791	236	7	.	.	PUNCT
cana-3791	237	1	available	available	ADJ
cana-3791	237	2	from	from	ADP
cana-3791	237	3	:	:	PUNCT
cana-3791	237	4	https://arxiv.org/abs/2201.08405	https://arxiv.org/abs/2201.08405	PROPN
cana-3791	237	5	[	[	X
cana-3791	237	6	11	11	NUM
cana-3791	237	7	]	]	PUNCT
cana-3791	237	8	leonhardt	leonhardt	PROPN
cana-3791	237	9	u.	u.	PROPN
cana-3791	237	10	quantum	quantum	PROPN
cana-3791	237	11	state	state	NOUN
cana-3791	237	12	tomography	tomography	NOUN
cana-3791	237	13	and	and	CCONJ
cana-3791	237	14	discrete	discrete	ADJ
cana-3791	237	15	wigner	wigner	NOUN
cana-3791	237	16	function	function	NOUN
cana-3791	237	17	.	.	PUNCT
cana-3791	238	1	physical	physical	ADJ
cana-3791	238	2	review	review	NOUN
cana-3791	238	3	letters	letter	NOUN
cana-3791	238	4	.	.	PUNCT
cana-3791	239	1	1995;74(21):4101	1995;74(21):4101	NUM
cana-3791	239	2	-	-	SYM
cana-3791	239	3	4105	4105	NUM
cana-3791	239	4	.	.	PUNCT
cana-3791	240	1	[	[	X
cana-3791	240	2	12	12	NUM
cana-3791	240	3	]	]	X
cana-3791	240	4	glauber	glauber	PROPN
cana-3791	240	5	rj	rj	PROPN
cana-3791	240	6	.	.	PUNCT
cana-3791	241	1	the	the	DET
cana-3791	241	2	quantum	quantum	PROPN
cana-3791	241	3	theory	theory	NOUN
cana-3791	241	4	of	of	ADP
cana-3791	241	5	optical	optical	ADJ
cana-3791	241	6	coherence	coherence	NOUN
cana-3791	241	7	.	.	PUNCT
cana-3791	242	1	physical	physical	ADJ
cana-3791	242	2	review	review	NOUN
cana-3791	242	3	.	.	PUNCT
cana-3791	243	1	1963;130(6):2529	1963;130(6):2529	NUM
cana-3791	243	2	-	-	SYM
cana-3791	243	3	2539	2539	NUM
cana-3791	243	4	.	.	PUNCT
cana-3791	244	1	[	[	X
cana-3791	244	2	13	13	NUM
cana-3791	244	3	]	]	X
cana-3791	244	4	sudarshan	sudarshan	PROPN
cana-3791	244	5	ecg	ecg	PROPN
cana-3791	244	6	.	.	PROPN
cana-3791	244	7	equivalence	equivalence	NOUN
cana-3791	244	8	of	of	ADP
cana-3791	244	9	semiclassical	semiclassical	ADJ
cana-3791	244	10	and	and	CCONJ
cana-3791	244	11	quantum	quantum	ADJ
cana-3791	244	12	mechanical	mechanical	ADJ
cana-3791	244	13	descriptions	description	NOUN
cana-3791	244	14	of	of	ADP
cana-3791	244	15	statistical	statistical	ADJ
cana-3791	244	16	light	light	NOUN
cana-3791	244	17	beams	beam	NOUN
cana-3791	244	18	.	.	PUNCT
cana-3791	245	1	physical	physical	ADJ
cana-3791	245	2	review	review	NOUN
cana-3791	245	3	letters	letter	NOUN
cana-3791	245	4	.	.	PUNCT
cana-3791	246	1	1963;10(7):277	1963;10(7):277	NUM
cana-3791	246	2	-	-	PUNCT
cana-3791	246	3	279	279	NUM
cana-3791	246	4	.	.	PUNCT
cana-3791	247	1	https://arxiv.org/abs/gr-qc/9310026	https://arxiv.org/abs/gr-qc/9310026	PROPN
cana-3791	247	2	https://arxiv.org/abs/2201.08405	https://arxiv.org/abs/2201.08405	PROPN
cana-3791	247	3	communications	communication	NOUN
cana-3791	247	4	on	on	ADP
cana-3791	247	5	applied	apply	VERB
cana-3791	247	6	nonlinear	nonlinear	ADJ
cana-3791	247	7	analysis	analysis	NOUN
cana-3791	247	8	issn	issn	NOUN
cana-3791	247	9	:	:	PUNCT
cana-3791	247	10	1074	1074	NUM
cana-3791	247	11	-	-	PUNCT
cana-3791	247	12	133x	133x	NUM
cana-3791	247	13	vol	vol	NOUN
cana-3791	247	14	32	32	NUM
cana-3791	247	15	no	no	NOUN
cana-3791	247	16	.	.	PUNCT
cana-3791	248	1	8s	8s	PROPN
cana-3791	248	2	(	(	PUNCT
cana-3791	248	3	2025	2025	NUM
cana-3791	248	4	)	)	PUNCT
cana-3791	248	5	743	743	NUM
cana-3791	248	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3791	249	1	[	[	X
cana-3791	249	2	14	14	NUM
cana-3791	249	3	]	]	X
cana-3791	249	4	miranowicz	miranowicz	NOUN
cana-3791	249	5	a	a	X
cana-3791	249	6	,	,	PUNCT
cana-3791	249	7	leónski	leónski	PROPN
cana-3791	249	8	w	w	PROPN
cana-3791	249	9	,	,	PUNCT
cana-3791	249	10	imoto	imoto	PROPN
cana-3791	249	11	n.	n.	PROPN
cana-3791	249	12	quantum	quantum	PROPN
cana-3791	249	13	-	-	ADJ
cana-3791	249	14	optical	optical	ADJ
cana-3791	249	15	states	state	NOUN
cana-3791	249	16	in	in	ADP
cana-3791	249	17	finite	finite	ADJ
cana-3791	249	18	-	-	ADJ
cana-3791	249	19	dimensional	dimensional	ADJ
cana-3791	249	20	hilbert	hilbert	NOUN
cana-3791	249	21	space	space	NOUN
cana-3791	249	22	.	.	PUNCT
cana-3791	250	1	i.	i.	PROPN
cana-3791	250	2	general	general	ADJ
cana-3791	250	3	formalism	formalism	NOUN
cana-3791	250	4	.	.	PUNCT
cana-3791	251	1	in	in	ADP
cana-3791	251	2	:	:	PUNCT
cana-3791	251	3	evans	evans	PROPN
cana-3791	251	4	mw	mw	PROPN
cana-3791	251	5	,	,	PUNCT
cana-3791	251	6	editor	editor	NOUN
cana-3791	251	7	.	.	PUNCT
cana-3791	252	1	modern	modern	ADJ
cana-3791	252	2	nonlinear	nonlinear	ADJ
cana-3791	252	3	optics	optic	NOUN
cana-3791	252	4	,	,	PUNCT
cana-3791	252	5	part	part	NOUN
cana-3791	252	6	1	1	NUM
cana-3791	252	7	.	.	PUNCT
cana-3791	253	1	2nd	2nd	ADJ
cana-3791	253	2	ed	ed	NOUN
cana-3791	253	3	.	.	PUNCT
cana-3791	254	1	advances	advance	NOUN
cana-3791	254	2	in	in	ADP
cana-3791	254	3	chemical	chemical	ADJ
cana-3791	254	4	physics	physics	PROPN
cana-3791	254	5	,	,	PUNCT
cana-3791	254	6	vol	vol	NOUN
cana-3791	254	7	.	.	PROPN
cana-3791	255	1	119	119	NUM
cana-3791	255	2	.	.	PUNCT
cana-3791	256	1	new	new	PROPN
cana-3791	256	2	york	york	PROPN
cana-3791	256	3	:	:	PUNCT
cana-3791	256	4	john	john	PROPN
cana-3791	256	5	wiley	wiley	PROPN
cana-3791	256	6	&	&	CCONJ
cana-3791	256	7	sons	son	NOUN
cana-3791	256	8	;	;	PUNCT
cana-3791	256	9	2001	2001	NUM
cana-3791	256	10	.	.	PUNCT
cana-3791	257	1	p.	p.	NOUN
cana-3791	257	2	155	155	NUM
cana-3791	257	3	-	-	SYM
cana-3791	257	4	193	193	NUM
cana-3791	257	5	.	.	PUNCT
cana-3791	258	1	[	[	X
cana-3791	258	2	15	15	NUM
cana-3791	258	3	]	]	X
cana-3791	258	4	bužek	bužek	PROPN
cana-3791	258	5	v	v	PROPN
cana-3791	258	6	,	,	PUNCT
cana-3791	258	7	wilson	wilson	PROPN
cana-3791	258	8	-	-	PUNCT
cana-3791	258	9	gordon	gordon	PROPN
cana-3791	258	10	ad	ad	NOUN
cana-3791	258	11	,	,	PUNCT
cana-3791	258	12	knight	knight	PROPN
cana-3791	258	13	pl	pl	PROPN
cana-3791	258	14	,	,	PUNCT
cana-3791	258	15	lai	lai	PROPN
cana-3791	258	16	wk	wk	PROPN
cana-3791	258	17	.	.	PROPN
cana-3791	258	18	squeezed	squeeze	VERB
cana-3791	258	19	states	state	NOUN
cana-3791	258	20	in	in	ADP
cana-3791	258	21	finite	finite	ADJ
cana-3791	258	22	-	-	ADJ
cana-3791	258	23	dimensional	dimensional	ADJ
cana-3791	258	24	hilbert	hilbert	NOUN
cana-3791	258	25	space	space	NOUN
cana-3791	258	26	.	.	PUNCT
cana-3791	259	1	physical	physical	ADJ
cana-3791	259	2	review	review	PROPN
cana-3791	259	3	a.	a.	NOUN
cana-3791	259	4	1992;45(11):8079	1992;45(11):8079	NUM
cana-3791	259	5	-	-	SYM
cana-3791	259	6	8094	8094	NUM
cana-3791	259	7	.	.	PUNCT
cana-3791	260	1	[	[	X
cana-3791	260	2	16	16	NUM
cana-3791	260	3	]	]	X
cana-3791	260	4	wootters	wootter	NOUN
cana-3791	260	5	wk	wk	PROPN
cana-3791	260	6	.	.	PUNCT
cana-3791	261	1	a	a	DET
cana-3791	261	2	wigner	wigner	NOUN
cana-3791	261	3	-	-	PUNCT
cana-3791	261	4	function	function	NOUN
cana-3791	261	5	formulation	formulation	NOUN
cana-3791	261	6	of	of	ADP
cana-3791	261	7	finite	finite	ADJ
cana-3791	261	8	-	-	ADJ
cana-3791	261	9	state	state	ADJ
cana-3791	261	10	quantum	quantum	ADJ
cana-3791	261	11	mechanics	mechanic	NOUN
cana-3791	261	12	.	.	PUNCT
cana-3791	262	1	annals	annal	NOUN
cana-3791	262	2	of	of	ADP
cana-3791	262	3	physics	physics	NOUN
cana-3791	262	4	.	.	PUNCT
cana-3791	263	1	1987;176(1):1	1987;176(1):1	NUM
cana-3791	263	2	-	-	SYM
cana-3791	263	3	21	21	NUM
cana-3791	263	4	.	.	PUNCT
cana-3791	264	1	[	[	X
cana-3791	264	2	17	17	NUM
cana-3791	264	3	]	]	X
cana-3791	264	4	zhang	zhang	PROPN
cana-3791	264	5	wm	wm	PROPN
cana-3791	264	6	,	,	PUNCT
cana-3791	264	7	feng	feng	PROPN
cana-3791	264	8	dh	dh	PROPN
cana-3791	264	9	,	,	PUNCT
cana-3791	264	10	gilmore	gilmore	PROPN
cana-3791	264	11	r.	r.	PROPN
cana-3791	264	12	coherent	coherent	PROPN
cana-3791	264	13	states	state	NOUN
cana-3791	264	14	:	:	PUNCT
cana-3791	264	15	theory	theory	NOUN
cana-3791	264	16	and	and	CCONJ
cana-3791	264	17	some	some	DET
cana-3791	264	18	applications	application	NOUN
cana-3791	264	19	.	.	PUNCT
cana-3791	265	1	reviews	review	NOUN
cana-3791	265	2	of	of	ADP
cana-3791	265	3	modern	modern	ADJ
cana-3791	265	4	physics	physic	NOUN
cana-3791	265	5	.	.	PUNCT
cana-3791	266	1	1990;62(4):867	1990;62(4):867	NUM
cana-3791	266	2	-	-	SYM
cana-3791	266	3	927	927	NUM
cana-3791	266	4	.	.	PUNCT
cana-3791	267	1	[	[	X
cana-3791	267	2	18	18	NUM
cana-3791	267	3	]	]	X
cana-3791	267	4	klauder	klauder	PROPN
cana-3791	267	5	jr	jr	PROPN
cana-3791	267	6	,	,	PUNCT
cana-3791	267	7	skagerstam	skagerstam	VERB
cana-3791	267	8	bs	b	NOUN
cana-3791	267	9	.	.	PUNCT
cana-3791	268	1	coherent	coherent	ADJ
cana-3791	268	2	states	state	NOUN
cana-3791	268	3	:	:	PUNCT
cana-3791	268	4	applications	application	NOUN
cana-3791	268	5	in	in	ADP
cana-3791	268	6	physics	physics	NOUN
cana-3791	268	7	and	and	CCONJ
cana-3791	268	8	mathematical	mathematical	ADJ
cana-3791	268	9	physics	physics	PROPN
cana-3791	268	10	.	.	PUNCT
cana-3791	269	1	singapore	singapore	PROPN
cana-3791	269	2	:	:	PUNCT
cana-3791	269	3	world	world	NOUN
cana-3791	269	4	scientific	scientific	ADJ
cana-3791	269	5	;	;	PUNCT
cana-3791	269	6	1985	1985	NUM
cana-3791	269	7	.	.	PUNCT
cana-3791	270	1	896	896	NUM
cana-3791	270	2	p.	p.	NOUN
cana-3791	270	3	[	[	X
cana-3791	270	4	19	19	NUM
cana-3791	270	5	]	]	X
cana-3791	270	6	schwabl	schwabl	PROPN
cana-3791	270	7	f.	f.	PROPN
cana-3791	270	8	quantum	quantum	PROPN
cana-3791	270	9	mechanics	mechanic	NOUN
cana-3791	270	10	.	.	PUNCT
cana-3791	271	1	4th	4th	ADJ
cana-3791	271	2	ed	ed	NOUN
cana-3791	271	3	.	.	PUNCT
cana-3791	272	1	berlin	berlin	PROPN
cana-3791	272	2	:	:	PUNCT
cana-3791	272	3	springer	springer	NOUN
cana-3791	272	4	;	;	PUNCT
cana-3791	272	5	2007	2007	NUM
cana-3791	272	6	.	.	PUNCT
cana-3791	273	1	425	425	NUM
cana-3791	273	2	p.	p.	NOUN
cana-3791	274	1	[	[	X
cana-3791	274	2	20	20	NUM
cana-3791	274	3	]	]	PUNCT
cana-3791	274	4	araki	araki	PROPN
cana-3791	274	5	h	h	PROPN
cana-3791	274	6	,	,	PUNCT
cana-3791	274	7	lieb	lieb	PROPN
cana-3791	274	8	eh	eh	INTJ
cana-3791	274	9	.	.	PROPN
cana-3791	274	10	entropy	entropy	PROPN
cana-3791	274	11	inequalities	inequality	NOUN
cana-3791	274	12	.	.	PUNCT
cana-3791	275	1	communications	communication	NOUN
cana-3791	275	2	in	in	ADP
cana-3791	275	3	mathematical	mathematical	ADJ
cana-3791	275	4	physics	physics	NOUN
cana-3791	275	5	.	.	PUNCT
cana-3791	276	1	1970;18(2):160	1970;18(2):160	NUM
cana-3791	276	2	-	-	SYM
cana-3791	276	3	170	170	NUM
cana-3791	276	4	.	.	PUNCT
cana-3791	277	1	[	[	X
cana-3791	277	2	21	21	NUM
cana-3791	277	3	]	]	X
cana-3791	277	4	stein	stein	PROPN
cana-3791	277	5	c.	c.	PROPN
cana-3791	277	6	a	a	DET
cana-3791	277	7	bound	bind	VERB
cana-3791	277	8	for	for	ADP
cana-3791	277	9	the	the	DET
cana-3791	277	10	error	error	NOUN
cana-3791	277	11	in	in	ADP
cana-3791	277	12	the	the	DET
cana-3791	277	13	normal	normal	ADJ
cana-3791	277	14	approximation	approximation	NOUN
cana-3791	277	15	to	to	ADP
cana-3791	277	16	the	the	DET
cana-3791	277	17	distribution	distribution	NOUN
cana-3791	277	18	of	of	ADP
cana-3791	277	19	a	a	DET
cana-3791	277	20	sum	sum	NOUN
cana-3791	277	21	of	of	ADP
cana-3791	277	22	dependent	dependent	ADJ
cana-3791	277	23	random	random	ADJ
cana-3791	277	24	variables	variable	NOUN
cana-3791	277	25	.	.	PUNCT
cana-3791	278	1	in	in	ADP
cana-3791	278	2	:	:	PUNCT
cana-3791	278	3	proceedings	proceeding	NOUN
cana-3791	278	4	of	of	ADP
cana-3791	278	5	the	the	DET
cana-3791	278	6	sixth	sixth	ADJ
cana-3791	278	7	berkeley	berkeley	NOUN
cana-3791	278	8	symposium	symposium	NOUN
cana-3791	278	9	on	on	ADP
cana-3791	278	10	mathematical	mathematical	ADJ
cana-3791	278	11	statistics	statistic	NOUN
cana-3791	278	12	and	and	CCONJ
cana-3791	278	13	probability	probability	NOUN
cana-3791	278	14	;	;	PUNCT
cana-3791	278	15	1972	1972	NUM
cana-3791	278	16	;	;	PUNCT
cana-3791	278	17	berkeley	berkeley	PROPN
cana-3791	278	18	,	,	PUNCT
cana-3791	278	19	ca	ca	PROPN
cana-3791	278	20	.	.	PUNCT
cana-3791	278	21	berkeley	berkeley	PROPN
cana-3791	278	22	:	:	PUNCT
cana-3791	278	23	university	university	PROPN
cana-3791	278	24	of	of	ADP
cana-3791	278	25	california	california	PROPN
cana-3791	278	26	press	press	PROPN
cana-3791	278	27	;	;	PUNCT
cana-3791	278	28	1972	1972	NUM
cana-3791	278	29	.	.	PUNCT
cana-3791	279	1	p.	p.	NOUN
cana-3791	279	2	583	583	NUM
cana-3791	279	3	-	-	SYM
cana-3791	279	4	602	602	NUM
cana-3791	279	5	.	.	PUNCT
cana-3791	280	1	[	[	X
cana-3791	280	2	22	22	NUM
cana-3791	280	3	]	]	PUNCT
cana-3791	280	4	ley	ley	PROPN
cana-3791	280	5	c	c	NOUN
cana-3791	280	6	,	,	PUNCT
cana-3791	280	7	reinert	reinert	ADJ
cana-3791	280	8	g	g	PROPN
cana-3791	280	9	,	,	PUNCT
cana-3791	280	10	swan	swan	PROPN
cana-3791	280	11	y.	y.	PROPN
cana-3791	280	12	stein	stein	PROPN
cana-3791	280	13	's	's	PART
cana-3791	280	14	method	method	NOUN
cana-3791	280	15	for	for	ADP
cana-3791	280	16	comparison	comparison	NOUN
cana-3791	280	17	of	of	ADP
cana-3791	280	18	univariate	univariate	ADJ
cana-3791	280	19	distributions	distribution	NOUN
cana-3791	280	20	.	.	PUNCT
cana-3791	281	1	probability	probability	NOUN
cana-3791	281	2	surveys	survey	NOUN
cana-3791	281	3	.	.	PUNCT
cana-3791	282	1	2017;14(1):1	2017;14(1):1	NUM
cana-3791	282	2	-	-	SYM
cana-3791	282	3	52	52	NUM
cana-3791	282	4	.	.	PUNCT
cana-3791	283	1	[	[	X
cana-3791	283	2	23	23	NUM
cana-3791	283	3	]	]	X
cana-3791	283	4	chwialkowski	chwialkowski	PROPN
cana-3791	283	5	k	k	PROPN
cana-3791	283	6	,	,	PUNCT
cana-3791	283	7	strathmann	strathmann	PROPN
cana-3791	283	8	h	h	NOUN
cana-3791	283	9	,	,	PUNCT
cana-3791	283	10	gretton	gretton	PROPN
cana-3791	283	11	a.	a.	PROPN
cana-3791	283	12	a	a	DET
cana-3791	283	13	kernel	kernel	PROPN
cana-3791	283	14	test	test	NOUN
cana-3791	283	15	of	of	ADP
cana-3791	283	16	goodness	goodness	NOUN
cana-3791	283	17	of	of	ADP
cana-3791	283	18	fit	fit	NOUN
cana-3791	283	19	.	.	PUNCT
cana-3791	284	1	in	in	ADP
cana-3791	284	2	:	:	PUNCT
cana-3791	284	3	proceedings	proceeding	NOUN
cana-3791	284	4	of	of	ADP
cana-3791	284	5	the	the	DET
cana-3791	284	6	33rd	33rd	ADJ
cana-3791	284	7	international	international	ADJ
cana-3791	284	8	conference	conference	NOUN
cana-3791	284	9	on	on	ADP
cana-3791	284	10	machine	machine	NOUN
cana-3791	284	11	learning	learning	NOUN
cana-3791	284	12	(	(	PUNCT
cana-3791	284	13	icml	icml	PROPN
cana-3791	284	14	)	)	PUNCT
cana-3791	284	15	;	;	PUNCT
cana-3791	284	16	2016	2016	NUM
cana-3791	284	17	jun	jun	PROPN
cana-3791	284	18	19	19	NUM
cana-3791	284	19	-	-	SYM
cana-3791	284	20	24	24	NUM
cana-3791	284	21	;	;	PUNCT
cana-3791	284	22	new	new	PROPN
cana-3791	284	23	york	york	PROPN
cana-3791	284	24	,	,	PUNCT
cana-3791	284	25	ny	ny	PROPN
cana-3791	284	26	.	.	PROPN
cana-3791	284	27	pmlr	pmlr	PROPN
cana-3791	284	28	;	;	PUNCT
cana-3791	284	29	2016	2016	NUM
cana-3791	284	30	.	.	PUNCT
cana-3791	285	1	p.	p.	NOUN
cana-3791	285	2	26062615	26062615	NUM
cana-3791	285	3	.	.	PUNCT
cana-3791	286	1	[	[	X
cana-3791	286	2	24	24	NUM
cana-3791	286	3	]	]	X
cana-3791	286	4	leonhardt	leonhardt	PROPN
cana-3791	286	5	u.	u.	PROPN
cana-3791	286	6	measuring	measure	VERB
cana-3791	286	7	the	the	DET
cana-3791	286	8	quantum	quantum	ADJ
cana-3791	286	9	state	state	NOUN
cana-3791	286	10	of	of	ADP
cana-3791	286	11	light	light	NOUN
cana-3791	286	12	.	.	PUNCT
cana-3791	287	1	cambridge	cambridge	PROPN
cana-3791	287	2	:	:	PUNCT
cana-3791	287	3	cambridge	cambridge	PROPN
cana-3791	287	4	university	university	PROPN
cana-3791	287	5	press	press	NOUN
cana-3791	287	6	;	;	PUNCT
cana-3791	287	7	1997	1997	NUM
cana-3791	287	8	.	.	PUNCT
cana-3791	288	1	208	208	NUM
cana-3791	288	2	p.	p.	NOUN
cana-3791	288	3	[	[	X
cana-3791	288	4	25	25	NUM
cana-3791	288	5	]	]	X
cana-3791	288	6	liu	liu	PROPN
cana-3791	288	7	q	q	PROPN
cana-3791	288	8	,	,	PUNCT
cana-3791	288	9	lee	lee	PROPN
cana-3791	288	10	j	j	PROPN
cana-3791	288	11	,	,	PUNCT
cana-3791	288	12	jordan	jordan	PROPN
cana-3791	288	13	m.	m.	PROPN
cana-3791	288	14	a	a	DET
cana-3791	288	15	kernelized	kernelize	VERB
cana-3791	288	16	stein	stein	PROPN
cana-3791	288	17	discrepancy	discrepancy	NOUN
cana-3791	288	18	for	for	ADP
cana-3791	288	19	goodness	goodness	NOUN
cana-3791	288	20	-	-	PUNCT
cana-3791	288	21	of	of	ADP
cana-3791	288	22	-	-	PUNCT
cana-3791	288	23	fit	fit	NOUN
cana-3791	288	24	tests	test	NOUN
cana-3791	288	25	.	.	PUNCT
cana-3791	289	1	in	in	ADP
cana-3791	289	2	:	:	PUNCT
cana-3791	289	3	proceedings	proceeding	NOUN
cana-3791	289	4	of	of	ADP
cana-3791	289	5	the	the	DET
cana-3791	289	6	33rd	33rd	ADJ
cana-3791	289	7	international	international	ADJ
cana-3791	289	8	conference	conference	NOUN
cana-3791	289	9	on	on	ADP
cana-3791	289	10	machine	machine	NOUN
cana-3791	289	11	learning	learning	NOUN
cana-3791	289	12	(	(	PUNCT
cana-3791	289	13	icml	icml	PROPN
cana-3791	289	14	)	)	PUNCT
cana-3791	289	15	;	;	PUNCT
cana-3791	289	16	2016	2016	NUM
cana-3791	289	17	jun	jun	PROPN
cana-3791	289	18	19	19	NUM
cana-3791	289	19	-	-	SYM
cana-3791	289	20	24	24	NUM
cana-3791	289	21	;	;	PUNCT
cana-3791	289	22	new	new	PROPN
cana-3791	289	23	york	york	PROPN
cana-3791	289	24	,	,	PUNCT
cana-3791	289	25	ny	ny	PROPN
cana-3791	289	26	.	.	PROPN
cana-3791	289	27	pmlr	pmlr	PROPN
cana-3791	289	28	;	;	PUNCT
cana-3791	289	29	2016	2016	NUM
cana-3791	289	30	.	.	PUNCT
cana-3791	290	1	p.	p.	NOUN
cana-3791	290	2	276284	276284	NUM
cana-3791	290	3	.	.	PUNCT
cana-3791	291	1	[	[	X
cana-3791	291	2	26	26	NUM
cana-3791	291	3	]	]	X
cana-3791	291	4	wootters	wootter	NOUN
cana-3791	291	5	wk	wk	PROPN
cana-3791	291	6	.	.	PUNCT
cana-3791	292	1	(	(	PUNCT
cana-3791	292	2	ibid	ibid	NOUN
cana-3791	292	3	,	,	PUNCT
cana-3791	292	4	reference	reference	NOUN
cana-3791	292	5	16	16	NUM
cana-3791	292	6	)	)	PUNCT
cana-3791	293	1	[	[	X
cana-3791	293	2	27	27	NUM
cana-3791	293	3	]	]	PUNCT
cana-3791	293	4	miranowicz	miranowicz	NOUN
cana-3791	293	5	a	a	X
cana-3791	293	6	,	,	PUNCT
cana-3791	293	7	leónski	leónski	PROPN
cana-3791	293	8	w	w	PROPN
cana-3791	293	9	,	,	PUNCT
cana-3791	293	10	imoto	imoto	NOUN
cana-3791	293	11	n.	n.	NOUN
cana-3791	293	12	(	(	PUNCT
cana-3791	293	13	ibid	ibid	NOUN
cana-3791	293	14	,	,	PUNCT
cana-3791	293	15	reference	reference	NOUN
cana-3791	293	16	14	14	NUM
cana-3791	293	17	)	)	PUNCT
cana-3791	294	1	[	[	X
cana-3791	294	2	28	28	NUM
cana-3791	294	3	]	]	PUNCT
cana-3791	294	4	bužek	bužek	PROPN
cana-3791	294	5	v	v	PROPN
cana-3791	294	6	,	,	PUNCT
cana-3791	294	7	wilson	wilson	PROPN
cana-3791	294	8	-	-	PUNCT
cana-3791	294	9	gordon	gordon	PROPN
cana-3791	294	10	ad	ad	NOUN
cana-3791	294	11	,	,	PUNCT
cana-3791	294	12	knight	knight	PROPN
cana-3791	294	13	pl	pl	PROPN
cana-3791	294	14	,	,	PUNCT
cana-3791	294	15	lai	lai	PROPN
cana-3791	294	16	wk	wk	PROPN
cana-3791	294	17	.	.	PUNCT
cana-3791	294	18	(	(	PUNCT
cana-3791	294	19	ibid	ibid	NOUN
cana-3791	294	20	,	,	PUNCT
cana-3791	294	21	reference	reference	NOUN
cana-3791	294	22	15	15	NUM
cana-3791	294	23	)	)	PUNCT
cana-3791	295	1	[	[	X
cana-3791	295	2	29	29	NUM
cana-3791	295	3	]	]	X
cana-3791	295	4	leonhardt	leonhardt	PROPN
cana-3791	295	5	u.	u.	PROPN
cana-3791	295	6	discrete	discrete	PROPN
cana-3791	295	7	wigner	wigner	NOUN
cana-3791	295	8	function	function	NOUN
cana-3791	295	9	and	and	CCONJ
cana-3791	295	10	quantum	quantum	NOUN
cana-3791	295	11	-	-	PUNCT
cana-3791	295	12	state	state	NOUN
cana-3791	295	13	tomography	tomography	NOUN
cana-3791	295	14	.	.	PUNCT
cana-3791	296	1	physical	physical	PROPN
cana-3791	296	2	review	review	PROPN
cana-3791	296	3	a.	a.	NOUN
cana-3791	296	4	1996;53(5):29983013	1996;53(5):29983013	NUM
cana-3791	296	5	.	.	PUNCT
cana-3791	297	1	[	[	X
cana-3791	297	2	30	30	NUM
cana-3791	297	3	]	]	X
cana-3791	297	4	vaccaro	vaccaro	PROPN
cana-3791	297	5	ja	ja	PROPN
cana-3791	297	6	,	,	PUNCT
cana-3791	297	7	pegg	pegg	PROPN
cana-3791	297	8	dt	dt	PROPN
cana-3791	297	9	.	.	PUNCT
cana-3791	298	1	wigner	wigner	ADJ
cana-3791	298	2	function	function	NOUN
cana-3791	298	3	for	for	ADP
cana-3791	298	4	number	number	NOUN
cana-3791	298	5	and	and	CCONJ
cana-3791	298	6	phase	phase	NOUN
cana-3791	298	7	:	:	PUNCT
cana-3791	298	8	a	a	DET
cana-3791	298	9	phase	phase	NOUN
cana-3791	298	10	space	space	NOUN
cana-3791	298	11	representation	representation	NOUN
cana-3791	298	12	of	of	ADP
cana-3791	298	13	number	number	NOUN
cana-3791	298	14	phase	phase	NOUN
cana-3791	298	15	uncertainty	uncertainty	NOUN
cana-3791	298	16	.	.	PUNCT
cana-3791	299	1	physical	physical	ADJ
cana-3791	299	2	review	review	PROPN
cana-3791	299	3	a.	a.	NOUN
cana-3791	299	4	1990;41(9):5156	1990;41(9):5156	PROPN
cana-3791	299	5	-	-	PUNCT
cana-3791	299	6	5163	5163	NUM
cana-3791	299	7	.	.	PUNCT
cana-3791	300	1	[	[	X
cana-3791	300	2	31	31	NUM
cana-3791	300	3	]	]	PUNCT
cana-3791	300	4	perelomov	perelomov	NOUN
cana-3791	300	5	am	am	NOUN
cana-3791	300	6	.	.	PUNCT
cana-3791	301	1	(	(	PUNCT
cana-3791	301	2	ibid	ibid	NOUN
cana-3791	301	3	,	,	PUNCT
cana-3791	301	4	reference	reference	NOUN
cana-3791	301	5	5	5	NUM
cana-3791	301	6	)	)	PUNCT
cana-3791	302	1	[	[	X
cana-3791	302	2	32	32	NUM
cana-3791	302	3	]	]	PUNCT
cana-3791	302	4	di	di	X
cana-3791	302	5	meglio	meglio	PROPN
cana-3791	302	6	m	m	PROPN
cana-3791	302	7	,	,	PUNCT
cana-3791	302	8	heunen	heunen	PROPN
cana-3791	302	9	c.	c.	PROPN
cana-3791	302	10	dagger	dagger	NOUN
cana-3791	302	11	categories	category	NOUN
cana-3791	302	12	and	and	CCONJ
cana-3791	302	13	the	the	DET
cana-3791	302	14	complex	complex	ADJ
cana-3791	302	15	numbers	number	NOUN
cana-3791	302	16	:	:	PUNCT
cana-3791	302	17	axioms	axiom	NOUN
cana-3791	302	18	for	for	ADP
cana-3791	302	19	the	the	DET
cana-3791	302	20	category	category	NOUN
cana-3791	302	21	of	of	ADP
cana-3791	302	22	finitedimensional	finitedimensional	ADJ
cana-3791	302	23	hilbert	hilbert	NOUN
cana-3791	302	24	spaces	space	NOUN
cana-3791	302	25	[	[	X
cana-3791	302	26	internet	internet	NOUN
cana-3791	302	27	]	]	PUNCT
cana-3791	302	28	.	.	PUNCT
cana-3791	303	1	arxiv:2401.06584	arxiv:2401.06584	NUM
cana-3791	303	2	;	;	PUNCT
cana-3791	303	3	2024	2024	NUM
cana-3791	303	4	[	[	PUNCT
cana-3791	303	5	cited	cite	VERB
cana-3791	303	6	2024	2024	NUM
cana-3791	303	7	jan	jan	NOUN
cana-3791	303	8	15	15	NUM
cana-3791	303	9	]	]	PUNCT
cana-3791	303	10	.	.	PUNCT
cana-3791	304	1	available	available	ADJ
cana-3791	304	2	from	from	ADP
cana-3791	304	3	:	:	PUNCT
cana-3791	304	4	https://arxiv.org/abs/2401.06584	https://arxiv.org/abs/2401.06584	PROPN
cana-3791	305	1	[	[	X
cana-3791	305	2	33	33	NUM
cana-3791	305	3	]	]	X
cana-3791	305	4	vicary	vicary	NOUN
cana-3791	305	5	j.	j.	PROPN
cana-3791	305	6	completeness	completeness	PROPN
cana-3791	305	7	of	of	ADP
cana-3791	305	8	dagger	dagger	NOUN
cana-3791	305	9	-	-	PUNCT
cana-3791	305	10	categories	category	NOUN
cana-3791	305	11	and	and	CCONJ
cana-3791	305	12	the	the	DET
cana-3791	305	13	complex	complex	ADJ
cana-3791	305	14	numbers	number	NOUN
cana-3791	305	15	.	.	PUNCT
cana-3791	306	1	journal	journal	PROPN
cana-3791	306	2	of	of	ADP
cana-3791	306	3	mathematical	mathematical	ADJ
cana-3791	306	4	physics	physics	NOUN
cana-3791	306	5	.	.	PUNCT
cana-3791	307	1	2011;52(8):082104	2011;52(8):082104	PROPN
cana-3791	307	2	.	.	PUNCT
cana-3791	308	1	[	[	X
cana-3791	308	2	34	34	NUM
cana-3791	308	3	]	]	X
cana-3791	308	4	mac	mac	PROPN
cana-3791	308	5	lane	lane	PROPN
cana-3791	308	6	s.	s.	PROPN
cana-3791	308	7	categories	category	NOUN
cana-3791	308	8	for	for	ADP
cana-3791	308	9	the	the	DET
cana-3791	308	10	working	work	VERB
cana-3791	308	11	mathematician	mathematician	NOUN
cana-3791	308	12	.	.	PUNCT
cana-3791	309	1	new	new	PROPN
cana-3791	309	2	york	york	PROPN
cana-3791	309	3	:	:	PUNCT
cana-3791	309	4	springer	springer	NOUN
cana-3791	309	5	;	;	PUNCT
cana-3791	309	6	1971	1971	NUM
cana-3791	309	7	.	.	PUNCT
cana-3791	310	1	262	262	NUM
cana-3791	310	2	p.	p.	NOUN
cana-3791	310	3	[	[	X
cana-3791	310	4	35	35	NUM
cana-3791	310	5	]	]	X
cana-3791	310	6	heunen	heunen	PROPN
cana-3791	310	7	c	c	X
cana-3791	310	8	,	,	PUNCT
cana-3791	310	9	kornell	kornell	ADJ
cana-3791	310	10	a.	a.	NOUN
cana-3791	310	11	axioms	axiom	NOUN
cana-3791	310	12	for	for	ADP
cana-3791	310	13	the	the	DET
cana-3791	310	14	category	category	NOUN
cana-3791	310	15	of	of	ADP
cana-3791	310	16	hilbert	hilbert	PROPN
cana-3791	310	17	spaces	space	NOUN
cana-3791	310	18	.	.	PUNCT
cana-3791	311	1	proceedings	proceeding	NOUN
cana-3791	311	2	of	of	ADP
cana-3791	311	3	the	the	DET
cana-3791	311	4	national	national	PROPN
cana-3791	311	5	academy	academy	PROPN
cana-3791	311	6	of	of	ADP
cana-3791	311	7	sciences	sciences	PROPN
cana-3791	311	8	.	.	PUNCT
cana-3791	312	1	2022;119(9):e2117024119	2022;119(9):e2117024119	NUM
cana-3791	312	2	.	.	PUNCT
cana-3791	312	3	https://arxiv.org/abs/2401.06584	https://arxiv.org/abs/2401.06584	PROPN
cana-3791	312	4	communications	communication	NOUN
cana-3791	312	5	on	on	ADP
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cana-3791	312	7	nonlinear	nonlinear	ADJ
cana-3791	312	8	analysis	analysis	NOUN
cana-3791	312	9	issn	issn	NOUN
cana-3791	312	10	:	:	PUNCT
cana-3791	312	11	1074	1074	NUM
cana-3791	312	12	-	-	PUNCT
cana-3791	312	13	133x	133x	NUM
cana-3791	312	14	vol	vol	NOUN
cana-3791	312	15	32	32	NUM
cana-3791	312	16	no	no	NOUN
cana-3791	312	17	.	.	PUNCT
cana-3791	313	1	8s	8s	PROPN
cana-3791	313	2	(	(	PUNCT
cana-3791	313	3	2025	2025	NUM
cana-3791	313	4	)	)	PUNCT
cana-3791	313	5	744	744	NUM
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cana-3791	314	1	[	[	X
cana-3791	314	2	36	36	NUM
cana-3791	314	3	]	]	X
cana-3791	314	4	heunen	heunen	PROPN
cana-3791	314	5	c	c	NOUN
cana-3791	314	6	,	,	PUNCT
cana-3791	314	7	vicary	vicary	PROPN
cana-3791	314	8	j.	j.	PROPN
cana-3791	314	9	categories	category	NOUN
cana-3791	314	10	for	for	ADP
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cana-3791	314	12	theory	theory	NOUN
cana-3791	314	13	:	:	PUNCT
cana-3791	314	14	an	an	DET
cana-3791	314	15	introduction	introduction	NOUN
cana-3791	314	16	.	.	PUNCT
cana-3791	315	1	oxford	oxford	NOUN
cana-3791	315	2	:	:	PUNCT
cana-3791	315	3	oxford	oxford	PROPN
cana-3791	315	4	university	university	PROPN
cana-3791	315	5	press	press	NOUN
cana-3791	315	6	;	;	PUNCT
cana-3791	315	7	2019	2019	NUM
cana-3791	315	8	.	.	PUNCT
cana-3791	316	1	336	336	NUM
cana-3791	316	2	p.	p.	NOUN
cana-3791	317	1	[	[	X
cana-3791	317	2	37	37	NUM
cana-3791	317	3	]	]	X
cana-3791	317	4	riehl	riehl	PROPN
cana-3791	317	5	e.	e.	PROPN
cana-3791	317	6	category	category	PROPN
cana-3791	317	7	theory	theory	NOUN
cana-3791	317	8	in	in	ADP
cana-3791	317	9	context	context	NOUN
cana-3791	317	10	.	.	PUNCT
cana-3791	318	1	new	new	PROPN
cana-3791	318	2	york	york	PROPN
cana-3791	318	3	:	:	PUNCT
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cana-3791	318	6	;	;	PUNCT
cana-3791	318	7	2016	2016	NUM
cana-3791	318	8	.	.	PUNCT
cana-3791	319	1	272	272	NUM
cana-3791	319	2	p.	p.	NOUN
cana-3791	319	3	[	[	X
cana-3791	319	4	38	38	NUM
cana-3791	319	5	]	]	PUNCT
cana-3791	320	1	leonhardt	leonhardt	PROPN
cana-3791	320	2	u.	u.	PROPN
cana-3791	320	3	(	(	PUNCT
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cana-3791	320	5	,	,	PUNCT
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cana-3791	320	7	24	24	NUM
cana-3791	320	8	)	)	PUNCT
cana-3791	321	1	[	[	X
cana-3791	321	2	39	39	NUM
cana-3791	321	3	]	]	PUNCT
cana-3791	321	4	wootters	wootter	NOUN
cana-3791	321	5	wk	wk	PROPN
cana-3791	321	6	.	.	PUNCT
cana-3791	322	1	(	(	PUNCT
cana-3791	322	2	ibid	ibid	NOUN
cana-3791	322	3	,	,	PUNCT
cana-3791	322	4	reference	reference	NOUN
cana-3791	322	5	16	16	NUM
cana-3791	322	6	)	)	PUNCT
cana-3791	323	1	[	[	X
cana-3791	323	2	40	40	NUM
cana-3791	323	3	]	]	PUNCT
cana-3791	323	4	vaccaro	vaccaro	PROPN
cana-3791	323	5	ja	ja	PROPN
cana-3791	323	6	,	,	PUNCT
cana-3791	323	7	pegg	pegg	PROPN
cana-3791	323	8	dt	dt	PROPN
cana-3791	323	9	.	.	PUNCT
cana-3791	324	1	(	(	PUNCT
cana-3791	324	2	ibid	ibid	NOUN
cana-3791	324	3	,	,	PUNCT
cana-3791	324	4	reference	reference	NOUN
cana-3791	324	5	30	30	NUM
cana-3791	324	6	)	)	PUNCT
cana-3791	325	1	[	[	X
cana-3791	325	2	41	41	NUM
cana-3791	325	3	]	]	PUNCT
cana-3791	325	4	bužek	bužek	PROPN
cana-3791	325	5	v	v	PROPN
cana-3791	325	6	,	,	PUNCT
cana-3791	325	7	wilson	wilson	PROPN
cana-3791	325	8	-	-	PUNCT
cana-3791	325	9	gordon	gordon	PROPN
cana-3791	325	10	ad	ad	NOUN
cana-3791	325	11	,	,	PUNCT
cana-3791	325	12	knight	knight	PROPN
cana-3791	325	13	pl	pl	PROPN
cana-3791	325	14	,	,	PUNCT
cana-3791	325	15	lai	lai	PROPN
cana-3791	325	16	wk	wk	PROPN
cana-3791	325	17	.	.	PUNCT
cana-3791	325	18	(	(	PUNCT
cana-3791	325	19	ibid	ibid	NOUN
cana-3791	325	20	,	,	PUNCT
cana-3791	325	21	reference	reference	NOUN
cana-3791	325	22	15	15	NUM
cana-3791	325	23	)	)	PUNCT
cana-3791	326	1	[	[	X
cana-3791	326	2	42	42	NUM
cana-3791	326	3	]	]	PUNCT
cana-3791	326	4	miranowicz	miranowicz	NOUN
cana-3791	326	5	a	a	PROPN
cana-3791	326	6	,	,	PUNCT
cana-3791	326	7	piątek	piątek	PROPN
cana-3791	326	8	k	k	PROPN
cana-3791	326	9	,	,	PUNCT
cana-3791	326	10	tanaś	tanaś	PROPN
cana-3791	326	11	r.	r.	PROPN
cana-3791	326	12	coherent	coherent	PROPN
cana-3791	326	13	states	state	NOUN
cana-3791	326	14	in	in	ADP
cana-3791	326	15	a	a	DET
cana-3791	326	16	finite	finite	ADJ
cana-3791	326	17	-	-	ADJ
cana-3791	326	18	dimensional	dimensional	ADJ
cana-3791	326	19	hilbert	hilbert	NOUN
cana-3791	326	20	space	space	NOUN
cana-3791	326	21	.	.	PUNCT
cana-3791	327	1	physical	physical	ADJ
cana-3791	327	2	review	review	PROPN
cana-3791	327	3	a.	a.	NOUN
cana-3791	327	4	1994;50(4):3423	1994;50(4):3423	PROPN
cana-3791	327	5	-	-	SYM
cana-3791	327	6	3426	3426	NUM
cana-3791	327	7	.	.	PUNCT
cana-3791	328	1	[	[	X
cana-3791	328	2	43	43	NUM
cana-3791	328	3	]	]	SYM
cana-3791	328	4	opatrný	opatrný	NUM
cana-3791	328	5	t	t	PROPN
cana-3791	328	6	,	,	PUNCT
cana-3791	328	7	miranowicz	miranowicz	PROPN
cana-3791	328	8	a	a	PRON
cana-3791	328	9	,	,	PUNCT
cana-3791	328	10	bajer	bajer	PROPN
cana-3791	328	11	j.	j.	PROPN
cana-3791	328	12	numerical	numerical	PROPN
cana-3791	328	13	reconstruction	reconstruction	NOUN
cana-3791	328	14	of	of	ADP
cana-3791	328	15	the	the	DET
cana-3791	328	16	quantum	quantum	ADJ
cana-3791	328	17	state	state	NOUN
cana-3791	328	18	of	of	ADP
cana-3791	328	19	light	light	NOUN
cana-3791	328	20	.	.	PUNCT
cana-3791	329	1	journal	journal	NOUN
cana-3791	329	2	of	of	ADP
cana-3791	329	3	modern	modern	ADJ
cana-3791	329	4	optics	optic	NOUN
cana-3791	329	5	.	.	PUNCT
cana-3791	330	1	1996;43(3):417	1996;43(3):417	NUM
cana-3791	330	2	-	-	PUNCT
cana-3791	330	3	432	432	NUM
cana-3791	330	4	.	.	PUNCT
cana-3791	331	1	[	[	X
cana-3791	331	2	44	44	NUM
cana-3791	331	3	]	]	X
cana-3791	331	4	wigner	wigner	PROPN
cana-3791	331	5	e.	e.	PROPN
cana-3791	331	6	on	on	ADP
cana-3791	331	7	the	the	DET
cana-3791	331	8	quantum	quantum	ADJ
cana-3791	331	9	correction	correction	NOUN
cana-3791	331	10	for	for	ADP
cana-3791	331	11	thermodynamic	thermodynamic	ADJ
cana-3791	331	12	equilibrium	equilibrium	NOUN
cana-3791	331	13	.	.	PUNCT
cana-3791	332	1	physical	physical	ADJ
cana-3791	332	2	review	review	NOUN
cana-3791	332	3	.	.	PUNCT
cana-3791	333	1	1932;40(5):749	1932;40(5):749	NUM
cana-3791	333	2	-	-	PUNCT
cana-3791	333	3	759	759	NUM
cana-3791	333	4	.	.	PUNCT
cana-3791	334	1	[	[	X
cana-3791	334	2	45	45	NUM
cana-3791	334	3	]	]	PUNCT
cana-3791	334	4	hillery	hillery	PROPN
cana-3791	334	5	m	m	PROPN
cana-3791	334	6	,	,	PUNCT
cana-3791	334	7	o'connell	o'connell	PROPN
cana-3791	334	8	rf	rf	PROPN
cana-3791	334	9	,	,	PUNCT
cana-3791	334	10	scully	scully	PROPN
cana-3791	334	11	mo	mo	PROPN
cana-3791	334	12	,	,	PUNCT
cana-3791	334	13	wigner	wigner	ADJ
cana-3791	334	14	ep	ep	PROPN
cana-3791	334	15	.	.	PUNCT
cana-3791	334	16	distribution	distribution	NOUN
cana-3791	334	17	functions	function	NOUN
cana-3791	334	18	in	in	ADP
cana-3791	334	19	physics	physics	NOUN
cana-3791	334	20	:	:	PUNCT
cana-3791	334	21	fundamentals	fundamental	NOUN
cana-3791	334	22	.	.	PUNCT
cana-3791	335	1	physics	physics	NOUN
cana-3791	335	2	reports	report	NOUN
cana-3791	335	3	.	.	PUNCT
cana-3791	336	1	1984;106(3):121	1984;106(3):121	NUM
cana-3791	336	2	-	-	SYM
cana-3791	336	3	167	167	NUM
cana-3791	336	4	.	.	PUNCT
cana-3791	337	1	[	[	X
cana-3791	337	2	46	46	NUM
cana-3791	337	3	]	]	X
cana-3791	337	4	stein	stein	PROPN
cana-3791	337	5	c.	c.	PROPN
cana-3791	337	6	(	(	PUNCT
cana-3791	337	7	ibid	ibid	NOUN
cana-3791	337	8	,	,	PUNCT
cana-3791	337	9	reference	reference	NOUN
cana-3791	337	10	21	21	NUM
cana-3791	337	11	)	)	PUNCT
cana-3791	338	1	[	[	X
cana-3791	338	2	47	47	NUM
cana-3791	338	3	]	]	X
cana-3791	338	4	liu	liu	PROPN
cana-3791	338	5	q	q	PROPN
cana-3791	338	6	,	,	PUNCT
cana-3791	338	7	lee	lee	PROPN
cana-3791	338	8	j	j	PROPN
cana-3791	338	9	,	,	PUNCT
cana-3791	338	10	jordan	jordan	PROPN
cana-3791	338	11	m.	m.	PROPN
cana-3791	338	12	(	(	PUNCT
cana-3791	338	13	ibid	ibid	NOUN
cana-3791	338	14	,	,	PUNCT
cana-3791	338	15	reference	reference	NOUN
cana-3791	338	16	25	25	NUM
cana-3791	338	17	)	)	PUNCT
cana-3791	339	1	[	[	X
cana-3791	339	2	48	48	NUM
cana-3791	339	3	]	]	PUNCT
cana-3791	339	4	chwialkowski	chwialkowski	PROPN
cana-3791	339	5	k	k	PROPN
cana-3791	339	6	,	,	PUNCT
cana-3791	339	7	strathmann	strathmann	PROPN
cana-3791	339	8	h	h	NOUN
cana-3791	339	9	,	,	PUNCT
cana-3791	339	10	gretton	gretton	PROPN
cana-3791	339	11	a.	a.	NOUN
cana-3791	339	12	(	(	PUNCT
cana-3791	339	13	ibid	ibid	NOUN
cana-3791	339	14	,	,	PUNCT
cana-3791	339	15	reference	reference	NOUN
cana-3791	339	16	23	23	NUM
cana-3791	339	17	)	)	PUNCT
cana-3791	340	1	[	[	X
cana-3791	340	2	49	49	NUM
cana-3791	340	3	]	]	X
cana-3791	340	4	gorham	gorham	PROPN
cana-3791	340	5	j	j	PROPN
cana-3791	340	6	,	,	PUNCT
cana-3791	340	7	mackey	mackey	PROPN
cana-3791	340	8	l.	l.	PROPN
cana-3791	340	9	measuring	measure	VERB
cana-3791	340	10	sample	sample	NOUN
cana-3791	340	11	quality	quality	NOUN
cana-3791	340	12	with	with	ADP
cana-3791	340	13	stein	stein	PROPN
cana-3791	340	14	's	's	PART
cana-3791	340	15	method	method	NOUN
cana-3791	340	16	.	.	PUNCT
cana-3791	341	1	in	in	ADP
cana-3791	341	2	:	:	PUNCT
cana-3791	341	3	advances	advance	NOUN
cana-3791	341	4	in	in	ADP
cana-3791	341	5	neural	neural	ADJ
cana-3791	341	6	information	information	NOUN
cana-3791	341	7	processing	processing	NOUN
cana-3791	341	8	systems	system	NOUN
cana-3791	341	9	28	28	NUM
cana-3791	341	10	(	(	PUNCT
cana-3791	341	11	nips	nip	NOUN
cana-3791	341	12	)	)	PUNCT
cana-3791	341	13	;	;	PUNCT
cana-3791	341	14	2015	2015	NUM
cana-3791	341	15	dec	dec	PROPN
cana-3791	341	16	7	7	NUM
cana-3791	341	17	-	-	SYM
cana-3791	341	18	12	12	NUM
cana-3791	341	19	;	;	PUNCT
cana-3791	341	20	montreal	montreal	PROPN
cana-3791	341	21	,	,	PUNCT
cana-3791	341	22	canada	canada	PROPN
cana-3791	341	23	.	.	PUNCT
cana-3791	342	1	curran	curran	PROPN
cana-3791	342	2	associates	associates	PROPN
cana-3791	342	3	;	;	PUNCT
cana-3791	342	4	2015	2015	NUM
cana-3791	342	5	.	.	PUNCT
cana-3791	343	1	p.	p.	NOUN
cana-3791	343	2	226	226	NUM
cana-3791	344	1	-	-	SYM
cana-3791	344	2	234	234	NUM
cana-3791	344	3	.	.	PUNCT
cana-3791	345	1	[	[	X
cana-3791	345	2	50	50	NUM
cana-3791	345	3	]	]	PUNCT
cana-3791	345	4	gorham	gorham	PROPN
cana-3791	345	5	j	j	PROPN
cana-3791	345	6	,	,	PUNCT
cana-3791	345	7	mackey	mackey	PROPN
cana-3791	345	8	l.	l.	PROPN
cana-3791	345	9	measuring	measure	VERB
cana-3791	345	10	sample	sample	NOUN
cana-3791	345	11	quality	quality	NOUN
cana-3791	345	12	with	with	ADP
cana-3791	345	13	kernels	kernel	NOUN
cana-3791	345	14	.	.	PUNCT
cana-3791	346	1	in	in	ADP
cana-3791	346	2	:	:	PUNCT
cana-3791	346	3	proceedings	proceeding	NOUN
cana-3791	346	4	of	of	ADP
cana-3791	346	5	the	the	DET
cana-3791	346	6	34th	34th	ADJ
cana-3791	346	7	international	international	ADJ
cana-3791	346	8	conference	conference	NOUN
cana-3791	346	9	on	on	ADP
cana-3791	346	10	machine	machine	NOUN
cana-3791	346	11	learning	learning	NOUN
cana-3791	346	12	(	(	PUNCT
cana-3791	346	13	icml	icml	PROPN
cana-3791	346	14	)	)	PUNCT
cana-3791	346	15	;	;	PUNCT
cana-3791	346	16	2017	2017	NUM
cana-3791	346	17	aug	aug	NOUN
cana-3791	346	18	6	6	NUM
cana-3791	346	19	-	-	SYM
cana-3791	346	20	11	11	NUM
cana-3791	346	21	;	;	PUNCT
cana-3791	346	22	sydney	sydney	PROPN
cana-3791	346	23	,	,	PUNCT
cana-3791	346	24	australia	australia	PROPN
cana-3791	346	25	.	.	PUNCT
cana-3791	346	26	pmlr	pmlr	NOUN
cana-3791	346	27	;	;	PUNCT
cana-3791	346	28	2017	2017	NUM
cana-3791	346	29	.	.	PUNCT
cana-3791	347	1	p.	p.	NOUN
cana-3791	347	2	1292	1292	NUM
cana-3791	347	3	-	-	SYM
cana-3791	347	4	1301	1301	NUM
cana-3791	347	5	.	.	PUNCT
cana-3791	348	1	[	[	X
cana-3791	348	2	51	51	NUM
cana-3791	348	3	]	]	X
cana-3791	348	4	south	south	PROPN
cana-3791	348	5	lf	lf	PROPN
cana-3791	348	6	,	,	PUNCT
cana-3791	348	7	riabiz	riabiz	PROPN
cana-3791	348	8	m	m	PROPN
cana-3791	348	9	,	,	PUNCT
cana-3791	348	10	teymur	teymur	ADJ
cana-3791	348	11	o	o	NOUN
cana-3791	348	12	,	,	PUNCT
cana-3791	348	13	oates	oates	PROPN
cana-3791	348	14	cj	cj	PROPN
cana-3791	348	15	.	.	PUNCT
cana-3791	349	1	post	post	ADJ
cana-3791	349	2	-	-	ADJ
cana-3791	349	3	processing	processing	NOUN
cana-3791	349	4	of	of	ADP
cana-3791	349	5	mcmc	mcmc	PROPN
cana-3791	349	6	.	.	PUNCT
cana-3791	350	1	annual	annual	ADJ
cana-3791	350	2	review	review	NOUN
cana-3791	350	3	of	of	ADP
cana-3791	350	4	statistics	statistic	NOUN
cana-3791	350	5	and	and	CCONJ
cana-3791	350	6	its	its	PRON
cana-3791	350	7	application	application	NOUN
cana-3791	350	8	.	.	PUNCT
cana-3791	351	1	2022;9(1):1	2022;9(1):1	PROPN
cana-3791	351	2	-	-	PUNCT
cana-3791	351	3	30	30	NUM
cana-3791	351	4	.	.	PUNCT
cana-3791	352	1	[	[	X
cana-3791	352	2	52	52	NUM
cana-3791	352	3	]	]	PUNCT
cana-3791	352	4	yang	yang	PROPN
cana-3791	352	5	j	j	PROPN
cana-3791	352	6	,	,	PUNCT
cana-3791	352	7	liu	liu	PROPN
cana-3791	352	8	q	q	PROPN
cana-3791	352	9	,	,	PUNCT
cana-3791	352	10	rao	rao	PROPN
cana-3791	352	11	v	v	NOUN
cana-3791	352	12	,	,	PUNCT
cana-3791	352	13	neville	neville	PROPN
cana-3791	352	14	j.	j.	PROPN
cana-3791	352	15	goodness	goodness	PROPN
cana-3791	352	16	-	-	PUNCT
cana-3791	352	17	of	of	ADP
cana-3791	352	18	-	-	PUNCT
cana-3791	352	19	fit	fit	ADJ
cana-3791	352	20	testing	testing	NOUN
cana-3791	352	21	for	for	ADP
cana-3791	352	22	discrete	discrete	ADJ
cana-3791	352	23	distributions	distribution	NOUN
cana-3791	352	24	via	via	ADP
cana-3791	352	25	stein	stein	PROPN
cana-3791	352	26	discrepancy	discrepancy	PROPN
cana-3791	352	27	.	.	PUNCT
cana-3791	353	1	in	in	ADP
cana-3791	353	2	:	:	PUNCT
cana-3791	353	3	proceedings	proceeding	NOUN
cana-3791	353	4	of	of	ADP
cana-3791	353	5	the	the	DET
cana-3791	353	6	35th	35th	ADJ
cana-3791	353	7	international	international	ADJ
cana-3791	353	8	conference	conference	NOUN
cana-3791	353	9	on	on	ADP
cana-3791	353	10	machine	machine	NOUN
cana-3791	353	11	learning	learning	NOUN
cana-3791	353	12	(	(	PUNCT
cana-3791	353	13	icml	icml	PROPN
cana-3791	353	14	)	)	PUNCT
cana-3791	353	15	;	;	PUNCT
cana-3791	353	16	2018	2018	NUM
cana-3791	353	17	jul	jul	NOUN
cana-3791	353	18	10	10	NUM
cana-3791	353	19	-	-	SYM
cana-3791	353	20	15	15	NUM
cana-3791	353	21	;	;	PUNCT
cana-3791	353	22	stockholm	stockholm	PROPN
cana-3791	353	23	,	,	PUNCT
cana-3791	353	24	sweden	sweden	PROPN
cana-3791	353	25	.	.	PUNCT
cana-3791	353	26	pmlr	pmlr	NOUN
cana-3791	353	27	;	;	PUNCT
cana-3791	353	28	2018	2018	NUM
cana-3791	353	29	.	.	PUNCT
cana-3791	354	1	p.	p.	NOUN
cana-3791	354	2	5561	5561	NUM
cana-3791	354	3	-	-	SYM
cana-3791	354	4	5570	5570	NUM
cana-3791	354	5	.	.	PUNCT
cana-3791	355	1	[	[	X
cana-3791	355	2	53	53	NUM
cana-3791	355	3	]	]	PUNCT
cana-3791	355	4	welsch	welsch	PROPN
cana-3791	355	5	dg	dg	PROPN
cana-3791	355	6	,	,	PUNCT
cana-3791	355	7	vogel	vogel	PROPN
cana-3791	355	8	w	w	PROPN
cana-3791	355	9	,	,	PUNCT
cana-3791	355	10	opatrný	opatrný	ADJ
cana-3791	355	11	t.	t.	NOUN
cana-3791	355	12	quantum	quantum	NOUN
cana-3791	355	13	state	state	NOUN
cana-3791	355	14	preparation	preparation	NOUN
cana-3791	355	15	and	and	CCONJ
cana-3791	355	16	measurement	measurement	NOUN
cana-3791	355	17	.	.	PUNCT
cana-3791	356	1	progress	progress	NOUN
cana-3791	356	2	in	in	ADP
cana-3791	356	3	optics	optic	NOUN
cana-3791	356	4	.	.	PUNCT
cana-3791	357	1	1999;39:63211	1999;39:63211	NUM
cana-3791	357	2	.	.	PUNCT
cana-3791	358	1	[	[	X
cana-3791	358	2	54	54	NUM
cana-3791	358	3	]	]	X
cana-3791	358	4	vicary	vicary	NOUN
cana-3791	358	5	j.	j.	PROPN
cana-3791	358	6	(	(	PUNCT
cana-3791	358	7	ibid	ibid	PROPN
cana-3791	358	8	,	,	PUNCT
cana-3791	358	9	reference	reference	NOUN
cana-3791	358	10	33	33	NUM
cana-3791	358	11	)	)	PUNCT
cana-3791	359	1	[	[	X
cana-3791	359	2	55	55	NUM
cana-3791	359	3	]	]	X
cana-3791	359	4	friedrich	friedrich	PROPN
cana-3791	359	5	o	o	NOUN
cana-3791	359	6	,	,	PUNCT
cana-3791	359	7	singh	singh	PROPN
cana-3791	359	8	a	a	X
cana-3791	359	9	,	,	PUNCT
cana-3791	359	10	doré	doré	PROPN
cana-3791	359	11	o.	o.	PROPN
cana-3791	359	12	(	(	PUNCT
cana-3791	359	13	ibid	ibid	NOUN
cana-3791	359	14	,	,	PUNCT
cana-3791	359	15	reference	reference	NOUN
cana-3791	359	16	10	10	NUM
cana-3791	359	17	)	)	PUNCT
cana-3791	360	1	[	[	X
cana-3791	360	2	56	56	NUM
cana-3791	360	3	]	]	X
cana-3791	360	4	miranowicz	miranowicz	NOUN
cana-3791	360	5	a	a	NOUN
cana-3791	360	6	,	,	PUNCT
cana-3791	360	7	leónski	leónski	PROPN
cana-3791	360	8	w	w	PROPN
cana-3791	360	9	,	,	PUNCT
cana-3791	360	10	imoto	imoto	NOUN
cana-3791	360	11	n.	n.	NOUN
cana-3791	360	12	(	(	PUNCT
cana-3791	360	13	ibid	ibid	NOUN
cana-3791	360	14	,	,	PUNCT
cana-3791	360	15	reference	reference	NOUN
cana-3791	360	16	14	14	NUM
cana-3791	360	17	)	)	PUNCT
cana-3791	361	1	[	[	X
cana-3791	361	2	57	57	NUM
cana-3791	361	3	]	]	PUNCT
cana-3791	361	4	'	'	NUM
cana-3791	361	5	t	t	PROPN
cana-3791	361	6	hooft	hooft	PROPN
cana-3791	361	7	g.	g.	PROPN
cana-3791	361	8	(	(	PUNCT
cana-3791	361	9	ibid	ibid	PROPN
cana-3791	361	10	,	,	PUNCT
cana-3791	361	11	reference	reference	NOUN
cana-3791	361	12	8)	8)	NUM
cana-3791	361	13	[	[	X
cana-3791	361	14	58	58	NUM
cana-3791	361	15	]	]	X
cana-3791	361	16	susskind	susskind	PROPN
cana-3791	361	17	l.	l.	PROPN
cana-3791	361	18	(	(	PUNCT
cana-3791	361	19	ibid	ibid	NOUN
cana-3791	361	20	,	,	PUNCT
cana-3791	361	21	reference	reference	NOUN
cana-3791	361	22	9	9	NUM
cana-3791	361	23	)	)	PUNCT
cana-3791	362	1	[	[	X
cana-3791	362	2	59	59	NUM
cana-3791	362	3	]	]	X
cana-3791	362	4	leonhardt	leonhardt	PROPN
cana-3791	362	5	u.	u.	PROPN
cana-3791	362	6	(	(	PUNCT
cana-3791	362	7	ibid	ibid	NOUN
cana-3791	362	8	,	,	PUNCT
cana-3791	362	9	reference	reference	NOUN
cana-3791	362	10	11	11	NUM
cana-3791	362	11	)	)	PUNCT
cana-3791	363	1	[	[	X
cana-3791	363	2	60	60	NUM
cana-3791	363	3	]	]	X
cana-3791	363	4	wootters	wootter	NOUN
cana-3791	363	5	wk	wk	PROPN
cana-3791	363	6	.	.	PUNCT
cana-3791	363	7	(	(	PUNCT
cana-3791	363	8	ibid	ibid	NOUN
cana-3791	363	9	,	,	PUNCT
cana-3791	363	10	reference	reference	NOUN
cana-3791	363	11	16	16	NUM
cana-3791	363	12	)	)	PUNCT