id	sid	tid	token	lemma	pos
ijassa-2040	1	1	adv	adv	PROPN
ijassa-2040	1	2	syst	syst	PROPN
ijassa-2040	1	3	sci	sci	PROPN
ijassa-2040	1	4	appl	appl	PROPN
ijassa-2040	1	5	2025	2025	NUM
ijassa-2040	1	6	;	;	PUNCT
ijassa-2040	1	7	1:12–21	1:12–21	NUM
ijassa-2040	1	8	published	publish	VERB
ijassa-2040	1	9	online	online	ADV
ijassa-2040	1	10	at	at	ADP
ijassa-2040	1	11	https://ijassa.ipu.ru	https://ijassa.ipu.ru	ADV
ijassa-2040	1	12	.	.	PUNCT
ijassa-2040	2	1	on	on	ADP
ijassa-2040	2	2	linear	linear	PROPN
ijassa-2040	2	3	differential	differential	ADJ
ijassa-2040	2	4	equations	equation	NOUN
ijassa-2040	2	5	on	on	ADP
ijassa-2040	2	6	the	the	DET
ijassa-2040	2	7	torus	torus	NOUN
ijassa-2040	2	8	and	and	CCONJ
ijassa-2040	2	9	non	non	ADJ
ijassa-2040	2	10	-	-	ADJ
ijassa-2040	2	11	standard	standard	ADJ
ijassa-2040	2	12	analysis	analysis	NOUN
ijassa-2040	2	13	vladimir	vladimir	PROPN
ijassa-2040	2	14	p.	p.	PROPN
ijassa-2040	2	15	burskii1	burskii1	PROPN
ijassa-2040	2	16	*	*	PROPN
ijassa-2040	2	17	1moscow	1moscow	NUM
ijassa-2040	2	18	institute	institute	PROPN
ijassa-2040	2	19	of	of	ADP
ijassa-2040	2	20	physics	physics	PROPN
ijassa-2040	2	21	and	and	CCONJ
ijassa-2040	2	22	technology	technology	NOUN
ijassa-2040	2	23	(	(	PUNCT
ijassa-2040	2	24	state	state	NOUN
ijassa-2040	2	25	university	university	NOUN
ijassa-2040	2	26	)	)	PUNCT
ijassa-2040	2	27	,	,	PUNCT
ijassa-2040	2	28	dolgoprudnyi	dolgoprudnyi	PROPN
ijassa-2040	2	29	,	,	PUNCT
ijassa-2040	2	30	russia	russia	PROPN
ijassa-2040	2	31	abstract	abstract	NOUN
ijassa-2040	2	32	:	:	PUNCT
ijassa-2040	2	33	in	in	ADP
ijassa-2040	2	34	this	this	DET
ijassa-2040	2	35	paper	paper	NOUN
ijassa-2040	2	36	,	,	PUNCT
ijassa-2040	2	37	we	we	PRON
ijassa-2040	2	38	consider	consider	VERB
ijassa-2040	2	39	periodic	periodic	ADJ
ijassa-2040	2	40	boundary	boundary	ADJ
ijassa-2040	2	41	value	value	NOUN
ijassa-2040	2	42	problems	problem	NOUN
ijassa-2040	2	43	for	for	ADP
ijassa-2040	2	44	differential	differential	ADJ
ijassa-2040	2	45	equations	equation	NOUN
ijassa-2040	2	46	whose	whose	DET
ijassa-2040	2	47	coefficients	coefficient	NOUN
ijassa-2040	2	48	are	be	AUX
ijassa-2040	2	49	trigonometric	trigonometric	ADJ
ijassa-2040	2	50	polynomials	polynomial	NOUN
ijassa-2040	2	51	.	.	PUNCT
ijassa-2040	3	1	we	we	PRON
ijassa-2040	3	2	construct	construct	VERB
ijassa-2040	3	3	the	the	DET
ijassa-2040	3	4	spaces	space	NOUN
ijassa-2040	3	5	of	of	ADP
ijassa-2040	3	6	generalized	generalized	ADJ
ijassa-2040	3	7	functions	function	NOUN
ijassa-2040	3	8	,	,	PUNCT
ijassa-2040	3	9	where	where	SCONJ
ijassa-2040	3	10	such	such	ADJ
ijassa-2040	3	11	problems	problem	NOUN
ijassa-2040	3	12	have	have	VERB
ijassa-2040	3	13	solutions	solution	NOUN
ijassa-2040	3	14	.	.	PUNCT
ijassa-2040	4	1	in	in	ADP
ijassa-2040	4	2	particular	particular	ADJ
ijassa-2040	4	3	,	,	PUNCT
ijassa-2040	4	4	the	the	DET
ijassa-2040	4	5	solvability	solvability	NOUN
ijassa-2040	4	6	space	space	NOUN
ijassa-2040	4	7	of	of	ADP
ijassa-2040	4	8	a	a	DET
ijassa-2040	4	9	periodic	periodic	ADJ
ijassa-2040	4	10	analogue	analogue	NOUN
ijassa-2040	4	11	of	of	ADP
ijassa-2040	4	12	the	the	DET
ijassa-2040	4	13	mizohata	mizohata	ADJ
ijassa-2040	4	14	equation	equation	NOUN
ijassa-2040	4	15	is	be	AUX
ijassa-2040	4	16	constructed	construct	VERB
ijassa-2040	4	17	.	.	PUNCT
ijassa-2040	5	1	we	we	PRON
ijassa-2040	5	2	build	build	VERB
ijassa-2040	5	3	also	also	ADV
ijassa-2040	5	4	a	a	DET
ijassa-2040	5	5	periodic	periodic	ADJ
ijassa-2040	5	6	analogue	analogue	NOUN
ijassa-2040	5	7	and	and	CCONJ
ijassa-2040	5	8	a	a	DET
ijassa-2040	5	9	generalization	generalization	NOUN
ijassa-2040	5	10	of	of	ADP
ijassa-2040	5	11	the	the	DET
ijassa-2040	5	12	construction	construction	NOUN
ijassa-2040	5	13	of	of	ADP
ijassa-2040	5	14	the	the	DET
ijassa-2040	5	15	nonstandard	nonstandard	ADJ
ijassa-2040	5	16	analysis	analysis	NOUN
ijassa-2040	5	17	,	,	PUNCT
ijassa-2040	5	18	where	where	SCONJ
ijassa-2040	5	19	infinitely	infinitely	ADV
ijassa-2040	5	20	small	small	ADJ
ijassa-2040	5	21	are	be	AUX
ijassa-2040	5	22	not	not	PART
ijassa-2040	5	23	only	only	ADJ
ijassa-2040	5	24	functions	function	NOUN
ijassa-2040	5	25	,	,	PUNCT
ijassa-2040	5	26	but	but	CCONJ
ijassa-2040	5	27	also	also	ADV
ijassa-2040	5	28	functional	functional	ADJ
ijassa-2040	5	29	spaces	space	NOUN
ijassa-2040	5	30	.	.	PUNCT
ijassa-2040	6	1	to	to	PART
ijassa-2040	6	2	show	show	VERB
ijassa-2040	6	3	that	that	SCONJ
ijassa-2040	6	4	not	not	PART
ijassa-2040	6	5	all	all	DET
ijassa-2040	6	6	constructions	construction	NOUN
ijassa-2040	6	7	on	on	ADP
ijassa-2040	6	8	the	the	DET
ijassa-2040	6	9	torus	torus	NOUN
ijassa-2040	6	10	lead	lead	NOUN
ijassa-2040	6	11	to	to	ADP
ijassa-2040	6	12	a	a	DET
ijassa-2040	6	13	simplification	simplification	NOUN
ijassa-2040	6	14	in	in	ADP
ijassa-2040	6	15	compare	compare	NOUN
ijassa-2040	6	16	with	with	ADP
ijassa-2040	6	17	the	the	DET
ijassa-2040	6	18	plane	plane	NOUN
ijassa-2040	6	19	,	,	PUNCT
ijassa-2040	6	20	we	we	PRON
ijassa-2040	6	21	consider	consider	VERB
ijassa-2040	6	22	a	a	DET
ijassa-2040	6	23	periodic	periodic	ADJ
ijassa-2040	6	24	analogue	analogue	NOUN
ijassa-2040	6	25	of	of	ADP
ijassa-2040	6	26	the	the	DET
ijassa-2040	6	27	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	6	28	differential	differential	NOUN
ijassa-2040	6	29	operator	operator	NOUN
ijassa-2040	6	30	and	and	CCONJ
ijassa-2040	6	31	show	show	VERB
ijassa-2040	6	32	that	that	SCONJ
ijassa-2040	6	33	its	its	PRON
ijassa-2040	6	34	number	number	NOUN
ijassa-2040	6	35	-	-	PUNCT
ijassa-2040	6	36	theoretic	theoretic	NOUN
ijassa-2040	6	37	properties	property	NOUN
ijassa-2040	6	38	are	be	AUX
ijassa-2040	6	39	significant	significant	ADJ
ijassa-2040	6	40	.	.	PUNCT
ijassa-2040	7	1	in	in	ADP
ijassa-2040	7	2	particular	particular	ADJ
ijassa-2040	7	3	,	,	PUNCT
ijassa-2040	7	4	it	it	PRON
ijassa-2040	7	5	turns	turn	VERB
ijassa-2040	7	6	out	out	ADP
ijassa-2040	7	7	that	that	SCONJ
ijassa-2040	7	8	if	if	SCONJ
ijassa-2040	7	9	a	a	DET
ijassa-2040	7	10	polynomial	polynomial	NOUN
ijassa-2040	7	11	with	with	ADP
ijassa-2040	7	12	integer	integer	NOUN
ijassa-2040	7	13	coefficients	coefficient	NOUN
ijassa-2040	7	14	is	be	AUX
ijassa-2040	7	15	irreducible	irreducible	ADJ
ijassa-2040	7	16	in	in	ADP
ijassa-2040	7	17	the	the	DET
ijassa-2040	7	18	rational	rational	ADJ
ijassa-2040	7	19	field	field	NOUN
ijassa-2040	7	20	,	,	PUNCT
ijassa-2040	7	21	then	then	ADV
ijassa-2040	7	22	the	the	DET
ijassa-2040	7	23	corresponding	corresponding	ADJ
ijassa-2040	7	24	differential	differential	NOUN
ijassa-2040	7	25	operator	operator	NOUN
ijassa-2040	7	26	is	be	AUX
ijassa-2040	7	27	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	7	28	on	on	ADP
ijassa-2040	7	29	the	the	DET
ijassa-2040	7	30	torus	torus	NOUN
ijassa-2040	7	31	.	.	PUNCT
ijassa-2040	8	1	keywords	keyword	NOUN
ijassa-2040	8	2	:	:	PUNCT
ijassa-2040	8	3	differential	differential	ADJ
ijassa-2040	8	4	operator	operator	NOUN
ijassa-2040	8	5	on	on	ADP
ijassa-2040	8	6	the	the	DET
ijassa-2040	8	7	torus	torus	NOUN
ijassa-2040	8	8	,	,	PUNCT
ijassa-2040	8	9	linear	linear	ADJ
ijassa-2040	8	10	differential	differential	NOUN
ijassa-2040	8	11	equation	equation	NOUN
ijassa-2040	8	12	on	on	ADP
ijassa-2040	8	13	the	the	DET
ijassa-2040	8	14	torus	torus	NOUN
ijassa-2040	8	15	,	,	PUNCT
ijassa-2040	8	16	mizohata	mizohata	ADJ
ijassa-2040	8	17	equation	equation	NOUN
ijassa-2040	8	18	,	,	PUNCT
ijassa-2040	8	19	nonstandard	nonstandard	ADJ
ijassa-2040	8	20	analysis	analysis	NOUN
ijassa-2040	8	21	,	,	PUNCT
ijassa-2040	8	22	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	8	23	1	1	NUM
ijassa-2040	8	24	.	.	PUNCT
ijassa-2040	9	1	introduction	introduction	NOUN
ijassa-2040	9	2	periodic	periodic	ADJ
ijassa-2040	9	3	boundary	boundary	ADJ
ijassa-2040	9	4	value	value	NOUN
ijassa-2040	9	5	problems	problem	NOUN
ijassa-2040	9	6	for	for	ADP
ijassa-2040	9	7	differential	differential	ADJ
ijassa-2040	9	8	equations	equation	NOUN
ijassa-2040	9	9	is	be	AUX
ijassa-2040	9	10	a	a	DET
ijassa-2040	9	11	famous	famous	ADJ
ijassa-2040	9	12	object	object	NOUN
ijassa-2040	9	13	both	both	CCONJ
ijassa-2040	9	14	in	in	ADP
ijassa-2040	9	15	mathematical	mathematical	ADJ
ijassa-2040	9	16	education	education	NOUN
ijassa-2040	9	17	and	and	CCONJ
ijassa-2040	9	18	research	research	NOUN
ijassa-2040	9	19	(	(	PUNCT
ijassa-2040	9	20	see	see	VERB
ijassa-2040	9	21	,	,	PUNCT
ijassa-2040	9	22	for	for	ADP
ijassa-2040	9	23	example	example	NOUN
ijassa-2040	9	24	,	,	PUNCT
ijassa-2040	9	25	[	[	X
ijassa-2040	9	26	1	1	NUM
ijassa-2040	9	27	]	]	PUNCT
ijassa-2040	9	28	,	,	PUNCT
ijassa-2040	9	29	[	[	X
ijassa-2040	9	30	2	2	NUM
ijassa-2040	9	31	]	]	PUNCT
ijassa-2040	9	32	,	,	PUNCT
ijassa-2040	9	33	[	[	X
ijassa-2040	9	34	3	3	NUM
ijassa-2040	9	35	]	]	NUM
ijassa-2040	9	36	)	)	PUNCT
ijassa-2040	9	37	.	.	PUNCT
ijassa-2040	10	1	as	as	SCONJ
ijassa-2040	10	2	lax	lax	PROPN
ijassa-2040	10	3	noted	note	VERB
ijassa-2040	10	4	in	in	ADP
ijassa-2040	10	5	[	[	X
ijassa-2040	10	6	4	4	NUM
ijassa-2040	10	7	]	]	PUNCT
ijassa-2040	10	8	,	,	PUNCT
ijassa-2040	10	9	in	in	ADP
ijassa-2040	10	10	the	the	DET
ijassa-2040	10	11	periodic	periodic	ADJ
ijassa-2040	10	12	theory	theory	NOUN
ijassa-2040	10	13	of	of	ADP
ijassa-2040	10	14	differential	differential	ADJ
ijassa-2040	10	15	equations	equation	NOUN
ijassa-2040	10	16	is	be	AUX
ijassa-2040	10	17	free	free	ADJ
ijassa-2040	10	18	of	of	ADP
ijassa-2040	10	19	some	some	DET
ijassa-2040	10	20	technical	technical	ADJ
ijassa-2040	10	21	difficulties	difficulty	NOUN
ijassa-2040	10	22	that	that	PRON
ijassa-2040	10	23	arise	arise	VERB
ijassa-2040	10	24	in	in	ADP
ijassa-2040	10	25	non	non	ADJ
ijassa-2040	10	26	-	-	ADJ
ijassa-2040	10	27	periodic	periodic	ADJ
ijassa-2040	10	28	theory	theory	NOUN
ijassa-2040	10	29	.	.	PUNCT
ijassa-2040	11	1	this	this	PRON
ijassa-2040	11	2	makes	make	VERB
ijassa-2040	11	3	it	it	PRON
ijassa-2040	11	4	possible	possible	ADJ
ijassa-2040	11	5	to	to	PART
ijassa-2040	11	6	create	create	VERB
ijassa-2040	11	7	a	a	DET
ijassa-2040	11	8	more	more	ADV
ijassa-2040	11	9	beautiful	beautiful	ADJ
ijassa-2040	11	10	theory	theory	NOUN
ijassa-2040	11	11	.	.	PUNCT
ijassa-2040	12	1	a	a	DET
ijassa-2040	12	2	study	study	NOUN
ijassa-2040	12	3	of	of	ADP
ijassa-2040	12	4	periodic	periodic	ADJ
ijassa-2040	12	5	boundary	boundary	ADJ
ijassa-2040	12	6	value	value	NOUN
ijassa-2040	12	7	problems	problem	NOUN
ijassa-2040	12	8	for	for	ADP
ijassa-2040	12	9	linear	linear	PROPN
ijassa-2040	12	10	differential	differential	ADJ
ijassa-2040	12	11	equations	equation	NOUN
ijassa-2040	12	12	brings	bring	VERB
ijassa-2040	12	13	us	we	PRON
ijassa-2040	12	14	to	to	ADP
ijassa-2040	12	15	the	the	DET
ijassa-2040	12	16	wonderful	wonderful	ADJ
ijassa-2040	12	17	world	world	NOUN
ijassa-2040	12	18	of	of	ADP
ijassa-2040	12	19	functions	function	NOUN
ijassa-2040	12	20	on	on	ADP
ijassa-2040	12	21	the	the	DET
ijassa-2040	12	22	torus	torus	NOUN
ijassa-2040	12	23	.	.	PUNCT
ijassa-2040	13	1	since	since	SCONJ
ijassa-2040	13	2	the	the	DET
ijassa-2040	13	3	torus	torus	NOUN
ijassa-2040	13	4	as	as	ADP
ijassa-2040	13	5	the	the	DET
ijassa-2040	13	6	product	product	NOUN
ijassa-2040	13	7	of	of	ADP
ijassa-2040	13	8	a	a	DET
ijassa-2040	13	9	finite	finite	ADJ
ijassa-2040	13	10	number	number	NOUN
ijassa-2040	13	11	of	of	ADP
ijassa-2040	13	12	circles	circle	NOUN
ijassa-2040	13	13	,	,	PUNCT
ijassa-2040	13	14	dealing	deal	VERB
ijassa-2040	13	15	with	with	ADP
ijassa-2040	13	16	the	the	DET
ijassa-2040	13	17	torus	torus	NOUN
ijassa-2040	13	18	simplifies	simplifie	NOUN
ijassa-2040	13	19	studying	study	VERB
ijassa-2040	13	20	the	the	DET
ijassa-2040	13	21	behavior	behavior	NOUN
ijassa-2040	13	22	of	of	ADP
ijassa-2040	13	23	functions	function	NOUN
ijassa-2040	13	24	of	of	ADP
ijassa-2040	13	25	several	several	ADJ
ijassa-2040	13	26	variables	variable	NOUN
ijassa-2040	13	27	by	by	ADP
ijassa-2040	13	28	each	each	DET
ijassa-2040	13	29	variable	variable	NOUN
ijassa-2040	13	30	separately	separately	ADV
ijassa-2040	13	31	.	.	PUNCT
ijassa-2040	14	1	moreover	moreover	ADV
ijassa-2040	14	2	,	,	PUNCT
ijassa-2040	14	3	the	the	DET
ijassa-2040	14	4	basis	basis	NOUN
ijassa-2040	14	5	elements	element	NOUN
ijassa-2040	14	6	are	be	AUX
ijassa-2040	14	7	eigenfunctions	eigenfunction	NOUN
ijassa-2040	14	8	of	of	ADP
ijassa-2040	14	9	linear	linear	PROPN
ijassa-2040	14	10	differential	differential	NOUN
ijassa-2040	14	11	operators	operator	NOUN
ijassa-2040	14	12	with	with	ADP
ijassa-2040	14	13	constant	constant	ADJ
ijassa-2040	14	14	coefficients	coefficient	NOUN
ijassa-2040	14	15	.	.	PUNCT
ijassa-2040	15	1	in	in	ADP
ijassa-2040	15	2	addition	addition	NOUN
ijassa-2040	15	3	,	,	PUNCT
ijassa-2040	15	4	the	the	DET
ijassa-2040	15	5	topology	topology	NOUN
ijassa-2040	15	6	of	of	ADP
ijassa-2040	15	7	the	the	DET
ijassa-2040	15	8	torus	torus	NOUN
ijassa-2040	15	9	allows	allow	VERB
ijassa-2040	15	10	us	we	PRON
ijassa-2040	15	11	to	to	PART
ijassa-2040	15	12	forget	forget	VERB
ijassa-2040	15	13	about	about	ADP
ijassa-2040	15	14	the	the	DET
ijassa-2040	15	15	boundary	boundary	NOUN
ijassa-2040	15	16	and	and	CCONJ
ijassa-2040	15	17	the	the	DET
ijassa-2040	15	18	behavior	behavior	NOUN
ijassa-2040	15	19	at	at	ADP
ijassa-2040	15	20	infinity	infinity	NOUN
ijassa-2040	15	21	.	.	PUNCT
ijassa-2040	16	1	this	this	PRON
ijassa-2040	16	2	allows	allow	VERB
ijassa-2040	16	3	us	we	PRON
ijassa-2040	16	4	to	to	PART
ijassa-2040	16	5	focus	focus	VERB
ijassa-2040	16	6	our	our	PRON
ijassa-2040	16	7	attention	attention	NOUN
ijassa-2040	16	8	on	on	ADP
ijassa-2040	16	9	the	the	DET
ijassa-2040	16	10	only	only	ADJ
ijassa-2040	16	11	infinity	infinity	NOUN
ijassa-2040	16	12	that	that	SCONJ
ijassa-2040	16	13	we	we	PRON
ijassa-2040	16	14	face	face	VERB
ijassa-2040	16	15	up	up	ADP
ijassa-2040	16	16	in	in	ADP
ijassa-2040	16	17	this	this	DET
ijassa-2040	16	18	way	way	NOUN
ijassa-2040	16	19	:	:	PUNCT
ijassa-2040	16	20	the	the	DET
ijassa-2040	16	21	infinite	infinite	ADJ
ijassa-2040	16	22	dimension	dimension	NOUN
ijassa-2040	16	23	of	of	ADP
ijassa-2040	16	24	functional	functional	ADJ
ijassa-2040	16	25	spaces	space	NOUN
ijassa-2040	16	26	.	.	PUNCT
ijassa-2040	17	1	there	there	PRON
ijassa-2040	17	2	are	be	VERB
ijassa-2040	17	3	many	many	ADJ
ijassa-2040	17	4	works	work	NOUN
ijassa-2040	17	5	devoted	devote	VERB
ijassa-2040	17	6	to	to	ADP
ijassa-2040	17	7	periodic	periodic	ADJ
ijassa-2040	17	8	boundary	boundary	ADJ
ijassa-2040	17	9	value	value	NOUN
ijassa-2040	17	10	problems	problem	NOUN
ijassa-2040	17	11	for	for	ADP
ijassa-2040	17	12	differential	differential	ADJ
ijassa-2040	17	13	equations	equation	NOUN
ijassa-2040	17	14	considered	consider	VERB
ijassa-2040	17	15	as	as	ADP
ijassa-2040	17	16	equations	equation	NOUN
ijassa-2040	17	17	on	on	ADP
ijassa-2040	17	18	the	the	DET
ijassa-2040	17	19	torus	torus	NOUN
ijassa-2040	17	20	.	.	PUNCT
ijassa-2040	18	1	see	see	VERB
ijassa-2040	18	2	,	,	PUNCT
ijassa-2040	18	3	for	for	ADP
ijassa-2040	18	4	example	example	NOUN
ijassa-2040	18	5	,	,	PUNCT
ijassa-2040	18	6	the	the	DET
ijassa-2040	18	7	books	book	NOUN
ijassa-2040	19	1	[	[	X
ijassa-2040	19	2	1	1	NUM
ijassa-2040	19	3	]	]	PUNCT
ijassa-2040	19	4	,	,	PUNCT
ijassa-2040	19	5	[	[	X
ijassa-2040	19	6	2	2	NUM
ijassa-2040	19	7	]	]	PUNCT
ijassa-2040	19	8	,	,	PUNCT
ijassa-2040	19	9	[	[	X
ijassa-2040	19	10	3	3	NUM
ijassa-2040	19	11	]	]	PUNCT
ijassa-2040	19	12	and	and	CCONJ
ijassa-2040	19	13	the	the	DET
ijassa-2040	19	14	references	reference	NOUN
ijassa-2040	19	15	therein	therein	ADV
ijassa-2040	19	16	.	.	PUNCT
ijassa-2040	20	1	however	however	ADV
ijassa-2040	20	2	,	,	PUNCT
ijassa-2040	20	3	the	the	DET
ijassa-2040	20	4	present	present	ADJ
ijassa-2040	20	5	paper	paper	NOUN
ijassa-2040	20	6	has	have	VERB
ijassa-2040	20	7	no	no	DET
ijassa-2040	20	8	intersection	intersection	NOUN
ijassa-2040	20	9	with	with	ADP
ijassa-2040	20	10	them	they	PRON
ijassa-2040	20	11	.	.	PUNCT
ijassa-2040	21	1	a	a	DET
ijassa-2040	21	2	viewpoint	viewpoint	NOUN
ijassa-2040	21	3	to	to	ADP
ijassa-2040	21	4	the	the	DET
ijassa-2040	21	5	structure	structure	NOUN
ijassa-2040	21	6	of	of	ADP
ijassa-2040	21	7	the	the	DET
ijassa-2040	21	8	set	set	NOUN
ijassa-2040	21	9	of	of	ADP
ijassa-2040	21	10	infinitesimals	infinitesimal	NOUN
ijassa-2040	21	11	presented	present	VERB
ijassa-2040	21	12	in	in	ADP
ijassa-2040	21	13	this	this	DET
ijassa-2040	21	14	paper	paper	NOUN
ijassa-2040	21	15	is	be	AUX
ijassa-2040	21	16	essentially	essentially	ADV
ijassa-2040	21	17	different	different	ADJ
ijassa-2040	21	18	from	from	ADP
ijassa-2040	21	19	those	those	PRON
ijassa-2040	21	20	in	in	ADP
ijassa-2040	21	21	the	the	DET
ijassa-2040	21	22	non	non	ADJ
ijassa-2040	21	23	-	-	ADJ
ijassa-2040	21	24	standard	standard	ADJ
ijassa-2040	21	25	analysis	analysis	NOUN
ijassa-2040	21	26	(	(	PUNCT
ijassa-2040	21	27	see	see	VERB
ijassa-2040	21	28	,	,	PUNCT
ijassa-2040	21	29	for	for	ADP
ijassa-2040	21	30	example	example	NOUN
ijassa-2040	21	31	,	,	PUNCT
ijassa-2040	21	32	[	[	X
ijassa-2040	21	33	14	14	NUM
ijassa-2040	21	34	]	]	SYM
ijassa-2040	21	35	)	)	PUNCT
ijassa-2040	21	36	.	.	PUNCT
ijassa-2040	22	1	the	the	DET
ijassa-2040	22	2	first	first	ADJ
ijassa-2040	22	3	mention	mention	NOUN
ijassa-2040	22	4	of	of	ADP
ijassa-2040	22	5	the	the	DET
ijassa-2040	22	6	presented	present	VERB
ijassa-2040	22	7	results	result	NOUN
ijassa-2040	22	8	was	be	AUX
ijassa-2040	22	9	published	publish	VERB
ijassa-2040	22	10	in	in	ADP
ijassa-2040	22	11	russian	russian	NOUN
ijassa-2040	22	12	in	in	ADV
ijassa-2040	22	13	in	in	ADP
ijassa-2040	22	14	hard	hard	ADJ
ijassa-2040	22	15	-	-	PUNCT
ijassa-2040	22	16	to	to	PART
ijassa-2040	22	17	-	-	PUNCT
ijassa-2040	22	18	find	find	VERB
ijassa-2040	22	19	publications	publication	NOUN
ijassa-2040	23	1	[	[	X
ijassa-2040	23	2	15],[16	15],[16	X
ijassa-2040	23	3	]	]	X
ijassa-2040	23	4	and	and	CCONJ
ijassa-2040	23	5	it	it	PRON
ijassa-2040	23	6	not	not	PART
ijassa-2040	23	7	published	publish	VERB
ijassa-2040	23	8	in	in	ADP
ijassa-2040	23	9	other	other	ADJ
ijassa-2040	23	10	languages	language	NOUN
ijassa-2040	23	11	.	.	PUNCT
ijassa-2040	24	1	in	in	ADP
ijassa-2040	24	2	the	the	DET
ijassa-2040	24	3	future	future	NOUN
ijassa-2040	24	4	it	it	PRON
ijassa-2040	24	5	would	would	AUX
ijassa-2040	24	6	be	be	AUX
ijassa-2040	24	7	interesting	interesting	ADJ
ijassa-2040	24	8	to	to	PART
ijassa-2040	24	9	consider	consider	VERB
ijassa-2040	24	10	,	,	PUNCT
ijassa-2040	24	11	in	in	ADP
ijassa-2040	24	12	particular	particular	ADJ
ijassa-2040	24	13	,	,	PUNCT
ijassa-2040	24	14	the	the	DET
ijassa-2040	24	15	spectral	spectral	ADJ
ijassa-2040	24	16	properties	property	NOUN
ijassa-2040	24	17	of	of	ADP
ijassa-2040	24	18	differential	differential	ADJ
ijassa-2040	24	19	operators	operator	NOUN
ijassa-2040	24	20	on	on	ADP
ijassa-2040	24	21	a	a	DET
ijassa-2040	24	22	torus	torus	NOUN
ijassa-2040	24	23	in	in	ADP
ijassa-2040	24	24	the	the	DET
ijassa-2040	24	25	spirit	spirit	NOUN
ijassa-2040	24	26	of	of	ADP
ijassa-2040	24	27	elliptic	elliptic	ADJ
ijassa-2040	24	28	theory	theory	NOUN
ijassa-2040	24	29	.	.	PUNCT
ijassa-2040	25	1	the	the	DET
ijassa-2040	25	2	author	author	NOUN
ijassa-2040	25	3	expresses	express	VERB
ijassa-2040	25	4	his	his	PRON
ijassa-2040	25	5	gratitude	gratitude	NOUN
ijassa-2040	25	6	to	to	PART
ijassa-2040	25	7	remizov	remizov	VERB
ijassa-2040	25	8	a.o	a.o	PROPN
ijassa-2040	25	9	.	.	PROPN
ijassa-2040	25	10	for	for	ADP
ijassa-2040	25	11	his	his	PRON
ijassa-2040	25	12	assistance	assistance	NOUN
ijassa-2040	25	13	.	.	PUNCT
ijassa-2040	26	1	∗corresponding	∗corresponde	VERB
ijassa-2040	26	2	author	author	NOUN
ijassa-2040	26	3	:	:	PUNCT
ijassa-2040	26	4	burskii.vp@phystech.edu	burskii.vp@phystech.edu	PROPN
ijassa-2040	26	5	linear	linear	ADJ
ijassa-2040	26	6	differential	differential	ADJ
ijassa-2040	26	7	equations	equation	NOUN
ijassa-2040	26	8	on	on	ADP
ijassa-2040	26	9	the	the	DET
ijassa-2040	26	10	torus	torus	NOUN
ijassa-2040	26	11	...	...	PUNCT
ijassa-2040	26	12	13	13	NUM
ijassa-2040	26	13	2	2	NUM
ijassa-2040	26	14	.	.	PUNCT
ijassa-2040	27	1	spaces	space	NOUN
ijassa-2040	27	2	of	of	ADP
ijassa-2040	27	3	periodic	periodic	ADJ
ijassa-2040	27	4	functions	function	NOUN
ijassa-2040	27	5	2.1	2.1	NUM
ijassa-2040	27	6	.	.	PUNCT
ijassa-2040	28	1	spaces	space	NOUN
ijassa-2040	28	2	of	of	ADP
ijassa-2040	28	3	periodic	periodic	ADJ
ijassa-2040	28	4	functions	function	NOUN
ijassa-2040	28	5	the	the	DET
ijassa-2040	28	6	number	number	NOUN
ijassa-2040	28	7	of	of	ADP
ijassa-2040	28	8	variables	variable	NOUN
ijassa-2040	28	9	in	in	ADP
ijassa-2040	28	10	this	this	DET
ijassa-2040	28	11	problem	problem	NOUN
ijassa-2040	28	12	is	be	AUX
ijassa-2040	28	13	not	not	PART
ijassa-2040	28	14	significant	significant	ADJ
ijassa-2040	28	15	,	,	PUNCT
ijassa-2040	28	16	and	and	CCONJ
ijassa-2040	28	17	without	without	ADP
ijassa-2040	28	18	loss	loss	NOUN
ijassa-2040	28	19	of	of	ADP
ijassa-2040	28	20	generality	generality	NOUN
ijassa-2040	28	21	we	we	PRON
ijassa-2040	28	22	shall	shall	AUX
ijassa-2040	28	23	consider	consider	VERB
ijassa-2040	28	24	the	the	DET
ijassa-2040	28	25	case	case	NOUN
ijassa-2040	28	26	of	of	ADP
ijassa-2040	28	27	two	two	NUM
ijassa-2040	28	28	variables	variable	NOUN
ijassa-2040	28	29	.	.	PUNCT
ijassa-2040	29	1	it	it	PRON
ijassa-2040	29	2	is	be	AUX
ijassa-2040	29	3	well	well	ADV
ijassa-2040	29	4	known	know	VERB
ijassa-2040	29	5	that	that	SCONJ
ijassa-2040	29	6	every	every	DET
ijassa-2040	29	7	fourier	fourier	NOUN
ijassa-2040	29	8	series	series	NOUN
ijassa-2040	29	9	(	(	PUNCT
ijassa-2040	29	10	trigonometric	trigonometric	NOUN
ijassa-2040	29	11	series)∑	series)∑	NOUN
ijassa-2040	29	12	k∈z⊕z	k∈z⊕z	PROPN
ijassa-2040	29	13	ake	ake	NOUN
ijassa-2040	29	14	ikx	ikx	PROPN
ijassa-2040	29	15	,	,	PUNCT
ijassa-2040	29	16	∑	∑	PROPN
ijassa-2040	29	17	k	k	X
ijassa-2040	29	18	|ak|2	|ak|2	PUNCT
ijassa-2040	29	19	<	<	X
ijassa-2040	29	20	∞	∞	PROPN
ijassa-2040	29	21	,	,	PUNCT
ijassa-2040	29	22	where	where	SCONJ
ijassa-2040	29	23	ak	ak	PROPN
ijassa-2040	29	24	∈	∈	PROPN
ijassa-2040	29	25	c	c	PROPN
ijassa-2040	29	26	,	,	PUNCT
ijassa-2040	29	27	k	k	X
ijassa-2040	29	28	=	=	PRON
ijassa-2040	29	29	(	(	PUNCT
ijassa-2040	29	30	k1	k1	PROPN
ijassa-2040	29	31	,	,	PUNCT
ijassa-2040	29	32	k2	k2	NOUN
ijassa-2040	29	33	)	)	PUNCT
ijassa-2040	29	34	,	,	PUNCT
ijassa-2040	29	35	x	x	PUNCT
ijassa-2040	29	36	=	=	PRON
ijassa-2040	29	37	(	(	PUNCT
ijassa-2040	29	38	x1	x1	PROPN
ijassa-2040	29	39	,	,	PUNCT
ijassa-2040	29	40	x2	x2	ADJ
ijassa-2040	29	41	)	)	PUNCT
ijassa-2040	29	42	∈	∈	NOUN
ijassa-2040	29	43	r2	r2	NOUN
ijassa-2040	29	44	,	,	PUNCT
ijassa-2040	29	45	and	and	CCONJ
ijassa-2040	29	46	kx	kx	PROPN
ijassa-2040	29	47	=	=	SYM
ijassa-2040	29	48	k1x1	k1x1	PROPN
ijassa-2040	29	49	+	+	CCONJ
ijassa-2040	29	50	k1x2	k1x2	X
ijassa-2040	29	51	,	,	PUNCT
ijassa-2040	29	52	defines	define	VERB
ijassa-2040	29	53	a	a	DET
ijassa-2040	29	54	periodic	periodic	ADJ
ijassa-2040	29	55	square	square	ADJ
ijassa-2040	29	56	integrable	integrable	ADJ
ijassa-2040	29	57	function	function	NOUN
ijassa-2040	29	58	on	on	ADP
ijassa-2040	29	59	x.	x.	PROPN
ijassa-2040	29	60	thus	thus	ADV
ijassa-2040	29	61	,	,	PUNCT
ijassa-2040	29	62	such	such	ADJ
ijassa-2040	29	63	series	series	NOUN
ijassa-2040	29	64	form	form	VERB
ijassa-2040	29	65	the	the	DET
ijassa-2040	29	66	space	space	NOUN
ijassa-2040	29	67	l2(t	l2(t	NOUN
ijassa-2040	29	68	2	2	NUM
ijassa-2040	29	69	)	)	PUNCT
ijassa-2040	29	70	,	,	PUNCT
ijassa-2040	29	71	where	where	SCONJ
ijassa-2040	29	72	t	t	PROPN
ijassa-2040	29	73	2	2	NUM
ijassa-2040	29	74	=	=	SYM
ijassa-2040	29	75	r2	r2	PROPN
ijassa-2040	29	76	/	/	SYM
ijassa-2040	29	77	z2	z2	PROPN
ijassa-2040	29	78	is	be	AUX
ijassa-2040	29	79	the	the	DET
ijassa-2040	29	80	torus	torus	NOUN
ijassa-2040	29	81	of	of	ADP
ijassa-2040	29	82	dimension	dimension	NOUN
ijassa-2040	29	83	2	2	NUM
ijassa-2040	29	84	.	.	PUNCT
ijassa-2040	30	1	for	for	ADP
ijassa-2040	30	2	every	every	DET
ijassa-2040	30	3	m	m	PROPN
ijassa-2040	30	4	∈	∈	PROPN
ijassa-2040	30	5	z	z	PROPN
ijassa-2040	30	6	,	,	PUNCT
ijassa-2040	30	7	m	m	VERB
ijassa-2040	30	8	≥	≥	NOUN
ijassa-2040	30	9	2	2	NUM
ijassa-2040	30	10	,	,	PUNCT
ijassa-2040	30	11	define	define	VERB
ijassa-2040	30	12	by	by	ADP
ijassa-2040	30	13	hm	hm	INTJ
ijassa-2040	30	14	the	the	DET
ijassa-2040	30	15	space	space	NOUN
ijassa-2040	30	16	of	of	ADP
ijassa-2040	30	17	2π	2π	NOUN
ijassa-2040	30	18	-	-	ADJ
ijassa-2040	30	19	periodic	periodic	ADJ
ijassa-2040	30	20	complex	complex	ADV
ijassa-2040	30	21	-	-	PUNCT
ijassa-2040	30	22	valued	value	VERB
ijassa-2040	30	23	functions	function	NOUN
ijassa-2040	30	24	in	in	ADP
ijassa-2040	30	25	r2	r2	PROPN
ijassa-2040	30	26	such	such	ADJ
ijassa-2040	30	27	that	that	SCONJ
ijassa-2040	30	28	∥u∥2	∥u∥2	NOUN
ijassa-2040	30	29	m	m	VERB
ijassa-2040	30	30	=	=	ADJ
ijassa-2040	30	31	∫	∫	PROPN
ijassa-2040	30	32	t	t	PROPN
ijassa-2040	30	33	2	2	NUM
ijassa-2040	30	34	u(x)(1−∆)mu(x)dx	u(x)(1−∆)mu(x)dx	NOUN
ijassa-2040	30	35	<	<	X
ijassa-2040	30	36	∞	∞	PROPN
ijassa-2040	30	37	,	,	PUNCT
ijassa-2040	30	38	where	where	SCONJ
ijassa-2040	30	39	∆	∆	PROPN
ijassa-2040	30	40	=	=	SYM
ijassa-2040	30	41	∂2	∂2	PROPN
ijassa-2040	30	42	∂x2	∂x2	NOUN
ijassa-2040	30	43	1	1	NUM
ijassa-2040	30	44	+	+	CCONJ
ijassa-2040	30	45	∂2	∂2	NUM
ijassa-2040	30	46	∂x2	∂x2	NOUN
ijassa-2040	30	47	2	2	NUM
ijassa-2040	30	48	is	be	AUX
ijassa-2040	30	49	the	the	DET
ijassa-2040	30	50	laplace	laplace	NOUN
ijassa-2040	30	51	operator	operator	NOUN
ijassa-2040	30	52	.	.	PUNCT
ijassa-2040	31	1	all	all	PRON
ijassa-2040	31	2	hm	hm	INTJ
ijassa-2040	31	3	are	be	AUX
ijassa-2040	31	4	hilbert	hilbert	NOUN
ijassa-2040	31	5	spaces	space	NOUN
ijassa-2040	31	6	,	,	PUNCT
ijassa-2040	31	7	namely	namely	ADV
ijassa-2040	31	8	,	,	PUNCT
ijassa-2040	31	9	the	the	DET
ijassa-2040	31	10	famous	famous	ADJ
ijassa-2040	31	11	sobolev	sobolev	NOUN
ijassa-2040	31	12	spaces	space	VERB
ijassa-2040	31	13	.	.	PUNCT
ijassa-2040	32	1	it	it	PRON
ijassa-2040	32	2	is	be	AUX
ijassa-2040	32	3	known	know	VERB
ijassa-2040	32	4	(	(	PUNCT
ijassa-2040	32	5	see	see	VERB
ijassa-2040	32	6	,	,	PUNCT
ijassa-2040	32	7	for	for	ADP
ijassa-2040	32	8	example	example	NOUN
ijassa-2040	32	9	,	,	PUNCT
ijassa-2040	32	10	[	[	X
ijassa-2040	32	11	1	1	NUM
ijassa-2040	32	12	]	]	PUNCT
ijassa-2040	32	13	)	)	PUNCT
ijassa-2040	32	14	that	that	SCONJ
ijassa-2040	32	15	functions	function	VERB
ijassa-2040	32	16	exp(ikx	exp(ikx	PROPN
ijassa-2040	32	17	)	)	PUNCT
ijassa-2040	32	18	,	,	PUNCT
ijassa-2040	32	19	where	where	SCONJ
ijassa-2040	32	20	kx	kx	PROPN
ijassa-2040	32	21	=	=	X
ijassa-2040	32	22	k1x1	k1x1	PROPN
ijassa-2040	32	23	+	+	CCONJ
ijassa-2040	32	24	k2x2	k2x2	AUX
ijassa-2040	32	25	,	,	PUNCT
ijassa-2040	32	26	form	form	VERB
ijassa-2040	32	27	an	an	DET
ijassa-2040	32	28	orthogonal	orthogonal	ADJ
ijassa-2040	32	29	basis	basis	NOUN
ijassa-2040	32	30	in	in	ADP
ijassa-2040	32	31	hm	hm	INTJ
ijassa-2040	32	32	and	and	CCONJ
ijassa-2040	32	33	,	,	PUNCT
ijassa-2040	32	34	consequently	consequently	ADV
ijassa-2040	32	35	,	,	PUNCT
ijassa-2040	32	36	every	every	DET
ijassa-2040	32	37	function	function	NOUN
ijassa-2040	32	38	f	f	PROPN
ijassa-2040	32	39	∈	∈	PROPN
ijassa-2040	33	1	hm	hm	INTJ
ijassa-2040	33	2	is	be	AUX
ijassa-2040	33	3	expandable	expandable	ADJ
ijassa-2040	33	4	into	into	ADP
ijassa-2040	33	5	the	the	DET
ijassa-2040	33	6	fourier	fourier	NOUN
ijassa-2040	33	7	series	series	NOUN
ijassa-2040	33	8	f	f	PROPN
ijassa-2040	34	1	=	=	PUNCT
ijassa-2040	34	2	∑	∑	PUNCT
ijassa-2040	34	3	k∈z⊕z	k∈z⊕z	VERB
ijassa-2040	34	4	fke	fke	PROPN
ijassa-2040	34	5	ikx	ikx	NOUN
ijassa-2040	34	6	converging	converge	VERB
ijassa-2040	34	7	to	to	ADP
ijassa-2040	34	8	f	f	PROPN
ijassa-2040	34	9	in	in	ADP
ijassa-2040	34	10	the	the	DET
ijassa-2040	34	11	topology	topology	NOUN
ijassa-2040	34	12	of	of	ADP
ijassa-2040	34	13	hm	hm	INTJ
ijassa-2040	34	14	.	.	PUNCT
ijassa-2040	35	1	further	far	ADV
ijassa-2040	35	2	we	we	PRON
ijassa-2040	35	3	shall	shall	AUX
ijassa-2040	35	4	consider	consider	VERB
ijassa-2040	35	5	trigonometric	trigonometric	ADJ
ijassa-2040	35	6	series	series	NOUN
ijassa-2040	35	7	with	with	ADP
ijassa-2040	35	8	arbitrary	arbitrary	ADJ
ijassa-2040	35	9	real	real	ADJ
ijassa-2040	35	10	coefficients	coefficient	NOUN
ijassa-2040	35	11	,	,	PUNCT
ijassa-2040	35	12	not	not	PART
ijassa-2040	35	13	necessarily	necessarily	ADV
ijassa-2040	35	14	converging	converge	VERB
ijassa-2040	35	15	(	(	PUNCT
ijassa-2040	35	16	such	such	ADJ
ijassa-2040	35	17	series	serie	NOUN
ijassa-2040	35	18	are	be	AUX
ijassa-2040	35	19	usually	usually	ADV
ijassa-2040	35	20	called	call	VERB
ijassa-2040	35	21	formal	formal	ADJ
ijassa-2040	35	22	)	)	PUNCT
ijassa-2040	35	23	.	.	PUNCT
ijassa-2040	36	1	one	one	PRON
ijassa-2040	36	2	can	can	AUX
ijassa-2040	36	3	consider	consider	VERB
ijassa-2040	36	4	hm	hm	INTJ
ijassa-2040	36	5	as	as	ADP
ijassa-2040	36	6	the	the	DET
ijassa-2040	36	7	vector	vector	NOUN
ijassa-2040	36	8	space	space	NOUN
ijassa-2040	36	9	consisting	consist	VERB
ijassa-2040	36	10	of	of	ADP
ijassa-2040	36	11	formal	formal	ADJ
ijassa-2040	36	12	fourier	fourier	NOUN
ijassa-2040	36	13	series	series	NOUN
ijassa-2040	36	14	with	with	ADP
ijassa-2040	36	15	the	the	DET
ijassa-2040	36	16	finite	finite	PROPN
ijassa-2040	36	17	norm	norm	PROPN
ijassa-2040	36	18	∥f∥2	∥f∥2	PROPN
ijassa-2040	36	19	m	m	PROPN
ijassa-2040	36	20	=	=	SYM
ijassa-2040	36	21	∑	∑	PUNCT
ijassa-2040	36	22	k	k	X
ijassa-2040	36	23	(	(	PUNCT
ijassa-2040	36	24	1	1	NUM
ijassa-2040	36	25	+	+	CCONJ
ijassa-2040	36	26	k	k	PROPN
ijassa-2040	36	27	·	·	PUNCT
ijassa-2040	36	28	k)m	k)m	X
ijassa-2040	36	29	·	·	PUNCT
ijassa-2040	37	1	|fk|2	|fk|2	PROPN
ijassa-2040	37	2	.	.	PUNCT
ijassa-2040	38	1	this	this	DET
ijassa-2040	38	2	formula	formula	NOUN
ijassa-2040	38	3	defines	define	VERB
ijassa-2040	38	4	a	a	DET
ijassa-2040	38	5	norm	norm	NOUN
ijassa-2040	38	6	in	in	ADP
ijassa-2040	38	7	the	the	DET
ijassa-2040	38	8	space	space	NOUN
ijassa-2040	38	9	hm	hm	INTJ
ijassa-2040	38	10	with	with	ADP
ijassa-2040	38	11	any	any	DET
ijassa-2040	38	12	real	real	ADJ
ijassa-2040	38	13	m.	m.	NOUN
ijassa-2040	38	14	consider	consider	VERB
ijassa-2040	38	15	the	the	DET
ijassa-2040	38	16	vector	vector	NOUN
ijassa-2040	38	17	space	space	NOUN
ijassa-2040	38	18	hm	hm	INTJ
ijassa-2040	38	19	with	with	ADP
ijassa-2040	38	20	the	the	DET
ijassa-2040	38	21	topology	topology	NOUN
ijassa-2040	38	22	of	of	ADP
ijassa-2040	38	23	the	the	DET
ijassa-2040	38	24	space	space	NOUN
ijassa-2040	38	25	rz	rz	NOUN
ijassa-2040	38	26	,	,	PUNCT
ijassa-2040	38	27	in	in	ADP
ijassa-2040	38	28	which	which	PRON
ijassa-2040	38	29	hm	hm	INTJ
ijassa-2040	38	30	is	be	AUX
ijassa-2040	38	31	continuously	continuously	ADV
ijassa-2040	38	32	embedded	embed	VERB
ijassa-2040	38	33	.	.	PUNCT
ijassa-2040	39	1	for	for	ADP
ijassa-2040	39	2	m	m	PROPN
ijassa-2040	39	3	<	<	X
ijassa-2040	39	4	0	0	PROPN
ijassa-2040	39	5	,	,	PUNCT
ijassa-2040	39	6	the	the	DET
ijassa-2040	39	7	space	space	NOUN
ijassa-2040	39	8	hm	hm	INTJ
ijassa-2040	39	9	is	be	AUX
ijassa-2040	39	10	conjugate	conjugate	ADJ
ijassa-2040	39	11	to	to	ADP
ijassa-2040	39	12	the	the	DET
ijassa-2040	39	13	space	space	NOUN
ijassa-2040	39	14	h−m	h−m	PROPN
ijassa-2040	39	15	in	in	ADP
ijassa-2040	39	16	the	the	DET
ijassa-2040	39	17	topology	topology	NOUN
ijassa-2040	39	18	of	of	ADP
ijassa-2040	39	19	the	the	DET
ijassa-2040	39	20	space	space	NOUN
ijassa-2040	39	21	h0	h0	NOUN
ijassa-2040	39	22	=	=	SYM
ijassa-2040	39	23	l2(t	l2(t	PROPN
ijassa-2040	39	24	2	2	NUM
ijassa-2040	39	25	)	)	PUNCT
ijassa-2040	39	26	.	.	PUNCT
ijassa-2040	40	1	moreover	moreover	ADV
ijassa-2040	40	2	,	,	PUNCT
ijassa-2040	40	3	can	can	AUX
ijassa-2040	40	4	prove	prove	VERB
ijassa-2040	40	5	a	a	DET
ijassa-2040	40	6	more	more	ADV
ijassa-2040	40	7	general	general	ADJ
ijassa-2040	40	8	statement	statement	NOUN
ijassa-2040	40	9	:	:	PUNCT
ijassa-2040	40	10	proposition	proposition	NOUN
ijassa-2040	40	11	2.1	2.1	NUM
ijassa-2040	40	12	:	:	PUNCT
ijassa-2040	40	13	let	let	VERB
ijassa-2040	40	14	e	e	PRON
ijassa-2040	40	15	be	be	AUX
ijassa-2040	40	16	a	a	DET
ijassa-2040	40	17	barreled	barrel	VERB
ijassa-2040	40	18	vector	vector	NOUN
ijassa-2040	40	19	topological	topological	ADJ
ijassa-2040	40	20	space	space	NOUN
ijassa-2040	40	21	of	of	ADP
ijassa-2040	40	22	2π	2π	NOUN
ijassa-2040	40	23	-	-	ADJ
ijassa-2040	40	24	periodic	periodic	ADJ
ijassa-2040	40	25	functions	function	NOUN
ijassa-2040	40	26	continuously	continuously	ADV
ijassa-2040	40	27	embedded	embed	VERB
ijassa-2040	40	28	in	in	ADP
ijassa-2040	40	29	h0	h0	PROPN
ijassa-2040	40	30	.	.	PUNCT
ijassa-2040	41	1	let	let	VERB
ijassa-2040	41	2	the	the	DET
ijassa-2040	41	3	system	system	NOUN
ijassa-2040	41	4	{	{	PUNCT
ijassa-2040	41	5	eikx	eikx	PROPN
ijassa-2040	41	6	}	}	PUNCT
ijassa-2040	41	7	be	be	AUX
ijassa-2040	41	8	a	a	DET
ijassa-2040	41	9	basis	basis	NOUN
ijassa-2040	41	10	in	in	ADP
ijassa-2040	41	11	e	e	NOUN
ijassa-2040	41	12	,	,	PUNCT
ijassa-2040	41	13	that	that	ADV
ijassa-2040	41	14	is	is	ADV
ijassa-2040	41	15	,	,	PUNCT
ijassa-2040	41	16	for	for	ADP
ijassa-2040	41	17	every	every	DET
ijassa-2040	41	18	v	v	NOUN
ijassa-2040	41	19	∈	∈	NOUN
ijassa-2040	41	20	e	e	NOUN
ijassa-2040	41	21	there	there	PRON
ijassa-2040	41	22	exists	exist	VERB
ijassa-2040	41	23	a	a	DET
ijassa-2040	41	24	unique	unique	ADJ
ijassa-2040	41	25	sequence	sequence	NOUN
ijassa-2040	41	26	{	{	PUNCT
ijassa-2040	41	27	vk	vk	NOUN
ijassa-2040	41	28	}	}	PUNCT
ijassa-2040	41	29	⊂	⊂	PROPN
ijassa-2040	42	1	e	e	NOUN
ijassa-2040	42	2	such	such	ADJ
ijassa-2040	42	3	that	that	SCONJ
ijassa-2040	42	4	∑	∑	PUNCT
ijassa-2040	42	5	k2≤n	k2≤n	PROPN
ijassa-2040	42	6	vke	vke	PROPN
ijassa-2040	42	7	ikx	ikx	PROPN
ijassa-2040	42	8	→	→	SYM
ijassa-2040	42	9	v	v	NOUN
ijassa-2040	42	10	as	as	ADP
ijassa-2040	42	11	n	n	PROPN
ijassa-2040	42	12	→	→	SYM
ijassa-2040	42	13	∞.	∞.	PROPN
ijassa-2040	42	14	then	then	ADV
ijassa-2040	42	15	the	the	DET
ijassa-2040	42	16	dual	dual	ADJ
ijassa-2040	42	17	space	space	NOUN
ijassa-2040	42	18	e∗	e∗	NOUN
ijassa-2040	42	19	is	be	AUX
ijassa-2040	42	20	naturally	naturally	ADV
ijassa-2040	42	21	isomorphic	isomorphic	ADJ
ijassa-2040	42	22	to	to	ADP
ijassa-2040	42	23	a	a	DET
ijassa-2040	42	24	subspace	subspace	NOUN
ijassa-2040	42	25	of	of	ADP
ijassa-2040	42	26	the	the	DET
ijassa-2040	42	27	space	space	NOUN
ijassa-2040	43	1	f	f	NOUN
ijassa-2040	43	2	=	=	PRON
ijassa-2040	43	3	{	{	PUNCT
ijassa-2040	43	4	u	u	NOUN
ijassa-2040	43	5	=	=	SYM
ijassa-2040	43	6	∑	∑	PROPN
ijassa-2040	43	7	k	k	PROPN
ijassa-2040	43	8	uke	uke	PROPN
ijassa-2040	43	9	ikx	ikx	PROPN
ijassa-2040	43	10	:	:	PUNCT
ijassa-2040	43	11	∑	∑	PUNCT
ijassa-2040	43	12	vke	vke	X
ijassa-2040	43	13	ikx	ikx	PROPN
ijassa-2040	43	14	∈	∈	PROPN
ijassa-2040	43	15	e	e	PROPN
ijassa-2040	43	16	,	,	PUNCT
ijassa-2040	43	17	⟨u	⟨u	NOUN
ijassa-2040	43	18	,	,	PUNCT
ijassa-2040	43	19	v⟩	v⟩	NOUN
ijassa-2040	43	20	=	=	PUNCT
ijassa-2040	43	21	∑	∑	PUNCT
ijassa-2040	43	22	ukvk	ukvk	VERB
ijassa-2040	43	23	<	<	X
ijassa-2040	43	24	∞	∞	NUM
ijassa-2040	43	25	∀v	∀v	PROPN
ijassa-2040	43	26	∈	∈	PROPN
ijassa-2040	43	27	e	e	NOUN
ijassa-2040	43	28	}	}	PUNCT
ijassa-2040	43	29	.	.	PUNCT
ijassa-2040	44	1	here	here	ADV
ijassa-2040	44	2	the	the	DET
ijassa-2040	44	3	paring	paring	NOUN
ijassa-2040	44	4	⟨	⟨	VERB
ijassa-2040	44	5	·	·	PUNCT
ijassa-2040	44	6	,	,	PUNCT
ijassa-2040	44	7	·	·	PUNCT
ijassa-2040	44	8	⟩	⟩	NOUN
ijassa-2040	44	9	gives	give	VERB
ijassa-2040	44	10	rise	rise	NOUN
ijassa-2040	44	11	to	to	ADP
ijassa-2040	44	12	the	the	DET
ijassa-2040	44	13	duality	duality	NOUN
ijassa-2040	44	14	.	.	PUNCT
ijassa-2040	45	1	copyright	copyright	NOUN
ijassa-2040	45	2	©	©	PROPN
ijassa-2040	45	3	2025	2025	NUM
ijassa-2040	45	4	assa	assa	NOUN
ijassa-2040	45	5	.	.	PUNCT
ijassa-2040	46	1	adv	adv	PROPN
ijassa-2040	46	2	syst	syst	PROPN
ijassa-2040	46	3	sci	sci	PROPN
ijassa-2040	46	4	appl	appl	PROPN
ijassa-2040	46	5	(	(	PUNCT
ijassa-2040	46	6	2025	2025	NUM
ijassa-2040	46	7	)	)	PUNCT
ijassa-2040	46	8	14	14	NUM
ijassa-2040	46	9	v.	v.	ADP
ijassa-2040	46	10	p.	p.	NOUN
ijassa-2040	46	11	burskii	burskii	PROPN
ijassa-2040	46	12	proof	proof	NOUN
ijassa-2040	46	13	this	this	DET
ijassa-2040	46	14	statement	statement	NOUN
ijassa-2040	46	15	follows	follow	VERB
ijassa-2040	46	16	from	from	ADP
ijassa-2040	46	17	the	the	DET
ijassa-2040	46	18	known	know	VERB
ijassa-2040	46	19	fact	fact	NOUN
ijassa-2040	46	20	that	that	SCONJ
ijassa-2040	46	21	the	the	DET
ijassa-2040	46	22	mackey	mackey	PROPN
ijassa-2040	46	23	topology	topology	PROPN
ijassa-2040	46	24	in	in	ADP
ijassa-2040	46	25	barrel	barrel	NOUN
ijassa-2040	46	26	spaces	space	NOUN
ijassa-2040	46	27	coincides	coincide	VERB
ijassa-2040	46	28	with	with	ADP
ijassa-2040	46	29	the	the	DET
ijassa-2040	46	30	original	original	ADJ
ijassa-2040	46	31	topology	topology	NOUN
ijassa-2040	46	32	,	,	PUNCT
ijassa-2040	46	33	since	since	SCONJ
ijassa-2040	46	34	it	it	PRON
ijassa-2040	46	35	is	be	AUX
ijassa-2040	46	36	the	the	DET
ijassa-2040	46	37	strongest	strong	ADJ
ijassa-2040	46	38	among	among	ADP
ijassa-2040	46	39	all	all	DET
ijassa-2040	46	40	topologies	topology	NOUN
ijassa-2040	46	41	consistent	consistent	ADJ
ijassa-2040	46	42	with	with	ADP
ijassa-2040	46	43	duality	duality	NOUN
ijassa-2040	46	44	(	(	PUNCT
ijassa-2040	46	45	see	see	VERB
ijassa-2040	46	46	[	[	X
ijassa-2040	46	47	5	5	NUM
ijassa-2040	46	48	]	]	PUNCT
ijassa-2040	46	49	)	)	PUNCT
ijassa-2040	46	50	.	.	PUNCT
ijassa-2040	47	1	there	there	PRON
ijassa-2040	47	2	are	be	VERB
ijassa-2040	47	3	several	several	ADJ
ijassa-2040	47	4	examples	example	NOUN
ijassa-2040	47	5	,	,	PUNCT
ijassa-2040	47	6	which	which	PRON
ijassa-2040	47	7	we	we	PRON
ijassa-2040	47	8	shall	shall	AUX
ijassa-2040	47	9	use	use	VERB
ijassa-2040	47	10	below	below	ADV
ijassa-2040	47	11	:	:	PUNCT
ijassa-2040	47	12	example	example	NOUN
ijassa-2040	47	13	2.1	2.1	NUM
ijassa-2040	47	14	:	:	PUNCT
ijassa-2040	47	15	the	the	DET
ijassa-2040	47	16	space	space	NOUN
ijassa-2040	47	17	of	of	ADP
ijassa-2040	47	18	infinitely	infinitely	ADV
ijassa-2040	47	19	differentiable	differentiable	ADJ
ijassa-2040	47	20	periodic	periodic	ADJ
ijassa-2040	47	21	functions	function	NOUN
ijassa-2040	47	22	h∞	h∞	X
ijassa-2040	48	1	=	=	SYM
ijassa-2040	48	2	∩	∩	NOUN
ijassa-2040	48	3	m	m	VERB
ijassa-2040	48	4	hm	hm	INTJ
ijassa-2040	48	5	,	,	PUNCT
ijassa-2040	48	6	whose	whose	DET
ijassa-2040	48	7	total	total	ADJ
ijassa-2040	48	8	element	element	NOUN
ijassa-2040	48	9	has	have	VERB
ijassa-2040	48	10	the	the	DET
ijassa-2040	48	11	form	form	NOUN
ijassa-2040	48	12	u	u	NOUN
ijassa-2040	48	13	=	=	NOUN
ijassa-2040	48	14	∑	∑	PROPN
ijassa-2040	48	15	uke	uke	PROPN
ijassa-2040	48	16	ikx	ikx	PROPN
ijassa-2040	48	17	,	,	PUNCT
ijassa-2040	48	18	k2luk	k2luk	PROPN
ijassa-2040	48	19	k→∞−→	k→∞−→	PROPN
ijassa-2040	48	20	0	0	NUM
ijassa-2040	48	21	(	(	PUNCT
ijassa-2040	48	22	∀	∀	X
ijassa-2040	48	23	l	l	NOUN
ijassa-2040	48	24	)	)	PUNCT
ijassa-2040	48	25	.	.	PUNCT
ijassa-2040	49	1	also	also	ADV
ijassa-2040	49	2	the	the	DET
ijassa-2040	49	3	conjugate	conjugate	ADJ
ijassa-2040	49	4	space	space	NOUN
ijassa-2040	49	5	(	(	PUNCT
ijassa-2040	49	6	h∞)∗	h∞)∗	NOUN
ijassa-2040	49	7	=	=	PRON
ijassa-2040	49	8	h−∞	h−∞	NOUN
ijassa-2040	49	9	=	=	SYM
ijassa-2040	49	10	∪	∪	ADP
ijassa-2040	49	11	m	m	VERB
ijassa-2040	49	12	hm	hm	INTJ
ijassa-2040	49	13	,	,	PUNCT
ijassa-2040	49	14	which	which	PRON
ijassa-2040	49	15	is	be	AUX
ijassa-2040	49	16	the	the	DET
ijassa-2040	49	17	space	space	NOUN
ijassa-2040	49	18	of	of	ADP
ijassa-2040	49	19	periodic	periodic	ADJ
ijassa-2040	49	20	distributions	distribution	NOUN
ijassa-2040	49	21	whose	whose	DET
ijassa-2040	49	22	fourier	fourier	NOUN
ijassa-2040	49	23	coefficients	coefficient	NOUN
ijassa-2040	49	24	tend	tend	VERB
ijassa-2040	49	25	to	to	PART
ijassa-2040	49	26	infinity	infinity	VERB
ijassa-2040	49	27	no	no	ADV
ijassa-2040	49	28	faster	fast	ADV
ijassa-2040	49	29	than	than	ADP
ijassa-2040	49	30	some	some	DET
ijassa-2040	49	31	power	power	NOUN
ijassa-2040	49	32	of	of	ADP
ijassa-2040	49	33	k2l	k2l	PROPN
ijassa-2040	49	34	.	.	PROPN
ijassa-2040	49	35	example	example	NOUN
ijassa-2040	49	36	2.2	2.2	NUM
ijassa-2040	49	37	:	:	PUNCT
ijassa-2040	49	38	the	the	DET
ijassa-2040	49	39	space	space	NOUN
ijassa-2040	49	40	e0	e0	PROPN
ijassa-2040	49	41	of	of	ADP
ijassa-2040	49	42	2π	2π	NOUN
ijassa-2040	49	43	-	-	ADJ
ijassa-2040	49	44	periodic	periodic	ADJ
ijassa-2040	49	45	functions	function	NOUN
ijassa-2040	49	46	u	u	NOUN
ijassa-2040	49	47	=	=	NOUN
ijassa-2040	49	48	∑	∑	PROPN
ijassa-2040	49	49	uke	uke	PROPN
ijassa-2040	49	50	ikx	ikx	PROPN
ijassa-2040	49	51	,	,	PUNCT
ijassa-2040	49	52	∃δ1	∃δ1	PROPN
ijassa-2040	49	53	>	>	X
ijassa-2040	49	54	0	0	PROPN
ijassa-2040	49	55	,	,	PUNCT
ijassa-2040	49	56	δ2	δ2	VERB
ijassa-2040	49	57	>	>	X
ijassa-2040	49	58	0	0	NUM
ijassa-2040	49	59	:	:	PUNCT
ijassa-2040	49	60	∑	∑	PUNCT
ijassa-2040	49	61	e|k1|δ1+|k2|δ2|uk|	e|k1|δ1+|k2|δ2|uk|	X
ijassa-2040	49	62	<	<	X
ijassa-2040	49	63	∞	∞	PROPN
ijassa-2040	49	64	,	,	PUNCT
ijassa-2040	49	65	included	include	VERB
ijassa-2040	49	66	in	in	ADP
ijassa-2040	49	67	the	the	DET
ijassa-2040	49	68	space	space	NOUN
ijassa-2040	49	69	of	of	ADP
ijassa-2040	49	70	periodic	periodic	ADJ
ijassa-2040	49	71	real	real	ADJ
ijassa-2040	49	72	analytic	analytic	ADJ
ijassa-2040	49	73	functions	function	NOUN
ijassa-2040	49	74	.	.	PUNCT
ijassa-2040	50	1	another	another	DET
ijassa-2040	50	2	examples	example	NOUN
ijassa-2040	50	3	is	be	AUX
ijassa-2040	50	4	the	the	DET
ijassa-2040	50	5	conjugate	conjugate	ADJ
ijassa-2040	50	6	space	space	NOUN
ijassa-2040	50	7	e∗	e∗	PROPN
ijassa-2040	50	8	0	0	NUM
ijassa-2040	50	9	,	,	PUNCT
ijassa-2040	50	10	which	which	PRON
ijassa-2040	50	11	consists	consist	VERB
ijassa-2040	50	12	of	of	ADP
ijassa-2040	50	13	series	series	PROPN
ijassa-2040	50	14	∑	∑	PROPN
ijassa-2040	50	15	uke	uke	PROPN
ijassa-2040	50	16	ikx	ikx	PROPN
ijassa-2040	50	17	whose	whose	DET
ijassa-2040	50	18	coefficients	coefficient	NOUN
ijassa-2040	50	19	uk	uk	PROPN
ijassa-2040	50	20	k→∞−→	k→∞−→	VERB
ijassa-2040	50	21	∞	∞	PROPN
ijassa-2040	50	22	slower	slow	ADJ
ijassa-2040	50	23	than	than	ADP
ijassa-2040	50	24	any	any	DET
ijassa-2040	50	25	exponential	exponential	ADJ
ijassa-2040	50	26	e|k1|δ1+|k2|δ2	e|k1|δ1+|k2|δ2	NOUN
ijassa-2040	50	27	.	.	PUNCT
ijassa-2040	51	1	this	this	DET
ijassa-2040	51	2	space	space	NOUN
ijassa-2040	51	3	contains	contain	VERB
ijassa-2040	51	4	the	the	DET
ijassa-2040	51	5	space	space	NOUN
ijassa-2040	51	6	of	of	ADP
ijassa-2040	51	7	hyperfunctions	hyperfunction	NOUN
ijassa-2040	51	8	(	(	PUNCT
ijassa-2040	51	9	[	[	X
ijassa-2040	51	10	6	6	NUM
ijassa-2040	51	11	]	]	NUM
ijassa-2040	51	12	)	)	PUNCT
ijassa-2040	51	13	.	.	PUNCT
ijassa-2040	52	1	example	example	NOUN
ijassa-2040	53	1	2.3	2.3	NUM
ijassa-2040	53	2	:	:	PUNCT
ijassa-2040	53	3	the	the	DET
ijassa-2040	53	4	space	space	NOUN
ijassa-2040	53	5	l1(|k1|	l1(|k1|	NOUN
ijassa-2040	53	6	!	!	PUNCT
ijassa-2040	53	7	)	)	PUNCT
ijassa-2040	53	8	of	of	ADP
ijassa-2040	53	9	functions	function	NOUN
ijassa-2040	53	10	u	u	NOUN
ijassa-2040	53	11	=	=	NOUN
ijassa-2040	53	12	∑	∑	PROPN
ijassa-2040	53	13	uke	uke	PROPN
ijassa-2040	53	14	ikx	ikx	PROPN
ijassa-2040	53	15	,	,	PUNCT
ijassa-2040	53	16	∑	∑	PROPN
ijassa-2040	53	17	k	k	PROPN
ijassa-2040	53	18	|k1|	|k1|	PROPN
ijassa-2040	53	19	!	!	PROPN
ijassa-2040	53	20	|uk|	|uk|	VERB
ijassa-2040	53	21	<	<	X
ijassa-2040	53	22	∞.	∞.	PROPN
ijassa-2040	53	23	also	also	ADV
ijassa-2040	53	24	the	the	DET
ijassa-2040	53	25	conjugate	conjugate	ADJ
ijassa-2040	53	26	space	space	NOUN
ijassa-2040	53	27	l∗1(|k1|	l∗1(|k1|	PROPN
ijassa-2040	53	28	!	!	PUNCT
ijassa-2040	53	29	)	)	PUNCT
ijassa-2040	53	30	of	of	ADP
ijassa-2040	53	31	series	series	PROPN
ijassa-2040	53	32	∑	∑	PROPN
ijassa-2040	53	33	k	k	PROPN
ijassa-2040	53	34	vke	vke	PROPN
ijassa-2040	53	35	ikx	ikx	PROPN
ijassa-2040	53	36	,	,	PUNCT
ijassa-2040	53	37	vk	vk	ADP
ijassa-2040	53	38	=	=	SYM
ijassa-2040	53	39	o(|k1|	o(|k1|	PROPN
ijassa-2040	53	40	!	!	PUNCT
ijassa-2040	53	41	)	)	PUNCT
ijassa-2040	53	42	.	.	PUNCT
ijassa-2040	54	1	let	let	VERB
ijassa-2040	54	2	p	p	NOUN
ijassa-2040	54	3	(	(	PUNCT
ijassa-2040	54	4	x1	x1	PROPN
ijassa-2040	54	5	,	,	PUNCT
ijassa-2040	54	6	x2	x2	PROPN
ijassa-2040	54	7	)	)	PUNCT
ijassa-2040	54	8	be	be	VERB
ijassa-2040	54	9	a	a	DET
ijassa-2040	54	10	homogeneous	homogeneous	ADJ
ijassa-2040	54	11	polynomial	polynomial	NOUN
ijassa-2040	54	12	of	of	ADP
ijassa-2040	54	13	degree	degree	NOUN
ijassa-2040	54	14	p	p	NOUN
ijassa-2040	54	15	with	with	ADP
ijassa-2040	54	16	constant	constant	ADJ
ijassa-2040	54	17	coefficients	coefficient	NOUN
ijassa-2040	54	18	.	.	PUNCT
ijassa-2040	55	1	consider	consider	VERB
ijassa-2040	55	2	the	the	DET
ijassa-2040	55	3	differential	differential	ADJ
ijassa-2040	55	4	operator	operator	NOUN
ijassa-2040	55	5	p̂	p̂	NOUN
ijassa-2040	55	6	:	:	PUNCT
ijassa-2040	55	7	hm	hm	INTJ
ijassa-2040	55	8	→	→	PUNCT
ijassa-2040	55	9	hm−p	hm−p	PROPN
ijassa-2040	55	10	generated	generate	VERB
ijassa-2040	55	11	by	by	ADP
ijassa-2040	55	12	the	the	DET
ijassa-2040	55	13	polynomial	polynomial	ADJ
ijassa-2040	55	14	p	p	X
ijassa-2040	55	15	:	:	PUNCT
ijassa-2040	55	16	p̂	p̂	X
ijassa-2040	55	17	u	u	NOUN
ijassa-2040	55	18	=	=	NOUN
ijassa-2040	55	19	p	p	X
ijassa-2040	55	20	(	(	PUNCT
ijassa-2040	55	21	−i	−i	PROPN
ijassa-2040	55	22	∂	∂	X
ijassa-2040	55	23	∂x1	∂x1	NOUN
ijassa-2040	55	24	,	,	PUNCT
ijassa-2040	55	25	−i	−i	PROPN
ijassa-2040	55	26	∂	∂	ADJ
ijassa-2040	55	27	∂x2	∂x2	NOUN
ijassa-2040	55	28	)	)	PUNCT
ijassa-2040	55	29	u.	u.	VERB
ijassa-2040	55	30	the	the	DET
ijassa-2040	55	31	obvious	obvious	ADJ
ijassa-2040	55	32	formula	formula	NOUN
ijassa-2040	55	33	p̂	p̂	NOUN
ijassa-2040	55	34	(	(	PUNCT
ijassa-2040	55	35	∑	∑	ADV
ijassa-2040	55	36	k∈z⊕z	k∈z⊕z	VERB
ijassa-2040	55	37	fke	fke	PROPN
ijassa-2040	55	38	ikx	ikx	NOUN
ijassa-2040	55	39	)	)	PUNCT
ijassa-2040	56	1	=	=	PUNCT
ijassa-2040	56	2	∑	∑	PUNCT
ijassa-2040	56	3	k∈z⊕z	k∈z⊕z	VERB
ijassa-2040	56	4	p	p	X
ijassa-2040	56	5	(	(	PUNCT
ijassa-2040	56	6	k1	k1	PROPN
ijassa-2040	56	7	,	,	PUNCT
ijassa-2040	56	8	k2)fke	k2)fke	PROPN
ijassa-2040	56	9	ikx	ikx	PROPN
ijassa-2040	56	10	allows	allow	VERB
ijassa-2040	56	11	us	we	PRON
ijassa-2040	56	12	to	to	PART
ijassa-2040	56	13	consider	consider	VERB
ijassa-2040	56	14	the	the	DET
ijassa-2040	56	15	operator	operator	NOUN
ijassa-2040	56	16	p̂	p̂	NOUN
ijassa-2040	56	17	on	on	ADP
ijassa-2040	56	18	the	the	DET
ijassa-2040	56	19	space	space	NOUN
ijassa-2040	56	20	f	f	NOUN
ijassa-2040	56	21	of	of	ADP
ijassa-2040	56	22	formal	formal	ADJ
ijassa-2040	56	23	trigonometric	trigonometric	ADJ
ijassa-2040	56	24	series	series	NOUN
ijassa-2040	56	25	.	.	PUNCT
ijassa-2040	57	1	2.2	2.2	NUM
ijassa-2040	57	2	.	.	PUNCT
ijassa-2040	57	3	solvability	solvability	NOUN
ijassa-2040	57	4	of	of	ADP
ijassa-2040	57	5	the	the	DET
ijassa-2040	57	6	mizohata	mizohata	ADJ
ijassa-2040	57	7	equation	equation	NOUN
ijassa-2040	57	8	in	in	ADP
ijassa-2040	57	9	[	[	X
ijassa-2040	57	10	7	7	NUM
ijassa-2040	57	11	]	]	PUNCT
ijassa-2040	57	12	,	,	PUNCT
ijassa-2040	57	13	g.	g.	PROPN
ijassa-2040	57	14	levy	levy	PROPN
ijassa-2040	57	15	gave	give	VERB
ijassa-2040	57	16	an	an	DET
ijassa-2040	57	17	example	example	NOUN
ijassa-2040	57	18	of	of	ADP
ijassa-2040	57	19	a	a	DET
ijassa-2040	57	20	linear	linear	ADJ
ijassa-2040	57	21	differential	differential	ADJ
ijassa-2040	57	22	equation	equation	NOUN
ijassa-2040	57	23	of	of	ADP
ijassa-2040	57	24	the	the	DET
ijassa-2040	57	25	first	first	ADJ
ijassa-2040	57	26	order	order	NOUN
ijassa-2040	57	27	with	with	ADP
ijassa-2040	57	28	infinitely	infinitely	ADV
ijassa-2040	57	29	differentiable	differentiable	ADJ
ijassa-2040	57	30	coefficients	coefficient	NOUN
ijassa-2040	57	31	that	that	PRON
ijassa-2040	57	32	has	have	VERB
ijassa-2040	57	33	no	no	DET
ijassa-2040	57	34	solutions	solution	NOUN
ijassa-2040	57	35	in	in	ADP
ijassa-2040	57	36	the	the	DET
ijassa-2040	57	37	space	space	NOUN
ijassa-2040	57	38	of	of	ADP
ijassa-2040	57	39	distributions	distribution	NOUN
ijassa-2040	57	40	in	in	ADP
ijassa-2040	57	41	threedimensional	threedimensional	ADJ
ijassa-2040	57	42	space	space	NOUN
ijassa-2040	57	43	.	.	PUNCT
ijassa-2040	58	1	developing	develop	VERB
ijassa-2040	58	2	the	the	DET
ijassa-2040	58	3	ideas	idea	NOUN
ijassa-2040	58	4	of	of	ADP
ijassa-2040	58	5	g.	g.	PROPN
ijassa-2040	58	6	levy	levy	PROPN
ijassa-2040	58	7	and	and	CCONJ
ijassa-2040	58	8	p.	p.	NOUN
ijassa-2040	58	9	garabedyan	garabedyan	VERB
ijassa-2040	59	1	[	[	X
ijassa-2040	59	2	9	9	NUM
ijassa-2040	59	3	]	]	PUNCT
ijassa-2040	59	4	,	,	PUNCT
ijassa-2040	59	5	v.	v.	CCONJ
ijassa-2040	59	6	v.	v.	ADP
ijassa-2040	59	7	grushin	grushin	NOUN
ijassa-2040	59	8	in	in	ADP
ijassa-2040	59	9	[	[	X
ijassa-2040	59	10	8	8	NUM
ijassa-2040	59	11	]	]	PUNCT
ijassa-2040	59	12	gave	give	VERB
ijassa-2040	59	13	an	an	DET
ijassa-2040	59	14	example	example	NOUN
ijassa-2040	59	15	of	of	ADP
ijassa-2040	59	16	a	a	DET
ijassa-2040	59	17	first	first	ADJ
ijassa-2040	59	18	order	order	NOUN
ijassa-2040	59	19	differential	differential	ADJ
ijassa-2040	59	20	equation	equation	NOUN
ijassa-2040	59	21	with	with	ADP
ijassa-2040	59	22	infinity	infinity	NOUN
ijassa-2040	59	23	differentiable	differentiable	ADJ
ijassa-2040	59	24	coefficients	coefficient	NOUN
ijassa-2040	59	25	that	that	PRON
ijassa-2040	59	26	has	have	VERB
ijassa-2040	59	27	no	no	DET
ijassa-2040	59	28	solutions	solution	NOUN
ijassa-2040	59	29	in	in	ADP
ijassa-2040	59	30	the	the	DET
ijassa-2040	59	31	space	space	NOUN
ijassa-2040	59	32	of	of	ADP
ijassa-2040	59	33	distributions	distribution	NOUN
ijassa-2040	59	34	on	on	ADP
ijassa-2040	59	35	the	the	DET
ijassa-2040	59	36	plane	plane	NOUN
ijassa-2040	59	37	:	:	PUNCT
ijassa-2040	59	38	∂u	∂u	PROPN
ijassa-2040	59	39	∂x	∂x	PROPN
ijassa-2040	60	1	+	+	CCONJ
ijassa-2040	60	2	ix	ix	ADP
ijassa-2040	60	3	∂u	∂u	PROPN
ijassa-2040	60	4	∂y	∂y	PROPN
ijassa-2040	60	5	=	=	SYM
ijassa-2040	60	6	f(x	f(x	PROPN
ijassa-2040	60	7	,	,	PUNCT
ijassa-2040	60	8	y	y	PROPN
ijassa-2040	60	9	)	)	PUNCT
ijassa-2040	60	10	.	.	PUNCT
ijassa-2040	61	1	(	(	PUNCT
ijassa-2040	61	2	2.1	2.1	NUM
ijassa-2040	61	3	)	)	PUNCT
ijassa-2040	61	4	copyright	copyright	NOUN
ijassa-2040	61	5	©	©	PROPN
ijassa-2040	61	6	2025	2025	NUM
ijassa-2040	61	7	assa	assa	NOUN
ijassa-2040	61	8	.	.	PUNCT
ijassa-2040	62	1	adv	adv	PROPN
ijassa-2040	62	2	syst	syst	PROPN
ijassa-2040	62	3	sci	sci	PROPN
ijassa-2040	62	4	appl	appl	PROPN
ijassa-2040	62	5	(	(	PUNCT
ijassa-2040	62	6	2025	2025	NUM
ijassa-2040	62	7	)	)	PUNCT
ijassa-2040	62	8	linear	linear	ADJ
ijassa-2040	62	9	differential	differential	ADJ
ijassa-2040	62	10	equations	equation	NOUN
ijassa-2040	62	11	on	on	ADP
ijassa-2040	62	12	the	the	DET
ijassa-2040	62	13	torus	torus	NOUN
ijassa-2040	62	14	...	...	PUNCT
ijassa-2040	62	15	15	15	NUM
ijassa-2040	62	16	the	the	DET
ijassa-2040	62	17	operator	operator	NOUN
ijassa-2040	62	18	in	in	ADP
ijassa-2040	62	19	the	the	DET
ijassa-2040	62	20	left	left	ADJ
ijassa-2040	62	21	-	-	PUNCT
ijassa-2040	62	22	hand	hand	NOUN
ijassa-2040	62	23	side	side	NOUN
ijassa-2040	62	24	of	of	ADP
ijassa-2040	62	25	equation	equation	NOUN
ijassa-2040	62	26	(	(	PUNCT
ijassa-2040	62	27	2.1	2.1	NUM
ijassa-2040	62	28	)	)	PUNCT
ijassa-2040	62	29	is	be	AUX
ijassa-2040	62	30	one	one	NUM
ijassa-2040	62	31	of	of	ADP
ijassa-2040	62	32	the	the	DET
ijassa-2040	62	33	mizohata	mizohata	ADJ
ijassa-2040	62	34	operators	operator	NOUN
ijassa-2040	62	35	,	,	PUNCT
ijassa-2040	62	36	considered	consider	VERB
ijassa-2040	62	37	in	in	ADP
ijassa-2040	62	38	[	[	X
ijassa-2040	62	39	10	10	NUM
ijassa-2040	62	40	]	]	PUNCT
ijassa-2040	62	41	.	.	PUNCT
ijassa-2040	63	1	the	the	DET
ijassa-2040	63	2	function	function	NOUN
ijassa-2040	63	3	f	f	PROPN
ijassa-2040	63	4	∈	∈	PROPN
ijassa-2040	63	5	c∞	c∞	PROPN
ijassa-2040	63	6	0	0	NUM
ijassa-2040	63	7	(	(	PUNCT
ijassa-2040	63	8	r2	r2	PROPN
ijassa-2040	63	9	)	)	PUNCT
ijassa-2040	63	10	is	be	AUX
ijassa-2040	63	11	even	even	ADV
ijassa-2040	63	12	by	by	ADP
ijassa-2040	63	13	x	x	NOUN
ijassa-2040	63	14	,	,	PUNCT
ijassa-2040	63	15	it	it	PRON
ijassa-2040	63	16	was	be	AUX
ijassa-2040	63	17	constructed	construct	VERB
ijassa-2040	63	18	by	by	ADP
ijassa-2040	63	19	grushin	grushin	NOUN
ijassa-2040	63	20	in	in	ADP
ijassa-2040	63	21	a	a	DET
ijassa-2040	63	22	special	special	ADJ
ijassa-2040	63	23	way	way	NOUN
ijassa-2040	63	24	.	.	PUNCT
ijassa-2040	64	1	consider	consider	VERB
ijassa-2040	64	2	a	a	DET
ijassa-2040	64	3	periodic	periodic	ADJ
ijassa-2040	64	4	modification	modification	NOUN
ijassa-2040	64	5	of	of	ADP
ijassa-2040	64	6	equation	equation	NOUN
ijassa-2040	64	7	(	(	PUNCT
ijassa-2040	64	8	2.1	2.1	NUM
ijassa-2040	64	9	):	):	PUNCT
ijassa-2040	64	10	∂u	∂u	PROPN
ijassa-2040	64	11	∂x	∂x	PROPN
ijassa-2040	65	1	+	+	CCONJ
ijassa-2040	66	1	i	i	PROPN
ijassa-2040	66	2	sinx	sinx	NOUN
ijassa-2040	66	3	∂u	∂u	PROPN
ijassa-2040	66	4	∂y	∂y	SYM
ijassa-2040	66	5	=	=	SYM
ijassa-2040	66	6	f̃(x	f̃(x	PROPN
ijassa-2040	66	7	,	,	PUNCT
ijassa-2040	66	8	y	y	NOUN
ijassa-2040	66	9	)	)	PUNCT
ijassa-2040	66	10	,	,	PUNCT
ijassa-2040	66	11	(	(	PUNCT
ijassa-2040	66	12	2.2	2.2	NUM
ijassa-2040	66	13	)	)	PUNCT
ijassa-2040	66	14	where	where	SCONJ
ijassa-2040	66	15	f̃	f̃	PROPN
ijassa-2040	66	16	is	be	AUX
ijassa-2040	66	17	2π	2π	NOUN
ijassa-2040	66	18	-	-	ADJ
ijassa-2040	66	19	periodic	periodic	ADJ
ijassa-2040	66	20	continuation	continuation	NOUN
ijassa-2040	66	21	of	of	ADP
ijassa-2040	66	22	the	the	DET
ijassa-2040	66	23	function	function	NOUN
ijassa-2040	66	24	f	f	PROPN
ijassa-2040	66	25	mentioned	mention	VERB
ijassa-2040	66	26	above	above	ADV
ijassa-2040	66	27	.	.	PUNCT
ijassa-2040	67	1	it	it	PRON
ijassa-2040	67	2	can	can	AUX
ijassa-2040	67	3	be	be	AUX
ijassa-2040	67	4	checked	check	VERB
ijassa-2040	67	5	that	that	SCONJ
ijassa-2040	67	6	the	the	DET
ijassa-2040	67	7	grushin	grushin	NOUN
ijassa-2040	67	8	’s	’s	PART
ijassa-2040	67	9	reasonings	reasoning	NOUN
ijassa-2040	67	10	are	be	AUX
ijassa-2040	67	11	also	also	ADV
ijassa-2040	67	12	applicable	applicable	ADJ
ijassa-2040	67	13	to	to	ADP
ijassa-2040	67	14	equation	equation	NOUN
ijassa-2040	67	15	(	(	PUNCT
ijassa-2040	67	16	2.2	2.2	NUM
ijassa-2040	67	17	)	)	PUNCT
ijassa-2040	67	18	.	.	PUNCT
ijassa-2040	68	1	we	we	PRON
ijassa-2040	68	2	shall	shall	AUX
ijassa-2040	68	3	prove	prove	VERB
ijassa-2040	68	4	that	that	SCONJ
ijassa-2040	68	5	equation	equation	NOUN
ijassa-2040	68	6	(	(	PUNCT
ijassa-2040	68	7	2.2	2.2	NUM
ijassa-2040	68	8	)	)	PUNCT
ijassa-2040	68	9	has	have	VERB
ijassa-2040	68	10	a	a	DET
ijassa-2040	68	11	solution	solution	NOUN
ijassa-2040	68	12	in	in	ADP
ijassa-2040	68	13	a	a	DET
ijassa-2040	68	14	wider	wide	ADJ
ijassa-2040	68	15	space	space	NOUN
ijassa-2040	68	16	of	of	ADP
ijassa-2040	68	17	generalized	generalized	ADJ
ijassa-2040	68	18	functions	function	NOUN
ijassa-2040	68	19	than	than	ADP
ijassa-2040	68	20	the	the	DET
ijassa-2040	68	21	space	space	NOUN
ijassa-2040	68	22	of	of	ADP
ijassa-2040	68	23	schwartz	schwartz	PROPN
ijassa-2040	68	24	distributions	distribution	NOUN
ijassa-2040	68	25	.	.	PUNCT
ijassa-2040	69	1	proposition	proposition	NOUN
ijassa-2040	69	2	2.2	2.2	NUM
ijassa-2040	69	3	:	:	PUNCT
ijassa-2040	69	4	for	for	ADP
ijassa-2040	69	5	any	any	DET
ijassa-2040	69	6	even	even	ADV
ijassa-2040	69	7	right	right	ADJ
ijassa-2040	69	8	-	-	PUNCT
ijassa-2040	69	9	hand	hand	NOUN
ijassa-2040	69	10	side	side	NOUN
ijassa-2040	69	11	f̃	f̃	PROPN
ijassa-2040	69	12	∈	∈	PROPN
ijassa-2040	69	13	h−∞	h−∞	NOUN
ijassa-2040	69	14	equation	equation	NOUN
ijassa-2040	69	15	(	(	PUNCT
ijassa-2040	69	16	2.2	2.2	NUM
ijassa-2040	69	17	)	)	PUNCT
ijassa-2040	69	18	has	have	VERB
ijassa-2040	69	19	a	a	DET
ijassa-2040	69	20	unique	unique	ADJ
ijassa-2040	69	21	periodic	periodic	ADJ
ijassa-2040	69	22	solution	solution	NOUN
ijassa-2040	69	23	u(x1	u(x1	ADJ
ijassa-2040	69	24	,	,	PUNCT
ijassa-2040	69	25	x2	x2	PROPN
ijassa-2040	69	26	)	)	PUNCT
ijassa-2040	69	27	odd	odd	ADJ
ijassa-2040	69	28	in	in	ADP
ijassa-2040	69	29	the	the	DET
ijassa-2040	69	30	variable	variable	NOUN
ijassa-2040	69	31	x1	x1	PROPN
ijassa-2040	69	32	,	,	PUNCT
ijassa-2040	69	33	which	which	PRON
ijassa-2040	69	34	belongs	belong	VERB
ijassa-2040	69	35	to	to	ADP
ijassa-2040	69	36	the	the	DET
ijassa-2040	69	37	space	space	NOUN
ijassa-2040	69	38	l∗1(|k1|	l∗1(|k1|	PROPN
ijassa-2040	69	39	!	!	PUNCT
ijassa-2040	69	40	)	)	PUNCT
ijassa-2040	69	41	.	.	PUNCT
ijassa-2040	70	1	proof	proof	NOUN
ijassa-2040	70	2	let	let	VERB
ijassa-2040	70	3	us	we	PRON
ijassa-2040	70	4	write	write	VERB
ijassa-2040	70	5	the	the	DET
ijassa-2040	70	6	equation	equation	NOUN
ijassa-2040	70	7	(	(	PUNCT
ijassa-2040	70	8	2.2	2.2	NUM
ijassa-2040	70	9	)	)	PUNCT
ijassa-2040	70	10	in	in	ADP
ijassa-2040	70	11	the	the	DET
ijassa-2040	70	12	form	form	NOUN
ijassa-2040	70	13	∂u	∂u	PROPN
ijassa-2040	70	14	∂x1	∂x1	NOUN
ijassa-2040	70	15	+	+	CCONJ
ijassa-2040	70	16	eix1	eix1	NOUN
ijassa-2040	70	17	−	−	ADP
ijassa-2040	70	18	e−ix1	e−ix1	NOUN
ijassa-2040	70	19	2	2	NUM
ijassa-2040	70	20	∂u	∂u	PROPN
ijassa-2040	70	21	∂x2	∂x2	NOUN
ijassa-2040	70	22	=	=	SYM
ijassa-2040	70	23	f̃	f̃	PROPN
ijassa-2040	70	24	,	,	PUNCT
ijassa-2040	70	25	which	which	PRON
ijassa-2040	70	26	yields	yield	VERB
ijassa-2040	70	27	k1uk1,k2	k1uk1,k2	VERB
ijassa-2040	70	28	+	+	CCONJ
ijassa-2040	70	29	k2	k2	X
ijassa-2040	70	30	2	2	NUM
ijassa-2040	70	31	(	(	PUNCT
ijassa-2040	70	32	uk1−1,k2	uk1−1,k2	NOUN
ijassa-2040	70	33	−	−	PROPN
ijassa-2040	71	1	uk1	uk1	PROPN
ijassa-2040	71	2	+	+	NOUN
ijassa-2040	71	3	1,k2	1,k2	NUM
ijassa-2040	71	4	)	)	PUNCT
ijassa-2040	72	1	=	=	SYM
ijassa-2040	72	2	fk	fk	INTJ
ijassa-2040	72	3	.	.	PUNCT
ijassa-2040	72	4	(	(	PUNCT
ijassa-2040	72	5	2.3	2.3	NUM
ijassa-2040	72	6	)	)	PUNCT
ijassa-2040	72	7	for	for	ADP
ijassa-2040	72	8	a	a	DET
ijassa-2040	72	9	fixed	fix	VERB
ijassa-2040	72	10	k2	k2	NOUN
ijassa-2040	72	11	̸=	̸=	PROPN
ijassa-2040	72	12	0	0	NUM
ijassa-2040	72	13	we	we	PRON
ijassa-2040	72	14	obtain	obtain	VERB
ijassa-2040	72	15	the	the	DET
ijassa-2040	72	16	recurrent	recurrent	ADJ
ijassa-2040	72	17	formula	formula	NOUN
ijassa-2040	72	18	with	with	ADP
ijassa-2040	72	19	respect	respect	NOUN
ijassa-2040	72	20	to	to	ADP
ijassa-2040	72	21	k1	k1	PROPN
ijassa-2040	72	22	uk1	uk1	PROPN
ijassa-2040	72	23	+	+	PROPN
ijassa-2040	72	24	1,k2	1,k2	PROPN
ijassa-2040	72	25	=	=	SYM
ijassa-2040	72	26	2	2	NUM
ijassa-2040	72	27	k2	k2	X
ijassa-2040	72	28	(	(	PUNCT
ijassa-2040	72	29	k1uk1,k2	k1uk1,k2	NUM
ijassa-2040	72	30	−	−	PROPN
ijassa-2040	72	31	fk)uk2−1,k2	fk)uk2−1,k2	NOUN
ijassa-2040	72	32	.	.	PUNCT
ijassa-2040	73	1	(	(	PUNCT
ijassa-2040	73	2	2.4	2.4	NUM
ijassa-2040	73	3	)	)	PUNCT
ijassa-2040	73	4	since	since	SCONJ
ijassa-2040	73	5	the	the	DET
ijassa-2040	73	6	function	function	NOUN
ijassa-2040	73	7	u(x1	u(x1	ADJ
ijassa-2040	73	8	,	,	PUNCT
ijassa-2040	73	9	x2	x2	PROPN
ijassa-2040	73	10	)	)	PUNCT
ijassa-2040	73	11	is	be	AUX
ijassa-2040	73	12	odd	odd	ADJ
ijassa-2040	73	13	in	in	ADP
ijassa-2040	73	14	the	the	DET
ijassa-2040	73	15	variable	variable	NOUN
ijassa-2040	73	16	x1	x1	PROPN
ijassa-2040	73	17	,	,	PUNCT
ijassa-2040	73	18	we	we	PRON
ijassa-2040	73	19	have	have	VERB
ijassa-2040	73	20	u0,k2	u0,k2	PROPN
ijassa-2040	73	21	=	=	SYM
ijassa-2040	73	22	0	0	NUM
ijassa-2040	73	23	,	,	PUNCT
ijassa-2040	73	24	u−k1,k2	u−k1,k2	PUNCT
ijassa-2040	73	25	=	=	PUNCT
ijassa-2040	73	26	−uk1,k2	−uk1,k2	NOUN
ijassa-2040	73	27	.	.	PUNCT
ijassa-2040	74	1	therefore	therefore	ADV
ijassa-2040	74	2	,	,	PUNCT
ijassa-2040	74	3	the	the	DET
ijassa-2040	74	4	coefficients	coefficient	NOUN
ijassa-2040	74	5	uk	uk	PROPN
ijassa-2040	74	6	are	be	AUX
ijassa-2040	74	7	uniquely	uniquely	ADV
ijassa-2040	74	8	determined	determine	VERB
ijassa-2040	74	9	by	by	ADP
ijassa-2040	74	10	(	(	PUNCT
ijassa-2040	74	11	2.4	2.4	NUM
ijassa-2040	74	12	)	)	PUNCT
ijassa-2040	74	13	.	.	PUNCT
ijassa-2040	75	1	from	from	ADP
ijassa-2040	75	2	(	(	PUNCT
ijassa-2040	75	3	2.4	2.4	NUM
ijassa-2040	75	4	)	)	PUNCT
ijassa-2040	75	5	we	we	PRON
ijassa-2040	75	6	have	have	VERB
ijassa-2040	75	7	the	the	DET
ijassa-2040	75	8	following	follow	VERB
ijassa-2040	75	9	estimation	estimation	NOUN
ijassa-2040	75	10	:	:	PUNCT
ijassa-2040	75	11	|uk1	|uk1	PROPN
ijassa-2040	75	12	+	+	PROPN
ijassa-2040	75	13	1,k2	1,k2	NUM
ijassa-2040	75	14	|	|	ADV
ijassa-2040	75	15	<	<	X
ijassa-2040	75	16	∑	∑	PUNCT
ijassa-2040	75	17	j=0	j=0	PROPN
ijassa-2040	75	18	(	(	PUNCT
ijassa-2040	75	19	j	j	PROPN
ijassa-2040	75	20	+	+	CCONJ
ijassa-2040	75	21	1)!fk1−j	1)!fk1−j	NUM
ijassa-2040	75	22	<	<	X
ijassa-2040	75	23	(	(	PUNCT
ijassa-2040	75	24	k1	k1	NOUN
ijassa-2040	75	25	+	+	CCONJ
ijassa-2040	75	26	1	1	NUM
ijassa-2040	75	27	)	)	PUNCT
ijassa-2040	75	28	!	!	PUNCT
ijassa-2040	76	1	∑	∑	PUNCT
ijassa-2040	77	1	k	k	PROPN
ijassa-2040	77	2	fk	fk	INTJ
ijassa-2040	77	3	=	=	SYM
ijassa-2040	77	4	c(k1	c(k1	PROPN
ijassa-2040	77	5	+	+	CCONJ
ijassa-2040	77	6	1	1	NUM
ijassa-2040	77	7	)	)	PUNCT
ijassa-2040	77	8	!	!	PUNCT
ijassa-2040	78	1	thus	thus	ADV
ijassa-2040	78	2	,	,	PUNCT
ijassa-2040	78	3	the	the	DET
ijassa-2040	78	4	solution	solution	NOUN
ijassa-2040	78	5	u	u	NOUN
ijassa-2040	78	6	=	=	NOUN
ijassa-2040	78	7	∑	∑	PROPN
ijassa-2040	78	8	uke	uke	PROPN
ijassa-2040	78	9	ikx	ikx	PROPN
ijassa-2040	78	10	belongs	belong	VERB
ijassa-2040	78	11	to	to	ADP
ijassa-2040	78	12	the	the	DET
ijassa-2040	78	13	space	space	NOUN
ijassa-2040	78	14	l∗1(|k1|	l∗1(|k1|	PROPN
ijassa-2040	78	15	!	!	PUNCT
ijassa-2040	78	16	)	)	PUNCT
ijassa-2040	78	17	.	.	PUNCT
ijassa-2040	79	1	note	note	VERB
ijassa-2040	79	2	that	that	SCONJ
ijassa-2040	79	3	the	the	DET
ijassa-2040	79	4	operations	operation	NOUN
ijassa-2040	79	5	of	of	ADP
ijassa-2040	79	6	differentiation	differentiation	NOUN
ijassa-2040	79	7	and	and	CCONJ
ijassa-2040	79	8	multiplication	multiplication	NOUN
ijassa-2040	79	9	by	by	ADP
ijassa-2040	79	10	a	a	DET
ijassa-2040	79	11	trigonometric	trigonometric	ADJ
ijassa-2040	79	12	polynomial	polynomial	NOUN
ijassa-2040	79	13	defined	define	VERB
ijassa-2040	79	14	formally	formally	ADV
ijassa-2040	79	15	in	in	ADP
ijassa-2040	79	16	the	the	DET
ijassa-2040	79	17	space	space	NOUN
ijassa-2040	79	18	f	f	NOUN
ijassa-2040	79	19	,	,	PUNCT
ijassa-2040	79	20	coincide	coincide	VERB
ijassa-2040	79	21	with	with	ADP
ijassa-2040	79	22	the	the	DET
ijassa-2040	79	23	analogues	analogue	NOUN
ijassa-2040	79	24	operations	operation	NOUN
ijassa-2040	79	25	in	in	ADP
ijassa-2040	79	26	the	the	DET
ijassa-2040	79	27	banach	banach	NOUN
ijassa-2040	79	28	space	space	NOUN
ijassa-2040	79	29	l∗1(|k1|	l∗1(|k1|	PROPN
ijassa-2040	79	30	!	!	PUNCT
ijassa-2040	79	31	)	)	PUNCT
ijassa-2040	80	1	defined	define	VERB
ijassa-2040	80	2	as	as	ADP
ijassa-2040	80	3	usual	usual	ADJ
ijassa-2040	80	4	in	in	ADP
ijassa-2040	80	5	spaces	space	NOUN
ijassa-2040	80	6	of	of	ADP
ijassa-2040	80	7	generalized	generalized	ADJ
ijassa-2040	80	8	functions	function	NOUN
ijassa-2040	80	9	through	through	ADP
ijassa-2040	80	10	pairing	pairing	NOUN
ijassa-2040	80	11	.	.	PUNCT
ijassa-2040	81	1	proposition	proposition	NOUN
ijassa-2040	81	2	2.3	2.3	NUM
ijassa-2040	81	3	:	:	PUNCT
ijassa-2040	81	4	every	every	DET
ijassa-2040	81	5	periodic	periodic	ADJ
ijassa-2040	81	6	solution	solution	NOUN
ijassa-2040	81	7	u(x1	u(x1	ADJ
ijassa-2040	81	8	,	,	PUNCT
ijassa-2040	81	9	x2	x2	PROPN
ijassa-2040	81	10	)	)	PUNCT
ijassa-2040	81	11	of	of	ADP
ijassa-2040	81	12	homogeneous	homogeneous	ADJ
ijassa-2040	81	13	equation	equation	NOUN
ijassa-2040	81	14	(	(	PUNCT
ijassa-2040	81	15	2.2	2.2	NUM
ijassa-2040	81	16	)	)	PUNCT
ijassa-2040	81	17	is	be	AUX
ijassa-2040	81	18	even	even	ADV
ijassa-2040	81	19	in	in	ADP
ijassa-2040	81	20	x1	x1	PROPN
ijassa-2040	81	21	and	and	CCONJ
ijassa-2040	81	22	it	it	PRON
ijassa-2040	81	23	is	be	AUX
ijassa-2040	81	24	uniquely	uniquely	ADV
ijassa-2040	81	25	determined	determine	VERB
ijassa-2040	81	26	by	by	ADP
ijassa-2040	81	27	the	the	DET
ijassa-2040	81	28	functions	function	NOUN
ijassa-2040	81	29	u0(x2	u0(x2	NOUN
ijassa-2040	81	30	)	)	PUNCT
ijassa-2040	81	31	:	:	PUNCT
ijassa-2040	82	1	=	=	PUNCT
ijassa-2040	82	2	∫	∫	PROPN
ijassa-2040	82	3	2π	2π	PROPN
ijassa-2040	82	4	0	0	NUM
ijassa-2040	82	5	u(x1	u(x1	ADJ
ijassa-2040	82	6	,	,	PUNCT
ijassa-2040	82	7	x2)dx1	x2)dx1	X
ijassa-2040	83	1	:	:	PUNCT
ijassa-2040	83	2	=	=	SYM
ijassa-2040	83	3	⟨u	⟨u	NOUN
ijassa-2040	83	4	,	,	PUNCT
ijassa-2040	83	5	1⟩x1	1⟩x1	NUM
ijassa-2040	83	6	,	,	PUNCT
ijassa-2040	83	7	u1(x2	u1(x2	NOUN
ijassa-2040	83	8	)	)	PUNCT
ijassa-2040	83	9	:	:	PUNCT
ijassa-2040	84	1	=	=	PUNCT
ijassa-2040	84	2	∫	∫	PROPN
ijassa-2040	84	3	2π	2π	PROPN
ijassa-2040	84	4	0	0	NUM
ijassa-2040	84	5	u(x1	u(x1	ADJ
ijassa-2040	84	6	,	,	PUNCT
ijassa-2040	84	7	x2)e	x2)e	PROPN
ijassa-2040	84	8	−ix1dx1	−ix1dx1	NOUN
ijassa-2040	84	9	:	:	PUNCT
ijassa-2040	84	10	=	=	SYM
ijassa-2040	84	11	⟨u	⟨u	NOUN
ijassa-2040	84	12	,	,	PUNCT
ijassa-2040	84	13	eix1⟩x1	eix1⟩x1	PROPN
ijassa-2040	84	14	.	.	PUNCT
ijassa-2040	85	1	it	it	PRON
ijassa-2040	85	2	belongs	belong	VERB
ijassa-2040	85	3	to	to	ADP
ijassa-2040	85	4	the	the	DET
ijassa-2040	85	5	space	space	NOUN
ijassa-2040	85	6	l∗1(|k1|	l∗1(|k1|	NOUN
ijassa-2040	85	7	!	!	PUNCT
ijassa-2040	85	8	)	)	PUNCT
ijassa-2040	86	1	if	if	SCONJ
ijassa-2040	86	2	u0	u0	ADJ
ijassa-2040	86	3	and	and	CCONJ
ijassa-2040	86	4	u1	u1	PROPN
ijassa-2040	86	5	have	have	AUX
ijassa-2040	86	6	bounded	bound	VERB
ijassa-2040	86	7	sequences	sequence	NOUN
ijassa-2040	86	8	of	of	ADP
ijassa-2040	86	9	coefficients	coefficient	NOUN
ijassa-2040	86	10	.	.	PUNCT
ijassa-2040	87	1	copyright	copyright	NOUN
ijassa-2040	87	2	©	©	PROPN
ijassa-2040	87	3	2025	2025	NUM
ijassa-2040	87	4	assa	assa	NOUN
ijassa-2040	87	5	.	.	PUNCT
ijassa-2040	88	1	adv	adv	PROPN
ijassa-2040	88	2	syst	syst	PROPN
ijassa-2040	88	3	sci	sci	PROPN
ijassa-2040	88	4	appl	appl	PROPN
ijassa-2040	88	5	(	(	PUNCT
ijassa-2040	88	6	2025	2025	NUM
ijassa-2040	88	7	)	)	PUNCT
ijassa-2040	88	8	16	16	NUM
ijassa-2040	88	9	v.	v.	ADP
ijassa-2040	88	10	p.	p.	PROPN
ijassa-2040	88	11	burskii	burskii	NOUN
ijassa-2040	88	12	the	the	DET
ijassa-2040	88	13	proof	proof	NOUN
ijassa-2040	88	14	follows	follow	VERB
ijassa-2040	88	15	from	from	ADP
ijassa-2040	88	16	formula	formula	NOUN
ijassa-2040	88	17	(	(	PUNCT
ijassa-2040	88	18	2.3	2.3	NUM
ijassa-2040	88	19	)	)	PUNCT
ijassa-2040	88	20	.	.	PUNCT
ijassa-2040	89	1	here	here	ADV
ijassa-2040	89	2	we	we	PRON
ijassa-2040	89	3	consider	consider	VERB
ijassa-2040	89	4	the	the	DET
ijassa-2040	89	5	function	function	NOUN
ijassa-2040	89	6	u	u	NOUN
ijassa-2040	89	7	as	as	ADP
ijassa-2040	89	8	a	a	DET
ijassa-2040	89	9	formal	formal	ADJ
ijassa-2040	89	10	trigonometric	trigonometric	ADJ
ijassa-2040	89	11	series	series	NOUN
ijassa-2040	89	12	,	,	PUNCT
ijassa-2040	89	13	and	and	CCONJ
ijassa-2040	89	14	pairing	pair	VERB
ijassa-2040	89	15	along	along	ADP
ijassa-2040	89	16	one	one	NUM
ijassa-2040	89	17	coordinate	coordinate	NOUN
ijassa-2040	89	18	is	be	AUX
ijassa-2040	89	19	defined	define	VERB
ijassa-2040	89	20	in	in	ADP
ijassa-2040	89	21	the	the	DET
ijassa-2040	89	22	standard	standard	ADJ
ijassa-2040	89	23	way	way	NOUN
ijassa-2040	89	24	:	:	PUNCT
ijassa-2040	89	25	⟨u	⟨u	NOUN
ijassa-2040	89	26	,	,	PUNCT
ijassa-2040	89	27	v⟩x1	v⟩x1	NUM
ijassa-2040	89	28	:	:	PUNCT
ijassa-2040	89	29	=	=	SYM
ijassa-2040	89	30	∑	∑	PROPN
ijassa-2040	89	31	n	n	PRON
ijassa-2040	89	32	〈	〈	PROPN
ijassa-2040	89	33	∑	∑	PROPN
ijassa-2040	89	34	k	k	PROPN
ijassa-2040	89	35	ukne	ukne	PROPN
ijassa-2040	89	36	ikx1	ikx1	PROPN
ijassa-2040	89	37	,	,	PUNCT
ijassa-2040	89	38	∑	∑	PROPN
ijassa-2040	89	39	m	m	PROPN
ijassa-2040	89	40	vmne	vmne	PROPN
ijassa-2040	89	41	imx1	imx1	NOUN
ijassa-2040	89	42	〉	〉	NOUN
ijassa-2040	89	43	einx2	einx2	NOUN
ijassa-2040	90	1	=	=	PUNCT
ijassa-2040	91	1	=	=	PUNCT
ijassa-2040	91	2	∑	∑	PROPN
ijassa-2040	91	3	n	n	PROPN
ijassa-2040	91	4	(	(	PUNCT
ijassa-2040	91	5	∑	∑	PROPN
ijassa-2040	91	6	m	m	PROPN
ijassa-2040	91	7	umnv−mn	umnv−mn	NUM
ijassa-2040	91	8	)	)	PUNCT
ijassa-2040	91	9	einx2	einx2	NOUN
ijassa-2040	91	10	.	.	PUNCT
ijassa-2040	92	1	it	it	PRON
ijassa-2040	92	2	is	be	AUX
ijassa-2040	92	3	clear	clear	ADJ
ijassa-2040	92	4	that	that	SCONJ
ijassa-2040	92	5	pairing	pair	VERB
ijassa-2040	92	6	on	on	ADP
ijassa-2040	92	7	x1	x1	PROPN
ijassa-2040	92	8	does	do	AUX
ijassa-2040	92	9	not	not	PART
ijassa-2040	92	10	always	always	ADV
ijassa-2040	92	11	exist	exist	VERB
ijassa-2040	92	12	,	,	PUNCT
ijassa-2040	92	13	but	but	CCONJ
ijassa-2040	92	14	if	if	SCONJ
ijassa-2040	92	15	t	t	PROPN
ijassa-2040	92	16	is	be	AUX
ijassa-2040	92	17	a	a	DET
ijassa-2040	92	18	trigonometric	trigonometric	ADJ
ijassa-2040	92	19	polynomial	polynomial	NOUN
ijassa-2040	92	20	,	,	PUNCT
ijassa-2040	92	21	then	then	ADV
ijassa-2040	92	22	the	the	DET
ijassa-2040	92	23	function	function	NOUN
ijassa-2040	92	24	⟨t	⟨t	NOUN
ijassa-2040	92	25	,	,	PUNCT
ijassa-2040	92	26	v⟩x1(x2	v⟩x1(x2	NOUN
ijassa-2040	92	27	)	)	PUNCT
ijassa-2040	92	28	exists	exist	VERB
ijassa-2040	92	29	.	.	PUNCT
ijassa-2040	93	1	2.3	2.3	NUM
ijassa-2040	93	2	.	.	PUNCT
ijassa-2040	93	3	solvability	solvability	NOUN
ijassa-2040	93	4	of	of	ADP
ijassa-2040	93	5	general	general	ADJ
ijassa-2040	93	6	equations	equation	NOUN
ijassa-2040	93	7	now	now	ADV
ijassa-2040	93	8	let	let	VERB
ijassa-2040	93	9	us	we	PRON
ijassa-2040	93	10	consider	consider	VERB
ijassa-2040	93	11	the	the	DET
ijassa-2040	93	12	general	general	ADJ
ijassa-2040	93	13	operator	operator	NOUN
ijassa-2040	93	14	l	l	NOUN
ijassa-2040	93	15	:	:	PUNCT
ijassa-2040	93	16	∑	∑	PUNCT
ijassa-2040	93	17	|α|≤m	|α|≤m	PROPN
ijassa-2040	93	18	tα(x)d	tα(x)d	PROPN
ijassa-2040	93	19	α	α	NOUN
ijassa-2040	93	20	,	,	PUNCT
ijassa-2040	93	21	where	where	SCONJ
ijassa-2040	93	22	tα(x	tα(x	NUM
ijassa-2040	93	23	)	)	PUNCT
ijassa-2040	93	24	is	be	AUX
ijassa-2040	93	25	a	a	DET
ijassa-2040	93	26	trigonometric	trigonometric	ADJ
ijassa-2040	93	27	polynomial	polynomial	NOUN
ijassa-2040	93	28	of	of	ADP
ijassa-2040	93	29	degree	degree	NOUN
ijassa-2040	93	30	(	(	PUNCT
ijassa-2040	93	31	s1α	s1α	PROPN
ijassa-2040	93	32	,	,	PUNCT
ijassa-2040	93	33	s	s	PART
ijassa-2040	93	34	2	2	NUM
ijassa-2040	93	35	α	α	NOUN
ijassa-2040	93	36	)	)	PUNCT
ijassa-2040	93	37	.	.	PUNCT
ijassa-2040	94	1	it	it	PRON
ijassa-2040	94	2	can	can	AUX
ijassa-2040	94	3	also	also	ADV
ijassa-2040	94	4	be	be	AUX
ijassa-2040	94	5	written	write	VERB
ijassa-2040	94	6	in	in	ADP
ijassa-2040	94	7	the	the	DET
ijassa-2040	94	8	form	form	NOUN
ijassa-2040	94	9	l	l	NOUN
ijassa-2040	94	10	=	=	PUNCT
ijassa-2040	94	11	s1∑	s1∑	PROPN
ijassa-2040	94	12	n1=−s1	n1=−s1	PROPN
ijassa-2040	94	13	s2∑	s2∑	PROPN
ijassa-2040	94	14	n2=−s2	n2=−s2	PROPN
ijassa-2040	94	15	einxpn(d	einxpn(d	PROPN
ijassa-2040	94	16	)	)	PUNCT
ijassa-2040	94	17	,	,	PUNCT
ijassa-2040	94	18	s1	s1	NOUN
ijassa-2040	94	19	:	:	PUNCT
ijassa-2040	94	20	=	=	PUNCT
ijassa-2040	94	21	max	max	PROPN
ijassa-2040	94	22	α	α	PROPN
ijassa-2040	94	23	s1α	s1α	PROPN
ijassa-2040	94	24	,	,	PUNCT
ijassa-2040	94	25	s2	s2	NOUN
ijassa-2040	94	26	:	:	PUNCT
ijassa-2040	95	1	=	=	SYM
ijassa-2040	95	2	max	max	PROPN
ijassa-2040	95	3	α	α	INTJ
ijassa-2040	95	4	s2α	s2α	PROPN
ijassa-2040	95	5	.	.	PUNCT
ijassa-2040	96	1	let	let	VERB
ijassa-2040	96	2	us	we	PRON
ijassa-2040	96	3	assume	assume	VERB
ijassa-2040	96	4	that	that	SCONJ
ijassa-2040	96	5	the	the	DET
ijassa-2040	96	6	operator	operator	NOUN
ijassa-2040	96	7	l	l	NOUN
ijassa-2040	96	8	satisfies	satisfy	VERB
ijassa-2040	96	9	the	the	DET
ijassa-2040	96	10	following	follow	VERB
ijassa-2040	96	11	condition	condition	NOUN
ijassa-2040	96	12	:	:	PUNCT
ijassa-2040	96	13	assumption	assumption	NOUN
ijassa-2040	96	14	2.1	2.1	NUM
ijassa-2040	96	15	:	:	PUNCT
ijassa-2040	96	16	for	for	ADP
ijassa-2040	96	17	every	every	DET
ijassa-2040	96	18	n	n	DET
ijassa-2040	96	19	the	the	DET
ijassa-2040	96	20	equation	equation	NOUN
ijassa-2040	96	21	pn(x	pn(x	PUNCT
ijassa-2040	96	22	)	)	PUNCT
ijassa-2040	96	23	=	=	SYM
ijassa-2040	96	24	0	0	PUNCT
ijassa-2040	96	25	has	have	VERB
ijassa-2040	96	26	no	no	DET
ijassa-2040	96	27	solutions	solution	NOUN
ijassa-2040	96	28	in	in	ADP
ijassa-2040	96	29	integers	integer	NOUN
ijassa-2040	96	30	.	.	PUNCT
ijassa-2040	97	1	then	then	ADV
ijassa-2040	97	2	the	the	DET
ijassa-2040	97	3	following	follow	VERB
ijassa-2040	97	4	generalization	generalization	NOUN
ijassa-2040	97	5	of	of	ADP
ijassa-2040	97	6	proposition	proposition	NOUN
ijassa-2040	97	7	2.3	2.3	NUM
ijassa-2040	97	8	is	be	AUX
ijassa-2040	97	9	true	true	ADJ
ijassa-2040	97	10	.	.	PUNCT
ijassa-2040	98	1	proposition	proposition	NOUN
ijassa-2040	98	2	2.4	2.4	NUM
ijassa-2040	98	3	:	:	PUNCT
ijassa-2040	98	4	under	under	ADP
ijassa-2040	98	5	assumption	assumption	NOUN
ijassa-2040	98	6	2.1	2.1	NUM
ijassa-2040	98	7	,	,	PUNCT
ijassa-2040	98	8	every	every	DET
ijassa-2040	98	9	formal	formal	ADJ
ijassa-2040	98	10	periodic	periodic	ADJ
ijassa-2040	98	11	solution	solution	NOUN
ijassa-2040	98	12	u(x1	u(x1	ADJ
ijassa-2040	98	13	,	,	PUNCT
ijassa-2040	98	14	x2	x2	PROPN
ijassa-2040	98	15	)	)	PUNCT
ijassa-2040	98	16	of	of	ADP
ijassa-2040	98	17	equation	equation	NOUN
ijassa-2040	98	18	lu	lu	NOUN
ijassa-2040	99	1	=	=	NOUN
ijassa-2040	99	2	0	0	NUM
ijassa-2040	99	3	is	be	AUX
ijassa-2040	99	4	uniquely	uniquely	ADV
ijassa-2040	99	5	determined	determine	VERB
ijassa-2040	99	6	by	by	ADP
ijassa-2040	99	7	the	the	DET
ijassa-2040	99	8	functions	function	NOUN
ijassa-2040	99	9	u02(x2	u02(x2	PROPN
ijassa-2040	99	10	)	)	PUNCT
ijassa-2040	100	1	:	:	PUNCT
ijassa-2040	100	2	=	=	SYM
ijassa-2040	100	3	⟨u	⟨u	NOUN
ijassa-2040	100	4	,	,	PUNCT
ijassa-2040	100	5	1⟩x1	1⟩x1	NUM
ijassa-2040	100	6	,	,	PUNCT
ijassa-2040	100	7	u12(x2	u12(x2	NOUN
ijassa-2040	100	8	)	)	PUNCT
ijassa-2040	100	9	:	:	PUNCT
ijassa-2040	100	10	=	=	SYM
ijassa-2040	100	11	⟨u	⟨u	NOUN
ijassa-2040	100	12	,	,	PUNCT
ijassa-2040	100	13	eix1⟩x1	eix1⟩x1	PROPN
ijassa-2040	100	14	,	,	PUNCT
ijassa-2040	100	15	.	.	PUNCT
ijassa-2040	100	16	.	.	PUNCT
ijassa-2040	101	1	.	.	PUNCT
ijassa-2040	102	1	,	,	PUNCT
ijassa-2040	102	2	us12(x2	us12(x2	NOUN
ijassa-2040	102	3	)	)	PUNCT
ijassa-2040	102	4	:	:	PUNCT
ijassa-2040	102	5	=	=	SYM
ijassa-2040	102	6	⟨u	⟨u	NOUN
ijassa-2040	102	7	,	,	PUNCT
ijassa-2040	102	8	eis1x1⟩x1	eis1x1⟩x1	PROPN
ijassa-2040	102	9	,	,	PUNCT
ijassa-2040	102	10	u01(x1	u01(x1	PROPN
ijassa-2040	102	11	)	)	PUNCT
ijassa-2040	102	12	:	:	PUNCT
ijassa-2040	102	13	=	=	SYM
ijassa-2040	102	14	⟨u	⟨u	X
ijassa-2040	102	15	,	,	PUNCT
ijassa-2040	102	16	1⟩x2	1⟩x2	NUM
ijassa-2040	102	17	,	,	PUNCT
ijassa-2040	102	18	u11(x1	u11(x1	PROPN
ijassa-2040	102	19	)	)	PUNCT
ijassa-2040	102	20	:	:	PUNCT
ijassa-2040	102	21	=	=	SYM
ijassa-2040	102	22	⟨u	⟨u	NOUN
ijassa-2040	102	23	,	,	PUNCT
ijassa-2040	102	24	eix2⟩x2	eix2⟩x2	PROPN
ijassa-2040	102	25	,	,	PUNCT
ijassa-2040	102	26	.	.	PUNCT
ijassa-2040	102	27	.	.	PUNCT
ijassa-2040	102	28	.	.	PUNCT
ijassa-2040	103	1	,	,	PUNCT
ijassa-2040	103	2	us21(x1	us21(x1	NOUN
ijassa-2040	103	3	)	)	PUNCT
ijassa-2040	103	4	:	:	PUNCT
ijassa-2040	103	5	=	=	SYM
ijassa-2040	103	6	⟨u	⟨u	NOUN
ijassa-2040	103	7	,	,	PUNCT
ijassa-2040	103	8	eis2x2⟩x2	eis2x2⟩x2	PROPN
ijassa-2040	103	9	.	.	PUNCT
ijassa-2040	104	1	if	if	SCONJ
ijassa-2040	104	2	they	they	PRON
ijassa-2040	104	3	satisfy	satisfy	VERB
ijassa-2040	104	4	the	the	DET
ijassa-2040	104	5	following	follow	VERB
ijassa-2040	104	6	conditions	condition	NOUN
ijassa-2040	104	7	:	:	PUNCT
ijassa-2040	104	8	⟨uq1	⟨uq1	NOUN
ijassa-2040	104	9	,	,	PUNCT
ijassa-2040	104	10	e	e	X
ijassa-2040	104	11	ipx1⟩x1	ipx1⟩x1	PROPN
ijassa-2040	104	12	=	=	PROPN
ijassa-2040	104	13	⟨up2	⟨up2	PROPN
ijassa-2040	104	14	,	,	PUNCT
ijassa-2040	104	15	e	e	X
ijassa-2040	104	16	iqx2⟩x2	iqx2⟩x2	PROPN
ijassa-2040	104	17	,	,	PUNCT
ijassa-2040	104	18	∀	∀	X
ijassa-2040	104	19	p	p	NOUN
ijassa-2040	104	20	,	,	PUNCT
ijassa-2040	104	21	q	q	NOUN
ijassa-2040	104	22	=	=	NOUN
ijassa-2040	104	23	1	1	NUM
ijassa-2040	104	24	,	,	PUNCT
ijassa-2040	104	25	.	.	PUNCT
ijassa-2040	104	26	.	.	PUNCT
ijassa-2040	105	1	.	.	PUNCT
ijassa-2040	106	1	,	,	PUNCT
ijassa-2040	106	2	min(s1	min(s1	PROPN
ijassa-2040	106	3	,	,	PUNCT
ijassa-2040	106	4	s2	s2	PROPN
ijassa-2040	106	5	)	)	PUNCT
ijassa-2040	106	6	.	.	PUNCT
ijassa-2040	107	1	the	the	DET
ijassa-2040	107	2	proof	proof	NOUN
ijassa-2040	107	3	is	be	AUX
ijassa-2040	107	4	by	by	ADP
ijassa-2040	107	5	the	the	DET
ijassa-2040	107	6	direct	direct	ADJ
ijassa-2040	107	7	substitution	substitution	NOUN
ijassa-2040	107	8	of	of	ADP
ijassa-2040	107	9	formal	formal	ADJ
ijassa-2040	107	10	series	series	NOUN
ijassa-2040	107	11	into	into	ADP
ijassa-2040	107	12	the	the	DET
ijassa-2040	107	13	equation	equation	NOUN
ijassa-2040	107	14	.	.	PUNCT
ijassa-2040	108	1	2.4	2.4	NUM
ijassa-2040	108	2	.	.	PUNCT
ijassa-2040	109	1	linear	linear	ADJ
ijassa-2040	109	2	sections	section	NOUN
ijassa-2040	109	3	as	as	ADP
ijassa-2040	109	4	objects	object	NOUN
ijassa-2040	109	5	of	of	ADP
ijassa-2040	109	6	non	non	ADJ
ijassa-2040	109	7	-	-	ADJ
ijassa-2040	109	8	standard	standard	ADJ
ijassa-2040	109	9	analysis	analysis	NOUN
ijassa-2040	109	10	consider	consider	VERB
ijassa-2040	109	11	the	the	DET
ijassa-2040	109	12	vector	vector	NOUN
ijassa-2040	109	13	space	space	NOUN
ijassa-2040	109	14	of	of	ADP
ijassa-2040	109	15	formal	formal	ADJ
ijassa-2040	109	16	trigonometric	trigonometric	NOUN
ijassa-2040	109	17	series	series	NOUN
ijassa-2040	109	18	f	f	PROPN
ijassa-2040	109	19	and	and	CCONJ
ijassa-2040	109	20	define	define	VERB
ijassa-2040	109	21	a	a	DET
ijassa-2040	109	22	relation	relation	NOUN
ijassa-2040	109	23	of	of	ADP
ijassa-2040	109	24	order	order	NOUN
ijassa-2040	109	25	u	u	NOUN
ijassa-2040	109	26	≤	≤	X
ijassa-2040	109	27	v	v	NOUN
ijassa-2040	109	28	in	in	ADP
ijassa-2040	109	29	the	the	DET
ijassa-2040	109	30	following	following	ADJ
ijassa-2040	109	31	way	way	NOUN
ijassa-2040	109	32	.	.	PUNCT
ijassa-2040	110	1	an	an	DET
ijassa-2040	110	2	element	element	NOUN
ijassa-2040	110	3	u	u	NOUN
ijassa-2040	110	4	∈	∈	PROPN
ijassa-2040	110	5	f	f	NOUN
ijassa-2040	110	6	is	be	AUX
ijassa-2040	110	7	more	more	ADV
ijassa-2040	110	8	regular	regular	ADJ
ijassa-2040	110	9	than	than	ADP
ijassa-2040	110	10	v	v	NOUN
ijassa-2040	110	11	∈	∈	NOUN
ijassa-2040	110	12	f	f	NOUN
ijassa-2040	110	13	or	or	CCONJ
ijassa-2040	110	14	,	,	PUNCT
ijassa-2040	110	15	equivalently	equivalently	ADV
ijassa-2040	110	16	,	,	PUNCT
ijassa-2040	110	17	v	v	NOUN
ijassa-2040	110	18	is	be	AUX
ijassa-2040	110	19	more	more	ADV
ijassa-2040	110	20	singular	singular	ADJ
ijassa-2040	110	21	than	than	ADP
ijassa-2040	110	22	u	u	PRON
ijassa-2040	110	23	if	if	SCONJ
ijassa-2040	110	24	there	there	PRON
ijassa-2040	110	25	exists	exist	VERB
ijassa-2040	110	26	an	an	DET
ijassa-2040	110	27	element	element	NOUN
ijassa-2040	110	28	h	h	NOUN
ijassa-2040	110	29	∈	∈	PROPN
ijassa-2040	110	30	f	f	PROPN
ijassa-2040	110	31	with	with	ADP
ijassa-2040	110	32	bounded	bound	VERB
ijassa-2040	110	33	positive	positive	ADJ
ijassa-2040	110	34	coefficients	coefficient	NOUN
ijassa-2040	110	35	hk	hk	INTJ
ijassa-2040	110	36	such	such	ADJ
ijassa-2040	110	37	that	that	SCONJ
ijassa-2040	110	38	u	u	PROPN
ijassa-2040	110	39	is	be	AUX
ijassa-2040	110	40	the	the	DET
ijassa-2040	110	41	convolution	convolution	NOUN
ijassa-2040	110	42	of	of	ADP
ijassa-2040	110	43	v	v	NOUN
ijassa-2040	110	44	and	and	CCONJ
ijassa-2040	110	45	h	h	NOUN
ijassa-2040	110	46	:	:	PUNCT
ijassa-2040	110	47	u	u	NOUN
ijassa-2040	110	48	=	=	PROPN
ijassa-2040	110	49	v	v	ADP
ijassa-2040	110	50	∗	∗	NOUN
ijassa-2040	110	51	h	h	NOUN
ijassa-2040	110	52	:	:	PUNCT
ijassa-2040	110	53	=	=	SYM
ijassa-2040	110	54	∑	∑	PUNCT
ijassa-2040	110	55	k	k	PROPN
ijassa-2040	110	56	vkhke	vkhke	PROPN
ijassa-2040	110	57	ikx	ikx	PROPN
ijassa-2040	110	58	.	.	PUNCT
ijassa-2040	111	1	this	this	PRON
ijassa-2040	111	2	is	be	AUX
ijassa-2040	111	3	obviously	obviously	ADV
ijassa-2040	111	4	equivalent	equivalent	ADJ
ijassa-2040	111	5	to	to	ADP
ijassa-2040	111	6	the	the	DET
ijassa-2040	111	7	condition	condition	NOUN
ijassa-2040	111	8	∃c	∃c	PROPN
ijassa-2040	111	9	>	>	X
ijassa-2040	111	10	0	0	NUM
ijassa-2040	111	11	:	:	PUNCT
ijassa-2040	111	12	∣∣∣un	∣∣∣un	NOUN
ijassa-2040	111	13	vn	vn	VERB
ijassa-2040	111	14	∣∣∣	∣∣∣	ADJ
ijassa-2040	111	15	<	<	X
ijassa-2040	111	16	c	c	X
ijassa-2040	111	17	∀n	∀n	PUNCT
ijassa-2040	111	18	∈	∈	PROPN
ijassa-2040	111	19	z.	z.	PROPN
ijassa-2040	111	20	copyright	copyright	NOUN
ijassa-2040	111	21	©	©	PROPN
ijassa-2040	111	22	2025	2025	NUM
ijassa-2040	111	23	assa	assa	NOUN
ijassa-2040	111	24	.	.	PUNCT
ijassa-2040	112	1	adv	adv	PROPN
ijassa-2040	112	2	syst	syst	PROPN
ijassa-2040	112	3	sci	sci	PROPN
ijassa-2040	112	4	appl	appl	PROPN
ijassa-2040	112	5	(	(	PUNCT
ijassa-2040	112	6	2025	2025	NUM
ijassa-2040	112	7	)	)	PUNCT
ijassa-2040	112	8	linear	linear	ADJ
ijassa-2040	112	9	differential	differential	ADJ
ijassa-2040	112	10	equations	equation	NOUN
ijassa-2040	112	11	on	on	ADP
ijassa-2040	112	12	the	the	DET
ijassa-2040	112	13	torus	torus	NOUN
ijassa-2040	112	14	...	...	PUNCT
ijassa-2040	112	15	17	17	NUM
ijassa-2040	112	16	similar	similar	ADJ
ijassa-2040	112	17	order	order	NOUN
ijassa-2040	112	18	relations	relation	NOUN
ijassa-2040	112	19	are	be	AUX
ijassa-2040	112	20	used	use	VERB
ijassa-2040	112	21	in	in	ADP
ijassa-2040	112	22	asymptotic	asymptotic	ADJ
ijassa-2040	112	23	expansions	expansion	NOUN
ijassa-2040	112	24	[	[	X
ijassa-2040	112	25	11	11	NUM
ijassa-2040	112	26	]	]	PUNCT
ijassa-2040	112	27	.	.	PUNCT
ijassa-2040	113	1	we	we	PRON
ijassa-2040	113	2	shall	shall	AUX
ijassa-2040	113	3	the	the	DET
ijassa-2040	113	4	following	follow	VERB
ijassa-2040	113	5	definition	definition	NOUN
ijassa-2040	113	6	:	:	PUNCT
ijassa-2040	113	7	a	a	DET
ijassa-2040	113	8	subspace	subspace	NOUN
ijassa-2040	113	9	α	α	PROPN
ijassa-2040	114	1	⊂	⊂	PROPN
ijassa-2040	115	1	f	f	PROPN
ijassa-2040	115	2	is	be	AUX
ijassa-2040	115	3	called	call	VERB
ijassa-2040	115	4	a	a	DET
ijassa-2040	115	5	linear	linear	ADJ
ijassa-2040	115	6	section	section	NOUN
ijassa-2040	115	7	of	of	ADP
ijassa-2040	115	8	f	f	PROPN
ijassa-2040	115	9	if	if	SCONJ
ijassa-2040	115	10	from	from	ADP
ijassa-2040	115	11	v	v	NUM
ijassa-2040	115	12	∈	∈	NOUN
ijassa-2040	115	13	α	α	NOUN
ijassa-2040	115	14	it	it	PRON
ijassa-2040	115	15	follows	follow	VERB
ijassa-2040	115	16	that	that	SCONJ
ijassa-2040	115	17	u	u	PROPN
ijassa-2040	115	18	∈	∈	PROPN
ijassa-2040	115	19	α	α	NOUN
ijassa-2040	115	20	for	for	ADP
ijassa-2040	115	21	every	every	DET
ijassa-2040	115	22	u	u	NOUN
ijassa-2040	115	23	≤	≤	X
ijassa-2040	115	24	v.	v.	ADP
ijassa-2040	115	25	a	a	DET
ijassa-2040	115	26	trivial	trivial	ADJ
ijassa-2040	115	27	example	example	NOUN
ijassa-2040	115	28	:	:	PUNCT
ijassa-2040	115	29	if	if	SCONJ
ijassa-2040	115	30	v	v	NUM
ijassa-2040	115	31	∈	∈	PROPN
ijassa-2040	115	32	f	f	NOUN
ijassa-2040	115	33	,	,	PUNCT
ijassa-2040	115	34	then	then	ADV
ijassa-2040	115	35	the	the	DET
ijassa-2040	115	36	set	set	NOUN
ijassa-2040	115	37	of	of	ADP
ijassa-2040	115	38	u	u	PROPN
ijassa-2040	115	39	∈	∈	PROPN
ijassa-2040	115	40	f	f	PROPN
ijassa-2040	115	41	such	such	ADJ
ijassa-2040	115	42	that	that	SCONJ
ijassa-2040	115	43	u	u	PROPN
ijassa-2040	115	44	≤	≤	NOUN
ijassa-2040	115	45	v	v	NOUN
ijassa-2040	115	46	is	be	AUX
ijassa-2040	115	47	a	a	DET
ijassa-2040	115	48	linear	linear	ADJ
ijassa-2040	115	49	section	section	NOUN
ijassa-2040	115	50	of	of	ADP
ijassa-2040	115	51	f	f	PROPN
ijassa-2040	115	52	.	.	PUNCT
ijassa-2040	116	1	it	it	PRON
ijassa-2040	116	2	is	be	AUX
ijassa-2040	116	3	called	call	VERB
ijassa-2040	116	4	the	the	DET
ijassa-2040	116	5	principal	principal	ADJ
ijassa-2040	116	6	linear	linear	PROPN
ijassa-2040	116	7	section	section	NOUN
ijassa-2040	116	8	of	of	ADP
ijassa-2040	116	9	f	f	PROPN
ijassa-2040	116	10	.	.	PUNCT
ijassa-2040	117	1	all	all	DET
ijassa-2040	117	2	the	the	DET
ijassa-2040	117	3	subspaces	subspace	NOUN
ijassa-2040	117	4	considered	consider	VERB
ijassa-2040	117	5	above	above	ADV
ijassa-2040	117	6	are	be	AUX
ijassa-2040	117	7	also	also	ADV
ijassa-2040	117	8	linear	linear	ADJ
ijassa-2040	117	9	sections	section	NOUN
ijassa-2040	117	10	.	.	PUNCT
ijassa-2040	118	1	note	note	VERB
ijassa-2040	118	2	that	that	SCONJ
ijassa-2040	118	3	the	the	DET
ijassa-2040	118	4	pairing	pairing	NOUN
ijassa-2040	118	5	considered	consider	VERB
ijassa-2040	118	6	in	in	ADP
ijassa-2040	118	7	section	section	NOUN
ijassa-2040	118	8	1	1	NUM
ijassa-2040	118	9	naturally	naturally	ADV
ijassa-2040	118	10	generates	generate	VERB
ijassa-2040	118	11	a	a	DET
ijassa-2040	118	12	hausdorff	hausdorff	NOUN
ijassa-2040	118	13	topology	topology	NOUN
ijassa-2040	118	14	on	on	ADP
ijassa-2040	118	15	every	every	DET
ijassa-2040	118	16	vector	vector	NOUN
ijassa-2040	118	17	subspace	subspace	NOUN
ijassa-2040	118	18	of	of	ADP
ijassa-2040	118	19	f	f	PROPN
ijassa-2040	118	20	(	(	PUNCT
ijassa-2040	118	21	see	see	VERB
ijassa-2040	118	22	[	[	X
ijassa-2040	118	23	5	5	NUM
ijassa-2040	118	24	]	]	NUM
ijassa-2040	118	25	)	)	PUNCT
ijassa-2040	118	26	.	.	PUNCT
ijassa-2040	119	1	therefore	therefore	ADV
ijassa-2040	119	2	,	,	PUNCT
ijassa-2040	119	3	every	every	DET
ijassa-2040	119	4	linear	linear	NOUN
ijassa-2040	119	5	section	section	NOUN
ijassa-2040	119	6	α	α	PRON
ijassa-2040	119	7	can	can	AUX
ijassa-2040	119	8	be	be	AUX
ijassa-2040	119	9	considered	consider	VERB
ijassa-2040	119	10	as	as	ADP
ijassa-2040	119	11	a	a	DET
ijassa-2040	119	12	complete	complete	ADJ
ijassa-2040	119	13	topological	topological	ADJ
ijassa-2040	119	14	vector	vector	NOUN
ijassa-2040	119	15	space	space	NOUN
ijassa-2040	119	16	,	,	PUNCT
ijassa-2040	119	17	and	and	CCONJ
ijassa-2040	119	18	the	the	DET
ijassa-2040	119	19	space	space	NOUN
ijassa-2040	119	20	α∗	α∗	NOUN
ijassa-2040	119	21	consisting	consist	VERB
ijassa-2040	119	22	of	of	ADP
ijassa-2040	119	23	g	g	PROPN
ijassa-2040	119	24	∈	∈	PROPN
ijassa-2040	119	25	f	f	PROPN
ijassa-2040	119	26	such	such	ADJ
ijassa-2040	119	27	that	that	PRON
ijassa-2040	119	28	⟨f	⟨f	PROPN
ijassa-2040	119	29	,	,	PUNCT
ijassa-2040	119	30	g⟩	g⟩	VERB
ijassa-2040	119	31	<	<	X
ijassa-2040	119	32	∞	∞	PROPN
ijassa-2040	119	33	for	for	ADP
ijassa-2040	119	34	all	all	DET
ijassa-2040	119	35	f	f	PROPN
ijassa-2040	119	36	∈	∈	PROPN
ijassa-2040	119	37	α	α	NOUN
ijassa-2040	119	38	is	be	AUX
ijassa-2040	119	39	the	the	DET
ijassa-2040	119	40	dual	dual	ADJ
ijassa-2040	119	41	space	space	NOUN
ijassa-2040	119	42	of	of	ADP
ijassa-2040	119	43	α	α	NOUN
ijassa-2040	119	44	.	.	PUNCT
ijassa-2040	120	1	the	the	DET
ijassa-2040	120	2	set	set	NOUN
ijassa-2040	120	3	m	m	NOUN
ijassa-2040	120	4	of	of	ADP
ijassa-2040	120	5	all	all	DET
ijassa-2040	120	6	linear	linear	ADJ
ijassa-2040	120	7	sections	section	NOUN
ijassa-2040	120	8	of	of	ADP
ijassa-2040	120	9	f	f	PROPN
ijassa-2040	120	10	is	be	AUX
ijassa-2040	120	11	ordered	order	VERB
ijassa-2040	120	12	by	by	ADP
ijassa-2040	120	13	the	the	DET
ijassa-2040	120	14	inclusion	inclusion	NOUN
ijassa-2040	120	15	.	.	PUNCT
ijassa-2040	121	1	for	for	ADP
ijassa-2040	121	2	every	every	DET
ijassa-2040	121	3	two	two	NUM
ijassa-2040	121	4	sections	section	NOUN
ijassa-2040	121	5	α	α	NOUN
ijassa-2040	121	6	and	and	CCONJ
ijassa-2040	121	7	β	β	NOUN
ijassa-2040	121	8	there	there	PRON
ijassa-2040	121	9	exist	exist	VERB
ijassa-2040	121	10	the	the	DET
ijassa-2040	121	11	linear	linear	ADJ
ijassa-2040	121	12	sections	section	NOUN
ijassa-2040	121	13	sup(α	sup(α	PROPN
ijassa-2040	121	14	,	,	PUNCT
ijassa-2040	121	15	β	β	X
ijassa-2040	121	16	)	)	PUNCT
ijassa-2040	122	1	=	=	SYM
ijassa-2040	122	2	α	α	PROPN
ijassa-2040	122	3	+	+	X
ijassa-2040	122	4	β	β	X
ijassa-2040	122	5	,	,	PUNCT
ijassa-2040	122	6	inf(α	inf(α	PROPN
ijassa-2040	122	7	,	,	PUNCT
ijassa-2040	122	8	β	β	NOUN
ijassa-2040	122	9	)	)	PUNCT
ijassa-2040	122	10	=	=	SYM
ijassa-2040	122	11	α	α	PROPN
ijassa-2040	122	12	∩	∩	NOUN
ijassa-2040	123	1	β	β	PRON
ijassa-2040	123	2	defined	define	VERB
ijassa-2040	123	3	as	as	ADP
ijassa-2040	123	4	minimal	minimal	ADJ
ijassa-2040	123	5	(	(	PUNCT
ijassa-2040	123	6	by	by	ADP
ijassa-2040	123	7	the	the	DET
ijassa-2040	123	8	inclusion	inclusion	NOUN
ijassa-2040	123	9	)	)	PUNCT
ijassa-2040	123	10	linear	linear	ADJ
ijassa-2040	123	11	sections	section	NOUN
ijassa-2040	123	12	that	that	PRON
ijassa-2040	123	13	contain	contain	VERB
ijassa-2040	123	14	respectively	respectively	ADV
ijassa-2040	123	15	α	α	NOUN
ijassa-2040	123	16	∪	∪	NOUN
ijassa-2040	123	17	β	β	X
ijassa-2040	123	18	or	or	CCONJ
ijassa-2040	123	19	α	α	PRON
ijassa-2040	123	20	∩	∩	NOUN
ijassa-2040	123	21	β	β	X
ijassa-2040	123	22	.	.	PUNCT
ijassa-2040	124	1	it	it	PRON
ijassa-2040	124	2	is	be	AUX
ijassa-2040	124	3	easy	easy	ADJ
ijassa-2040	124	4	to	to	PART
ijassa-2040	124	5	see	see	VERB
ijassa-2040	124	6	that	that	SCONJ
ijassa-2040	124	7	the	the	DET
ijassa-2040	124	8	distributivity	distributivity	NOUN
ijassa-2040	124	9	relations	relation	NOUN
ijassa-2040	124	10	are	be	AUX
ijassa-2040	124	11	satisfied	satisfied	ADJ
ijassa-2040	124	12	:	:	PUNCT
ijassa-2040	124	13	α	α	NOUN
ijassa-2040	124	14	∩	∩	NOUN
ijassa-2040	124	15	(	(	PUNCT
ijassa-2040	124	16	β	β	X
ijassa-2040	124	17	+	+	X
ijassa-2040	124	18	γ	γ	X
ijassa-2040	124	19	)	)	PUNCT
ijassa-2040	124	20	=	=	SYM
ijassa-2040	124	21	α	α	PROPN
ijassa-2040	124	22	∩	∩	NOUN
ijassa-2040	124	23	β	β	X
ijassa-2040	124	24	+	+	ADJ
ijassa-2040	124	25	α	α	PROPN
ijassa-2040	124	26	∩	∩	ADJ
ijassa-2040	124	27	γ	γ	X
ijassa-2040	124	28	,	,	PUNCT
ijassa-2040	124	29	α	α	NOUN
ijassa-2040	124	30	+	+	X
ijassa-2040	124	31	(	(	PUNCT
ijassa-2040	124	32	β	β	X
ijassa-2040	124	33	∩	∩	ADJ
ijassa-2040	124	34	γ	γ	X
ijassa-2040	124	35	)	)	PUNCT
ijassa-2040	124	36	=	=	SYM
ijassa-2040	124	37	(	(	PUNCT
ijassa-2040	124	38	α	α	NOUN
ijassa-2040	124	39	+	+	NOUN
ijassa-2040	124	40	β	β	NOUN
ijassa-2040	124	41	)	)	PUNCT
ijassa-2040	124	42	∩	∩	NOUN
ijassa-2040	124	43	(	(	PUNCT
ijassa-2040	124	44	α	α	NOUN
ijassa-2040	124	45	+	+	X
ijassa-2040	124	46	γ	γ	NOUN
ijassa-2040	124	47	)	)	PUNCT
ijassa-2040	124	48	.	.	PUNCT
ijassa-2040	125	1	thus	thus	ADV
ijassa-2040	125	2	,	,	PUNCT
ijassa-2040	125	3	in	in	ADP
ijassa-2040	125	4	the	the	DET
ijassa-2040	125	5	set	set	NOUN
ijassa-2040	125	6	m	m	VERB
ijassa-2040	125	7	a	a	DET
ijassa-2040	125	8	certain	certain	ADJ
ijassa-2040	125	9	structure	structure	NOUN
ijassa-2040	125	10	of	of	ADP
ijassa-2040	125	11	the	the	DET
ijassa-2040	125	12	distributive	distributive	ADJ
ijassa-2040	125	13	lattice	lattice	NOUN
ijassa-2040	125	14	with	with	ADP
ijassa-2040	125	15	additive	additive	ADJ
ijassa-2040	125	16	and	and	CCONJ
ijassa-2040	125	17	multiplicative	multiplicative	ADJ
ijassa-2040	125	18	identity	identity	NOUN
ijassa-2040	125	19	elements	element	NOUN
ijassa-2040	125	20	is	be	AUX
ijassa-2040	125	21	introduced	introduce	VERB
ijassa-2040	125	22	.	.	PUNCT
ijassa-2040	126	1	if	if	SCONJ
ijassa-2040	126	2	a	a	PRON
ijassa-2040	126	3	is	be	AUX
ijassa-2040	126	4	an	an	DET
ijassa-2040	126	5	arbitrary	arbitrary	ADJ
ijassa-2040	126	6	set	set	NOUN
ijassa-2040	126	7	and	and	CCONJ
ijassa-2040	126	8	{	{	PUNCT
ijassa-2040	126	9	δa|	δa|	PROPN
ijassa-2040	126	10	a	a	PROPN
ijassa-2040	126	11	∈	∈	PROPN
ijassa-2040	126	12	a	a	PRON
ijassa-2040	126	13	}	}	PUNCT
ijassa-2040	126	14	is	be	AUX
ijassa-2040	126	15	a	a	DET
ijassa-2040	126	16	family	family	NOUN
ijassa-2040	126	17	of	of	ADP
ijassa-2040	126	18	linear	linear	PROPN
ijassa-2040	126	19	sections	section	NOUN
ijassa-2040	126	20	,	,	PUNCT
ijassa-2040	126	21	then	then	ADV
ijassa-2040	126	22	the	the	DET
ijassa-2040	126	23	supremum	supremum	NOUN
ijassa-2040	126	24	of	of	ADP
ijassa-2040	126	25	sup	sup	NOUN
ijassa-2040	126	26	δa	δa	PROPN
ijassa-2040	126	27	is	be	AUX
ijassa-2040	126	28	minimal	minimal	ADJ
ijassa-2040	126	29	linear	linear	ADJ
ijassa-2040	126	30	section	section	NOUN
ijassa-2040	126	31	that	that	PRON
ijassa-2040	126	32	contains	contain	VERB
ijassa-2040	126	33	all	all	DET
ijassa-2040	126	34	δa	δa	PROPN
ijassa-2040	126	35	.	.	PUNCT
ijassa-2040	127	1	by	by	ADP
ijassa-2040	127	2	the	the	DET
ijassa-2040	127	3	zorn	zorn	PROPN
ijassa-2040	127	4	lemma	lemma	PROPN
ijassa-2040	127	5	,	,	PUNCT
ijassa-2040	127	6	the	the	DET
ijassa-2040	127	7	supremum	supremum	NOUN
ijassa-2040	127	8	of	of	ADP
ijassa-2040	127	9	any	any	DET
ijassa-2040	127	10	family	family	NOUN
ijassa-2040	127	11	exists	exist	VERB
ijassa-2040	127	12	.	.	PUNCT
ijassa-2040	128	1	2.5	2.5	NUM
ijassa-2040	128	2	.	.	PUNCT
ijassa-2040	129	1	linear	linear	ADJ
ijassa-2040	129	2	sections	section	NOUN
ijassa-2040	129	3	and	and	CCONJ
ijassa-2040	129	4	solvability	solvability	NOUN
ijassa-2040	129	5	of	of	ADP
ijassa-2040	129	6	general	general	ADJ
ijassa-2040	129	7	equations	equation	NOUN
ijassa-2040	129	8	proposition	proposition	VERB
ijassa-2040	129	9	2.5	2.5	NUM
ijassa-2040	129	10	:	:	SYM
ijassa-2040	130	1	1	1	X
ijassa-2040	130	2	.	.	X
ijassa-2040	130	3	the	the	DET
ijassa-2040	130	4	operator	operator	NOUN
ijassa-2040	130	5	l	l	NOUN
ijassa-2040	130	6	=	=	PUNCT
ijassa-2040	130	7	∑	∑	PUNCT
ijassa-2040	130	8	|α|≤m	|α|≤m	PROPN
ijassa-2040	130	9	tα(x)d	tα(x)d	PROPN
ijassa-2040	130	10	α	α	PROPN
ijassa-2040	130	11	sends	send	VERB
ijassa-2040	130	12	any	any	DET
ijassa-2040	130	13	linear	linear	ADJ
ijassa-2040	130	14	section	section	NOUN
ijassa-2040	130	15	to	to	ADP
ijassa-2040	130	16	a	a	DET
ijassa-2040	130	17	linear	linear	ADJ
ijassa-2040	130	18	section	section	NOUN
ijassa-2040	130	19	an	an	PRON
ijassa-2040	130	20	,	,	PUNCT
ijassa-2040	130	21	consequently	consequently	ADV
ijassa-2040	130	22	,	,	PUNCT
ijassa-2040	130	23	it	it	PRON
ijassa-2040	130	24	induces	induce	VERB
ijassa-2040	130	25	a	a	DET
ijassa-2040	130	26	mapping	mapping	NOUN
ijassa-2040	130	27	(	(	PUNCT
ijassa-2040	130	28	endomorphism	endomorphism	PROPN
ijassa-2040	130	29	)	)	PUNCT
ijassa-2040	130	30	l̃	l̃	PROPN
ijassa-2040	130	31	in	in	ADP
ijassa-2040	130	32	the	the	DET
ijassa-2040	130	33	set	set	NOUN
ijassa-2040	130	34	m	m	PROPN
ijassa-2040	130	35	,	,	PUNCT
ijassa-2040	130	36	which	which	PRON
ijassa-2040	130	37	preserves	preserve	VERB
ijassa-2040	130	38	the	the	DET
ijassa-2040	130	39	lattice	lattice	NOUN
ijassa-2040	130	40	structure	structure	NOUN
ijassa-2040	130	41	.	.	PUNCT
ijassa-2040	131	1	2	2	X
ijassa-2040	131	2	.	.	X
ijassa-2040	131	3	under	under	ADP
ijassa-2040	131	4	assumption	assumption	NOUN
ijassa-2040	131	5	2.1	2.1	NUM
ijassa-2040	131	6	,	,	PUNCT
ijassa-2040	131	7	the	the	DET
ijassa-2040	131	8	mapping	mapping	NOUN
ijassa-2040	131	9	l̃	l̃	PROPN
ijassa-2040	131	10	is	be	AUX
ijassa-2040	131	11	an	an	DET
ijassa-2040	131	12	epimorphism	epimorphism	NOUN
ijassa-2040	131	13	of	of	ADP
ijassa-2040	131	14	the	the	DET
ijassa-2040	131	15	lattice	lattice	NOUN
ijassa-2040	131	16	m	m	PROPN
ijassa-2040	131	17	and	and	CCONJ
ijassa-2040	131	18	for	for	ADP
ijassa-2040	131	19	every	every	DET
ijassa-2040	131	20	linear	linear	ADJ
ijassa-2040	131	21	section	section	NOUN
ijassa-2040	131	22	g	g	PROPN
ijassa-2040	131	23	there	there	PRON
ijassa-2040	131	24	exists	exist	VERB
ijassa-2040	131	25	the	the	DET
ijassa-2040	131	26	maximum	maximum	ADJ
ijassa-2040	131	27	β	β	X
ijassa-2040	131	28	among	among	ADP
ijassa-2040	131	29	those	those	DET
ijassa-2040	131	30	linear	linear	PROPN
ijassa-2040	131	31	sections	section	NOUN
ijassa-2040	131	32	α	α	VERB
ijassa-2040	131	33	for	for	ADP
ijassa-2040	131	34	which	which	PRON
ijassa-2040	131	35	l̃α	l̃α	VERB
ijassa-2040	131	36	≤	≤	ADJ
ijassa-2040	131	37	g	g	NOUN
ijassa-2040	131	38	,	,	PUNCT
ijassa-2040	131	39	and	and	CCONJ
ijassa-2040	131	40	β	β	X
ijassa-2040	131	41	̸=	̸=	PROPN
ijassa-2040	131	42	f	f	PROPN
ijassa-2040	131	43	.	.	PUNCT
ijassa-2040	132	1	proof	proof	NOUN
ijassa-2040	132	2	since	since	SCONJ
ijassa-2040	132	3	operations	operation	NOUN
ijassa-2040	132	4	of	of	ADP
ijassa-2040	132	5	differentiation	differentiation	NOUN
ijassa-2040	132	6	,	,	PUNCT
ijassa-2040	132	7	multiplication	multiplication	NOUN
ijassa-2040	132	8	by	by	ADP
ijassa-2040	132	9	a	a	DET
ijassa-2040	132	10	scalar	scalar	ADJ
ijassa-2040	132	11	,	,	PUNCT
ijassa-2040	132	12	addition	addition	NOUN
ijassa-2040	132	13	,	,	PUNCT
ijassa-2040	132	14	and	and	CCONJ
ijassa-2040	132	15	shifts	shift	NOUN
ijassa-2040	132	16	{	{	PUNCT
ijassa-2040	132	17	un	un	PROPN
ijassa-2040	132	18	}	}	PUNCT
ijassa-2040	132	19	→	→	SYM
ijassa-2040	132	20	{	{	PUNCT
ijassa-2040	132	21	un+k	un+k	NOUN
ijassa-2040	132	22	}	}	PUNCT
ijassa-2040	132	23	preserve	preserve	VERB
ijassa-2040	132	24	the	the	DET
ijassa-2040	132	25	relation	relation	NOUN
ijassa-2040	132	26	of	of	ADP
ijassa-2040	132	27	order	order	NOUN
ijassa-2040	132	28	≤	≤	NOUN
ijassa-2040	132	29	,	,	PUNCT
ijassa-2040	132	30	the	the	DET
ijassa-2040	132	31	mapping	mapping	NOUN
ijassa-2040	132	32	l	l	NOUN
ijassa-2040	132	33	sends	send	VERB
ijassa-2040	132	34	every	every	DET
ijassa-2040	132	35	linear	linear	ADJ
ijassa-2040	132	36	section	section	NOUN
ijassa-2040	132	37	to	to	ADP
ijassa-2040	132	38	a	a	DET
ijassa-2040	132	39	linear	linear	ADJ
ijassa-2040	132	40	section	section	NOUN
ijassa-2040	132	41	and	and	CCONJ
ijassa-2040	132	42	the	the	DET
ijassa-2040	132	43	induced	induced	ADJ
ijassa-2040	132	44	mapping	mapping	NOUN
ijassa-2040	132	45	preserves	preserve	VERB
ijassa-2040	132	46	the	the	DET
ijassa-2040	132	47	relation	relation	NOUN
ijassa-2040	132	48	of	of	ADP
ijassa-2040	132	49	order	order	NOUN
ijassa-2040	132	50	and	and	CCONJ
ijassa-2040	132	51	the	the	DET
ijassa-2040	132	52	lattice	lattice	PROPN
ijassa-2040	132	53	operations	operation	NOUN
ijassa-2040	132	54	.	.	PUNCT
ijassa-2040	133	1	from	from	ADP
ijassa-2040	133	2	assumption	assumption	NOUN
ijassa-2040	133	3	2.1	2.1	NUM
ijassa-2040	133	4	,	,	PUNCT
ijassa-2040	133	5	it	it	PRON
ijassa-2040	133	6	follows	follow	VERB
ijassa-2040	133	7	that	that	SCONJ
ijassa-2040	133	8	every	every	DET
ijassa-2040	133	9	equation	equation	NOUN
ijassa-2040	133	10	lu	lu	NOUN
ijassa-2040	133	11	=	=	NOUN
ijassa-2040	133	12	einx	einx	PROPN
ijassa-2040	133	13	has	have	VERB
ijassa-2040	133	14	a	a	DET
ijassa-2040	133	15	solution	solution	NOUN
ijassa-2040	133	16	un	un	NOUN
ijassa-2040	133	17	among	among	ADP
ijassa-2040	133	18	trigonometric	trigonometric	ADJ
ijassa-2040	133	19	polynomials	polynomial	NOUN
ijassa-2040	133	20	.	.	PUNCT
ijassa-2040	134	1	further	far	ADV
ijassa-2040	134	2	,	,	PUNCT
ijassa-2040	134	3	if	if	SCONJ
ijassa-2040	134	4	f	f	PROPN
ijassa-2040	134	5	is	be	AUX
ijassa-2040	134	6	a	a	DET
ijassa-2040	134	7	formal	formal	ADJ
ijassa-2040	134	8	series	series	NOUN
ijassa-2040	134	9	and	and	CCONJ
ijassa-2040	134	10	the	the	DET
ijassa-2040	134	11	operator	operator	NOUN
ijassa-2040	134	12	l	l	NOUN
ijassa-2040	134	13	satisfies	satisfie	NOUN
ijassa-2040	134	14	condition	condition	NOUN
ijassa-2040	134	15	2.1	2.1	NUM
ijassa-2040	134	16	,	,	PUNCT
ijassa-2040	134	17	then	then	ADV
ijassa-2040	134	18	the	the	DET
ijassa-2040	134	19	equation	equation	NOUN
ijassa-2040	134	20	lu	lu	NOUN
ijassa-2040	135	1	=	=	SYM
ijassa-2040	135	2	f	f	PROPN
ijassa-2040	135	3	=	=	PUNCT
ijassa-2040	135	4	∑	∑	PROPN
ijassa-2040	135	5	n	n	PRON
ijassa-2040	135	6	fne	fne	NOUN
ijassa-2040	135	7	inx	inx	NOUN
ijassa-2040	135	8	is	be	AUX
ijassa-2040	135	9	solvable	solvable	ADJ
ijassa-2040	135	10	in	in	ADP
ijassa-2040	135	11	the	the	DET
ijassa-2040	135	12	class	class	NOUN
ijassa-2040	135	13	of	of	ADP
ijassa-2040	135	14	formal	formal	ADJ
ijassa-2040	135	15	series	series	NOUN
ijassa-2040	135	16	.	.	PUNCT
ijassa-2040	136	1	let	let	VERB
ijassa-2040	136	2	such	such	DET
ijassa-2040	136	3	a	a	DET
ijassa-2040	136	4	solution	solution	NOUN
ijassa-2040	136	5	be	be	AUX
ijassa-2040	136	6	the	the	DET
ijassa-2040	136	7	formal	formal	ADJ
ijassa-2040	136	8	series	series	NOUN
ijassa-2040	137	1	w	w	PROPN
ijassa-2040	137	2	=	=	PUNCT
ijassa-2040	137	3	∑	∑	PROPN
ijassa-2040	137	4	n	n	CCONJ
ijassa-2040	137	5	,	,	PUNCT
ijassa-2040	137	6	k	k	PROPN
ijassa-2040	137	7	fnunke	fnunke	PROPN
ijassa-2040	137	8	ikx	ikx	PROPN
ijassa-2040	137	9	,	,	PUNCT
ijassa-2040	137	10	where	where	SCONJ
ijassa-2040	137	11	for	for	ADP
ijassa-2040	137	12	each	each	DET
ijassa-2040	137	13	k	k	NOUN
ijassa-2040	137	14	the	the	DET
ijassa-2040	137	15	sum	sum	NOUN
ijassa-2040	137	16	over	over	ADP
ijassa-2040	137	17	n	n	PROPN
ijassa-2040	137	18	is	be	AUX
ijassa-2040	137	19	finite	finite	ADJ
ijassa-2040	137	20	,	,	PUNCT
ijassa-2040	137	21	and	and	CCONJ
ijassa-2040	137	22	the	the	DET
ijassa-2040	137	23	coefficients	coefficient	NOUN
ijassa-2040	137	24	unk	unk	NOUN
ijassa-2040	137	25	are	be	AUX
ijassa-2040	137	26	uniquely	uniquely	ADV
ijassa-2040	137	27	defined	define	VERB
ijassa-2040	137	28	by	by	ADP
ijassa-2040	137	29	condition	condition	NOUN
ijassa-2040	137	30	2.1	2.1	NUM
ijassa-2040	137	31	.	.	PUNCT
ijassa-2040	138	1	given	give	VERB
ijassa-2040	138	2	linear	linear	PROPN
ijassa-2040	138	3	section	section	NOUN
ijassa-2040	138	4	g	g	NOUN
ijassa-2040	138	5	,	,	PUNCT
ijassa-2040	138	6	for	for	ADP
ijassa-2040	138	7	every	every	DET
ijassa-2040	138	8	f	f	PROPN
ijassa-2040	138	9	∈	∈	PROPN
ijassa-2040	138	10	g	g	PROPN
ijassa-2040	138	11	consider	consider	VERB
ijassa-2040	138	12	the	the	DET
ijassa-2040	138	13	set	set	ADJ
ijassa-2040	138	14	wf	wf	PROPN
ijassa-2040	138	15	of	of	ADP
ijassa-2040	138	16	solutions	solution	NOUN
ijassa-2040	138	17	to	to	ADP
ijassa-2040	138	18	copyright	copyright	NOUN
ijassa-2040	138	19	©	©	PROPN
ijassa-2040	138	20	2025	2025	NUM
ijassa-2040	138	21	assa	assa	NOUN
ijassa-2040	138	22	.	.	PUNCT
ijassa-2040	139	1	adv	adv	PROPN
ijassa-2040	139	2	syst	syst	PROPN
ijassa-2040	139	3	sci	sci	PROPN
ijassa-2040	139	4	appl	appl	PROPN
ijassa-2040	139	5	(	(	PUNCT
ijassa-2040	139	6	2025	2025	NUM
ijassa-2040	139	7	)	)	PUNCT
ijassa-2040	139	8	18	18	NUM
ijassa-2040	139	9	v.	v.	ADP
ijassa-2040	139	10	p.	p.	NOUN
ijassa-2040	139	11	burskii	burskii	VERB
ijassa-2040	139	12	the	the	DET
ijassa-2040	139	13	equation	equation	NOUN
ijassa-2040	139	14	lu	lu	NOUN
ijassa-2040	139	15	=	=	SYM
ijassa-2040	139	16	f	f	PROPN
ijassa-2040	139	17	and	and	CCONJ
ijassa-2040	139	18	the	the	DET
ijassa-2040	139	19	set	set	NOUN
ijassa-2040	139	20	of	of	ADP
ijassa-2040	139	21	the	the	DET
ijassa-2040	139	22	corresponding	corresponding	ADJ
ijassa-2040	139	23	principal	principal	ADJ
ijassa-2040	139	24	linear	linear	PROPN
ijassa-2040	139	25	sections	section	NOUN
ijassa-2040	139	26	.	.	PUNCT
ijassa-2040	140	1	by	by	ADP
ijassa-2040	140	2	the	the	DET
ijassa-2040	140	3	zorn	zorn	PROPN
ijassa-2040	140	4	lemma	lemma	PROPN
ijassa-2040	140	5	,	,	PUNCT
ijassa-2040	140	6	there	there	PRON
ijassa-2040	140	7	exists	exist	VERB
ijassa-2040	140	8	the	the	DET
ijassa-2040	140	9	supremum	supremum	NOUN
ijassa-2040	140	10	s	s	X
ijassa-2040	140	11	∈	∈	PROPN
ijassa-2040	140	12	w̃f	w̃f	NOUN
ijassa-2040	140	13	,	,	PUNCT
ijassa-2040	140	14	which	which	PRON
ijassa-2040	140	15	is	be	AUX
ijassa-2040	140	16	the	the	DET
ijassa-2040	140	17	desired	desire	VERB
ijassa-2040	140	18	linear	linear	PROPN
ijassa-2040	140	19	section	section	NOUN
ijassa-2040	140	20	β	β	NOUN
ijassa-2040	140	21	.	.	PUNCT
ijassa-2040	141	1	obviously	obviously	ADV
ijassa-2040	141	2	,	,	PUNCT
ijassa-2040	141	3	β	β	PROPN
ijassa-2040	141	4	does	do	AUX
ijassa-2040	141	5	not	not	PART
ijassa-2040	141	6	coincide	coincide	VERB
ijassa-2040	141	7	with	with	ADP
ijassa-2040	141	8	f	f	PROPN
ijassa-2040	141	9	,	,	PUNCT
ijassa-2040	141	10	since	since	SCONJ
ijassa-2040	141	11	otherwise	otherwise	ADV
ijassa-2040	141	12	g	g	PROPN
ijassa-2040	141	13	=	=	SYM
ijassa-2040	141	14	f	f	PROPN
ijassa-2040	141	15	.	.	PUNCT
ijassa-2040	142	1	this	this	PRON
ijassa-2040	142	2	completes	complete	VERB
ijassa-2040	142	3	the	the	DET
ijassa-2040	142	4	proof	proof	NOUN
ijassa-2040	142	5	.	.	PUNCT
ijassa-2040	143	1	remark	remark	VERB
ijassa-2040	143	2	2.1	2.1	NUM
ijassa-2040	143	3	:	:	PUNCT
ijassa-2040	143	4	assumption	assumption	NOUN
ijassa-2040	143	5	2.1	2.1	NUM
ijassa-2040	143	6	can	can	AUX
ijassa-2040	143	7	be	be	AUX
ijassa-2040	143	8	replaced	replace	VERB
ijassa-2040	143	9	with	with	ADP
ijassa-2040	143	10	be	be	AUX
ijassa-2040	143	11	replaced	replace	VERB
ijassa-2040	143	12	with	with	ADP
ijassa-2040	143	13	a	a	DET
ijassa-2040	143	14	weaker	weak	ADJ
ijassa-2040	143	15	condition	condition	NOUN
ijassa-2040	143	16	:	:	PUNCT
ijassa-2040	143	17	∀m	∀m	PROPN
ijassa-2040	143	18	∈	∈	PROPN
ijassa-2040	143	19	z2	z2	PROPN
ijassa-2040	143	20	∃n	∃n	X
ijassa-2040	143	21	:	:	PUNCT
ijassa-2040	143	22	pn(m	pn(m	NOUN
ijassa-2040	143	23	)	)	PUNCT
ijassa-2040	144	1	̸=	̸=	PROPN
ijassa-2040	144	2	0	0	NUM
ijassa-2040	144	3	.	.	PUNCT
ijassa-2040	144	4	definition	definition	NOUN
ijassa-2040	144	5	2.1	2.1	NUM
ijassa-2040	144	6	:	:	PUNCT
ijassa-2040	144	7	the	the	DET
ijassa-2040	144	8	linear	linear	PROPN
ijassa-2040	144	9	section	section	NOUN
ijassa-2040	144	10	β	β	PROPN
ijassa-2040	144	11	constructed	construct	VERB
ijassa-2040	144	12	in	in	ADP
ijassa-2040	144	13	proposition	proposition	NOUN
ijassa-2040	144	14	2.5	2.5	NUM
ijassa-2040	144	15	is	be	AUX
ijassa-2040	144	16	called	call	VERB
ijassa-2040	144	17	a	a	DET
ijassa-2040	144	18	solution	solution	NOUN
ijassa-2040	144	19	of	of	ADP
ijassa-2040	144	20	the	the	DET
ijassa-2040	144	21	equation	equation	NOUN
ijassa-2040	145	1	lu	lu	NOUN
ijassa-2040	145	2	=	=	NOUN
ijassa-2040	145	3	g	g	NOUN
ijassa-2040	145	4	with	with	ADP
ijassa-2040	145	5	linear	linear	PROPN
ijassa-2040	145	6	section	section	NOUN
ijassa-2040	145	7	g	g	NOUN
ijassa-2040	145	8	in	in	ADP
ijassa-2040	145	9	the	the	DET
ijassa-2040	145	10	right	right	ADJ
ijassa-2040	145	11	-	-	PUNCT
ijassa-2040	145	12	hand	hand	NOUN
ijassa-2040	145	13	side	side	NOUN
ijassa-2040	145	14	.	.	PUNCT
ijassa-2040	146	1	for	for	ADP
ijassa-2040	146	2	example	example	NOUN
ijassa-2040	146	3	,	,	PUNCT
ijassa-2040	146	4	from	from	ADP
ijassa-2040	146	5	what	what	PRON
ijassa-2040	146	6	we	we	PRON
ijassa-2040	146	7	proved	prove	VERB
ijassa-2040	146	8	above	above	ADV
ijassa-2040	146	9	,	,	PUNCT
ijassa-2040	146	10	it	it	PRON
ijassa-2040	146	11	follows	follow	VERB
ijassa-2040	146	12	that	that	SCONJ
ijassa-2040	146	13	solution	solution	NOUN
ijassa-2040	146	14	of	of	ADP
ijassa-2040	146	15	equation	equation	NOUN
ijassa-2040	146	16	(	(	PUNCT
ijassa-2040	146	17	2.2	2.2	NUM
ijassa-2040	146	18	)	)	PUNCT
ijassa-2040	146	19	with	with	ADP
ijassa-2040	146	20	the	the	DET
ijassa-2040	146	21	right	right	ADJ
ijassa-2040	146	22	-	-	PUNCT
ijassa-2040	146	23	hand	hand	NOUN
ijassa-2040	146	24	side	side	NOUN
ijassa-2040	146	25	g	g	NOUN
ijassa-2040	147	1	=	=	PUNCT
ijassa-2040	147	2	hm	hm	INTJ
ijassa-2040	147	3	is	be	AUX
ijassa-2040	147	4	the	the	DET
ijassa-2040	147	5	section	section	NOUN
ijassa-2040	147	6	β	β	PROPN
ijassa-2040	147	7	⊂	⊂	PROPN
ijassa-2040	147	8	l∗1(|n1|	l∗1(|n1|	PROPN
ijassa-2040	147	9	!	!	PUNCT
ijassa-2040	147	10	)	)	PUNCT
ijassa-2040	147	11	.	.	PUNCT
ijassa-2040	148	1	remark	remark	VERB
ijassa-2040	148	2	2.2	2.2	NUM
ijassa-2040	148	3	:	:	PUNCT
ijassa-2040	148	4	the	the	DET
ijassa-2040	148	5	term	term	NOUN
ijassa-2040	148	6	“	"	PUNCT
ijassa-2040	148	7	section	section	NOUN
ijassa-2040	148	8	”	"	PUNCT
ijassa-2040	148	9	was	be	AUX
ijassa-2040	148	10	chosen	choose	VERB
ijassa-2040	148	11	due	due	ADP
ijassa-2040	148	12	to	to	ADP
ijassa-2040	148	13	the	the	DET
ijassa-2040	148	14	obvious	obvious	ADJ
ijassa-2040	148	15	analogy	analogy	NOUN
ijassa-2040	148	16	with	with	ADP
ijassa-2040	148	17	dedekind	dedekind	NOUN
ijassa-2040	148	18	sections	section	NOUN
ijassa-2040	148	19	when	when	SCONJ
ijassa-2040	148	20	constructing	construct	VERB
ijassa-2040	148	21	the	the	DET
ijassa-2040	148	22	field	field	NOUN
ijassa-2040	148	23	of	of	ADP
ijassa-2040	148	24	real	real	ADJ
ijassa-2040	148	25	numbers	number	NOUN
ijassa-2040	148	26	.	.	PUNCT
ijassa-2040	149	1	note	note	VERB
ijassa-2040	149	2	also	also	ADV
ijassa-2040	149	3	that	that	SCONJ
ijassa-2040	149	4	the	the	DET
ijassa-2040	149	5	presented	present	VERB
ijassa-2040	149	6	construction	construction	NOUN
ijassa-2040	149	7	is	be	AUX
ijassa-2040	149	8	consonant	consonant	ADJ
ijassa-2040	149	9	with	with	ADP
ijassa-2040	149	10	some	some	DET
ijassa-2040	149	11	constructions	construction	NOUN
ijassa-2040	149	12	of	of	ADP
ijassa-2040	149	13	non	non	ADJ
ijassa-2040	149	14	-	-	ADJ
ijassa-2040	149	15	standard	standard	ADJ
ijassa-2040	149	16	analysis	analysis	NOUN
ijassa-2040	149	17	related	relate	VERB
ijassa-2040	149	18	to	to	ADP
ijassa-2040	149	19	the	the	DET
ijassa-2040	149	20	extension	extension	NOUN
ijassa-2040	149	21	of	of	ADP
ijassa-2040	149	22	the	the	DET
ijassa-2040	149	23	field	field	NOUN
ijassa-2040	149	24	of	of	ADP
ijassa-2040	149	25	reals	real	NOUN
ijassa-2040	149	26	:	:	PUNCT
ijassa-2040	149	27	the	the	DET
ijassa-2040	149	28	infinitesimal	infinitesimal	ADJ
ijassa-2040	149	29	germs	germ	NOUN
ijassa-2040	149	30	of	of	ADP
ijassa-2040	149	31	functions	function	NOUN
ijassa-2040	149	32	from	from	ADP
ijassa-2040	149	33	both	both	CCONJ
ijassa-2040	149	34	a	a	DET
ijassa-2040	149	35	field	field	NOUN
ijassa-2040	149	36	and	and	CCONJ
ijassa-2040	149	37	an	an	DET
ijassa-2040	149	38	ultrafilter	ultrafilter	NOUN
ijassa-2040	149	39	.	.	PUNCT
ijassa-2040	150	1	the	the	DET
ijassa-2040	150	2	set	set	NOUN
ijassa-2040	150	3	m	m	PROPN
ijassa-2040	150	4	of	of	ADP
ijassa-2040	150	5	linear	linear	PROPN
ijassa-2040	150	6	sections	section	NOUN
ijassa-2040	150	7	is	be	AUX
ijassa-2040	150	8	only	only	ADV
ijassa-2040	150	9	partially	partially	ADV
ijassa-2040	150	10	ordered	order	VERB
ijassa-2040	150	11	,	,	PUNCT
ijassa-2040	150	12	but	but	CCONJ
ijassa-2040	150	13	it	it	PRON
ijassa-2040	150	14	contains	contain	VERB
ijassa-2040	150	15	all	all	DET
ijassa-2040	150	16	the	the	DET
ijassa-2040	150	17	germs	germ	NOUN
ijassa-2040	150	18	of	of	ADP
ijassa-2040	150	19	sequences	sequence	NOUN
ijassa-2040	150	20	as	as	ADP
ijassa-2040	150	21	principal	principal	ADJ
ijassa-2040	150	22	sections	section	NOUN
ijassa-2040	150	23	.	.	PUNCT
ijassa-2040	151	1	in	in	ADP
ijassa-2040	151	2	the	the	DET
ijassa-2040	151	3	set	set	NOUN
ijassa-2040	151	4	m	m	VERB
ijassa-2040	151	5	,	,	PUNCT
ijassa-2040	151	6	there	there	PRON
ijassa-2040	151	7	exists	exist	VERB
ijassa-2040	151	8	the	the	DET
ijassa-2040	151	9	operation	operation	NOUN
ijassa-2040	151	10	of	of	ADP
ijassa-2040	151	11	convolution	convolution	NOUN
ijassa-2040	151	12	,	,	PUNCT
ijassa-2040	151	13	which	which	PRON
ijassa-2040	151	14	is	be	AUX
ijassa-2040	151	15	associative	associative	ADJ
ijassa-2040	151	16	,	,	PUNCT
ijassa-2040	151	17	commutative	commutative	ADJ
ijassa-2040	151	18	and	and	CCONJ
ijassa-2040	151	19	distributive	distributive	ADJ
ijassa-2040	151	20	with	with	ADP
ijassa-2040	151	21	respect	respect	NOUN
ijassa-2040	151	22	to	to	ADP
ijassa-2040	151	23	addition	addition	NOUN
ijassa-2040	151	24	and	and	CCONJ
ijassa-2040	151	25	intersection	intersection	NOUN
ijassa-2040	151	26	(	(	PUNCT
ijassa-2040	151	27	however	however	ADV
ijassa-2040	151	28	,	,	PUNCT
ijassa-2040	151	29	the	the	DET
ijassa-2040	151	30	inverse	inverse	NOUN
ijassa-2040	151	31	exists	exist	VERB
ijassa-2040	151	32	not	not	PART
ijassa-2040	151	33	for	for	ADP
ijassa-2040	151	34	all	all	DET
ijassa-2040	151	35	elements	element	NOUN
ijassa-2040	151	36	)	)	PUNCT
ijassa-2040	151	37	.	.	PUNCT
ijassa-2040	152	1	moreover	moreover	ADV
ijassa-2040	152	2	,	,	PUNCT
ijassa-2040	152	3	in	in	ADP
ijassa-2040	152	4	m	m	NOUN
ijassa-2040	152	5	there	there	PRON
ijassa-2040	152	6	exists	exist	VERB
ijassa-2040	152	7	the	the	DET
ijassa-2040	152	8	conjugation	conjugation	NOUN
ijassa-2040	152	9	and	and	CCONJ
ijassa-2040	152	10	all	all	DET
ijassa-2040	152	11	linear	linear	ADJ
ijassa-2040	152	12	sections	section	NOUN
ijassa-2040	152	13	are	be	AUX
ijassa-2040	152	14	reflexive	reflexive	ADJ
ijassa-2040	152	15	spaces	space	NOUN
ijassa-2040	152	16	.	.	PUNCT
ijassa-2040	153	1	in	in	ADP
ijassa-2040	153	2	the	the	DET
ijassa-2040	153	3	set	set	NOUN
ijassa-2040	153	4	m	m	VERB
ijassa-2040	153	5	one	one	PRON
ijassa-2040	153	6	can	can	AUX
ijassa-2040	153	7	also	also	ADV
ijassa-2040	153	8	introduce	introduce	VERB
ijassa-2040	153	9	an	an	DET
ijassa-2040	153	10	associative	associative	ADJ
ijassa-2040	153	11	commutative	commutative	ADJ
ijassa-2040	153	12	product	product	NOUN
ijassa-2040	153	13	,	,	PUNCT
ijassa-2040	153	14	which	which	PRON
ijassa-2040	153	15	is	be	AUX
ijassa-2040	153	16	not	not	PART
ijassa-2040	153	17	always	always	ADV
ijassa-2040	153	18	defined	define	VERB
ijassa-2040	153	19	,	,	PUNCT
ijassa-2040	153	20	but	but	CCONJ
ijassa-2040	153	21	covering	cover	VERB
ijassa-2040	153	22	the	the	DET
ijassa-2040	153	23	product	product	NOUN
ijassa-2040	153	24	of	of	ADP
ijassa-2040	153	25	smooth	smooth	ADJ
ijassa-2040	153	26	functions	function	NOUN
ijassa-2040	153	27	and	and	CCONJ
ijassa-2040	153	28	the	the	DET
ijassa-2040	153	29	product	product	NOUN
ijassa-2040	153	30	of	of	ADP
ijassa-2040	153	31	distributions	distribution	NOUN
ijassa-2040	153	32	according	accord	VERB
ijassa-2040	153	33	to	to	ADP
ijassa-2040	153	34	mikusinsky	mikusinsky	ADJ
ijassa-2040	153	35	–	–	PUNCT
ijassa-2040	153	36	hirata	hirata	NOUN
ijassa-2040	153	37	–	–	PUNCT
ijassa-2040	153	38	ogawa	ogawa	PROPN
ijassa-2040	153	39	,	,	PUNCT
ijassa-2040	153	40	and	and	CCONJ
ijassa-2040	153	41	,	,	PUNCT
ijassa-2040	153	42	in	in	ADP
ijassa-2040	153	43	addition	addition	NOUN
ijassa-2040	153	44	,	,	PUNCT
ijassa-2040	153	45	the	the	DET
ijassa-2040	153	46	inverse	inverse	NOUN
ijassa-2040	153	47	element	element	NOUN
ijassa-2040	153	48	not	not	PART
ijassa-2040	153	49	lying	lie	VERB
ijassa-2040	153	50	in	in	ADP
ijassa-2040	153	51	m	m	NOUN
ijassa-2040	153	52	as	as	SCONJ
ijassa-2040	153	53	,	,	PUNCT
ijassa-2040	153	54	so	so	ADV
ijassa-2040	153	55	to	to	PART
ijassa-2040	153	56	speak	speak	VERB
ijassa-2040	153	57	,	,	PUNCT
ijassa-2040	153	58	a	a	DET
ijassa-2040	153	59	singular	singular	ADJ
ijassa-2040	153	60	section	section	NOUN
ijassa-2040	153	61	,	,	PUNCT
ijassa-2040	153	62	containing	contain	VERB
ijassa-2040	153	63	not	not	PART
ijassa-2040	153	64	all	all	PRON
ijassa-2040	153	65	more	more	ADV
ijassa-2040	153	66	regular	regular	ADJ
ijassa-2040	153	67	sequences	sequence	NOUN
ijassa-2040	153	68	,	,	PUNCT
ijassa-2040	153	69	but	but	CCONJ
ijassa-2040	153	70	all	all	PRON
ijassa-2040	153	71	more	more	ADV
ijassa-2040	153	72	singular	singular	ADJ
ijassa-2040	153	73	,	,	PUNCT
ijassa-2040	153	74	and	and	CCONJ
ijassa-2040	153	75	so	so	ADV
ijassa-2040	153	76	on	on	ADV
ijassa-2040	153	77	.	.	PUNCT
ijassa-2040	154	1	the	the	DET
ijassa-2040	154	2	most	most	ADV
ijassa-2040	154	3	important	important	ADJ
ijassa-2040	154	4	thing	thing	NOUN
ijassa-2040	154	5	here	here	ADV
ijassa-2040	154	6	is	be	AUX
ijassa-2040	154	7	that	that	SCONJ
ijassa-2040	154	8	this	this	DET
ijassa-2040	154	9	set	set	NOUN
ijassa-2040	154	10	contains	contain	VERB
ijassa-2040	154	11	not	not	PART
ijassa-2040	154	12	only	only	ADV
ijassa-2040	154	13	all	all	DET
ijassa-2040	154	14	functions	function	NOUN
ijassa-2040	154	15	,	,	PUNCT
ijassa-2040	154	16	but	but	CCONJ
ijassa-2040	154	17	also	also	ADV
ijassa-2040	154	18	all	all	DET
ijassa-2040	154	19	functional	functional	ADJ
ijassa-2040	154	20	spaces	space	NOUN
ijassa-2040	154	21	,	,	PUNCT
ijassa-2040	154	22	which	which	PRON
ijassa-2040	154	23	,	,	PUNCT
ijassa-2040	154	24	along	along	ADP
ijassa-2040	154	25	with	with	ADP
ijassa-2040	154	26	each	each	PRON
ijassa-2040	154	27	of	of	ADP
ijassa-2040	154	28	their	their	PRON
ijassa-2040	154	29	elements	element	NOUN
ijassa-2040	154	30	,	,	PUNCT
ijassa-2040	154	31	also	also	ADV
ijassa-2040	154	32	contain	contain	VERB
ijassa-2040	154	33	increasingly	increasingly	ADV
ijassa-2040	154	34	smooth	smooth	ADJ
ijassa-2040	154	35	elements	element	NOUN
ijassa-2040	154	36	.	.	PUNCT
ijassa-2040	155	1	and	and	CCONJ
ijassa-2040	155	2	now	now	ADV
ijassa-2040	155	3	we	we	PRON
ijassa-2040	155	4	can	can	AUX
ijassa-2040	155	5	consider	consider	VERB
ijassa-2040	155	6	the	the	DET
ijassa-2040	155	7	question	question	NOUN
ijassa-2040	155	8	of	of	ADP
ijassa-2040	155	9	solving	solve	VERB
ijassa-2040	155	10	the	the	DET
ijassa-2040	155	11	differential	differential	ADJ
ijassa-2040	155	12	equation	equation	NOUN
ijassa-2040	155	13	in	in	ADP
ijassa-2040	155	14	a	a	DET
ijassa-2040	155	15	class	class	NOUN
ijassa-2040	155	16	of	of	ADP
ijassa-2040	155	17	function	function	NOUN
ijassa-2040	155	18	spaces	space	NOUN
ijassa-2040	155	19	,	,	PUNCT
ijassa-2040	155	20	if	if	SCONJ
ijassa-2040	155	21	the	the	DET
ijassa-2040	155	22	given	give	VERB
ijassa-2040	155	23	right	right	ADJ
ijassa-2040	155	24	-	-	PUNCT
ijassa-2040	155	25	hand	hand	NOUN
ijassa-2040	155	26	side	side	NOUN
ijassa-2040	155	27	is	be	AUX
ijassa-2040	155	28	a	a	DET
ijassa-2040	155	29	space	space	NOUN
ijassa-2040	155	30	as	as	SCONJ
ijassa-2040	155	31	it	it	PRON
ijassa-2040	155	32	was	be	AUX
ijassa-2040	155	33	in	in	ADP
ijassa-2040	155	34	statement	statement	NOUN
ijassa-2040	155	35	5	5	NUM
ijassa-2040	155	36	.	.	NOUN
ijassa-2040	156	1	3	3	NUM
ijassa-2040	156	2	.	.	X
ijassa-2040	157	1	on	on	ADP
ijassa-2040	157	2	the	the	DET
ijassa-2040	157	3	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	157	4	of	of	ADP
ijassa-2040	157	5	differential	differential	ADJ
ijassa-2040	157	6	operators	operator	NOUN
ijassa-2040	157	7	on	on	ADP
ijassa-2040	157	8	the	the	DET
ijassa-2040	157	9	torus	torus	NOUN
ijassa-2040	157	10	as	as	SCONJ
ijassa-2040	157	11	we	we	PRON
ijassa-2040	157	12	noted	note	VERB
ijassa-2040	157	13	above	above	ADV
ijassa-2040	157	14	,	,	PUNCT
ijassa-2040	157	15	according	accord	VERB
ijassa-2040	157	16	to	to	ADP
ijassa-2040	157	17	lax	lax	PROPN
ijassa-2040	157	18	’s	’s	PART
ijassa-2040	157	19	statement	statement	NOUN
ijassa-2040	157	20	(	(	PUNCT
ijassa-2040	157	21	[	[	X
ijassa-2040	157	22	4	4	NUM
ijassa-2040	157	23	]	]	NUM
ijassa-2040	157	24	)	)	PUNCT
ijassa-2040	157	25	,	,	PUNCT
ijassa-2040	157	26	in	in	ADP
ijassa-2040	157	27	the	the	DET
ijassa-2040	157	28	periodic	periodic	ADJ
ijassa-2040	157	29	case	case	NOUN
ijassa-2040	157	30	it	it	PRON
ijassa-2040	157	31	is	be	AUX
ijassa-2040	157	32	possible	possible	ADJ
ijassa-2040	157	33	to	to	PART
ijassa-2040	157	34	construct	construct	VERB
ijassa-2040	157	35	a	a	DET
ijassa-2040	157	36	more	more	ADV
ijassa-2040	157	37	beautiful	beautiful	ADJ
ijassa-2040	157	38	theory	theory	NOUN
ijassa-2040	157	39	.	.	PUNCT
ijassa-2040	158	1	however	however	ADV
ijassa-2040	158	2	,	,	PUNCT
ijassa-2040	158	3	the	the	DET
ijassa-2040	158	4	periodic	periodic	ADJ
ijassa-2040	158	5	case	case	NOUN
ijassa-2040	158	6	is	be	AUX
ijassa-2040	158	7	not	not	PART
ijassa-2040	158	8	always	always	ADV
ijassa-2040	158	9	simpler	simple	ADJ
ijassa-2040	158	10	that	that	SCONJ
ijassa-2040	158	11	the	the	DET
ijassa-2040	158	12	non	non	ADJ
ijassa-2040	158	13	-	-	ADJ
ijassa-2040	158	14	periodic	periodic	ADJ
ijassa-2040	158	15	one	one	NUM
ijassa-2040	158	16	.	.	PUNCT
ijassa-2040	159	1	in	in	ADP
ijassa-2040	159	2	this	this	DET
ijassa-2040	159	3	section	section	NOUN
ijassa-2040	159	4	,	,	PUNCT
ijassa-2040	159	5	we	we	PRON
ijassa-2040	159	6	characterize	characterize	VERB
ijassa-2040	159	7	homogeneous	homogeneous	ADJ
ijassa-2040	159	8	differential	differential	NOUN
ijassa-2040	159	9	operators	operator	NOUN
ijassa-2040	159	10	with	with	ADP
ijassa-2040	159	11	constant	constant	ADJ
ijassa-2040	159	12	coefficients	coefficient	NOUN
ijassa-2040	159	13	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	159	14	in	in	ADP
ijassa-2040	159	15	the	the	DET
ijassa-2040	159	16	space	space	NOUN
ijassa-2040	159	17	of	of	ADP
ijassa-2040	159	18	periodic	periodic	ADJ
ijassa-2040	159	19	functions	function	NOUN
ijassa-2040	159	20	on	on	ADP
ijassa-2040	159	21	the	the	DET
ijassa-2040	159	22	plane	plane	NOUN
ijassa-2040	159	23	.	.	PUNCT
ijassa-2040	160	1	this	this	PRON
ijassa-2040	160	2	is	be	AUX
ijassa-2040	160	3	one	one	NUM
ijassa-2040	160	4	of	of	ADP
ijassa-2040	160	5	the	the	DET
ijassa-2040	160	6	cases	case	NOUN
ijassa-2040	160	7	when	when	SCONJ
ijassa-2040	160	8	the	the	DET
ijassa-2040	160	9	lax	lax	PROPN
ijassa-2040	160	10	’s	’s	PART
ijassa-2040	160	11	statement	statement	NOUN
ijassa-2040	160	12	is	be	AUX
ijassa-2040	160	13	quite	quite	ADV
ijassa-2040	160	14	controversial	controversial	ADJ
ijassa-2040	160	15	.	.	PUNCT
ijassa-2040	161	1	let	let	VERB
ijassa-2040	161	2	hm	hm	INTJ
ijassa-2040	161	3	,	,	PUNCT
ijassa-2040	161	4	m	m	PROPN
ijassa-2040	161	5	≥	≥	NOUN
ijassa-2040	161	6	0	0	NUM
ijassa-2040	161	7	,	,	PUNCT
ijassa-2040	161	8	be	be	AUX
ijassa-2040	161	9	the	the	DET
ijassa-2040	161	10	space	space	NOUN
ijassa-2040	161	11	of	of	ADP
ijassa-2040	161	12	complex	complex	ADJ
ijassa-2040	161	13	functions	function	NOUN
ijassa-2040	161	14	on	on	ADP
ijassa-2040	161	15	the	the	DET
ijassa-2040	161	16	plane	plane	NOUN
ijassa-2040	161	17	,	,	PUNCT
ijassa-2040	161	18	2π	2π	NOUN
ijassa-2040	161	19	-	-	NOUN
ijassa-2040	161	20	periodic	periodic	ADJ
ijassa-2040	161	21	in	in	ADP
ijassa-2040	161	22	the	the	DET
ijassa-2040	161	23	both	both	DET
ijassa-2040	161	24	arguments	argument	NOUN
ijassa-2040	162	1	such	such	ADJ
ijassa-2040	162	2	that	that	SCONJ
ijassa-2040	162	3	∥u∥2	∥u∥2	NOUN
ijassa-2040	162	4	m	m	VERB
ijassa-2040	162	5	=	=	ADJ
ijassa-2040	162	6	∫	∫	PROPN
ijassa-2040	162	7	t	t	PROPN
ijassa-2040	162	8	2	2	NUM
ijassa-2040	162	9	u(x)(1−∆)mu(x)dx	u(x)(1−∆)mu(x)dx	NOUN
ijassa-2040	162	10	<	<	X
ijassa-2040	162	11	∞	∞	PROPN
ijassa-2040	162	12	,	,	PUNCT
ijassa-2040	162	13	∆	∆	X
ijassa-2040	162	14	=	=	SYM
ijassa-2040	162	15	∂2	∂2	ADJ
ijassa-2040	162	16	∂x2	∂x2	NOUN
ijassa-2040	162	17	1	1	NUM
ijassa-2040	162	18	+	+	NUM
ijassa-2040	162	19	∂2	∂2	NUM
ijassa-2040	162	20	∂x2	∂x2	NOUN
ijassa-2040	162	21	2	2	NUM
ijassa-2040	162	22	.	.	PUNCT
ijassa-2040	163	1	the	the	DET
ijassa-2040	163	2	space	space	NOUN
ijassa-2040	163	3	h−m	h−m	PROPN
ijassa-2040	163	4	is	be	AUX
ijassa-2040	163	5	defined	define	VERB
ijassa-2040	163	6	as	as	ADP
ijassa-2040	163	7	the	the	DET
ijassa-2040	163	8	space	space	NOUN
ijassa-2040	163	9	dual	dual	ADJ
ijassa-2040	163	10	to	to	ADP
ijassa-2040	163	11	hm	hm	INTJ
ijassa-2040	163	12	in	in	ADP
ijassa-2040	163	13	the	the	DET
ijassa-2040	163	14	h0	h0	NOUN
ijassa-2040	163	15	-	-	PUNCT
ijassa-2040	163	16	topology	topology	NOUN
ijassa-2040	163	17	.	.	PUNCT
ijassa-2040	164	1	in	in	ADP
ijassa-2040	164	2	section	section	NOUN
ijassa-2040	164	3	1	1	NUM
ijassa-2040	164	4	,	,	PUNCT
ijassa-2040	164	5	we	we	PRON
ijassa-2040	164	6	noted	note	VERB
ijassa-2040	164	7	that	that	SCONJ
ijassa-2040	164	8	the	the	DET
ijassa-2040	164	9	functions	function	NOUN
ijassa-2040	164	10	exp(inx	exp(inx	NOUN
ijassa-2040	164	11	)	)	PUNCT
ijassa-2040	164	12	,	,	PUNCT
ijassa-2040	164	13	nx	nx	X
ijassa-2040	164	14	=	=	SYM
ijassa-2040	164	15	n1x1	n1x1	PROPN
ijassa-2040	164	16	+	+	SYM
ijassa-2040	164	17	n2x2	n2x2	PROPN
ijassa-2040	164	18	,	,	PUNCT
ijassa-2040	164	19	copyright	copyright	NOUN
ijassa-2040	164	20	©	©	PROPN
ijassa-2040	164	21	2025	2025	NUM
ijassa-2040	164	22	assa	assa	NOUN
ijassa-2040	164	23	.	.	PUNCT
ijassa-2040	165	1	adv	adv	PROPN
ijassa-2040	165	2	syst	syst	PROPN
ijassa-2040	165	3	sci	sci	PROPN
ijassa-2040	165	4	appl	appl	PROPN
ijassa-2040	165	5	(	(	PUNCT
ijassa-2040	165	6	2025	2025	NUM
ijassa-2040	165	7	)	)	PUNCT
ijassa-2040	165	8	linear	linear	ADJ
ijassa-2040	165	9	differential	differential	ADJ
ijassa-2040	165	10	equations	equation	NOUN
ijassa-2040	165	11	on	on	ADP
ijassa-2040	165	12	the	the	DET
ijassa-2040	165	13	torus	torus	NOUN
ijassa-2040	165	14	...	...	PUNCT
ijassa-2040	165	15	19	19	NUM
ijassa-2040	165	16	are	be	AUX
ijassa-2040	165	17	an	an	DET
ijassa-2040	165	18	orthogonal	orthogonal	ADJ
ijassa-2040	165	19	basis	basis	NOUN
ijassa-2040	165	20	in	in	ADP
ijassa-2040	165	21	the	the	DET
ijassa-2040	165	22	hilbert	hilbert	NOUN
ijassa-2040	165	23	space	space	NOUN
ijassa-2040	166	1	hm	hm	INTJ
ijassa-2040	166	2	and	and	CCONJ
ijassa-2040	166	3	,	,	PUNCT
ijassa-2040	166	4	consequently	consequently	ADV
ijassa-2040	166	5	,	,	PUNCT
ijassa-2040	166	6	every	every	DET
ijassa-2040	166	7	function	function	NOUN
ijassa-2040	166	8	f	f	PROPN
ijassa-2040	166	9	∈	∈	PROPN
ijassa-2040	167	1	hm	hm	INTJ
ijassa-2040	167	2	is	be	AUX
ijassa-2040	167	3	presented	present	VERB
ijassa-2040	167	4	as	as	ADP
ijassa-2040	167	5	fourier	fouri	ADJ
ijassa-2040	167	6	series	series	NOUN
ijassa-2040	167	7	f	f	PROPN
ijassa-2040	168	1	=	=	PUNCT
ijassa-2040	168	2	∑	∑	PUNCT
ijassa-2040	168	3	n∈z⊕z	n∈z⊕z	ADV
ijassa-2040	168	4	fne	fne	INTJ
ijassa-2040	168	5	inx	inx	AUX
ijassa-2040	168	6	converging	converge	VERB
ijassa-2040	168	7	to	to	ADP
ijassa-2040	168	8	f	f	PROPN
ijassa-2040	168	9	in	in	ADP
ijassa-2040	168	10	the	the	DET
ijassa-2040	168	11	topology	topology	NOUN
ijassa-2040	168	12	of	of	ADP
ijassa-2040	168	13	hm	hm	INTJ
ijassa-2040	168	14	.	.	PUNCT
ijassa-2040	169	1	thus	thus	ADV
ijassa-2040	169	2	,	,	PUNCT
ijassa-2040	169	3	one	one	PRON
ijassa-2040	169	4	can	can	AUX
ijassa-2040	169	5	consider	consider	VERB
ijassa-2040	169	6	hm	hm	INTJ
ijassa-2040	169	7	,	,	PUNCT
ijassa-2040	169	8	m	m	PROPN
ijassa-2040	169	9	∈	∈	NOUN
ijassa-2040	169	10	r	r	NOUN
ijassa-2040	169	11	,	,	PUNCT
ijassa-2040	169	12	as	as	ADP
ijassa-2040	169	13	the	the	DET
ijassa-2040	169	14	space	space	NOUN
ijassa-2040	169	15	of	of	ADP
ijassa-2040	169	16	formal	formal	ADJ
ijassa-2040	169	17	fourier	fourier	NOUN
ijassa-2040	169	18	series	series	NOUN
ijassa-2040	169	19	with	with	ADP
ijassa-2040	169	20	the	the	DET
ijassa-2040	169	21	finite	finite	PROPN
ijassa-2040	169	22	norm	norm	PROPN
ijassa-2040	169	23	∥f∥2	∥f∥2	PROPN
ijassa-2040	169	24	m	m	PROPN
ijassa-2040	169	25	=	=	SYM
ijassa-2040	169	26	∑	∑	PROPN
ijassa-2040	169	27	n	n	PROPN
ijassa-2040	169	28	(	(	PUNCT
ijassa-2040	169	29	1	1	NUM
ijassa-2040	169	30	+	+	CCONJ
ijassa-2040	169	31	n	n	CCONJ
ijassa-2040	169	32	·	·	PUNCT
ijassa-2040	169	33	n)m	n)m	PUNCT
ijassa-2040	169	34	·	·	PUNCT
ijassa-2040	170	1	|fn|2	|fn|2	ADP
ijassa-2040	170	2	(	(	PUNCT
ijassa-2040	170	3	see	see	VERB
ijassa-2040	170	4	section	section	NOUN
ijassa-2040	170	5	1	1	NUM
ijassa-2040	170	6	)	)	PUNCT
ijassa-2040	170	7	.	.	PUNCT
ijassa-2040	171	1	these	these	PRON
ijassa-2040	171	2	are	be	AUX
ijassa-2040	171	3	the	the	DET
ijassa-2040	171	4	famous	famous	ADJ
ijassa-2040	171	5	sobolev	sobolev	NOUN
ijassa-2040	171	6	spaces	space	VERB
ijassa-2040	171	7	.	.	PUNCT
ijassa-2040	172	1	let	let	VERB
ijassa-2040	172	2	p	p	NOUN
ijassa-2040	172	3	(	(	PUNCT
ijassa-2040	172	4	x1	x1	PROPN
ijassa-2040	172	5	,	,	PUNCT
ijassa-2040	172	6	x2	x2	PROPN
ijassa-2040	172	7	)	)	PUNCT
ijassa-2040	172	8	be	be	VERB
ijassa-2040	172	9	a	a	DET
ijassa-2040	172	10	homogeneous	homogeneous	ADJ
ijassa-2040	172	11	polynomial	polynomial	NOUN
ijassa-2040	172	12	of	of	ADP
ijassa-2040	172	13	the	the	DET
ijassa-2040	172	14	degree	degree	NOUN
ijassa-2040	172	15	p	p	X
ijassa-2040	172	16	≥	≥	NOUN
ijassa-2040	172	17	2	2	NUM
ijassa-2040	172	18	with	with	ADP
ijassa-2040	172	19	constant	constant	ADJ
ijassa-2040	172	20	coefficients	coefficient	NOUN
ijassa-2040	172	21	.	.	PUNCT
ijassa-2040	173	1	consider	consider	VERB
ijassa-2040	173	2	the	the	DET
ijassa-2040	173	3	differential	differential	ADJ
ijassa-2040	173	4	operator	operator	NOUN
ijassa-2040	173	5	l	l	NOUN
ijassa-2040	173	6	:	:	PUNCT
ijassa-2040	173	7	hm	hm	INTJ
ijassa-2040	173	8	→	→	PUNCT
ijassa-2040	173	9	hm−p	hm−p	NOUN
ijassa-2040	173	10	acting	act	VERB
ijassa-2040	173	11	according	accord	VERB
ijassa-2040	173	12	to	to	ADP
ijassa-2040	173	13	the	the	DET
ijassa-2040	173	14	rule	rule	NOUN
ijassa-2040	173	15	lu	lu	NOUN
ijassa-2040	173	16	=	=	NOUN
ijassa-2040	173	17	p	p	X
ijassa-2040	173	18	(	(	PUNCT
ijassa-2040	173	19	−	−	PROPN
ijassa-2040	173	20	i	i	PRON
ijassa-2040	173	21	∂	∂	ADV
ijassa-2040	173	22	∂x1	∂x1	NOUN
ijassa-2040	173	23	,	,	PUNCT
ijassa-2040	173	24	−i	−i	PROPN
ijassa-2040	173	25	∂	∂	ADJ
ijassa-2040	173	26	∂x2	∂x2	NOUN
ijassa-2040	173	27	)	)	PUNCT
ijassa-2040	174	1	u.	u.	VERB
ijassa-2040	174	2	according	accord	VERB
ijassa-2040	174	3	to	to	ADP
ijassa-2040	174	4	[	[	X
ijassa-2040	174	5	1	1	NUM
ijassa-2040	174	6	]	]	PUNCT
ijassa-2040	174	7	,	,	PUNCT
ijassa-2040	174	8	define	define	VERB
ijassa-2040	174	9	the	the	DET
ijassa-2040	174	10	spaces	space	NOUN
ijassa-2040	174	11	of	of	ADP
ijassa-2040	174	12	infinitely	infinitely	ADV
ijassa-2040	174	13	differentiable	differentiable	ADJ
ijassa-2040	174	14	and	and	CCONJ
ijassa-2040	174	15	generalized	generalized	ADJ
ijassa-2040	174	16	functions	function	NOUN
ijassa-2040	174	17	:	:	PUNCT
ijassa-2040	174	18	h∞	h∞	X
ijassa-2040	174	19	=	=	SYM
ijassa-2040	174	20	∩	∩	NOUN
ijassa-2040	174	21	m	m	VERB
ijassa-2040	174	22	hm	hm	INTJ
ijassa-2040	174	23	,	,	PUNCT
ijassa-2040	174	24	h−∞	h−∞	NOUN
ijassa-2040	174	25	=	=	SYM
ijassa-2040	174	26	∪	∪	ADP
ijassa-2040	174	27	m	m	VERB
ijassa-2040	174	28	hm	hm	INTJ
ijassa-2040	174	29	.	.	PUNCT
ijassa-2040	174	30	definition	definition	NOUN
ijassa-2040	174	31	3.1	3.1	NUM
ijassa-2040	174	32	:	:	PUNCT
ijassa-2040	174	33	we	we	PRON
ijassa-2040	174	34	shall	shall	AUX
ijassa-2040	174	35	call	call	VERB
ijassa-2040	174	36	the	the	DET
ijassa-2040	174	37	operator	operator	NOUN
ijassa-2040	174	38	l	l	PROPN
ijassa-2040	174	39	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	174	40	,	,	PUNCT
ijassa-2040	174	41	if	if	SCONJ
ijassa-2040	174	42	for	for	ADP
ijassa-2040	174	43	any	any	DET
ijassa-2040	174	44	u	u	PROPN
ijassa-2040	174	45	∈	∈	NOUN
ijassa-2040	174	46	h−∞	h−∞	NOUN
ijassa-2040	174	47	the	the	DET
ijassa-2040	174	48	inclusion	inclusion	NOUN
ijassa-2040	174	49	lu	lu	PROPN
ijassa-2040	174	50	∈	∈	PROPN
ijassa-2040	174	51	h∞	h∞	PROPN
ijassa-2040	174	52	implies	imply	VERB
ijassa-2040	174	53	u	u	PROPN
ijassa-2040	174	54	∈	∈	PROPN
ijassa-2040	175	1	h∞.	h∞.	X
ijassa-2040	175	2	lemma	lemma	PROPN
ijassa-2040	175	3	3.1	3.1	NUM
ijassa-2040	175	4	:	:	PUNCT
ijassa-2040	175	5	the	the	DET
ijassa-2040	175	6	operator	operator	NOUN
ijassa-2040	175	7	l	l	NOUN
ijassa-2040	175	8	is	be	AUX
ijassa-2040	175	9	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	175	10	if	if	SCONJ
ijassa-2040	176	1	and	and	CCONJ
ijassa-2040	176	2	only	only	ADV
ijassa-2040	176	3	if	if	SCONJ
ijassa-2040	176	4	the	the	DET
ijassa-2040	176	5	exist	exist	NOUN
ijassa-2040	176	6	constants	constant	NOUN
ijassa-2040	176	7	c	c	NOUN
ijassa-2040	176	8	>	>	X
ijassa-2040	176	9	0	0	PUNCT
ijassa-2040	177	1	and	and	CCONJ
ijassa-2040	177	2	k1	k1	NOUN
ijassa-2040	177	3	such	such	ADJ
ijassa-2040	177	4	that	that	SCONJ
ijassa-2040	177	5	|p	|p	PROPN
ijassa-2040	177	6	(	(	PUNCT
ijassa-2040	177	7	n)|	n)|	X
ijassa-2040	177	8	>	>	X
ijassa-2040	177	9	c(n2)k1	c(n2)k1	PROPN
ijassa-2040	177	10	∀n	∀n	X
ijassa-2040	177	11	.	.	PUNCT
ijassa-2040	178	1	(	(	PUNCT
ijassa-2040	178	2	3.5	3.5	NUM
ijassa-2040	178	3	)	)	PUNCT
ijassa-2040	178	4	proof	proof	NOUN
ijassa-2040	178	5	assume	assume	VERB
ijassa-2040	178	6	that	that	SCONJ
ijassa-2040	178	7	there	there	PRON
ijassa-2040	178	8	exist	exist	VERB
ijassa-2040	178	9	constants	constant	NOUN
ijassa-2040	178	10	c	c	NOUN
ijassa-2040	178	11	>	>	X
ijassa-2040	178	12	0	0	PUNCT
ijassa-2040	178	13	and	and	CCONJ
ijassa-2040	178	14	k1	k1	NOUN
ijassa-2040	178	15	such	such	ADJ
ijassa-2040	178	16	that	that	SCONJ
ijassa-2040	178	17	(	(	PUNCT
ijassa-2040	178	18	5	5	NUM
ijassa-2040	178	19	)	)	PUNCT
ijassa-2040	178	20	holds	hold	VERB
ijassa-2040	178	21	true	true	ADJ
ijassa-2040	178	22	and	and	CCONJ
ijassa-2040	178	23	f	f	X
ijassa-2040	179	1	=	=	SYM
ijassa-2040	179	2	∑	∑	PROPN
ijassa-2040	179	3	n	n	PRON
ijassa-2040	179	4	fne	fne	INTJ
ijassa-2040	180	1	inx	inx	PROPN
ijassa-2040	180	2	∈	∈	PROPN
ijassa-2040	181	1	h∞.	h∞.	NOUN
ijassa-2040	181	2	then	then	ADV
ijassa-2040	181	3	fn	fn	NOUN
ijassa-2040	181	4	/	/	SYM
ijassa-2040	181	5	p	p	X
ijassa-2040	181	6	(	(	PUNCT
ijassa-2040	181	7	n	n	CCONJ
ijassa-2040	181	8	)	)	PUNCT
ijassa-2040	181	9	decreases	decrease	VERB
ijassa-2040	181	10	faster	fast	ADV
ijassa-2040	181	11	than	than	ADP
ijassa-2040	181	12	any	any	DET
ijassa-2040	181	13	power	power	NOUN
ijassa-2040	181	14	of	of	ADP
ijassa-2040	181	15	n.	n.	NOUN
ijassa-2040	181	16	if	if	SCONJ
ijassa-2040	181	17	there	there	PRON
ijassa-2040	181	18	exists	exist	VERB
ijassa-2040	181	19	a	a	DET
ijassa-2040	181	20	sequence	sequence	NOUN
ijassa-2040	181	21	of	of	ADP
ijassa-2040	181	22	pairs	pair	NOUN
ijassa-2040	181	23	nj	nj	VERB
ijassa-2040	181	24	such	such	ADJ
ijassa-2040	181	25	that	that	SCONJ
ijassa-2040	181	26	|p	|p	PROPN
ijassa-2040	181	27	(	(	PUNCT
ijassa-2040	181	28	nj)|	nj)|	ADJ
ijassa-2040	181	29	→	→	SYM
ijassa-2040	181	30	0	0	NUM
ijassa-2040	181	31	for	for	ADP
ijassa-2040	181	32	j	j	PROPN
ijassa-2040	181	33	→	→	SYM
ijassa-2040	181	34	∞	∞	PROPN
ijassa-2040	181	35	faster	fast	ADV
ijassa-2040	181	36	than	than	ADP
ijassa-2040	181	37	any	any	DET
ijassa-2040	181	38	power	power	NOUN
ijassa-2040	181	39	of	of	ADP
ijassa-2040	181	40	|n|	|n|	NOUN
ijassa-2040	181	41	,	,	PUNCT
ijassa-2040	181	42	then	then	ADV
ijassa-2040	181	43	,	,	PUNCT
ijassa-2040	181	44	for	for	ADP
ijassa-2040	181	45	example	example	NOUN
ijassa-2040	181	46	,	,	PUNCT
ijassa-2040	181	47	for	for	ADP
ijassa-2040	181	48	the	the	DET
ijassa-2040	181	49	functions	function	NOUN
ijassa-2040	182	1	f	f	X
ijassa-2040	183	1	=	=	PUNCT
ijassa-2040	183	2	∑	∑	PUNCT
ijassa-2040	183	3	j	j	PROPN
ijassa-2040	183	4	p	p	X
ijassa-2040	183	5	(	(	PUNCT
ijassa-2040	183	6	nj)ein	nj)ein	ADV
ijassa-2040	183	7	jx	jx	PROPN
ijassa-2040	183	8	∈	∈	PROPN
ijassa-2040	183	9	h∞	h∞	PROPN
ijassa-2040	183	10	,	,	PUNCT
ijassa-2040	183	11	but	but	CCONJ
ijassa-2040	183	12	solution	solution	NOUN
ijassa-2040	183	13	∑	∑	PUNCT
ijassa-2040	183	14	j	j	PROPN
ijassa-2040	183	15	e	e	PROPN
ijassa-2040	183	16	injx	injx	VERB
ijassa-2040	183	17	∈	∈	PROPN
ijassa-2040	183	18	h−2	h−2	PROPN
ijassa-2040	183	19	,	,	PUNCT
ijassa-2040	183	20	and	and	CCONJ
ijassa-2040	183	21	there	there	PRON
ijassa-2040	183	22	is	be	VERB
ijassa-2040	183	23	no	no	DET
ijassa-2040	183	24	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	183	25	.	.	PUNCT
ijassa-2040	184	1	recall	recall	VERB
ijassa-2040	184	2	that	that	SCONJ
ijassa-2040	184	3	in	in	ADP
ijassa-2040	184	4	the	the	DET
ijassa-2040	184	5	case	case	NOUN
ijassa-2040	184	6	of	of	ADP
ijassa-2040	184	7	two	two	NUM
ijassa-2040	184	8	variables	variable	NOUN
ijassa-2040	184	9	one	one	NUM
ijassa-2040	184	10	of	of	ADP
ijassa-2040	184	11	necessary	necessary	ADJ
ijassa-2040	184	12	and	and	CCONJ
ijassa-2040	184	13	sufficient	sufficient	ADJ
ijassa-2040	184	14	conditions	condition	NOUN
ijassa-2040	184	15	of	of	ADP
ijassa-2040	184	16	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	184	17	is	be	AUX
ijassa-2040	184	18	the	the	DET
ijassa-2040	184	19	following	follow	VERB
ijassa-2040	184	20	(	(	PUNCT
ijassa-2040	184	21	see	see	INTJ
ijassa-2040	184	22	,	,	PUNCT
ijassa-2040	184	23	e.g.	e.g.	ADV
ijassa-2040	184	24	,	,	PUNCT
ijassa-2040	184	25	[	[	X
ijassa-2040	184	26	12	12	NUM
ijassa-2040	184	27	]	]	PUNCT
ijassa-2040	184	28	):	):	PUNCT
ijassa-2040	184	29	there	there	PRON
ijassa-2040	184	30	exist	exist	VERB
ijassa-2040	184	31	constants	constant	NOUN
ijassa-2040	184	32	c	c	PROPN
ijassa-2040	184	33	and	and	CCONJ
ijassa-2040	184	34	c	c	PROPN
ijassa-2040	184	35	such	such	ADJ
ijassa-2040	184	36	that∣∣∣p	that∣∣∣p	PROPN
ijassa-2040	184	37	(	(	PUNCT
ijassa-2040	184	38	α)(ξ	α)(ξ	NUM
ijassa-2040	184	39	)	)	PUNCT
ijassa-2040	184	40	p	p	X
ijassa-2040	184	41	(	(	PUNCT
ijassa-2040	184	42	ξ	ξ	NOUN
ijassa-2040	184	43	)	)	PUNCT
ijassa-2040	184	44	∣∣∣	∣∣∣	ADJ
ijassa-2040	184	45	≤	≤	PROPN
ijassa-2040	184	46	c	c	NOUN
ijassa-2040	184	47	|ξ|−|α|c	|ξ|−|α|c	NOUN
ijassa-2040	184	48	for	for	ADP
ijassa-2040	184	49	every	every	DET
ijassa-2040	184	50	multi	multi	ADJ
ijassa-2040	184	51	-	-	NOUN
ijassa-2040	184	52	index	index	ADJ
ijassa-2040	184	53	α	α	NOUN
ijassa-2040	184	54	and	and	CCONJ
ijassa-2040	184	55	any	any	DET
ijassa-2040	184	56	ξ	ξ	PROPN
ijassa-2040	184	57	∈	∈	PROPN
ijassa-2040	184	58	r2	r2	NOUN
ijassa-2040	184	59	large	large	ADJ
ijassa-2040	184	60	enough	enough	ADV
ijassa-2040	184	61	.	.	PUNCT
ijassa-2040	185	1	for	for	ADP
ijassa-2040	185	2	α	α	NOUN
ijassa-2040	185	3	=	=	SYM
ijassa-2040	185	4	p	p	PROPN
ijassa-2040	185	5	,	,	PUNCT
ijassa-2040	185	6	the	the	DET
ijassa-2040	185	7	latter	latter	ADJ
ijassa-2040	185	8	inequality	inequality	NOUN
ijassa-2040	185	9	gives	give	VERB
ijassa-2040	185	10	the	the	DET
ijassa-2040	185	11	inequality	inequality	NOUN
ijassa-2040	185	12	|p	|p	PROPN
ijassa-2040	185	13	(	(	PUNCT
ijassa-2040	185	14	ξ)|	ξ)|	INTJ
ijassa-2040	185	15	≥	≥	NUM
ijassa-2040	185	16	c|ξ|pc	c|ξ|pc	PROPN
ijassa-2040	185	17	,	,	PUNCT
ijassa-2040	185	18	which	which	PRON
ijassa-2040	185	19	coincides	coincide	VERB
ijassa-2040	185	20	with	with	ADP
ijassa-2040	185	21	inequality	inequality	NOUN
ijassa-2040	185	22	(	(	PUNCT
ijassa-2040	185	23	3.5	3.5	NUM
ijassa-2040	185	24	)	)	PUNCT
ijassa-2040	185	25	on	on	ADP
ijassa-2040	185	26	the	the	DET
ijassa-2040	185	27	integer	integer	PROPN
ijassa-2040	185	28	lattice	lattice	PROPN
ijassa-2040	185	29	.	.	PUNCT
ijassa-2040	186	1	this	this	PRON
ijassa-2040	186	2	means	mean	VERB
ijassa-2040	186	3	that	that	SCONJ
ijassa-2040	186	4	every	every	DET
ijassa-2040	186	5	operator	operator	NOUN
ijassa-2040	186	6	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	186	7	on	on	ADP
ijassa-2040	186	8	the	the	DET
ijassa-2040	186	9	plane	plane	NOUN
ijassa-2040	186	10	is	be	AUX
ijassa-2040	186	11	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	186	12	on	on	ADP
ijassa-2040	186	13	the	the	DET
ijassa-2040	186	14	torus	torus	NOUN
ijassa-2040	186	15	,	,	PUNCT
ijassa-2040	186	16	but	but	CCONJ
ijassa-2040	186	17	not	not	PART
ijassa-2040	186	18	vice	vice	ADV
ijassa-2040	186	19	versa	versa	ADV
ijassa-2040	186	20	.	.	PUNCT
ijassa-2040	187	1	proposition	proposition	NOUN
ijassa-2040	187	2	3.1	3.1	NUM
ijassa-2040	187	3	:	:	PUNCT
ijassa-2040	187	4	the	the	DET
ijassa-2040	187	5	operator	operator	NOUN
ijassa-2040	187	6	l	l	NOUN
ijassa-2040	187	7	is	be	AUX
ijassa-2040	187	8	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	187	9	if	if	SCONJ
ijassa-2040	187	10	and	and	CCONJ
ijassa-2040	187	11	only	only	ADV
ijassa-2040	187	12	if	if	SCONJ
ijassa-2040	187	13	for	for	ADP
ijassa-2040	187	14	every	every	DET
ijassa-2040	187	15	real	real	ADJ
ijassa-2040	187	16	root	root	NOUN
ijassa-2040	187	17	α	α	NOUN
ijassa-2040	187	18	of	of	ADP
ijassa-2040	187	19	the	the	DET
ijassa-2040	187	20	polynomial	polynomial	ADJ
ijassa-2040	187	21	p	p	NOUN
ijassa-2040	187	22	(	(	PUNCT
ijassa-2040	187	23	x	x	X
ijassa-2040	187	24	,	,	PUNCT
ijassa-2040	187	25	1	1	X
ijassa-2040	187	26	)	)	PUNCT
ijassa-2040	187	27	copyright	copyright	NOUN
ijassa-2040	187	28	©	©	PROPN
ijassa-2040	187	29	2025	2025	NUM
ijassa-2040	187	30	assa	assa	NOUN
ijassa-2040	187	31	.	.	PUNCT
ijassa-2040	188	1	adv	adv	PROPN
ijassa-2040	188	2	syst	syst	PROPN
ijassa-2040	188	3	sci	sci	PROPN
ijassa-2040	188	4	appl	appl	PROPN
ijassa-2040	188	5	(	(	PUNCT
ijassa-2040	188	6	2025	2025	NUM
ijassa-2040	188	7	)	)	PUNCT
ijassa-2040	188	8	20	20	NUM
ijassa-2040	188	9	v.	v.	ADP
ijassa-2040	188	10	p.	p.	NOUN
ijassa-2040	188	11	burskii	burskii	NOUN
ijassa-2040	188	12	there	there	PRON
ijassa-2040	188	13	exist	exist	VERB
ijassa-2040	188	14	constants	constant	NOUN
ijassa-2040	188	15	c	c	NOUN
ijassa-2040	188	16	>	>	X
ijassa-2040	188	17	0	0	PUNCT
ijassa-2040	189	1	and	and	CCONJ
ijassa-2040	189	2	k	k	ADP
ijassa-2040	189	3	such	such	ADJ
ijassa-2040	189	4	that∣∣∣α−	that∣∣∣α−	NOUN
ijassa-2040	189	5	p	p	NOUN
ijassa-2040	189	6	q	q	NOUN
ijassa-2040	189	7	∣∣∣	∣∣∣	NOUN
ijassa-2040	189	8	>	>	X
ijassa-2040	189	9	c	c	PROPN
ijassa-2040	189	10	qk	qk	NOUN
ijassa-2040	189	11	(	(	PUNCT
ijassa-2040	189	12	3.6	3.6	NUM
ijassa-2040	189	13	)	)	PUNCT
ijassa-2040	189	14	for	for	ADP
ijassa-2040	189	15	every	every	DET
ijassa-2040	189	16	rational	rational	ADJ
ijassa-2040	189	17	p	p	X
ijassa-2040	189	18	/	/	SYM
ijassa-2040	189	19	q	q	NOUN
ijassa-2040	189	20	sufficiently	sufficiently	ADV
ijassa-2040	189	21	close	close	ADJ
ijassa-2040	189	22	to	to	ADP
ijassa-2040	189	23	α	α	PRON
ijassa-2040	189	24	.	.	PUNCT
ijassa-2040	190	1	proof	proof	NOUN
ijassa-2040	190	2	let	let	VERB
ijassa-2040	190	3	u	u	PRON
ijassa-2040	190	4	∈	∈	NOUN
ijassa-2040	190	5	h−∞	h−∞	NOUN
ijassa-2040	190	6	,	,	PUNCT
ijassa-2040	190	7	then	then	ADV
ijassa-2040	190	8	u	u	PROPN
ijassa-2040	190	9	∈	∈	PROPN
ijassa-2040	190	10	hm	hm	INTJ
ijassa-2040	190	11	with	with	ADP
ijassa-2040	190	12	some	some	DET
ijassa-2040	190	13	m.	m.	NOUN
ijassa-2040	190	14	therefore	therefore	ADV
ijassa-2040	190	15	,	,	PUNCT
ijassa-2040	190	16	u	u	PROPN
ijassa-2040	190	17	=	=	SYM
ijassa-2040	190	18	∑	∑	PROPN
ijassa-2040	190	19	n	n	CCONJ
ijassa-2040	190	20	une	une	PROPN
ijassa-2040	190	21	inx	inx	PROPN
ijassa-2040	190	22	,	,	PUNCT
ijassa-2040	190	23	lu	lu	PROPN
ijassa-2040	190	24	=	=	SYM
ijassa-2040	190	25	∑	∑	PROPN
ijassa-2040	190	26	n	n	DET
ijassa-2040	190	27	unp	unp	PROPN
ijassa-2040	190	28	(	(	PUNCT
ijassa-2040	190	29	n)einx	n)einx	PROPN
ijassa-2040	190	30	.	.	PUNCT
ijassa-2040	191	1	let	let	VERB
ijassa-2040	191	2	f	f	PROPN
ijassa-2040	191	3	=	=	SYM
ijassa-2040	191	4	∑	∑	PROPN
ijassa-2040	191	5	n	n	PRON
ijassa-2040	191	6	fne	fne	NOUN
ijassa-2040	191	7	inx	inx	VERB
ijassa-2040	191	8	.	.	PUNCT
ijassa-2040	192	1	it	it	PRON
ijassa-2040	192	2	is	be	AUX
ijassa-2040	192	3	clear	clear	ADJ
ijassa-2040	192	4	that	that	SCONJ
ijassa-2040	192	5	f	f	PROPN
ijassa-2040	192	6	∈	∈	PROPN
ijassa-2040	192	7	h∞	h∞	X
ijassa-2040	192	8	if	if	SCONJ
ijassa-2040	192	9	and	and	CCONJ
ijassa-2040	192	10	only	only	ADV
ijassa-2040	192	11	if	if	SCONJ
ijassa-2040	192	12	|fn|	|fn|	PROPN
ijassa-2040	192	13	→	→	SYM
ijassa-2040	192	14	0	0	NUM
ijassa-2040	192	15	for	for	ADP
ijassa-2040	192	16	n2	n2	ADJ
ijassa-2040	192	17	→	→	SYM
ijassa-2040	192	18	∞	∞	NOUN
ijassa-2040	192	19	faster	fast	ADV
ijassa-2040	192	20	than	than	ADP
ijassa-2040	192	21	any	any	DET
ijassa-2040	192	22	power	power	NOUN
ijassa-2040	192	23	of	of	ADP
ijassa-2040	192	24	n2	n2	ADJ
ijassa-2040	192	25	.	.	PUNCT
ijassa-2040	193	1	note	note	VERB
ijassa-2040	193	2	that	that	SCONJ
ijassa-2040	193	3	if	if	SCONJ
ijassa-2040	193	4	the	the	DET
ijassa-2040	193	5	operator	operator	NOUN
ijassa-2040	193	6	l	l	NOUN
ijassa-2040	193	7	is	be	AUX
ijassa-2040	193	8	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	193	9	,	,	PUNCT
ijassa-2040	193	10	then	then	ADV
ijassa-2040	193	11	the	the	DET
ijassa-2040	193	12	equation	equation	NOUN
ijassa-2040	193	13	p	p	X
ijassa-2040	193	14	(	(	PUNCT
ijassa-2040	193	15	n	n	CCONJ
ijassa-2040	193	16	)	)	PUNCT
ijassa-2040	193	17	=	=	SYM
ijassa-2040	193	18	0	0	PROPN
ijassa-2040	193	19	has	have	VERB
ijassa-2040	193	20	a	a	DET
ijassa-2040	193	21	unique	unique	ADJ
ijassa-2040	193	22	integer	integer	NOUN
ijassa-2040	193	23	solution	solution	NOUN
ijassa-2040	193	24	n	n	NOUN
ijassa-2040	193	25	=	=	SYM
ijassa-2040	193	26	0	0	X
ijassa-2040	193	27	.	.	PUNCT
ijassa-2040	194	1	indeed	indeed	ADV
ijassa-2040	194	2	,	,	PUNCT
ijassa-2040	194	3	assume	assume	VERB
ijassa-2040	194	4	that	that	SCONJ
ijassa-2040	194	5	ν	ν	NOUN
ijassa-2040	194	6	=	=	SYM
ijassa-2040	194	7	(	(	PUNCT
ijassa-2040	194	8	ν1	ν1	NOUN
ijassa-2040	194	9	,	,	PUNCT
ijassa-2040	194	10	ν2	ν2	NOUN
ijassa-2040	194	11	)	)	PUNCT
ijassa-2040	194	12	is	be	AUX
ijassa-2040	194	13	another	another	DET
ijassa-2040	194	14	solution	solution	NOUN
ijassa-2040	194	15	,	,	PUNCT
ijassa-2040	194	16	then	then	ADV
ijassa-2040	194	17	the	the	DET
ijassa-2040	194	18	function	function	NOUN
ijassa-2040	195	1	+	+	PROPN
ijassa-2040	195	2	∞∑	∞∑	ADJ
ijassa-2040	195	3	k=−∞	k=−∞	NOUN
ijassa-2040	195	4	eikνx	eikνx	VERB
ijassa-2040	195	5	∈	∈	PROPN
ijassa-2040	195	6	h−∞	h−∞	NOUN
ijassa-2040	195	7	belongs	belong	VERB
ijassa-2040	195	8	to	to	ADP
ijassa-2040	195	9	the	the	DET
ijassa-2040	195	10	kernel	kernel	NOUN
ijassa-2040	195	11	of	of	ADP
ijassa-2040	195	12	l	l	PROPN
ijassa-2040	195	13	,	,	PUNCT
ijassa-2040	195	14	which	which	PRON
ijassa-2040	195	15	contradicts	contradict	VERB
ijassa-2040	195	16	the	the	DET
ijassa-2040	195	17	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	195	18	.	.	PUNCT
ijassa-2040	196	1	therefore	therefore	ADV
ijassa-2040	196	2	,	,	PUNCT
ijassa-2040	196	3	the	the	DET
ijassa-2040	196	4	solution	solution	NOUN
ijassa-2040	196	5	of	of	ADP
ijassa-2040	196	6	the	the	DET
ijassa-2040	196	7	equation	equation	NOUN
ijassa-2040	196	8	lu	lu	NOUN
ijassa-2040	196	9	=	=	NOUN
ijassa-2040	196	10	f	f	PROPN
ijassa-2040	196	11	can	can	AUX
ijassa-2040	196	12	be	be	AUX
ijassa-2040	196	13	formally	formally	ADV
ijassa-2040	196	14	written	write	VERB
ijassa-2040	196	15	as	as	ADP
ijassa-2040	196	16	u	u	NOUN
ijassa-2040	196	17	=	=	PUNCT
ijassa-2040	196	18	∑	∑	PUNCT
ijassa-2040	196	19	n̸=0	n̸=0	PROPN
ijassa-2040	196	20	fn	fn	PROPN
ijassa-2040	196	21	p	p	X
ijassa-2040	196	22	(	(	PUNCT
ijassa-2040	196	23	n	n	CCONJ
ijassa-2040	196	24	)	)	PUNCT
ijassa-2040	196	25	einx	einx	NOUN
ijassa-2040	196	26	+	+	X
ijassa-2040	196	27	c0	c0	NOUN
ijassa-2040	196	28	.	.	PUNCT
ijassa-2040	197	1	the	the	DET
ijassa-2040	197	2	condition	condition	NOUN
ijassa-2040	197	3	f0	f0	PROPN
ijassa-2040	197	4	=	=	SYM
ijassa-2040	197	5	0	0	NUM
ijassa-2040	197	6	is	be	AUX
ijassa-2040	197	7	obviously	obviously	ADV
ijassa-2040	197	8	a	a	DET
ijassa-2040	197	9	condition	condition	NOUN
ijassa-2040	197	10	for	for	ADP
ijassa-2040	197	11	the	the	DET
ijassa-2040	197	12	solvability	solvability	NOUN
ijassa-2040	197	13	of	of	ADP
ijassa-2040	197	14	the	the	DET
ijassa-2040	197	15	equation	equation	NOUN
ijassa-2040	198	1	lu	lu	NOUN
ijassa-2040	198	2	=	=	SYM
ijassa-2040	198	3	f	f	PROPN
ijassa-2040	198	4	.	.	PUNCT
ijassa-2040	199	1	let	let	VERB
ijassa-2040	199	2	us	we	PRON
ijassa-2040	199	3	apply	apply	VERB
ijassa-2040	199	4	lemma	lemma	PROPN
ijassa-2040	199	5	3.1	3.1	NUM
ijassa-2040	199	6	.	.	PUNCT
ijassa-2040	200	1	let	let	VERB
ijassa-2040	200	2	α	α	PRON
ijassa-2040	200	3	be	be	AUX
ijassa-2040	200	4	a	a	DET
ijassa-2040	200	5	real	real	ADJ
ijassa-2040	200	6	root	root	NOUN
ijassa-2040	200	7	of	of	ADP
ijassa-2040	200	8	the	the	DET
ijassa-2040	200	9	polynomial	polynomial	ADJ
ijassa-2040	200	10	p	p	NOUN
ijassa-2040	200	11	(	(	PUNCT
ijassa-2040	200	12	x	x	X
ijassa-2040	200	13	,	,	PUNCT
ijassa-2040	200	14	1	1	NUM
ijassa-2040	200	15	)	)	PUNCT
ijassa-2040	200	16	of	of	ADP
ijassa-2040	200	17	multiplicity	multiplicity	NOUN
ijassa-2040	200	18	r.	r.	PROPN
ijassa-2040	200	19	the	the	DET
ijassa-2040	200	20	inequality	inequality	PROPN
ijassa-2040	200	21	|p	|p	PROPN
ijassa-2040	200	22	(	(	PUNCT
ijassa-2040	200	23	n)|	n)|	X
ijassa-2040	200	24	>	>	X
ijassa-2040	200	25	cn2k1	cn2k1	PROPN
ijassa-2040	200	26	is	be	AUX
ijassa-2040	200	27	equivalent	equivalent	ADJ
ijassa-2040	200	28	to∣∣p	to∣∣p	ADV
ijassa-2040	200	29	(	(	PUNCT
ijassa-2040	200	30	n1	n1	PROPN
ijassa-2040	200	31	n2	n2	NOUN
ijassa-2040	200	32	,	,	PUNCT
ijassa-2040	200	33	1	1	NUM
ijassa-2040	200	34	)	)	PUNCT
ijassa-2040	201	1	∣∣	∣∣	AUX
ijassa-2040	201	2	>	>	X
ijassa-2040	201	3	c(n2)(k1−	c(n2)(k1−	VERB
ijassa-2040	201	4	p	p	NOUN
ijassa-2040	201	5	2	2	NUM
ijassa-2040	201	6	)	)	PUNCT
ijassa-2040	201	7	,	,	PUNCT
ijassa-2040	201	8	since	since	SCONJ
ijassa-2040	201	9	p	p	PROPN
ijassa-2040	201	10	(	(	PUNCT
ijassa-2040	201	11	n1	n1	PROPN
ijassa-2040	201	12	n2	n2	NOUN
ijassa-2040	201	13	,	,	PUNCT
ijassa-2040	201	14	1	1	NUM
ijassa-2040	201	15	)	)	PUNCT
ijassa-2040	201	16	tends	tend	VERB
ijassa-2040	201	17	to	to	ADP
ijassa-2040	201	18	zero	zero	NUM
ijassa-2040	201	19	as	as	SCONJ
ijassa-2040	201	20	n1	n1	PROPN
ijassa-2040	201	21	n2	n2	NOUN
ijassa-2040	201	22	tends	tend	VERB
ijassa-2040	201	23	to	to	ADP
ijassa-2040	201	24	one	one	NUM
ijassa-2040	201	25	of	of	ADP
ijassa-2040	201	26	the	the	DET
ijassa-2040	201	27	roots	root	NOUN
ijassa-2040	201	28	.	.	PUNCT
ijassa-2040	202	1	thus	thus	ADV
ijassa-2040	202	2	,	,	PUNCT
ijassa-2040	202	3	we	we	PRON
ijassa-2040	202	4	get	get	VERB
ijassa-2040	202	5	the	the	DET
ijassa-2040	202	6	inequality∣∣p	inequality∣∣p	NOUN
ijassa-2040	202	7	(	(	PUNCT
ijassa-2040	202	8	n1	n1	PROPN
ijassa-2040	202	9	n2	n2	NOUN
ijassa-2040	202	10	,	,	PUNCT
ijassa-2040	202	11	1	1	NUM
ijassa-2040	202	12	)	)	PUNCT
ijassa-2040	202	13	∣∣	∣∣	X
ijassa-2040	202	14	=	=	SYM
ijassa-2040	202	15	∣∣n1	∣∣n1	PROPN
ijassa-2040	202	16	n2	n2	PROPN
ijassa-2040	202	17	α	α	X
ijassa-2040	202	18	∣∣r∣∣p	∣∣r∣∣p	PROPN
ijassa-2040	202	19	(	(	PUNCT
ijassa-2040	202	20	r	r	NOUN
ijassa-2040	202	21	)	)	PUNCT
ijassa-2040	202	22	x1	x1	PROPN
ijassa-2040	202	23	(	(	PUNCT
ijassa-2040	202	24	n1	n1	PROPN
ijassa-2040	202	25	n2	n2	NOUN
ijassa-2040	202	26	+	+	CCONJ
ijassa-2040	202	27	τ	τ	X
ijassa-2040	202	28	(	(	PUNCT
ijassa-2040	202	29	α−	α−	ADP
ijassa-2040	202	30	n1	n1	ADJ
ijassa-2040	202	31	n2	n2	NOUN
ijassa-2040	202	32	)	)	PUNCT
ijassa-2040	202	33	∣∣	∣∣	X
ijassa-2040	202	34	>	>	X
ijassa-2040	202	35	c(n2)(k1−	c(n2)(k1−	VERB
ijassa-2040	202	36	p	p	NOUN
ijassa-2040	202	37	2	2	NUM
ijassa-2040	202	38	)	)	PUNCT
ijassa-2040	202	39	.	.	PUNCT
ijassa-2040	203	1	for	for	ADP
ijassa-2040	203	2	n1	n1	PROPN
ijassa-2040	203	3	n2	n2	NOUN
ijassa-2040	203	4	close	close	ADJ
ijassa-2040	203	5	enough	enough	ADV
ijassa-2040	203	6	to	to	ADP
ijassa-2040	203	7	α	α	PRON
ijassa-2040	203	8	,	,	PUNCT
ijassa-2040	203	9	we	we	PRON
ijassa-2040	203	10	have∣∣n1	have∣∣n1	AUX
ijassa-2040	203	11	n2	n2	NOUN
ijassa-2040	203	12	−	−	PROPN
ijassa-2040	203	13	α	α	PROPN
ijassa-2040	203	14	∣∣	∣∣	X
ijassa-2040	203	15	>	>	PUNCT
ijassa-2040	203	16	cn2k1	cn2k1	PROPN
ijassa-2040	203	17	>	>	X
ijassa-2040	203	18	cn2k	cn2k	PROPN
ijassa-2040	203	19	2	2	NUM
ijassa-2040	203	20	.	.	PUNCT
ijassa-2040	204	1	a	a	DET
ijassa-2040	204	2	direct	direct	ADJ
ijassa-2040	204	3	calculation	calculation	NOUN
ijassa-2040	204	4	gives	give	VERB
ijassa-2040	204	5	k	k	NOUN
ijassa-2040	204	6	=	=	SYM
ijassa-2040	204	7	1	1	NUM
ijassa-2040	204	8	r	r	NOUN
ijassa-2040	204	9	(	(	PUNCT
ijassa-2040	204	10	k1	k1	NOUN
ijassa-2040	204	11	−	−	PROPN
ijassa-2040	204	12	p	p	NOUN
ijassa-2040	204	13	2	2	NUM
ijassa-2040	204	14	)	)	PUNCT
ijassa-2040	204	15	.	.	PUNCT
ijassa-2040	205	1	on	on	ADP
ijassa-2040	205	2	the	the	DET
ijassa-2040	205	3	contrary	contrary	NOUN
ijassa-2040	205	4	,	,	PUNCT
ijassa-2040	205	5	if	if	SCONJ
ijassa-2040	205	6	∣∣n1	∣∣n1	PROPN
ijassa-2040	205	7	n2	n2	NOUN
ijassa-2040	205	8	−	−	PROPN
ijassa-2040	205	9	α	α	PROPN
ijassa-2040	205	10	∣∣	∣∣	X
ijassa-2040	205	11	>	>	PUNCT
ijassa-2040	205	12	cn2k	cn2k	PROPN
ijassa-2040	205	13	2	2	NUM
ijassa-2040	205	14	,	,	PUNCT
ijassa-2040	205	15	then	then	ADV
ijassa-2040	205	16	∣∣n1	∣∣n1	PROPN
ijassa-2040	205	17	n2	n2	PROPN
ijassa-2040	205	18	−	−	PROPN
ijassa-2040	205	19	α	α	PROPN
ijassa-2040	205	20	∣∣	∣∣	X
ijassa-2040	205	21	>	>	PUNCT
ijassa-2040	205	22	cn2k(k	cn2k(k	X
ijassa-2040	205	23	<	<	X
ijassa-2040	205	24	0	0	NUM
ijassa-2040	205	25	)	)	PUNCT
ijassa-2040	205	26	,	,	PUNCT
ijassa-2040	205	27	whence	whence	SCONJ
ijassa-2040	205	28	we	we	PRON
ijassa-2040	205	29	obtain	obtain	VERB
ijassa-2040	205	30	|p	|p	X
ijassa-2040	205	31	(	(	PUNCT
ijassa-2040	205	32	n)|	n)|	X
ijassa-2040	205	33	>	>	X
ijassa-2040	205	34	cn2k1	cn2k1	PROPN
ijassa-2040	205	35	.	.	PUNCT
ijassa-2040	206	1	the	the	DET
ijassa-2040	206	2	proof	proof	NOUN
ijassa-2040	206	3	is	be	AUX
ijassa-2040	206	4	complete	complete	ADJ
ijassa-2040	206	5	.	.	PUNCT
ijassa-2040	207	1	remark	remark	PROPN
ijassa-2040	207	2	3.1	3.1	NUM
ijassa-2040	207	3	:	:	PUNCT
ijassa-2040	208	1	inequality	inequality	NOUN
ijassa-2040	208	2	(	(	PUNCT
ijassa-2040	208	3	3.6	3.6	NUM
ijassa-2040	208	4	)	)	PUNCT
ijassa-2040	208	5	is	be	AUX
ijassa-2040	208	6	not	not	PART
ijassa-2040	208	7	valid	valid	ADJ
ijassa-2040	208	8	for	for	ADP
ijassa-2040	208	9	some	some	DET
ijassa-2040	208	10	transcendental	transcendental	ADJ
ijassa-2040	208	11	numbers	number	NOUN
ijassa-2040	208	12	α	α	NOUN
ijassa-2040	208	13	,	,	PUNCT
ijassa-2040	208	14	for	for	ADP
ijassa-2040	208	15	example	example	NOUN
ijassa-2040	208	16	,	,	PUNCT
ijassa-2040	208	17	∑∞	∑∞	X
ijassa-2040	208	18	ν=1	ν=1	X
ijassa-2040	208	19	1	1	NUM
ijassa-2040	208	20	10ν	10ν	NOUN
ijassa-2040	208	21	!	!	PUNCT
ijassa-2040	209	1	(	(	PUNCT
ijassa-2040	209	2	see	see	VERB
ijassa-2040	209	3	[	[	X
ijassa-2040	209	4	13	13	NUM
ijassa-2040	209	5	]	]	NUM
ijassa-2040	209	6	)	)	PUNCT
ijassa-2040	209	7	.	.	PUNCT
ijassa-2040	210	1	on	on	ADP
ijassa-2040	210	2	the	the	DET
ijassa-2040	210	3	other	other	ADJ
ijassa-2040	210	4	hand	hand	NOUN
ijassa-2040	210	5	,	,	PUNCT
ijassa-2040	210	6	the	the	DET
ijassa-2040	210	7	liouville	liouville	NOUN
ijassa-2040	210	8	theorem	theorem	VERB
ijassa-2040	210	9	states	state	NOUN
ijassa-2040	210	10	that	that	SCONJ
ijassa-2040	210	11	for	for	ADP
ijassa-2040	210	12	every	every	DET
ijassa-2040	210	13	algebraic	algebraic	ADJ
ijassa-2040	210	14	number	number	NOUN
ijassa-2040	210	15	α	α	NOUN
ijassa-2040	210	16	of	of	ADP
ijassa-2040	210	17	degree	degree	NOUN
ijassa-2040	210	18	ν	ν	NOUN
ijassa-2040	210	19	inequality	inequality	NOUN
ijassa-2040	210	20	(	(	PUNCT
ijassa-2040	210	21	3.6	3.6	NUM
ijassa-2040	210	22	)	)	PUNCT
ijassa-2040	210	23	holds	hold	VERB
ijassa-2040	210	24	true	true	ADJ
ijassa-2040	210	25	with	with	ADP
ijassa-2040	210	26	k	k	PROPN
ijassa-2040	210	27	=	=	PUNCT
ijassa-2040	210	28	ν	ν	NOUN
ijassa-2040	210	29	(	(	PUNCT
ijassa-2040	210	30	see	see	VERB
ijassa-2040	210	31	[	[	X
ijassa-2040	210	32	13	13	NUM
ijassa-2040	210	33	]	]	NUM
ijassa-2040	210	34	)	)	PUNCT
ijassa-2040	210	35	.	.	PUNCT
ijassa-2040	211	1	in	in	ADP
ijassa-2040	211	2	particular	particular	ADJ
ijassa-2040	211	3	,	,	PUNCT
ijassa-2040	211	4	we	we	PRON
ijassa-2040	211	5	obtain	obtain	VERB
ijassa-2040	211	6	the	the	DET
ijassa-2040	211	7	following	follow	VERB
ijassa-2040	211	8	proposition	proposition	NOUN
ijassa-2040	211	9	3.2	3.2	NUM
ijassa-2040	211	10	:	:	PUNCT
ijassa-2040	211	11	if	if	SCONJ
ijassa-2040	211	12	a	a	DET
ijassa-2040	211	13	polynomial	polynomial	ADJ
ijassa-2040	211	14	p	p	NOUN
ijassa-2040	211	15	with	with	ADP
ijassa-2040	211	16	integer	integer	NOUN
ijassa-2040	211	17	(	(	PUNCT
ijassa-2040	211	18	or	or	CCONJ
ijassa-2040	211	19	rational	rational	ADJ
ijassa-2040	211	20	)	)	PUNCT
ijassa-2040	211	21	coefficients	coefficient	NOUN
ijassa-2040	211	22	is	be	AUX
ijassa-2040	211	23	irreducible	irreducible	ADJ
ijassa-2040	211	24	in	in	ADP
ijassa-2040	211	25	the	the	DET
ijassa-2040	211	26	field	field	NOUN
ijassa-2040	212	1	q	q	NOUN
ijassa-2040	212	2	,	,	PUNCT
ijassa-2040	212	3	then	then	ADV
ijassa-2040	212	4	the	the	DET
ijassa-2040	212	5	operator	operator	NOUN
ijassa-2040	212	6	l	l	NOUN
ijassa-2040	212	7	is	be	AUX
ijassa-2040	212	8	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	212	9	.	.	PUNCT
ijassa-2040	213	1	copyright	copyright	NOUN
ijassa-2040	213	2	©	©	PROPN
ijassa-2040	213	3	2025	2025	NUM
ijassa-2040	213	4	assa	assa	NOUN
ijassa-2040	213	5	.	.	PUNCT
ijassa-2040	214	1	adv	adv	PROPN
ijassa-2040	214	2	syst	syst	PROPN
ijassa-2040	214	3	sci	sci	PROPN
ijassa-2040	214	4	appl	appl	PROPN
ijassa-2040	214	5	(	(	PUNCT
ijassa-2040	214	6	2025	2025	NUM
ijassa-2040	214	7	)	)	PUNCT
ijassa-2040	214	8	linear	linear	ADJ
ijassa-2040	214	9	differential	differential	ADJ
ijassa-2040	214	10	equations	equation	NOUN
ijassa-2040	214	11	on	on	ADP
ijassa-2040	214	12	the	the	DET
ijassa-2040	214	13	torus	torus	NOUN
ijassa-2040	214	14	...	...	PUNCT
ijassa-2040	214	15	21	21	NUM
ijassa-2040	214	16	it	it	PRON
ijassa-2040	214	17	is	be	AUX
ijassa-2040	214	18	easy	easy	ADJ
ijassa-2040	214	19	to	to	PART
ijassa-2040	214	20	see	see	VERB
ijassa-2040	214	21	that	that	SCONJ
ijassa-2040	214	22	the	the	DET
ijassa-2040	214	23	hypoelliptic	hypoelliptic	ADJ
ijassa-2040	214	24	operator	operator	NOUN
ijassa-2040	214	25	l	l	NOUN
ijassa-2040	214	26	:	:	PUNCT
ijassa-2040	214	27	h∞	h∞	X
ijassa-2040	214	28	→	→	PUNCT
ijassa-2040	214	29	h∞	h∞	X
ijassa-2040	214	30	is	be	AUX
ijassa-2040	214	31	reversible	reversible	ADJ
ijassa-2040	214	32	.	.	PUNCT
ijassa-2040	215	1	here	here	ADV
ijassa-2040	215	2	and	and	CCONJ
ijassa-2040	215	3	below	below	ADV
ijassa-2040	215	4	,	,	PUNCT
ijassa-2040	215	5	all	all	DET
ijassa-2040	215	6	spaces	space	NOUN
ijassa-2040	215	7	are	be	AUX
ijassa-2040	215	8	assumed	assume	VERB
ijassa-2040	215	9	to	to	PART
ijassa-2040	215	10	be	be	AUX
ijassa-2040	215	11	quotient	quotient	NOUN
ijassa-2040	215	12	by	by	ADP
ijassa-2040	215	13	the	the	DET
ijassa-2040	215	14	subspace	subspace	NOUN
ijassa-2040	215	15	of	of	ADP
ijassa-2040	215	16	constants	constant	NOUN
ijassa-2040	215	17	.	.	PUNCT
ijassa-2040	216	1	the	the	DET
ijassa-2040	216	2	inverse	inverse	NOUN
ijassa-2040	216	3	operator	operator	NOUN
ijassa-2040	216	4	l−1	l−1	PROPN
ijassa-2040	216	5	acts	act	VERB
ijassa-2040	216	6	from	from	ADP
ijassa-2040	216	7	hm	hm	INTJ
ijassa-2040	216	8	into	into	ADP
ijassa-2040	216	9	hk	hk	PROPN
ijassa-2040	216	10	with	with	ADP
ijassa-2040	216	11	some	some	DET
ijassa-2040	216	12	k.	k.	NOUN
ijassa-2040	216	13	according	accord	VERB
ijassa-2040	216	14	to	to	ADP
ijassa-2040	216	15	the	the	DET
ijassa-2040	216	16	thue	thue	PROPN
ijassa-2040	216	17	–	–	PUNCT
ijassa-2040	216	18	siegel	siegel	PROPN
ijassa-2040	216	19	–	–	PUNCT
ijassa-2040	216	20	roth	roth	PROPN
ijassa-2040	216	21	theorem	theorem	VERB
ijassa-2040	216	22	[	[	PUNCT
ijassa-2040	216	23	13	13	NUM
ijassa-2040	216	24	]	]	PUNCT
ijassa-2040	216	25	,	,	PUNCT
ijassa-2040	216	26	for	for	ADP
ijassa-2040	216	27	every	every	DET
ijassa-2040	216	28	algebraic	algebraic	ADJ
ijassa-2040	216	29	number	number	NOUN
ijassa-2040	216	30	α	α	NOUN
ijassa-2040	216	31	of	of	ADP
ijassa-2040	216	32	degree	degree	NOUN
ijassa-2040	216	33	r	r	NOUN
ijassa-2040	216	34	≥	≥	NOUN
ijassa-2040	216	35	2	2	NUM
ijassa-2040	216	36	and	and	CCONJ
ijassa-2040	216	37	any	any	DET
ijassa-2040	216	38	ε	ε	PROPN
ijassa-2040	216	39	>	>	X
ijassa-2040	216	40	0	0	PUNCT
ijassa-2040	217	1	there	there	PRON
ijassa-2040	217	2	exists	exist	VERB
ijassa-2040	217	3	c	c	NOUN
ijassa-2040	217	4	>	>	X
ijassa-2040	217	5	0	0	NUM
ijassa-2040	217	6	such	such	ADJ
ijassa-2040	217	7	that	that	PRON
ijassa-2040	217	8	for	for	ADP
ijassa-2040	217	9	every	every	DET
ijassa-2040	217	10	rational	rational	ADJ
ijassa-2040	217	11	number	number	NOUN
ijassa-2040	217	12	p	p	NOUN
ijassa-2040	217	13	/	/	SYM
ijassa-2040	217	14	q	q	NOUN
ijassa-2040	217	15	the	the	DET
ijassa-2040	217	16	inequality∣∣∣α−	inequality∣∣∣α−	NOUN
ijassa-2040	217	17	p	p	NOUN
ijassa-2040	217	18	q	q	PROPN
ijassa-2040	217	19	∣∣∣	∣∣∣	NOUN
ijassa-2040	217	20	>	>	X
ijassa-2040	217	21	c	c	X
ijassa-2040	217	22	q2+ε	q2+ε	PROPN
ijassa-2040	217	23	.	.	PUNCT
ijassa-2040	218	1	holds	hold	VERB
ijassa-2040	218	2	true	true	ADJ
ijassa-2040	218	3	.	.	PUNCT
ijassa-2040	219	1	then	then	ADV
ijassa-2040	219	2	,	,	PUNCT
ijassa-2040	219	3	using	use	VERB
ijassa-2040	219	4	calculation	calculation	NOUN
ijassa-2040	219	5	of	of	ADP
ijassa-2040	219	6	the	the	DET
ijassa-2040	219	7	exponents	exponent	NOUN
ijassa-2040	219	8	from	from	ADP
ijassa-2040	219	9	the	the	DET
ijassa-2040	219	10	proof	proof	NOUN
ijassa-2040	219	11	of	of	ADP
ijassa-2040	219	12	proposition	proposition	NOUN
ijassa-2040	219	13	3.2	3.2	NUM
ijassa-2040	219	14	,	,	PUNCT
ijassa-2040	219	15	we	we	PRON
ijassa-2040	219	16	obtain	obtain	VERB
ijassa-2040	219	17	the	the	DET
ijassa-2040	219	18	following	follow	VERB
ijassa-2040	219	19	proposition	proposition	NOUN
ijassa-2040	219	20	3.3	3.3	NUM
ijassa-2040	219	21	:	:	PUNCT
ijassa-2040	219	22	let	let	VERB
ijassa-2040	219	23	r	r	NOUN
ijassa-2040	219	24	be	be	AUX
ijassa-2040	219	25	the	the	DET
ijassa-2040	219	26	greatest	great	ADJ
ijassa-2040	219	27	multiplicity	multiplicity	NOUN
ijassa-2040	219	28	of	of	ADP
ijassa-2040	219	29	real	real	ADJ
ijassa-2040	219	30	roots	root	NOUN
ijassa-2040	219	31	of	of	ADP
ijassa-2040	219	32	the	the	DET
ijassa-2040	219	33	irreducible	irreducible	ADJ
ijassa-2040	219	34	polynomial	polynomial	ADJ
ijassa-2040	219	35	p	p	NOUN
ijassa-2040	219	36	.	.	PUNCT
ijassa-2040	220	1	then	then	ADV
ijassa-2040	220	2	for	for	ADP
ijassa-2040	220	3	every	every	DET
ijassa-2040	220	4	ε	ε	PROPN
ijassa-2040	220	5	>	>	X
ijassa-2040	220	6	0	0	NUM
ijassa-2040	221	1	the	the	DET
ijassa-2040	221	2	operator	operator	NOUN
ijassa-2040	221	3	l−1	l−1	PROPN
ijassa-2040	221	4	acts	act	VERB
ijassa-2040	221	5	from	from	ADP
ijassa-2040	221	6	hm	hm	INTJ
ijassa-2040	221	7	into	into	ADP
ijassa-2040	221	8	hp/2+m−r−ε	hp/2+m−r−ε	NOUN
ijassa-2040	221	9	continuously	continuously	ADV
ijassa-2040	221	10	.	.	PUNCT
ijassa-2040	222	1	remark	remark	VERB
ijassa-2040	222	2	3.2	3.2	NUM
ijassa-2040	222	3	:	:	PUNCT
ijassa-2040	222	4	for	for	ADP
ijassa-2040	222	5	p	p	NOUN
ijassa-2040	222	6	=	=	SYM
ijassa-2040	222	7	2	2	NUM
ijassa-2040	222	8	,	,	PUNCT
ijassa-2040	222	9	by	by	ADP
ijassa-2040	222	10	the	the	DET
ijassa-2040	222	11	liouville	liouville	NOUN
ijassa-2040	222	12	theorem	theorem	NOUN
ijassa-2040	222	13	,	,	PUNCT
ijassa-2040	222	14	one	one	PRON
ijassa-2040	222	15	can	can	AUX
ijassa-2040	222	16	put	put	VERB
ijassa-2040	222	17	ε	ε	PROPN
ijassa-2040	222	18	=	=	SYM
ijassa-2040	222	19	0	0	PROPN
ijassa-2040	222	20	.	.	PUNCT
ijassa-2040	223	1	references	reference	NOUN
ijassa-2040	223	2	1	1	NUM
ijassa-2040	223	3	.	.	PUNCT
ijassa-2040	223	4	bers	ber	NOUN
ijassa-2040	223	5	,	,	PUNCT
ijassa-2040	223	6	l.	l.	PROPN
ijassa-2040	223	7	,	,	PUNCT
ijassa-2040	223	8	john	john	PROPN
ijassa-2040	223	9	,	,	PUNCT
ijassa-2040	223	10	f.	f.	PROPN
ijassa-2040	223	11	,	,	PUNCT
ijassa-2040	223	12	&	&	CCONJ
ijassa-2040	223	13	schechter	schechter	PROPN
ijassa-2040	223	14	,	,	PUNCT
ijassa-2040	223	15	m.	m.	NOUN
ijassa-2040	223	16	(	(	PUNCT
ijassa-2040	223	17	1964	1964	NUM
ijassa-2040	223	18	)	)	PUNCT
ijassa-2040	223	19	partial	partial	ADJ
ijassa-2040	223	20	differential	differential	NOUN
ijassa-2040	223	21	equations	equation	NOUN
ijassa-2040	223	22	,	,	PUNCT
ijassa-2040	223	23	am	be	AUX
ijassa-2040	223	24	.	.	PUNCT
ijassa-2040	224	1	math	math	NOUN
ijassa-2040	224	2	.	.	PUNCT
ijassa-2040	225	1	soc	soc	PROPN
ijassa-2040	225	2	.	.	PUNCT
ijassa-2040	225	3	,	,	PUNCT
ijassa-2040	225	4	196	196	NUM
ijassa-2040	225	5	.	.	X
ijassa-2040	226	1	2	2	X
ijassa-2040	226	2	.	.	X
ijassa-2040	226	3	bourbaki	bourbaki	NOUN
ijassa-2040	226	4	,	,	PUNCT
ijassa-2040	226	5	n.	n.	PROPN
ijassa-2040	226	6	(	(	PUNCT
ijassa-2040	226	7	1987	1987	NUM
ijassa-2040	226	8	)	)	PUNCT
ijassa-2040	226	9	topological	topological	ADJ
ijassa-2040	226	10	vector	vector	NOUN
ijassa-2040	226	11	spaces	space	NOUN
ijassa-2040	226	12	,	,	PUNCT
ijassa-2040	226	13	springer	springer	NOUN
ijassa-2040	226	14	-	-	PUNCT
ijassa-2040	226	15	verlag	verlag	PROPN
ijassa-2040	226	16	.	.	PUNCT
ijassa-2040	227	1	3	3	X
ijassa-2040	227	2	.	.	X
ijassa-2040	227	3	burskii	burskii	PROPN
ijassa-2040	227	4	,	,	PUNCT
ijassa-2040	227	5	v.	v.	ADP
ijassa-2040	228	1	p.	p.	NOUN
ijassa-2040	228	2	(	(	PUNCT
ijassa-2040	228	3	1980	1980	NUM
ijassa-2040	228	4	)	)	PUNCT
ijassa-2040	228	5	on	on	ADP
ijassa-2040	228	6	the	the	DET
ijassa-2040	228	7	solvability	solvability	NOUN
ijassa-2040	228	8	of	of	ADP
ijassa-2040	228	9	the	the	DET
ijassa-2040	228	10	garabedian	garabedian	NOUN
ijassa-2040	228	11	-	-	PUNCT
ijassa-2040	228	12	grushin	grushin	NOUN
ijassa-2040	228	13	equation	equation	NOUN
ijassa-2040	228	14	,	,	PUNCT
ijassa-2040	228	15	in	in	ADP
ijassa-2040	228	16	:	:	PUNCT
ijassa-2040	228	17	collection	collection	NOUN
ijassa-2040	228	18	of	of	ADP
ijassa-2040	228	19	scientific	scientific	ADJ
ijassa-2040	228	20	articles	article	NOUN
ijassa-2040	228	21	boundary	boundary	ADJ
ijassa-2040	228	22	value	value	NOUN
ijassa-2040	228	23	problems	problem	NOUN
ijassa-2040	228	24	for	for	ADP
ijassa-2040	228	25	differential	differential	ADJ
ijassa-2040	228	26	equations	equation	NOUN
ijassa-2040	228	27	,	,	PUNCT
ijassa-2040	228	28	kiev	kiev	PROPN
ijassa-2040	228	29	:	:	PUNCT
ijassa-2040	228	30	naukova	naukova	PROPN
ijassa-2040	228	31	dumka	dumka	PROPN
ijassa-2040	228	32	,	,	PUNCT
ijassa-2040	228	33	35–39	35–39	NUM
ijassa-2040	229	1	[	[	X
ijassa-2040	229	2	in	in	ADP
ijassa-2040	229	3	russian	russian	PROPN
ijassa-2040	229	4	]	]	PUNCT
ijassa-2040	229	5	.	.	PUNCT
ijassa-2040	229	6	4	4	X
ijassa-2040	229	7	.	.	X
ijassa-2040	229	8	burskii	burskii	PROPN
ijassa-2040	229	9	,	,	PUNCT
ijassa-2040	229	10	v.	v.	ADP
ijassa-2040	230	1	p.	p.	NOUN
ijassa-2040	230	2	(	(	PUNCT
ijassa-2040	230	3	2018	2018	NUM
ijassa-2040	230	4	)	)	PUNCT
ijassa-2040	230	5	on	on	ADP
ijassa-2040	230	6	differential	differential	ADJ
ijassa-2040	230	7	operators	operator	NOUN
ijassa-2040	230	8	and	and	CCONJ
ijassa-2040	230	9	differential	differential	ADJ
ijassa-2040	230	10	equations	equation	NOUN
ijassa-2040	230	11	on	on	ADP
ijassa-2040	230	12	torus	torus	PROPN
ijassa-2040	230	13	,	,	PUNCT
ijassa-2040	230	14	j.	j.	PROPN
ijassa-2040	230	15	samara	samara	PROPN
ijassa-2040	230	16	state	state	PROPN
ijassa-2040	230	17	techn	techn	PROPN
ijassa-2040	230	18	.	.	PUNCT
ijassa-2040	231	1	univ	univ	PROPN
ijassa-2040	231	2	.	.	PROPN
ijassa-2040	231	3	,	,	PUNCT
ijassa-2040	231	4	ser	ser	PROPN
ijassa-2040	231	5	.	.	PUNCT
ijassa-2040	232	1	phys	phys	PROPN
ijassa-2040	232	2	.	.	PUNCT
ijassa-2040	233	1	math	math	NOUN
ijassa-2040	233	2	.	.	PUNCT
ijassa-2040	234	1	sci	sci	PROPN
ijassa-2040	234	2	.	.	PROPN
ijassa-2040	234	3	,	,	PUNCT
ijassa-2040	234	4	22:4	22:4	NUM
ijassa-2040	234	5	,	,	PUNCT
ijassa-2040	234	6	607–619	607–619	NUM
ijassa-2040	234	7	[	[	PUNCT
ijassa-2040	234	8	in	in	ADP
ijassa-2040	234	9	russian	russian	PROPN
ijassa-2040	234	10	]	]	PUNCT
ijassa-2040	234	11	.	.	PUNCT
ijassa-2040	235	1	5	5	X
ijassa-2040	235	2	.	.	X
ijassa-2040	235	3	davis	davis	PROPN
ijassa-2040	235	4	,	,	PUNCT
ijassa-2040	235	5	m.	m.	NOUN
ijassa-2040	235	6	(	(	PUNCT
ijassa-2040	235	7	1977	1977	NUM
ijassa-2040	235	8	)	)	PUNCT
ijassa-2040	235	9	applied	apply	VERB
ijassa-2040	235	10	nonstandard	nonstandard	ADJ
ijassa-2040	235	11	analysis	analysis	NOUN
ijassa-2040	235	12	,	,	PUNCT
ijassa-2040	235	13	ny	ny	PROPN
ijassa-2040	235	14	:	:	PUNCT
ijassa-2040	235	15	wiley	wiley	PROPN
ijassa-2040	235	16	publication	publication	NOUN
ijassa-2040	235	17	.	.	PUNCT
ijassa-2040	236	1	6	6	NUM
ijassa-2040	236	2	.	.	X
ijassa-2040	236	3	dezin	dezin	NOUN
ijassa-2040	236	4	,	,	PUNCT
ijassa-2040	236	5	a.	a.	NOUN
ijassa-2040	236	6	a.	a.	NOUN
ijassa-2040	236	7	(	(	PUNCT
ijassa-2040	236	8	1987	1987	NUM
ijassa-2040	236	9	)	)	PUNCT
ijassa-2040	236	10	partial	partial	ADJ
ijassa-2040	236	11	differential	differential	NOUN
ijassa-2040	236	12	equations	equation	NOUN
ijassa-2040	236	13	.	.	PUNCT
ijassa-2040	237	1	an	an	DET
ijassa-2040	237	2	introduction	introduction	NOUN
ijassa-2040	237	3	to	to	ADP
ijassa-2040	237	4	a	a	DET
ijassa-2040	237	5	general	general	ADJ
ijassa-2040	237	6	theory	theory	NOUN
ijassa-2040	237	7	of	of	ADP
ijassa-2040	237	8	linear	linear	PROPN
ijassa-2040	237	9	boundary	boundary	ADJ
ijassa-2040	237	10	value	value	NOUN
ijassa-2040	237	11	problems	problem	NOUN
ijassa-2040	237	12	,	,	PUNCT
ijassa-2040	237	13	springer	springer	NOUN
ijassa-2040	237	14	-	-	PUNCT
ijassa-2040	237	15	verlag	verlag	PROPN
ijassa-2040	237	16	.	.	PUNCT
ijassa-2040	238	1	7	7	X
ijassa-2040	238	2	.	.	X
ijassa-2040	238	3	fedoryuk	fedoryuk	NOUN
ijassa-2040	238	4	,	,	PUNCT
ijassa-2040	238	5	m.	m.	NOUN
ijassa-2040	238	6	v.	v.	PROPN
ijassa-2040	238	7	(	(	PUNCT
ijassa-2040	238	8	1977	1977	NUM
ijassa-2040	238	9	)	)	PUNCT
ijassa-2040	238	10	metod	metod	VERB
ijassa-2040	238	11	perevala	perevala	NOUN
ijassa-2040	238	12	,	,	PUNCT
ijassa-2040	238	13	moscow	moscow	PROPN
ijassa-2040	238	14	:	:	PUNCT
ijassa-2040	238	15	nauka	nauka	PROPN
ijassa-2040	239	1	[	[	X
ijassa-2040	239	2	in	in	ADP
ijassa-2040	239	3	russian	russian	PROPN
ijassa-2040	239	4	]	]	PUNCT
ijassa-2040	239	5	.	.	PUNCT
ijassa-2040	239	6	8	8	X
ijassa-2040	239	7	.	.	X
ijassa-2040	239	8	garabedian	garabedian	PROPN
ijassa-2040	239	9	,	,	PUNCT
ijassa-2040	239	10	p.	p.	PROPN
ijassa-2040	239	11	r.	r.	PROPN
ijassa-2040	239	12	(	(	PUNCT
ijassa-2040	239	13	1970	1970	NUM
ijassa-2040	239	14	)	)	PUNCT
ijassa-2040	239	15	an	an	DET
ijassa-2040	239	16	unsolvable	unsolvable	ADJ
ijassa-2040	239	17	equation	equation	NOUN
ijassa-2040	239	18	,	,	PUNCT
ijassa-2040	239	19	proc	proc	NOUN
ijassa-2040	239	20	.	.	PUNCT
ijassa-2040	239	21	am	be	AUX
ijassa-2040	239	22	.	.	PUNCT
ijassa-2040	240	1	math	math	NOUN
ijassa-2040	240	2	.	.	PUNCT
ijassa-2040	241	1	soc	soc	PROPN
ijassa-2040	241	2	.	.	PROPN
ijassa-2040	241	3	,	,	PUNCT
ijassa-2040	241	4	25	25	NUM
ijassa-2040	241	5	,	,	PUNCT
ijassa-2040	241	6	207–208	207–208	NUM
ijassa-2040	241	7	.	.	PUNCT
ijassa-2040	242	1	9	9	NUM
ijassa-2040	242	2	.	.	X
ijassa-2040	242	3	gelfond	gelfond	NOUN
ijassa-2040	242	4	,	,	PUNCT
ijassa-2040	242	5	a.	a.	NOUN
ijassa-2040	242	6	o.	o.	PROPN
ijassa-2040	242	7	(	(	PUNCT
ijassa-2040	242	8	1960	1960	NUM
ijassa-2040	242	9	)	)	PUNCT
ijassa-2040	242	10	transcendental	transcendental	ADJ
ijassa-2040	242	11	and	and	CCONJ
ijassa-2040	242	12	algebraic	algebraic	ADJ
ijassa-2040	242	13	numbers	number	NOUN
ijassa-2040	242	14	,	,	PUNCT
ijassa-2040	242	15	new	new	PROPN
ijassa-2040	242	16	york	york	PROPN
ijassa-2040	242	17	:	:	PUNCT
ijassa-2040	242	18	dover	dover	PROPN
ijassa-2040	242	19	publications	publications	PROPN
ijassa-2040	242	20	,	,	PUNCT
ijassa-2040	242	21	inc	inc	PROPN
ijassa-2040	242	22	.	.	PROPN
ijassa-2040	242	23	vii	vii	PROPN
ijassa-2040	242	24	.	.	PROPN
ijassa-2040	243	1	10	10	NUM
ijassa-2040	243	2	.	.	PUNCT
ijassa-2040	244	1	grushin	grushin	NOUN
ijassa-2040	244	2	,	,	PUNCT
ijassa-2040	244	3	v.	v.	PROPN
ijassa-2040	244	4	v.	v.	PROPN
ijassa-2040	244	5	(	(	PUNCT
ijassa-2040	244	6	1971	1971	NUM
ijassa-2040	244	7	)	)	PUNCT
ijassa-2040	244	8	a	a	DET
ijassa-2040	244	9	differential	differential	ADJ
ijassa-2040	244	10	equation	equation	NOUN
ijassa-2040	244	11	without	without	ADP
ijassa-2040	244	12	a	a	DET
ijassa-2040	244	13	solution	solution	NOUN
ijassa-2040	244	14	,	,	PUNCT
ijassa-2040	244	15	math	math	NOUN
ijassa-2040	244	16	.	.	PUNCT
ijassa-2040	245	1	notes	note	NOUN
ijassa-2040	245	2	,	,	PUNCT
ijassa-2040	245	3	10:2	10:2	NUM
ijassa-2040	245	4	,	,	PUNCT
ijassa-2040	245	5	499–501	499–501	NUM
ijassa-2040	245	6	.	.	PUNCT
ijassa-2040	246	1	11	11	NUM
ijassa-2040	246	2	.	.	PUNCT
ijassa-2040	247	1	hörmander	hörmander	PROPN
ijassa-2040	247	2	,	,	PUNCT
ijassa-2040	247	3	l.	l.	PROPN
ijassa-2040	247	4	(	(	PUNCT
ijassa-2040	247	5	1963	1963	NUM
ijassa-2040	247	6	)	)	PUNCT
ijassa-2040	247	7	linear	linear	VERB
ijassa-2040	247	8	partial	partial	ADJ
ijassa-2040	247	9	differential	differential	NOUN
ijassa-2040	247	10	operators	operator	NOUN
ijassa-2040	247	11	,	,	PUNCT
ijassa-2040	247	12	springer	springer	NOUN
ijassa-2040	247	13	-	-	PUNCT
ijassa-2040	247	14	verlag	verlag	PROPN
ijassa-2040	247	15	.	.	PUNCT
ijassa-2040	248	1	12	12	NUM
ijassa-2040	248	2	.	.	PUNCT
ijassa-2040	249	1	hörmander	hörmander	PROPN
ijassa-2040	249	2	,	,	PUNCT
ijassa-2040	249	3	l.	l.	PROPN
ijassa-2040	249	4	(	(	PUNCT
ijassa-2040	249	5	1983	1983	NUM
ijassa-2040	249	6	)	)	PUNCT
ijassa-2040	249	7	the	the	DET
ijassa-2040	249	8	analysis	analysis	NOUN
ijassa-2040	249	9	of	of	ADP
ijassa-2040	249	10	linear	linear	ADJ
ijassa-2040	249	11	partial	partial	ADJ
ijassa-2040	249	12	differential	differential	NOUN
ijassa-2040	249	13	operators	operator	NOUN
ijassa-2040	249	14	ii	ii	PROPN
ijassa-2040	249	15	:	:	PUNCT
ijassa-2040	249	16	differential	differential	ADJ
ijassa-2040	249	17	operators	operator	NOUN
ijassa-2040	249	18	with	with	ADP
ijassa-2040	249	19	constant	constant	ADJ
ijassa-2040	249	20	coefficients	coefficient	NOUN
ijassa-2040	249	21	,	,	PUNCT
ijassa-2040	249	22	springer	springer	NOUN
ijassa-2040	249	23	-	-	PUNCT
ijassa-2040	249	24	verlag	verlag	PROPN
ijassa-2040	249	25	.	.	PUNCT
ijassa-2040	250	1	13	13	NUM
ijassa-2040	250	2	.	.	X
ijassa-2040	251	1	lax	lax	PROPN
ijassa-2040	251	2	,	,	PUNCT
ijassa-2040	251	3	p.	p.	NOUN
ijassa-2040	251	4	d.	d.	PROPN
ijassa-2040	251	5	(	(	PUNCT
ijassa-2040	251	6	1955	1955	NUM
ijassa-2040	251	7	)	)	PUNCT
ijassa-2040	251	8	on	on	ADP
ijassa-2040	251	9	cauchy	cauchy	PROPN
ijassa-2040	251	10	’s	’s	PART
ijassa-2040	251	11	problem	problem	NOUN
ijassa-2040	251	12	for	for	ADP
ijassa-2040	251	13	hyperbolic	hyperbolic	ADJ
ijassa-2040	251	14	equations	equation	NOUN
ijassa-2040	251	15	and	and	CCONJ
ijassa-2040	251	16	the	the	DET
ijassa-2040	251	17	differentiability	differentiability	NOUN
ijassa-2040	251	18	of	of	ADP
ijassa-2040	251	19	elliptic	elliptic	ADJ
ijassa-2040	251	20	equations	equation	NOUN
ijassa-2040	251	21	,	,	PUNCT
ijassa-2040	251	22	comm	comm	NOUN
ijassa-2040	251	23	.	.	PUNCT
ijassa-2040	252	1	pure	pure	ADJ
ijassa-2040	252	2	appl	appl	PROPN
ijassa-2040	252	3	.	.	PUNCT
ijassa-2040	252	4	math	math	PROPN
ijassa-2040	252	5	.	.	PUNCT
ijassa-2040	253	1	,	,	PUNCT
ijassa-2040	253	2	6	6	NUM
ijassa-2040	253	3	,	,	PUNCT
ijassa-2040	253	4	43–59	43–59	NOUN
ijassa-2040	253	5	.	.	PUNCT
ijassa-2040	254	1	14	14	NUM
ijassa-2040	254	2	.	.	PUNCT
ijassa-2040	255	1	lewy	lewy	PROPN
ijassa-2040	255	2	,	,	PUNCT
ijassa-2040	255	3	h.	h.	PROPN
ijassa-2040	255	4	(	(	PUNCT
ijassa-2040	255	5	1957	1957	NUM
ijassa-2040	255	6	)	)	PUNCT
ijassa-2040	255	7	an	an	DET
ijassa-2040	255	8	example	example	NOUN
ijassa-2040	255	9	of	of	ADP
ijassa-2040	255	10	a	a	DET
ijassa-2040	255	11	smooth	smooth	ADJ
ijassa-2040	255	12	linear	linear	ADJ
ijassa-2040	255	13	partial	partial	ADJ
ijassa-2040	255	14	differential	differential	NOUN
ijassa-2040	255	15	equation	equation	NOUN
ijassa-2040	255	16	without	without	ADP
ijassa-2040	255	17	solution	solution	NOUN
ijassa-2040	255	18	,	,	PUNCT
ijassa-2040	255	19	ann	ann	PROPN
ijassa-2040	255	20	.	.	PROPN
ijassa-2040	255	21	of	of	ADP
ijassa-2040	255	22	math	math	NOUN
ijassa-2040	255	23	.	.	PUNCT
ijassa-2040	255	24	,	,	PUNCT
ijassa-2040	255	25	2:66	2:66	NUM
ijassa-2040	255	26	,	,	PUNCT
ijassa-2040	255	27	155–158	155–158	NUM
ijassa-2040	255	28	.	.	NOUN
ijassa-2040	255	29	15	15	NUM
ijassa-2040	255	30	.	.	PUNCT
ijassa-2040	256	1	mizohata	mizohata	PROPN
ijassa-2040	256	2	,	,	PUNCT
ijassa-2040	256	3	s.	s.	PROPN
ijassa-2040	256	4	(	(	PUNCT
ijassa-2040	256	5	1962	1962	NUM
ijassa-2040	256	6	)	)	PUNCT
ijassa-2040	256	7	solutions	solution	NOUN
ijassa-2040	256	8	nulles	nulle	NOUN
ijassa-2040	256	9	et	et	NOUN
ijassa-2040	256	10	solution	solution	NOUN
ijassa-2040	256	11	non	non	PROPN
ijassa-2040	256	12	analytiques	analytique	NOUN
ijassa-2040	256	13	,	,	PUNCT
ijassa-2040	256	14	j.	j.	PROPN
ijassa-2040	256	15	math	math	PROPN
ijassa-2040	256	16	.	.	PUNCT
ijassa-2040	257	1	kyoto	kyoto	PROPN
ijassa-2040	257	2	unlv	unlv	PROPN
ijassa-2040	257	3	.	.	PROPN
ijassa-2040	257	4	,	,	PUNCT
ijassa-2040	257	5	1	1	NUM
ijassa-2040	257	6	,	,	PUNCT
ijassa-2040	257	7	271–302	271–302	NUM
ijassa-2040	257	8	.	.	PUNCT
ijassa-2040	258	1	16	16	NUM
ijassa-2040	258	2	.	.	PUNCT
ijassa-2040	259	1	ptashnik	ptashnik	PROPN
ijassa-2040	259	2	,	,	PUNCT
ijassa-2040	259	3	b.	b.	PROPN
ijassa-2040	259	4	i.	i.	PROPN
ijassa-2040	259	5	(	(	PUNCT
ijassa-2040	259	6	1984	1984	NUM
ijassa-2040	259	7	)	)	PUNCT
ijassa-2040	259	8	ill	ill	ADV
ijassa-2040	259	9	-	-	PUNCT
ijassa-2040	259	10	posed	pose	VERB
ijassa-2040	259	11	boundary	boundary	ADJ
ijassa-2040	259	12	value	value	NOUN
ijassa-2040	259	13	problems	problem	NOUN
ijassa-2040	259	14	for	for	ADP
ijassa-2040	259	15	partial	partial	ADJ
ijassa-2040	259	16	differential	differential	NOUN
ijassa-2040	259	17	equations	equation	NOUN
ijassa-2040	259	18	,	,	PUNCT
ijassa-2040	259	19	kiev	kiev	PROPN
ijassa-2040	259	20	:	:	PUNCT
ijassa-2040	259	21	naukova	naukova	PROPN
ijassa-2040	259	22	dumka	dumka	PROPN
ijassa-2040	260	1	[	[	X
ijassa-2040	260	2	in	in	ADP
ijassa-2040	260	3	russian	russian	PROPN
ijassa-2040	260	4	]	]	PUNCT
ijassa-2040	260	5	.	.	PUNCT
ijassa-2040	261	1	copyright	copyright	NOUN
ijassa-2040	261	2	©	©	PROPN
ijassa-2040	261	3	2025	2025	NUM
ijassa-2040	261	4	assa	assa	NOUN
ijassa-2040	261	5	.	.	PUNCT
ijassa-2040	262	1	adv	adv	PROPN
ijassa-2040	262	2	syst	syst	PROPN
ijassa-2040	262	3	sci	sci	PROPN
ijassa-2040	262	4	appl	appl	PROPN
ijassa-2040	262	5	(	(	PUNCT
ijassa-2040	262	6	2025	2025	NUM
ijassa-2040	262	7	)	)	PUNCT
ijassa-2040	262	8	introduction	introduction	NOUN
ijassa-2040	262	9	spaces	space	NOUN
ijassa-2040	262	10	of	of	ADP
ijassa-2040	262	11	periodic	periodic	ADJ
ijassa-2040	262	12	functions	function	NOUN
ijassa-2040	262	13	spaces	space	NOUN
ijassa-2040	262	14	of	of	ADP
ijassa-2040	262	15	periodic	periodic	ADJ
ijassa-2040	262	16	functions	function	NOUN
ijassa-2040	262	17	solvability	solvability	NOUN
ijassa-2040	262	18	of	of	ADP
ijassa-2040	262	19	the	the	DET
ijassa-2040	262	20	mizohata	mizohata	ADJ
ijassa-2040	262	21	equation	equation	NOUN
ijassa-2040	262	22	solvability	solvability	NOUN
ijassa-2040	262	23	of	of	ADP
ijassa-2040	262	24	general	general	ADJ
ijassa-2040	262	25	equations	equation	NOUN
ijassa-2040	262	26	linear	linear	PROPN
ijassa-2040	262	27	sections	section	NOUN
ijassa-2040	262	28	as	as	ADP
ijassa-2040	262	29	objects	object	NOUN
ijassa-2040	262	30	of	of	ADP
ijassa-2040	262	31	non	non	ADJ
ijassa-2040	262	32	-	-	ADJ
ijassa-2040	262	33	standard	standard	ADJ
ijassa-2040	262	34	analysis	analysis	NOUN
ijassa-2040	262	35	linear	linear	ADJ
ijassa-2040	262	36	sections	section	NOUN
ijassa-2040	262	37	and	and	CCONJ
ijassa-2040	262	38	solvability	solvability	NOUN
ijassa-2040	262	39	of	of	ADP
ijassa-2040	262	40	general	general	ADJ
ijassa-2040	262	41	equations	equation	NOUN
ijassa-2040	262	42	on	on	ADP
ijassa-2040	262	43	the	the	DET
ijassa-2040	262	44	hypoellipticity	hypoellipticity	NOUN
ijassa-2040	262	45	of	of	ADP
ijassa-2040	262	46	differential	differential	ADJ
ijassa-2040	262	47	operators	operator	NOUN
ijassa-2040	262	48	on	on	ADP
ijassa-2040	262	49	the	the	DET
ijassa-2040	262	50	torus	torus	NOUN