id	sid	tid	token	lemma	pos
cana-1956	1	1	communications	communication	NOUN
cana-1956	1	2	on	on	ADP
cana-1956	1	3	applied	apply	VERB
cana-1956	1	4	nonlinear	nonlinear	ADJ
cana-1956	1	5	analysis	analysis	NOUN
cana-1956	1	6	issn	issn	NOUN
cana-1956	1	7	:	:	PUNCT
cana-1956	1	8	1074	1074	NUM
cana-1956	1	9	-	-	PUNCT
cana-1956	1	10	133x	133x	NUM
cana-1956	1	11	vol	vol	NOUN
cana-1956	1	12	32	32	NUM
cana-1956	1	13	no	no	NOUN
cana-1956	1	14	.	.	NOUN
cana-1956	1	15	3	3	NUM
cana-1956	1	16	(	(	PUNCT
cana-1956	1	17	2025	2025	NUM
cana-1956	1	18	)	)	PUNCT
cana-1956	2	1	229	229	NUM
cana-1956	2	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	2	3	logic	logic	NOUN
cana-1956	2	4	density	density	NOUN
cana-1956	2	5	in	in	ADP
cana-1956	2	6	gabor	gabor	PROPN
cana-1956	2	7	frame	frame	PROPN
cana-1956	2	8	structures	structure	NOUN
cana-1956	2	9	manal	manal	ADJ
cana-1956	2	10	yagoub	yagoub	PROPN
cana-1956	2	11	ahmed	ahmed	PROPN
cana-1956	2	12	juma	juma	PROPN
cana-1956	2	13	m.juma@qu.edu.sa	m.juma@qu.edu.sa	PROPN
cana-1956	2	14	department	department	PROPN
cana-1956	2	15	of	of	ADP
cana-1956	2	16	mathematic	mathematic	PROPN
cana-1956	2	17	,	,	PUNCT
cana-1956	2	18	college	college	NOUN
cana-1956	2	19	of	of	ADP
cana-1956	2	20	science	science	NOUN
cana-1956	2	21	,	,	PUNCT
cana-1956	2	22	qassim	qassim	PROPN
cana-1956	2	23	university	university	PROPN
cana-1956	2	24	,	,	PUNCT
cana-1956	2	25	buraidah	buraidah	PROPN
cana-1956	2	26	,	,	PUNCT
cana-1956	2	27	saudi	saudi	PROPN
cana-1956	2	28	arabia	arabia	PROPN
cana-1956	2	29	article	article	NOUN
cana-1956	2	30	history	history	NOUN
cana-1956	2	31	:	:	PUNCT
cana-1956	2	32	received	receive	VERB
cana-1956	2	33	:	:	PUNCT
cana-1956	2	34	28	28	NUM
cana-1956	2	35	-	-	SYM
cana-1956	2	36	07	07	NUM
cana-1956	2	37	-	-	PUNCT
cana-1956	2	38	2024	2024	NUM
cana-1956	2	39	revised	revise	VERB
cana-1956	2	40	:	:	PUNCT
cana-1956	2	41	17	17	NUM
cana-1956	2	42	-	-	SYM
cana-1956	2	43	09	09	NUM
cana-1956	2	44	-	-	PUNCT
cana-1956	2	45	2024	2024	NUM
cana-1956	2	46	accepted	accept	VERB
cana-1956	2	47	:	:	PUNCT
cana-1956	2	48	01	01	NUM
cana-1956	2	49	-	-	SYM
cana-1956	2	50	10	10	NUM
cana-1956	2	51	-	-	PUNCT
cana-1956	2	52	2024	2024	NUM
cana-1956	2	53	abstract	abstract	NOUN
cana-1956	2	54	:	:	PUNCT
cana-1956	2	55	the	the	DET
cana-1956	2	56	gabor	gabor	PROPN
cana-1956	2	57	system	system	NOUN
cana-1956	2	58	g(g,1+ε	g(g,1+ε	PROPN
cana-1956	2	59	,	,	PUNCT
cana-1956	2	60	ε-1)={e^2iπ(ε-1)nt	ε-1)={e^2iπ(ε-1)nt	PROPN
cana-1956	2	61	g(t-(1+ε)m):m	g(t-(1+ε)m):m	PROPN
cana-1956	2	62	,	,	PUNCT
cana-1956	2	63	n∈z}is	n∈z}i	NOUN
cana-1956	2	64	examined	examine	VERB
cana-1956	2	65	with	with	ADP
cana-1956	2	66	respect	respect	NOUN
cana-1956	2	67	to	to	ADP
cana-1956	2	68	its	its	PRON
cana-1956	2	69	frame	frame	NOUN
cana-1956	2	70	property	property	NOUN
cana-1956	2	71	,	,	PUNCT
cana-1956	2	72	uunder	uunder	NOUN
cana-1956	2	73	reasonable	reasonable	ADJ
cana-1956	2	74	oversampling	oversampling	ADJ
cana-1956	2	75	conditions	condition	NOUN
cana-1956	2	76	,	,	PUNCT
cana-1956	2	77	that	that	ADV
cana-1956	2	78	is	is	ADV
cana-1956	2	79	,	,	PUNCT
cana-1956	2	80	when	when	SCONJ
cana-1956	2	81	(	(	PUNCT
cana-1956	2	82	ε+1,ε-1∈q	ε+1,ε-1∈q	NOUN
cana-1956	2	83	)	)	PUNCT
cana-1956	2	84	.	.	PUNCT
cana-1956	3	1	an	an	DET
cana-1956	3	2	appropriate	appropriate	ADJ
cana-1956	3	3	"	"	PUNCT
cana-1956	3	4	rational	rational	ADJ
cana-1956	3	5	"	"	PUNCT
cana-1956	3	6	counterpart	counterpart	NOUN
cana-1956	3	7	of	of	ADP
cana-1956	3	8	the	the	DET
cana-1956	3	9	ron	ron	PROPN
cana-1956	3	10	-	-	PUNCT
cana-1956	3	11	shen	shen	PROPN
cana-1956	3	12	gramian	gramian	PROPN
cana-1956	3	13	is	be	AUX
cana-1956	3	14	developed	develop	VERB
cana-1956	3	15	,	,	PUNCT
cana-1956	3	16	establishing	establish	VERB
cana-1956	3	17	that	that	PRON
cana-1956	3	18	for	for	ADP
cana-1956	3	19	every	every	DET
cana-1956	3	20	peculiar	peculiar	ADJ
cana-1956	3	21	pane	pane	NOUN
cana-1956	3	22	function	function	NOUN
cana-1956	3	23	(	(	PUNCT
cana-1956	3	24	g	g	NOUN
cana-1956	3	25	)	)	PUNCT
cana-1956	3	26	,	,	PUNCT
cana-1956	3	27	the	the	DET
cana-1956	3	28	structure	structure	NOUN
cana-1956	3	29	:	:	PUNCT
cana-1956	3	30	g(g	g(g	PROPN
cana-1956	3	31	,	,	PUNCT
cana-1956	3	32	ε+1,ε-1	ε+1,ε-1	NUM
cana-1956	3	33	)	)	PUNCT
cana-1956	3	34	fails	fail	VERB
cana-1956	3	35	to	to	PART
cana-1956	3	36	produce	produce	VERB
cana-1956	3	37	a	a	DET
cana-1956	3	38	frame	frame	NOUN
cana-1956	3	39	if	if	SCONJ
cana-1956	3	40	ε^2=(2n-1)/(n-1	ε^2=(2n-1)/(n-1	NOUN
cana-1956	3	41	)	)	PUNCT
cana-1956	3	42	.	.	PUNCT
cana-1956	4	1	a	a	DET
cana-1956	4	2	particular	particular	ADJ
cana-1956	4	3	focus	focus	NOUN
cana-1956	4	4	is	be	AUX
cana-1956	4	5	placed	place	VERB
cana-1956	4	6	on	on	ADP
cana-1956	4	7	the	the	DET
cana-1956	4	8	initial	initial	ADJ
cana-1956	4	9	hermite	hermite	ADJ
cana-1956	4	10	function	function	NOUN
cana-1956	4	11	,	,	PUNCT
cana-1956	4	12	h_1	h_1	PROPN
cana-1956	4	13	=	=	PRON
cana-1956	4	14	te^(-πt^2	te^(-πt^2	NOUN
cana-1956	4	15	)	)	PUNCT
cana-1956	4	16	.	.	PUNCT
cana-1956	5	1	key	key	ADJ
cana-1956	5	2	word	word	NOUN
cana-1956	5	3	:	:	PUNCT
cana-1956	5	4	gabor	gabor	NOUN
cana-1956	5	5	system	system	NOUN
cana-1956	5	6	,	,	PUNCT
cana-1956	5	7	rational	rational	ADJ
cana-1956	5	8	,	,	PUNCT
cana-1956	5	9	window	window	NOUN
cana-1956	5	10	function	function	NOUN
cana-1956	5	11	,	,	PUNCT
cana-1956	5	12	initial	initial	ADJ
cana-1956	5	13	hermite	hermite	ADJ
cana-1956	5	14	function	function	NOUN
cana-1956	5	15	.	.	PUNCT
cana-1956	6	1	1	1	X
cana-1956	6	2	.	.	X
cana-1956	6	3	introduction	introduction	NOUN
cana-1956	6	4	one	one	NUM
cana-1956	6	5	of	of	ADP
cana-1956	6	6	the	the	DET
cana-1956	6	7	primary	primary	ADJ
cana-1956	6	8	concerns	concern	NOUN
cana-1956	6	9	with	with	ADP
cana-1956	6	10	gabor	gabor	PROPN
cana-1956	6	11	analysis	analysis	NOUN
cana-1956	6	12	is	be	AUX
cana-1956	6	13	the	the	DET
cana-1956	6	14	following	following	NOUN
cana-1956	6	15	.	.	PUNCT
cana-1956	7	1	determine	determine	VERB
cana-1956	7	2	the	the	DET
cana-1956	7	3	set	set	NOUN
cana-1956	7	4	of	of	ADP
cana-1956	7	5	lattice	lattice	NOUN
cana-1956	7	6	parameters	parameter	NOUN
cana-1956	7	7	(	(	PUNCT
cana-1956	7	8	𝜀	𝜀	X
cana-1956	7	9	>	>	X
cana-1956	7	10	1	1	NUM
cana-1956	7	11	)	)	PUNCT
cana-1956	7	12	for	for	ADP
cana-1956	7	13	the	the	DET
cana-1956	7	14	gabor	gabor	PROPN
cana-1956	7	15	system	system	NOUN
cana-1956	7	16	:	:	PUNCT
cana-1956	7	17	𝒢(𝑔	𝒢(𝑔	NOUN
cana-1956	7	18	,	,	PUNCT
cana-1956	7	19	𝜀	𝜀	X
cana-1956	7	20	+	+	NOUN
cana-1956	7	21	1	1	NUM
cana-1956	7	22	,	,	PUNCT
cana-1956	7	23	𝜀	𝜀	VERB
cana-1956	7	24	−	−	NOUN
cana-1956	7	25	1	1	NUM
cana-1956	7	26	)	)	PUNCT
cana-1956	7	27	=	=	PRON
cana-1956	7	28	{	{	PUNCT
cana-1956	7	29	𝑒2𝑖𝜋(𝜀−1)𝑛𝑡𝑔(𝑡	𝑒2𝑖𝜋(𝜀−1)𝑛𝑡𝑔(𝑡	PROPN
cana-1956	7	30	−	−	PROPN
cana-1956	7	31	(	(	PUNCT
cana-1956	7	32	𝜀	𝜀	X
cana-1956	7	33	+	+	CCONJ
cana-1956	7	34	1)𝑚):𝑚	1)𝑚):𝑚	NUM
cana-1956	7	35	,	,	PUNCT
cana-1956	7	36	𝑛	𝑛	DET
cana-1956	7	37	∈	∈	PROPN
cana-1956	7	38	ℤ	ℤ	PROPN
cana-1956	7	39	}	}	PUNCT
cana-1956	7	40	;	;	PUNCT
cana-1956	7	41	forms	form	VERB
cana-1956	7	42	a	a	DET
cana-1956	7	43	frame	frame	NOUN
cana-1956	7	44	in	in	ADP
cana-1956	7	45	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	7	46	)	)	PUNCT
cana-1956	7	47	given	give	VERB
cana-1956	7	48	a	a	DET
cana-1956	7	49	window	window	NOUN
cana-1956	7	50	function	function	NOUN
cana-1956	7	51	𝑔	𝑔	PROPN
cana-1956	7	52	∈	∈	PROPN
cana-1956	7	53	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	7	54	):	):	PUNCT
cana-1956	7	55	a	a	DET
cana-1956	7	56	frame	frame	NOUN
cana-1956	7	57	for	for	ADP
cana-1956	7	58	𝐿2(ℝ)is	𝐿2(ℝ)is	PUNCT
cana-1956	7	59	given	give	VERB
cana-1956	7	60	by	by	ADP
cana-1956	7	61	:	:	PUNCT
cana-1956	7	62	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	7	63	)	)	PUNCT
cana-1956	7	64	≔	≔	NOUN
cana-1956	7	65	{	{	PUNCT
cana-1956	7	66	(	(	PUNCT
cana-1956	7	67	𝜀	𝜀	X
cana-1956	7	68	+	+	ADJ
cana-1956	7	69	1	1	NUM
cana-1956	7	70	,	,	PUNCT
cana-1956	7	71	𝜀	𝜀	VERB
cana-1956	7	72	−	−	NUM
cana-1956	7	73	1	1	NUM
cana-1956	7	74	)	)	PUNCT
cana-1956	7	75	∈	∈	NOUN
cana-1956	7	76	ℝ+	ℝ+	PUNCT
cana-1956	7	77	2	2	NUM
cana-1956	7	78	:	:	PUNCT
cana-1956	7	79	𝒢(𝑔	𝒢(𝑔	NOUN
cana-1956	7	80	,	,	PUNCT
cana-1956	7	81	𝜀	𝜀	X
cana-1956	7	82	+	+	NOUN
cana-1956	7	83	1	1	NUM
cana-1956	7	84	,	,	PUNCT
cana-1956	7	85	𝜀	𝜀	X
cana-1956	7	86	−	−	NOUN
cana-1956	7	87	1	1	NUM
cana-1956	7	88	)	)	PUNCT
cana-1956	7	89	}	}	PUNCT
cana-1956	7	90	.	.	PUNCT
cana-1956	8	1	we	we	PRON
cana-1956	8	2	evaluate	evaluate	VERB
cana-1956	8	3	some	some	PRON
cana-1956	8	4	of	of	ADP
cana-1956	8	5	the	the	DET
cana-1956	8	6	available	available	ADJ
cana-1956	8	7	information	information	NOUN
cana-1956	8	8	on	on	ADP
cana-1956	8	9	the	the	DET
cana-1956	8	10	structure	structure	NOUN
cana-1956	8	11	set	set	VERB
cana-1956	8	12	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	8	13	)	)	PUNCT
cana-1956	8	14	,	,	PUNCT
cana-1956	8	15	which	which	PRON
cana-1956	8	16	comes	come	VERB
cana-1956	8	17	after	after	ADP
cana-1956	8	18	[	[	X
cana-1956	8	19	1	1	NUM
cana-1956	8	20	]	]	PUNCT
cana-1956	8	21	and	and	CCONJ
cana-1956	8	22	[	[	X
cana-1956	8	23	2	2	NUM
cana-1956	8	24	]	]	PUNCT
cana-1956	8	25	(	(	PUNCT
cana-1956	8	26	in	in	ADP
cana-1956	8	27	a	a	DET
cana-1956	8	28	more	more	ADV
cana-1956	8	29	condensed	condense	VERB
cana-1956	8	30	format	format	NOUN
cana-1956	8	31	)	)	PUNCT
cana-1956	8	32	.	.	PUNCT
cana-1956	9	1	in	in	ADP
cana-1956	9	2	more	more	ADV
cana-1956	9	3	relaxed	relaxed	ADJ
cana-1956	9	4	circumstances	circumstance	NOUN
cana-1956	9	5	,	,	PUNCT
cana-1956	9	6	if	if	SCONJ
cana-1956	9	7	𝑔	𝑔	PROPN
cana-1956	9	8	∈	∈	PROPN
cana-1956	9	9	feichtinger	feichtinger	NOUN
cana-1956	9	10	algebra	algebra	PROPN
cana-1956	9	11	𝑀1	𝑀1	PROPN
cana-1956	9	12	,	,	PUNCT
cana-1956	9	13	the	the	DET
cana-1956	9	14	set	set	NOUN
cana-1956	9	15	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	9	16	)	)	PUNCT
cana-1956	9	17	is	be	AUX
cana-1956	9	18	open	open	ADJ
cana-1956	9	19	in	in	ADP
cana-1956	9	20	ℝ+	ℝ+	X
cana-1956	9	21	2	2	NUM
cana-1956	9	22	and	and	CCONJ
cana-1956	9	23	includes	include	VERB
cana-1956	9	24	the	the	DET
cana-1956	9	25	region	region	NOUN
cana-1956	9	26	around	around	ADP
cana-1956	9	27	the	the	DET
cana-1956	9	28	origin	origin	NOUN
cana-1956	9	29	.	.	PUNCT
cana-1956	10	1	if	if	SCONJ
cana-1956	10	2	𝑔	𝑔	PROPN
cana-1956	10	3	∈	∈	PROPN
cana-1956	10	4	feichtinger	feichtinger	NOUN
cana-1956	10	5	algebra	algebra	PROPN
cana-1956	10	6	𝑀1	𝑀1	PROPN
cana-1956	10	7	,	,	PUNCT
cana-1956	10	8	the	the	DET
cana-1956	10	9	open	open	ADJ
cana-1956	10	10	set	set	NOUN
cana-1956	10	11	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	10	12	)	)	PUNCT
cana-1956	10	13	in	in	ADP
cana-1956	10	14	ℝ+	ℝ+	X
cana-1956	10	15	2	2	NUM
cana-1956	10	16	includes	include	VERB
cana-1956	10	17	the	the	DET
cana-1956	10	18	region	region	NOUN
cana-1956	10	19	around	around	ADP
cana-1956	10	20	the	the	DET
cana-1956	10	21	origin	origin	NOUN
cana-1956	10	22	.	.	PUNCT
cana-1956	11	1	in	in	ADP
cana-1956	11	2	addition	addition	NOUN
cana-1956	11	3	,	,	PUNCT
cana-1956	11	4	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	11	5	)	)	PUNCT
cana-1956	11	6	⊂	⊂	PROPN
cana-1956	11	7	∏+	∏+	PROPN
cana-1956	11	8	≔	≔	VERB
cana-1956	11	9	{	{	PUNCT
cana-1956	11	10	(	(	PUNCT
cana-1956	11	11	𝜀	𝜀	X
cana-1956	11	12	+	+	ADJ
cana-1956	11	13	1	1	NUM
cana-1956	11	14	,	,	PUNCT
cana-1956	11	15	𝜀	𝜀	VERB
cana-1956	11	16	−	−	NUM
cana-1956	11	17	1	1	NUM
cana-1956	11	18	)	)	PUNCT
cana-1956	11	19	∈	∈	NOUN
cana-1956	11	20	ℝ+	ℝ+	PUNCT
cana-1956	11	21	2	2	NUM
cana-1956	11	22	:	:	PUNCT
cana-1956	11	23	𝜀	𝜀	X
cana-1956	11	24	<	<	X
cana-1956	11	25	√2	√2	PROPN
cana-1956	11	26	}	}	PUNCT
cana-1956	11	27	for	for	ADP
cana-1956	11	28	𝑔	𝑔	PROPN
cana-1956	11	29	∈	∈	PROPN
cana-1956	11	30	𝑀1	𝑀1	PROPN
cana-1956	11	31	,	,	PUNCT
cana-1956	11	32	according	accord	VERB
cana-1956	11	33	to	to	ADP
cana-1956	11	34	fundamental	fundamental	ADJ
cana-1956	11	35	density	density	NOUN
cana-1956	11	36	theorems	theorem	NOUN
cana-1956	11	37	[	[	X
cana-1956	11	38	3,4,5	3,4,5	NUM
cana-1956	11	39	]	]	X
cana-1956	11	40	and	and	CCONJ
cana-1956	11	41	an	an	DET
cana-1956	11	42	adaptation	adaptation	NOUN
cana-1956	11	43	of	of	ADP
cana-1956	11	44	the	the	DET
cana-1956	11	45	uncertainly	uncertainly	ADJ
cana-1956	11	46	principle	principle	NOUN
cana-1956	11	47	[	[	X
cana-1956	11	48	6	6	NUM
cana-1956	11	49	,	,	PUNCT
cana-1956	11	50	7	7	NUM
cana-1956	11	51	]	]	PUNCT
cana-1956	11	52	.	.	PUNCT
cana-1956	12	1	only	only	ADV
cana-1956	12	2	a	a	DET
cana-1956	12	3	small	small	ADJ
cana-1956	12	4	number	number	NOUN
cana-1956	12	5	of	of	ADP
cana-1956	12	6	functions	function	NOUN
cana-1956	12	7	for	for	ADP
cana-1956	12	8	which	which	PRON
cana-1956	12	9	ℱ	ℱ	PROPN
cana-1956	12	10	=	=	PUNCT
cana-1956	12	11	∏+have	∏+have	NOUN
cana-1956	12	12	been	be	AUX
cana-1956	12	13	identified	identify	VERB
cana-1956	12	14	up	up	ADP
cana-1956	12	15	until	until	ADP
cana-1956	12	16	quite	quite	ADV
cana-1956	12	17	recently	recently	ADV
cana-1956	12	18	.	.	PUNCT
cana-1956	13	1	the	the	DET
cana-1956	13	2	hyperbolic	hyperbolic	ADJ
cana-1956	13	3	secant	secant	NOUN
cana-1956	13	4	𝑔(𝑡	𝑔(𝑡	PROPN
cana-1956	13	5	)	)	PUNCT
cana-1956	13	6	=	=	PUNCT
cana-1956	13	7	(	(	PUNCT
cana-1956	13	8	𝑒𝑡	𝑒𝑡	NOUN
cana-1956	13	9	+	+	CCONJ
cana-1956	13	10	𝑒−𝑡)−1	𝑒−𝑡)−1	PROPN
cana-1956	13	11	,	,	PUNCT
cana-1956	13	12	while	while	SCONJ
cana-1956	13	13	the	the	DET
cana-1956	13	14	gaussian	gaussian	NOUN
cana-1956	13	15	𝑔(𝑡	𝑔(𝑡	PROPN
cana-1956	13	16	)	)	PUNCT
cana-1956	13	17	=	=	PUNCT
cana-1956	13	18	𝑒−𝜋𝑡	𝑒−𝜋𝑡	ADV
cana-1956	13	19	2	2	NUM
cana-1956	13	20	[	[	X
cana-1956	13	21	8,9,10	8,9,10	NUM
cana-1956	13	22	]	]	PUNCT
cana-1956	13	23	,	,	PUNCT
cana-1956	13	24	and	and	CCONJ
cana-1956	13	25	the	the	DET
cana-1956	13	26	oneand	oneand	ADJ
cana-1956	13	27	two	two	NUM
cana-1956	13	28	-	-	PUNCT
cana-1956	13	29	sided	sided	ADJ
cana-1956	13	30	exponential	exponential	ADJ
cana-1956	13	31	functions	function	NOUN
cana-1956	13	32	𝑔(𝑡	𝑔(𝑡	PROPN
cana-1956	13	33	)	)	PUNCT
cana-1956	13	34	=	=	SYM
cana-1956	13	35	𝑒−𝑖𝔦ℝ+	𝑒−𝑖𝔦ℝ+	NOUN
cana-1956	13	36	(	(	PUNCT
cana-1956	13	37	here	here	ADV
cana-1956	13	38	𝜀	𝜀	X
cana-1956	13	39	=	=	SYM
cana-1956	13	40	√2	√2	PROPN
cana-1956	13	41	)	)	PUNCT
cana-1956	13	42	also	also	ADV
cana-1956	13	43	produces	produce	VERB
cana-1956	13	44	a	a	DET
cana-1956	13	45	frame	frame	NOUN
cana-1956	13	46	)	)	PUNCT
cana-1956	13	47	and	and	CCONJ
cana-1956	13	48	𝑔(𝑡	𝑔(𝑡	NUM
cana-1956	13	49	)	)	PUNCT
cana-1956	13	50	=	=	PUNCT
cana-1956	13	51	𝑒−𝜋𝑡	𝑒−𝜋𝑡	ADV
cana-1956	13	52	2	2	X
cana-1956	13	53	)	)	PUNCT
cana-1956	14	1	[	[	X
cana-1956	14	2	11,12	11,12	NUM
cana-1956	14	3	]	]	PUNCT
cana-1956	14	4	are	be	AUX
cana-1956	14	5	included	include	VERB
cana-1956	14	6	in	in	ADP
cana-1956	14	7	the	the	DET
cana-1956	14	8	list	list	NOUN
cana-1956	14	9	,	,	PUNCT
cana-1956	14	10	along	along	ADP
cana-1956	14	11	with	with	ADP
cana-1956	14	12	their	their	PRON
cana-1956	14	13	shifts	shift	NOUN
cana-1956	14	14	and	and	CCONJ
cana-1956	14	15	fourier	fouri	ADJ
cana-1956	14	16	transformations	transformation	NOUN
cana-1956	14	17	.	.	PUNCT
cana-1956	15	1	in	in	ADP
cana-1956	15	2	[	[	X
cana-1956	15	3	2	2	NUM
cana-1956	15	4	]	]	PUNCT
cana-1956	15	5	,	,	PUNCT
cana-1956	15	6	a	a	DET
cana-1956	15	7	significant	significant	ADJ
cana-1956	15	8	discovery	discovery	NOUN
cana-1956	15	9	was	be	AUX
cana-1956	15	10	made	make	VERB
cana-1956	15	11	when	when	SCONJ
cana-1956	15	12	the	the	DET
cana-1956	15	13	authors	author	NOUN
cana-1956	15	14	established	establish	VERB
cana-1956	15	15	that	that	SCONJ
cana-1956	15	16	any	any	DET
cana-1956	15	17	fully	fully	ADV
cana-1956	15	18	positive	positive	ADJ
cana-1956	15	19	function	function	NOUN
cana-1956	15	20	of	of	ADP
cana-1956	15	21	finite	finite	ADJ
cana-1956	15	22	type	type	NOUN
cana-1956	15	23	had	have	VERB
cana-1956	15	24	this	this	DET
cana-1956	15	25	feature	feature	NOUN
cana-1956	15	26	,	,	PUNCT
cana-1956	15	27	this	this	PRON
cana-1956	15	28	results	result	VERB
cana-1956	15	29	in	in	ADP
cana-1956	15	30	an	an	DET
cana-1956	15	31	infinite	infinite	ADJ
cana-1956	15	32	family	family	NOUN
cana-1956	15	33	of	of	ADP
cana-1956	15	34	functions	function	NOUN
cana-1956	15	35	where	where	SCONJ
cana-1956	15	36	ℱ(𝑔	ℱ(𝑔	AUX
cana-1956	15	37	)	)	PUNCT
cana-1956	15	38	=	=	PUNCT
cana-1956	15	39	∏+	∏+	NOUN
cana-1956	15	40	.	.	PUNCT
cana-1956	16	1	communications	communication	NOUN
cana-1956	16	2	on	on	ADP
cana-1956	16	3	applied	apply	VERB
cana-1956	16	4	nonlinear	nonlinear	ADJ
cana-1956	16	5	analysis	analysis	NOUN
cana-1956	16	6	issn	issn	NOUN
cana-1956	16	7	:	:	PUNCT
cana-1956	16	8	1074	1074	NUM
cana-1956	16	9	-	-	PUNCT
cana-1956	16	10	133x	133x	NUM
cana-1956	16	11	vol	vol	NOUN
cana-1956	16	12	32	32	NUM
cana-1956	16	13	no	no	NOUN
cana-1956	16	14	.	.	NOUN
cana-1956	16	15	3	3	NUM
cana-1956	16	16	(	(	PUNCT
cana-1956	16	17	2025	2025	NUM
cana-1956	16	18	)	)	PUNCT
cana-1956	16	19	230	230	NUM
cana-1956	16	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	16	21	however	however	ADV
cana-1956	16	22	,	,	PUNCT
cana-1956	16	23	[	[	X
cana-1956	16	24	13	13	NUM
cana-1956	16	25	]	]	PUNCT
cana-1956	16	26	shows	show	VERB
cana-1956	16	27	that	that	SCONJ
cana-1956	16	28	even	even	ADV
cana-1956	16	29	for	for	ADP
cana-1956	16	30	a	a	DET
cana-1956	16	31	"	"	PUNCT
cana-1956	16	32	simple	simple	ADJ
cana-1956	16	33	"	"	PUNCT
cana-1956	16	34	function	function	NOUN
cana-1956	16	35	like	like	ADP
cana-1956	16	36	the	the	DET
cana-1956	16	37	characteristic	characteristic	ADJ
cana-1956	16	38	function	function	NOUN
cana-1956	16	39	𝑔	𝑔	PROPN
cana-1956	16	40	=	=	PROPN
cana-1956	16	41	1𝑙	1𝑙	PROPN
cana-1956	16	42	of	of	ADP
cana-1956	16	43	an	an	DET
cana-1956	16	44	interval	interval	NOUN
cana-1956	16	45	,	,	PUNCT
cana-1956	16	46	the	the	DET
cana-1956	16	47	set	set	NOUN
cana-1956	16	48	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	16	49	)	)	PUNCT
cana-1956	16	50	may	may	AUX
cana-1956	16	51	have	have	VERB
cana-1956	16	52	a	a	DET
cana-1956	16	53	somewhat	somewhat	ADV
cana-1956	16	54	complex	complex	ADJ
cana-1956	16	55	shape	shape	NOUN
cana-1956	16	56	.	.	PUNCT
cana-1956	17	1	when	when	SCONJ
cana-1956	17	2	𝑔	𝑔	PROPN
cana-1956	17	3	is	be	AUX
cana-1956	17	4	concentrated	concentrate	VERB
cana-1956	17	5	in	in	ADP
cana-1956	17	6	both	both	DET
cana-1956	17	7	time	time	NOUN
cana-1956	17	8	and	and	CCONJ
cana-1956	17	9	frequency	frequency	NOUN
cana-1956	17	10	and	and	CCONJ
cana-1956	17	11	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	17	12	)	)	PUNCT
cana-1956	17	13	≠	≠	PROPN
cana-1956	17	14	∏+	∏+	NOUN
cana-1956	17	15	,	,	PUNCT
cana-1956	17	16	there	there	PRON
cana-1956	17	17	are	be	VERB
cana-1956	17	18	certain	certain	ADJ
cana-1956	17	19	instances	instance	NOUN
cana-1956	17	20	.	.	PUNCT
cana-1956	18	1	first	first	ADV
cana-1956	18	2	,	,	PUNCT
cana-1956	18	3	we	we	PRON
cana-1956	18	4	want	want	VERB
cana-1956	18	5	to	to	PART
cana-1956	18	6	get	get	VERB
cana-1956	18	7	the	the	DET
cana-1956	18	8	hermite	hermite	ADJ
cana-1956	18	9	function	function	NOUN
cana-1956	18	10	ℎ1(𝑡	ℎ1(𝑡	NOUN
cana-1956	18	11	)	)	PUNCT
cana-1956	18	12	=	=	NOUN
cana-1956	18	13	𝑡𝑒	𝑡𝑒	ADP
cana-1956	18	14	−𝜋𝑡2	−𝜋𝑡2	PROPN
cana-1956	18	15	.	.	PUNCT
cana-1956	19	1	this	this	DET
cana-1956	19	2	conclusion	conclusion	NOUN
cana-1956	19	3	is	be	AUX
cana-1956	19	4	motivated	motivate	VERB
cana-1956	19	5	by	by	ADP
cana-1956	19	6	the	the	DET
cana-1956	19	7	uncertainty	uncertainty	NOUN
cana-1956	19	8	principle	principle	NOUN
cana-1956	19	9	,	,	PUNCT
cana-1956	19	10	which	which	PRON
cana-1956	19	11	states	state	VERB
cana-1956	19	12	that	that	SCONJ
cana-1956	19	13	ℎ1	ℎ1	PROPN
cana-1956	19	14	lowers	lower	VERB
cana-1956	19	15	the	the	DET
cana-1956	19	16	heisenberg	heisenberg	PROPN
cana-1956	19	17	uncertainty	uncertainty	NOUN
cana-1956	19	18	of	of	ADP
cana-1956	19	19	all	all	DET
cana-1956	19	20	functions	function	NOUN
cana-1956	19	21	orthogonal	orthogonal	ADJ
cana-1956	19	22	to	to	ADP
cana-1956	19	23	the	the	DET
cana-1956	19	24	gaussian	gaussian	NOUN
cana-1956	19	25	,	,	PUNCT
cana-1956	19	26	as	as	ADV
cana-1956	19	27	well	well	ADV
cana-1956	19	28	as	as	ADP
cana-1956	19	29	present	present	ADJ
cana-1956	19	30	studies	study	NOUN
cana-1956	19	31	on	on	ADP
cana-1956	19	32	vector	vector	NOUN
cana-1956	19	33	-	-	PUNCT
cana-1956	19	34	valued	value	VERB
cana-1956	19	35	gabor	gabor	NOUN
cana-1956	19	36	frames	frame	NOUN
cana-1956	19	37	[	[	X
cana-1956	19	38	14	14	NUM
cana-1956	19	39	]	]	PUNCT
cana-1956	19	40	.	.	PUNCT
cana-1956	20	1	the	the	DET
cana-1956	20	2	paper	paper	NOUN
cana-1956	20	3	is	be	AUX
cana-1956	20	4	organized	organize	VERB
cana-1956	20	5	as	as	ADP
cana-1956	20	6	following	follow	VERB
cana-1956	20	7	.	.	PUNCT
cana-1956	21	1	basic	basic	ADJ
cana-1956	21	2	information	information	NOUN
cana-1956	21	3	from	from	ADP
cana-1956	21	4	gabor	gabor	PROPN
cana-1956	21	5	analysis	analysis	NOUN
cana-1956	21	6	and	and	CCONJ
cana-1956	21	7	notation	notation	NOUN
cana-1956	21	8	are	be	AUX
cana-1956	21	9	presented	present	VERB
cana-1956	21	10	in	in	ADP
cana-1956	21	11	the	the	DET
cana-1956	21	12	next	next	ADJ
cana-1956	21	13	section	section	NOUN
cana-1956	21	14	.	.	PUNCT
cana-1956	22	1	we	we	PRON
cana-1956	22	2	examine	examine	VERB
cana-1956	22	3	the	the	DET
cana-1956	22	4	scenario	scenario	NOUN
cana-1956	22	5	of	of	ADP
cana-1956	22	6	logical	logical	ADJ
cana-1956	22	7	oversampling	oversampling	NOUN
cana-1956	22	8	,	,	PUNCT
cana-1956	22	9	where	where	SCONJ
cana-1956	22	10	𝜀2	𝜀2	PROPN
cana-1956	22	11	=	=	SYM
cana-1956	22	12	𝑝	𝑝	PROPN
cana-1956	22	13	𝑞	𝑞	NOUN
cana-1956	22	14	+	+	CCONJ
cana-1956	22	15	1	1	NUM
cana-1956	22	16	∈	∈	PROPN
cana-1956	22	17	ℚ	ℚ	PROPN
cana-1956	22	18	.the	.the	PRON
cana-1956	22	19	union	union	PROPN
cana-1956	22	20	of	of	ADP
cana-1956	22	21	hyperbolas	hyperbolas	PROPN
cana-1956	22	22	is	be	AUX
cana-1956	22	23	not	not	PART
cana-1956	22	24	contained	contain	VERB
cana-1956	22	25	in	in	ADP
cana-1956	22	26	the	the	DET
cana-1956	22	27	set	set	NOUN
cana-1956	22	28	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	22	29	)	)	PUNCT
cana-1956	22	30	for	for	ADP
cana-1956	22	31	any	any	DET
cana-1956	22	32	odd	odd	ADJ
cana-1956	22	33	function	function	NOUN
cana-1956	22	34	𝑔	𝑔	PROPN
cana-1956	22	35	∈	∈	PROPN
cana-1956	22	36	𝑀1	𝑀1	NOUN
cana-1956	22	37	(	(	PUNCT
cana-1956	22	38	especially	especially	ADV
cana-1956	22	39	ℎ1	ℎ1	PROPN
cana-1956	22	40	):	):	PUNCT
cana-1956	22	41	𝜀2	𝜀2	PROPN
cana-1956	22	42	=	=	SYM
cana-1956	22	43	2𝑛	2𝑛	PROPN
cana-1956	22	44	−	−	PROPN
cana-1956	22	45	1	1	NUM
cana-1956	22	46	𝑛	𝑛	DET
cana-1956	22	47	−	−	NUM
cana-1956	22	48	1	1	NUM
cana-1956	22	49	⇒	⇒	NOUN
cana-1956	22	50	(	(	PUNCT
cana-1956	22	51	𝜀	𝜀	X
cana-1956	22	52	+	+	ADJ
cana-1956	22	53	1	1	NUM
cana-1956	22	54	,	,	PUNCT
cana-1956	22	55	𝜀	𝜀	VERB
cana-1956	22	56	−	−	NOUN
cana-1956	22	57	1	1	NUM
cana-1956	22	58	)	)	PUNCT
cana-1956	22	59	∉	∉	PROPN
cana-1956	22	60	ℱ(𝑔	ℱ(𝑔	NUM
cana-1956	22	61	)	)	PUNCT
cana-1956	22	62	,	,	PUNCT
cana-1956	22	63	𝑛	𝑛	PROPN
cana-1956	22	64	=	=	SYM
cana-1956	22	65	1,2	1,2	NUM
cana-1956	22	66	,	,	PUNCT
cana-1956	22	67	…	…	PUNCT
cana-1956	22	68	.	.	PUNCT
cana-1956	23	1	(	(	PUNCT
cana-1956	23	2	1	1	X
cana-1956	23	3	)	)	PUNCT
cana-1956	23	4	the	the	DET
cana-1956	23	5	vector	vector	NOUN
cana-1956	23	6	-	-	PUNCT
cana-1956	23	7	valued	value	VERB
cana-1956	23	8	zak	zak	PROPN
cana-1956	23	9	transform	transform	NOUN
cana-1956	23	10	,	,	PUNCT
cana-1956	23	11	which	which	PRON
cana-1956	23	12	encodes	encode	VERB
cana-1956	23	13	the	the	DET
cana-1956	23	14	frame	frame	NOUN
cana-1956	23	15	operator	operator	NOUN
cana-1956	23	16	as	as	ADP
cana-1956	23	17	matrix	matrix	NOUN
cana-1956	23	18	multiplication	multiplication	NOUN
cana-1956	23	19	in	in	ADP
cana-1956	23	20	a	a	DET
cana-1956	23	21	space	space	NOUN
cana-1956	23	22	of	of	ADP
cana-1956	23	23	vector	vector	NOUN
cana-1956	23	24	-	-	PUNCT
cana-1956	23	25	valued	value	VERB
cana-1956	23	26	functions	function	NOUN
cana-1956	23	27	,	,	PUNCT
cana-1956	23	28	is	be	AUX
cana-1956	23	29	examined	examine	VERB
cana-1956	23	30	in	in	ADP
cana-1956	23	31	[	[	X
cana-1956	23	32	15	15	NUM
cana-1956	23	33	]	]	PUNCT
cana-1956	23	34	and	and	CCONJ
cana-1956	23	35	[	[	X
cana-1956	23	36	4	4	NUM
cana-1956	23	37	,	,	PUNCT
cana-1956	23	38	ch	ch	NOUN
cana-1956	23	39	.	.	NOUN
cana-1956	23	40	8	8	NUM
cana-1956	23	41	]	]	PUNCT
cana-1956	23	42	to	to	PART
cana-1956	23	43	lay	lay	VERB
cana-1956	23	44	the	the	DET
cana-1956	23	45	groundwork	groundwork	NOUN
cana-1956	23	46	for	for	ADP
cana-1956	23	47	the	the	DET
cana-1956	23	48	proof	proof	NOUN
cana-1956	23	49	.	.	PUNCT
cana-1956	24	1	notice	notice	VERB
cana-1956	24	2	the	the	DET
cana-1956	24	3	following	following	ADJ
cana-1956	24	4	part	part	NOUN
cana-1956	24	5	,	,	PUNCT
cana-1956	24	6	we	we	PRON
cana-1956	24	7	factorize	factorize	VERB
cana-1956	24	8	the	the	DET
cana-1956	24	9	vector	vector	NOUN
cana-1956	24	10	-	-	PUNCT
cana-1956	24	11	valued	value	VERB
cana-1956	24	12	zak	zak	PROPN
cana-1956	24	13	transform	transform	NOUN
cana-1956	24	14	matrix	matrix	NOUN
cana-1956	24	15	and	and	CCONJ
cana-1956	24	16	extract	extract	VERB
cana-1956	24	17	a	a	DET
cana-1956	24	18	component	component	NOUN
cana-1956	24	19	that	that	PRON
cana-1956	24	20	is	be	AUX
cana-1956	24	21	a	a	DET
cana-1956	24	22	rational	rational	ADJ
cana-1956	24	23	version	version	NOUN
cana-1956	24	24	of	of	ADP
cana-1956	24	25	the	the	DET
cana-1956	24	26	well	well	ADV
cana-1956	24	27	-	-	PUNCT
cana-1956	24	28	known	know	VERB
cana-1956	24	29	ron	ron	PROPN
cana-1956	24	30	-	-	PUNCT
cana-1956	24	31	shen	shen	PROPN
cana-1956	24	32	gramian	gramian	PROPN
cana-1956	24	33	(	(	PUNCT
cana-1956	24	34	see	see	VERB
cana-1956	24	35	[	[	X
cana-1956	24	36	16	16	NUM
cana-1956	24	37	]	]	PUNCT
cana-1956	24	38	and	and	CCONJ
cana-1956	24	39	[	[	X
cana-1956	24	40	4	4	NUM
cana-1956	24	41	]	]	NUM
cana-1956	24	42	)	)	PUNCT
cana-1956	24	43	.	.	PUNCT
cana-1956	25	1	we	we	PRON
cana-1956	25	2	hypothesize	hypothesize	VERB
cana-1956	25	3	that	that	SCONJ
cana-1956	25	4	the	the	DET
cana-1956	25	5	only	only	ADJ
cana-1956	25	6	limitation	limitation	NOUN
cana-1956	25	7	on	on	ADP
cana-1956	25	8	the	the	DET
cana-1956	25	9	set	set	NOUN
cana-1956	25	10	ℱ(ℎ1	ℱ(ℎ1	PROPN
cana-1956	25	11	)	)	PUNCT
cana-1956	25	12	is	be	AUX
cana-1956	25	13	condition	condition	NOUN
cana-1956	25	14	(	(	PUNCT
cana-1956	25	15	1	1	NUM
cana-1956	25	16	)	)	PUNCT
cana-1956	25	17	.	.	PUNCT
cana-1956	26	1	ℱ(ℎ1	ℱ(ℎ1	X
cana-1956	26	2	)	)	PUNCT
cana-1956	26	3	=	=	PRON
cana-1956	26	4	{	{	PUNCT
cana-1956	26	5	(	(	PUNCT
cana-1956	26	6	𝜀	𝜀	X
cana-1956	26	7	+	+	ADJ
cana-1956	26	8	1	1	NUM
cana-1956	26	9	,	,	PUNCT
cana-1956	26	10	𝜀	𝜀	VERB
cana-1956	26	11	−	−	NUM
cana-1956	26	12	1	1	NUM
cana-1956	26	13	)	)	PUNCT
cana-1956	26	14	∈	∈	PROPN
cana-1956	27	1	∏+	∏+	NOUN
cana-1956	27	2	:	:	PUNCT
cana-1956	27	3	𝜀	𝜀	PROPN
cana-1956	27	4	2	2	NUM
cana-1956	27	5	≠	≠	PROPN
cana-1956	27	6	2𝑛	2𝑛	PROPN
cana-1956	27	7	−	−	PROPN
cana-1956	27	8	1	1	NUM
cana-1956	27	9	𝑛	𝑛	PRON
cana-1956	27	10	−	−	PROPN
cana-1956	27	11	1	1	NUM
cana-1956	27	12	,	,	PUNCT
cana-1956	27	13	𝑛	𝑛	NOUN
cana-1956	27	14	=	=	SYM
cana-1956	27	15	1,2	1,2	NUM
cana-1956	27	16	,	,	PUNCT
cana-1956	27	17	…	…	PUNCT
cana-1956	27	18	.	.	PUNCT
cana-1956	27	19	}	}	PUNCT
cana-1956	27	20	.	.	PUNCT
cana-1956	28	1	regretfully	regretfully	ADV
cana-1956	28	2	,	,	PUNCT
cana-1956	28	3	we	we	PRON
cana-1956	28	4	are	be	AUX
cana-1956	28	5	unable	unable	ADJ
cana-1956	28	6	to	to	PART
cana-1956	28	7	fully	fully	ADV
cana-1956	28	8	verify	verify	VERB
cana-1956	28	9	this	this	DET
cana-1956	28	10	conjecture	conjecture	NOUN
cana-1956	28	11	.	.	PUNCT
cana-1956	29	1	we	we	PRON
cana-1956	29	2	provide	provide	VERB
cana-1956	29	3	numerical	numerical	ADJ
cana-1956	29	4	verification	verification	NOUN
cana-1956	29	5	for	for	ADP
cana-1956	29	6	a	a	DET
cana-1956	29	7	larger	large	ADJ
cana-1956	29	8	collection	collection	NOUN
cana-1956	29	9	of	of	ADP
cana-1956	29	10	points	point	NOUN
cana-1956	29	11	and	and	CCONJ
cana-1956	29	12	show	show	VERB
cana-1956	29	13	it	it	PRON
cana-1956	29	14	analytically	analytically	ADV
cana-1956	29	15	only	only	ADV
cana-1956	29	16	for	for	ADP
cana-1956	29	17	a	a	DET
cana-1956	29	18	subset	subset	NOUN
cana-1956	29	19	of	of	ADP
cana-1956	29	20	∏+	∏+	NOUN
cana-1956	29	21	.	.	PUNCT
cana-1956	30	1	the	the	DET
cana-1956	30	2	rational	rational	ADJ
cana-1956	30	3	equivalent	equivalent	NOUN
cana-1956	30	4	of	of	ADP
cana-1956	30	5	the	the	DET
cana-1956	30	6	gramain	gramain	NOUN
cana-1956	30	7	provided	provide	VERB
cana-1956	30	8	in	in	ADP
cana-1956	30	9	section	section	NOUN
cana-1956	30	10	4	4	NUM
cana-1956	30	11	serves	serve	VERB
cana-1956	30	12	as	as	ADP
cana-1956	30	13	the	the	DET
cana-1956	30	14	foundation	foundation	NOUN
cana-1956	30	15	for	for	ADP
cana-1956	30	16	these	these	DET
cana-1956	30	17	creations	creation	NOUN
cana-1956	30	18	.	.	PUNCT
cana-1956	31	1	2.preliminaries	2.preliminaries	NUM
cana-1956	31	2	the	the	DET
cana-1956	31	3	fundamental	fundamental	ADJ
cana-1956	31	4	information	information	NOUN
cana-1956	31	5	from	from	ADP
cana-1956	31	6	the	the	DET
cana-1956	31	7	gabor	gabor	NOUN
cana-1956	31	8	analysis	analysis	NOUN
cana-1956	31	9	is	be	AUX
cana-1956	31	10	reviewed	review	VERB
cana-1956	31	11	in	in	ADP
cana-1956	31	12	this	this	DET
cana-1956	31	13	part	part	NOUN
cana-1956	31	14	and	and	CCONJ
cana-1956	31	15	will	will	AUX
cana-1956	31	16	be	be	AUX
cana-1956	31	17	applied	apply	VERB
cana-1956	31	18	later	later	ADV
cana-1956	31	19	.	.	PUNCT
cana-1956	32	1	for	for	ADP
cana-1956	32	2	a	a	DET
cana-1956	32	3	more	more	ADV
cana-1956	32	4	thorough	thorough	ADJ
cana-1956	32	5	explanation	explanation	NOUN
cana-1956	32	6	and	and	CCONJ
cana-1956	32	7	the	the	DET
cana-1956	32	8	background	background	NOUN
cana-1956	32	9	information	information	NOUN
cana-1956	32	10	on	on	ADP
cana-1956	32	11	the	the	DET
cana-1956	32	12	topic	topic	NOUN
cana-1956	32	13	,	,	PUNCT
cana-1956	32	14	we	we	PRON
cana-1956	32	15	direct	direct	VERB
cana-1956	32	16	the	the	DET
cana-1956	32	17	reader	reader	NOUN
cana-1956	32	18	to	to	ADP
cana-1956	32	19	[	[	X
cana-1956	32	20	4	4	NUM
cana-1956	32	21	]	]	PUNCT
cana-1956	32	22	.	.	PUNCT
cana-1956	33	1	we	we	PRON
cana-1956	33	2	investigate	investigate	VERB
cana-1956	33	3	the	the	DET
cana-1956	33	4	lattice	lattice	NOUN
cana-1956	33	5	∧=	∧=	NOUN
cana-1956	33	6	(	(	PUNCT
cana-1956	33	7	𝜀	𝜀	X
cana-1956	33	8	+	+	CCONJ
cana-1956	33	9	1)ℤ	1)ℤ	NUM
cana-1956	33	10	×	×	NOUN
cana-1956	33	11	(	(	PUNCT
cana-1956	33	12	𝜀	𝜀	X
cana-1956	33	13	−	−	PROPN
cana-1956	33	14	1)ℤ	1)ℤ	NUM
cana-1956	33	15	,	,	PUNCT
cana-1956	33	16	the	the	DET
cana-1956	33	17	gabor	gabor	NOUN
cana-1956	33	18	system	system	NOUN
cana-1956	33	19	has	have	VERB
cana-1956	33	20	an	an	DET
cana-1956	33	21	area	area	NOUN
cana-1956	33	22	function	function	NOUN
cana-1956	33	23	g	g	NOUN
cana-1956	33	24	and	and	CCONJ
cana-1956	33	25	a	a	DET
cana-1956	33	26	value	value	NOUN
cana-1956	33	27	𝜀	𝜀	X
cana-1956	33	28	>	>	X
cana-1956	33	29	1	1	NUM
cana-1956	33	30	.	.	PUNCT
cana-1956	34	1	𝒢(𝑔	𝒢(𝑔	NOUN
cana-1956	34	2	,	,	PUNCT
cana-1956	34	3	𝜀	𝜀	X
cana-1956	34	4	+	+	NOUN
cana-1956	34	5	1	1	NUM
cana-1956	34	6	,	,	PUNCT
cana-1956	34	7	𝜀	𝜀	VERB
cana-1956	34	8	−	−	NOUN
cana-1956	34	9	1	1	NUM
cana-1956	34	10	)	)	PUNCT
cana-1956	34	11	=	=	PRON
cana-1956	34	12	{	{	PUNCT
cana-1956	34	13	𝜋𝜆	𝜋𝜆	NOUN
cana-1956	34	14	:	:	PUNCT
cana-1956	34	15	𝜆	𝜆	DET
cana-1956	34	16	∈∧	∈∧	NOUN
cana-1956	34	17	}	}	PUNCT
cana-1956	34	18	the	the	DET
cana-1956	34	19	standard	standard	ADJ
cana-1956	34	20	time	time	NOUN
cana-1956	34	21	-	-	PUNCT
cana-1956	34	22	frequency	frequency	NOUN
cana-1956	34	23	shift	shift	NOUN
cana-1956	34	24	is	be	AUX
cana-1956	34	25	indicated	indicate	VERB
cana-1956	34	26	by	by	ADP
cana-1956	34	27	the	the	DET
cana-1956	34	28	symbol	symbol	NOUN
cana-1956	34	29	:	:	PUNCT
cana-1956	34	30	𝜋𝜆	𝜋𝜆	NOUN
cana-1956	34	31	:	:	PUNCT
cana-1956	34	32	𝑔	𝑔	X
cana-1956	34	33	→	→	SYM
cana-1956	34	34	𝑒2𝜋𝑖𝑏𝑡𝑔(𝑡	𝑒2𝜋𝑖𝑏𝑡𝑔(𝑡	ADP
cana-1956	34	35	−	−	NUM
cana-1956	34	36	𝑎	𝑎	NOUN
cana-1956	34	37	)	)	PUNCT
cana-1956	34	38	for	for	ADP
cana-1956	34	39	𝜆	𝜆	NOUN
cana-1956	34	40	=	=	SYM
cana-1956	34	41	(	(	PUNCT
cana-1956	34	42	𝑎	𝑎	PROPN
cana-1956	34	43	,	,	PUNCT
cana-1956	34	44	𝑏	𝑏	NOUN
cana-1956	34	45	)	)	PUNCT
cana-1956	34	46	.	.	PUNCT
cana-1956	35	1	the	the	DET
cana-1956	35	2	definition	definition	NOUN
cana-1956	35	3	of	of	ADP
cana-1956	35	4	the	the	DET
cana-1956	35	5	gabor	gabor	PROPN
cana-1956	35	6	frame	frame	NOUN
cana-1956	35	7	operator	operator	NOUN
cana-1956	35	8	𝑆𝑔.∧	𝑆𝑔.∧	NOUN
cana-1956	35	9	:	:	PUNCT
cana-1956	35	10	𝐿	𝐿	PROPN
cana-1956	35	11	2(ℝ	2(ℝ	NOUN
cana-1956	35	12	)	)	PUNCT
cana-1956	35	13	→	→	SYM
cana-1956	35	14	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	35	15	)	)	PUNCT
cana-1956	35	16	is	be	AUX
cana-1956	35	17	:	:	PUNCT
cana-1956	35	18	∑𝑆𝑔,∧𝑓𝑖(𝑡	∑𝑆𝑔,∧𝑓𝑖(𝑡	NUM
cana-1956	35	19	)	)	PUNCT
cana-1956	35	20	𝑖	𝑖	NOUN
cana-1956	36	1	=	=	PUNCT
cana-1956	36	2	∑(∑〈𝑓𝑖	∑(∑〈𝑓𝑖	X
cana-1956	36	3	,	,	PUNCT
cana-1956	36	4	𝜋𝜆𝑔〉𝐿2(ℝ	𝜋𝜆𝑔〉𝐿2(ℝ	NOUN
cana-1956	36	5	)	)	PUNCT
cana-1956	36	6	𝜆∈∧	𝜆∈∧	NOUN
cana-1956	36	7	𝜋𝜆𝑔(𝑡	𝜋𝜆𝑔(𝑡	PROPN
cana-1956	36	8	)	)	PUNCT
cana-1956	36	9	)	)	PUNCT
cana-1956	36	10	𝑖	𝑖	NOUN
cana-1956	36	11	,	,	PUNCT
cana-1956	36	12	𝑓𝑖	𝑓𝑖	PROPN
cana-1956	36	13	∈	∈	PROPN
cana-1956	36	14	𝐿	𝐿	PROPN
cana-1956	36	15	2(ℝ	2(ℝ	NOUN
cana-1956	36	16	)	)	PUNCT
cana-1956	36	17	communications	communication	NOUN
cana-1956	36	18	on	on	ADP
cana-1956	36	19	applied	apply	VERB
cana-1956	36	20	nonlinear	nonlinear	ADJ
cana-1956	36	21	analysis	analysis	NOUN
cana-1956	36	22	issn	issn	NOUN
cana-1956	36	23	:	:	PUNCT
cana-1956	36	24	1074	1074	NUM
cana-1956	36	25	-	-	PUNCT
cana-1956	36	26	133x	133x	NUM
cana-1956	36	27	vol	vol	NOUN
cana-1956	36	28	32	32	NUM
cana-1956	36	29	no	no	NOUN
cana-1956	36	30	.	.	NOUN
cana-1956	36	31	3	3	NUM
cana-1956	36	32	(	(	PUNCT
cana-1956	36	33	2025	2025	NUM
cana-1956	36	34	)	)	PUNCT
cana-1956	36	35	231	231	NUM
cana-1956	36	36	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	36	37	the	the	DET
cana-1956	36	38	operation	operation	NOUN
cana-1956	36	39	𝒢(𝑔,∧	𝒢(𝑔,∧	NOUN
cana-1956	36	40	)	)	PUNCT
cana-1956	36	41	is	be	AUX
cana-1956	36	42	bounded	bound	VERB
cana-1956	36	43	if	if	SCONJ
cana-1956	36	44	and	and	CCONJ
cana-1956	36	45	only	only	ADV
cana-1956	36	46	if	if	SCONJ
cana-1956	36	47	the	the	DET
cana-1956	36	48	gabor	gabor	PROPN
cana-1956	36	49	frame	frame	NOUN
cana-1956	36	50	operator	operator	NOUN
cana-1956	36	51	is	be	AUX
cana-1956	36	52	invertible	invertible	ADJ
cana-1956	36	53	and	and	CCONJ
cana-1956	36	54	𝑔	𝑔	PROPN
cana-1956	36	55	is	be	AUX
cana-1956	36	56	in	in	ADP
cana-1956	36	57	the	the	DET
cana-1956	36	58	modulation	modulation	NOUN
cana-1956	36	59	space	space	NOUN
cana-1956	36	60	𝑀1(ℝ	𝑀1(ℝ	PROPN
cana-1956	36	61	)	)	PUNCT
cana-1956	36	62	,	,	PUNCT
cana-1956	36	63	as	as	SCONJ
cana-1956	36	64	defined	define	VERB
cana-1956	36	65	later	later	ADV
cana-1956	36	66	in	in	ADP
cana-1956	36	67	this	this	DET
cana-1956	36	68	section	section	NOUN
cana-1956	36	69	.	.	PUNCT
cana-1956	37	1	in	in	ADP
cana-1956	37	2	this	this	DET
cana-1956	37	3	case	case	NOUN
cana-1956	37	4	,	,	PUNCT
cana-1956	37	5	the	the	DET
cana-1956	37	6	operator	operator	NOUN
cana-1956	37	7	𝑆𝑔,∧	𝑆𝑔,∧	PROPN
cana-1956	37	8	can	can	AUX
cana-1956	37	9	be	be	AUX
cana-1956	37	10	written	write	VERB
cana-1956	37	11	as	as	ADP
cana-1956	37	12	a	a	DET
cana-1956	37	13	multiplication	multiplication	NOUN
cana-1956	37	14	operator	operator	NOUN
cana-1956	37	15	in	in	ADP
cana-1956	37	16	a	a	DET
cana-1956	37	17	vector	vector	NOUN
cana-1956	37	18	-	-	PUNCT
cana-1956	37	19	valued	value	VERB
cana-1956	37	20	function	function	NOUN
cana-1956	37	21	space	space	NOUN
cana-1956	37	22	.	.	PUNCT
cana-1956	38	1	assume	assume	VERB
cana-1956	38	2	that	that	SCONJ
cana-1956	38	3	for	for	ADP
cana-1956	38	4	reasonably	reasonably	ADV
cana-1956	38	5	prime	prime	ADJ
cana-1956	38	6	𝑝	𝑝	NOUN
cana-1956	38	7	,	,	PUNCT
cana-1956	38	8	𝑞	𝑞	PROPN
cana-1956	38	9	∈	∈	PROPN
cana-1956	38	10	ℕ	ℕ	PROPN
cana-1956	38	11	,	,	PUNCT
cana-1956	38	12	𝜀2	𝜀2	NOUN
cana-1956	38	13	=	=	SYM
cana-1956	38	14	𝑝	𝑝	PROPN
cana-1956	38	15	𝑞	𝑞	X
cana-1956	38	16	+	+	NOUN
cana-1956	38	17	1	1	X
cana-1956	38	18	.	.	X
cana-1956	38	19	examine	examine	VERB
cana-1956	38	20	the	the	DET
cana-1956	38	21	following	following	NOUN
cana-1956	38	22	:	:	PUNCT
cana-1956	38	23	the	the	DET
cana-1956	38	24	dimension	dimension	NOUN
cana-1956	38	25	of	of	ADP
cana-1956	38	26	vector	vector	NOUN
cana-1956	38	27	-	-	PUNCT
cana-1956	38	28	valued	value	VERB
cana-1956	38	29	function	function	NOUN
cana-1956	38	30	ℋ𝜀+1,𝑝	ℋ𝜀+1,𝑝	NOUN
cana-1956	38	31	=	=	SYM
cana-1956	38	32	𝐿	𝐿	PROPN
cana-1956	38	33	2(𝑄𝜀+1,𝑝	2(𝑄𝜀+1,𝑝	NUM
cana-1956	38	34	,	,	PUNCT
cana-1956	38	35	ℂ	ℂ	PROPN
cana-1956	38	36	𝑝	𝑝	PROPN
cana-1956	38	37	)	)	PUNCT
cana-1956	38	38	,	,	PUNCT
cana-1956	38	39	and	and	CCONJ
cana-1956	38	40	the	the	DET
cana-1956	38	41	rectangle	rectangle	NOUN
cana-1956	38	42	𝑄𝜀+1,𝑝	𝑄𝜀+1,𝑝	ADJ
cana-1956	38	43	=	=	PUNCT
cana-1956	39	1	[	[	X
cana-1956	39	2	0	0	NUM
cana-1956	39	3	,	,	PUNCT
cana-1956	39	4	𝜀	𝜀	X
cana-1956	39	5	+	+	CCONJ
cana-1956	39	6	1	1	NUM
cana-1956	39	7	𝑝⁄	𝑝⁄	NOUN
cana-1956	39	8	)	)	PUNCT
cana-1956	39	9	×	×	NOUN
cana-1956	40	1	[	[	X
cana-1956	40	2	0	0	NUM
cana-1956	40	3	,	,	PUNCT
cana-1956	40	4	[	[	X
cana-1956	40	5	0	0	NUM
cana-1956	40	6	,	,	PUNCT
cana-1956	40	7	1	1	NUM
cana-1956	40	8	𝜀	𝜀	NOUN
cana-1956	40	9	+	+	NOUN
cana-1956	40	10	1⁄	1⁄	NUM
cana-1956	40	11	)	)	PUNCT
cana-1956	40	12	.	.	PUNCT
cana-1956	41	1	recall	recall	VERB
cana-1956	41	2	that	that	SCONJ
cana-1956	41	3	a	a	DET
cana-1956	41	4	function	function	NOUN
cana-1956	41	5	𝑓𝑖	𝑓𝑖	PROPN
cana-1956	41	6	∈	∈	PROPN
cana-1956	41	7	𝐿	𝐿	PROPN
cana-1956	41	8	2(ℝ)has	2(ℝ)has	NUM
cana-1956	41	9	a	a	DET
cana-1956	41	10	zak	zak	PROPN
cana-1956	41	11	transform	transform	NOUN
cana-1956	41	12	defined	define	VERB
cana-1956	41	13	as	as	ADP
cana-1956	41	14	:	:	PUNCT
cana-1956	41	15	∑𝒵𝜀+1𝑓𝑖(𝑥	∑𝒵𝜀+1𝑓𝑖(𝑥	ADJ
cana-1956	41	16	,	,	PUNCT
cana-1956	41	17	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	41	18	)	)	PUNCT
cana-1956	41	19	𝑖	𝑖	PUNCT
cana-1956	42	1	=	=	NOUN
cana-1956	42	2	∑∑𝑓𝑖	∑∑𝑓𝑖	NOUN
cana-1956	42	3	𝑛∈ℤ	𝑛∈ℤ	PROPN
cana-1956	42	4	(	(	PUNCT
cana-1956	42	5	𝑥	𝑥	X
cana-1956	42	6	−	−	PROPN
cana-1956	42	7	(	(	PUNCT
cana-1956	42	8	𝜀	𝜀	X
cana-1956	42	9	+	+	X
cana-1956	42	10	1)𝑛)𝑒2𝜋𝑖𝑛(𝜀+1)𝑤𝑖	1)𝑛)𝑒2𝜋𝑖𝑛(𝜀+1)𝑤𝑖	NUM
cana-1956	42	11	.	.	PUNCT
cana-1956	43	1	𝑖	𝑖	X
cana-1956	43	2	(	(	PUNCT
cana-1956	43	3	2	2	X
cana-1956	43	4	)	)	PUNCT
cana-1956	43	5	we	we	PRON
cana-1956	43	6	study	study	VERB
cana-1956	43	7	the	the	DET
cana-1956	43	8	vector	vector	NOUN
cana-1956	43	9	-	-	PUNCT
cana-1956	43	10	valued	value	VERB
cana-1956	43	11	zak	zak	PROPN
cana-1956	43	12	transform	transform	VERB
cana-1956	43	13	�	�	PROPN
cana-1956	43	14	⃗	⃗	NOUN
cana-1956	43	15	�	�	NOUN
cana-1956	43	16	𝜀+1	𝜀+1	NOUN
cana-1956	43	17	:	:	PUNCT
cana-1956	43	18	𝐿	𝐿	PROPN
cana-1956	43	19	2(ℝ	2(ℝ	NOUN
cana-1956	43	20	)	)	PUNCT
cana-1956	43	21	→	→	SYM
cana-1956	43	22	ℋ𝜀+1,𝑝	ℋ𝜀+1,𝑝	ADJ
cana-1956	43	23	defind	defind	NOUN
cana-1956	43	24	as	as	SCONJ
cana-1956	43	25	follows	follow	VERB
cana-1956	43	26	,	,	PUNCT
cana-1956	43	27	in	in	ADP
cana-1956	43	28	accordance	accordance	NOUN
cana-1956	43	29	with	with	ADP
cana-1956	43	30	[	[	X
cana-1956	43	31	4,ch.8	4,ch.8	NOUN
cana-1956	43	32	]	]	X
cana-1956	43	33	:	:	PUNCT
cana-1956	43	34	�	�	PROPN
cana-1956	43	35	⃗	⃗	PROPN
cana-1956	43	36	�	�	NOUN
cana-1956	43	37	𝜀+1∑𝑓𝑖(𝑥	𝜀+1∑𝑓𝑖(𝑥	PROPN
cana-1956	43	38	,	,	PUNCT
cana-1956	43	39	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	43	40	)	)	PUNCT
cana-1956	43	41	𝑖	𝑖	NOUN
cana-1956	44	1	=	=	PUNCT
cana-1956	44	2	𝒵𝜀+1∑((𝑥	𝒵𝜀+1∑((𝑥	NUM
cana-1956	44	3	+	+	NOUN
cana-1956	44	4	𝜀	𝜀	X
cana-1956	44	5	+	+	CCONJ
cana-1956	44	6	1	1	NUM
cana-1956	44	7	𝑝	𝑝	PROPN
cana-1956	44	8	𝑟	𝑟	NOUN
cana-1956	44	9	,	,	PUNCT
cana-1956	44	10	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	44	11	)	)	PUNCT
cana-1956	44	12	)	)	PUNCT
cana-1956	45	1	𝑖	𝑖	X
cana-1956	45	2	𝑟=1	𝑟=1	PROPN
cana-1956	45	3	𝑝	𝑝	PROPN
cana-1956	45	4	,	,	PUNCT
cana-1956	45	5	(	(	PUNCT
cana-1956	45	6	𝑥	𝑥	NOUN
cana-1956	45	7	,	,	PUNCT
cana-1956	45	8	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	45	9	)	)	PUNCT
cana-1956	45	10	∈	∈	NOUN
cana-1956	45	11	𝑄𝜀+1,𝑝.	𝑄𝜀+1,𝑝.	VERB
cana-1956	45	12	the	the	DET
cana-1956	45	13	unitary	unitary	ADJ
cana-1956	45	14	mapping	mapping	NOUN
cana-1956	45	15	between	between	ADP
cana-1956	45	16	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	45	17	)	)	PUNCT
cana-1956	45	18	and	and	CCONJ
cana-1956	45	19	ℋ𝜀+1,𝑝	ℋ𝜀+1,𝑝	PROPN
cana-1956	45	20	is	be	AUX
cana-1956	45	21	the	the	DET
cana-1956	45	22	vector	vector	NOUN
cana-1956	45	23	-	-	PUNCT
cana-1956	45	24	valued	value	VERB
cana-1956	45	25	zak	zak	PROPN
cana-1956	45	26	transform	transform	NOUN
cana-1956	45	27	,	,	PUNCT
cana-1956	45	28	subject	subject	ADJ
cana-1956	45	29	to	to	ADP
cana-1956	45	30	normalization	normalization	NOUN
cana-1956	45	31	.	.	PUNCT
cana-1956	46	1	additionally	additionally	ADV
cana-1956	46	2	,	,	PUNCT
cana-1956	46	3	note	note	VERB
cana-1956	46	4	:	:	PUNCT
cana-1956	46	5	∑𝐴𝑟	∑𝐴𝑟	PROPN
cana-1956	46	6	𝑠(𝑥	𝑠(𝑥	PROPN
cana-1956	46	7	,	,	PUNCT
cana-1956	46	8	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	46	9	)	)	PUNCT
cana-1956	46	10	𝑖	𝑖	SYM
cana-1956	47	1	=	=	PUNCT
cana-1956	47	2	(	(	PUNCT
cana-1956	47	3	𝜀	𝜀	X
cana-1956	47	4	+	+	PROPN
cana-1956	47	5	1)(∑∑𝒵𝜀+1𝑔	1)(∑∑𝒵𝜀+1𝑔	NUM
cana-1956	47	6	(	(	PUNCT
cana-1956	47	7	𝑥	𝑥	PROPN
cana-1956	47	8	+	+	X
cana-1956	47	9	𝜀	𝜀	X
cana-1956	47	10	+	+	CCONJ
cana-1956	47	11	1	1	PROPN
cana-1956	47	12	𝑝	𝑝	PROPN
cana-1956	47	13	𝑠,𝑤𝑖	𝑠,𝑤𝑖	NUM
cana-1956	47	14	−	−	PROPN
cana-1956	47	15	(	(	PUNCT
cana-1956	47	16	𝜀	𝜀	X
cana-1956	47	17	−	−	NUM
cana-1956	47	18	1)𝑗	1)𝑗	NUM
cana-1956	47	19	)	)	PUNCT
cana-1956	47	20	̅̅	̅̅	PROPN
cana-1956	47	21	̅̅	̅̅	PROPN
cana-1956	47	22	̅̅	̅̅	PROPN
cana-1956	47	23	̅̅	̅̅	PROPN
cana-1956	47	24	̅̅	̅̅	PROPN
cana-1956	47	25	̅̅	̅̅	PROPN
cana-1956	47	26	̅̅	̅̅	PROPN
cana-1956	47	27	̅̅	̅̅	PROPN
cana-1956	47	28	̅̅	̅̅	PROPN
cana-1956	47	29	̅̅	̅̅	PROPN
cana-1956	47	30	̅̅	̅̅	PROPN
cana-1956	47	31	̅̅	̅̅	PROPN
cana-1956	47	32	̅̅	̅̅	PROPN
cana-1956	47	33	̅̅	̅̅	PROPN
cana-1956	47	34	̅̅	̅̅	PROPN
cana-1956	47	35	̅̅	̅̅	PROPN
cana-1956	47	36	̅̅	̅̅	PROPN
cana-1956	47	37	̅̅	̅̅	PROPN
cana-1956	47	38	̅̅	̅̅	PROPN
cana-1956	47	39	̅̅	̅̅	PROPN
cana-1956	47	40	̅̅	̅̅	PROPN
cana-1956	47	41	̅	̅	PROPN
cana-1956	47	42	𝑞−1	𝑞−1	PROPN
cana-1956	47	43	𝑗=0	𝑗=0	PROPN
cana-1956	48	1	𝒵𝜀+1𝑔	𝒵𝜀+1𝑔	PROPN
cana-1956	48	2	(	(	PUNCT
cana-1956	48	3	𝑥	𝑥	PROPN
cana-1956	48	4	+	+	X
cana-1956	49	1	𝜀	𝜀	X
cana-1956	49	2	+	+	CCONJ
cana-1956	49	3	1	1	PROPN
cana-1956	49	4	𝑝	𝑝	PROPN
cana-1956	49	5	𝑠,𝑤𝑖	𝑠,𝑤𝑖	NUM
cana-1956	49	6	−	−	PROPN
cana-1956	49	7	(	(	PUNCT
cana-1956	49	8	𝜀	𝜀	X
cana-1956	49	9	𝑖	𝑖	SYM
cana-1956	49	10	−	−	NOUN
cana-1956	49	11	1)𝑗	1)𝑗	NUM
cana-1956	49	12	)	)	PUNCT
cana-1956	49	13	𝑒2𝜋𝑖𝑗(𝑟−𝑠	𝑒2𝜋𝑖𝑗(𝑟−𝑠	NOUN
cana-1956	49	14	)	)	PUNCT
cana-1956	50	1	𝑞⁄	𝑞⁄	PROPN
cana-1956	50	2	)	)	PUNCT
cana-1956	50	3	,	,	PUNCT
cana-1956	50	4	as	as	ADP
cana-1956	50	5	an	an	DET
cana-1956	50	6	example	example	NOUN
cana-1956	50	7	,	,	PUNCT
cana-1956	50	8	take	take	VERB
cana-1956	50	9	into	into	ADP
cana-1956	50	10	consideration	consideration	NOUN
cana-1956	50	11	the	the	DET
cana-1956	50	12	𝑝	𝑝	PROPN
cana-1956	50	13	×	×	NOUN
cana-1956	50	14	𝑝	𝑝	PROPN
cana-1956	50	15	matrix	matrix	NOUN
cana-1956	50	16	function	function	NOUN
cana-1956	50	17	𝒜(𝑥,𝑤𝑖)=∑	𝒜(𝑥,𝑤𝑖)=∑	NOUN
cana-1956	50	18	(	(	PUNCT
cana-1956	50	19	(	(	PUNCT
cana-1956	50	20	𝐴𝑟	𝐴𝑟	PROPN
cana-1956	50	21	𝑠(𝑥	𝑠(𝑥	PROPN
cana-1956	50	22	,	,	PUNCT
cana-1956	50	23	𝑤𝑖))𝑟,𝑠=0	𝑤𝑖))𝑟,𝑠=0	ADJ
cana-1956	50	24	𝑝−1	𝑝−1	PROPN
cana-1956	50	25	)	)	PUNCT
cana-1956	50	26	𝑖	𝑖	SYM
cana-1956	50	27	,	,	PUNCT
cana-1956	50	28	where	where	SCONJ
cana-1956	50	29	(	(	PUNCT
cana-1956	50	30	𝑥	𝑥	NOUN
cana-1956	50	31	,	,	PUNCT
cana-1956	50	32	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	50	33	)	)	PUNCT
cana-1956	50	34	∈	∈	PROPN
cana-1956	50	35	𝑄𝜀+1,𝑝.	𝑄𝜀+1,𝑝.	NOUN
cana-1956	50	36	theorem	theorem	NOUN
cana-1956	50	37	(	(	PUNCT
cana-1956	50	38	1.1	1.1	NUM
cana-1956	50	39	):	):	PUNCT
cana-1956	50	40	zibulski	zibulski	ADJ
cana-1956	50	41	and	and	CCONJ
cana-1956	50	42	zeevi	zeevi	PROPN
cana-1956	50	43	.	.	PUNCT
cana-1956	51	1	we	we	PRON
cana-1956	51	2	obtain	obtain	VERB
cana-1956	51	3	:	:	PUNCT
cana-1956	51	4	�	�	PROPN
cana-1956	51	5	⃗	⃗	NOUN
cana-1956	51	6	�	�	PROPN
cana-1956	51	7	𝜀+1∑	𝜀+1∑	PROPN
cana-1956	51	8	(	(	PUNCT
cana-1956	51	9	(	(	PUNCT
cana-1956	51	10	𝑆𝑔,𝜀+1,(𝜀−1)𝑓𝑖)(𝑥	𝑆𝑔,𝜀+1,(𝜀−1)𝑓𝑖)(𝑥	NOUN
cana-1956	51	11	,	,	PUNCT
cana-1956	51	12	𝑤𝑖))𝑖	𝑤𝑖))𝑖	PROPN
cana-1956	51	13	=	=	PUNCT
cana-1956	51	14	∑	∑	PROPN
cana-1956	51	15	(	(	PUNCT
cana-1956	51	16	𝒜(𝑥,𝑤𝑖)	𝒜(𝑥,𝑤𝑖)	PROPN
cana-1956	51	17	�	�	NOUN
cana-1956	51	18	⃗	⃗	NOUN
cana-1956	51	19	�	�	NOUN
cana-1956	51	20	(𝜀+1)𝑓𝑖(𝑥	(𝜀+1)𝑓𝑖(𝑥	NOUN
cana-1956	51	21	,	,	PUNCT
cana-1956	51	22	𝑤𝑖))𝑖	𝑤𝑖))𝑖	PROPN
cana-1956	51	23	,	,	PUNCT
cana-1956	51	24	for	for	ADP
cana-1956	51	25	nearly	nearly	ADV
cana-1956	51	26	all	all	PRON
cana-1956	51	27	(	(	PUNCT
cana-1956	51	28	𝑥	𝑥	NOUN
cana-1956	51	29	,	,	PUNCT
cana-1956	51	30	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	51	31	)	)	PUNCT
cana-1956	51	32	∈	∈	PROPN
cana-1956	51	33	𝑄𝜀+1,𝑝	𝑄𝜀+1,𝑝	NOUN
cana-1956	51	34	,	,	PUNCT
cana-1956	51	35	based	base	VERB
cana-1956	51	36	on	on	ADP
cana-1956	51	37	the	the	DET
cana-1956	51	38	aforementioned	aforementioned	ADJ
cana-1956	51	39	assumptions	assumption	NOUN
cana-1956	51	40	.	.	PUNCT
cana-1956	52	1	we	we	PRON
cana-1956	52	2	always	always	ADV
cana-1956	52	3	assume	assume	VERB
cana-1956	52	4	that	that	SCONJ
cana-1956	52	5	g	g	PROPN
cana-1956	52	6	is	be	AUX
cana-1956	52	7	a	a	DET
cana-1956	52	8	member	member	NOUN
cana-1956	52	9	of	of	ADP
cana-1956	52	10	the	the	DET
cana-1956	52	11	modulation	modulation	NOUN
cana-1956	52	12	space	space	NOUN
cana-1956	52	13	𝑀1(ℝ	𝑀1(ℝ	PROPN
cana-1956	52	14	)	)	PUNCT
cana-1956	52	15	,	,	PUNCT
cana-1956	52	16	in	in	ADP
cana-1956	52	17	the	the	DET
cana-1956	52	18	following	following	NOUN
cana-1956	52	19	.	.	PUNCT
cana-1956	53	1	we	we	PRON
cana-1956	53	2	merely	merely	ADV
cana-1956	53	3	restate	restate	VERB
cana-1956	53	4	the	the	DET
cana-1956	53	5	definition	definition	NOUN
cana-1956	53	6	here	here	ADV
cana-1956	53	7	,	,	PUNCT
cana-1956	53	8	and	and	CCONJ
cana-1956	53	9	readers	reader	NOUN
cana-1956	53	10	are	be	AUX
cana-1956	53	11	directed	direct	VERB
cana-1956	53	12	to	to	ADP
cana-1956	53	13	[	[	X
cana-1956	53	14	4	4	X
cana-1956	53	15	]	]	PUNCT
cana-1956	53	16	for	for	ADP
cana-1956	53	17	a	a	DET
cana-1956	53	18	more	more	ADV
cana-1956	53	19	thorough	thorough	ADJ
cana-1956	53	20	presentation	presentation	NOUN
cana-1956	53	21	.	.	PUNCT
cana-1956	54	1	examine	examine	VERB
cana-1956	54	2	the	the	DET
cana-1956	54	3	immediate	immediate	ADJ
cana-1956	54	4	fourier	fourier	NOUN
cana-1956	54	5	transform	transform	NOUN
cana-1956	54	6	,	,	PUNCT
cana-1956	54	7	where	where	SCONJ
cana-1956	54	8	𝑓𝑖	𝑓𝑖	PROPN
cana-1956	54	9	acts	act	VERB
cana-1956	54	10	as	as	ADP
cana-1956	54	11	the	the	DET
cana-1956	54	12	aperture	aperture	ADJ
cana-1956	54	13	function	function	NOUN
cana-1956	54	14	,	,	PUNCT
cana-1956	54	15	given	give	VERB
cana-1956	54	16	a	a	DET
cana-1956	54	17	functions	function	NOUN
cana-1956	54	18	𝑓𝑖	𝑓𝑖	VERB
cana-1956	54	19	in	in	ADP
cana-1956	54	20	the	the	DET
cana-1956	54	21	schwartz	schwartz	PROPN
cana-1956	54	22	class	class	PROPN
cana-1956	54	23	𝑆(ℝ	𝑆(ℝ	PROPN
cana-1956	54	24	)	)	PUNCT
cana-1956	54	25	∑(𝑉𝑓𝑖𝑔(𝑥,𝑤𝑖	∑(𝑉𝑓𝑖𝑔(𝑥,𝑤𝑖	NOUN
cana-1956	54	26	)	)	PUNCT
cana-1956	54	27	)	)	PUNCT
cana-1956	55	1	𝑖	𝑖	X
cana-1956	56	1	=	=	PUNCT
cana-1956	56	2	lim	lim	PROPN
cana-1956	56	3	𝑏→−∞	𝑏→−∞	PROPN
cana-1956	56	4	(	(	PUNCT
cana-1956	56	5	∑(∫	∑(∫	NUM
cana-1956	56	6	𝑔(𝑡	𝑔(𝑡	NUM
cana-1956	56	7	)	)	PUNCT
cana-1956	57	1	∞	∞	NUM
cana-1956	57	2	𝑏	𝑏	NOUN
cana-1956	57	3	𝑓𝑖(𝑥	𝑓𝑖(𝑥	NUM
cana-1956	57	4	−	−	NOUN
cana-1956	57	5	𝑡)𝑒	𝑡)𝑒	ADJ
cana-1956	57	6	2𝜋𝑖𝑤𝑖𝑡𝑑𝑡	2𝜋𝑖𝑤𝑖𝑡𝑑𝑡	NUM
cana-1956	57	7	)	)	PUNCT
cana-1956	57	8	𝑖	𝑖	NOUN
cana-1956	57	9	)	)	PUNCT
cana-1956	57	10	.	.	PUNCT
cana-1956	58	1	definition	definition	NOUN
cana-1956	58	2	(	(	PUNCT
cana-1956	58	3	1.2	1.2	NUM
cana-1956	58	4	)	)	PUNCT
cana-1956	58	5	:	:	PUNCT
cana-1956	58	6	the	the	DET
cana-1956	58	7	modulation	modulation	NOUN
cana-1956	58	8	space	space	NOUN
cana-1956	58	9	𝑀1(ℝ	𝑀1(ℝ	NOUN
cana-1956	58	10	)	)	PUNCT
cana-1956	58	11	includes	include	VERB
cana-1956	58	12	functions	function	NOUN
cana-1956	58	13	𝑔	𝑔	PROPN
cana-1956	58	14	∈	∈	PROPN
cana-1956	58	15	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	58	16	)	)	PUNCT
cana-1956	58	17	with	with	ADP
cana-1956	58	18	∑(∫∫|𝑉𝑓𝑖𝑔(𝑥,𝑤𝑖)|	∑(∫∫|𝑉𝑓𝑖𝑔(𝑥,𝑤𝑖)|	NOUN
cana-1956	58	19	ℝ	ℝ	PROPN
cana-1956	58	20	ℝ	ℝ	PROPN
cana-1956	58	21	𝑑𝑥𝑑𝑤𝑖	𝑑𝑥𝑑𝑤𝑖	ADV
cana-1956	58	22	)	)	PUNCT
cana-1956	58	23	𝑖	𝑖	X
cana-1956	58	24	<	<	X
cana-1956	58	25	∞	∞	PROPN
cana-1956	58	26	for	for	ADP
cana-1956	58	27	certain	certain	ADJ
cana-1956	58	28	(	(	PUNCT
cana-1956	58	29	or	or	CCONJ
cana-1956	58	30	all	all	PRON
cana-1956	58	31	)	)	PUNCT
cana-1956	58	32	non	non	ADJ
cana-1956	58	33	-	-	ADJ
cana-1956	58	34	trivial	trivial	ADJ
cana-1956	58	35	functions	function	NOUN
cana-1956	58	36	,	,	PUNCT
cana-1956	58	37	𝑓𝑖is	𝑓𝑖is	NOUN
cana-1956	58	38	in	in	ADP
cana-1956	58	39	𝒮(ℝ	𝒮(ℝ	PROPN
cana-1956	58	40	)	)	PUNCT
cana-1956	58	41	.	.	PUNCT
cana-1956	59	1	the	the	DET
cana-1956	59	2	following	follow	VERB
cana-1956	59	3	statement	statement	NOUN
cana-1956	59	4	is	be	AUX
cana-1956	59	5	the	the	DET
cana-1956	59	6	result	result	NOUN
cana-1956	59	7	of	of	ADP
cana-1956	59	8	theorem	theorem	NOUN
cana-1956	59	9	(	(	PUNCT
cana-1956	59	10	1.1	1.1	NUM
cana-1956	59	11	)	)	PUNCT
cana-1956	60	1	[	[	X
cana-1956	60	2	1	1	NUM
cana-1956	60	3	]	]	PUNCT
cana-1956	60	4	.	.	PUNCT
cana-1956	61	1	communications	communication	NOUN
cana-1956	61	2	on	on	ADP
cana-1956	61	3	applied	apply	VERB
cana-1956	61	4	nonlinear	nonlinear	ADJ
cana-1956	61	5	analysis	analysis	NOUN
cana-1956	61	6	issn	issn	NOUN
cana-1956	61	7	:	:	PUNCT
cana-1956	61	8	1074	1074	NUM
cana-1956	61	9	-	-	PUNCT
cana-1956	61	10	133x	133x	NUM
cana-1956	61	11	vol	vol	NOUN
cana-1956	61	12	32	32	NUM
cana-1956	61	13	no	no	NOUN
cana-1956	61	14	.	.	NOUN
cana-1956	61	15	3	3	NUM
cana-1956	61	16	(	(	PUNCT
cana-1956	61	17	2025	2025	NUM
cana-1956	61	18	)	)	PUNCT
cana-1956	61	19	232	232	NUM
cana-1956	62	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	62	2	corollary	corollary	NOUN
cana-1956	62	3	(	(	PUNCT
cana-1956	62	4	1.3	1.3	NUM
cana-1956	62	5	)	)	PUNCT
cana-1956	62	6	:	:	PUNCT
cana-1956	62	7	the	the	DET
cana-1956	62	8	gabor	gabor	PROPN
cana-1956	62	9	set	set	VERB
cana-1956	62	10	𝒢(𝑔,∧	𝒢(𝑔,∧	PROPN
cana-1956	62	11	)	)	PUNCT
cana-1956	62	12	is	be	AUX
cana-1956	62	13	a	a	DET
cana-1956	62	14	structure	structure	NOUN
cana-1956	62	15	in	in	ADP
cana-1956	62	16	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	62	17	)	)	PUNCT
cana-1956	62	18	only	only	ADV
cana-1956	62	19	if	if	SCONJ
cana-1956	62	20	:	:	PUNCT
cana-1956	62	21	𝑑𝑒𝑡∑𝒜(𝑥,𝑤𝑖	𝑑𝑒𝑡∑𝒜(𝑥,𝑤𝑖	X
cana-1956	62	22	)	)	PUNCT
cana-1956	62	23	𝑖	𝑖	SYM
cana-1956	62	24	≠	≠	PROPN
cana-1956	62	25	0	0	NUM
cana-1956	62	26	,	,	PUNCT
cana-1956	62	27	(	(	PUNCT
cana-1956	62	28	𝑥	𝑥	NOUN
cana-1956	62	29	,	,	PUNCT
cana-1956	62	30	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	62	31	)	)	PUNCT
cana-1956	62	32	∈	∈	PROPN
cana-1956	62	33	𝑄𝜀+1,𝑝.	𝑄𝜀+1,𝑝.	NOUN
cana-1956	62	34	(	(	PUNCT
cana-1956	62	35	3	3	X
cana-1956	62	36	)	)	PUNCT
cana-1956	62	37	to	to	PART
cana-1956	62	38	make	make	VERB
cana-1956	62	39	this	this	DET
cana-1956	62	40	condition	condition	NOUN
cana-1956	62	41	more	more	ADV
cana-1956	62	42	understandable	understandable	ADJ
cana-1956	62	43	,	,	PUNCT
cana-1956	62	44	we	we	PRON
cana-1956	62	45	factorize	factorize	VERB
cana-1956	62	46	matrix	matrix	NOUN
cana-1956	62	47	𝒜	𝒜	NOUN
cana-1956	62	48	a.	a.	NOUN
cana-1956	62	49	consider	consider	VERB
cana-1956	62	50	column	column	NOUN
cana-1956	62	51	vectors	vector	NOUN
cana-1956	62	52	:	:	PUNCT
cana-1956	62	53	∑𝑋𝑗(𝑥	∑𝑋𝑗(𝑥	PUNCT
cana-1956	62	54	,	,	PUNCT
cana-1956	62	55	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	62	56	)	)	PUNCT
cana-1956	62	57	𝑖	𝑖	PUNCT
cana-1956	63	1	=	=	PRON
cana-1956	63	2	∑(𝑋𝑟	∑(𝑋𝑟	X
cana-1956	63	3	𝑗(𝑥	𝑗(𝑥	X
cana-1956	63	4	,	,	PUNCT
cana-1956	63	5	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	63	6	)	)	PUNCT
cana-1956	63	7	)	)	PUNCT
cana-1956	64	1	𝑟=0	𝑟=0	PUNCT
cana-1956	65	1	𝑝−1	𝑝−1	PROPN
cana-1956	65	2	𝑖	𝑖	SYM
cana-1956	65	3	,	,	PUNCT
cana-1956	65	4	𝑗	𝑗	NOUN
cana-1956	65	5	=	=	SYM
cana-1956	65	6	0,1	0,1	NUM
cana-1956	65	7	,	,	PUNCT
cana-1956	65	8	…	…	PUNCT
cana-1956	65	9	…	…	PUNCT
cana-1956	65	10	.	.	PUNCT
cana-1956	65	11	,	,	PUNCT
cana-1956	65	12	𝑞	𝑞	X
cana-1956	65	13	−	−	PROPN
cana-1956	65	14	1	1	NUM
cana-1956	65	15	in	in	ADP
cana-1956	65	16	which	which	PRON
cana-1956	65	17	:	:	PUNCT
cana-1956	65	18	∑𝑋𝑟	∑𝑋𝑟	X
cana-1956	65	19	𝑗(𝑥	𝑗(𝑥	X
cana-1956	65	20	,	,	PUNCT
cana-1956	65	21	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	65	22	)	)	PUNCT
cana-1956	65	23	𝑖	𝑖	NOUN
cana-1956	65	24	=	=	PRON
cana-1956	65	25	𝒵(𝜀+1)𝑔∑(𝑡	𝒵(𝜀+1)𝑔∑(𝑡	X
cana-1956	65	26	+	+	CCONJ
cana-1956	65	27	(	(	PUNCT
cana-1956	65	28	𝜀	𝜀	X
cana-1956	65	29	+	+	CCONJ
cana-1956	65	30	1)𝑟	1)𝑟	NUM
cana-1956	65	31	𝑝	𝑝	NOUN
cana-1956	65	32	,	,	PUNCT
cana-1956	65	33	𝑤𝑖	𝑤𝑖	ADP
cana-1956	65	34	−	−	PROPN
cana-1956	65	35	(	(	PUNCT
cana-1956	65	36	𝜀	𝜀	X
cana-1956	65	37	−	−	NOUN
cana-1956	65	38	1)𝑗	1)𝑗	NUM
cana-1956	65	39	)	)	PUNCT
cana-1956	65	40	𝑒	𝑒	ADP
cana-1956	65	41	2𝜋𝑖𝑗𝑟	2𝜋𝑖𝑗𝑟	NUM
cana-1956	65	42	𝑞⁄	𝑞⁄	SYM
cana-1956	65	43	𝑖	𝑖	PROPN
cana-1956	65	44	,	,	PUNCT
cana-1956	65	45	(	(	PUNCT
cana-1956	65	46	4	4	X
cana-1956	65	47	)	)	PUNCT
cana-1956	65	48	including	include	VERB
cana-1956	65	49	𝑝	𝑝	NUM
cana-1956	65	50	×	×	NOUN
cana-1956	65	51	𝑞	𝑞	X
cana-1956	65	52	matrix	matrix	NOUN
cana-1956	65	53	:	:	PUNCT
cana-1956	65	54	∑𝒬(𝑡,𝑤𝑖	∑𝒬(𝑡,𝑤𝑖	NOUN
cana-1956	65	55	)	)	PUNCT
cana-1956	65	56	𝑖	𝑖	SYM
cana-1956	66	1	=	=	SYM
cana-1956	67	1	(	(	PUNCT
cana-1956	67	2	𝑋𝑗	𝑋𝑗	PROPN
cana-1956	67	3	)	)	PUNCT
cana-1956	67	4	𝑗=0	𝑗=0	PUNCT
cana-1956	68	1	𝑞−1	𝑞−1	PROPN
cana-1956	68	2	=	=	PUNCT
cana-1956	68	3	∑(𝒵(𝜀+1)𝑔	∑(𝒵(𝜀+1)𝑔	PROPN
cana-1956	68	4	(	(	PUNCT
cana-1956	68	5	𝑡	𝑡	X
cana-1956	68	6	+	+	X
cana-1956	68	7	(	(	PUNCT
cana-1956	68	8	𝜀	𝜀	X
cana-1956	68	9	+	+	CCONJ
cana-1956	68	10	1)𝑟	1)𝑟	NUM
cana-1956	68	11	𝑝	𝑝	NOUN
cana-1956	68	12	,	,	PUNCT
cana-1956	68	13	𝑤𝑖	𝑤𝑖	ADP
cana-1956	68	14	−	−	PROPN
cana-1956	68	15	(	(	PUNCT
cana-1956	68	16	𝜀	𝜀	X
cana-1956	68	17	−	−	NOUN
cana-1956	68	18	1)𝑗	1)𝑗	NUM
cana-1956	68	19	)	)	PUNCT
cana-1956	68	20	𝑒	𝑒	ADP
cana-1956	68	21	2𝜋𝑖𝑗𝑟	2𝜋𝑖𝑗𝑟	NUM
cana-1956	68	22	𝑞⁄	𝑞⁄	NUM
cana-1956	68	23	)	)	PUNCT
cana-1956	68	24	𝑟=0,𝑗=0	𝑟=0,𝑗=0	NOUN
cana-1956	69	1	𝑝−1,𝑞−1	𝑝−1,𝑞−1	PROPN
cana-1956	69	2	𝑖	𝑖	PROPN
cana-1956	69	3	clearly	clearly	ADV
cana-1956	69	4	∑𝒜(𝑥,𝑤𝑖	∑𝒜(𝑥,𝑤𝑖	PROPN
cana-1956	69	5	)	)	PUNCT
cana-1956	69	6	𝑖	𝑖	VERB
cana-1956	70	1	=	=	NOUN
cana-1956	70	2	∑𝒬(𝑥,𝑤𝑖)𝒬	∑𝒬(𝑥,𝑤𝑖)𝒬	VERB
cana-1956	70	3	𝑇(𝑥	𝑇(𝑥	NOUN
cana-1956	70	4	,	,	PUNCT
cana-1956	70	5	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	70	6	)	)	PUNCT
cana-1956	70	7	𝑖	𝑖	X
cana-1956	70	8	.	.	PUNCT
cana-1956	71	1	here	here	ADV
cana-1956	71	2	,	,	PUNCT
cana-1956	71	3	as	as	SCONJ
cana-1956	71	4	always,𝒬𝑇t	always,𝒬𝑇t	PROPN
cana-1956	71	5	represents	represent	VERB
cana-1956	71	6	the	the	DET
cana-1956	71	7	adjoint	adjoint	NOUN
cana-1956	71	8	matrix	matrix	NOUN
cana-1956	71	9	of	of	ADP
cana-1956	71	10	𝒬.	𝒬.	PROPN
cana-1956	71	11	�	�	PROPN
cana-1956	71	12	⃗	⃗	NOUN
cana-1956	71	13	�	�	NOUN
cana-1956	71	14	(𝜀+1)∑𝒜(𝑥,𝑤𝑖)𝑓𝑖(𝑥	(𝜀+1)∑𝒜(𝑥,𝑤𝑖)𝑓𝑖(𝑥	SYM
cana-1956	71	15	,	,	PUNCT
cana-1956	71	16	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	71	17	)	)	PUNCT
cana-1956	71	18	𝑖	𝑖	PRON
cana-1956	72	1	=	=	NOUN
cana-1956	72	2	∑(∑〈𝑋𝑗(𝑥	∑(∑〈𝑋𝑗(𝑥	NOUN
cana-1956	72	3	,	,	PUNCT
cana-1956	72	4	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	72	5	)	)	PUNCT
cana-1956	72	6	,	,	PUNCT
cana-1956	72	7	�	�	PROPN
cana-1956	72	8	⃗	⃗	NOUN
cana-1956	72	9	�	�	NOUN
cana-1956	72	10	(𝜀+1)𝑓𝑖(𝑥	(𝜀+1)𝑓𝑖(𝑥	ADJ
cana-1956	72	11	,	,	PUNCT
cana-1956	72	12	𝑤𝑖	𝑤𝑖	PRON
cana-1956	72	13	)	)	PUNCT
cana-1956	72	14	〉	〉	NOUN
cana-1956	73	1	𝑞−1	𝑞−1	PROPN
cana-1956	73	2	𝑗=0	𝑗=0	PUNCT
cana-1956	74	1	〈	〈	X
cana-1956	74	2	𝑋𝑗(𝑥	𝑋𝑗(𝑥	PROPN
cana-1956	74	3	,	,	PUNCT
cana-1956	74	4	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	74	5	)	)	PUNCT
cana-1956	74	6	,	,	PUNCT
cana-1956	74	7	�	�	PROPN
cana-1956	74	8	⃗	⃗	NOUN
cana-1956	74	9	�	�	NOUN
cana-1956	74	10	(𝜀+1)𝑓𝑖(𝑥	(𝜀+1)𝑓𝑖(𝑥	ADJ
cana-1956	74	11	,	,	PUNCT
cana-1956	74	12	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	74	13	)	)	PUNCT
cana-1956	74	14	〉	〉	NOUN
cana-1956	74	15	)	)	PUNCT
cana-1956	74	16	𝑖	𝑖	NOUN
cana-1956	74	17	,	,	PUNCT
cana-1956	74	18	for	for	ADP
cana-1956	74	19	each	each	DET
cana-1956	74	20	(	(	PUNCT
cana-1956	74	21	𝑥	𝑥	PROPN
cana-1956	74	22	,	,	PUNCT
cana-1956	74	23	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	74	24	)	)	PUNCT
cana-1956	74	25	∈	∈	PROPN
cana-1956	74	26	𝑄𝜀+1,𝑝	𝑄𝜀+1,𝑝	PROPN
cana-1956	74	27	,	,	PUNCT
cana-1956	74	28	the	the	DET
cana-1956	74	29	𝑋𝑗(𝑥	𝑋𝑗(𝑥	NOUN
cana-1956	74	30	,	,	PUNCT
cana-1956	74	31	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	74	32	)	)	PUNCT
cana-1956	74	33	,	,	PUNCT
cana-1956	74	34	𝑗	𝑗	NOUN
cana-1956	74	35	=	=	SYM
cana-1956	74	36	0,1	0,1	NUM
cana-1956	74	37	,	,	PUNCT
cana-1956	74	38	…	…	PUNCT
cana-1956	74	39	…	…	PUNCT
cana-1956	74	40	.	.	PUNCT
cana-1956	74	41	.	.	PUNCT
cana-1956	75	1	𝑞	𝑞	X
cana-1956	75	2	−	−	PROPN
cana-1956	75	3	1	1	NUM
cana-1956	75	4	,	,	PUNCT
cana-1956	75	5	must	must	AUX
cana-1956	75	6	span	span	VERB
cana-1956	75	7	the	the	DET
cana-1956	75	8	entire	entire	ADJ
cana-1956	75	9	ℂ𝑝,to	ℂ𝑝,to	PROPN
cana-1956	75	10	meet	meet	VERB
cana-1956	75	11	condition	condition	NOUN
cana-1956	75	12	(	(	PUNCT
cana-1956	75	13	3	3	NUM
cana-1956	75	14	)	)	PUNCT
cana-1956	75	15	.	.	PUNCT
cana-1956	76	1	corollary	corollary	ADJ
cana-1956	76	2	(	(	PUNCT
cana-1956	76	3	1.4	1.4	NUM
cana-1956	76	4	):	):	PUNCT
cana-1956	76	5	let	let	VERB
cana-1956	76	6	𝜀2	𝜀2	NOUN
cana-1956	76	7	=	=	SYM
cana-1956	76	8	𝑝	𝑝	PROPN
cana-1956	76	9	𝑞	𝑞	NOUN
cana-1956	76	10	+	+	CCONJ
cana-1956	76	11	1	1	NUM
cana-1956	76	12	∈	∈	PROPN
cana-1956	76	13	ℚ	ℚ	NOUN
cana-1956	76	14	,	,	PUNCT
cana-1956	76	15	and	and	CCONJ
cana-1956	76	16	𝑔	𝑔	PROPN
cana-1956	76	17	∈	∈	PROPN
cana-1956	76	18	𝑀1(ℝ	𝑀1(ℝ	PROPN
cana-1956	76	19	)	)	PUNCT
cana-1956	76	20	.	.	PUNCT
cana-1956	77	1	if	if	SCONJ
cana-1956	77	2	𝒢(𝑔,∧	𝒢(𝑔,∧	PROPN
cana-1956	77	3	)	)	PUNCT
cana-1956	77	4	is	be	AUX
cana-1956	77	5	a	a	DET
cana-1956	77	6	farm	farm	NOUN
cana-1956	77	7	in	in	ADP
cana-1956	77	8	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	77	9	)	)	PUNCT
cana-1956	77	10	,	,	PUNCT
cana-1956	77	11	rank	rank	VERB
cana-1956	77	12	∑	∑	SYM
cana-1956	77	13	𝒬(𝑥,𝑤𝑖)𝑖	𝒬(𝑥,𝑤𝑖)𝑖	PROPN
cana-1956	77	14	=	=	PRON
cana-1956	77	15	must	must	AUX
cana-1956	77	16	be	be	AUX
cana-1956	77	17	equal	equal	ADJ
cana-1956	77	18	to	to	ADP
cana-1956	77	19	𝑝	𝑝	NOUN
cana-1956	77	20	for	for	ADP
cana-1956	77	21	any	any	DET
cana-1956	77	22	(	(	PUNCT
cana-1956	77	23	𝑥	𝑥	NOUN
cana-1956	77	24	,	,	PUNCT
cana-1956	77	25	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	77	26	)	)	PUNCT
cana-1956	77	27	)	)	PUNCT
cana-1956	78	1	in	in	ADP
cana-1956	78	2	𝑄𝜀+1,𝑝.	𝑄𝜀+1,𝑝.	ADV
cana-1956	78	3	in	in	ADP
cana-1956	78	4	the	the	DET
cana-1956	78	5	following	follow	VERB
cana-1956	78	6	section	section	NOUN
cana-1956	78	7	,	,	PUNCT
cana-1956	78	8	we	we	PRON
cana-1956	78	9	apply	apply	VERB
cana-1956	78	10	this	this	DET
cana-1956	78	11	requirement	requirement	NOUN
cana-1956	78	12	to	to	PART
cana-1956	78	13	investigate	investigate	VERB
cana-1956	78	14	gabor	gabor	PROPN
cana-1956	78	15	systems	system	NOUN
cana-1956	78	16	formed	form	VERB
cana-1956	78	17	by	by	ADP
cana-1956	78	18	odd	odd	ADJ
cana-1956	78	19	functions	function	NOUN
cana-1956	78	20	.	.	PUNCT
cana-1956	79	1	3	3	X
cana-1956	79	2	.	.	X
cana-1956	79	3	the	the	DET
cana-1956	79	4	odd	odd	ADJ
cana-1956	79	5	functions	function	NOUN
cana-1956	79	6	that	that	PRON
cana-1956	79	7	is	be	AUX
cana-1956	79	8	generate	generate	VERB
cana-1956	79	9	gabor	gabor	NOUN
cana-1956	79	10	frames	frame	VERB
cana-1956	79	11	the	the	DET
cana-1956	79	12	next	next	ADJ
cana-1956	79	13	part	part	NOUN
cana-1956	79	14	includes	include	VERB
cana-1956	79	15	what	what	PRON
cana-1956	79	16	comes	come	VERB
cana-1956	79	17	next	next	ADV
cana-1956	79	18	.	.	PUNCT
cana-1956	80	1	theorem	theorem	NOUN
cana-1956	80	2	(	(	PUNCT
cana-1956	80	3	3.1	3.1	NUM
cana-1956	80	4	):	):	PUNCT
cana-1956	80	5	assume	assume	VERB
cana-1956	80	6	𝑔	𝑔	PROPN
cana-1956	80	7	is	be	AUX
cana-1956	80	8	an	an	DET
cana-1956	80	9	odd	odd	ADJ
cana-1956	80	10	function	function	NOUN
cana-1956	80	11	in	in	ADP
cana-1956	80	12	𝑀1(ℝ	𝑀1(ℝ	NOUN
cana-1956	80	13	)	)	PUNCT
cana-1956	80	14	)	)	PUNCT
cana-1956	80	15	and	and	CCONJ
cana-1956	80	16	𝜀2	𝜀2	PROPN
cana-1956	80	17	=	=	SYM
cana-1956	80	18	2𝑛−1	2𝑛−1	NUM
cana-1956	80	19	𝑛	𝑛	NOUN
cana-1956	80	20	,	,	PUNCT
cana-1956	80	21	where	where	SCONJ
cana-1956	80	22	𝑛	𝑛	PRON
cana-1956	80	23	=	=	SYM
cana-1956	80	24	1,2	1,2	NUM
cana-1956	80	25	,	,	PUNCT
cana-1956	80	26	…	…	PUNCT
cana-1956	80	27	…	…	PUNCT
cana-1956	80	28	,	,	PUNCT
cana-1956	80	29	then	then	ADV
cana-1956	80	30	,	,	PUNCT
cana-1956	80	31	𝒢(𝑔,∧	𝒢(𝑔,∧	PROPN
cana-1956	80	32	)	)	PUNCT
cana-1956	80	33	does	do	AUX
cana-1956	80	34	not	not	PART
cana-1956	80	35	constitute	constitute	VERB
cana-1956	80	36	a	a	DET
cana-1956	80	37	frame	frame	NOUN
cana-1956	80	38	in	in	ADP
cana-1956	80	39	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	80	40	)	)	PUNCT
cana-1956	80	41	.	.	PUNCT
cana-1956	81	1	proof	proof	NOUN
cana-1956	81	2	:	:	PUNCT
cana-1956	81	3	it	it	PRON
cana-1956	81	4	is	be	AUX
cana-1956	81	5	sufficient	sufficient	ADJ
cana-1956	81	6	to	to	PART
cana-1956	81	7	establish	establish	VERB
cana-1956	81	8	that	that	SCONJ
cana-1956	81	9	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	ADJ
cana-1956	81	10	𝒬(0,0	𝒬(0,0	NOUN
cana-1956	81	11	)	)	PUNCT
cana-1956	81	12	<	<	X
cana-1956	81	13	2𝑛	2𝑛	PROPN
cana-1956	82	1	−	−	NOUN
cana-1956	82	2	1	1	NUM
cana-1956	82	3	(	(	PUNCT
cana-1956	82	4	5	5	NUM
cana-1956	82	5	)	)	PUNCT
cana-1956	82	6	the	the	DET
cana-1956	82	7	conclusion	conclusion	NOUN
cana-1956	82	8	follows	follow	VERB
cana-1956	82	9	from	from	ADP
cana-1956	82	10	theorem	theorem	NOUN
cana-1956	82	11	(	(	PUNCT
cana-1956	82	12	1.1	1.1	NUM
cana-1956	82	13	)	)	PUNCT
cana-1956	83	1	[	[	X
cana-1956	83	2	2	2	NUM
cana-1956	83	3	]	]	PUNCT
cana-1956	83	4	.	.	PUNCT
cana-1956	84	1	equation	equation	NOUN
cana-1956	84	2	(	(	PUNCT
cana-1956	84	3	5	5	NUM
cana-1956	84	4	)	)	PUNCT
cana-1956	84	5	is	be	AUX
cana-1956	84	6	calculated	calculate	VERB
cana-1956	84	7	from	from	ADP
cana-1956	84	8	the	the	DET
cana-1956	84	9	findings	finding	NOUN
cana-1956	84	10	that	that	PRON
cana-1956	84	11	for	for	ADP
cana-1956	84	12	odd	odd	ADJ
cana-1956	84	13	open	open	ADJ
cana-1956	84	14	spaces	space	NOUN
cana-1956	84	15	,	,	PUNCT
cana-1956	84	16	the	the	DET
cana-1956	84	17	elements	element	NOUN
cana-1956	84	18	of	of	ADP
cana-1956	84	19	the	the	DET
cana-1956	84	20	matrices	matrix	NOUN
cana-1956	84	21	𝒬(0,0)have	𝒬(0,0)have	VERB
cana-1956	84	22	additional	additional	ADJ
cana-1956	84	23	symmetries	symmetry	NOUN
cana-1956	84	24	.	.	PUNCT
cana-1956	85	1	we	we	PRON
cana-1956	85	2	consider	consider	VERB
cana-1956	85	3	𝜀	𝜀	NOUN
cana-1956	85	4	=	=	PUNCT
cana-1956	85	5	communications	communication	NOUN
cana-1956	85	6	on	on	ADP
cana-1956	85	7	applied	apply	VERB
cana-1956	85	8	nonlinear	nonlinear	ADJ
cana-1956	85	9	analysis	analysis	NOUN
cana-1956	85	10	issn	issn	NOUN
cana-1956	85	11	:	:	PUNCT
cana-1956	85	12	1074	1074	NUM
cana-1956	85	13	-	-	PUNCT
cana-1956	85	14	133x	133x	NUM
cana-1956	85	15	vol	vol	NOUN
cana-1956	85	16	32	32	NUM
cana-1956	85	17	no	no	NOUN
cana-1956	85	18	.	.	NOUN
cana-1956	85	19	3	3	NUM
cana-1956	85	20	(	(	PUNCT
cana-1956	85	21	2025	2025	NUM
cana-1956	85	22	)	)	PUNCT
cana-1956	85	23	233	233	NUM
cana-1956	85	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	85	25	0	0	NUM
cana-1956	85	26	,	,	PUNCT
cana-1956	85	27	this	this	PRON
cana-1956	85	28	may	may	AUX
cana-1956	85	29	always	always	ADV
cana-1956	85	30	be	be	AUX
cana-1956	85	31	accomplished	accomplish	VERB
cana-1956	85	32	by	by	ADP
cana-1956	85	33	appropriately	appropriately	ADV
cana-1956	85	34	scaling	scale	VERB
cana-1956	85	35	𝑔.	𝑔.	NOUN
cana-1956	86	1	we	we	PRON
cana-1956	86	2	assume	assume	VERB
cana-1956	86	3	𝜀	𝜀	PROPN
cana-1956	86	4	=	=	SYM
cana-1956	86	5	0	0	PROPN
cana-1956	86	6	,	,	PUNCT
cana-1956	86	7	which	which	PRON
cana-1956	86	8	can	can	AUX
cana-1956	86	9	be	be	AUX
cana-1956	86	10	done	do	VERB
cana-1956	86	11	by	by	ADP
cana-1956	86	12	appropriately	appropriately	ADV
cana-1956	86	13	scaling	scale	VERB
cana-1956	86	14	g.	g.	PROPN
cana-1956	86	15	lemma	lemma	PROPN
cana-1956	86	16	(	(	PUNCT
cana-1956	86	17	3.2	3.2	NUM
cana-1956	86	18	):	):	PUNCT
cana-1956	86	19	let	let	VERB
cana-1956	86	20	𝑔	𝑔	PRON
cana-1956	86	21	be	be	AUX
cana-1956	86	22	an	an	DET
cana-1956	86	23	odd	odd	ADJ
cana-1956	86	24	function	function	NOUN
cana-1956	86	25	in	in	ADP
cana-1956	86	26	𝑀1(ℝ)with	𝑀1(ℝ)with	PROPN
cana-1956	86	27	𝜀	𝜀	NOUN
cana-1956	86	28	=	=	SYM
cana-1956	86	29	0	0	NUM
cana-1956	86	30	,	,	PUNCT
cana-1956	86	31	𝜀	𝜀	X
cana-1956	86	32	=	=	SYM
cana-1956	86	33	𝑝	𝑝	PROPN
cana-1956	86	34	𝑞	𝑞	NOUN
cana-1956	86	35	+	+	CCONJ
cana-1956	86	36	1	1	NUM
cana-1956	86	37	∈	∈	PROPN
cana-1956	86	38	ℚ	ℚ	PROPN
cana-1956	86	39	and	and	CCONJ
cana-1956	86	40	𝑀𝑠	𝑀𝑠	PROPN
cana-1956	86	41	𝑗	𝑗	NOUN
cana-1956	86	42	=	=	SYM
cana-1956	86	43	𝑀𝑠	𝑀𝑠	PROPN
cana-1956	86	44	𝑗(0,0	𝑗(0,0	NOUN
cana-1956	86	45	)	)	PUNCT
cana-1956	86	46	,	,	PUNCT
cana-1956	86	47	the	the	DET
cana-1956	86	48	functions	function	NOUN
cana-1956	86	49	such	such	ADJ
cana-1956	86	50	as	as	ADP
cana-1956	86	51	𝑀𝑠	𝑀𝑠	PROPN
cana-1956	86	52	𝑗(𝑥	𝑗(𝑥	X
cana-1956	86	53	,	,	PUNCT
cana-1956	86	54	𝑤𝑖	𝑤𝑖	PRON
cana-1956	86	55	)	)	PUNCT
cana-1956	86	56	are	be	AUX
cana-1956	86	57	defined	define	VERB
cana-1956	86	58	in	in	ADP
cana-1956	86	59	[	[	X
cana-1956	86	60	4	4	NUM
cana-1956	86	61	]	]	PUNCT
cana-1956	86	62	.	.	PUNCT
cana-1956	87	1	then	then	ADV
cana-1956	87	2	:	:	PUNCT
cana-1956	87	3	𝑀𝑠	𝑀𝑠	PROPN
cana-1956	87	4	𝑗	𝑗	ADJ
cana-1956	87	5	=	=	SYM
cana-1956	87	6	𝑀𝑝−𝑠	𝑀𝑝−𝑠	NOUN
cana-1956	87	7	𝑞−𝑗	𝑞−𝑗	NOUN
cana-1956	87	8	,	,	PUNCT
cana-1956	87	9	𝑠	𝑠	PROPN
cana-1956	87	10	=	=	SYM
cana-1956	87	11	0,1	0,1	NUM
cana-1956	87	12	,	,	PUNCT
cana-1956	87	13	…	…	PUNCT
cana-1956	87	14	.	.	PUNCT
cana-1956	87	15	.	.	PUNCT
cana-1956	88	1	𝑝	𝑝	X
cana-1956	88	2	−	−	NOUN
cana-1956	88	3	1	1	NUM
cana-1956	88	4	,	,	PUNCT
cana-1956	88	5	𝑗	𝑗	NOUN
cana-1956	88	6	=	=	SYM
cana-1956	88	7	0,1	0,1	NUM
cana-1956	88	8	,	,	PUNCT
cana-1956	88	9	…	…	PUNCT
cana-1956	88	10	.	.	PUNCT
cana-1956	88	11	.	.	PUNCT
cana-1956	89	1	𝑞	𝑞	X
cana-1956	89	2	−	−	PROPN
cana-1956	89	3	1	1	NUM
cana-1956	89	4	(	(	PUNCT
cana-1956	89	5	6	6	NUM
cana-1956	89	6	)	)	PUNCT
cana-1956	89	7	the	the	DET
cana-1956	89	8	explanation	explanation	NOUN
cana-1956	89	9	of	of	ADP
cana-1956	89	10	the	the	DET
cana-1956	89	11	zak	zak	PROPN
cana-1956	89	12	transformation	transformation	NOUN
cana-1956	89	13	,	,	PUNCT
cana-1956	89	14	along	along	ADP
cana-1956	89	15	with	with	ADP
cana-1956	89	16	the	the	DET
cana-1956	89	17	fact	fact	NOUN
cana-1956	89	18	that	that	SCONJ
cana-1956	89	19	g	g	PROPN
cana-1956	89	20	is	be	AUX
cana-1956	89	21	odd	odd	ADJ
cana-1956	89	22	,	,	PUNCT
cana-1956	89	23	makes	make	VERB
cana-1956	89	24	the	the	DET
cana-1956	89	25	lemma	lemma	PROPN
cana-1956	89	26	simple	simple	NOUN
cana-1956	89	27	to	to	PART
cana-1956	89	28	prove	prove	VERB
cana-1956	89	29	.	.	PUNCT
cana-1956	90	1	we	we	PRON
cana-1956	90	2	𝑞	𝑞	VERB
cana-1956	90	3	−	−	PROPN
cana-1956	90	4	𝑝	𝑝	PROPN
cana-1956	90	5	=	=	SYM
cana-1956	90	6	1	1	X
cana-1956	90	7	.	.	X
cana-1956	90	8	assume	assume	VERB
cana-1956	90	9	q	q	PROPN
cana-1956	90	10	is	be	AUX
cana-1956	90	11	an	an	DET
cana-1956	90	12	even	even	ADJ
cana-1956	90	13	number	number	NOUN
cana-1956	90	14	;	;	PUNCT
cana-1956	90	15	𝑞	𝑞	X
cana-1956	90	16	=	=	X
cana-1956	90	17	2𝑘	2𝑘	NUM
cana-1956	91	1	+	+	CCONJ
cana-1956	91	2	2	2	NUM
cana-1956	91	3	,	,	PUNCT
cana-1956	91	4	𝑝	𝑝	NOUN
cana-1956	91	5	=	=	SYM
cana-1956	91	6	2𝑘	2𝑘	NUM
cana-1956	92	1	+	+	NOUN
cana-1956	92	2	1	1	X
cana-1956	92	3	.	.	X
cana-1956	92	4	q(0,0	q(0,0	NOUN
cana-1956	92	5	)	)	PUNCT
cana-1956	92	6	denotes	denote	VERB
cana-1956	92	7	the	the	DET
cana-1956	92	8	(	(	PUNCT
cana-1956	92	9	2𝑘	2𝑘	NUM
cana-1956	92	10	+	+	NOUN
cana-1956	92	11	1	1	X
cana-1956	92	12	)	)	PUNCT
cana-1956	92	13	×	×	NOUN
cana-1956	92	14	(	(	PUNCT
cana-1956	92	15	2𝑘	2𝑘	NUM
cana-1956	92	16	+	+	CCONJ
cana-1956	92	17	2	2	NUM
cana-1956	92	18	)	)	PUNCT
cana-1956	92	19	matrix	matrix	NOUN
cana-1956	92	20	.	.	PUNCT
cana-1956	93	1	additional	additional	ADJ
cana-1956	93	2	relations	relation	NOUN
cana-1956	93	3	must	must	AUX
cana-1956	93	4	exist	exist	VERB
cana-1956	93	5	for	for	ADP
cana-1956	93	6	the	the	DET
cana-1956	93	7	zero	zero	NUM
cana-1956	93	8	row	row	NOUN
cana-1956	93	9	,	,	PUNCT
cana-1956	93	10	zero	zero	NUM
cana-1956	93	11	column	column	NOUN
cana-1956	93	12	,	,	PUNCT
cana-1956	93	13	and	and	CCONJ
cana-1956	93	14	(	(	PUNCT
cana-1956	93	15	𝑘	𝑘	PROPN
cana-1956	93	16	+	+	NOUN
cana-1956	93	17	1	1	NUM
cana-1956	93	18	)	)	PUNCT
cana-1956	93	19	−	−	NOUN
cana-1956	93	20	𝑡ℎ(𝑘	𝑡ℎ(𝑘	PUNCT
cana-1956	94	1	+	+	CCONJ
cana-1956	94	2	1	1	X
cana-1956	94	3	)	)	PUNCT
cana-1956	94	4	-th	-th	NOUN
cana-1956	94	5	column	column	NOUN
cana-1956	94	6	of	of	ADP
cana-1956	94	7	𝒬(0,0	𝒬(0,0	NOUN
cana-1956	94	8	)	)	PUNCT
cana-1956	94	9	.	.	PUNCT
cana-1956	95	1	in	in	ADP
cana-1956	95	2	particular	particular	ADJ
cana-1956	95	3	:	:	PUNCT
cana-1956	95	4	𝑋0	𝑋0	NOUN
cana-1956	95	5	0	0	NUM
cana-1956	96	1	=	=	SYM
cana-1956	96	2	0	0	NUM
cana-1956	96	3	,	,	PUNCT
cana-1956	96	4	𝑋0	𝑋0	VERB
cana-1956	96	5	𝑘+1	𝑘+1	PRON
cana-1956	96	6	=	=	SYM
cana-1956	96	7	0	0	NUM
cana-1956	96	8	,	,	PUNCT
cana-1956	96	9	𝑋0	𝑋0	VERB
cana-1956	96	10	𝑗	𝑗	PROPN
cana-1956	96	11	=	=	SYM
cana-1956	96	12	−𝑋0	−𝑋0	PROPN
cana-1956	96	13	𝑞−𝑗	𝑞−𝑗	PROPN
cana-1956	96	14	-zero	-zero	PROPN
cana-1956	96	15	row	row	NOUN
cana-1956	96	16	;	;	PUNCT
cana-1956	96	17	(	(	PUNCT
cana-1956	96	18	7	7	X
cana-1956	96	19	)	)	PUNCT
cana-1956	96	20	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	96	21	0	0	NUM
cana-1956	96	22	=	=	SYM
cana-1956	96	23	−𝑋𝑝−𝑠	−𝑋𝑝−𝑠	NUM
cana-1956	96	24	0	0	NUM
cana-1956	96	25	,	,	PUNCT
cana-1956	96	26	𝑠	𝑠	PROPN
cana-1956	96	27	=	=	SYM
cana-1956	96	28	1	1	NUM
cana-1956	96	29	,	,	PUNCT
cana-1956	96	30	……	……	NOUN
cana-1956	96	31	,	,	PUNCT
cana-1956	96	32	𝑝	𝑝	NOUN
cana-1956	96	33	−	−	NOUN
cana-1956	96	34	1	1	NUM
cana-1956	96	35	-zero	-zero	NOUN
cana-1956	96	36	column	column	NOUN
cana-1956	96	37	;	;	PUNCT
cana-1956	96	38	(	(	PUNCT
cana-1956	96	39	8)	8)	NUM
cana-1956	96	40	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	96	41	𝑘+1	𝑘+1	NOUN
cana-1956	96	42	=	=	PUNCT
cana-1956	96	43	−𝑋𝑝−𝑠	−𝑋𝑝−𝑠	NUM
cana-1956	96	44	𝑘+1	𝑘+1	NUM
cana-1956	96	45	,	,	PUNCT
cana-1956	96	46	𝑠	𝑠	PROPN
cana-1956	96	47	=	=	SYM
cana-1956	96	48	1	1	NUM
cana-1956	96	49	,	,	PUNCT
cana-1956	96	50	…	…	PUNCT
cana-1956	96	51	…	…	PUNCT
cana-1956	96	52	,	,	PUNCT
cana-1956	96	53	𝑝	𝑝	NOUN
cana-1956	96	54	−	−	PROPN
cana-1956	96	55	1	1	NUM
cana-1956	96	56	-(𝑘	-(𝑘	NOUN
cana-1956	96	57	+	+	CCONJ
cana-1956	96	58	1)-th	1)-th	NUM
cana-1956	96	59	column	column	NOUN
cana-1956	96	60	.	.	PUNCT
cana-1956	97	1	(	(	PUNCT
cana-1956	97	2	9	9	NUM
cana-1956	97	3	)	)	PUNCT
cana-1956	97	4	as	as	ADP
cana-1956	97	5	in	in	ADP
cana-1956	97	6	theorem	theorem	NOUN
cana-1956	97	7	(	(	PUNCT
cana-1956	97	8	1.1	1.1	NUM
cana-1956	97	9	)	)	PUNCT
cana-1956	98	1	[	[	X
cana-1956	98	2	1	1	NUM
cana-1956	98	3	]	]	PUNCT
cana-1956	98	4	,	,	PUNCT
cana-1956	98	5	the	the	DET
cana-1956	98	6	connections	connection	NOUN
cana-1956	98	7	are	be	AUX
cana-1956	98	8	simply	simply	ADV
cana-1956	98	9	established	establish	VERB
cana-1956	98	10	by	by	ADP
cana-1956	98	11	the	the	DET
cana-1956	98	12	zak	zak	PROPN
cana-1956	98	13	transform	transform	VERB
cana-1956	98	14	formula	formula	NOUN
cana-1956	98	15	and	and	CCONJ
cana-1956	98	16	the	the	DET
cana-1956	98	17	fact	fact	NOUN
cana-1956	98	18	that	that	SCONJ
cana-1956	98	19	𝑔	𝑔	PROPN
cana-1956	98	20	is	be	AUX
cana-1956	98	21	odd	odd	ADJ
cana-1956	98	22	.	.	PUNCT
cana-1956	99	1	r_s	r_s	PUNCT
cana-1956	99	2	denotes	denote	VERB
cana-1956	99	3	the	the	DET
cana-1956	99	4	s	s	NOUN
cana-1956	99	5	-	-	PUNCT
cana-1956	99	6	th	th	VERB
cana-1956	99	7	row	row	NOUN
cana-1956	99	8	of	of	ADP
cana-1956	99	9	𝒬(0,0	𝒬(0,0	NOUN
cana-1956	99	10	)	)	PUNCT
cana-1956	99	11	..	..	PUNCT
cana-1956	99	12	consider	consider	VERB
cana-1956	99	13	the	the	DET
cana-1956	99	14	row	row	NOUN
cana-1956	99	15	vectors	vector	NOUN
cana-1956	99	16	:	:	PUNCT
cana-1956	100	1	𝑒𝑙	𝑒𝑙	PROPN
cana-1956	100	2	=	=	X
cana-1956	100	3	(	(	PUNCT
cana-1956	100	4	𝑒𝑙	𝑒𝑙	INTJ
cana-1956	100	5	𝑗	𝑗	PROPN
cana-1956	100	6	)	)	PUNCT
cana-1956	100	7	𝑗=0,1,	𝑗=0,1,	NOUN
cana-1956	100	8	…	…	PROPN
cana-1956	100	9	…	…	SYM
cana-1956	100	10	,2𝑘+2	,2𝑘+2	PUNCT
cana-1956	100	11	𝑙	𝑙	X
cana-1956	100	12	=	=	SYM
cana-1956	100	13	1,2	1,2	NUM
cana-1956	100	14	,	,	PUNCT
cana-1956	100	15	…	…	PUNCT
cana-1956	100	16	.	.	PUNCT
cana-1956	100	17	,	,	PUNCT
cana-1956	100	18	𝑘	𝑘	X
cana-1956	100	19	,	,	PUNCT
cana-1956	100	20	where	where	SCONJ
cana-1956	100	21	𝑒𝑙	𝑒𝑙	INTJ
cana-1956	100	22	𝑗	𝑗	NOUN
cana-1956	100	23	=	=	NOUN
cana-1956	100	24	0	0	NUM
cana-1956	100	25	,	,	PUNCT
cana-1956	100	26	for	for	ADP
cana-1956	100	27	𝑗	𝑗	PRON
cana-1956	100	28	≠	≠	PROPN
cana-1956	100	29	𝑙	𝑙	X
cana-1956	101	1	+	+	PUNCT
cana-1956	101	2	1,2𝑘	1,2𝑘	NUM
cana-1956	101	3	+	+	SYM
cana-1956	101	4	2	2	NUM
cana-1956	101	5	−	−	NOUN
cana-1956	101	6	1	1	NUM
cana-1956	101	7	,	,	PUNCT
cana-1956	101	8	𝑒𝑙	𝑒𝑙	ADP
cana-1956	101	9	2𝑘+2−𝑙	2𝑘+2−𝑙	NUM
cana-1956	101	10	=	=	SYM
cana-1956	101	11	−1	−1	NOUN
cana-1956	101	12	.	.	PUNCT
cana-1956	102	1	according	accord	VERB
cana-1956	102	2	to	to	ADP
cana-1956	102	3	the	the	DET
cana-1956	102	4	relations	relation	NOUN
cana-1956	102	5	[	[	X
cana-1956	102	6	6	6	NUM
cana-1956	102	7	]	]	PUNCT
cana-1956	102	8	,	,	PUNCT
cana-1956	102	9	[	[	X
cana-1956	102	10	7	7	NUM
cana-1956	102	11	]	]	PUNCT
cana-1956	102	12	,	,	PUNCT
cana-1956	102	13	[	[	X
cana-1956	102	14	8	8	NUM
cana-1956	102	15	]	]	PUNCT
cana-1956	102	16	,	,	PUNCT
cana-1956	102	17	and	and	CCONJ
cana-1956	102	18	[	[	X
cana-1956	102	19	9	9	NUM
cana-1956	102	20	]	]	PUNCT
cana-1956	102	21	,	,	PUNCT
cana-1956	102	22	all	all	DET
cana-1956	102	23	rows	row	NOUN
cana-1956	102	24	of	of	ADP
cana-1956	102	25	𝒬	𝒬	PROPN
cana-1956	102	26	belong	belong	VERB
cana-1956	102	27	to	to	ADP
cana-1956	102	28	𝑆	𝑆	PROPN
cana-1956	102	29	=	=	SYM
cana-1956	102	30	𝑠𝑝𝑎𝑛{{𝑅𝑠}𝑠=1	𝑠𝑝𝑎𝑛{{𝑅𝑠}𝑠=1	VERB
cana-1956	102	31	𝑘	𝑘	PRON
cana-1956	102	32	∪	∪	X
cana-1956	102	33	{	{	PUNCT
cana-1956	102	34	𝑒𝑙}𝑙=0	𝑒𝑙}𝑙=0	NOUN
cana-1956	102	35	𝑘	𝑘	X
cana-1956	102	36	}	}	PUNCT
cana-1956	102	37	.	.	PUNCT
cana-1956	103	1	yes	yes	INTJ
cana-1956	103	2	,	,	PUNCT
cana-1956	103	3	the	the	DET
cana-1956	103	4	row	row	NOUN
cana-1956	103	5	𝑅0	𝑅0	NOUN
cana-1956	103	6	has	have	VERB
cana-1956	103	7	the	the	DET
cana-1956	103	8	from	from	ADP
cana-1956	103	9	:	:	PUNCT
cana-1956	103	10	𝑅	𝑅	PROPN
cana-1956	103	11	=	=	SYM
cana-1956	103	12	(	(	PUNCT
cana-1956	103	13	0	0	NUM
cana-1956	103	14	,	,	PUNCT
cana-1956	103	15	(	(	PUNCT
cana-1956	103	16	𝜀	𝜀	X
cana-1956	103	17	+	+	ADJ
cana-1956	103	18	1)1	1)1	NUM
cana-1956	103	19	,	,	PUNCT
cana-1956	103	20	…	…	PUNCT
cana-1956	103	21	.	.	PUNCT
cana-1956	103	22	.	.	PUNCT
cana-1956	104	1	,	,	PUNCT
cana-1956	104	2	(	(	PUNCT
cana-1956	104	3	𝜀	𝜀	X
cana-1956	104	4	+	+	NUM
cana-1956	104	5	1)𝑘	1)𝑘	NOUN
cana-1956	104	6	,	,	PUNCT
cana-1956	104	7	0	0	NUM
cana-1956	104	8	,	,	PUNCT
cana-1956	104	9	−(𝜀	−(𝜀	NOUN
cana-1956	104	10	+	+	CCONJ
cana-1956	104	11	1)𝑘	1)𝑘	NOUN
cana-1956	104	12	,	,	PUNCT
cana-1956	104	13	…	…	PUNCT
cana-1956	104	14	…	…	PUNCT
cana-1956	104	15	,	,	PUNCT
cana-1956	104	16	−(𝜀	−(𝜀	NOUN
cana-1956	104	17	+	+	CCONJ
cana-1956	104	18	1)1	1)1	NUM
cana-1956	104	19	)	)	PUNCT
cana-1956	104	20	(	(	PUNCT
cana-1956	104	21	10	10	NUM
cana-1956	104	22	)	)	PUNCT
cana-1956	104	23	this	this	DET
cana-1956	104	24	vector	vector	NOUN
cana-1956	104	25	can	can	AUX
cana-1956	104	26	be	be	AUX
cana-1956	104	27	spanned	span	VERB
cana-1956	104	28	by	by	ADP
cana-1956	104	29	{	{	PUNCT
cana-1956	104	30	𝑒𝑙}𝑙=0	𝑒𝑙}𝑙=0	PROPN
cana-1956	104	31	𝑘	𝑘	PROPN
cana-1956	104	32	for	for	ADP
cana-1956	104	33	certain	certain	ADJ
cana-1956	104	34	(	(	PUNCT
cana-1956	104	35	𝜀	𝜀	X
cana-1956	104	36	+	+	ADJ
cana-1956	104	37	1)1	1)1	NUM
cana-1956	104	38	,	,	PUNCT
cana-1956	104	39	…	…	PUNCT
cana-1956	104	40	.	.	PUNCT
cana-1956	104	41	.	.	PUNCT
cana-1956	105	1	,	,	PUNCT
cana-1956	105	2	(	(	PUNCT
cana-1956	105	3	𝜀	𝜀	X
cana-1956	105	4	+	+	CCONJ
cana-1956	105	5	1)𝑘.	1)𝑘.	NUM
cana-1956	105	6	the	the	DET
cana-1956	105	7	rows	row	NOUN
cana-1956	106	1	𝑅𝑠	𝑅𝑠	PROPN
cana-1956	106	2	,	,	PUNCT
cana-1956	106	3	𝑠	𝑠	PROPN
cana-1956	106	4	=	=	SYM
cana-1956	106	5	1	1	NUM
cana-1956	106	6	,	,	PUNCT
cana-1956	106	7	…	…	PUNCT
cana-1956	106	8	,	,	PUNCT
cana-1956	106	9	𝑘	𝑘	X
cana-1956	106	10	,	,	PUNCT
cana-1956	106	11	belong	belong	VERB
cana-1956	106	12	to	to	ADP
cana-1956	106	13	the	the	DET
cana-1956	106	14	spanning	span	VERB
cana-1956	106	15	set	set	NOUN
cana-1956	106	16	.	.	PUNCT
cana-1956	107	1	to	to	PART
cana-1956	107	2	show	show	VERB
cana-1956	107	3	that	that	SCONJ
cana-1956	107	4	the	the	DET
cana-1956	107	5	rows	row	NOUN
cana-1956	107	6	𝑅𝑝−𝑠	𝑅𝑝−𝑠	PROPN
cana-1956	107	7	,	,	PUNCT
cana-1956	107	8	𝑠	𝑠	PROPN
cana-1956	107	9	=	=	SYM
cana-1956	107	10	1,	1,	NUM
cana-1956	107	11	…	…	SYM
cana-1956	107	12	…	…	SYM
cana-1956	107	13	𝑘	𝑘	NOUN
cana-1956	107	14	,	,	PUNCT
cana-1956	107	15	and	and	CCONJ
cana-1956	107	16	the	the	DET
cana-1956	107	17	vectors	vector	NOUN
cana-1956	107	18	:	:	PUNCT
cana-1956	107	19	𝑅𝑠	𝑅𝑠	ADP
cana-1956	107	20	+	+	CCONJ
cana-1956	107	21	𝑅𝑝−𝑠	𝑅𝑝−𝑠	ADJ
cana-1956	107	22	belong	belong	NOUN
cana-1956	107	23	to	to	ADP
cana-1956	107	24	𝑠	𝑠	PROPN
cana-1956	107	25	.	.	PUNCT
cana-1956	108	1	the	the	DET
cana-1956	108	2	latter	latter	ADJ
cana-1956	108	3	is	be	AUX
cana-1956	108	4	clear	clear	ADJ
cana-1956	108	5	because	because	SCONJ
cana-1956	108	6	[	[	X
cana-1956	108	7	6	6	NUM
cana-1956	108	8	]	]	PUNCT
cana-1956	108	9	and	and	CCONJ
cana-1956	109	1	[	[	X
cana-1956	109	2	8	8	X
cana-1956	109	3	]	]	X
cana-1956	109	4	state	state	NOUN
cana-1956	109	5	that	that	SCONJ
cana-1956	109	6	these	these	DET
cana-1956	109	7	vectors	vector	NOUN
cana-1956	109	8	also	also	ADV
cana-1956	109	9	contain	contain	VERB
cana-1956	109	10	[	[	X
cana-1956	109	11	10	10	NUM
cana-1956	109	12	]	]	PUNCT
cana-1956	109	13	.	.	PUNCT
cana-1956	110	1	this	this	PRON
cana-1956	110	2	completes	complete	VERB
cana-1956	110	3	the	the	DET
cana-1956	110	4	example	example	NOUN
cana-1956	110	5	of	of	ADP
cana-1956	110	6	the	the	DET
cana-1956	110	7	theorem	theorem	NOUN
cana-1956	110	8	for	for	ADP
cana-1956	110	9	the	the	DET
cana-1956	110	10	case	case	NOUN
cana-1956	110	11	𝑞	𝑞	X
cana-1956	110	12	=	=	SYM
cana-1956	110	13	2𝑘	2𝑘	NUM
cana-1956	111	1	+	+	CCONJ
cana-1956	111	2	2	2	NUM
cana-1956	111	3	,	,	PUNCT
cana-1956	111	4	𝑝	𝑝	NOUN
cana-1956	111	5	=	=	SYM
cana-1956	111	6	2𝑘	2𝑘	NUM
cana-1956	112	1	+	+	NOUN
cana-1956	112	2	1	1	X
cana-1956	112	3	.	.	X
cana-1956	112	4	let	let	VERB
cana-1956	112	5	q	q	PART
cana-1956	112	6	be	be	AUX
cana-1956	112	7	an	an	DET
cana-1956	112	8	odd	odd	ADJ
cana-1956	112	9	number	number	NOUN
cana-1956	112	10	,	,	PUNCT
cana-1956	112	11	𝑝	𝑝	NOUN
cana-1956	112	12	=	=	SYM
cana-1956	112	13	2𝑘	2𝑘	NUM
cana-1956	112	14	and	and	CCONJ
cana-1956	112	15	𝑞	𝑞	X
cana-1956	112	16	=	=	NOUN
cana-1956	112	17	2𝑘	2𝑘	NUM
cana-1956	113	1	+	+	NOUN
cana-1956	113	2	1	1	X
cana-1956	113	3	.	.	PUNCT
cana-1956	113	4	again	again	ADV
cana-1956	113	5	,	,	PUNCT
cana-1956	113	6	in	in	ADP
cana-1956	113	7	addition	addition	NOUN
cana-1956	113	8	to	to	ADP
cana-1956	113	9	the	the	DET
cana-1956	113	10	basic	basic	ADJ
cana-1956	113	11	relation	relation	NOUN
cana-1956	113	12	[	[	X
cana-1956	113	13	6	6	NUM
cana-1956	113	14	]	]	PUNCT
cana-1956	113	15	,	,	PUNCT
cana-1956	113	16	we	we	PRON
cana-1956	113	17	must	must	AUX
cana-1956	113	18	have	have	VERB
cana-1956	113	19	the	the	DET
cana-1956	113	20	relations	relation	NOUN
cana-1956	113	21	for	for	ADP
cana-1956	113	22	specific	specific	ADJ
cana-1956	113	23	rows	row	NOUN
cana-1956	113	24	and	and	CCONJ
cana-1956	113	25	columns	column	NOUN
cana-1956	113	26	:	:	PUNCT
cana-1956	113	27	𝑋0	𝑋0	VERB
cana-1956	113	28	𝑗	𝑗	PROPN
cana-1956	113	29	=	=	SYM
cana-1956	113	30	−𝑋0	−𝑋0	PROPN
cana-1956	113	31	𝑞−𝑗	𝑞−𝑗	NOUN
cana-1956	113	32	,	,	PUNCT
cana-1956	113	33	-zero	-zero	NOUN
cana-1956	113	34	row	row	NOUN
cana-1956	113	35	;	;	PUNCT
cana-1956	113	36	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	113	37	0	0	NUM
cana-1956	113	38	=	=	SYM
cana-1956	113	39	−𝑋𝑝−𝑠	−𝑋𝑝−𝑠	NUM
cana-1956	113	40	0	0	NUM
cana-1956	113	41	,	,	PUNCT
cana-1956	113	42	-zero	-zero	ADJ
cana-1956	113	43	column	column	NOUN
cana-1956	113	44	;	;	PUNCT
cana-1956	113	45	𝑋𝑘	𝑋𝑘	PROPN
cana-1956	113	46	𝑗	𝑗	NOUN
cana-1956	113	47	=	=	X
cana-1956	113	48	−𝑋𝑘	−𝑋𝑘	ADJ
cana-1956	113	49	𝑞−𝑗	𝑞−𝑗	NOUN
cana-1956	113	50	,	,	PUNCT
cana-1956	113	51	-𝑘-th	-𝑘-th	NOUN
cana-1956	113	52	row	row	NOUN
cana-1956	113	53	.	.	PUNCT
cana-1956	114	1	take	take	VERB
cana-1956	114	2	the	the	DET
cana-1956	114	3	rows	row	NOUN
cana-1956	114	4	𝑒𝑙	𝑒𝑙	NOUN
cana-1956	114	5	=	=	X
cana-1956	114	6	(	(	PUNCT
cana-1956	114	7	𝑒𝑙	𝑒𝑙	INTJ
cana-1956	114	8	𝑗	𝑗	PROPN
cana-1956	114	9	)	)	PUNCT
cana-1956	114	10	𝑗=0,1,	𝑗=0,1,	PROPN
cana-1956	114	11	…	…	PROPN
cana-1956	114	12	.,2𝑘	.,2𝑘	ADJ
cana-1956	114	13	,	,	PUNCT
cana-1956	114	14	𝑙	𝑙	X
cana-1956	114	15	=	=	SYM
cana-1956	114	16	1,2	1,2	NUM
cana-1956	114	17	,	,	PUNCT
cana-1956	114	18	…	…	PUNCT
cana-1956	114	19	.	.	PUNCT
cana-1956	114	20	.	.	PUNCT
cana-1956	115	1	,	,	PUNCT
cana-1956	115	2	𝑘	𝑘	X
cana-1956	115	3	,	,	PUNCT
cana-1956	115	4	with	with	ADP
cana-1956	115	5	𝑒𝑙	𝑒𝑙	PROPN
cana-1956	115	6	𝑗	𝑗	NOUN
cana-1956	115	7	=	=	NOUN
cana-1956	115	8	0	0	NUM
cana-1956	115	9	,	,	PUNCT
cana-1956	115	10	if	if	SCONJ
cana-1956	115	11	:	:	PUNCT
cana-1956	115	12	𝑗	𝑗	NOUN
cana-1956	115	13	≠	≠	PROPN
cana-1956	115	14	1	1	NUM
cana-1956	115	15	,	,	PUNCT
cana-1956	115	16	𝑒𝑙	𝑒𝑙	PRON
cana-1956	115	17	𝑗	𝑗	NOUN
cana-1956	115	18	=	=	SYM
cana-1956	115	19	1	1	NUM
cana-1956	115	20	and	and	CCONJ
cana-1956	115	21	𝑒𝑙	𝑒𝑙	VERB
cana-1956	115	22	2𝑘−𝑙+1	2𝑘−𝑙+1	NUM
cana-1956	115	23	=	=	SYM
cana-1956	115	24	−1	−1	NOUN
cana-1956	115	25	.	.	PUNCT
cana-1956	116	1	communications	communication	NOUN
cana-1956	116	2	on	on	ADP
cana-1956	116	3	applied	apply	VERB
cana-1956	116	4	nonlinear	nonlinear	ADJ
cana-1956	116	5	analysis	analysis	NOUN
cana-1956	116	6	issn	issn	NOUN
cana-1956	116	7	:	:	PUNCT
cana-1956	116	8	1074	1074	NUM
cana-1956	116	9	-	-	PUNCT
cana-1956	116	10	133x	133x	NUM
cana-1956	116	11	vol	vol	NOUN
cana-1956	116	12	32	32	NUM
cana-1956	116	13	no	no	NOUN
cana-1956	116	14	.	.	NOUN
cana-1956	116	15	3	3	NUM
cana-1956	116	16	(	(	PUNCT
cana-1956	116	17	2025	2025	NUM
cana-1956	116	18	)	)	PUNCT
cana-1956	116	19	234	234	NUM
cana-1956	116	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	116	21	using	use	VERB
cana-1956	116	22	the	the	DET
cana-1956	116	23	exact	exact	ADJ
cana-1956	116	24	same	same	ADJ
cana-1956	116	25	values	value	NOUN
cana-1956	116	26	as	as	ADP
cana-1956	116	27	before	before	ADV
cana-1956	116	28	,	,	PUNCT
cana-1956	116	29	we	we	PRON
cana-1956	116	30	can	can	AUX
cana-1956	116	31	observe	observe	VERB
cana-1956	116	32	that	that	SCONJ
cana-1956	116	33	the	the	DET
cana-1956	116	34	set	set	NOUN
cana-1956	116	35	of	of	ADP
cana-1956	116	36	2𝑘	2𝑘	NUM
cana-1956	116	37	−	−	PROPN
cana-1956	116	38	1	1	NUM
cana-1956	116	39	vectors	vector	NOUN
cana-1956	116	40	{	{	PUNCT
cana-1956	116	41	𝑅𝑠}𝑠=1,	𝑅𝑠}𝑠=1,	PROPN
cana-1956	116	42	…	…	NUM
cana-1956	116	43	..	..	PUNCT
cana-1956	116	44	,𝑘−1	,𝑘−1	X
cana-1956	116	45	∪	∪	X
cana-1956	116	46	{	{	PUNCT
cana-1956	116	47	𝑒𝑙}𝑙=1,	𝑒𝑙}𝑙=1,	NOUN
cana-1956	116	48	…	…	PUNCT
cana-1956	116	49	..	..	PUNCT
cana-1956	116	50	,𝑘	,𝑘	PUNCT
cana-1956	116	51	spans	span	VERB
cana-1956	116	52	all	all	DET
cana-1956	116	53	rows	row	NOUN
cana-1956	116	54	of	of	ADP
cana-1956	116	55	the	the	DET
cana-1956	116	56	matrix	matrix	NOUN
cana-1956	116	57	𝒬(0,0	𝒬(0,0	NOUN
cana-1956	116	58	)	)	PUNCT
cana-1956	116	59	this	this	PRON
cana-1956	116	60	concludes	conclude	VERB
cana-1956	116	61	the	the	DET
cana-1956	116	62	proving	proving	NOUN
cana-1956	116	63	of	of	ADP
cana-1956	116	64	theorem	theorem	NOUN
cana-1956	116	65	(	(	PUNCT
cana-1956	116	66	1.1	1.1	NUM
cana-1956	116	67	)	)	PUNCT
cana-1956	117	1	[	[	X
cana-1956	117	2	1	1	NUM
cana-1956	117	3	]	]	PUNCT
cana-1956	117	4	.	.	PUNCT
cana-1956	118	1	4	4	X
cana-1956	118	2	.	.	X
cana-1956	118	3	factorizing	factorizing	NOUN
cana-1956	118	4	of	of	ADP
cana-1956	118	5	zibulskii	zibulskii	PRON
cana-1956	118	6	-	-	PUNCT
cana-1956	118	7	zeevi	zeevi	NOUN
cana-1956	118	8	matrix	matrix	NOUN
cana-1956	118	9	this	this	DET
cana-1956	118	10	section	section	NOUN
cana-1956	118	11	focuses	focus	VERB
cana-1956	118	12	on	on	ADP
cana-1956	118	13	the	the	DET
cana-1956	118	14	zibulski	zibulski	ADJ
cana-1956	118	15	-	-	PUNCT
cana-1956	118	16	zeevi	zeevi	NOUN
cana-1956	118	17	matrix	matrix	NOUN
cana-1956	118	18	𝒬(𝑥	𝒬(𝑥	PROPN
cana-1956	118	19	,	,	PUNCT
cana-1956	118	20	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	118	21	)	)	PUNCT
cana-1956	118	22	.	.	PUNCT
cana-1956	119	1	our	our	PRON
cana-1956	119	2	objective	objective	NOUN
cana-1956	119	3	is	be	AUX
cana-1956	119	4	to	to	PART
cana-1956	119	5	reduce	reduce	VERB
cana-1956	119	6	it	it	PRON
cana-1956	119	7	to	to	ADP
cana-1956	119	8	a	a	DET
cana-1956	119	9	simple	simple	ADJ
cana-1956	119	10	𝑝	𝑝	PROPN
cana-1956	119	11	×	×	NOUN
cana-1956	119	12	𝑞	𝑞	X
cana-1956	119	13	matrix	matrix	NOUN
cana-1956	119	14	with	with	ADP
cana-1956	119	15	the	the	DET
cana-1956	119	16	same	same	ADJ
cana-1956	119	17	rank	rank	NOUN
cana-1956	119	18	as	as	ADP
cana-1956	119	19	𝒬.	𝒬.	NOUN
cana-1956	119	20	this	this	DET
cana-1956	119	21	simpler	simple	ADJ
cana-1956	119	22	matrix	matrix	NOUN
cana-1956	119	23	can	can	AUX
cana-1956	119	24	be	be	AUX
cana-1956	119	25	thought	think	VERB
cana-1956	119	26	of	of	ADP
cana-1956	119	27	as	as	ADP
cana-1956	119	28	an	an	DET
cana-1956	119	29	analogue	analogue	NOUN
cana-1956	119	30	of	of	ADP
cana-1956	119	31	the	the	DET
cana-1956	119	32	ron	ron	PROPN
cana-1956	119	33	-	-	PUNCT
cana-1956	119	34	shen	shen	PROPN
cana-1956	119	35	gramian	gramian	PROPN
cana-1956	120	1	[	[	X
cana-1956	120	2	12	12	NUM
cana-1956	120	3	]	]	PUNCT
cana-1956	120	4	for	for	ADP
cana-1956	120	5	rationally	rationally	ADV
cana-1956	120	6	oversampled	oversample	VERB
cana-1956	120	7	gabor	gabor	NOUN
cana-1956	120	8	systems	system	NOUN
cana-1956	120	9	:	:	PUNCT
cana-1956	120	10	similarly	similarly	ADV
cana-1956	120	11	,	,	PUNCT
cana-1956	120	12	to	to	ADP
cana-1956	120	13	[	[	X
cana-1956	120	14	12	12	NUM
cana-1956	120	15	]	]	PUNCT
cana-1956	120	16	,	,	PUNCT
cana-1956	120	17	the	the	DET
cana-1956	120	18	study	study	NOUN
cana-1956	120	19	of	of	ADP
cana-1956	120	20	the	the	DET
cana-1956	120	21	frame	frame	NOUN
cana-1956	120	22	property	property	NOUN
cana-1956	120	23	can	can	AUX
cana-1956	120	24	be	be	AUX
cana-1956	120	25	reduced	reduce	VERB
cana-1956	120	26	to	to	ADP
cana-1956	120	27	the	the	DET
cana-1956	120	28	study	study	NOUN
cana-1956	120	29	of	of	ADP
cana-1956	120	30	a	a	DET
cana-1956	120	31	specific	specific	ADJ
cana-1956	120	32	matrix	matrix	NOUN
cana-1956	120	33	generated	generate	VERB
cana-1956	120	34	by	by	ADP
cana-1956	120	35	translates	translate	NOUN
cana-1956	120	36	of	of	ADP
cana-1956	120	37	the	the	DET
cana-1956	120	38	generating	generate	VERB
cana-1956	120	39	function	function	NOUN
cana-1956	120	40	;	;	PUNCT
cana-1956	120	41	in	in	ADP
cana-1956	120	42	our	our	PRON
cana-1956	120	43	case	case	NOUN
cana-1956	120	44	,	,	PUNCT
cana-1956	120	45	we	we	PRON
cana-1956	120	46	consider	consider	VERB
cana-1956	120	47	the	the	DET
cana-1956	120	48	zak	zak	PROPN
cana-1956	120	49	transform	transform	NOUN
cana-1956	120	50	of	of	ADP
cana-1956	120	51	the	the	DET
cana-1956	120	52	generating	generate	VERB
cana-1956	120	53	function	function	NOUN
cana-1956	120	54	.	.	PUNCT
cana-1956	121	1	we	we	PRON
cana-1956	121	2	think	think	VERB
cana-1956	121	3	this	this	DET
cana-1956	121	4	reduction	reduction	NOUN
cana-1956	121	5	is	be	AUX
cana-1956	121	6	fascinating	fascinating	ADJ
cana-1956	121	7	on	on	ADP
cana-1956	121	8	its	its	PRON
cana-1956	121	9	own	own	ADJ
cana-1956	121	10	,	,	PUNCT
cana-1956	121	11	but	but	CCONJ
cana-1956	121	12	it	it	PRON
cana-1956	121	13	will	will	AUX
cana-1956	121	14	also	also	ADV
cana-1956	121	15	be	be	AUX
cana-1956	121	16	utilized	utilize	VERB
cana-1956	121	17	in	in	ADP
cana-1956	121	18	the	the	DET
cana-1956	121	19	next	next	ADJ
cana-1956	121	20	part	part	NOUN
cana-1956	121	21	of	of	ADP
cana-1956	121	22	our	our	PRON
cana-1956	121	23	study	study	NOUN
cana-1956	121	24	gabor	gabor	PROPN
cana-1956	121	25	frames	frame	NOUN
cana-1956	121	26	generated	generate	VERB
cana-1956	121	27	by	by	ADP
cana-1956	121	28	the	the	DET
cana-1956	121	29	first	first	ADJ
cana-1956	121	30	harmonic	harmonic	ADJ
cana-1956	121	31	function	function	NOUN
cana-1956	121	32	.	.	PUNCT
cana-1956	122	1	theorem	theorem	NOUN
cana-1956	122	2	(	(	PUNCT
cana-1956	122	3	4.1	4.1	NUM
cana-1956	122	4	):	):	PUNCT
cana-1956	122	5	assume	assume	VERB
cana-1956	122	6	the	the	DET
cana-1956	122	7	area	area	NOUN
cana-1956	122	8	of	of	ADP
cana-1956	122	9	the	the	DET
cana-1956	122	10	function	function	NOUN
cana-1956	122	11	g	g	NOUN
cana-1956	122	12	from	from	ADP
cana-1956	122	13	𝑀1(ℝ	𝑀1(ℝ	ADV
cana-1956	122	14	)	)	PUNCT
cana-1956	122	15	with	with	ADP
cana-1956	122	16	𝜀2	𝜀2	PROPN
cana-1956	122	17	=	=	SYM
cana-1956	122	18	𝑝	𝑝	PROPN
cana-1956	122	19	𝑞	𝑞	NOUN
cana-1956	122	20	+	+	CCONJ
cana-1956	122	21	1	1	NUM
cana-1956	122	22	∈	∈	PROPN
cana-1956	122	23	ℚ	ℚ	PROPN
cana-1956	122	24	,	,	PUNCT
cana-1956	122	25	where	where	SCONJ
cana-1956	122	26	p	p	NOUN
cana-1956	122	27	and	and	CCONJ
cana-1956	122	28	q	q	NOUN
cana-1956	122	29	are	be	AUX
cana-1956	122	30	essentially	essentially	ADV
cana-1956	122	31	prime	prime	ADJ
cana-1956	122	32	.	.	PUNCT
cana-1956	123	1	the	the	DET
cana-1956	123	2	system	system	NOUN
cana-1956	123	3	𝒢(𝑔	𝒢(𝑔	PROPN
cana-1956	123	4	,	,	PUNCT
cana-1956	123	5	𝜀	𝜀	X
cana-1956	123	6	+	+	NOUN
cana-1956	123	7	1	1	NUM
cana-1956	123	8	,	,	PUNCT
cana-1956	123	9	𝜀	𝜀	VERB
cana-1956	123	10	−	−	NOUN
cana-1956	123	11	1	1	NUM
cana-1956	123	12	)	)	PUNCT
cana-1956	123	13	)	)	PUNCT
cana-1956	123	14	yields	yield	VERB
cana-1956	123	15	a	a	DET
cana-1956	123	16	frame	frame	NOUN
cana-1956	123	17	in	in	ADP
cana-1956	123	18	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	123	19	)	)	PUNCT
cana-1956	123	20	if	if	SCONJ
cana-1956	123	21	and	and	CCONJ
cana-1956	123	22	only	only	ADV
cana-1956	123	23	if	if	SCONJ
cana-1956	123	24	the	the	DET
cana-1956	123	25	matrix	matrix	NOUN
cana-1956	123	26	:	:	PUNCT
cana-1956	123	27	∑	∑	PROPN
cana-1956	123	28	𝒫(𝑥	𝒫(𝑥	NOUN
cana-1956	123	29	,	,	PUNCT
cana-1956	123	30	𝑤𝑖)𝑖	𝑤𝑖)𝑖	PROPN
cana-1956	123	31	=	=	SYM
cana-1956	123	32	∑	∑	PUNCT
cana-1956	123	33	(	(	PUNCT
cana-1956	123	34	(	(	PUNCT
cana-1956	123	35	𝑍(𝜀+1)𝑞𝑔	𝑍(𝜀+1)𝑞𝑔	NOUN
cana-1956	123	36	(	(	PUNCT
cana-1956	123	37	𝑥	𝑥	PROPN
cana-1956	123	38	+	+	CCONJ
cana-1956	123	39	𝜀+1	𝜀+1	PROPN
cana-1956	123	40	𝑝	𝑝	PROPN
cana-1956	123	41	(	(	PUNCT
cana-1956	123	42	𝑡𝑝	𝑡𝑝	PROPN
cana-1956	123	43	+	+	NUM
cana-1956	123	44	𝑠𝑞	𝑠𝑞	NOUN
cana-1956	123	45	)	)	PUNCT
cana-1956	123	46	,	,	PUNCT
cana-1956	123	47	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	123	48	)	)	PUNCT
cana-1956	123	49	)	)	PUNCT
cana-1956	123	50	𝑠=0,𝑡=0	𝑠=0,𝑡=0	NUM
cana-1956	123	51	𝑝−1,𝑞−1	𝑝−1,𝑞−1	NOUN
cana-1956	123	52	)	)	PUNCT
cana-1956	124	1	𝑖	𝑖	X
cana-1956	124	2	(	(	PUNCT
cana-1956	124	3	11	11	NUM
cana-1956	124	4	)	)	PUNCT
cana-1956	124	5	has	have	VERB
cana-1956	124	6	𝑟𝑎𝑛𝑘	𝑟𝑎𝑛𝑘	NOUN
cana-1956	124	7	=	=	SYM
cana-1956	124	8	𝑝	𝑝	NOUN
cana-1956	124	9	for	for	ADP
cana-1956	124	10	all	all	DET
cana-1956	124	11	(	(	PUNCT
cana-1956	124	12	𝑥	𝑥	NOUN
cana-1956	124	13	,	,	PUNCT
cana-1956	124	14	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	124	15	)	)	PUNCT
cana-1956	124	16	∈	∈	PROPN
cana-1956	124	17	𝑄𝜀+1,𝑞.	𝑄𝜀+1,𝑞.	NOUN
cana-1956	124	18	proof	proof	NOUN
cana-1956	124	19	:	:	PUNCT
cana-1956	124	20	fix	fix	NOUN
cana-1956	124	21	(	(	PUNCT
cana-1956	124	22	𝑥	𝑥	NOUN
cana-1956	124	23	,	,	PUNCT
cana-1956	124	24	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	124	25	)	)	PUNCT
cana-1956	124	26	∈	∈	NOUN
cana-1956	124	27	𝑄𝜀+1,𝑞	𝑄𝜀+1,𝑞	NOUN
cana-1956	124	28	and	and	CCONJ
cana-1956	124	29	let	let	VERB
cana-1956	124	30	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	124	31	𝑗	𝑗	NOUN
cana-1956	124	32	=	=	PRON
cana-1956	124	33	∑𝒵(𝜀+1)𝑔	∑𝒵(𝜀+1)𝑔	NOUN
cana-1956	124	34	(	(	PUNCT
cana-1956	124	35	𝑥	𝑥	PROPN
cana-1956	125	1	+	+	X
cana-1956	125	2	𝜀	𝜀	X
cana-1956	125	3	+	+	CCONJ
cana-1956	125	4	1	1	PROPN
cana-1956	125	5	𝑝	𝑝	PROPN
cana-1956	125	6	𝑠	𝑠	PROPN
cana-1956	125	7	,	,	PUNCT
cana-1956	125	8	𝑤𝑖	𝑤𝑖	ADP
cana-1956	125	9	−	−	PROPN
cana-1956	125	10	(	(	PUNCT
cana-1956	125	11	𝜀	𝜀	X
cana-1956	125	12	−	−	NOUN
cana-1956	125	13	1)𝑗	1)𝑗	NUM
cana-1956	125	14	)	)	PUNCT
cana-1956	125	15	𝑒	𝑒	PROPN
cana-1956	125	16	2𝑖𝜋	2𝑖𝜋	NOUN
cana-1956	125	17	𝑗𝑠	𝑗𝑠	VERB
cana-1956	125	18	𝑞	𝑞	X
cana-1956	125	19	𝑖	𝑖	PROPN
cana-1956	126	1	=	=	PROPN
cana-1956	126	2	∑𝑔(𝑥	∑𝑔(𝑥	PROPN
cana-1956	126	3	+	+	SYM
cana-1956	126	4	𝜀	𝜀	X
cana-1956	126	5	+	+	CCONJ
cana-1956	126	6	1	1	PROPN
cana-1956	126	7	𝑝	𝑝	NOUN
cana-1956	126	8	(	(	PUNCT
cana-1956	126	9	𝑠	𝑠	INTJ
cana-1956	126	10	−	−	NOUN
cana-1956	126	11	𝑝𝑛	𝑝𝑛	PROPN
cana-1956	126	12	)	)	PUNCT
cana-1956	126	13	)	)	PUNCT
cana-1956	126	14	𝑒	𝑒	PROPN
cana-1956	126	15	2𝑖𝜋𝑗	2𝑖𝜋𝑗	PROPN
cana-1956	126	16	(	(	PUNCT
cana-1956	126	17	𝑠	𝑠	PROPN
cana-1956	126	18	𝑞	𝑞	X
cana-1956	126	19	−(𝜀2−1)𝑛	−(𝜀2−1)𝑛	NOUN
cana-1956	126	20	)	)	PUNCT
cana-1956	126	21	𝑛	𝑛	DET
cana-1956	126	22	𝑒	𝑒	PROPN
cana-1956	126	23	2𝑖𝜋𝑗	2𝑖𝜋𝑗	PROPN
cana-1956	126	24	(	(	PUNCT
cana-1956	126	25	𝑠	𝑠	PROPN
cana-1956	126	26	𝑞	𝑞	PROPN
cana-1956	126	27	−(𝜀2−1)𝑛	−(𝜀2−1)𝑛	PROPN
cana-1956	126	28	)	)	PUNCT
cana-1956	126	29	(	(	PUNCT
cana-1956	126	30	12	12	X
cana-1956	126	31	)	)	PUNCT
cana-1956	126	32	be	be	AUX
cana-1956	126	33	the	the	DET
cana-1956	126	34	appropriate	appropriate	ADJ
cana-1956	126	35	entry	entry	NOUN
cana-1956	126	36	in	in	ADP
cana-1956	126	37	the	the	DET
cana-1956	126	38	matrix	matrix	NOUN
cana-1956	126	39	𝒬(𝑥,𝑤𝑖	𝒬(𝑥,𝑤𝑖	NOUN
cana-1956	126	40	)	)	PUNCT
cana-1956	126	41	.	.	PUNCT
cana-1956	127	1	let	let	VERB
cana-1956	127	2	𝐿(𝑠	𝐿(𝑠	PRON
cana-1956	127	3	)	)	PUNCT
cana-1956	127	4	≔	≔	VERB
cana-1956	127	5	{	{	PUNCT
cana-1956	127	6	𝑙	𝑙	NOUN
cana-1956	127	7	:	:	PUNCT
cana-1956	127	8	𝑙	𝑙	X
cana-1956	127	9	=	=	PUNCT
cana-1956	127	10	𝑠	𝑠	INTJ
cana-1956	127	11	−	−	PROPN
cana-1956	127	12	𝑝	𝑝	PROPN
cana-1956	127	13	−	−	PROPN
cana-1956	127	14	1	1	NUM
cana-1956	127	15	,	,	PUNCT
cana-1956	127	16	𝑛	𝑛	DET
cana-1956	127	17	∈	∈	NOUN
cana-1956	127	18	𝕫	𝕫	NOUN
cana-1956	127	19	}	}	PUNCT
cana-1956	127	20	,	,	PUNCT
cana-1956	127	21	𝐿(𝑠	𝐿(𝑠	PROPN
cana-1956	127	22	,	,	PUNCT
cana-1956	127	23	𝑡	𝑡	X
cana-1956	127	24	)	)	PUNCT
cana-1956	127	25	=	=	SYM
cana-1956	127	26	{	{	PUNCT
cana-1956	127	27	𝑙	𝑙	NOUN
cana-1956	127	28	:	:	PUNCT
cana-1956	127	29	𝐿(𝑠	𝐿(𝑠	NUM
cana-1956	127	30	):	):	PUNCT
cana-1956	127	31	𝑙	𝑙	X
cana-1956	127	32	=	=	SYM
cana-1956	127	33	𝑡	𝑡	PROPN
cana-1956	127	34	+	+	CCONJ
cana-1956	127	35	𝑚𝑞,𝑚	𝑚𝑞,𝑚	PUNCT
cana-1956	127	36	∈	∈	PROPN
cana-1956	127	37	𝕫}and	𝕫}and	NOUN
cana-1956	127	38	1	1	NUM
cana-1956	127	39	,	,	PUNCT
cana-1956	127	40	…	…	PUNCT
cana-1956	127	41	.	.	PUNCT
cana-1956	128	1	,	,	PUNCT
cana-1956	128	2	𝑞	𝑞	X
cana-1956	128	3	−	−	PROPN
cana-1956	128	4	1	1	X
cana-1956	128	5	.	.	PUNCT
cana-1956	128	6	setting	set	VERB
cana-1956	128	7	𝑙	𝑙	X
cana-1956	128	8	=	=	PUNCT
cana-1956	128	9	𝑠	𝑠	PART
cana-1956	128	10	−	−	NOUN
cana-1956	128	11	𝑝𝑛	𝑝𝑛	NOUN
cana-1956	128	12	in	in	ADP
cana-1956	128	13	[	[	X
cana-1956	128	14	12	12	NUM
cana-1956	128	15	]	]	PUNCT
cana-1956	128	16	yields	yield	NOUN
cana-1956	128	17	:	:	PUNCT
cana-1956	128	18	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	128	19	𝑗	𝑗	VERB
cana-1956	128	20	=	=	PRON
cana-1956	128	21	∑∑𝑔(𝑥	∑∑𝑔(𝑥	PROPN
cana-1956	128	22	+	+	X
cana-1956	128	23	𝜀	𝜀	X
cana-1956	128	24	+	+	CCONJ
cana-1956	128	25	1	1	PROPN
cana-1956	128	26	𝑝	𝑝	PROPN
cana-1956	128	27	𝑙	𝑙	NOUN
cana-1956	128	28	)	)	PUNCT
cana-1956	128	29	𝑒	𝑒	PROPN
cana-1956	128	30	2𝑖𝜋	2𝑖𝜋	NOUN
cana-1956	129	1	𝑗𝑙	𝑗𝑙	PROPN
cana-1956	129	2	𝑞	𝑞	PROPN
cana-1956	129	3	+2𝑖𝜋(𝜀+1)𝑤𝑖	+2𝑖𝜋(𝜀+1)𝑤𝑖	PROPN
cana-1956	129	4	𝑠−1	𝑠−1	PROPN
cana-1956	129	5	𝑝	𝑝	PROPN
cana-1956	129	6	𝑛𝑖	𝑛𝑖	PROPN
cana-1956	130	1	=	=	NOUN
cana-1956	130	2	∑(∑𝑒	∑(∑𝑒	PROPN
cana-1956	130	3	2𝑖𝜋(𝜀+1)𝑤𝑖	2𝑖𝜋(𝜀+1)𝑤𝑖	NUM
cana-1956	130	4	𝑠	𝑠	SYM
cana-1956	130	5	𝑝	𝑝	PROPN
cana-1956	130	6	𝑞−1	𝑞−1	PROPN
cana-1956	130	7	𝑡=0	𝑡=0	PUNCT
cana-1956	130	8	∑	∑	PUNCT
cana-1956	130	9	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1956	130	10	+	+	CCONJ
cana-1956	130	11	𝜀	𝜀	X
cana-1956	131	1	+	+	CCONJ
cana-1956	131	2	1	1	NUM
cana-1956	131	3	𝑝	𝑝	PROPN
cana-1956	131	4	𝑙	𝑙	PROPN
cana-1956	131	5	)	)	PUNCT
cana-1956	131	6	𝑙∈𝐿(𝑠,𝑡	𝑙∈𝐿(𝑠,𝑡	PROPN
cana-1956	131	7	)	)	PUNCT
cana-1956	131	8	𝑒	𝑒	PROPN
cana-1956	131	9	−2𝑖𝜋(𝜀+1)𝑤𝑖	−2𝑖𝜋(𝜀+1)𝑤𝑖	NOUN
cana-1956	131	10	1	1	NUM
cana-1956	131	11	𝑝𝑒	𝑝𝑒	NOUN
cana-1956	131	12	2𝑖𝜋	2𝑖𝜋	PROPN
cana-1956	131	13	𝑗𝑡	𝑗𝑡	PROPN
cana-1956	131	14	𝑞	𝑞	PROPN
cana-1956	131	15	)	)	PUNCT
cana-1956	131	16	𝑖	𝑖	PROPN
cana-1956	131	17	(	(	PUNCT
cana-1956	131	18	13	13	NUM
cana-1956	131	19	)	)	PUNCT
cana-1956	131	20	we	we	PRON
cana-1956	131	21	picked	pick	VERB
cana-1956	131	22	𝑘𝑡	𝑘𝑡	PROPN
cana-1956	131	23	∈	∈	PROPN
cana-1956	131	24	{	{	PUNCT
cana-1956	131	25	0	0	NUM
cana-1956	131	26	,	,	PUNCT
cana-1956	131	27	…	…	PUNCT
cana-1956	131	28	,	,	PUNCT
cana-1956	131	29	𝑞	𝑞	X
cana-1956	131	30	−	−	PROPN
cana-1956	131	31	1	1	NUM
cana-1956	131	32	}	}	PUNCT
cana-1956	131	33	and	and	CCONJ
cana-1956	131	34	𝑚𝑠	𝑚𝑠	ADP
cana-1956	131	35	∈	∈	PROPN
cana-1956	131	36	{	{	PUNCT
cana-1956	131	37	0	0	NUM
cana-1956	131	38	,	,	PUNCT
cana-1956	131	39	…	…	PUNCT
cana-1956	131	40	,	,	PUNCT
cana-1956	131	41	𝑝	𝑝	NOUN
cana-1956	131	42	−	−	PROPN
cana-1956	131	43	1	1	NUM
cana-1956	131	44	}	}	PUNCT
cana-1956	131	45	for	for	ADP
cana-1956	131	46	each	each	DET
cana-1956	131	47	𝑡	𝑡	PROPN
cana-1956	131	48	∈	∈	PROPN
cana-1956	131	49	{	{	PUNCT
cana-1956	131	50	0	0	NUM
cana-1956	131	51	,	,	PUNCT
cana-1956	131	52	…	…	PUNCT
cana-1956	131	53	,	,	PUNCT
cana-1956	131	54	𝑞	𝑞	X
cana-1956	131	55	−	−	PROPN
cana-1956	131	56	1	1	NUM
cana-1956	131	57	}	}	PUNCT
cana-1956	131	58	and	and	CCONJ
cana-1956	131	59	𝑠	𝑠	PROPN
cana-1956	131	60	∈	∈	PROPN
cana-1956	131	61	{	{	PUNCT
cana-1956	131	62	0	0	NUM
cana-1956	131	63	,	,	PUNCT
cana-1956	131	64	…	…	PUNCT
cana-1956	131	65	,	,	PUNCT
cana-1956	131	66	𝑝	𝑝	NOUN
cana-1956	131	67	−	−	PROPN
cana-1956	131	68	1	1	NUM
cana-1956	131	69	}	}	PUNCT
cana-1956	131	70	,	,	PUNCT
cana-1956	131	71	respectively	respectively	ADV
cana-1956	131	72	,	,	PUNCT
cana-1956	131	73	so	so	SCONJ
cana-1956	131	74	that	that	SCONJ
cana-1956	131	75	:	:	PUNCT
cana-1956	131	76	𝑘𝑡𝑝	𝑘𝑡𝑝	NOUN
cana-1956	131	77	=	=	PUNCT
cana-1956	131	78	𝑡	𝑡	PROPN
cana-1956	131	79	(	(	PUNCT
cana-1956	131	80	𝑚𝑜𝑑𝑞	𝑚𝑜𝑑𝑞	PROPN
cana-1956	131	81	)	)	PUNCT
cana-1956	131	82	,	,	PUNCT
cana-1956	131	83	𝑚𝑠𝑝	𝑚𝑠𝑝	ADV
cana-1956	131	84	=	=	SYM
cana-1956	131	85	𝑠	𝑠	PROPN
cana-1956	131	86	(	(	PUNCT
cana-1956	131	87	𝑚𝑜𝑑𝑝	𝑚𝑜𝑑𝑝	NOUN
cana-1956	131	88	)	)	PUNCT
cana-1956	131	89	.	.	PUNCT
cana-1956	132	1	since	since	SCONJ
cana-1956	132	2	𝐿(𝑠	𝐿(𝑠	PRON
cana-1956	132	3	,	,	PUNCT
cana-1956	132	4	𝑡	𝑡	PROPN
cana-1956	132	5	)	)	PUNCT
cana-1956	132	6	=	=	PRON
cana-1956	132	7	{	{	PUNCT
cana-1956	132	8	𝑘𝑡𝑝	𝑘𝑡𝑝	NOUN
cana-1956	132	9	+	+	NOUN
cana-1956	132	10	𝑚𝑠𝑞	𝑚𝑠𝑞	PROPN
cana-1956	132	11	−	−	PROPN
cana-1956	132	12	𝑝𝑞𝑚	𝑝𝑞𝑚	NOUN
cana-1956	132	13	:	:	PUNCT
cana-1956	132	14	𝑚	𝑚	PROPN
cana-1956	132	15	∈	∈	PROPN
cana-1956	132	16	𝕫	𝕫	X
cana-1956	132	17	}	}	PUNCT
cana-1956	132	18	,	,	PUNCT
cana-1956	132	19	(	(	PUNCT
cana-1956	132	20	13	13	NUM
cana-1956	132	21	)	)	PUNCT
cana-1956	132	22	can	can	AUX
cana-1956	132	23	be	be	AUX
cana-1956	132	24	rewritten	rewrite	VERB
cana-1956	132	25	as	as	ADP
cana-1956	132	26	:	:	PUNCT
cana-1956	132	27	communications	communication	NOUN
cana-1956	132	28	on	on	ADP
cana-1956	132	29	applied	apply	VERB
cana-1956	132	30	nonlinear	nonlinear	ADJ
cana-1956	132	31	analysis	analysis	NOUN
cana-1956	132	32	issn	issn	NOUN
cana-1956	132	33	:	:	PUNCT
cana-1956	132	34	1074	1074	NUM
cana-1956	132	35	-	-	PUNCT
cana-1956	132	36	133x	133x	NUM
cana-1956	132	37	vol	vol	NOUN
cana-1956	132	38	32	32	NUM
cana-1956	132	39	no	no	NOUN
cana-1956	132	40	.	.	NOUN
cana-1956	132	41	3	3	NUM
cana-1956	132	42	(	(	PUNCT
cana-1956	132	43	2025	2025	NUM
cana-1956	132	44	)	)	PUNCT
cana-1956	132	45	235	235	NUM
cana-1956	132	46	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	132	47	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	132	48	𝑗	𝑗	PROPN
cana-1956	133	1	=	=	X
cana-1956	133	2	∑	∑	PROPN
cana-1956	133	3	𝑒	𝑒	PROPN
cana-1956	133	4	2𝑖𝜋(𝜀+1)𝑤𝑖	2𝑖𝜋(𝜀+1)𝑤𝑖	NUM
cana-1956	133	5	𝑠−𝑚𝑠𝑞	𝑠−𝑚𝑠𝑞	ADV
cana-1956	133	6	𝑝	𝑝	PROPN
cana-1956	133	7	∑	∑	PROPN
cana-1956	133	8	𝑒2𝑖𝜋(𝜀+1)(𝜀+1)𝑘𝑡	𝑒2𝑖𝜋(𝜀+1)(𝜀+1)𝑘𝑡	PROPN
cana-1956	133	9	𝑞−1	𝑞−1	PROPN
cana-1956	133	10	𝑡=0	𝑡=0	PROPN
cana-1956	133	11	𝑒	𝑒	PROPN
cana-1956	133	12	2𝑖𝜋(𝜀+1)𝑤𝑖𝑘𝑡𝑗	2𝑖𝜋(𝜀+1)𝑤𝑖𝑘𝑡𝑗	NUM
cana-1956	133	13	𝑝	𝑝	PROPN
cana-1956	133	14	𝑞	𝑞	X
cana-1956	133	15	∑	∑	ADV
cana-1956	133	16	𝑔	𝑔	PROPN
cana-1956	133	17	(	(	PUNCT
cana-1956	133	18	𝑥	𝑥	PROPN
cana-1956	133	19	+	+	CCONJ
cana-1956	133	20	𝜀+1	𝜀+1	PROPN
cana-1956	133	21	𝑝	𝑝	PROPN
cana-1956	133	22	(	(	PUNCT
cana-1956	133	23	𝑘𝑡𝑝	𝑘𝑡𝑝	NOUN
cana-1956	133	24	+	+	CCONJ
cana-1956	133	25	𝑚𝑠𝑞	𝑚𝑠𝑞	NOUN
cana-1956	133	26	)	)	PUNCT
cana-1956	133	27	−	−	X
cana-1956	133	28	𝑚(𝜀	𝑚(𝜀	X
cana-1956	133	29	+	+	CCONJ
cana-1956	133	30	1)𝑞	1)𝑞	NUM
cana-1956	133	31	)	)	PUNCT
cana-1956	133	32	𝑒	𝑒	NOUN
cana-1956	133	33	2𝑖𝜋(𝜀+1)𝑤𝑖𝑚(𝜀+1)𝑞	2𝑖𝜋(𝜀+1)𝑤𝑖𝑚(𝜀+1)𝑞	NOUN
cana-1956	133	34	𝑚∈𝕫`⏟	𝑚∈𝕫`⏟	ADJ
cana-1956	133	35	=	=	PRON
cana-1956	133	36	∑	∑	PUNCT
cana-1956	133	37	(	(	PUNCT
cana-1956	133	38	𝑧(𝜀+1)𝑞𝑔(𝑥+	𝑧(𝜀+1)𝑞𝑔(𝑥+	PROPN
cana-1956	133	39	𝜀+1	𝜀+1	PROPN
cana-1956	133	40	𝑝	𝑝	PROPN
cana-1956	133	41	(	(	PUNCT
cana-1956	133	42	𝑘𝑡𝑝+𝑚𝑠𝑞),𝑤𝑖))𝑖	𝑘𝑡𝑝+𝑚𝑠𝑞),𝑤𝑖))𝑖	NOUN
cana-1956	133	43	𝑖	𝑖	PROPN
cana-1956	133	44	(	(	PUNCT
cana-1956	133	45	14	14	NUM
cana-1956	133	46	)	)	PUNCT
cana-1956	133	47	because	because	SCONJ
cana-1956	133	48	the	the	DET
cana-1956	133	49	integer	integer	NOUN
cana-1956	133	50	𝑘𝑡	𝑘𝑡	PROPN
cana-1956	133	51	passes	pass	VERB
cana-1956	133	52	through	through	ADP
cana-1956	133	53	the	the	DET
cana-1956	133	54	set	set	NOUN
cana-1956	133	55	{	{	PUNCT
cana-1956	133	56	0	0	NUM
cana-1956	133	57	,	,	PUNCT
cana-1956	133	58	…	…	PUNCT
cana-1956	133	59	,	,	PUNCT
cana-1956	133	60	𝑞	𝑞	X
cana-1956	133	61	−	−	PROPN
cana-1956	133	62	1	1	NUM
cana-1956	133	63	}	}	PUNCT
cana-1956	133	64	as	as	SCONJ
cana-1956	133	65	t	t	PROPN
cana-1956	133	66	runs	run	VERB
cana-1956	133	67	through	through	ADP
cana-1956	133	68	it	it	PRON
cana-1956	133	69	,	,	PUNCT
cana-1956	133	70	we	we	PRON
cana-1956	133	71	may	may	AUX
cana-1956	133	72	rewrite	rewrite	VERB
cana-1956	133	73	[	[	X
cana-1956	133	74	14	14	NUM
cana-1956	133	75	]	]	PUNCT
cana-1956	133	76	as	as	ADP
cana-1956	133	77	:	:	PUNCT
cana-1956	133	78	𝑋𝑠	𝑋𝑠	PROPN
cana-1956	133	79	𝑗	𝑗	PROPN
cana-1956	133	80	=	=	ADJ
cana-1956	133	81	∑𝑒	∑𝑒	PROPN
cana-1956	133	82	2𝑖𝜋(𝜀+1)𝑤𝑖	2𝑖𝜋(𝜀+1)𝑤𝑖	NUM
cana-1956	133	83	𝑠−𝑚𝑠𝑞	𝑠−𝑚𝑠𝑞	NOUN
cana-1956	133	84	𝑝	𝑝	PRON
cana-1956	133	85	∑𝑒2𝑖𝜋(𝜀+1)𝑤𝑖𝜏	∑𝑒2𝑖𝜋(𝜀+1)𝑤𝑖𝜏	PROPN
cana-1956	133	86	𝑞−1	𝑞−1	PROPN
cana-1956	133	87	𝜏=0	𝜏=0	PUNCT
cana-1956	133	88	𝑒	𝑒	PROPN
cana-1956	133	89	2𝑖𝜋(𝜀+1)𝑤𝑖𝜏𝑗	2𝑖𝜋(𝜀+1)𝑤𝑖𝜏𝑗	PROPN
cana-1956	133	90	𝑝	𝑝	PROPN
cana-1956	133	91	𝑞	𝑞	NOUN
cana-1956	133	92	∑	∑	ADP
cana-1956	133	93	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1956	133	94	+	+	CCONJ
cana-1956	133	95	𝜀	𝜀	X
cana-1956	133	96	+	+	CCONJ
cana-1956	133	97	1	1	PROPN
cana-1956	133	98	𝑝	𝑝	NOUN
cana-1956	133	99	(	(	PUNCT
cana-1956	133	100	𝜏𝑝	𝜏𝑝	ADP
cana-1956	133	101	+	+	NOUN
cana-1956	133	102	𝑚𝑠𝑞	𝑚𝑠𝑞	NOUN
cana-1956	133	103	)	)	PUNCT
cana-1956	133	104	−𝑚(𝜀	−𝑚(𝜀	NOUN
cana-1956	133	105	+	+	CCONJ
cana-1956	133	106	1)𝑞	1)𝑞	NUM
cana-1956	133	107	)	)	PUNCT
cana-1956	133	108	𝑒	𝑒	NOUN
cana-1956	133	109	2𝑖𝜋(𝜀+1)𝑤𝑖𝑚(𝜀+1)𝑞	2𝑖𝜋(𝜀+1)𝑤𝑖𝑚(𝜀+1)𝑞	NOUN
cana-1956	133	110	𝑚∈𝕫`⏟	𝑚∈𝕫`⏟	ADJ
cana-1956	133	111	=	=	ADJ
cana-1956	133	112	𝑧(𝜀+1)𝑞𝑔(𝑥+	𝑧(𝜀+1)𝑞𝑔(𝑥+	PROPN
cana-1956	133	113	𝜀+1	𝜀+1	PROPN
cana-1956	133	114	𝑝	𝑝	PROPN
cana-1956	133	115	(	(	PUNCT
cana-1956	133	116	𝜏𝑝+𝑚𝑠𝑞),𝑤𝑖	𝜏𝑝+𝑚𝑠𝑞),𝑤𝑖	PROPN
cana-1956	133	117	)	)	PUNCT
cana-1956	133	118	𝑖	𝑖	NOUN
cana-1956	133	119	or	or	CCONJ
cana-1956	133	120	∑	∑	ADV
cana-1956	133	121	𝒬(𝑥,𝑤𝑖)𝑖	𝒬(𝑥,𝑤𝑖)𝑖	PROPN
cana-1956	133	122	=	=	SYM
cana-1956	133	123	𝑑𝑖𝑎𝑔∑	𝑑𝑖𝑎𝑔∑	PROPN
cana-1956	133	124	{	{	PUNCT
cana-1956	133	125	𝑒	𝑒	PROPN
cana-1956	133	126	2𝑖𝜋(𝜀+1)𝑤𝑖	2𝑖𝜋(𝜀+1)𝑤𝑖	NUM
cana-1956	133	127	𝑠−𝑚𝑠𝑞	𝑠−𝑚𝑠𝑞	NOUN
cana-1956	133	128	𝑝	𝑝	PROPN
cana-1956	133	129	}	}	PUNCT
cana-1956	133	130	𝑠=0	𝑠=0	PROPN
cana-1956	133	131	𝑝−1	𝑝−1	PROPN
cana-1956	133	132	�	�	PROPN
cana-1956	133	133	̃	̃	PROPN
cana-1956	133	134	�	�	PROPN
cana-1956	133	135	(𝑥	(𝑥	NOUN
cana-1956	133	136	,	,	PUNCT
cana-1956	133	137	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	133	138	)	)	PUNCT
cana-1956	133	139	𝑑𝑖𝑎𝑔{𝑒	𝑑𝑖𝑎𝑔{𝑒	ADV
cana-1956	133	140	2𝑖𝜋(𝜀+1)𝑤𝑖𝜏	2𝑖𝜋(𝜀+1)𝑤𝑖𝜏	NUM
cana-1956	133	141	}	}	PUNCT
cana-1956	133	142	𝜏=0	𝜏=0	PUNCT
cana-1956	133	143	𝑞−1	𝑞−1	PROPN
cana-1956	133	144	𝑊𝑖	𝑊𝑖	PROPN
cana-1956	133	145	,	,	PUNCT
cana-1956	133	146	or	or	CCONJ
cana-1956	133	147	∑	∑	ADP
cana-1956	133	148	�	�	PROPN
cana-1956	133	149	̃	̃	PROPN
cana-1956	133	150	�	�	PROPN
cana-1956	133	151	(𝑥	(𝑥	PROPN
cana-1956	133	152	,	,	PUNCT
cana-1956	133	153	𝑤𝑖)𝑖	𝑤𝑖)𝑖	PROPN
cana-1956	133	154	=	=	SYM
cana-1956	133	155	∑	∑	PUNCT
cana-1956	133	156	(	(	PUNCT
cana-1956	133	157	𝑍(𝜀+1)𝑞𝑔	𝑍(𝜀+1)𝑞𝑔	NOUN
cana-1956	133	158	(	(	PUNCT
cana-1956	133	159	𝑥	𝑥	PROPN
cana-1956	133	160	+	+	CCONJ
cana-1956	133	161	𝜀+1	𝜀+1	PROPN
cana-1956	133	162	𝑝	𝑝	PROPN
cana-1956	133	163	(	(	PUNCT
cana-1956	133	164	𝜏𝑝	𝜏𝑝	NOUN
cana-1956	133	165	+	+	NUM
cana-1956	133	166	𝑚𝑠𝑞	𝑚𝑠𝑞	NOUN
cana-1956	133	167	)	)	PUNCT
cana-1956	133	168	,	,	PUNCT
cana-1956	133	169	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	133	170	)	)	PUNCT
cana-1956	133	171	)	)	PUNCT
cana-1956	134	1	𝑠=0,𝜏=0	𝑠=0,𝜏=0	PROPN
cana-1956	135	1	𝑝−1,𝑞−1	𝑝−1,𝑞−1	NOUN
cana-1956	135	2	,	,	PUNCT
cana-1956	135	3	𝑊	𝑊	PROPN
cana-1956	135	4	=	=	SYM
cana-1956	135	5	(	(	PUNCT
cana-1956	135	6	𝑒	𝑒	PROPN
cana-1956	135	7	2𝑖𝜋𝜏𝑗	2𝑖𝜋𝜏𝑗	NUM
cana-1956	135	8	𝑝	𝑝	PROPN
cana-1956	135	9	𝑞	𝑞	PROPN
cana-1956	135	10	)	)	PUNCT
cana-1956	135	11	𝜏,𝑗=0	𝜏,𝑗=0	NOUN
cana-1956	136	1	𝑞−1	𝑞−1	PROPN
cana-1956	136	2	𝑖	𝑖	PROPN
cana-1956	136	3	.	.	PUNCT
cana-1956	137	1	the	the	DET
cana-1956	137	2	matrix	matrix	NOUN
cana-1956	137	3	values	value	VERB
cana-1956	137	4	�	�	PROPN
cana-1956	137	5	̃	̃	PROPN
cana-1956	137	6	�	�	PROPN
cana-1956	137	7	(𝑥	(𝑥	NOUN
cana-1956	137	8	,	,	PUNCT
cana-1956	137	9	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	137	10	)	)	PUNCT
cana-1956	137	11	and	and	CCONJ
cana-1956	137	12	𝒬(𝑥,𝑤𝑖	𝒬(𝑥,𝑤𝑖	NOUN
cana-1956	137	13	)	)	PUNCT
cana-1956	137	14	have	have	VERB
cana-1956	137	15	the	the	DET
cana-1956	137	16	same	same	ADJ
cana-1956	137	17	rank	rank	NOUN
cana-1956	137	18	.	.	PUNCT
cana-1956	138	1	the	the	DET
cana-1956	138	2	matrices	matrix	NOUN
cana-1956	138	3	�	�	PROPN
cana-1956	138	4	̃	̃	PROPN
cana-1956	138	5	�	�	PROPN
cana-1956	138	6	(𝑥	(𝑥	NOUN
cana-1956	138	7	,	,	PUNCT
cana-1956	138	8	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	138	9	)	)	PUNCT
cana-1956	138	10	and	and	CCONJ
cana-1956	138	11	𝒬(𝑥,𝑤𝑖)difference	𝒬(𝑥,𝑤𝑖)difference	NOUN
cana-1956	138	12	only	only	ADV
cana-1956	138	13	by	by	ADP
cana-1956	138	14	row	row	NOUN
cana-1956	138	15	permutations	permutation	NOUN
cana-1956	138	16	,	,	PUNCT
cana-1956	138	17	resulting	result	VERB
cana-1956	138	18	in	in	ADP
cana-1956	138	19	the	the	DET
cana-1956	138	20	same	same	ADJ
cana-1956	138	21	rank	rank	NOUN
cana-1956	138	22	.	.	PUNCT
cana-1956	139	1	the	the	DET
cana-1956	139	2	number	number	NOUN
cana-1956	139	3	𝑚𝑠	𝑚𝑠	ADP
cana-1956	139	4	spans	span	VERB
cana-1956	139	5	the	the	DET
cana-1956	139	6	entire	entire	ADJ
cana-1956	139	7	set	set	NOUN
cana-1956	139	8	of	of	ADP
cana-1956	139	9	{	{	PUNCT
cana-1956	139	10	0,1	0,1	NUM
cana-1956	139	11	,	,	PUNCT
cana-1956	139	12	…	…	PUNCT
cana-1956	139	13	,	,	PUNCT
cana-1956	139	14	𝑝	𝑝	NOUN
cana-1956	139	15	−	−	PROPN
cana-1956	139	16	1	1	NUM
cana-1956	139	17	}	}	PUNCT
cana-1956	139	18	.	.	PUNCT
cana-1956	140	1	5.examples	5.examples	NUM
cana-1956	140	2	according	accord	VERB
cana-1956	140	3	to	to	ADP
cana-1956	140	4	[	[	X
cana-1956	140	5	5	5	NUM
cana-1956	140	6	]	]	PUNCT
cana-1956	140	7	,	,	PUNCT
cana-1956	140	8	the	the	DET
cana-1956	140	9	gabor	gabor	PROPN
cana-1956	140	10	system	system	PROPN
cana-1956	140	11	𝒢(ℎ1	𝒢(ℎ1	PROPN
cana-1956	140	12	,	,	PUNCT
cana-1956	140	13	𝜀	𝜀	X
cana-1956	140	14	+	+	ADJ
cana-1956	140	15	1	1	NUM
cana-1956	140	16	,	,	PUNCT
cana-1956	140	17	𝜀	𝜀	VERB
cana-1956	140	18	−	−	NOUN
cana-1956	140	19	1	1	NUM
cana-1956	140	20	)	)	PUNCT
cana-1956	140	21	is	be	AUX
cana-1956	140	22	a	a	DET
cana-1956	140	23	frame	frame	NOUN
cana-1956	140	24	if	if	SCONJ
cana-1956	140	25	𝜀2	𝜀2	VERB
cana-1956	140	26	<	<	X
cana-1956	140	27	3	3	NUM
cana-1956	140	28	2	2	NUM
cana-1956	140	29	,	,	PUNCT
cana-1956	140	30	but	but	CCONJ
cana-1956	140	31	not	not	PART
cana-1956	140	32	if	if	SCONJ
cana-1956	140	33	𝜀2	𝜀2	NOUN
cana-1956	140	34	=	=	PUNCT
cana-1956	140	35	3	3	NUM
cana-1956	140	36	2	2	NUM
cana-1956	140	37	.	.	PUNCT
cana-1956	141	1	also	also	ADV
cana-1956	141	2	,	,	PUNCT
cana-1956	141	3	the	the	DET
cana-1956	141	4	authors	author	NOUN
cana-1956	141	5	present	present	VERB
cana-1956	141	6	an	an	DET
cana-1956	141	7	example	example	NOUN
cana-1956	141	8	indicating	indicate	VERB
cana-1956	141	9	that	that	SCONJ
cana-1956	141	10	this	this	DET
cana-1956	141	11	result	result	NOUN
cana-1956	141	12	could	could	AUX
cana-1956	141	13	be	be	AUX
cana-1956	141	14	sharp	sharp	ADJ
cana-1956	141	15	.	.	PUNCT
cana-1956	142	1	this	this	DET
cana-1956	142	2	section	section	NOUN
cana-1956	142	3	demonstrates	demonstrate	VERB
cana-1956	142	4	that	that	SCONJ
cana-1956	142	5	the	the	DET
cana-1956	142	6	system	system	NOUN
cana-1956	142	7	𝒢(ℎ1	𝒢(ℎ1	PROPN
cana-1956	142	8	,	,	PUNCT
cana-1956	142	9	𝜀	𝜀	X
cana-1956	142	10	+	+	ADJ
cana-1956	142	11	1	1	NUM
cana-1956	142	12	,	,	PUNCT
cana-1956	142	13	𝜀	𝜀	VERB
cana-1956	142	14	−	−	ADP
cana-1956	142	15	1)yields	1)yields	NUM
cana-1956	142	16	a	a	DET
cana-1956	142	17	frame	frame	NOUN
cana-1956	142	18	in	in	ADP
cana-1956	142	19	𝐿2(ℝ),for	𝐿2(ℝ),for	ADP
cana-1956	142	20	𝜀	𝜀	NOUN
cana-1956	142	21	+	+	NOUN
cana-1956	142	22	1	1	NUM
cana-1956	142	23	,	,	PUNCT
cana-1956	142	24	𝜀	𝜀	VERB
cana-1956	142	25	−	−	PROPN
cana-1956	142	26	1,with	1,with	NUM
cana-1956	142	27	𝜀2	𝜀2	PROPN
cana-1956	142	28	>	>	X
cana-1956	142	29	3	3	NUM
cana-1956	142	30	2	2	NUM
cana-1956	142	31	.	.	PUNCT
cana-1956	143	1	the	the	DET
cana-1956	143	2	proof	proof	NOUN
cana-1956	143	3	is	be	AUX
cana-1956	143	4	based	base	VERB
cana-1956	143	5	on	on	ADP
cana-1956	143	6	the	the	DET
cana-1956	143	7	matrix	matrix	NOUN
cana-1956	143	8	-	-	PUNCT
cana-1956	143	9	function	function	NOUN
cana-1956	143	10	p	p	NOUN
cana-1956	143	11	from	from	ADP
cana-1956	143	12	the	the	DET
cana-1956	143	13	previous	previous	ADJ
cana-1956	143	14	paragraph	paragraph	NOUN
cana-1956	143	15	and	and	CCONJ
cana-1956	143	16	a	a	DET
cana-1956	143	17	result	result	NOUN
cana-1956	143	18	on	on	ADP
cana-1956	143	19	diagonally	diagonally	ADV
cana-1956	143	20	dominant	dominant	ADJ
cana-1956	143	21	matrices	matrix	NOUN
cana-1956	143	22	.	.	PUNCT
cana-1956	144	1	theorem	theorem	VERB
cana-1956	144	2	5.1	5.1	NUM
cana-1956	144	3	:	:	PUNCT
cana-1956	144	4	suppose	suppose	VERB
cana-1956	144	5	that	that	SCONJ
cana-1956	144	6	𝜀2	𝜀2	NOUN
cana-1956	144	7	=	=	SYM
cana-1956	144	8	8	8	NUM
cana-1956	144	9	5	5	NUM
cana-1956	144	10	and	and	CCONJ
cana-1956	144	11	ℎ1(𝑡	ℎ1(𝑡	NOUN
cana-1956	144	12	)	)	PUNCT
cana-1956	144	13	=	=	NOUN
cana-1956	144	14	𝑡𝑒	𝑡𝑒	ADP
cana-1956	144	15	−𝜋𝑡2	−𝜋𝑡2	PROPN
cana-1956	144	16	,	,	PUNCT
cana-1956	144	17	when	when	SCONJ
cana-1956	144	18	the	the	DET
cana-1956	144	19	system	system	NOUN
cana-1956	144	20	:	:	PUNCT
cana-1956	144	21	𝒢(ℎ1	𝒢(ℎ1	NOUN
cana-1956	144	22	,	,	PUNCT
cana-1956	144	23	𝜀	𝜀	X
cana-1956	144	24	+	+	ADJ
cana-1956	144	25	1	1	NUM
cana-1956	144	26	,	,	PUNCT
cana-1956	144	27	𝜀	𝜀	X
cana-1956	144	28	−	−	NOUN
cana-1956	144	29	1	1	NUM
cana-1956	144	30	)	)	PUNCT
cana-1956	144	31	a	a	DET
cana-1956	144	32	structure	structure	NOUN
cana-1956	144	33	in	in	ADP
cana-1956	144	34	𝐿2(ℝ	𝐿2(ℝ	NOUN
cana-1956	144	35	)	)	PUNCT
cana-1956	144	36	.	.	PUNCT
cana-1956	145	1	proof	proof	NOUN
cana-1956	145	2	:	:	PUNCT
cana-1956	145	3	the	the	DET
cana-1956	145	4	structure	structure	NOUN
cana-1956	145	5	of	of	ADP
cana-1956	145	6	the	the	DET
cana-1956	145	7	matrix	matrix	NOUN
cana-1956	145	8	�	�	PROPN
cana-1956	145	9	̃	̃	PROPN
cana-1956	145	10	�	�	PROPN
cana-1956	145	11	(𝑥	(𝑥	NOUN
cana-1956	145	12	,	,	PUNCT
cana-1956	145	13	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	145	14	)	)	PUNCT
cana-1956	145	15	is	be	AUX
cana-1956	145	16	defined	define	VERB
cana-1956	145	17	as	as	ADP
cana-1956	145	18	:	:	PUNCT
cana-1956	145	19	∑	∑	PUNCT
cana-1956	145	20	𝒫1(𝑥	𝒫1(𝑥	NOUN
cana-1956	145	21	,	,	PUNCT
cana-1956	145	22	𝑤𝑖)𝑖	𝑤𝑖)𝑖	PROPN
cana-1956	145	23	=	=	SYM
cana-1956	145	24	∑	∑	PUNCT
cana-1956	145	25	(	(	PUNCT
cana-1956	145	26	(	(	PUNCT
cana-1956	145	27	𝑍8(𝜀+1)ℎ1(𝑥	𝑍8(𝜀+1)ℎ1(𝑥	PUNCT
cana-1956	145	28	+	+	CCONJ
cana-1956	145	29	(	(	PUNCT
cana-1956	145	30	𝜀	𝜀	X
cana-1956	145	31	+	+	X
cana-1956	145	32	1)𝑡	1)𝑡	NUM
cana-1956	145	33	+	+	CCONJ
cana-1956	145	34	𝑠	𝑠	PROPN
cana-1956	145	35	5(𝜀+1	5(𝜀+1	NUM
cana-1956	145	36	)	)	PUNCT
cana-1956	145	37	8	8	NUM
cana-1956	145	38	,	,	PUNCT
cana-1956	145	39	𝑤𝑖)))𝑠,𝑡=0	𝑤𝑖)))𝑠,𝑡=0	DET
cana-1956	145	40	7,4	7,4	NUM
cana-1956	145	41	𝑖	𝑖	NOUN
cana-1956	145	42	.	.	PUNCT
cana-1956	146	1	to	to	PART
cana-1956	146	2	prove	prove	VERB
cana-1956	146	3	rank	rank	PROPN
cana-1956	146	4	∑	∑	PROPN
cana-1956	146	5	�	�	PROPN
cana-1956	146	6	̃	̃	PROPN
cana-1956	146	7	�	�	PROPN
cana-1956	146	8	(𝑥	(𝑥	NOUN
cana-1956	146	9	,	,	PUNCT
cana-1956	146	10	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	146	11	)	)	PUNCT
cana-1956	147	1	=	=	SYM
cana-1956	147	2	5𝑖	5𝑖	NOUN
cana-1956	147	3	,	,	PUNCT
cana-1956	147	4	for	for	ADP
cana-1956	147	5	all	all	PRON
cana-1956	147	6	(	(	PUNCT
cana-1956	147	7	𝑥	𝑥	NOUN
cana-1956	147	8	,	,	PUNCT
cana-1956	147	9	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	147	10	)	)	PUNCT
cana-1956	147	11	∈	∈	PROPN
cana-1956	147	12	𝑄𝜀+1,5	𝑄𝜀+1,5	PROPN
cana-1956	147	13	,	,	PUNCT
cana-1956	147	14	apply	apply	VERB
cana-1956	147	15	theorem	theorem	ADJ
cana-1956	147	16	2	2	NUM
cana-1956	147	17	and	and	CCONJ
cana-1956	147	18	corollary	corollary	ADJ
cana-1956	147	19	2	2	NUM
cana-1956	147	20	.	.	PUNCT
cana-1956	148	1	we	we	PRON
cana-1956	148	2	structured	structure	VERB
cana-1956	148	3	the	the	DET
cana-1956	148	4	proof	proof	NOUN
cana-1956	148	5	into	into	ADP
cana-1956	148	6	multiple	multiple	ADJ
cana-1956	148	7	steps	step	NOUN
cana-1956	148	8	.	.	PUNCT
cana-1956	149	1	communications	communication	NOUN
cana-1956	149	2	on	on	ADP
cana-1956	149	3	applied	apply	VERB
cana-1956	149	4	nonlinear	nonlinear	ADJ
cana-1956	149	5	analysis	analysis	NOUN
cana-1956	149	6	issn	issn	NOUN
cana-1956	149	7	:	:	PUNCT
cana-1956	149	8	1074	1074	NUM
cana-1956	149	9	-	-	PUNCT
cana-1956	149	10	133x	133x	NUM
cana-1956	149	11	vol	vol	NOUN
cana-1956	149	12	32	32	NUM
cana-1956	149	13	no	no	NOUN
cana-1956	149	14	.	.	NOUN
cana-1956	149	15	3	3	NUM
cana-1956	149	16	(	(	PUNCT
cana-1956	149	17	2025	2025	NUM
cana-1956	149	18	)	)	PUNCT
cana-1956	149	19	236	236	NUM
cana-1956	149	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	149	21	a	a	PRON
cana-1956	149	22	)	)	PUNCT
cana-1956	149	23	establish	establish	VERB
cana-1956	149	24	that	that	SCONJ
cana-1956	149	25	𝒢(ℎ1	𝒢(ℎ1	PROPN
cana-1956	149	26	,	,	PUNCT
cana-1956	149	27	𝜀	𝜀	X
cana-1956	149	28	+	+	ADJ
cana-1956	149	29	1	1	NUM
cana-1956	149	30	,	,	PUNCT
cana-1956	149	31	𝜀	𝜀	VERB
cana-1956	149	32	−	−	NOUN
cana-1956	149	33	1	1	NUM
cana-1956	149	34	)	)	PUNCT
cana-1956	149	35	)	)	PUNCT
cana-1956	149	36	is	be	AUX
cana-1956	149	37	a	a	DET
cana-1956	149	38	frame	frame	NOUN
cana-1956	149	39	for	for	ADP
cana-1956	149	40	𝜀2	𝜀2	NOUN
cana-1956	149	41	≥	≥	NOUN
cana-1956	149	42	2√2	2√2	NUM
cana-1956	149	43	5	5	NUM
cana-1956	149	44	+	+	CCONJ
cana-1956	149	45	1	1	NUM
cana-1956	149	46	,	,	PUNCT
cana-1956	149	47	simply	simply	ADV
cana-1956	149	48	show	show	VERB
cana-1956	149	49	that	that	SCONJ
cana-1956	149	50	𝜀2	𝜀2	NOUN
cana-1956	149	51	=	=	PUNCT
cana-1956	149	52	3	3	NUM
cana-1956	149	53	2	2	NUM
cana-1956	149	54	.	.	PUNCT
cana-1956	150	1	using	use	VERB
cana-1956	150	2	the	the	DET
cana-1956	150	3	fourier	fourier	NOUN
cana-1956	150	4	transform	transform	NOUN
cana-1956	150	5	,	,	PUNCT
cana-1956	150	6	the	the	DET
cana-1956	150	7	case	case	NOUN
cana-1956	150	8	𝜀2	𝜀2	VERB
cana-1956	150	9	≥	≥	NOUN
cana-1956	150	10	2√2	2√2	NUM
cana-1956	150	11	5	5	NUM
cana-1956	150	12	+	+	CCONJ
cana-1956	150	13	1	1	NUM
cana-1956	150	14	is	be	AUX
cana-1956	150	15	reduced	reduce	VERB
cana-1956	150	16	to	to	ADP
cana-1956	150	17	the	the	DET
cana-1956	150	18	prior	prior	ADJ
cana-1956	150	19	one	one	NUM
cana-1956	150	20	.	.	PUNCT
cana-1956	151	1	b	b	X
cana-1956	151	2	)	)	PUNCT
cana-1956	151	3	because	because	SCONJ
cana-1956	151	4	the	the	DET
cana-1956	151	5	function	function	NOUN
cana-1956	151	6	ℎ1	ℎ1	PROPN
cana-1956	151	7	decays	decay	VERB
cana-1956	151	8	quickly	quickly	ADV
cana-1956	151	9	,	,	PUNCT
cana-1956	151	10	we	we	PRON
cana-1956	151	11	can	can	AUX
cana-1956	151	12	approximate	approximate	VERB
cana-1956	151	13	𝑧5(𝜀+1)ℎ1(𝑥	𝑧5(𝜀+1)ℎ1(𝑥	NUM
cana-1956	151	14	,	,	PUNCT
cana-1956	151	15	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	151	16	)	)	PUNCT
cana-1956	151	17	using	use	VERB
cana-1956	151	18	the	the	DET
cana-1956	151	19	series	series	NOUN
cana-1956	151	20	'	'	PART
cana-1956	151	21	maximal	maximal	ADJ
cana-1956	151	22	term	term	NOUN
cana-1956	151	23	.	.	PUNCT
cana-1956	152	1	specifically	specifically	ADV
cana-1956	152	2	,	,	PUNCT
cana-1956	152	3	the	the	DET
cana-1956	152	4	following	follow	VERB
cana-1956	152	5	holds	hold	VERB
cana-1956	152	6	true	true	ADJ
cana-1956	152	7	.	.	PUNCT
cana-1956	153	1	lemma	lemma	PROPN
cana-1956	153	2	5.2	5.2	NUM
cana-1956	153	3	:	:	PUNCT
cana-1956	153	4	let	let	VERB
cana-1956	153	5	0	0	NUM
cana-1956	153	6	≤	≤	NUM
cana-1956	153	7	|𝑥|	|𝑥|	ADV
cana-1956	153	8	<	<	X
cana-1956	153	9	5(𝜀+1	5(𝜀+1	NUM
cana-1956	153	10	)	)	PUNCT
cana-1956	153	11	2	2	NUM
cana-1956	153	12	.	.	PUNCT
cana-1956	154	1	then	then	ADV
cana-1956	154	2	:	:	PUNCT
cana-1956	154	3	|∑	|∑	X
cana-1956	154	4	(	(	PUNCT
cana-1956	154	5	ℎ1(𝑥	ℎ1(𝑥	NOUN
cana-1956	154	6	)	)	PUNCT
cana-1956	154	7	)	)	PUNCT
cana-1956	155	1	−	−	ADP
cana-1956	155	2	𝑍5(𝜀+1)ℎ1(𝑥	𝑍5(𝜀+1)ℎ1(𝑥	NOUN
cana-1956	155	3	,	,	PUNCT
cana-1956	155	4	𝑤𝑖)𝑖	𝑤𝑖)𝑖	PROPN
cana-1956	155	5	|	|	ADV
cana-1956	155	6	≤	≤	NUM
cana-1956	155	7	ℎ1(5𝜀	ℎ1(5𝜀	NOUN
cana-1956	155	8	+	+	CCONJ
cana-1956	155	9	5	5	NUM
cana-1956	155	10	−	−	NOUN
cana-1956	155	11	|𝑥|	|𝑥|	NUM
cana-1956	155	12	)	)	PUNCT
cana-1956	155	13	,	,	PUNCT
cana-1956	155	14	where	where	SCONJ
cana-1956	155	15	:	:	PUNCT
cana-1956	155	16	𝐶5(𝜀+1	𝐶5(𝜀+1	X
cana-1956	155	17	)	)	PUNCT
cana-1956	155	18	=	=	SYM
cana-1956	155	19	2	2	NUM
cana-1956	155	20	+	+	SYM
cana-1956	155	21	1	1	NUM
cana-1956	155	22	ℎ(5𝜀	ℎ(5𝜀	NUM
cana-1956	155	23	+	+	X
cana-1956	155	24	5	5	NUM
cana-1956	155	25	)	)	PUNCT
cana-1956	155	26	∑ℎ1	∑ℎ1	VERB
cana-1956	155	27	𝑛≥2	𝑛≥2	NOUN
cana-1956	155	28	(	(	PUNCT
cana-1956	155	29	5𝑛𝜀	5𝑛𝜀	NOUN
cana-1956	155	30	+	+	CCONJ
cana-1956	155	31	5𝑛	5𝑛	NUM
cana-1956	155	32	)	)	PUNCT
cana-1956	156	1	+	+	CCONJ
cana-1956	156	2	ℎ1	ℎ1	PROPN
cana-1956	156	3	(	(	PUNCT
cana-1956	156	4	(	(	PUNCT
cana-1956	156	5	5𝜀	5𝜀	NOUN
cana-1956	156	6	+	+	NOUN
cana-1956	156	7	5)(2𝑛	5)(2𝑛	NUM
cana-1956	156	8	−	−	NOUN
cana-1956	156	9	1	1	NUM
cana-1956	156	10	)	)	PUNCT
cana-1956	156	11	2	2	NUM
cana-1956	156	12	)	)	PUNCT
cana-1956	156	13	this	this	DET
cana-1956	156	14	lemma	lemma	PROPN
cana-1956	156	15	may	may	AUX
cana-1956	156	16	be	be	AUX
cana-1956	156	17	confirmed	confirm	VERB
cana-1956	156	18	immediately	immediately	ADV
cana-1956	156	19	.	.	PUNCT
cana-1956	157	1	for	for	ADP
cana-1956	157	2	practical	practical	ADJ
cana-1956	157	3	purposes	purpose	NOUN
cana-1956	157	4	.	.	PUNCT
cana-1956	158	1	we	we	PRON
cana-1956	158	2	can	can	AUX
cana-1956	158	3	suppose	suppose	VERB
cana-1956	158	4	that	that	SCONJ
cana-1956	158	5	𝐶5(𝜀+1	𝐶5(𝜀+1	X
cana-1956	158	6	)	)	PUNCT
cana-1956	158	7	=	=	SYM
cana-1956	159	1	0	0	X
cana-1956	159	2	.	.	PUNCT
cana-1956	160	1	c	c	X
cana-1956	160	2	)	)	PUNCT
cana-1956	160	3	we	we	PRON
cana-1956	160	4	will	will	AUX
cana-1956	160	5	demonstrate	demonstrate	VERB
cana-1956	160	6	later	later	ADV
cana-1956	160	7	in	in	ADV
cana-1956	160	8	(	(	PUNCT
cana-1956	160	9	𝑔	𝑔	NOUN
cana-1956	160	10	)	)	PUNCT
cana-1956	160	11	,	,	PUNCT
cana-1956	160	12	it	it	PRON
cana-1956	160	13	is	be	AUX
cana-1956	160	14	necessary	necessary	ADJ
cana-1956	160	15	for	for	ADP
cana-1956	160	16	consideration	consideration	NOUN
cana-1956	160	17	:	:	PUNCT
cana-1956	160	18	0	0	NUM
cana-1956	160	19	≤	≤	NUM
cana-1956	160	20	𝑥	𝑥	PRON
cana-1956	160	21	≤	≤	NUM
cana-1956	160	22	𝜀+1	𝜀+1	ADP
cana-1956	160	23	6	6	NUM
cana-1956	160	24	.	.	PUNCT
cana-1956	161	1	the	the	DET
cana-1956	161	2	zak	zak	PROPN
cana-1956	161	3	transform	transform	PROPN
cana-1956	161	4	's	's	PART
cana-1956	161	5	quasi	quasi	NOUN
cana-1956	161	6	-	-	NOUN
cana-1956	161	7	periodicity	periodicity	NOUN
cana-1956	161	8	and	and	CCONJ
cana-1956	161	9	symmetry	symmetry	NOUN
cana-1956	161	10	with	with	ADP
cana-1956	161	11	ℎ1	ℎ1	PROPN
cana-1956	161	12	will	will	AUX
cana-1956	161	13	lead	lead	VERB
cana-1956	161	14	to	to	ADP
cana-1956	161	15	this	this	DET
cana-1956	161	16	result	result	NOUN
cana-1956	161	17	.	.	PUNCT
cana-1956	162	1	we	we	PRON
cana-1956	162	2	split	split	VERB
cana-1956	162	3	the	the	DET
cana-1956	162	4	interval	interval	NOUN
cana-1956	162	5	0	0	NUM
cana-1956	162	6	≤	≤	NUM
cana-1956	163	1	𝑥	𝑥	DET
cana-1956	163	2	≤	≤	NUM
cana-1956	163	3	𝜀+1	𝜀+1	ADP
cana-1956	163	4	6	6	NUM
cana-1956	163	5	into	into	ADP
cana-1956	163	6	two	two	NUM
cana-1956	163	7	parts	part	NOUN
cana-1956	163	8	:	:	PUNCT
cana-1956	163	9	:	:	PUNCT
cana-1956	163	10	0	0	NUM
cana-1956	163	11	≤	≤	NUM
cana-1956	164	1	𝑥	𝑥	ADP
cana-1956	164	2	<	<	X
cana-1956	164	3	𝜀+1	𝜀+1	PROPN
cana-1956	164	4	12	12	NUM
cana-1956	164	5	2	2	NUM
cana-1956	164	6	and	and	CCONJ
cana-1956	164	7	1	1	NUM
cana-1956	164	8	12	12	NUM
cana-1956	164	9	≤	≤	NUM
cana-1956	164	10	𝑥	𝑥	DET
cana-1956	164	11	≤	≤	NUM
cana-1956	164	12	𝜀+1	𝜀+1	ADP
cana-1956	164	13	6	6	NUM
cana-1956	164	14	.	.	PUNCT
cana-1956	165	1	d.	d.	PROPN
cana-1956	165	2	let	let	VERB
cana-1956	165	3	0	0	NUM
cana-1956	165	4	≤	≤	NUM
cana-1956	166	1	𝑥	𝑥	ADP
cana-1956	166	2	<	<	X
cana-1956	166	3	𝜀+1	𝜀+1	PROPN
cana-1956	166	4	12	12	NUM
cana-1956	166	5	.	.	PUNCT
cana-1956	167	1	assume	assume	VERB
cana-1956	167	2	the	the	DET
cana-1956	167	3	sub	sub	ADJ
cana-1956	167	4	-	-	NOUN
cana-1956	167	5	matrix	matrix	NOUN
cana-1956	167	6	𝒫1(𝑥	𝒫1(𝑥	NOUN
cana-1956	167	7	,	,	PUNCT
cana-1956	167	8	𝑤𝑖)for	𝑤𝑖)for	NOUN
cana-1956	167	9	:	:	PUNCT
cana-1956	167	10	𝑡	𝑡	X
cana-1956	167	11	=	=	NOUN
cana-1956	167	12	1,2,3	1,2,3	NUM
cana-1956	167	13	,	,	PUNCT
cana-1956	167	14	…	…	PUNCT
cana-1956	167	15	…	…	PUNCT
cana-1956	167	16	,	,	PUNCT
cana-1956	167	17	when	when	SCONJ
cana-1956	167	18	adjusting	adjust	VERB
cana-1956	167	19	the	the	DET
cana-1956	167	20	second	second	ADJ
cana-1956	167	21	and	and	CCONJ
cana-1956	167	22	third	third	ADJ
cana-1956	167	23	rows	row	NOUN
cana-1956	167	24	,	,	PUNCT
cana-1956	167	25	this	this	DET
cana-1956	167	26	matrix	matrix	NOUN
cana-1956	167	27	assumes	assume	VERB
cana-1956	167	28	the	the	DET
cana-1956	167	29	following	follow	VERB
cana-1956	167	30	type	type	NOUN
cana-1956	167	31	:	:	PUNCT
cana-1956	167	32	(	(	PUNCT
cana-1956	167	33	𝑍5𝜀+5ℎ1(𝑥	𝑍5𝜀+5ℎ1(𝑥	NOUN
cana-1956	168	1	+	+	ADP
cana-1956	168	2	𝜀	𝜀	X
cana-1956	168	3	+	+	CCONJ
cana-1956	168	4	1,𝑤𝑖	1,𝑤𝑖	NUM
cana-1956	168	5	)	)	PUNCT
cana-1956	168	6	𝑍5𝜀+5ℎ1(𝑥	𝑍5𝜀+5ℎ1(𝑥	PUNCT
cana-1956	169	1	+	+	NOUN
cana-1956	169	2	2𝜀	2𝜀	NOUN
cana-1956	169	3	+	+	CCONJ
cana-1956	169	4	2,𝑤𝑖	2,𝑤𝑖	NUM
cana-1956	169	5	)	)	PUNCT
cana-1956	169	6	𝑍5𝜀+5ℎ1(𝑥	𝑍5𝜀+5ℎ1(𝑥	PUNCT
cana-1956	170	1	+	+	NOUN
cana-1956	170	2	3𝜀	3𝜀	NUM
cana-1956	170	3	+	+	ADJ
cana-1956	170	4	3,𝑤𝑖	3,𝑤𝑖	NUM
cana-1956	170	5	)	)	PUNCT
cana-1956	170	6	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	7	(	(	PUNCT
cana-1956	170	8	𝑥	𝑥	X
cana-1956	170	9	+	+	ADJ
cana-1956	170	10	13𝜀	13𝜀	NOUN
cana-1956	170	11	+	+	CCONJ
cana-1956	170	12	13	13	NUM
cana-1956	170	13	3	3	NUM
cana-1956	170	14	,	,	PUNCT
cana-1956	170	15	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	16	)	)	PUNCT
cana-1956	170	17	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	18	(	(	PUNCT
cana-1956	170	19	𝑥	𝑥	PROPN
cana-1956	170	20	+	+	NUM
cana-1956	170	21	16𝜀	16𝜀	NOUN
cana-1956	170	22	+	+	CCONJ
cana-1956	170	23	16	16	NUM
cana-1956	170	24	3	3	NUM
cana-1956	170	25	,	,	PUNCT
cana-1956	170	26	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	27	)	)	PUNCT
cana-1956	170	28	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	29	(	(	PUNCT
cana-1956	170	30	𝑥	𝑥	X
cana-1956	170	31	+	+	NUM
cana-1956	170	32	19𝜀	19𝜀	NOUN
cana-1956	170	33	+	+	CCONJ
cana-1956	170	34	19	19	NUM
cana-1956	170	35	3	3	NUM
cana-1956	170	36	,	,	PUNCT
cana-1956	170	37	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	38	)	)	PUNCT
cana-1956	170	39	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	40	(	(	PUNCT
cana-1956	170	41	𝑥	𝑥	X
cana-1956	170	42	+	+	NUM
cana-1956	170	43	8𝜀	8𝜀	NOUN
cana-1956	170	44	+	+	CCONJ
cana-1956	170	45	8	8	NUM
cana-1956	170	46	3	3	NUM
cana-1956	170	47	,	,	PUNCT
cana-1956	170	48	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	49	)	)	PUNCT
cana-1956	170	50	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	51	(	(	PUNCT
cana-1956	170	52	𝑥	𝑥	PROPN
cana-1956	170	53	+	+	NUM
cana-1956	170	54	11𝜀	11𝜀	X
cana-1956	170	55	+	+	CCONJ
cana-1956	170	56	11	11	NUM
cana-1956	170	57	3	3	NUM
cana-1956	170	58	,	,	PUNCT
cana-1956	170	59	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	60	)	)	PUNCT
cana-1956	170	61	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	NOUN
cana-1956	170	62	(	(	PUNCT
cana-1956	170	63	𝑥	𝑥	PROPN
cana-1956	170	64	+	+	NUM
cana-1956	170	65	11𝜀	11𝜀	NOUN
cana-1956	170	66	+	+	CCONJ
cana-1956	170	67	14	14	NUM
cana-1956	170	68	3	3	NUM
cana-1956	170	69	,	,	PUNCT
cana-1956	170	70	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	170	71	)	)	PUNCT
cana-1956	170	72	)	)	PUNCT
cana-1956	171	1	we	we	PRON
cana-1956	171	2	shall	shall	AUX
cana-1956	171	3	apply	apply	VERB
cana-1956	171	4	the	the	DET
cana-1956	171	5	following	follow	VERB
cana-1956	171	6	theorem	theorem	VERB
cana-1956	171	7	to	to	PART
cana-1956	171	8	exponentially	exponentially	ADV
cana-1956	171	9	dominant	dominant	ADJ
cana-1956	171	10	matrices	matrix	NOUN
cana-1956	171	11	.	.	PUNCT
cana-1956	172	1	theorem	theorem	NOUN
cana-1956	172	2	(	(	PUNCT
cana-1956	172	3	5.3	5.3	NUM
cana-1956	172	4	):	):	PUNCT
cana-1956	172	5	(	(	PUNCT
cana-1956	172	6	refer	refer	VERB
cana-1956	172	7	to	to	ADP
cana-1956	172	8	[	[	X
cana-1956	172	9	15	15	NUM
cana-1956	172	10	]	]	PUNCT
cana-1956	172	11	)	)	PUNCT
cana-1956	172	12	.	.	PUNCT
cana-1956	173	1	if	if	SCONJ
cana-1956	173	2	(	(	PUNCT
cana-1956	173	3	𝑎𝑖	𝑎𝑖	ADP
cana-1956	173	4	𝑘	𝑘	NOUN
cana-1956	173	5	)	)	PUNCT
cana-1956	173	6	,	,	PUNCT
cana-1956	173	7	is	be	AUX
cana-1956	173	8	an	an	DET
cana-1956	173	9	𝑛	𝑛	DET
cana-1956	173	10	×	×	NOUN
cana-1956	173	11	𝑛-matrix	𝑛-matrix	NOUN
cana-1956	173	12	with	with	ADP
cana-1956	173	13	complex	complex	ADJ
cana-1956	173	14	members	member	NOUN
cana-1956	173	15	,	,	PUNCT
cana-1956	173	16	then	then	ADV
cana-1956	173	17	either	either	ADV
cana-1956	173	18	:	:	PUNCT
cana-1956	173	19	(	(	PUNCT
cana-1956	173	20	i	i	NOUN
cana-1956	173	21	)	)	PUNCT
cana-1956	173	22	|𝑎𝑖	|𝑎𝑖	PROPN
cana-1956	173	23	𝑖|	𝑖|	PROPN
cana-1956	173	24	>	>	X
cana-1956	173	25	∑	∑	PUNCT
cana-1956	173	26	|𝑎𝑘	|𝑎𝑘	PROPN
cana-1956	173	27	𝑖	𝑖	SYM
cana-1956	173	28	|𝑘≠𝑖	|𝑘≠𝑖	PROPN
cana-1956	173	29	,	,	PUNCT
cana-1956	173	30	1	1	NUM
cana-1956	173	31	≤	≤	NUM
cana-1956	173	32	𝑖	𝑖	PRON
cana-1956	173	33	≤	≤	NOUN
cana-1956	173	34	𝑛.	𝑛.	NOUN
cana-1956	173	35	(	(	PUNCT
cana-1956	173	36	ii	ii	NOUN
cana-1956	173	37	)	)	PUNCT
cana-1956	173	38	|𝑎𝑖	|𝑎𝑖	AUX
cana-1956	173	39	𝑖||𝑎𝑗	𝑖||𝑎𝑗	NOUN
cana-1956	173	40	𝑗	𝑗	INTJ
cana-1956	173	41	|	|	ADV
cana-1956	173	42	>	>	X
cana-1956	173	43	(	(	PUNCT
cana-1956	173	44	∑	∑	PROPN
cana-1956	173	45	|𝑎𝑘	|𝑎𝑘	PROPN
cana-1956	173	46	𝑖	𝑖	SYM
cana-1956	173	47	|𝑘≠𝑖	|𝑘≠𝑖	PROPN
cana-1956	173	48	)	)	PUNCT
cana-1956	173	49	(	(	PUNCT
cana-1956	173	50	∑	∑	PROPN
cana-1956	173	51	|𝑎𝑘	|𝑎𝑘	NUM
cana-1956	173	52	𝑗	𝑗	ADJ
cana-1956	173	53	|𝑘≠𝑗	|𝑘≠𝑗	NOUN
cana-1956	173	54	)	)	PUNCT
cana-1956	173	55	,	,	PUNCT
cana-1956	173	56	1	1	NUM
cana-1956	173	57	≤	≤	NUM
cana-1956	173	58	𝑖	𝑖	PUNCT
cana-1956	173	59	,	,	PUNCT
cana-1956	173	60	𝑗	𝑗	PROPN
cana-1956	173	61	≤	≤	NUM
cana-1956	173	62	𝑛	𝑛	NOUN
cana-1956	173	63	,	,	PUNCT
cana-1956	173	64	𝑖	𝑖	SYM
cana-1956	173	65	≠	≠	PROPN
cana-1956	173	66	𝑗	𝑗	INTJ
cana-1956	173	67	→	→	NOUN
cana-1956	173	68	|𝑎𝑖𝑘|	|𝑎𝑖𝑘|	ADJ
cana-1956	173	69	≠	≠	PROPN
cana-1956	173	70	0	0	NUM
cana-1956	173	71	.	.	PUNCT
cana-1956	173	72	a.	a.	NOUN
cana-1956	173	73	to	to	PART
cana-1956	173	74	establish	establish	VERB
cana-1956	173	75	the	the	DET
cana-1956	173	76	invertibility	invertibility	NOUN
cana-1956	173	77	of	of	ADP
cana-1956	173	78	the	the	DET
cana-1956	173	79	matrix	matrix	NOUN
cana-1956	173	80	in	in	ADP
cana-1956	173	81	(	(	PUNCT
cana-1956	173	82	d	d	NOUN
cana-1956	173	83	)	)	PUNCT
cana-1956	173	84	,	,	PUNCT
cana-1956	173	85	we	we	PRON
cana-1956	173	86	must	must	AUX
cana-1956	173	87	use	use	VERB
cana-1956	173	88	theorem	theorem	ADJ
cana-1956	173	89	5.3	5.3	NUM
cana-1956	173	90	.	.	PUNCT
cana-1956	174	1	we	we	PRON
cana-1956	174	2	emphasized	emphasize	VERB
cana-1956	174	3	the	the	DET
cana-1956	174	4	essential	essential	ADJ
cana-1956	174	5	steps	step	NOUN
cana-1956	174	6	of	of	ADP
cana-1956	174	7	the	the	DET
cana-1956	174	8	proof	proof	NOUN
cana-1956	174	9	.	.	PUNCT
cana-1956	175	1	•	•	VERB
cana-1956	175	2	given	give	VERB
cana-1956	175	3	that	that	PRON
cana-1956	175	4	|𝑍5𝜀+5ℎ1(𝑥	|𝑍5𝜀+5ℎ1(𝑥	NOUN
cana-1956	175	5	,	,	PUNCT
cana-1956	175	6	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	175	7	is	be	AUX
cana-1956	175	8	(	(	PUNCT
cana-1956	175	9	5𝜀	5𝜀	NOUN
cana-1956	175	10	+	+	NOUN
cana-1956	175	11	5	5	NUM
cana-1956	175	12	)	)	PUNCT
cana-1956	175	13	,	,	PUNCT
cana-1956	175	14	)	)	PUNCT
cana-1956	175	15	periodic	periodic	ADJ
cana-1956	175	16	function	function	NOUN
cana-1956	175	17	,	,	PUNCT
cana-1956	175	18	the	the	DET
cana-1956	175	19	absolute	absolute	ADJ
cana-1956	175	20	numbers	number	NOUN
cana-1956	175	21	of	of	ADP
cana-1956	175	22	the	the	DET
cana-1956	175	23	matrix	matrix	NOUN
cana-1956	175	24	participants	participant	NOUN
cana-1956	175	25	in	in	ADP
cana-1956	175	26	(	(	PUNCT
cana-1956	175	27	e	e	NOUN
cana-1956	175	28	)	)	PUNCT
cana-1956	175	29	can	can	AUX
cana-1956	175	30	be	be	AUX
cana-1956	175	31	written	write	VERB
cana-1956	175	32	as	as	ADP
cana-1956	175	33	:	:	PUNCT
cana-1956	175	34	(	(	PUNCT
cana-1956	175	35	|𝑍5𝜀+5ℎ1(𝑥	|𝑍5𝜀+5ℎ1(𝑥	PROPN
cana-1956	175	36	+	+	NUM
cana-1956	175	37	𝜀	𝜀	X
cana-1956	175	38	+	+	NOUN
cana-1956	175	39	1,𝑤𝑖	1,𝑤𝑖	NUM
cana-1956	175	40	)	)	PUNCT
cana-1956	176	1	|	|	ADV
cana-1956	176	2	|𝑍5𝜀+5ℎ1(𝑥	|𝑍5𝜀+5ℎ1(𝑥	PROPN
cana-1956	176	3	+	+	NUM
cana-1956	176	4	2𝜀	2𝜀	NOUN
cana-1956	176	5	+	+	CCONJ
cana-1956	176	6	2,𝑤𝑖	2,𝑤𝑖	X
cana-1956	176	7	)	)	PUNCT
cana-1956	177	1	|	|	ADV
cana-1956	177	2	|𝑍5𝜀+5ℎ1(𝑥	|𝑍5𝜀+5ℎ1(𝑥	PROPN
cana-1956	177	3	+	+	NUM
cana-1956	177	4	2𝜀	2𝜀	NUM
cana-1956	177	5	−	−	PROPN
cana-1956	177	6	2,𝑤𝑖	2,𝑤𝑖	NUM
cana-1956	177	7	)	)	PUNCT
cana-1956	177	8	|	|	ADV
cana-1956	177	9	|𝑍5𝜀+5ℎ1	|𝑍5𝜀+5ℎ1	PROPN
cana-1956	177	10	(	(	PUNCT
cana-1956	177	11	𝑥	𝑥	NOUN
cana-1956	177	12	−	−	PROPN
cana-1956	177	13	2𝜀	2𝜀	NOUN
cana-1956	177	14	+	+	CCONJ
cana-1956	177	15	2	2	NUM
cana-1956	177	16	3	3	NUM
cana-1956	177	17	,	,	PUNCT
cana-1956	177	18	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	177	19	|𝑍5𝜀+5ℎ1	|𝑍5𝜀+5ℎ1	PROPN
cana-1956	177	20	(	(	PUNCT
cana-1956	177	21	𝑥	𝑥	PROPN
cana-1956	177	22	+	+	X
cana-1956	177	23	𝜀	𝜀	X
cana-1956	177	24	+	+	CCONJ
cana-1956	177	25	1	1	NUM
cana-1956	177	26	3	3	NUM
cana-1956	177	27	,	,	PUNCT
cana-1956	177	28	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	177	29	)	)	PUNCT
cana-1956	177	30	|	|	ADV
cana-1956	177	31	|	|	ADV
cana-1956	177	32	𝑍5𝜀+5ℎ1	𝑍5𝜀+5ℎ1	VERB
cana-1956	177	33	(	(	PUNCT
cana-1956	177	34	𝑥	𝑥	NOUN
cana-1956	177	35	+	+	NUM
cana-1956	177	36	4𝜀	4𝜀	NOUN
cana-1956	177	37	+	+	CCONJ
cana-1956	177	38	4	4	NUM
cana-1956	177	39	3	3	NUM
cana-1956	177	40	,	,	PUNCT
cana-1956	177	41	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	177	42	|𝑍5𝜀+5ℎ1	|𝑍5𝜀+5ℎ1	PROPN
cana-1956	177	43	(	(	PUNCT
cana-1956	177	44	𝑥	𝑥	NOUN
cana-1956	177	45	−	−	NOUN
cana-1956	177	46	7𝜀	7𝜀	NOUN
cana-1956	178	1	+	+	CCONJ
cana-1956	178	2	7	7	NUM
cana-1956	178	3	3	3	NUM
cana-1956	178	4	,	,	PUNCT
cana-1956	178	5	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	178	6	|𝑍5𝜀+5ℎ1	|𝑍5𝜀+5ℎ1	PROPN
cana-1956	178	7	(	(	PUNCT
cana-1956	178	8	𝑥	𝑥	NOUN
cana-1956	178	9	−	−	NOUN
cana-1956	178	10	4𝜀	4𝜀	NOUN
cana-1956	178	11	+	+	CCONJ
cana-1956	178	12	4	4	NUM
cana-1956	178	13	3	3	NUM
cana-1956	178	14	,	,	PUNCT
cana-1956	178	15	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	178	16	)	)	PUNCT
cana-1956	178	17	|	|	ADV
cana-1956	178	18	|𝑍5𝜀+5ℎ1	|𝑍5𝜀+5ℎ1	PROPN
cana-1956	178	19	(	(	PUNCT
cana-1956	178	20	𝑥	𝑥	NOUN
cana-1956	178	21	−	−	PROPN
cana-1956	178	22	𝜀	𝜀	NOUN
cana-1956	178	23	+	+	CCONJ
cana-1956	178	24	1	1	NUM
cana-1956	178	25	3	3	NUM
cana-1956	178	26	,	,	PUNCT
cana-1956	178	27	𝑤𝑖)|	𝑤𝑖)|	ADP
cana-1956	178	28	)	)	PUNCT
cana-1956	178	29	communications	communication	NOUN
cana-1956	178	30	on	on	ADP
cana-1956	178	31	applied	apply	VERB
cana-1956	178	32	nonlinear	nonlinear	ADJ
cana-1956	178	33	analysis	analysis	NOUN
cana-1956	178	34	issn	issn	NOUN
cana-1956	178	35	:	:	PUNCT
cana-1956	178	36	1074	1074	NUM
cana-1956	178	37	-	-	PUNCT
cana-1956	178	38	133x	133x	NUM
cana-1956	178	39	vol	vol	NOUN
cana-1956	178	40	32	32	NUM
cana-1956	178	41	no	no	NOUN
cana-1956	178	42	.	.	NOUN
cana-1956	178	43	3	3	NUM
cana-1956	178	44	(	(	PUNCT
cana-1956	178	45	2025	2025	NUM
cana-1956	178	46	)	)	PUNCT
cana-1956	178	47	237	237	NUM
cana-1956	178	48	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	178	49	•	•	NOUN
cana-1956	178	50	for	for	ADP
cana-1956	178	51	0	0	NUM
cana-1956	178	52	≤	≤	NUM
cana-1956	178	53	𝑥	𝑥	DET
cana-1956	178	54	≤	≤	NUM
cana-1956	178	55	𝜀+1	𝜀+1	ADP
cana-1956	178	56	12	12	NUM
cana-1956	178	57	,	,	PUNCT
cana-1956	178	58	and	and	CCONJ
cana-1956	178	59	√	√	NUM
cana-1956	178	60	3	3	NUM
cana-1956	178	61	5	5	NUM
cana-1956	178	62	−	−	PROPN
cana-1956	178	63	1	1	NUM
cana-1956	178	64	≤	≤	NUM
cana-1956	178	65	𝜀	𝜀	NOUN
cana-1956	178	66	≤	≤	NOUN
cana-1956	178	67	0	0	NUM
cana-1956	178	68	,	,	PUNCT
cana-1956	178	69	the	the	DET
cana-1956	178	70	rule	rule	NOUN
cana-1956	178	71	(	(	PUNCT
cana-1956	178	72	ii	ii	NOUN
cana-1956	178	73	)	)	PUNCT
cana-1956	178	74	in	in	ADP
cana-1956	178	75	theorem	theorem	NOUN
cana-1956	178	76	(	(	PUNCT
cana-1956	178	77	5.3	5.3	NUM
cana-1956	178	78	)	)	PUNCT
cana-1956	178	79	can	can	AUX
cana-1956	178	80	now	now	ADV
cana-1956	178	81	be	be	AUX
cana-1956	178	82	verified	verify	VERB
cana-1956	178	83	effectively	effectively	ADV
cana-1956	178	84	.	.	PUNCT
cana-1956	179	1	•	•	NOUN
cana-1956	179	2	for	for	ADP
cana-1956	179	3	0	0	NUM
cana-1956	179	4	≤	≤	NUM
cana-1956	179	5	𝑥	𝑥	DET
cana-1956	179	6	≤	≤	NUM
cana-1956	179	7	𝜀+1	𝜀+1	ADP
cana-1956	179	8	12	12	NUM
cana-1956	179	9	,	,	PUNCT
cana-1956	179	10	and	and	CCONJ
cana-1956	179	11	𝜀	𝜀	X
cana-1956	179	12	≥	≥	X
cana-1956	179	13	0	0	NUM
cana-1956	179	14	,	,	PUNCT
cana-1956	179	15	we	we	PRON
cana-1956	179	16	consider	consider	VERB
cana-1956	179	17	the	the	DET
cana-1956	179	18	difference	difference	NOUN
cana-1956	179	19	:	:	PUNCT
cana-1956	180	1	𝐻𝑖	𝐻𝑖	PROPN
cana-1956	180	2	𝑖	𝑖	ADP
cana-1956	180	3	−∑𝐻𝑖	−∑𝐻𝑖	PROPN
cana-1956	180	4	𝑘	𝑘	PROPN
cana-1956	180	5	𝑘≠𝑖	𝑘≠𝑖	PROPN
cana-1956	180	6	,	,	PUNCT
cana-1956	180	7	1	1	NUM
cana-1956	180	8	≤	≤	NUM
cana-1956	180	9	𝑖	𝑖	SYM
cana-1956	180	10	≤	≤	NOUN
cana-1956	180	11	3	3	NUM
cana-1956	180	12	if	if	SCONJ
cana-1956	180	13	the	the	DET
cana-1956	180	14	above	above	ADJ
cana-1956	180	15	statement	statement	NOUN
cana-1956	180	16	is	be	AUX
cana-1956	180	17	positive	positive	ADJ
cana-1956	180	18	,	,	PUNCT
cana-1956	180	19	condition	condition	NOUN
cana-1956	180	20	(	(	PUNCT
cana-1956	180	21	i	i	NOUN
cana-1956	180	22	)	)	PUNCT
cana-1956	180	23	in	in	ADP
cana-1956	180	24	theorem	theorem	NOUN
cana-1956	180	25	(	(	PUNCT
cana-1956	180	26	5.3	5.3	NUM
cana-1956	180	27	)	)	PUNCT
cana-1956	180	28	is	be	AUX
cana-1956	180	29	satisfied	satisfied	ADJ
cana-1956	180	30	.	.	PUNCT
cana-1956	181	1	assume	assume	VERB
cana-1956	181	2	the	the	DET
cana-1956	181	3	first	first	ADJ
cana-1956	181	4	situation	situation	NOUN
cana-1956	181	5	,	,	PUNCT
cana-1956	181	6	when	when	SCONJ
cana-1956	181	7	𝑖	𝑖	X
cana-1956	181	8	=	=	SYM
cana-1956	181	9	1	1	X
cana-1956	181	10	.	.	PUNCT
cana-1956	182	1	then	then	ADV
cana-1956	182	2	we	we	PRON
cana-1956	182	3	have	have	VERB
cana-1956	182	4	to	to	PART
cana-1956	182	5	verify	verify	VERB
cana-1956	182	6	:	:	PUNCT
cana-1956	182	7	𝐻1	𝐻1	PROPN
cana-1956	182	8	1	1	NUM
cana-1956	182	9	−	−	NOUN
cana-1956	182	10	𝐻1	𝐻1	PROPN
cana-1956	182	11	2	2	NUM
cana-1956	182	12	−	−	NOUN
cana-1956	182	13	𝐻1	𝐻1	NOUN
cana-1956	182	14	3	3	NUM
cana-1956	182	15	>	>	X
cana-1956	182	16	0	0	PUNCT
cana-1956	183	1	(	(	PUNCT
cana-1956	183	2	15	15	NUM
cana-1956	183	3	)	)	PUNCT
cana-1956	183	4	let	let	VERB
cana-1956	183	5	𝑥	𝑥	NOUN
cana-1956	183	6	=	=	PUNCT
cana-1956	183	7	𝑦(𝜀	𝑦(𝜀	NOUN
cana-1956	183	8	+	+	CCONJ
cana-1956	183	9	1	1	NUM
cana-1956	183	10	)	)	PUNCT
cana-1956	183	11	,	,	PUNCT
cana-1956	183	12	for	for	ADP
cana-1956	183	13	0	0	NUM
cana-1956	183	14	≤	≤	NUM
cana-1956	183	15	𝑦	𝑦	SYM
cana-1956	183	16	≤	≤	NUM
cana-1956	183	17	1	1	NUM
cana-1956	183	18	12	12	NUM
cana-1956	183	19	.	.	PUNCT
cana-1956	184	1	then	then	ADV
cana-1956	184	2	(	(	PUNCT
cana-1956	184	3	15	15	X
cana-1956	184	4	)	)	PUNCT
cana-1956	184	5	is	be	AUX
cana-1956	184	6	equivalent	equivalent	ADJ
cana-1956	184	7	to	to	ADP
cana-1956	184	8	:	:	PUNCT
cana-1956	184	9	ℎ1((𝜀	ℎ1((𝜀	PROPN
cana-1956	184	10	+	+	CCONJ
cana-1956	184	11	1)(𝑦	1)(𝑦	NUM
cana-1956	184	12	+	+	NOUN
cana-1956	184	13	1	1	NUM
cana-1956	184	14	)	)	PUNCT
cana-1956	184	15	)	)	PUNCT
cana-1956	185	1	−	−	PROPN
cana-1956	186	1	3ℎ1((𝜀	3ℎ1((𝜀	NUM
cana-1956	187	1	+	+	NUM
cana-1956	187	2	1)(𝑦	1)(𝑦	NUM
cana-1956	187	3	+	+	CCONJ
cana-1956	187	4	2	2	NUM
cana-1956	187	5	)	)	PUNCT
cana-1956	187	6	)	)	PUNCT
cana-1956	188	1	−	−	PROPN
cana-1956	189	1	3ℎ1((𝜀	3ℎ1((𝜀	NUM
cana-1956	190	1	+	+	NUM
cana-1956	190	2	1)(𝑦	1)(𝑦	NUM
cana-1956	190	3	−	−	NOUN
cana-1956	190	4	2	2	NUM
cana-1956	190	5	)	)	PUNCT
cana-1956	190	6	)	)	PUNCT
cana-1956	191	1	>	>	PUNCT
cana-1956	192	1	ℎ1	ℎ1	PROPN
cana-1956	192	2	(	(	PUNCT
cana-1956	192	3	(	(	PUNCT
cana-1956	192	4	𝜀	𝜀	X
cana-1956	192	5	+	+	ADJ
cana-1956	192	6	1	1	NUM
cana-1956	192	7	)	)	PUNCT
cana-1956	192	8	(	(	PUNCT
cana-1956	192	9	1	1	NUM
cana-1956	192	10	2	2	NUM
cana-1956	192	11	+	+	NUM
cana-1956	192	12	1	1	NUM
cana-1956	192	13	)	)	PUNCT
cana-1956	192	14	)	)	PUNCT
cana-1956	193	1	−	−	PROPN
cana-1956	193	2	3ℎ1(2𝜀	3ℎ1(2𝜀	NUM
cana-1956	193	3	+	+	CCONJ
cana-1956	193	4	2	2	NUM
cana-1956	193	5	)	)	PUNCT
cana-1956	193	6	−	−	PROPN
cana-1956	193	7	3ℎ1	3ℎ1	NUM
cana-1956	193	8	(	(	PUNCT
cana-1956	193	9	(	(	PUNCT
cana-1956	193	10	𝜀	𝜀	X
cana-1956	193	11	+	+	ADJ
cana-1956	193	12	1	1	NUM
cana-1956	193	13	)	)	PUNCT
cana-1956	193	14	(	(	PUNCT
cana-1956	193	15	−	−	PROPN
cana-1956	193	16	1	1	NUM
cana-1956	193	17	12	12	NUM
cana-1956	193	18	+	+	CCONJ
cana-1956	193	19	2	2	NUM
cana-1956	193	20	)	)	PUNCT
cana-1956	193	21	)	)	PUNCT
cana-1956	193	22	>	>	PUNCT
cana-1956	194	1	ℎ1	ℎ1	PROPN
cana-1956	194	2	(	(	PUNCT
cana-1956	194	3	13𝜀	13𝜀	X
cana-1956	194	4	+	+	CCONJ
cana-1956	194	5	13	13	NUM
cana-1956	194	6	12	12	NUM
cana-1956	194	7	)	)	PUNCT
cana-1956	194	8	−	−	PROPN
cana-1956	194	9	6ℎ1	6ℎ1	NUM
cana-1956	194	10	(	(	PUNCT
cana-1956	194	11	23𝜀	23𝜀	NUM
cana-1956	194	12	+	+	CCONJ
cana-1956	194	13	23	23	NUM
cana-1956	194	14	12	12	NUM
cana-1956	194	15	)	)	PUNCT
cana-1956	194	16	−	−	PROPN
cana-1956	195	1	ℎ1	ℎ1	PROPN
cana-1956	195	2	(	(	PUNCT
cana-1956	195	3	13𝜀	13𝜀	X
cana-1956	195	4	+	+	CCONJ
cana-1956	195	5	13	13	NUM
cana-1956	195	6	12	12	NUM
cana-1956	195	7	)	)	PUNCT
cana-1956	195	8	(	(	PUNCT
cana-1956	195	9	1	1	NUM
cana-1956	195	10	−	−	NUM
cana-1956	195	11	138	138	NUM
cana-1956	195	12	13	13	NUM
cana-1956	195	13	𝑒−	𝑒−	NOUN
cana-1956	195	14	5𝜋(𝜀+1)2	5𝜋(𝜀+1)2	NUM
cana-1956	195	15	2	2	NUM
cana-1956	195	16	)	)	PUNCT
cana-1956	195	17	this	this	PRON
cana-1956	195	18	is	be	AUX
cana-1956	195	19	clearly	clearly	ADV
cana-1956	195	20	positive	positive	ADJ
cana-1956	195	21	for	for	ADP
cana-1956	195	22	any	any	DET
cana-1956	195	23	𝜀	𝜀	NOUN
cana-1956	195	24	values	value	NOUN
cana-1956	195	25	that	that	PRON
cana-1956	195	26	are	be	AUX
cana-1956	195	27	greater	great	ADJ
cana-1956	195	28	than	than	ADP
cana-1956	195	29	zero	zero	NUM
cana-1956	195	30	.	.	PUNCT
cana-1956	196	1	the	the	DET
cana-1956	196	2	proofs	proof	NOUN
cana-1956	196	3	for	for	ADP
cana-1956	196	4	𝑖	𝑖	NOUN
cana-1956	196	5	=	=	SYM
cana-1956	196	6	2	2	NUM
cana-1956	196	7	and	and	CCONJ
cana-1956	196	8	𝑖	𝑖	SYM
cana-1956	196	9	=	=	SYM
cana-1956	196	10	3	3	NUM
cana-1956	196	11	follow	follow	VERB
cana-1956	196	12	the	the	DET
cana-1956	196	13	same	same	ADJ
cana-1956	196	14	method	method	NOUN
cana-1956	196	15	.	.	PUNCT
cana-1956	197	1	f	f	X
cana-1956	197	2	)	)	PUNCT
cana-1956	197	3	the	the	DET
cana-1956	197	4	situations	situation	NOUN
cana-1956	197	5	1	1	NUM
cana-1956	197	6	12	12	NUM
cana-1956	197	7	≤	≤	NUM
cana-1956	197	8	𝑥	𝑥	PRON
cana-1956	197	9	≤	≤	NUM
cana-1956	197	10	1	1	NUM
cana-1956	197	11	6	6	NUM
cana-1956	197	12	can	can	AUX
cana-1956	197	13	be	be	AUX
cana-1956	197	14	solved	solve	VERB
cana-1956	197	15	similarly	similarly	ADV
cana-1956	197	16	to	to	ADP
cana-1956	197	17	the	the	DET
cana-1956	197	18	previous	previous	ADJ
cana-1956	197	19	one	one	NUM
cana-1956	197	20	.	.	PUNCT
cana-1956	198	1	the	the	DET
cana-1956	198	2	submatrix	submatrix	NOUN
cana-1956	198	3	of	of	ADP
cana-1956	198	4	𝒫1(𝑥	𝒫1(𝑥	PROPN
cana-1956	198	5	,	,	PUNCT
cana-1956	198	6	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	198	7	)	)	PUNCT
cana-1956	198	8	,	,	PUNCT
cana-1956	198	9	the	the	DET
cana-1956	198	10	condition	condition	NOUN
cana-1956	198	11	(	(	PUNCT
cana-1956	198	12	i	i	NOUN
cana-1956	198	13	)	)	PUNCT
cana-1956	198	14	of	of	ADP
cana-1956	198	15	theorem	theorem	NOUN
cana-1956	198	16	(	(	PUNCT
cana-1956	198	17	3.5	3.5	NUM
cana-1956	198	18	)	)	PUNCT
cana-1956	198	19	has	have	AUX
cana-1956	198	20	been	be	AUX
cana-1956	198	21	confirmed	confirm	VERB
cana-1956	198	22	by	by	ADP
cana-1956	198	23	columns	column	NOUN
cana-1956	198	24	𝑡	𝑡	X
cana-1956	198	25	=	=	SYM
cana-1956	198	26	0,2,3	0,2,3	NUM
cana-1956	198	27	,	,	PUNCT
cana-1956	198	28	…	…	PUNCT
cana-1956	198	29	.	.	PUNCT
cana-1956	199	1	g	g	NOUN
cana-1956	199	2	)	)	PUNCT
cana-1956	199	3	we	we	PRON
cana-1956	199	4	show	show	VERB
cana-1956	199	5	that	that	SCONJ
cana-1956	199	6	the	the	DET
cana-1956	199	7	present	present	ADJ
cana-1956	199	8	circumstances	circumstance	NOUN
cana-1956	199	9	𝜀+1	𝜀+1	ADP
cana-1956	199	10	6	6	NUM
cana-1956	199	11	≤	≤	NUM
cana-1956	199	12	𝑥	𝑥	PRON
cana-1956	199	13	≤	≤	NUM
cana-1956	199	14	𝜀+1	𝜀+1	ADP
cana-1956	199	15	3	3	NUM
cana-1956	199	16	,	,	PUNCT
cana-1956	199	17	can	can	AUX
cana-1956	199	18	be	be	AUX
cana-1956	199	19	simplified	simplify	VERB
cana-1956	199	20	to	to	ADP
cana-1956	199	21	the	the	DET
cana-1956	199	22	prior	prior	ADJ
cana-1956	199	23	ones	one	NOUN
cana-1956	199	24	.	.	PUNCT
cana-1956	200	1	by	by	ADP
cana-1956	200	2	inserting	insert	VERB
cana-1956	200	3	𝑥	𝑥	PROPN
cana-1956	200	4	=	=	SYM
cana-1956	200	5	𝜀+1	𝜀+1	PROPN
cana-1956	200	6	3	3	NUM
cana-1956	200	7	−	−	NOUN
cana-1956	200	8	𝑦	𝑦	NOUN
cana-1956	200	9	,	,	PUNCT
cana-1956	200	10	we	we	PRON
cana-1956	200	11	obtain	obtain	VERB
cana-1956	200	12	:	:	PUNCT
cana-1956	200	13	∑|𝑍5𝜀+5ℎ1	∑|𝑍5𝜀+5ℎ1	PROPN
cana-1956	200	14	(	(	PUNCT
cana-1956	200	15	𝜀	𝜀	VERB
cana-1956	200	16	+	+	NOUN
cana-1956	200	17	1	1	NUM
cana-1956	200	18	3	3	NUM
cana-1956	200	19	−	−	PROPN
cana-1956	200	20	𝑦	𝑦	NOUN
cana-1956	200	21	+	+	CCONJ
cana-1956	200	22	(	(	PUNCT
cana-1956	200	23	𝜀	𝜀	X
cana-1956	200	24	+	+	SYM
cana-1956	200	25	1)𝑡	1)𝑡	NUM
cana-1956	200	26	+	+	SYM
cana-1956	200	27	𝑟	𝑟	SYM
cana-1956	201	1	5𝜖	5𝜖	NOUN
cana-1956	201	2	+	+	CCONJ
cana-1956	201	3	5	5	NUM
cana-1956	201	4	3	3	NUM
cana-1956	201	5	,	,	PUNCT
cana-1956	201	6	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	201	7	𝑖	𝑖	PUNCT
cana-1956	201	8	=	=	X
cana-1956	201	9	∑|𝑍5𝜀+5ℎ1	∑|𝑍5𝜀+5ℎ1	PROPN
cana-1956	201	10	(	(	PUNCT
cana-1956	201	11	𝜀	𝜀	VERB
cana-1956	201	12	+	+	NOUN
cana-1956	201	13	1	1	NUM
cana-1956	201	14	3	3	NUM
cana-1956	201	15	−	−	PROPN
cana-1956	201	16	𝑦	𝑦	NOUN
cana-1956	201	17	+	+	CCONJ
cana-1956	201	18	(	(	PUNCT
cana-1956	201	19	𝜀	𝜀	X
cana-1956	201	20	+	+	ADJ
cana-1956	201	21	1)(𝑟	1)(𝑟	NUM
cana-1956	201	22	−	−	NOUN
cana-1956	201	23	1	1	NUM
cana-1956	201	24	)	)	PUNCT
cana-1956	201	25	5	5	NUM
cana-1956	201	26	3	3	NUM
cana-1956	201	27	+	+	NUM
cana-1956	201	28	5𝜖	5𝜖	NUM
cana-1956	201	29	+	+	CCONJ
cana-1956	201	30	5	5	NUM
cana-1956	201	31	3	3	NUM
cana-1956	201	32	,	,	PUNCT
cana-1956	201	33	𝑤𝑖)|	𝑤𝑖)|	ADP
cana-1956	201	34	𝑖	𝑖	PUNCT
cana-1956	201	35	=	=	NOUN
cana-1956	201	36	∑|𝑍5𝜀+5ℎ1(𝑦	∑|𝑍5𝜀+5ℎ1(𝑦	PROPN
cana-1956	201	37	−	−	PROPN
cana-1956	201	38	(	(	PUNCT
cana-1956	201	39	𝜀	𝜀	X
cana-1956	201	40	+	+	CCONJ
cana-1956	201	41	1)(𝑡	1)(𝑡	NUM
cana-1956	201	42	+	+	CCONJ
cana-1956	201	43	2	2	NUM
cana-1956	201	44	)	)	PUNCT
cana-1956	201	45	−	−	PROPN
cana-1956	201	46	(	(	PUNCT
cana-1956	201	47	𝜀	𝜀	X
cana-1956	201	48	+	+	ADJ
cana-1956	201	49	1)(𝑟	1)(𝑟	NUM
cana-1956	201	50	−	−	NOUN
cana-1956	201	51	1	1	NUM
cana-1956	201	52	)	)	PUNCT
cana-1956	201	53	5	5	NUM
cana-1956	201	54	3	3	NUM
cana-1956	201	55	,	,	PUNCT
cana-1956	201	56	𝑤𝑖)|	𝑤𝑖)|	ADV
cana-1956	201	57	𝑖	𝑖	X
cana-1956	201	58	.	.	PUNCT
cana-1956	202	1	the	the	DET
cana-1956	202	2	two	two	NUM
cana-1956	202	3	manifestations	manifestation	NOUN
cana-1956	202	4	are	be	AUX
cana-1956	202	5	clearly	clearly	ADV
cana-1956	202	6	exactly	exactly	ADV
cana-1956	202	7	the	the	DET
cana-1956	202	8	same	same	ADJ
cana-1956	202	9	in	in	ADP
cana-1956	202	10	terms	term	NOUN
cana-1956	202	11	of	of	ADP
cana-1956	202	12	row	row	NOUN
cana-1956	202	13	/	/	SYM
cana-1956	202	14	column	column	NOUN
cana-1956	202	15	permutations	permutation	NOUN
cana-1956	202	16	for	for	ADP
cana-1956	202	17	0	0	NUM
cana-1956	202	18	≤	≤	NUM
cana-1956	202	19	𝑥	𝑥	DET
cana-1956	202	20	≤	≤	NUM
cana-1956	202	21	𝜀+1	𝜀+1	ADP
cana-1956	202	22	6	6	NUM
cana-1956	202	23	and	and	CCONJ
cana-1956	202	24	𝜀+1	𝜀+1	NUM
cana-1956	202	25	6	6	NUM
cana-1956	202	26	≤	≤	NUM
cana-1956	202	27	𝑥	𝑥	DET
cana-1956	202	28	≤	≤	NUM
cana-1956	202	29	𝜀+1	𝜀+1	ADP
cana-1956	202	30	3	3	NUM
cana-1956	202	31	.	.	PUNCT
cana-1956	203	1	6	6	NUM
cana-1956	203	2	.	.	X
cana-1956	203	3	hypothesis	hypothesis	NOUN
cana-1956	203	4	equations	equation	NOUN
cana-1956	203	5	in	in	ADP
cana-1956	203	6	the	the	DET
cana-1956	203	7	previous	previous	ADJ
cana-1956	203	8	part	part	NOUN
cana-1956	203	9	become	become	VERB
cana-1956	203	10	tedious	tedious	ADJ
cana-1956	203	11	for	for	ADP
cana-1956	203	12	arbitrary	arbitrary	ADJ
cana-1956	203	13	𝜀	𝜀	NOUN
cana-1956	203	14	+	+	ADJ
cana-1956	203	15	1	1	NUM
cana-1956	203	16	,	,	PUNCT
cana-1956	203	17	𝜀	𝜀	X
cana-1956	203	18	−	−	PROPN
cana-1956	203	19	1with	1with	NUM
cana-1956	203	20	𝜀	𝜀	X
cana-1956	203	21	∈	∈	NOUN
cana-1956	203	22	𝑄,𝜀	𝑄,𝜀	ADP
cana-1956	203	23	<	<	X
cana-1956	203	24	√2	√2	PROPN
cana-1956	203	25	.	.	PUNCT
cana-1956	203	26	numerous	numerous	ADJ
cana-1956	203	27	numerical	numerical	ADJ
cana-1956	203	28	calculations	calculation	NOUN
cana-1956	203	29	support	support	VERB
cana-1956	203	30	the	the	DET
cana-1956	203	31	following	follow	VERB
cana-1956	203	32	statement	statement	NOUN
cana-1956	203	33	.	.	PUNCT
cana-1956	204	1	conjecture	conjecture	NOUN
cana-1956	204	2	(	(	PUNCT
cana-1956	204	3	6.1	6.1	NUM
cana-1956	204	4	):	):	PUNCT
cana-1956	204	5	suppose	suppose	VERB
cana-1956	204	6	<	<	X
cana-1956	204	7	√2	√2	NOUN
cana-1956	204	8	,	,	PUNCT
cana-1956	204	9	and	and	CCONJ
cana-1956	204	10	𝜀2	𝜀2	PROPN
cana-1956	204	11	≠	≠	PROPN
cana-1956	204	12	2𝑛−1	2𝑛−1	NUM
cana-1956	204	13	𝑛	𝑛	NOUN
cana-1956	204	14	,	,	PUNCT
cana-1956	204	15	𝑛	𝑛	PROPN
cana-1956	204	16	=	=	SYM
cana-1956	204	17	1	1	NUM
cana-1956	204	18	2	2	NUM
cana-1956	204	19	,	,	PUNCT
cana-1956	204	20	…	…	PUNCT
cana-1956	204	21	.	.	PUNCT
cana-1956	205	1	then	then	ADV
cana-1956	205	2	𝒢(𝐻1	𝒢(𝐻1	PROPN
cana-1956	205	3	,	,	PUNCT
cana-1956	205	4	𝜀	𝜀	X
cana-1956	205	5	+	+	ADJ
cana-1956	205	6	1	1	NUM
cana-1956	205	7	,	,	PUNCT
cana-1956	205	8	𝜀	𝜀	X
cana-1956	205	9	−	−	PROPN
cana-1956	205	10	1)in	1)in	NOUN
cana-1956	205	11	a	a	DET
cana-1956	205	12	frame	frame	NOUN
cana-1956	205	13	in	in	ADP
cana-1956	205	14	𝐿2(ℝ).however	𝐿2(ℝ).however	PROPN
cana-1956	205	15	,	,	PUNCT
cana-1956	205	16	the	the	DET
cana-1956	205	17	researchers	researcher	NOUN
cana-1956	205	18	are	be	AUX
cana-1956	205	19	at	at	ADP
cana-1956	205	20	present	present	ADJ
cana-1956	205	21	unable	unable	ADJ
cana-1956	205	22	to	to	PART
cana-1956	205	23	confirm	confirm	VERB
cana-1956	205	24	or	or	CCONJ
cana-1956	205	25	reject	reject	VERB
cana-1956	205	26	this	this	DET
cana-1956	205	27	conclusion	conclusion	NOUN
cana-1956	205	28	,	,	PUNCT
cana-1956	205	29	even	even	ADV
cana-1956	205	30	if	if	SCONJ
cana-1956	205	31	𝜀	𝜀	NOUN
cana-1956	205	32	is	be	AUX
cana-1956	205	33	communications	communication	NOUN
cana-1956	205	34	on	on	ADP
cana-1956	205	35	applied	apply	VERB
cana-1956	205	36	nonlinear	nonlinear	ADJ
cana-1956	205	37	analysis	analysis	NOUN
cana-1956	205	38	issn	issn	NOUN
cana-1956	205	39	:	:	PUNCT
cana-1956	205	40	1074	1074	NUM
cana-1956	205	41	-	-	PUNCT
cana-1956	205	42	133x	133x	NUM
cana-1956	205	43	vol	vol	NOUN
cana-1956	205	44	32	32	NUM
cana-1956	205	45	no	no	NOUN
cana-1956	205	46	.	.	NOUN
cana-1956	205	47	3	3	NUM
cana-1956	205	48	(	(	PUNCT
cana-1956	205	49	2025	2025	NUM
cana-1956	205	50	)	)	PUNCT
cana-1956	205	51	238	238	NUM
cana-1956	205	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-1956	205	53	in	in	ADP
cana-1956	205	54	𝑄.	𝑄.	PROPN
cana-1956	205	55	let	let	VERB
cana-1956	205	56	𝜀2	𝜀2	NOUN
cana-1956	205	57	=	=	SYM
cana-1956	205	58	2𝑛−1	2𝑛−1	NUM
cana-1956	205	59	𝑛	𝑛	NOUN
cana-1956	205	60	,	,	PUNCT
cana-1956	205	61	are	be	AUX
cana-1956	205	62	the	the	DET
cana-1956	205	63	only	only	ADJ
cana-1956	205	64	not	not	PART
cana-1956	205	65	common	common	ADJ
cana-1956	205	66	issues	issue	NOUN
cana-1956	205	67	.	.	PUNCT
cana-1956	206	1	figure	figure	VERB
cana-1956	206	2	1	1	NUM
cana-1956	206	3	presents	present	VERB
cana-1956	206	4	the	the	DET
cana-1956	206	5	most	most	ADV
cana-1956	206	6	modest	modest	ADJ
cana-1956	206	7	eigenvalue	eigenvalue	NOUN
cana-1956	206	8	of	of	ADP
cana-1956	206	9	�	�	PROPN
cana-1956	206	10	̃	̃	PROPN
cana-1956	206	11	�	�	PROPN
cana-1956	206	12	(𝑥	(𝑥	VERB
cana-1956	206	13	,	,	PUNCT
cana-1956	206	14	𝑤𝑖)	𝑤𝑖)	NOUN
cana-1956	206	15	�	�	PROPN
cana-1956	206	16	̃	̃	NOUN
cana-1956	206	17	�	�	PROPN
cana-1956	206	18	(𝑥	(𝑥	NOUN
cana-1956	206	19	,	,	PUNCT
cana-1956	206	20	𝑤𝑖	𝑤𝑖	NOUN
cana-1956	206	21	)	)	PUNCT
cana-1956	206	22	𝑇	𝑇	NOUN
cana-1956	206	23	for	for	ADP
cana-1956	206	24	0	0	NUM
cana-1956	206	25	≤	≤	NUM
cana-1956	206	26	𝑥	𝑥	DET
cana-1956	206	27	≤	≤	NUM
cana-1956	206	28	𝜀+1	𝜀+1	PRON
cana-1956	206	29	2𝑝	2𝑝	NOUN
cana-1956	206	30	,	,	PUNCT
cana-1956	206	31	and	and	CCONJ
cana-1956	206	32	0	0	NUM
cana-1956	206	33	≤	≤	NOUN
cana-1956	207	1	𝑤𝑖	𝑤𝑖	ADP
cana-1956	207	2	≤	≤	NUM
cana-1956	207	3	1	1	NUM
cana-1956	207	4	𝜀+1	𝜀+1	NOUN
cana-1956	207	5	for	for	ADP
cana-1956	207	6	𝜀2	𝜀2	NOUN
cana-1956	207	7	=	=	SYM
cana-1956	207	8	𝑛−𝑗	𝑛−𝑗	PROPN
cana-1956	207	9	𝑛	𝑛	PRON
cana-1956	207	10	+	+	X
cana-1956	207	11	1	1	NUM
cana-1956	207	12	,	,	PUNCT
cana-1956	207	13	for	for	ADP
cana-1956	207	14	5	5	NUM
cana-1956	207	15	≤	≤	NOUN
cana-1956	207	16	𝑛	𝑛	DET
cana-1956	207	17	≤	≤	NUM
cana-1956	207	18	201	201	NUM
cana-1956	207	19	,	,	PUNCT
cana-1956	207	20	and	and	CCONJ
cana-1956	207	21	1	1	NUM
cana-1956	207	22	≤	≤	NUM
cana-1956	207	23	𝑗	𝑗	PRON
cana-1956	207	24	≤	≤	NUM
cana-1956	207	25	𝑛	𝑛	DET
cana-1956	207	26	−	−	PROPN
cana-1956	207	27	1.using	1.using	NUM
cana-1956	207	28	these	these	DET
cana-1956	207	29	values	value	NOUN
cana-1956	207	30	for	for	ADP
cana-1956	207	31	n	n	PRON
cana-1956	207	32	and	and	CCONJ
cana-1956	207	33	𝑗	𝑗	NOUN
cana-1956	207	34	,	,	PUNCT
cana-1956	207	35	𝜀	𝜀	X
cana-1956	207	36	<	<	X
cana-1956	207	37	√1.995	√1.995	PROPN
cana-1956	207	38	..	..	PUNCT
cana-1956	207	39	in	in	ADP
cana-1956	207	40	the	the	DET
cana-1956	207	41	experiments	experiment	NOUN
cana-1956	207	42	,	,	PUNCT
cana-1956	207	43	we	we	PRON
cana-1956	207	44	assumed	assume	VERB
cana-1956	207	45	𝜀	𝜀	VERB
cana-1956	207	46	=	=	SYM
cana-1956	207	47	0	0	PROPN
cana-1956	207	48	.	.	PUNCT
cana-1956	208	1	the	the	DET
cana-1956	208	2	minimal	minimal	ADJ
cana-1956	208	3	eigenvalue	eigenvalue	NOUN
cana-1956	208	4	of	of	ADP
cana-1956	208	5	𝒫𝒫𝑇for	𝒫𝒫𝑇for	PROPN
cana-1956	208	6	(	(	PUNCT
cana-1956	208	7	a	a	NOUN
cana-1956	208	8	)	)	PUNCT
cana-1956	208	9	𝜀	𝜀	PROPN
cana-1956	208	10	<	<	X
cana-1956	208	11	√1.98and	√1.98and	NOUN
cana-1956	208	12	(	(	PUNCT
cana-1956	208	13	b)√1.98	b)√1.98	VERB
cana-1956	208	14	<	<	X
cana-1956	208	15	𝜀	𝜀	X
cana-1956	208	16	<	<	X
cana-1956	208	17	√1.995	√1.995	PROPN
cana-1956	208	18	.	.	PUNCT
cana-1956	209	1	in	in	ADP
cana-1956	209	2	the	the	DET
cana-1956	209	3	studies	study	NOUN
cana-1956	209	4	,	,	PUNCT
cana-1956	209	5	we	we	PRON
cana-1956	209	6	utilized	utilize	VERB
cana-1956	209	7	𝜀	𝜀	VERB
cana-1956	209	8	=	=	NOUN
cana-1956	209	9	0.it	0.it	NUM
cana-1956	209	10	should	should	AUX
cana-1956	209	11	be	be	AUX
cana-1956	209	12	noted	note	VERB
cana-1956	209	13	that	that	SCONJ
cana-1956	209	14	the	the	DET
cana-1956	209	15	y	y	PROPN
cana-1956	209	16	-	-	PUNCT
cana-1956	209	17	axis	axis	NOUN
cana-1956	209	18	has	have	AUX
cana-1956	209	19	been	be	AUX
cana-1956	209	20	scaled	scale	VERB
cana-1956	209	21	differently	differently	ADV
cana-1956	209	22	.	.	PUNCT
cana-1956	210	1	references	reference	NOUN
cana-1956	210	2	[	[	X
cana-1956	210	3	1	1	X
cana-1956	210	4	]	]	PUNCT
cana-1956	210	5	j.	j.	PROPN
cana-1956	210	6	j.	j.	PROPN
cana-1956	210	7	benedetto	benedetto	PROPN
cana-1956	210	8	,	,	PUNCT
cana-1956	210	9	c.	c.	PROPN
cana-1956	210	10	heil	heil	PROPN
cana-1956	210	11	,	,	PUNCT
cana-1956	210	12	and	and	CCONJ
cana-1956	210	13	d.	d.	PROPN
cana-1956	210	14	f.	f.	PROPN
cana-1956	210	15	walnut	walnut	PROPN
cana-1956	210	16	.	.	PUNCT
cana-1956	211	1	differentiation	differentiation	NOUN
cana-1956	211	2	and	and	CCONJ
cana-1956	211	3	the	the	DET
cana-1956	211	4	balian	balian	ADJ
cana-1956	211	5	-	-	PUNCT
cana-1956	211	6	low	low	NOUN
cana-1956	211	7	theorem	theorem	PROPN
cana-1956	211	8	.	.	PUNCT
cana-1956	212	1	j.	j.	PROPN
cana-1956	212	2	fourier	fourier	PROPN
cana-1956	212	3	anal	anal	PROPN
cana-1956	212	4	.	.	PUNCT
cana-1956	213	1	appl	appl	PROPN
cana-1956	213	2	,	,	PUNCT
cana-1956	213	3	1:355–402	1:355–402	PROPN
cana-1956	213	4	,	,	PUNCT
cana-1956	213	5	1995	1995	NUM
cana-1956	213	6	.	.	PUNCT
cana-1956	214	1	[	[	X
cana-1956	214	2	2	2	X
cana-1956	214	3	]	]	PUNCT
cana-1956	214	4	w.	w.	PROPN
cana-1956	214	5	czaja	czaja	PROPN
cana-1956	214	6	and	and	CCONJ
cana-1956	214	7	a.	a.	NOUN
cana-1956	214	8	m.	m.	NOUN
cana-1956	214	9	powell	powell	PROPN
cana-1956	214	10	.	.	PUNCT
cana-1956	215	1	recent	recent	ADJ
cana-1956	215	2	developments	development	NOUN
cana-1956	215	3	in	in	ADP
cana-1956	215	4	the	the	DET
cana-1956	215	5	balian	balian	ADJ
cana-1956	215	6	–	–	PUNCT
cana-1956	215	7	low	low	ADJ
cana-1956	215	8	theorem	theorem	NOUN
cana-1956	215	9	.	.	PUNCT
cana-1956	216	1	in	in	ADP
cana-1956	216	2	harmonic	harmonic	ADJ
cana-1956	216	3	analysis	analysis	NOUN
cana-1956	216	4	and	and	CCONJ
cana-1956	216	5	applications	application	NOUN
cana-1956	216	6	.	.	PUNCT
cana-1956	217	1	in	in	ADP
cana-1956	217	2	honor	honor	NOUN
cana-1956	217	3	of	of	ADP
cana-1956	217	4	john	john	PROPN
cana-1956	217	5	j.	j.	PROPN
cana-1956	217	6	benedetto	benedetto	PROPN
cana-1956	217	7	,	,	PUNCT
cana-1956	217	8	applied	apply	VERB
cana-1956	217	9	and	and	CCONJ
cana-1956	217	10	numerical	numerical	ADJ
cana-1956	217	11	harmonic	harmonic	ADJ
cana-1956	217	12	analysis	analysis	NOUN
cana-1956	217	13	.	.	PUNCT
cana-1956	218	1	birkh¨auser	birkh¨auser	PROPN
cana-1956	218	2	boston	boston	PROPN
cana-1956	218	3	,	,	PUNCT
cana-1956	218	4	2006	2006	NUM
cana-1956	218	5	.	.	PUNCT
cana-1956	219	1	[	[	X
cana-1956	219	2	3	3	NUM
cana-1956	219	3	]	]	X
cana-1956	219	4	i.	i.	PROPN
cana-1956	219	5	daubechies	daubechies	PROPN
cana-1956	219	6	.	.	PUNCT
cana-1956	220	1	ten	ten	NUM
cana-1956	220	2	lectures	lecture	NOUN
cana-1956	220	3	on	on	ADP
cana-1956	220	4	wavelets	wavelet	NOUN
cana-1956	220	5	.	.	PUNCT
cana-1956	221	1	society	society	NOUN
cana-1956	221	2	for	for	ADP
cana-1956	221	3	industrial	industrial	ADJ
cana-1956	221	4	and	and	CCONJ
cana-1956	221	5	applied	applied	ADJ
cana-1956	221	6	mathematics	mathematic	NOUN
cana-1956	221	7	,	,	PUNCT
cana-1956	221	8	philadelphia	philadelphia	PROPN
cana-1956	221	9	,	,	PUNCT
cana-1956	221	10	pa	pa	PROPN
cana-1956	221	11	,	,	PUNCT
cana-1956	221	12	usa	usa	PROPN
cana-1956	221	13	,	,	PUNCT
cana-1956	221	14	1992	1992	NUM
cana-1956	221	15	.	.	PUNCT
cana-1956	222	1	[	[	X
cana-1956	222	2	4	4	X
cana-1956	222	3	]	]	PUNCT
cana-1956	222	4	k.	k.	NOUN
cana-1956	222	5	gr¨ochenig	gr¨ochenig	PROPN
cana-1956	222	6	.	.	PUNCT
cana-1956	223	1	foundations	foundation	NOUN
cana-1956	223	2	of	of	ADP
cana-1956	223	3	time	time	NOUN
cana-1956	223	4	-	-	PUNCT
cana-1956	223	5	frequency	frequency	NOUN
cana-1956	223	6	analysis	analysis	NOUN
cana-1956	223	7	.	.	PUNCT
cana-1956	224	1	birkhuser	birkhuser	NOUN
cana-1956	224	2	,	,	PUNCT
cana-1956	224	3	boston	boston	PROPN
cana-1956	224	4	,	,	PUNCT
cana-1956	224	5	2001	2001	NUM
cana-1956	224	6	.	.	PUNCT
cana-1956	225	1	[	[	X
cana-1956	225	2	5	5	X
cana-1956	225	3	]	]	PUNCT
cana-1956	225	4	k.	k.	NOUN
cana-1956	225	5	gr¨ochenig	gr¨ochenig	PROPN
cana-1956	225	6	and	and	CCONJ
cana-1956	225	7	yu	yu	PROPN
cana-1956	225	8	.	.	PUNCT
cana-1956	225	9	lyubarskii	lyubarskii	PROPN
cana-1956	225	10	.	.	PUNCT
cana-1956	226	1	gabor	gabor	PROPN
cana-1956	226	2	(	(	PUNCT
cana-1956	226	3	super)frames	super)frame	VERB
cana-1956	226	4	with	with	ADP
cana-1956	226	5	hermite	hermite	ADJ
cana-1956	226	6	functions	function	NOUN
cana-1956	226	7	.	.	PUNCT
cana-1956	227	1	math	math	NOUN
cana-1956	227	2	.	.	PUNCT
cana-1956	228	1	ann	ann	PROPN
cana-1956	228	2	.	.	PROPN
cana-1956	228	3	,	,	PUNCT
cana-1956	228	4	345(2):267–286	345(2):267–286	PROPN
cana-1956	228	5	,	,	PUNCT
cana-1956	228	6	2009.13	2009.13	NUM
cana-1956	228	7	[	[	X
cana-1956	228	8	6	6	NUM
cana-1956	228	9	]	]	PUNCT
cana-1956	228	10	k.	k.	NOUN
cana-1956	228	11	gr¨ochenig	gr¨ochenig	PROPN
cana-1956	228	12	and	and	CCONJ
cana-1956	228	13	j.	j.	PROPN
cana-1956	228	14	st¨ockler	st¨ockler	PROPN
cana-1956	228	15	.	.	PROPN
cana-1956	228	16	gabor	gabor	PROPN
cana-1956	228	17	frames	frame	NOUN
cana-1956	228	18	and	and	CCONJ
cana-1956	228	19	totally	totally	ADV
cana-1956	228	20	positive	positive	ADJ
cana-1956	228	21	functions	function	NOUN
cana-1956	228	22	.	.	PUNCT
cana-1956	229	1	(	(	PUNCT
cana-1956	229	2	arxiv:1104.4894	arxiv:1104.4894	NUM
cana-1956	229	3	)	)	PUNCT
cana-1956	229	4	,	,	PUNCT
cana-1956	229	5	april	april	PROPN
cana-1956	229	6	2011	2011	NUM
cana-1956	229	7	.	.	PUNCT
cana-1956	230	1	[	[	X
cana-1956	230	2	7	7	X
cana-1956	230	3	]	]	X
cana-1956	230	4	c.	c.	PROPN
cana-1956	230	5	heil	heil	PROPN
cana-1956	230	6	.	.	PUNCT
cana-1956	231	1	history	history	NOUN
cana-1956	231	2	and	and	CCONJ
cana-1956	231	3	evolution	evolution	NOUN
cana-1956	231	4	of	of	ADP
cana-1956	231	5	the	the	DET
cana-1956	231	6	density	density	NOUN
cana-1956	231	7	theorem	theorem	VERB
cana-1956	231	8	for	for	ADP
cana-1956	231	9	gabor	gabor	PROPN
cana-1956	231	10	frames	frames	PROPN
cana-1956	231	11	.	.	PUNCT
cana-1956	232	1	journal	journal	PROPN
cana-1956	232	2	of	of	ADP
cana-1956	232	3	fourier	fourier	ADJ
cana-1956	232	4	analysis	analysis	NOUN
cana-1956	232	5	and	and	CCONJ
cana-1956	232	6	applications	application	NOUN
cana-1956	232	7	,	,	PUNCT
cana-1956	232	8	13:113–166	13:113–166	NUM
cana-1956	232	9	,	,	PUNCT
cana-1956	232	10	2007	2007	NUM
cana-1956	232	11	.	.	PUNCT
cana-1956	233	1	[	[	X
cana-1956	233	2	8	8	NUM
cana-1956	233	3	]	]	PUNCT
cana-1956	233	4	a.	a.	NOUN
cana-1956	233	5	j.	j.	PROPN
cana-1956	233	6	e.	e.	PROPN
cana-1956	233	7	m.	m.	PROPN
cana-1956	233	8	janssen	janssen	PROPN
cana-1956	233	9	.	.	PUNCT
cana-1956	234	1	some	some	DET
cana-1956	234	2	weyl	weyl	PROPN
cana-1956	234	3	-	-	PUNCT
cana-1956	234	4	heisenberg	heisenberg	PROPN
cana-1956	234	5	frame	frame	PROPN
cana-1956	234	6	bound	bind	VERB
cana-1956	234	7	calculations	calculation	NOUN
cana-1956	234	8	.	.	PUNCT
cana-1956	235	1	indagationes	indagatione	NOUN
cana-1956	235	2	mathematicae	mathematicae	PROPN
cana-1956	235	3	7(2	7(2	NUM
cana-1956	235	4	)	)	PUNCT
cana-1956	235	5	,	,	PUNCT
cana-1956	235	6	7:165–183	7:165–183	NUM
cana-1956	235	7	,	,	PUNCT
cana-1956	235	8	1996	1996	NUM
cana-1956	235	9	.	.	PUNCT
cana-1956	236	1	[	[	X
cana-1956	236	2	9	9	NUM
cana-1956	236	3	]	]	PUNCT
cana-1956	236	4	a.	a.	NOUN
cana-1956	236	5	j.	j.	PROPN
cana-1956	236	6	e.	e.	PROPN
cana-1956	236	7	m.	m.	PROPN
cana-1956	236	8	janssen	janssen	PROPN
cana-1956	236	9	.	.	PUNCT
cana-1956	237	1	zak	zak	PROPN
cana-1956	237	2	transforms	transform	VERB
cana-1956	237	3	with	with	ADP
cana-1956	237	4	few	few	ADJ
cana-1956	237	5	zeros	zero	NOUN
cana-1956	237	6	and	and	CCONJ
cana-1956	237	7	the	the	DET
cana-1956	237	8	tie	tie	NOUN
cana-1956	237	9	.	.	PUNCT
cana-1956	238	1	in	in	ADP
cana-1956	238	2	in	in	ADP
cana-1956	238	3	advances	advance	NOUN
cana-1956	238	4	in	in	ADP
cana-1956	238	5	gabor	gabor	PROPN
cana-1956	238	6	analysis	analysis	NOUN
cana-1956	238	7	,	,	PUNCT
cana-1956	238	8	pages	page	NOUN
cana-1956	238	9	31–70	31–70	NUM
cana-1956	238	10	.	.	PUNCT
cana-1956	239	1	birkhuser	birkhuser	PROPN
cana-1956	239	2	boston	boston	PROPN
cana-1956	239	3	,	,	PUNCT
cana-1956	239	4	2003	2003	NUM
cana-1956	239	5	.	.	PUNCT
cana-1956	240	1	[	[	X
cana-1956	240	2	10	10	NUM
cana-1956	240	3	]	]	X
cana-1956	240	4	a.	a.	NOUN
cana-1956	240	5	j.	j.	PROPN
cana-1956	240	6	e.	e.	PROPN
cana-1956	240	7	m.	m.	PROPN
cana-1956	240	8	janssen	janssen	PROPN
cana-1956	240	9	and	and	CCONJ
cana-1956	240	10	thomas	thomas	PROPN
cana-1956	240	11	strohmer	strohmer	PROPN
cana-1956	240	12	.	.	PUNCT
cana-1956	241	1	hyperbolic	hyperbolic	ADJ
cana-1956	241	2	secants	secant	NOUN
cana-1956	241	3	yield	yield	VERB
cana-1956	241	4	gabor	gabor	PROPN
cana-1956	241	5	frames	frames	PROPN
cana-1956	241	6	.	.	PUNCT
cana-1956	242	1	trans	trans	PROPN
cana-1956	242	2	.	.	PUNCT
cana-1956	243	1	amer	amer	PROPN
cana-1956	243	2	.	.	PUNCT
cana-1956	243	3	math	math	PROPN
cana-1956	243	4	.	.	PUNCT
cana-1956	244	1	soc	soc	PROPN
cana-1956	244	2	,	,	PUNCT
cana-1956	244	3	12:259–267	12:259–267	NUM
cana-1956	244	4	,	,	PUNCT
cana-1956	244	5	2002	2002	NUM
cana-1956	244	6	.	.	PUNCT
cana-1956	245	1	[	[	X
cana-1956	245	2	11	11	NUM
cana-1956	245	3	]	]	X
cana-1956	245	4	yu	yu	PROPN
cana-1956	245	5	.	.	PROPN
cana-1956	245	6	lyubarski˘ı	lyubarski˘ı	PROPN
cana-1956	245	7	.	.	PROPN
cana-1956	245	8	frames	frame	NOUN
cana-1956	245	9	in	in	ADP
cana-1956	245	10	the	the	DET
cana-1956	245	11	bargmann	bargmann	PROPN
cana-1956	245	12	space	space	NOUN
cana-1956	245	13	of	of	ADP
cana-1956	245	14	entire	entire	ADJ
cana-1956	245	15	functions	function	NOUN
cana-1956	245	16	.	.	PUNCT
cana-1956	246	1	in	in	ADP
cana-1956	246	2	entire	entire	ADJ
cana-1956	246	3	and	and	CCONJ
cana-1956	246	4	subharmonic	subharmonic	ADJ
cana-1956	246	5	functions	function	NOUN
cana-1956	246	6	,	,	PUNCT
cana-1956	246	7	volume	volume	NOUN
cana-1956	246	8	11	11	NUM
cana-1956	246	9	of	of	ADP
cana-1956	246	10	adv	adv	PROPN
cana-1956	246	11	.	.	PUNCT
cana-1956	247	1	soviet	soviet	ADJ
cana-1956	247	2	math	math	PROPN
cana-1956	247	3	.	.	PUNCT
cana-1956	248	1	,	,	PUNCT
cana-1956	248	2	pages	page	NOUN
cana-1956	248	3	167–180	167–180	NUM
cana-1956	248	4	.	.	PUNCT
cana-1956	249	1	amer	amer	PROPN
cana-1956	249	2	.	.	PUNCT
cana-1956	249	3	math	math	PROPN
cana-1956	249	4	.	.	PUNCT
cana-1956	250	1	soc	soc	PROPN
cana-1956	250	2	.	.	PUNCT
cana-1956	250	3	,	,	PUNCT
cana-1956	250	4	providence	providence	NOUN
cana-1956	250	5	,	,	PUNCT
cana-1956	250	6	ri	ri	NOUN
cana-1956	250	7	,	,	PUNCT
cana-1956	250	8	1992	1992	NUM
cana-1956	250	9	.	.	PUNCT
cana-1956	251	1	[	[	X
cana-1956	251	2	12	12	NUM
cana-1956	251	3	]	]	PUNCT
cana-1956	251	4	a.	a.	NOUN
cana-1956	251	5	ron	ron	PROPN
cana-1956	251	6	and	and	CCONJ
cana-1956	251	7	z.	z.	PROPN
cana-1956	251	8	shen	shen	PROPN
cana-1956	251	9	.	.	PUNCT
cana-1956	252	1	weyl	weyl	PROPN
cana-1956	252	2	-	-	PUNCT
cana-1956	252	3	heisenberg	heisenberg	PROPN
cana-1956	252	4	frames	frame	NOUN
cana-1956	252	5	and	and	CCONJ
cana-1956	252	6	riesz	riesz	VERB
cana-1956	252	7	bases	basis	NOUN
cana-1956	252	8	in	in	ADP
cana-1956	252	9	l2(rd	l2(rd	PROPN
cana-1956	252	10	)	)	PUNCT
cana-1956	252	11	.	.	PUNCT
cana-1956	253	1	duke	duke	PROPN
cana-1956	253	2	math	math	PROPN
cana-1956	253	3	j	j	PROPN
cana-1956	253	4	,	,	PUNCT
cana-1956	253	5	89:237–282	89:237–282	PROPN
cana-1956	253	6	,	,	PUNCT
cana-1956	253	7	1997	1997	NUM
cana-1956	253	8	.	.	PUNCT
cana-1956	254	1	[	[	X
cana-1956	254	2	13	13	NUM
cana-1956	254	3	]	]	PUNCT
cana-1956	254	4	k.	k.	NOUN
cana-1956	254	5	seip	seip	PROPN
cana-1956	254	6	.	.	PUNCT
cana-1956	255	1	density	density	NOUN
cana-1956	255	2	theorems	theorem	VERB
cana-1956	255	3	for	for	ADP
cana-1956	255	4	sampling	sampling	NOUN
cana-1956	255	5	and	and	CCONJ
cana-1956	255	6	interpolation	interpolation	NOUN
cana-1956	255	7	in	in	ADP
cana-1956	255	8	the	the	DET
cana-1956	255	9	bargmann	bargmann	NOUN
cana-1956	255	10	-	-	PUNCT
cana-1956	255	11	fock	fock	ADJ
cana-1956	255	12	space	space	NOUN
cana-1956	255	13	.	.	PUNCT
cana-1956	256	1	i.	i.	PROPN
cana-1956	256	2	j.	j.	PROPN
cana-1956	256	3	reine	reine	PROPN
cana-1956	256	4	angew	angew	PROPN
cana-1956	256	5	.	.	PUNCT
cana-1956	257	1	math	math	NOUN
cana-1956	257	2	.	.	PUNCT
cana-1956	257	3	,	,	PUNCT
cana-1956	257	4	429:91–106	429:91–106	NUM
cana-1956	257	5	,	,	PUNCT
cana-1956	257	6	1992	1992	NUM
cana-1956	257	7	.	.	PUNCT
cana-1956	258	1	[	[	X
cana-1956	258	2	14	14	NUM
cana-1956	258	3	]	]	PUNCT
cana-1956	258	4	k.	k.	NOUN
cana-1956	258	5	seip	seip	PROPN
cana-1956	258	6	and	and	CCONJ
cana-1956	258	7	r.	r.	PROPN
cana-1956	258	8	wallst´en	wallst´en	PROPN
cana-1956	258	9	.	.	PUNCT
cana-1956	259	1	density	density	NOUN
cana-1956	259	2	theorems	theorem	NOUN
cana-1956	259	3	for	for	ADP
cana-1956	259	4	sampling	sampling	NOUN
cana-1956	259	5	and	and	CCONJ
cana-1956	259	6	interpolation	interpolation	NOUN
cana-1956	259	7	in	in	ADP
cana-1956	259	8	the	the	DET
cana-1956	259	9	bargmann	bargmann	NOUN
cana-1956	259	10	-	-	PUNCT
cana-1956	259	11	fock	fock	ADJ
cana-1956	259	12	space	space	NOUN
cana-1956	259	13	.	.	PUNCT
cana-1956	260	1	ii	ii	PROPN
cana-1956	260	2	.	.	PUNCT
cana-1956	261	1	j.	j.	PROPN
cana-1956	261	2	reine	reine	PROPN
cana-1956	261	3	angew	angew	PROPN
cana-1956	261	4	.	.	PUNCT
cana-1956	262	1	math	math	PROPN
cana-1956	262	2	.	.	PUNCT
cana-1956	262	3	,	,	PUNCT
cana-1956	262	4	429:107–113	429:107–113	NUM
cana-1956	262	5	,	,	PUNCT
cana-1956	262	6	1992	1992	NUM
cana-1956	262	7	.	.	PUNCT
cana-1956	263	1	[	[	X
cana-1956	263	2	15	15	NUM
cana-1956	263	3	]	]	X
cana-1956	263	4	o.	o.	NOUN
cana-1956	263	5	taussky	taussky	NOUN
cana-1956	263	6	.	.	PUNCT
cana-1956	264	1	a	a	DET
cana-1956	264	2	recurring	recur	VERB
cana-1956	264	3	theorem	theorem	NOUN
cana-1956	264	4	on	on	ADP
cana-1956	264	5	determinants	determinant	NOUN
cana-1956	264	6	.	.	PUNCT
cana-1956	265	1	the	the	DET
cana-1956	265	2	american	american	PROPN
cana-1956	265	3	mathematical	mathematical	PROPN
cana-1956	265	4	monthly	monthly	ADV
cana-1956	265	5	,	,	PUNCT
cana-1956	265	6	56(10	56(10	NUM
cana-1956	265	7	):	):	PUNCT
cana-1956	265	8	pp	pp	ADJ
cana-1956	265	9	.	.	PUNCT
cana-1956	266	1	672–676	672–676	NUM
cana-1956	266	2	,	,	PUNCT
cana-1956	266	3	1949	1949	NUM
cana-1956	266	4	.	.	PUNCT
cana-1956	267	1	[	[	X
cana-1956	267	2	16	16	NUM
cana-1956	267	3	]	]	X
cana-1956	267	4	m.	m.	NOUN
cana-1956	267	5	zibulski	zibulski	PROPN
cana-1956	267	6	and	and	CCONJ
cana-1956	267	7	y.	y.	PROPN
cana-1956	267	8	y.	y.	PROPN
cana-1956	267	9	zeevi	zeevi	PROPN
cana-1956	267	10	.	.	PUNCT
cana-1956	268	1	analysis	analysis	NOUN
cana-1956	268	2	of	of	ADP
cana-1956	268	3	multiwindow	multiwindow	NOUN
cana-1956	268	4	gabor	gabor	PROPN
cana-1956	268	5	-	-	PUNCT
cana-1956	268	6	type	type	NOUN
cana-1956	268	7	schemes	scheme	NOUN
cana-1956	268	8	by	by	ADP
cana-1956	268	9	frame	frame	NOUN
cana-1956	268	10	methods	method	NOUN
cana-1956	268	11	.	.	PUNCT
cana-1956	269	1	applied	apply	VERB
cana-1956	269	2	and	and	CCONJ
cana-1956	269	3	computational	computational	ADJ
cana-1956	269	4	harmonic	harmonic	ADJ
cana-1956	269	5	analysis	analysis	NOUN
cana-1956	269	6	,	,	PUNCT
cana-1956	269	7	4(2):188	4(2):188	NUM
cana-1956	269	8	–	–	SYM
cana-1956	269	9	221	221	NUM
cana-1956	269	10	,	,	PUNCT
cana-1956	269	11	1997	1997	NUM
cana-1956	269	12	.	.	PUNCT