id	sid	tid	token	lemma	pos
ma-247	1	1	2024	2024	NUM
ma-247	1	2	ada	ada	PROPN
ma-247	1	3	academica	academica	PROPN
ma-247	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-247	1	5	.	.	PUNCT
ma-247	2	1	j.	j.	PROPN
ma-247	2	2	math	math	PROPN
ma-247	2	3	.	.	PUNCT
ma-247	3	1	anal	anal	ADJ
ma-247	3	2	.	.	PUNCT
ma-247	4	1	4	4	NUM
ma-247	4	2	(	(	PUNCT
ma-247	4	3	2024	2024	NUM
ma-247	4	4	)	)	PUNCT
ma-247	4	5	23doi	23doi	ADP
ma-247	4	6	:	:	PUNCT
ma-247	4	7	10.28924	10.28924	NUM
ma-247	4	8	/	/	SYM
ma-247	4	9	ada	ada	PROPN
ma-247	4	10	/	/	SYM
ma-247	4	11	ma.4.23	ma.4.23	NOUN
ma-247	4	12	slicing	slicing	NOUN
ma-247	4	13	of	of	ADP
ma-247	4	14	negative	negative	ADJ
ma-247	4	15	plurisubharmonic	plurisubharmonic	ADJ
ma-247	4	16	currents	current	NOUN
ma-247	4	17	arising	arise	VERB
ma-247	4	18	from	from	ADP
ma-247	4	19	analytic	analytic	ADJ
ma-247	4	20	subsets	subset	NOUN
ma-247	4	21	hedi	hedi	PROPN
ma-247	4	22	khedhiri	khedhiri	PROPN
ma-247	4	23	university	university	PROPN
ma-247	4	24	of	of	ADP
ma-247	4	25	monastir	monastir	PROPN
ma-247	4	26	,	,	PUNCT
ma-247	4	27	preparatory	preparatory	PROPN
ma-247	4	28	institute	institute	NOUN
ma-247	4	29	for	for	ADP
ma-247	4	30	engineering	engineering	NOUN
ma-247	4	31	studies	study	NOUN
ma-247	4	32	,	,	PUNCT
ma-247	4	33	research	research	NOUN
ma-247	4	34	laboratory	laboratory	NOUN
ma-247	4	35	lr18es16	lr18es16	PROPN
ma-247	4	36	,	,	PUNCT
ma-247	4	37	road	road	NOUN
ma-247	4	38	ibn	ibn	PROPN
ma-247	4	39	eljazzar	eljazzar	PROPN
ma-247	4	40	monastir	monastir	PROPN
ma-247	4	41	5019	5019	NUM
ma-247	4	42	,	,	PUNCT
ma-247	4	43	tunisia	tunisia	NOUN
ma-247	4	44	correspondence	correspondence	NOUN
ma-247	4	45	:	:	PUNCT
ma-247	4	46	khediri_h@yahoo.fr	khediri_h@yahoo.fr	X
ma-247	4	47	abstract	abstract	NOUN
ma-247	4	48	.	.	PUNCT
ma-247	5	1	in	in	ADP
ma-247	5	2	this	this	DET
ma-247	5	3	paper	paper	NOUN
ma-247	5	4	,	,	PUNCT
ma-247	5	5	first	first	ADV
ma-247	5	6	we	we	PRON
ma-247	5	7	focus	focus	VERB
ma-247	5	8	on	on	ADP
ma-247	5	9	the	the	DET
ma-247	5	10	slicing	slicing	NOUN
ma-247	5	11	of	of	ADP
ma-247	5	12	negative	negative	ADJ
ma-247	5	13	plurisubharmonic	plurisubharmonic	ADJ
ma-247	5	14	currents	current	NOUN
ma-247	5	15	whichare	whichare	VERB
ma-247	5	16	finite	finite	ADJ
ma-247	5	17	sums	sum	NOUN
ma-247	5	18	involving	involve	VERB
ma-247	5	19	currents	current	NOUN
ma-247	5	20	arising	arise	VERB
ma-247	5	21	from	from	ADP
ma-247	5	22	analytic	analytic	ADJ
ma-247	5	23	subsets	subset	NOUN
ma-247	5	24	with	with	ADP
ma-247	5	25	geometric	geometric	ADJ
ma-247	5	26	complete	complete	ADJ
ma-247	5	27	intersection.next	intersection.next	NUM
ma-247	5	28	,	,	PUNCT
ma-247	5	29	we	we	PRON
ma-247	5	30	provide	provide	VERB
ma-247	5	31	significant	significant	ADJ
ma-247	5	32	results	result	NOUN
ma-247	5	33	on	on	ADP
ma-247	5	34	the	the	DET
ma-247	5	35	integrability	integrability	NOUN
ma-247	5	36	across	across	ADP
ma-247	5	37	analytic	analytic	ADJ
ma-247	5	38	subsets	subset	NOUN
ma-247	5	39	,	,	PUNCT
ma-247	5	40	of	of	ADP
ma-247	5	41	the	the	DET
ma-247	5	42	coefficients	coefficient	NOUN
ma-247	5	43	ofsuch	ofsuch	ADJ
ma-247	5	44	currents	current	NOUN
ma-247	5	45	and	and	CCONJ
ma-247	5	46	of	of	ADP
ma-247	5	47	their	their	PRON
ma-247	5	48	slices	slice	NOUN
ma-247	5	49	.	.	PUNCT
ma-247	6	1	1	1	X
ma-247	6	2	.	.	X
ma-247	6	3	introduction	introduction	NOUN
ma-247	6	4	in	in	ADP
ma-247	6	5	this	this	DET
ma-247	6	6	paper	paper	NOUN
ma-247	6	7	,	,	PUNCT
ma-247	6	8	we	we	PRON
ma-247	6	9	let	let	VERB
ma-247	6	10	ω	ω	NOUN
ma-247	6	11	be	be	AUX
ma-247	6	12	a	a	DET
ma-247	6	13	domain	domain	NOUN
ma-247	6	14	in	in	ADP
ma-247	6	15	cn	cn	PROPN
ma-247	6	16	such	such	ADJ
ma-247	6	17	that	that	SCONJ
ma-247	6	18	the	the	DET
ma-247	6	19	unit	unit	NOUN
ma-247	6	20	polydisc	polydisc	NOUN
ma-247	6	21	∆n	∆n	PROPN
ma-247	6	22	satisfies	satisfy	VERB
ma-247	6	23	∆n	∆n	PROPN
ma-247	6	24	b	b	PROPN
ma-247	6	25	ω	ω	PROPN
ma-247	6	26	and	and	CCONJ
ma-247	6	27	ϕ	ϕ	PROPN
ma-247	6	28	be	be	AUX
ma-247	6	29	a	a	DET
ma-247	6	30	plurisubharmonic	plurisubharmonic	ADJ
ma-247	6	31	function	function	NOUN
ma-247	6	32	(	(	PUNCT
ma-247	6	33	psh	psh	NOUN
ma-247	6	34	for	for	ADP
ma-247	6	35	short	short	ADJ
ma-247	6	36	)	)	PUNCT
ma-247	6	37	,	,	PUNCT
ma-247	6	38	locally	locally	ADV
ma-247	6	39	bounded	bound	VERB
ma-247	6	40	on	on	ADP
ma-247	6	41	ω	ω	PROPN
ma-247	6	42	.	.	PUNCT
ma-247	7	1	we	we	PRON
ma-247	7	2	let	let	VERB
ma-247	7	3	n	n	PRON
ma-247	7	4	,	,	PUNCT
ma-247	7	5	k	k	PROPN
ma-247	7	6	,	,	PUNCT
ma-247	7	7	p	p	PROPN
ma-247	7	8	and	and	CCONJ
ma-247	7	9	n	n	PRON
ma-247	7	10	benonzero	benonzero	NOUN
ma-247	7	11	fixed	fix	VERB
ma-247	7	12	arbitrary	arbitrary	ADJ
ma-247	7	13	natural	natural	ADJ
ma-247	7	14	numbers	number	NOUN
ma-247	7	15	such	such	ADJ
ma-247	7	16	that	that	SCONJ
ma-247	7	17	k	k	PROPN
ma-247	7	18	≤	≤	PROPN
ma-247	7	19	p	p	NOUN
ma-247	7	20	≤	≤	NOUN
ma-247	7	21	n	n	CCONJ
ma-247	8	1	and	and	CCONJ
ma-247	8	2	we	we	PRON
ma-247	8	3	consider	consider	VERB
ma-247	8	4	the	the	DET
ma-247	8	5	n	n	CCONJ
ma-247	8	6	-	-	PUNCT
ma-247	8	7	complex	complex	ADJ
ma-247	8	8	space	space	NOUN
ma-247	8	9	cn	cn	NOUN
ma-247	8	10	with	with	ADP
ma-247	8	11	variables	variable	NOUN
ma-247	8	12	z	z	NOUN
ma-247	8	13	such	such	ADJ
ma-247	8	14	that	that	PRON
ma-247	8	15	cn	cn	PROPN
ma-247	9	1	=	=	NOUN
ma-247	10	1	ck	ck	PROPN
ma-247	11	1	×	×	PROPN
ma-247	11	2	cn−k	cn−k	NOUN
ma-247	11	3	,	,	PUNCT
ma-247	11	4	z	z	NOUN
ma-247	11	5	=	=	SYM
ma-247	11	6	(	(	PUNCT
ma-247	11	7	z	z	NOUN
ma-247	11	8	′	′	NOUN
ma-247	11	9	,	,	PUNCT
ma-247	11	10	z	z	PROPN
ma-247	11	11	′′	′′	PROPN
ma-247	11	12	)	)	PUNCT
ma-247	11	13	,	,	PUNCT
ma-247	12	1	z	z	NOUN
ma-247	12	2	′	′	NUM
ma-247	13	1	∈	∈	INTJ
ma-247	13	2	ck	ck	INTJ
ma-247	13	3	,	,	PUNCT
ma-247	13	4	z	z	PROPN
ma-247	13	5	′′	′′	PROPN
ma-247	13	6	∈	∈	PROPN
ma-247	13	7	cn−k	cn−k	NOUN
ma-247	13	8	.	.	PUNCT
ma-247	14	1	assuming	assume	VERB
ma-247	14	2	that	that	SCONJ
ma-247	14	3	ϕ	ϕ	NOUN
ma-247	14	4	depends	depend	VERB
ma-247	14	5	only	only	ADV
ma-247	14	6	on	on	ADP
ma-247	14	7	the	the	DET
ma-247	14	8	variable	variable	NOUN
ma-247	15	1	z	z	NOUN
ma-247	15	2	′	′	NUM
ma-247	16	1	∈	∈	INTJ
ma-247	16	2	ck	ck	NOUN
ma-247	16	3	and	and	CCONJ
ma-247	16	4	the	the	DET
ma-247	16	5	support	support	NOUN
ma-247	16	6	sϕ	sϕ	NOUN
ma-247	16	7	of	of	ADP
ma-247	16	8	the	the	DET
ma-247	16	9	associatedmong	associatedmong	NOUN
ma-247	16	10	-	-	PUNCT
ma-247	16	11	ampère	ampère	NOUN
ma-247	16	12	measure	measure	NOUN
ma-247	16	13	µϕ	µϕ	ADV
ma-247	16	14	=	=	PUNCT
ma-247	16	15	(	(	PUNCT
ma-247	16	16	ddcϕ)kis	ddcϕ)kis	X
ma-247	16	17	∆k	∆k	PROPN
ma-247	16	18	,	,	PUNCT
ma-247	16	19	then	then	ADV
ma-247	16	20	,	,	PUNCT
ma-247	16	21	the	the	DET
ma-247	16	22	slice	slice	NOUN
ma-247	16	23	(	(	PUNCT
ma-247	16	24	or	or	CCONJ
ma-247	16	25	the	the	DET
ma-247	16	26	ϕ-slice	ϕ-slice	NOUN
ma-247	16	27	)	)	PUNCT
ma-247	16	28	denoted	denote	VERB
ma-247	16	29	〈	〈	PROPN
ma-247	16	30	t	t	PROPN
ma-247	16	31	,	,	PUNCT
ma-247	16	32	π	π	PROPN
ma-247	16	33	,	,	PUNCT
ma-247	16	34	a〉ϕ	a〉ϕ	NOUN
ma-247	16	35	of	of	ADP
ma-247	16	36	a	a	DET
ma-247	16	37	current	current	ADJ
ma-247	16	38	t	t	NOUN
ma-247	16	39	∈	∈	PROPN
ma-247	16	40	d	d	X
ma-247	16	41	′p	′p	PROPN
ma-247	16	42	,	,	PUNCT
ma-247	16	43	p(ω	p(ω	PROPN
ma-247	16	44	)	)	PUNCT
ma-247	16	45	,	,	PUNCT
ma-247	16	46	at	at	ADP
ma-247	16	47	point	point	NOUN
ma-247	16	48	a	a	DET
ma-247	16	49	∈	∈	PROPN
ma-247	16	50	sϕ,is	sϕ,is	NOUN
ma-247	16	51	studied	study	VERB
ma-247	16	52	and	and	CCONJ
ma-247	16	53	well	well	ADV
ma-247	16	54	defined	define	VERB
ma-247	16	55	in	in	ADP
ma-247	16	56	[	[	X
ma-247	16	57	10	10	NUM
ma-247	16	58	]	]	X
ma-247	16	59	,	,	PUNCT
ma-247	16	60	such	such	ADJ
ma-247	16	61	that	that	SCONJ
ma-247	16	62	<	<	X
ma-247	16	63	t	t	PROPN
ma-247	16	64	,	,	PUNCT
ma-247	16	65	π	π	PROPN
ma-247	16	66	,	,	PUNCT
ma-247	16	67	a	a	PRON
ma-247	16	68	>	>	X
ma-247	16	69	ϕ	ϕ	X
ma-247	16	70	(	(	PUNCT
ma-247	16	71	ψ	ψ	NOUN
ma-247	16	72	)	)	PUNCT
ma-247	16	73	:	:	PUNCT
ma-247	16	74	=	=	PUNCT
ma-247	16	75	lim	lim	NOUN
ma-247	16	76	ε→0	ε→0	NOUN
ma-247	16	77	1	1	NUM
ma-247	16	78	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	16	79	,	,	PUNCT
ma-247	16	80	ε	ε	PROPN
ma-247	16	81	)	)	PUNCT
ma-247	16	82	)	)	PUNCT
ma-247	17	1	∫	∫	PROPN
ma-247	17	2	bk(a	bk(a	NOUN
ma-247	17	3	,	,	PUNCT
ma-247	17	4	ε)×cn−k	ε)×cn−k	PROPN
ma-247	17	5	t	t	PROPN
ma-247	17	6	∧	∧	PROPN
ma-247	17	7	(	(	PUNCT
ma-247	17	8	ddc	ddc	PROPN
ma-247	17	9	ϕ̃)k	ϕ̃)k	PROPN
ma-247	17	10	∧ψ	∧ψ	PROPN
ma-247	17	11	.	.	PUNCT
ma-247	18	1	(	(	PUNCT
ma-247	18	2	1.1	1.1	NUM
ma-247	18	3	)	)	PUNCT
ma-247	18	4	the	the	DET
ma-247	18	5	definition	definition	NOUN
ma-247	18	6	(	(	PUNCT
ma-247	18	7	1.1	1.1	NUM
ma-247	18	8	)	)	PUNCT
ma-247	18	9	makes	make	VERB
ma-247	18	10	sense	sense	NOUN
ma-247	18	11	as	as	ADV
ma-247	18	12	well	well	ADV
ma-247	18	13	as	as	SCONJ
ma-247	18	14	the	the	DET
ma-247	18	15	limit	limit	NOUN
ma-247	18	16	exists	exist	VERB
ma-247	18	17	in	in	ADP
ma-247	18	18	c	c	PROPN
ma-247	18	19	for	for	ADP
ma-247	18	20	any	any	DET
ma-247	18	21	test	test	NOUN
ma-247	18	22	form	form	NOUN
ma-247	18	23	ψ	ψ	ADP
ma-247	18	24	∈	∈	PROPN
ma-247	18	25	d(p−k	d(p−k	PROPN
ma-247	18	26	,	,	PUNCT
ma-247	18	27	p−k)(ω),where	p−k)(ω),where	NUM
ma-247	18	28	,	,	PUNCT
ma-247	18	29	ϕ̃	ϕ̃	PROPN
ma-247	19	1	=	=	SYM
ma-247	19	2	ϕ	ϕ	PROPN
ma-247	19	3	◦	◦	NOUN
ma-247	19	4	π	π	PROPN
ma-247	19	5	stands	stand	VERB
ma-247	19	6	for	for	ADP
ma-247	19	7	the	the	DET
ma-247	19	8	composite	composite	NOUN
ma-247	19	9	of	of	ADP
ma-247	19	10	ϕ	ϕ	NOUN
ma-247	19	11	with	with	ADP
ma-247	19	12	the	the	DET
ma-247	19	13	projection	projection	NOUN
ma-247	19	14	map	map	NOUN
ma-247	19	15	π	π	X
ma-247	19	16	:	:	PUNCT
ma-247	20	1	cn	cn	VERB
ma-247	20	2	−→	−→	ADJ
ma-247	20	3	ck	ck	PROPN
ma-247	20	4	(	(	PUNCT
ma-247	20	5	z	z	NOUN
ma-247	20	6	′	′	NUM
ma-247	20	7	,	,	PUNCT
ma-247	20	8	z	z	PROPN
ma-247	20	9	′′	′′	PROPN
ma-247	20	10	)	)	PUNCT
ma-247	20	11	7−→	7−→	PROPN
ma-247	20	12	z	z	NOUN
ma-247	20	13	′	′	NUM
ma-247	20	14	received	receive	VERB
ma-247	20	15	:	:	PUNCT
ma-247	20	16	6	6	NUM
ma-247	20	17	jun	jun	PROPN
ma-247	20	18	2024.2020	2024.2020	NUM
ma-247	20	19	mathematics	mathematic	NOUN
ma-247	20	20	subject	subject	ADJ
ma-247	20	21	classification	classification	NOUN
ma-247	20	22	.	.	PUNCT
ma-247	21	1	32u05	32u05	NUM
ma-247	21	2	,	,	PUNCT
ma-247	21	3	31c10	31c10	NUM
ma-247	21	4	,	,	PUNCT
ma-247	21	5	32u25	32u25	NUM
ma-247	21	6	,	,	PUNCT
ma-247	21	7	32c30	32c30	NUM
ma-247	21	8	,	,	PUNCT
ma-247	21	9	32u40	32u40	NUM
ma-247	21	10	.	.	PUNCT
ma-247	22	1	key	key	ADJ
ma-247	22	2	words	word	NOUN
ma-247	22	3	and	and	CCONJ
ma-247	22	4	phrases	phrase	NOUN
ma-247	22	5	.	.	PUNCT
ma-247	23	1	plurisubharmonic	plurisubharmonic	ADJ
ma-247	23	2	function	function	NOUN
ma-247	23	3	,	,	PUNCT
ma-247	23	4	plurisubharmonic	plurisubharmonic	ADJ
ma-247	23	5	current	current	ADJ
ma-247	23	6	,	,	PUNCT
ma-247	23	7	slice	slice	NOUN
ma-247	23	8	,	,	PUNCT
ma-247	23	9	analytic	analytic	ADJ
ma-247	23	10	set	set	NOUN
ma-247	23	11	,	,	PUNCT
ma-247	23	12	complete	complete	ADJ
ma-247	23	13	intersection.1	intersection.1	PROPN
ma-247	23	14	https://adac.ee	https://adac.ee	PROPN
ma-247	23	15	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	X
ma-247	23	16	eur	eur	PROPN
ma-247	23	17	.	.	PUNCT
ma-247	24	1	j.	j.	PROPN
ma-247	24	2	math	math	PROPN
ma-247	24	3	.	.	PUNCT
ma-247	25	1	anal	anal	PROPN
ma-247	25	2	.	.	PUNCT
ma-247	26	1	10.28924	10.28924	NUM
ma-247	26	2	/	/	SYM
ma-247	26	3	ada	ada	PROPN
ma-247	26	4	/	/	SYM
ma-247	26	5	ma.4.23	ma.4.23	PROPN
ma-247	27	1	2and	2and	NUM
ma-247	27	2	bk(a	bk(a	NOUN
ma-247	27	3	,	,	PUNCT
ma-247	27	4	ε	ε	PROPN
ma-247	27	5	)	)	PUNCT
ma-247	27	6	is	be	AUX
ma-247	27	7	the	the	DET
ma-247	27	8	open	open	ADJ
ma-247	27	9	ball	ball	NOUN
ma-247	27	10	in	in	ADP
ma-247	27	11	ck	ck	PROPN
ma-247	27	12	centered	center	VERB
ma-247	27	13	at	at	ADP
ma-247	27	14	point	point	NOUN
ma-247	27	15	a	a	PRON
ma-247	27	16	and	and	CCONJ
ma-247	27	17	of	of	ADP
ma-247	27	18	radius	radius	NOUN
ma-247	27	19	ε	ε	PROPN
ma-247	27	20	>	>	X
ma-247	27	21	0.in	0.in	NUM
ma-247	27	22	the	the	DET
ma-247	27	23	present	present	ADJ
ma-247	27	24	work	work	NOUN
ma-247	27	25	we	we	PRON
ma-247	27	26	are	be	AUX
ma-247	27	27	concerned	concern	VERB
ma-247	27	28	by	by	ADP
ma-247	27	29	topics	topic	NOUN
ma-247	27	30	studied	study	VERB
ma-247	27	31	in	in	ADP
ma-247	27	32	[	[	X
ma-247	27	33	9–12	9–12	NOUN
ma-247	27	34	]	]	PUNCT
ma-247	27	35	.	.	PUNCT
ma-247	28	1	particularly	particularly	ADV
ma-247	28	2	,	,	PUNCT
ma-247	28	3	we	we	PRON
ma-247	28	4	will	will	AUX
ma-247	28	5	studytopics	studytopic	NOUN
ma-247	28	6	based	base	VERB
ma-247	28	7	on	on	ADP
ma-247	28	8	the	the	DET
ma-247	28	9	definition	definition	NOUN
ma-247	28	10	(	(	PUNCT
ma-247	28	11	1.1	1.1	NUM
ma-247	28	12	)	)	PUNCT
ma-247	28	13	recently	recently	ADV
ma-247	28	14	investigated	investigate	VERB
ma-247	28	15	in	in	ADP
ma-247	28	16	[	[	X
ma-247	28	17	11	11	NUM
ma-247	28	18	]	]	PUNCT
ma-247	28	19	,	,	PUNCT
ma-247	28	20	for	for	ADP
ma-247	28	21	negative	negative	ADJ
ma-247	28	22	psh	psh	NOUN
ma-247	28	23	currents	current	NOUN
ma-247	28	24	of	of	ADP
ma-247	28	25	smallsupport	smallsupport	NOUN
ma-247	28	26	.	.	PUNCT
ma-247	29	1	a	a	DET
ma-247	29	2	fundamental	fundamental	ADJ
ma-247	29	3	result	result	NOUN
ma-247	29	4	was	be	AUX
ma-247	29	5	provided	provide	VERB
ma-247	29	6	showing	show	VERB
ma-247	29	7	that	that	SCONJ
ma-247	29	8	for	for	ADP
ma-247	29	9	any	any	DET
ma-247	29	10	psh	psh	NOUN
ma-247	29	11	function	function	NOUN
ma-247	29	12	v	v	ADP
ma-247	29	13	non	non	ADJ
ma-247	29	14	identicallyequals	identicallyequal	NOUN
ma-247	29	15	−∞	−∞	ADP
ma-247	29	16	,	,	PUNCT
ma-247	29	17	there	there	PRON
ma-247	29	18	exists	exist	VERB
ma-247	29	19	a	a	DET
ma-247	29	20	pluripolar	pluripolar	ADJ
ma-247	29	21	subset	subset	NOUN
ma-247	29	22	e	e	NOUN
ma-247	29	23	in	in	ADP
ma-247	29	24	ck	ck	PROPN
ma-247	29	25	,	,	PUNCT
ma-247	29	26	such	such	ADJ
ma-247	29	27	that	that	SCONJ
ma-247	29	28	the	the	DET
ma-247	29	29	slice	slice	NOUN
ma-247	29	30	<	<	X
ma-247	29	31	v	v	PROPN
ma-247	29	32	,	,	PUNCT
ma-247	29	33	π	π	PROPN
ma-247	29	34	,	,	PUNCT
ma-247	29	35	a	a	DET
ma-247	29	36	>	>	X
ma-247	29	37	ϕ	ϕ	NOUN
ma-247	29	38	of	of	ADP
ma-247	29	39	v	v	NOUN
ma-247	29	40	at	at	ADP
ma-247	29	41	point	point	NOUN
ma-247	29	42	a	a	PRON
ma-247	29	43	is	be	AUX
ma-247	29	44	well	well	ADV
ma-247	29	45	defined	define	VERB
ma-247	29	46	and	and	CCONJ
ma-247	29	47	is	be	AUX
ma-247	29	48	expressed	express	VERB
ma-247	29	49	explicitly	explicitly	ADV
ma-247	29	50	by	by	ADP
ma-247	29	51	v(a	v(a	NOUN
ma-247	29	52	,	,	PUNCT
ma-247	29	53	.	.	PUNCT
ma-247	29	54	)	)	PUNCT
ma-247	30	1	as	as	ADV
ma-247	30	2	well	well	ADV
ma-247	30	3	as	as	ADP
ma-247	30	4	the	the	DET
ma-247	30	5	point	point	NOUN
ma-247	30	6	a	a	DET
ma-247	30	7	lies	lie	NOUN
ma-247	30	8	outside	outside	ADP
ma-247	30	9	e.as	e.as	PROPN
ma-247	30	10	a	a	DET
ma-247	30	11	consequence	consequence	NOUN
ma-247	30	12	,	,	PUNCT
ma-247	30	13	an	an	DET
ma-247	30	14	existence	existence	NOUN
ma-247	30	15	slicing	slice	VERB
ma-247	30	16	result	result	NOUN
ma-247	30	17	was	be	AUX
ma-247	30	18	deduced	deduce	VERB
ma-247	30	19	for	for	ADP
ma-247	30	20	negative	negative	ADJ
ma-247	30	21	psh	psh	NOUN
ma-247	30	22	currents	current	NOUN
ma-247	30	23	having	have	VERB
ma-247	30	24	theirsupports	theirsupport	NOUN
ma-247	30	25	contained	contain	VERB
ma-247	30	26	in	in	ADP
ma-247	30	27	a	a	DET
ma-247	30	28	strip	strip	NOUN
ma-247	30	29	.	.	PUNCT
ma-247	31	1	in	in	ADP
ma-247	31	2	particular	particular	ADJ
ma-247	31	3	,	,	PUNCT
ma-247	31	4	for	for	ADP
ma-247	31	5	positive	positive	ADJ
ma-247	31	6	or	or	CCONJ
ma-247	31	7	negative	negative	ADJ
ma-247	31	8	closed	closed	ADJ
ma-247	31	9	currents	current	NOUN
ma-247	31	10	with	with	ADP
ma-247	31	11	supportcontained	supportcontaine	VERB
ma-247	31	12	in	in	ADP
ma-247	31	13	a	a	DET
ma-247	31	14	strip	strip	NOUN
ma-247	31	15	,	,	PUNCT
ma-247	31	16	it	it	PRON
ma-247	31	17	was	be	AUX
ma-247	31	18	established	establish	VERB
ma-247	31	19	that	that	SCONJ
ma-247	31	20	the	the	DET
ma-247	31	21	slices	slice	NOUN
ma-247	31	22	are	be	AUX
ma-247	31	23	well	well	ADV
ma-247	31	24	defined	define	VERB
ma-247	31	25	and	and	CCONJ
ma-247	31	26	are	be	AUX
ma-247	31	27	vanishing	vanish	VERB
ma-247	31	28	everywhereas	everywhereas	PROPN
ma-247	31	29	soon	soon	ADV
ma-247	31	30	as	as	SCONJ
ma-247	31	31	they	they	PRON
ma-247	31	32	are	be	AUX
ma-247	31	33	vanishing	vanish	VERB
ma-247	31	34	outside	outside	ADP
ma-247	31	35	a	a	DET
ma-247	31	36	pluripolar	pluripolar	ADJ
ma-247	31	37	subset.in	subset.in	PRON
ma-247	31	38	this	this	DET
ma-247	31	39	paper	paper	NOUN
ma-247	31	40	,	,	PUNCT
ma-247	31	41	we	we	PRON
ma-247	31	42	shall	shall	AUX
ma-247	31	43	develop	develop	VERB
ma-247	31	44	the	the	DET
ma-247	31	45	work	work	NOUN
ma-247	31	46	in	in	ADP
ma-247	31	47	[	[	X
ma-247	31	48	11	11	NUM
ma-247	31	49	]	]	PUNCT
ma-247	31	50	by	by	ADP
ma-247	31	51	providing	provide	VERB
ma-247	31	52	others	other	NOUN
ma-247	31	53	slicing	slice	VERB
ma-247	31	54	results	result	NOUN
ma-247	31	55	for	for	ADP
ma-247	31	56	negativepsh	negativepsh	ADJ
ma-247	31	57	currents	current	NOUN
ma-247	31	58	which	which	PRON
ma-247	31	59	are	be	AUX
ma-247	31	60	arising	arise	VERB
ma-247	31	61	from	from	ADP
ma-247	31	62	analytic	analytic	ADJ
ma-247	31	63	subsets	subset	NOUN
ma-247	31	64	having	have	VERB
ma-247	31	65	suitable	suitable	ADJ
ma-247	31	66	intersections.among	intersections.among	PRON
ma-247	31	67	several	several	ADJ
ma-247	31	68	currents	current	NOUN
ma-247	31	69	,	,	PUNCT
ma-247	31	70	we	we	PRON
ma-247	31	71	are	be	AUX
ma-247	31	72	interested	interested	ADJ
ma-247	31	73	on	on	ADP
ma-247	31	74	those	those	PRON
ma-247	31	75	written	write	VERB
ma-247	31	76	as	as	ADP
ma-247	31	77	finite	finite	ADJ
ma-247	31	78	sums	sum	NOUN
ma-247	31	79	of	of	ADP
ma-247	31	80	the	the	DET
ma-247	31	81	form∑	form∑	PROPN
ma-247	31	82	1≤j≤n	1≤j≤n	NUM
ma-247	31	83	(	(	PUNCT
ma-247	31	84	log	log	NOUN
ma-247	31	85	|fj	|fj	X
ma-247	31	86	|	|	NOUN
ma-247	31	87	)	)	PUNCT
ma-247	31	88	[	[	PUNCT
ma-247	31	89	xj	xj	X
ma-247	31	90	]	]	PUNCT
ma-247	31	91	,	,	PUNCT
ma-247	31	92	∑	∑	PROPN
ma-247	31	93	1≤j≤n	1≤j≤n	NUM
ma-247	31	94	vj	vj	PROPN
ma-247	32	1	[	[	X
ma-247	32	2	xj	xj	X
ma-247	32	3	]	]	X
ma-247	32	4	,	,	PUNCT
ma-247	32	5	∑	∑	PROPN
ma-247	32	6	1≤j≤n	1≤j≤n	NUM
ma-247	32	7	u[xj	u[xj	PROPN
ma-247	32	8	]	]	PUNCT
ma-247	32	9	∧	∧	PROPN
ma-247	33	1	[	[	X
ma-247	33	2	yj	yj	X
ma-247	33	3	]	]	X
ma-247	33	4	,	,	PUNCT
ma-247	33	5	or	or	CCONJ
ma-247	33	6	∑	∑	PROPN
ma-247	33	7	1≤j≤n	1≤j≤n	NUM
ma-247	33	8	[	[	X
ma-247	33	9	xj	xj	X
ma-247	33	10	]	]	X
ma-247	33	11	∧	∧	PROPN
ma-247	33	12	u[yj	u[yj	PROPN
ma-247	33	13	]	]	PUNCT
ma-247	33	14	,	,	PUNCT
ma-247	33	15	where	where	SCONJ
ma-247	33	16	(	(	PUNCT
ma-247	33	17	fj)1≤j≤n	fj)1≤j≤n	PROPN
ma-247	33	18	is	be	AUX
ma-247	33	19	a	a	DET
ma-247	33	20	finite	finite	ADJ
ma-247	33	21	family	family	NOUN
ma-247	33	22	of	of	ADP
ma-247	33	23	holomorphic	holomorphic	ADJ
ma-247	33	24	functions	function	NOUN
ma-247	33	25	non	non	X
ma-247	33	26	identically	identically	ADV
ma-247	33	27	vanishing	vanish	VERB
ma-247	33	28	on	on	ADP
ma-247	33	29	∆n	∆n	PROPN
ma-247	33	30	,	,	PUNCT
ma-247	33	31	(	(	PUNCT
ma-247	33	32	xj)1≤j≤nis	xj)1≤j≤nis	PROPN
ma-247	33	33	a	a	DET
ma-247	33	34	finite	finite	ADJ
ma-247	33	35	family	family	NOUN
ma-247	33	36	of	of	ADP
ma-247	33	37	analytic	analytic	ADJ
ma-247	33	38	subsets	subset	NOUN
ma-247	33	39	all	all	PRON
ma-247	33	40	of	of	ADP
ma-247	33	41	pure	pure	ADJ
ma-247	33	42	dimension	dimension	NOUN
ma-247	33	43	p	p	X
ma-247	33	44	,	,	PUNCT
ma-247	33	45	(	(	PUNCT
ma-247	33	46	yj)1≤j≤n	yj)1≤j≤n	PROPN
ma-247	33	47	is	be	AUX
ma-247	33	48	another	another	DET
ma-247	33	49	finite	finite	ADJ
ma-247	33	50	family	family	NOUN
ma-247	33	51	ofanalytic	ofanalytic	ADJ
ma-247	33	52	subsets	subset	NOUN
ma-247	33	53	such	such	ADJ
ma-247	33	54	that	that	SCONJ
ma-247	33	55	each	each	DET
ma-247	33	56	yj	yj	PROPN
ma-247	33	57	forms	form	VERB
ma-247	33	58	a	a	DET
ma-247	33	59	complete	complete	ADJ
ma-247	33	60	intersection	intersection	NOUN
ma-247	33	61	with	with	ADP
ma-247	33	62	xj	xj	PROPN
ma-247	33	63	in	in	ADP
ma-247	33	64	∆n	∆n	PROPN
ma-247	33	65	,	,	PUNCT
ma-247	33	66	(	(	PUNCT
ma-247	33	67	vj)1≤j≤n	vj)1≤j≤n	PROPN
ma-247	33	68	is	be	AUX
ma-247	33	69	afinite	afinite	ADJ
ma-247	33	70	family	family	NOUN
ma-247	33	71	of	of	ADP
ma-247	33	72	psh	psh	NOUN
ma-247	33	73	functions	function	NOUN
ma-247	33	74	non	non	X
ma-247	33	75	identically	identically	ADV
ma-247	33	76	−∞	−∞	X
ma-247	33	77	on	on	ADP
ma-247	33	78	∆n	∆n	PROPN
ma-247	33	79	and	and	CCONJ
ma-247	33	80	u[xj	u[xj	PROPN
ma-247	33	81	]	]	PUNCT
ma-247	33	82	is	be	AUX
ma-247	33	83	the	the	DET
ma-247	33	84	lelong	lelong	PROPN
ma-247	33	85	-	-	PUNCT
ma-247	33	86	skoda	skoda	PROPN
ma-247	33	87	potentialassociated	potentialassociate	VERB
ma-247	33	88	to	to	ADP
ma-247	33	89	the	the	DET
ma-247	33	90	current	current	NOUN
ma-247	33	91	[	[	X
ma-247	33	92	xj	xj	X
ma-247	33	93	]	]	X
ma-247	33	94	,	,	PUNCT
ma-247	33	95	(	(	PUNCT
ma-247	33	96	1	1	NUM
ma-247	33	97	≤	≤	NUM
ma-247	33	98	j	j	PROPN
ma-247	33	99	≤	≤	PROPN
ma-247	33	100	n	n	CCONJ
ma-247	33	101	)	)	PUNCT
ma-247	33	102	.	.	PUNCT
ma-247	34	1	the	the	DET
ma-247	34	2	first	first	ADJ
ma-247	34	3	result	result	NOUN
ma-247	34	4	describes	describe	VERB
ma-247	34	5	properly	properly	ADV
ma-247	34	6	the	the	DET
ma-247	34	7	slices	slice	NOUN
ma-247	34	8	of	of	ADP
ma-247	34	9	the	the	DET
ma-247	34	10	current	current	ADJ
ma-247	34	11	∑	∑	PROPN
ma-247	34	12	1≤j≤n	1≤j≤n	NUM
ma-247	34	13	(	(	PUNCT
ma-247	34	14	log	log	NOUN
ma-247	34	15	|fj	|fj	X
ma-247	34	16	|	|	NOUN
ma-247	34	17	)	)	PUNCT
ma-247	34	18	[	[	PUNCT
ma-247	34	19	xj	xj	X
ma-247	34	20	]	]	PUNCT
ma-247	34	21	.	.	PUNCT
ma-247	35	1	theorem	theorem	VERB
ma-247	35	2	1.1	1.1	NUM
ma-247	35	3	.	.	PUNCT
ma-247	36	1	assume	assume	VERB
ma-247	36	2	we	we	PRON
ma-247	36	3	have	have	VERB
ma-247	36	4	in	in	ADP
ma-247	36	5	∆n	∆n	PROPN
ma-247	36	6	,	,	PUNCT
ma-247	36	7	a	a	DET
ma-247	36	8	finite	finite	ADJ
ma-247	36	9	family	family	NOUN
ma-247	36	10	(	(	PUNCT
ma-247	36	11	xj)1≤j≤n	xj)1≤j≤n	PROPN
ma-247	36	12	of	of	ADP
ma-247	36	13	analytic	analytic	ADJ
ma-247	36	14	subsets	subset	NOUN
ma-247	36	15	all	all	PRON
ma-247	36	16	of	of	ADP
ma-247	36	17	the	the	DET
ma-247	36	18	same	same	ADJ
ma-247	36	19	pure	pure	ADJ
ma-247	36	20	dimension	dimension	NOUN
ma-247	37	1	q	q	PROPN
ma-247	37	2	>	>	X
ma-247	37	3	k	k	PROPN
ma-247	37	4	and	and	CCONJ
ma-247	37	5	a	a	DET
ma-247	37	6	finite	finite	ADJ
ma-247	37	7	family	family	NOUN
ma-247	37	8	(	(	PUNCT
ma-247	37	9	fj)1≤j≤n	fj)1≤j≤n	PROPN
ma-247	37	10	of	of	ADP
ma-247	37	11	holomorphic	holomorphic	ADJ
ma-247	37	12	functions	function	NOUN
ma-247	37	13	non	non	X
ma-247	37	14	identically	identically	ADV
ma-247	37	15	vanishing	vanish	VERB
ma-247	37	16	such	such	DET
ma-247	37	17	that	that	PRON
ma-247	37	18	for	for	ADP
ma-247	37	19	all	all	DET
ma-247	37	20	1	1	NUM
ma-247	37	21	≤	≤	NUM
ma-247	37	22	j	j	PROPN
ma-247	37	23	≤	≤	NOUN
ma-247	37	24	n	n	CCONJ
ma-247	37	25	the	the	DET
ma-247	37	26	hypersurface	hypersurface	NOUN
ma-247	37	27	yj	yj	PROPN
ma-247	37	28	=	=	PROPN
ma-247	37	29	{	{	PUNCT
ma-247	37	30	fj	fj	X
ma-247	37	31	=	=	SYM
ma-247	37	32	0	0	NUM
ma-247	37	33	}	}	PUNCT
ma-247	37	34	yields	yield	VERB
ma-247	37	35	a	a	DET
ma-247	37	36	complete	complete	ADJ
ma-247	37	37	intersection	intersection	NOUN
ma-247	37	38	near	near	ADP
ma-247	37	39	the	the	DET
ma-247	37	40	origin	origin	NOUN
ma-247	37	41	with	with	ADP
ma-247	37	42	xj	xj	PROPN
ma-247	37	43	.	.	PUNCT
ma-247	38	1	if	if	SCONJ
ma-247	38	2	σj	σj	VERB
ma-247	38	3	:	:	PUNCT
ma-247	38	4	xj	xj	PROPN
ma-247	38	5	∩	∩	PROPN
ma-247	38	6	π−1(a	π−1(a	PROPN
ma-247	38	7	)	)	PUNCT
ma-247	39	1	−→	−→	PROPN
ma-247	39	2	∆n	∆n	PROPN
ma-247	39	3	denotes	denote	VERB
ma-247	39	4	the	the	DET
ma-247	39	5	canonical	canonical	ADJ
ma-247	39	6	injection	injection	NOUN
ma-247	39	7	,	,	PUNCT
ma-247	39	8	then	then	ADV
ma-247	39	9	,	,	PUNCT
ma-247	39	10	there	there	PRON
ma-247	39	11	exist	exist	VERB
ma-247	39	12	,	,	PUNCT
ma-247	39	13	a	a	DET
ma-247	39	14	closed	closed	ADJ
ma-247	39	15	pluripolar	pluripolar	ADJ
ma-247	39	16	subset	subset	NOUN
ma-247	39	17	e	e	NOUN
ma-247	39	18	of	of	ADP
ma-247	39	19	∆k	∆k	PROPN
ma-247	39	20	,	,	PUNCT
ma-247	39	21	an	an	DET
ma-247	39	22	integer	integer	NOUN
ma-247	39	23	m	m	NOUN
ma-247	39	24	∈	∈	PROPN
ma-247	39	25	n	n	ADJ
ma-247	39	26	and	and	CCONJ
ma-247	39	27	homogenous	homogenous	ADJ
ma-247	39	28	polynomials	polynomial	NOUN
ma-247	39	29	qm	qm	PROPN
ma-247	39	30	,	,	PUNCT
ma-247	39	31	j	j	PROPN
ma-247	39	32	on	on	ADP
ma-247	39	33	xj	xj	PROPN
ma-247	39	34	,	,	PUNCT
ma-247	39	35	1	1	NUM
ma-247	39	36	≤	≤	NUM
ma-247	39	37	j	j	PROPN
ma-247	39	38	≤	≤	PROPN
ma-247	39	39	n	n	CCONJ
ma-247	39	40	,	,	PUNCT
ma-247	39	41	such	such	ADJ
ma-247	39	42	that	that	SCONJ
ma-247	39	43	for	for	ADP
ma-247	39	44	all	all	DET
ma-247	39	45	a	a	DET
ma-247	39	46	∈	∈	NOUN
ma-247	39	47	∆k	∆k	NOUN
ma-247	39	48	r	r	NOUN
ma-247	39	49	e	e	NOUN
ma-247	39	50	the	the	DET
ma-247	39	51	slice	slice	NOUN
ma-247	39	52	〈	〈	PROPN
ma-247	39	53	∑	∑	PROPN
ma-247	39	54	1≤j≤n	1≤j≤n	NUM
ma-247	39	55	(	(	PUNCT
ma-247	39	56	log|fj	log|fj	ADJ
ma-247	39	57	|)[xj	|)[xj	PROPN
ma-247	40	1	]	]	PUNCT
ma-247	40	2	,	,	PUNCT
ma-247	40	3	π	π	PROPN
ma-247	40	4	,	,	PUNCT
ma-247	40	5	a〉ϕ	a〉ϕ	PROPN
ma-247	40	6	is	be	AUX
ma-247	40	7	defined	define	VERB
ma-247	40	8	such	such	ADJ
ma-247	40	9	that(1	that(1	NOUN
ma-247	40	10	)	)	PUNCT
ma-247	40	11	if	if	SCONJ
ma-247	40	12	σj∗fj	σj∗fj	PROPN
ma-247	40	13	6≡	6≡	NUM
ma-247	40	14	0	0	NUM
ma-247	40	15	on	on	ADP
ma-247	40	16	xj	xj	PROPN
ma-247	40	17	∩	∩	PROPN
ma-247	40	18	π−1(a	π−1(a	PROPN
ma-247	40	19	)	)	PUNCT
ma-247	40	20	,	,	PUNCT
ma-247	40	21	∀	∀	NOUN
ma-247	40	22	1	1	NUM
ma-247	40	23	≤	≤	NUM
ma-247	40	24	j	j	PROPN
ma-247	40	25	≤	≤	NUM
ma-247	40	26	n	n	CCONJ
ma-247	40	27	,	,	PUNCT
ma-247	40	28	then	then	ADV
ma-247	40	29	〈	〈	PROPN
ma-247	40	30	∑	∑	PROPN
ma-247	40	31	1≤j≤n	1≤j≤n	NUM
ma-247	40	32	(	(	PUNCT
ma-247	40	33	log|fj	log|fj	ADJ
ma-247	40	34	|)[xj	|)[xj	PROPN
ma-247	40	35	]	]	PUNCT
ma-247	40	36	,	,	PUNCT
ma-247	40	37	π	π	PROPN
ma-247	40	38	,	,	PUNCT
ma-247	40	39	a〉ϕ	a〉ϕ	PROPN
ma-247	40	40	=	=	SYM
ma-247	40	41	∑	∑	PROPN
ma-247	40	42	1≤j≤n	1≤j≤n	NUM
ma-247	40	43	(	(	PUNCT
ma-247	40	44	log	log	NOUN
ma-247	40	45	|σj∗fj	|σj∗fj	NOUN
ma-247	40	46	|	|	ADV
ma-247	40	47	)	)	PUNCT
ma-247	40	48	[	[	PUNCT
ma-247	40	49	xj	xj	PROPN
ma-247	40	50	∩	∩	PROPN
ma-247	40	51	π−1(a	π−1(a	PROPN
ma-247	40	52	)	)	PUNCT
ma-247	40	53	]	]	PUNCT
ma-247	40	54	,	,	PUNCT
ma-247	40	55	(	(	PUNCT
ma-247	40	56	2	2	X
ma-247	40	57	)	)	PUNCT
ma-247	40	58	if	if	SCONJ
ma-247	40	59	σj∗fj	σj∗fj	PROPN
ma-247	40	60	≡	≡	PROPN
ma-247	40	61	0	0	PUNCT
ma-247	40	62	and	and	CCONJ
ma-247	40	63	qm	qm	PROPN
ma-247	40	64	,	,	PUNCT
ma-247	40	65	j	j	PROPN
ma-247	40	66	|xj∩π−1(a	|xj∩π−1(a	ADP
ma-247	40	67	)	)	PUNCT
ma-247	40	68	6≡	6≡	NUM
ma-247	40	69	0	0	NUM
ma-247	40	70	,	,	PUNCT
ma-247	40	71	∀	∀	X
ma-247	40	72	1	1	NUM
ma-247	40	73	≤	≤	NUM
ma-247	40	74	j	j	PROPN
ma-247	40	75	≤	≤	NUM
ma-247	40	76	n	n	CCONJ
ma-247	40	77	,	,	PUNCT
ma-247	40	78	then	then	ADV
ma-247	40	79	〈	〈	PROPN
ma-247	40	80	∑	∑	PROPN
ma-247	40	81	1≤j≤n	1≤j≤n	NUM
ma-247	40	82	(	(	PUNCT
ma-247	40	83	log|fj	log|fj	ADJ
ma-247	40	84	|)[xj	|)[xj	PROPN
ma-247	40	85	]	]	PUNCT
ma-247	40	86	,	,	PUNCT
ma-247	40	87	π	π	PROPN
ma-247	40	88	,	,	PUNCT
ma-247	40	89	a〉ϕ	a〉ϕ	PROPN
ma-247	40	90	=	=	SYM
ma-247	40	91	∑	∑	PROPN
ma-247	40	92	1≤j≤n	1≤j≤n	NUM
ma-247	40	93	(	(	PUNCT
ma-247	40	94	log	log	PROPN
ma-247	40	95	|σj∗qm	|σj∗qm	PROPN
ma-247	40	96	,	,	PUNCT
ma-247	40	97	j	j	PROPN
ma-247	40	98	|	|	ADV
ma-247	40	99	)	)	PUNCT
ma-247	40	100	[	[	PUNCT
ma-247	40	101	xj	xj	PROPN
ma-247	40	102	∩	∩	PROPN
ma-247	40	103	π−1(a	π−1(a	PROPN
ma-247	40	104	)	)	PUNCT
ma-247	40	105	]	]	PUNCT
ma-247	40	106	.	.	PUNCT
ma-247	41	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	41	2	eur	eur	PROPN
ma-247	41	3	.	.	PUNCT
ma-247	42	1	j.	j.	PROPN
ma-247	42	2	math	math	PROPN
ma-247	42	3	.	.	PUNCT
ma-247	43	1	anal	anal	PROPN
ma-247	43	2	.	.	PUNCT
ma-247	44	1	10.28924	10.28924	NUM
ma-247	44	2	/	/	SYM
ma-247	44	3	ada	ada	PROPN
ma-247	44	4	/	/	SYM
ma-247	44	5	ma.4.23	ma.4.23	NOUN
ma-247	44	6	3the	3the	DET
ma-247	44	7	second	second	ADJ
ma-247	44	8	result	result	NOUN
ma-247	44	9	generalizes	generalize	VERB
ma-247	44	10	theorem	theorem	VERB
ma-247	44	11	1.1	1.1	NUM
ma-247	44	12	to	to	ADP
ma-247	44	13	psh	psh	NOUN
ma-247	44	14	functions	function	NOUN
ma-247	44	15	.	.	PUNCT
ma-247	45	1	it	it	PRON
ma-247	45	2	provides	provide	VERB
ma-247	45	3	the	the	DET
ma-247	45	4	existence	existence	NOUN
ma-247	45	5	and	and	CCONJ
ma-247	45	6	theexpression	theexpression	NOUN
ma-247	45	7	of	of	ADP
ma-247	45	8	the	the	DET
ma-247	45	9	slices	slice	NOUN
ma-247	45	10	of	of	ADP
ma-247	45	11	a	a	DET
ma-247	45	12	negative	negative	ADJ
ma-247	45	13	psh	psh	NOUN
ma-247	45	14	current	current	NOUN
ma-247	45	15	of	of	ADP
ma-247	45	16	the	the	DET
ma-247	45	17	form	form	NOUN
ma-247	45	18	∑	∑	PROPN
ma-247	45	19	1≤j≤n	1≤j≤n	NUM
ma-247	45	20	vj	vj	PROPN
ma-247	46	1	[	[	X
ma-247	46	2	xj	xj	X
ma-247	46	3	]	]	PUNCT
ma-247	46	4	.	.	PUNCT
ma-247	47	1	theorem	theorem	VERB
ma-247	47	2	1.2	1.2	NUM
ma-247	47	3	.	.	PUNCT
ma-247	48	1	assume	assume	VERB
ma-247	48	2	we	we	PRON
ma-247	48	3	have	have	VERB
ma-247	48	4	in	in	ADP
ma-247	48	5	∆n	∆n	PROPN
ma-247	48	6	,	,	PUNCT
ma-247	48	7	a	a	DET
ma-247	48	8	finite	finite	ADJ
ma-247	48	9	family	family	NOUN
ma-247	48	10	(	(	PUNCT
ma-247	48	11	xj)1≤j≤n	xj)1≤j≤n	PROPN
ma-247	48	12	of	of	ADP
ma-247	48	13	analytic	analytic	ADJ
ma-247	48	14	subsets	subset	NOUN
ma-247	48	15	all	all	PRON
ma-247	48	16	of	of	ADP
ma-247	48	17	pure	pure	ADJ
ma-247	48	18	dimension	dimension	NOUN
ma-247	48	19	p	p	NOUN
ma-247	48	20	and	and	CCONJ
ma-247	48	21	a	a	DET
ma-247	48	22	finite	finite	ADJ
ma-247	48	23	family	family	NOUN
ma-247	48	24	(	(	PUNCT
ma-247	48	25	vj)1≤j≤n	vj)1≤j≤n	PROPN
ma-247	48	26	of	of	ADP
ma-247	48	27	negative	negative	ADJ
ma-247	48	28	psh	psh	NOUN
ma-247	48	29	functions	function	NOUN
ma-247	48	30	such	such	ADJ
ma-247	48	31	that	that	PRON
ma-247	48	32	for	for	ADP
ma-247	48	33	all	all	DET
ma-247	48	34	1	1	NUM
ma-247	48	35	≤	≤	NUM
ma-247	48	36	j	j	PROPN
ma-247	48	37	≤	≤	PROPN
ma-247	48	38	n	n	CCONJ
ma-247	48	39	the	the	DET
ma-247	48	40	set	set	NOUN
ma-247	48	41	of	of	ADP
ma-247	48	42	singular	singular	ADJ
ma-247	48	43	points	point	NOUN
ma-247	48	44	of	of	ADP
ma-247	48	45	vj	vj	PROPN
ma-247	48	46	is	be	AUX
ma-247	48	47	contained	contain	VERB
ma-247	48	48	in	in	ADP
ma-247	48	49	a	a	DET
ma-247	48	50	hypersurface	hypersurface	NOUN
ma-247	48	51	yj	yj	NOUN
ma-247	48	52	that	that	PRON
ma-247	48	53	yields	yield	VERB
ma-247	48	54	a	a	DET
ma-247	48	55	complete	complete	ADJ
ma-247	48	56	intersection	intersection	NOUN
ma-247	48	57	with	with	ADP
ma-247	48	58	xj	xj	PROPN
ma-247	48	59	.	.	PUNCT
ma-247	49	1	then	then	ADV
ma-247	49	2	,	,	PUNCT
ma-247	49	3	there	there	PRON
ma-247	49	4	is	be	VERB
ma-247	49	5	a	a	DET
ma-247	49	6	pluripolar	pluripolar	ADJ
ma-247	49	7	subset	subset	NOUN
ma-247	49	8	e	e	NOUN
ma-247	49	9	of	of	ADP
ma-247	49	10	∆k	∆k	PROPN
ma-247	49	11	,	,	PUNCT
ma-247	49	12	such	such	ADJ
ma-247	49	13	that	that	SCONJ
ma-247	49	14	for	for	ADP
ma-247	49	15	all	all	DET
ma-247	49	16	a	a	DET
ma-247	49	17	∈	∈	NOUN
ma-247	49	18	∆k	∆k	PROPN
ma-247	49	19	r	r	NOUN
ma-247	49	20	e	e	NOUN
ma-247	49	21	,	,	PUNCT
ma-247	49	22	the	the	DET
ma-247	49	23	current∑	current∑	PROPN
ma-247	49	24	1≤j≤n	1≤j≤n	NUM
ma-247	49	25	vj	vj	PROPN
ma-247	50	1	[	[	X
ma-247	50	2	xj	xj	X
ma-247	50	3	]	]	PUNCT
ma-247	50	4	admits	admit	VERB
ma-247	50	5	a	a	DET
ma-247	50	6	slice	slice	NOUN
ma-247	50	7	expressed	express	VERB
ma-247	50	8	by	by	ADP
ma-247	50	9	〈	〈	PROPN
ma-247	50	10	∑	∑	PROPN
ma-247	50	11	1≤j≤n	1≤j≤n	NUM
ma-247	50	12	vj	vj	NOUN
ma-247	50	13	.	.	PUNCT
ma-247	51	1	[	[	PUNCT
ma-247	51	2	xj	xj	X
ma-247	51	3	]	]	PUNCT
ma-247	51	4	,	,	PUNCT
ma-247	51	5	π	π	PROPN
ma-247	51	6	,	,	PUNCT
ma-247	51	7	a〉ϕ	a〉ϕ	PROPN
ma-247	51	8	=	=	SYM
ma-247	51	9	∑	∑	PROPN
ma-247	51	10	1≤j≤n	1≤j≤n	NUM
ma-247	51	11	vj	vj	NOUN
ma-247	51	12	|xj∩π−1(a	|xj∩π−1(a	ADP
ma-247	51	13	)	)	PUNCT
ma-247	51	14	.	.	PUNCT
ma-247	52	1	[	[	PUNCT
ma-247	52	2	xj	xj	PROPN
ma-247	52	3	∩	∩	PROPN
ma-247	52	4	π−1(a	π−1(a	PROPN
ma-247	52	5	)	)	PUNCT
ma-247	52	6	]	]	PUNCT
ma-247	52	7	.	.	PUNCT
ma-247	53	1	next	next	ADV
ma-247	53	2	,	,	PUNCT
ma-247	53	3	we	we	PRON
ma-247	53	4	let	let	VERB
ma-247	53	5	n(x	n(x	PRON
ma-247	53	6	)	)	PUNCT
ma-247	53	7	=	=	SYM
ma-247	54	1	−	−	PROPN
ma-247	54	2	1	1	NUM
ma-247	54	3	(	(	PUNCT
ma-247	54	4	n−1)(4π)n	n−1)(4π)n	PROPN
ma-247	54	5	1	1	NUM
ma-247	54	6	|x	|x	NOUN
ma-247	54	7	|2n−2	|2n−2	NOUN
ma-247	54	8	be	be	AUX
ma-247	54	9	the	the	DET
ma-247	54	10	newton	newton	PROPN
ma-247	54	11	kernel	kernel	PROPN
ma-247	54	12	in	in	ADP
ma-247	54	13	cn	cn	PROPN
ma-247	54	14	and	and	CCONJ
ma-247	54	15	we	we	PRON
ma-247	54	16	let	let	VERB
ma-247	54	17	η	η	PROPN
ma-247	54	18	be	be	AUX
ma-247	54	19	a	a	DET
ma-247	54	20	positivesmooth	positivesmooth	ADJ
ma-247	54	21	function	function	NOUN
ma-247	54	22	with	with	ADP
ma-247	54	23	compact	compact	ADJ
ma-247	54	24	support	support	NOUN
ma-247	54	25	in	in	ADP
ma-247	54	26	ω	ω	NUM
ma-247	54	27	such	such	ADJ
ma-247	54	28	that	that	SCONJ
ma-247	54	29	0	0	NUM
ma-247	54	30	≤	≤	NUM
ma-247	54	31	η	η	PROPN
ma-247	54	32	≤	≤	PROPN
ma-247	54	33	1	1	NUM
ma-247	54	34	,	,	PUNCT
ma-247	54	35	η	η	PROPN
ma-247	54	36	≡	≡	PROPN
ma-247	54	37	1	1	NUM
ma-247	54	38	on	on	ADP
ma-247	54	39	a	a	DET
ma-247	54	40	neighborhood	neighborhood	NOUN
ma-247	54	41	of	of	ADP
ma-247	54	42	∆n.for	∆n.for	ADP
ma-247	54	43	a	a	DET
ma-247	54	44	given	give	VERB
ma-247	54	45	analytic	analytic	ADJ
ma-247	54	46	subset	subset	NOUN
ma-247	54	47	x	x	PUNCT
ma-247	54	48	in	in	ADP
ma-247	54	49	∆n	∆n	PROPN
ma-247	54	50	,	,	PUNCT
ma-247	54	51	we	we	PRON
ma-247	54	52	take	take	VERB
ma-247	54	53	the	the	DET
ma-247	54	54	lelong	lelong	PROPN
ma-247	54	55	-	-	PUNCT
ma-247	54	56	skoda	skoda	PROPN
ma-247	54	57	potential	potential	NOUN
ma-247	54	58	u[x	u[x	X
ma-247	54	59	]	]	PUNCT
ma-247	54	60	associated	associate	VERB
ma-247	54	61	to	to	ADP
ma-247	54	62	thecurrent	thecurrent	NOUN
ma-247	54	63	[	[	X
ma-247	54	64	x	x	X
ma-247	54	65	]	]	X
ma-247	54	66	,	,	PUNCT
ma-247	54	67	as	as	ADP
ma-247	54	68	the	the	DET
ma-247	54	69	(	(	PUNCT
ma-247	54	70	n	n	NOUN
ma-247	54	71	−	−	PROPN
ma-247	54	72	p	p	NOUN
ma-247	54	73	−	−	PROPN
ma-247	54	74	1	1	NUM
ma-247	54	75	,	,	PUNCT
ma-247	54	76	n	n	PRON
ma-247	54	77	−	−	PROPN
ma-247	54	78	p	p	PRON
ma-247	54	79	−	−	PROPN
ma-247	54	80	1)-negative	1)-negative	NUM
ma-247	54	81	current	current	NOUN
ma-247	54	82	defined	define	VERB
ma-247	54	83	by	by	ADP
ma-247	54	84	the	the	DET
ma-247	54	85	following	follow	VERB
ma-247	54	86	integral	integral	ADJ
ma-247	54	87	u[x](z	u[x](z	NOUN
ma-247	54	88	)	)	PUNCT
ma-247	54	89	=	=	SYM
ma-247	55	1	∫	∫	PROPN
ma-247	55	2	ξ∈cn	ξ∈cn	PROPN
ma-247	55	3	η(ξ)n(z	η(ξ)n(z	VERB
ma-247	55	4	−	−	PROPN
ma-247	55	5	ξ)[x](ξ	ξ)[x](ξ	NOUN
ma-247	55	6	)	)	PUNCT
ma-247	55	7	∧	∧	PROPN
ma-247	55	8	(	(	PUNCT
ma-247	55	9	ddc	ddc	PROPN
ma-247	55	10	|z	|z	PROPN
ma-247	56	1	−	−	PROPN
ma-247	56	2	ξ|2	ξ|2	PROPN
ma-247	56	3	)	)	PUNCT
ma-247	56	4	n−1	n−1	PROPN
ma-247	56	5	.	.	PUNCT
ma-247	57	1	we	we	PRON
ma-247	57	2	prove	prove	VERB
ma-247	57	3	the	the	DET
ma-247	57	4	third	third	ADJ
ma-247	57	5	result	result	NOUN
ma-247	57	6	of	of	ADP
ma-247	57	7	this	this	DET
ma-247	57	8	paper	paper	NOUN
ma-247	57	9	on	on	ADP
ma-247	57	10	the	the	DET
ma-247	57	11	existence	existence	NOUN
ma-247	57	12	and	and	CCONJ
ma-247	57	13	the	the	DET
ma-247	57	14	expression	expression	NOUN
ma-247	57	15	of	of	ADP
ma-247	57	16	the	the	DET
ma-247	57	17	slices	slice	NOUN
ma-247	57	18	of	of	ADP
ma-247	57	19	thewedge	thewedge	NOUN
ma-247	57	20	product	product	NOUN
ma-247	57	21	current	current	ADJ
ma-247	57	22	∑	∑	PROPN
ma-247	57	23	1≤j≤n	1≤j≤n	NUM
ma-247	57	24	u[xj	u[xj	PROPN
ma-247	57	25	]	]	PUNCT
ma-247	58	1	∧	∧	PROPN
ma-247	59	1	[	[	X
ma-247	59	2	yj	yj	X
ma-247	59	3	]	]	PUNCT
ma-247	59	4	introduced	introduce	VERB
ma-247	59	5	and	and	CCONJ
ma-247	59	6	studied	study	VERB
ma-247	59	7	in	in	ADP
ma-247	59	8	[	[	PUNCT
ma-247	59	9	9	9	NUM
ma-247	59	10	]	]	PUNCT
ma-247	59	11	.	.	PUNCT
ma-247	60	1	theorem	theorem	VERB
ma-247	60	2	1.3	1.3	NUM
ma-247	60	3	.	.	PUNCT
ma-247	61	1	assume	assume	VERB
ma-247	61	2	we	we	PRON
ma-247	61	3	have	have	VERB
ma-247	61	4	in	in	ADP
ma-247	61	5	∆n	∆n	PROPN
ma-247	61	6	,	,	PUNCT
ma-247	61	7	finite	finite	ADJ
ma-247	61	8	families	family	NOUN
ma-247	61	9	(	(	PUNCT
ma-247	61	10	xj)1≤j≤n	xj)1≤j≤n	PROPN
ma-247	61	11	and	and	CCONJ
ma-247	61	12	(	(	PUNCT
ma-247	61	13	yl)1≤l≤n	yl)1≤l≤n	NOUN
ma-247	61	14	of	of	ADP
ma-247	61	15	analytic	analytic	ADJ
ma-247	61	16	subsets	subset	NOUN
ma-247	61	17	such	such	ADJ
ma-247	61	18	that	that	PRON
ma-247	61	19	for	for	ADP
ma-247	61	20	all	all	DET
ma-247	61	21	1	1	NUM
ma-247	61	22	≤	≤	NUM
ma-247	61	23	j	j	PROPN
ma-247	61	24	≤	≤	NUM
ma-247	61	25	n	n	CCONJ
ma-247	61	26	,	,	PUNCT
ma-247	61	27	,	,	PUNCT
ma-247	61	28	xj	xj	PROPN
ma-247	61	29	is	be	AUX
ma-247	61	30	of	of	ADP
ma-247	61	31	pure	pure	ADJ
ma-247	61	32	dimension	dimension	NOUN
ma-247	61	33	p	p	PROPN
ma-247	61	34	and	and	CCONJ
ma-247	61	35	yj	yj	PROPN
ma-247	61	36	is	be	AUX
ma-247	61	37	of	of	ADP
ma-247	61	38	pure	pure	ADJ
ma-247	61	39	dimension	dimension	NOUN
ma-247	61	40	q	q	PUNCT
ma-247	61	41	and	and	CCONJ
ma-247	61	42	xj	xj	PROPN
ma-247	61	43	yields	yield	VERB
ma-247	61	44	a	a	DET
ma-247	61	45	complete	complete	ADJ
ma-247	61	46	intersection	intersection	NOUN
ma-247	61	47	with	with	ADP
ma-247	61	48	yj	yj	PROPN
ma-247	61	49	.	.	PUNCT
ma-247	62	1	then	then	ADV
ma-247	62	2	,	,	PUNCT
ma-247	62	3	there	there	PRON
ma-247	62	4	exists	exist	VERB
ma-247	62	5	a	a	DET
ma-247	62	6	subset	subset	NOUN
ma-247	62	7	e	e	NOUN
ma-247	62	8	contained	contain	VERB
ma-247	62	9	in	in	ADP
ma-247	62	10	a	a	DET
ma-247	62	11	countable	countable	ADJ
ma-247	62	12	union	union	NOUN
ma-247	62	13	of	of	ADP
ma-247	62	14	analytic	analytic	ADJ
ma-247	62	15	subsets	subset	NOUN
ma-247	62	16	of	of	ADP
ma-247	62	17	∆k	∆k	PROPN
ma-247	62	18	of	of	ADP
ma-247	62	19	dimensions	dimension	NOUN
ma-247	62	20	≤	≤	PUNCT
ma-247	63	1	k	k	ADP
ma-247	64	1	−	−	PROPN
ma-247	64	2	1	1	NUM
ma-247	64	3	such	such	ADJ
ma-247	64	4	that	that	DET
ma-247	64	5	for	for	ADP
ma-247	64	6	all	all	DET
ma-247	64	7	a	a	DET
ma-247	64	8	∈	∈	NOUN
ma-247	64	9	∆k	∆k	PROPN
ma-247	64	10	r	r	NOUN
ma-247	64	11	e	e	NOUN
ma-247	64	12	,	,	PUNCT
ma-247	64	13	the	the	DET
ma-247	64	14	slice	slice	NOUN
ma-247	64	15	of∑	of∑	ADP
ma-247	64	16	1≤j≤n	1≤j≤n	NUM
ma-247	64	17	u[xj	u[xj	PROPN
ma-247	64	18	]	]	X
ma-247	65	1	∧	∧	PROPN
ma-247	65	2	[	[	X
ma-247	65	3	yj	yj	X
ma-247	65	4	]	]	X
ma-247	65	5	at	at	ADP
ma-247	65	6	point	point	NOUN
ma-247	65	7	a	a	PRON
ma-247	65	8	is	be	AUX
ma-247	65	9	given	give	VERB
ma-247	65	10	by	by	ADP
ma-247	65	11	〈	〈	PROPN
ma-247	65	12	∑	∑	PROPN
ma-247	65	13	1≤j≤n	1≤j≤n	NUM
ma-247	65	14	u[xj	u[xj	PROPN
ma-247	66	1	]	]	PUNCT
ma-247	66	2	∧	∧	PROPN
ma-247	67	1	[	[	X
ma-247	67	2	yj	yj	X
ma-247	67	3	]	]	X
ma-247	67	4	,	,	PUNCT
ma-247	67	5	π	π	PROPN
ma-247	67	6	,	,	PUNCT
ma-247	67	7	a〉ϕ	a〉ϕ	PROPN
ma-247	67	8	=	=	SYM
ma-247	67	9	∑	∑	PROPN
ma-247	67	10	1≤j≤n	1≤j≤n	NUM
ma-247	67	11	j∗au[xj	j∗au[xj	PROPN
ma-247	67	12	]	]	PUNCT
ma-247	67	13	∧	∧	PROPN
ma-247	67	14	[	[	PUNCT
ma-247	67	15	yj	yj	PROPN
ma-247	67	16	∩	∩	PROPN
ma-247	67	17	π−1(a	π−1(a	PROPN
ma-247	67	18	)	)	PUNCT
ma-247	67	19	]	]	PUNCT
ma-247	67	20	.	.	PUNCT
ma-247	68	1	in	in	ADP
ma-247	68	2	the	the	DET
ma-247	68	3	sequel	sequel	NOUN
ma-247	68	4	,	,	PUNCT
ma-247	68	5	we	we	PRON
ma-247	68	6	show	show	VERB
ma-247	68	7	that	that	SCONJ
ma-247	68	8	the	the	DET
ma-247	68	9	above	above	ADJ
ma-247	68	10	results	result	NOUN
ma-247	68	11	enjoy	enjoy	VERB
ma-247	68	12	interesting	interesting	ADJ
ma-247	68	13	properties	property	NOUN
ma-247	68	14	like	like	ADP
ma-247	68	15	the	the	DET
ma-247	68	16	integrability	integrability	NOUN
ma-247	68	17	ofcoefficients	ofcoefficient	NOUN
ma-247	68	18	of	of	ADP
ma-247	68	19	the	the	DET
ma-247	68	20	above	above	ADJ
ma-247	68	21	currents	current	NOUN
ma-247	68	22	and	and	CCONJ
ma-247	68	23	their	their	PRON
ma-247	68	24	slices	slice	NOUN
ma-247	68	25	.	.	PUNCT
ma-247	69	1	for	for	ADP
ma-247	69	2	instance	instance	NOUN
ma-247	69	3	,	,	PUNCT
ma-247	69	4	the	the	DET
ma-247	69	5	following	follow	VERB
ma-247	69	6	theorem	theorem	VERB
ma-247	69	7	1.4	1.4	NUM
ma-247	69	8	providesan	providesan	ADJ
ma-247	69	9	important	important	ADJ
ma-247	69	10	integrability	integrability	NOUN
ma-247	69	11	property	property	NOUN
ma-247	69	12	for	for	ADP
ma-247	69	13	a	a	DET
ma-247	69	14	given	give	VERB
ma-247	69	15	psh	psh	NOUN
ma-247	69	16	function	function	NOUN
ma-247	69	17	v	v	ADP
ma-247	69	18	not	not	PART
ma-247	69	19	identically	identically	ADV
ma-247	69	20	−∞	−∞	X
ma-247	69	21	on	on	ADP
ma-247	69	22	∆n	∆n	PROPN
ma-247	69	23	.	.	PUNCT
ma-247	70	1	in	in	ADP
ma-247	70	2	fact	fact	NOUN
ma-247	70	3	,	,	PUNCT
ma-247	70	4	itdescribes	itdescribe	VERB
ma-247	70	5	the	the	DET
ma-247	70	6	integrability	integrability	NOUN
ma-247	70	7	of	of	ADP
ma-247	70	8	the	the	DET
ma-247	70	9	function	function	NOUN
ma-247	70	10	exp(−v	exp(−v	PROPN
ma-247	70	11	)	)	PUNCT
ma-247	70	12	across	across	ADP
ma-247	70	13	an	an	DET
ma-247	70	14	analytic	analytic	ADJ
ma-247	70	15	subset	subset	NOUN
ma-247	70	16	x	x	X
ma-247	70	17	.	.	PUNCT
ma-247	71	1	consequently	consequently	ADV
ma-247	71	2	,	,	PUNCT
ma-247	71	3	weget	weget	VERB
ma-247	71	4	the	the	DET
ma-247	71	5	following	follow	VERB
ma-247	71	6	beautiful	beautiful	ADJ
ma-247	71	7	result	result	NOUN
ma-247	71	8	in	in	ADP
ma-247	71	9	terms	term	NOUN
ma-247	71	10	of	of	ADP
ma-247	71	11	slices	slice	NOUN
ma-247	71	12	,	,	PUNCT
ma-247	71	13	about	about	ADP
ma-247	71	14	the	the	DET
ma-247	71	15	integrability	integrability	NOUN
ma-247	71	16	of	of	ADP
ma-247	71	17	exp(−v	exp(−v	PROPN
ma-247	71	18	)	)	PUNCT
ma-247	71	19	across	across	ADP
ma-247	71	20	theintersection	theintersection	NOUN
ma-247	71	21	π−1(a	π−1(a	PROPN
ma-247	71	22	)	)	PUNCT
ma-247	71	23	∩x	∩x	NOUN
ma-247	71	24	.	.	PUNCT
ma-247	72	1	theorem	theorem	VERB
ma-247	72	2	1.4	1.4	NUM
ma-247	72	3	.	.	PUNCT
ma-247	73	1	assume	assume	VERB
ma-247	73	2	we	we	PRON
ma-247	73	3	have	have	VERB
ma-247	73	4	a	a	DET
ma-247	73	5	finite	finite	ADJ
ma-247	73	6	family	family	NOUN
ma-247	73	7	(	(	PUNCT
ma-247	73	8	vj)1≤j≤n	vj)1≤j≤n	PROPN
ma-247	73	9	of	of	ADP
ma-247	73	10	psh	psh	NOUN
ma-247	73	11	functions	function	NOUN
ma-247	73	12	in	in	ADP
ma-247	73	13	∆n	∆n	PROPN
ma-247	73	14	such	such	ADJ
ma-247	73	15	that	that	PRON
ma-247	73	16	for	for	ADP
ma-247	73	17	all	all	DET
ma-247	73	18	1	1	NUM
ma-247	73	19	≤	≤	NUM
ma-247	73	20	j	j	PROPN
ma-247	73	21	≤	≤	PROPN
ma-247	73	22	n	n	CCONJ
ma-247	73	23	the	the	DET
ma-247	73	24	set	set	NOUN
ma-247	73	25	of	of	ADP
ma-247	73	26	singular	singular	ADJ
ma-247	73	27	points	point	NOUN
ma-247	73	28	of	of	ADP
ma-247	73	29	vj	vj	PROPN
ma-247	73	30	is	be	AUX
ma-247	73	31	contained	contain	VERB
ma-247	73	32	in	in	ADP
ma-247	73	33	a	a	DET
ma-247	73	34	hypersurface	hypersurface	NOUN
ma-247	73	35	x	x	PUNCT
ma-247	73	36	that	that	PRON
ma-247	73	37	yields	yield	VERB
ma-247	73	38	a	a	DET
ma-247	73	39	complete	complete	ADJ
ma-247	73	40	intersection	intersection	NOUN
ma-247	73	41	with	with	ADP
ma-247	73	42	another	another	DET
ma-247	73	43	analytic	analytic	ADJ
ma-247	73	44	subset	subset	NOUN
ma-247	73	45	y	y	PROPN
ma-247	73	46	in	in	ADP
ma-247	73	47	∆n	∆n	PROPN
ma-247	73	48	.	.	PUNCT
ma-247	74	1	then	then	ADV
ma-247	74	2	,	,	PUNCT
ma-247	74	3	there	there	PRON
ma-247	74	4	exist	exist	VERB
ma-247	74	5	a	a	DET
ma-247	74	6	constant	constant	ADJ
ma-247	74	7	α	α	NOUN
ma-247	74	8	>	>	X
ma-247	74	9	0	0	PUNCT
ma-247	74	10	and	and	CCONJ
ma-247	74	11	a	a	DET
ma-247	74	12	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PROPN
ma-247	74	13	eur	eur	NOUN
ma-247	74	14	.	.	PUNCT
ma-247	75	1	j.	j.	PROPN
ma-247	75	2	math	math	PROPN
ma-247	75	3	.	.	PUNCT
ma-247	76	1	anal	anal	PROPN
ma-247	76	2	.	.	PUNCT
ma-247	77	1	10.28924	10.28924	NUM
ma-247	77	2	/	/	SYM
ma-247	77	3	ada	ada	PROPN
ma-247	77	4	/	/	SYM
ma-247	77	5	ma.4.23	ma.4.23	NOUN
ma-247	77	6	4	4	NUM
ma-247	77	7	pluripolar	pluripolar	NOUN
ma-247	77	8	subset	subset	NOUN
ma-247	77	9	e	e	NOUN
ma-247	77	10	of	of	ADP
ma-247	77	11	∆k	∆k	PROPN
ma-247	77	12	,	,	PUNCT
ma-247	77	13	such	such	ADJ
ma-247	77	14	that	that	PRON
ma-247	77	15	for	for	ADP
ma-247	77	16	any	any	DET
ma-247	77	17	point	point	NOUN
ma-247	77	18	a	a	DET
ma-247	77	19	∈	∈	NOUN
ma-247	77	20	∆kre	∆kre	NOUN
ma-247	77	21	,	,	PUNCT
ma-247	77	22	the	the	DET
ma-247	77	23	function	function	NOUN
ma-247	77	24	∑	∑	PROPN
ma-247	77	25	1≤j≤n	1≤j≤n	NUM
ma-247	77	26	exp	exp	NOUN
ma-247	77	27	(	(	PUNCT
ma-247	77	28	−αvj)|y	−αvj)|y	NOUN
ma-247	77	29	∩π−1(a	∩π−1(a	NOUN
ma-247	77	30	)	)	PUNCT
ma-247	77	31	lies	lie	VERB
ma-247	77	32	in	in	ADP
ma-247	77	33	l1	l1	PROPN
ma-247	77	34	loc	loc	PROPN
ma-247	77	35	(	(	PUNCT
ma-247	77	36	y	y	PROPN
ma-247	77	37	∩	∩	PROPN
ma-247	77	38	π−1(a	π−1(a	PROPN
ma-247	77	39	)	)	PUNCT
ma-247	77	40	)	)	PUNCT
ma-247	77	41	.	.	PUNCT
ma-247	78	1	finally	finally	ADV
ma-247	78	2	,	,	PUNCT
ma-247	78	3	as	as	ADP
ma-247	78	4	an	an	DET
ma-247	78	5	other	other	ADJ
ma-247	78	6	application	application	NOUN
ma-247	78	7	we	we	PRON
ma-247	78	8	establish	establish	VERB
ma-247	78	9	the	the	DET
ma-247	78	10	following	follow	VERB
ma-247	78	11	result	result	VERB
ma-247	78	12	valid	valid	ADJ
ma-247	78	13	for	for	ADP
ma-247	78	14	analytic	analytic	ADJ
ma-247	78	15	subsets	subset	NOUN
ma-247	78	16	xand	xand	CCONJ
ma-247	79	1	y	y	PROPN
ma-247	79	2	yielding	yield	VERB
ma-247	79	3	a	a	DET
ma-247	79	4	complete	complete	ADJ
ma-247	79	5	intersection	intersection	NOUN
ma-247	79	6	in	in	ADP
ma-247	79	7	ω	ω	PROPN
ma-247	79	8	.	.	PUNCT
ma-247	80	1	theorem	theorem	VERB
ma-247	80	2	1.5	1.5	NUM
ma-247	80	3	.	.	PUNCT
ma-247	81	1	assume	assume	VERB
ma-247	81	2	we	we	PRON
ma-247	81	3	have	have	VERB
ma-247	81	4	analytic	analytic	ADJ
ma-247	81	5	subsets	subset	NOUN
ma-247	81	6	x	x	PUNCT
ma-247	81	7	and	and	CCONJ
ma-247	81	8	y	y	PROPN
ma-247	81	9	of	of	ADP
ma-247	81	10	pure	pure	ADJ
ma-247	81	11	dimensions	dimension	NOUN
ma-247	81	12	p	p	NOUN
ma-247	81	13	and	and	CCONJ
ma-247	81	14	q	q	NOUN
ma-247	81	15	respectively	respectively	ADV
ma-247	81	16	,	,	PUNCT
ma-247	81	17	yielding	yield	VERB
ma-247	81	18	a	a	DET
ma-247	81	19	complete	complete	ADJ
ma-247	81	20	intersection	intersection	NOUN
ma-247	81	21	in	in	ADP
ma-247	81	22	∆n	∆n	PROPN
ma-247	81	23	.	.	PUNCT
ma-247	82	1	then	then	ADV
ma-247	82	2	,	,	PUNCT
ma-247	82	3	there	there	PRON
ma-247	82	4	exist	exist	VERB
ma-247	82	5	a	a	DET
ma-247	82	6	constant	constant	ADJ
ma-247	82	7	δ	δ	NOUN
ma-247	82	8	>	>	X
ma-247	82	9	0	0	PUNCT
ma-247	82	10	and	and	CCONJ
ma-247	82	11	a	a	DET
ma-247	82	12	pluripolar	pluripolar	ADJ
ma-247	82	13	subset	subset	NOUN
ma-247	82	14	e	e	NOUN
ma-247	82	15	in	in	ADP
ma-247	82	16	ck	ck	PROPN
ma-247	82	17	,	,	PUNCT
ma-247	82	18	such	such	ADJ
ma-247	82	19	that	that	SCONJ
ma-247	82	20	for	for	ADP
ma-247	82	21	all	all	DET
ma-247	82	22	a	a	DET
ma-247	82	23	∈	∈	PROPN
ma-247	82	24	∆k	∆k	PROPN
ma-247	82	25	re	re	NOUN
ma-247	82	26	,	,	PUNCT
ma-247	82	27	the	the	DET
ma-247	82	28	coefficients	coefficient	NOUN
ma-247	82	29	of	of	ADP
ma-247	82	30	the	the	DET
ma-247	82	31	current	current	ADJ
ma-247	82	32	j∗a	j∗a	PROPN
ma-247	82	33	(	(	PUNCT
ma-247	82	34	u[x])∧	u[x])∧	PROPN
ma-247	82	35	[	[	PUNCT
ma-247	82	36	y	y	PROPN
ma-247	82	37	∩	∩	PROPN
ma-247	82	38	π−1(a	π−1(a	PROPN
ma-247	82	39	)	)	PUNCT
ma-247	82	40	]	]	PUNCT
ma-247	83	1	lie	lie	VERB
ma-247	83	2	in	in	ADP
ma-247	83	3	l1+δ	l1+δ	PROPN
ma-247	83	4	loc	loc	PROPN
ma-247	83	5	(	(	PUNCT
ma-247	83	6	y	y	PROPN
ma-247	83	7	∩	∩	PROPN
ma-247	83	8	π−1(a	π−1(a	PROPN
ma-247	83	9	)	)	PUNCT
ma-247	83	10	)	)	PUNCT
ma-247	83	11	.	.	PUNCT
ma-247	84	1	2	2	X
ma-247	84	2	.	.	X
ma-247	84	3	preliminaries	preliminary	NOUN
ma-247	84	4	let	let	VERB
ma-247	84	5	ω	ω	NOUN
ma-247	84	6	be	be	AUX
ma-247	84	7	a	a	DET
ma-247	84	8	domain	domain	NOUN
ma-247	84	9	in	in	ADP
ma-247	84	10	cn	cn	PROPN
ma-247	84	11	,	,	PUNCT
ma-247	84	12	we	we	PRON
ma-247	84	13	use	use	VERB
ma-247	84	14	the	the	DET
ma-247	84	15	standard	standard	ADJ
ma-247	84	16	notations	notation	NOUN
ma-247	84	17	for	for	ADP
ma-247	84	18	the	the	DET
ma-247	84	19	operators	operator	NOUN
ma-247	85	1	d	d	NOUN
ma-247	85	2	=	=	SYM
ma-247	85	3	∂	∂	NUM
ma-247	85	4	+	+	X
ma-247	85	5	∂̄	∂̄	NOUN
ma-247	85	6	and	and	CCONJ
ma-247	85	7	dc	dc	PROPN
ma-247	85	8	=	=	PUNCT
ma-247	85	9	i(∂̄	i(∂̄	PROPN
ma-247	85	10	−	−	NOUN
ma-247	85	11	∂	∂	NUM
ma-247	85	12	)	)	PUNCT
ma-247	85	13	;	;	PUNCT
ma-247	86	1	the	the	DET
ma-247	86	2	operator	operator	NOUN
ma-247	86	3	ddc	ddc	NOUN
ma-247	86	4	is	be	AUX
ma-247	86	5	then	then	ADV
ma-247	86	6	defined	define	VERB
ma-247	86	7	by	by	ADP
ma-247	86	8	ddc	ddc	PROPN
ma-247	86	9	=	=	SYM
ma-247	86	10	2i∂∂.	2i∂∂.	NUM
ma-247	86	11	the	the	DET
ma-247	86	12	space	space	NOUN
ma-247	86	13	d	d	X
ma-247	86	14	′p	′p	NOUN
ma-247	86	15	,	,	PUNCT
ma-247	86	16	p(ω	p(ω	PROPN
ma-247	86	17	)	)	PUNCT
ma-247	86	18	of	of	ADP
ma-247	86	19	(	(	PUNCT
ma-247	86	20	n−	n−	NOUN
ma-247	86	21	p	p	NOUN
ma-247	86	22	,	,	PUNCT
ma-247	86	23	n−	n−	NOUN
ma-247	86	24	p)-currents	p)-current	NOUN
ma-247	86	25	(	(	PUNCT
ma-247	86	26	of	of	ADP
ma-247	86	27	bidimension	bidimension	NOUN
ma-247	86	28	(	(	PUNCT
ma-247	86	29	p	p	X
ma-247	86	30	,	,	PUNCT
ma-247	86	31	p	p	NOUN
ma-247	86	32	)	)	PUNCT
ma-247	86	33	)	)	PUNCT
ma-247	86	34	on	on	ADP
ma-247	86	35	ω	ω	NUM
ma-247	86	36	,	,	PUNCT
ma-247	86	37	is	be	AUX
ma-247	86	38	the	the	DET
ma-247	86	39	dual	dual	ADJ
ma-247	86	40	of	of	ADP
ma-247	86	41	the	the	DET
ma-247	86	42	space	space	NOUN
ma-247	86	43	dp	dp	NOUN
ma-247	86	44	,	,	PUNCT
ma-247	86	45	p(ω	p(ω	PROPN
ma-247	86	46	)	)	PUNCT
ma-247	86	47	of	of	ADP
ma-247	86	48	smooth	smooth	ADJ
ma-247	86	49	compactlysupported	compactlysupporte	VERB
ma-247	86	50	(	(	PUNCT
ma-247	86	51	p	p	NOUN
ma-247	86	52	,	,	PUNCT
ma-247	86	53	p)-forms	p)-form	NOUN
ma-247	86	54	on	on	ADP
ma-247	86	55	ω	ω	NUM
ma-247	86	56	.	.	PUNCT
ma-247	87	1	we	we	PRON
ma-247	87	2	say	say	VERB
ma-247	87	3	that	that	SCONJ
ma-247	87	4	a	a	DET
ma-247	87	5	current	current	ADJ
ma-247	87	6	t	t	X
ma-247	87	7	∈	∈	PROPN
ma-247	87	8	d	d	X
ma-247	87	9	′p	′p	PROPN
ma-247	87	10	,	,	PUNCT
ma-247	87	11	p(ω	p(ω	PROPN
ma-247	87	12	)	)	PUNCT
ma-247	87	13	is	be	AUX
ma-247	87	14	positive	positive	ADJ
ma-247	87	15	,	,	PUNCT
ma-247	87	16	if	if	SCONJ
ma-247	87	17	for	for	ADP
ma-247	87	18	all	all	DET
ma-247	87	19	smooth	smooth	ADJ
ma-247	87	20	(	(	PUNCT
ma-247	87	21	1	1	NUM
ma-247	87	22	,	,	PUNCT
ma-247	87	23	0)-forms	0)-forms	NUM
ma-247	87	24	α1	α1	NOUN
ma-247	87	25	,	,	PUNCT
ma-247	87	26	.	.	PUNCT
ma-247	87	27	.	.	PUNCT
ma-247	87	28	.	.	PUNCT
ma-247	88	1	,	,	PUNCT
ma-247	88	2	αp	αp	VERB
ma-247	88	3	on	on	ADP
ma-247	88	4	ω	ω	PROPN
ma-247	88	5	,	,	PUNCT
ma-247	88	6	the	the	DET
ma-247	88	7	product	product	NOUN
ma-247	88	8	t	t	PROPN
ma-247	88	9	∧	∧	PROPN
ma-247	88	10	iα1	iα1	ADP
ma-247	88	11	∧	∧	PROPN
ma-247	88	12	ᾱ1	ᾱ1	NOUN
ma-247	88	13	∧	∧	PROPN
ma-247	88	14	.	.	PUNCT
ma-247	88	15	.	.	PUNCT
ma-247	88	16	.	.	PUNCT
ma-247	89	1	∧	∧	NOUN
ma-247	89	2	iαp	iαp	NOUN
ma-247	89	3	∧	∧	PROPN
ma-247	89	4	ᾱp	ᾱp	X
ma-247	89	5	is	be	AUX
ma-247	89	6	a	a	DET
ma-247	89	7	positive	positive	ADJ
ma-247	89	8	measure	measure	NOUN
ma-247	89	9	.	.	PUNCT
ma-247	90	1	the	the	DET
ma-247	90	2	current	current	ADJ
ma-247	90	3	t	t	PROPN
ma-247	90	4	is	be	AUX
ma-247	90	5	said	say	VERB
ma-247	90	6	to	to	PART
ma-247	90	7	be	be	AUX
ma-247	90	8	closed	close	VERB
ma-247	90	9	if	if	SCONJ
ma-247	90	10	dt	dt	ADP
ma-247	90	11	=	=	SYM
ma-247	90	12	0	0	NUM
ma-247	90	13	and	and	CCONJ
ma-247	90	14	psh	psh	PROPN
ma-247	90	15	if	if	SCONJ
ma-247	90	16	ddct	ddct	VERB
ma-247	90	17	≥	≥	NOUN
ma-247	90	18	0.in	0.in	NUM
ma-247	90	19	particular	particular	ADJ
ma-247	90	20	,	,	PUNCT
ma-247	90	21	if	if	SCONJ
ma-247	90	22	p	p	NOUN
ma-247	90	23	=	=	SYM
ma-247	90	24	n	n	CCONJ
ma-247	90	25	,	,	PUNCT
ma-247	90	26	then	then	ADV
ma-247	90	27	t	t	PROPN
ma-247	90	28	is	be	AUX
ma-247	90	29	a	a	DET
ma-247	90	30	psh	psh	NOUN
ma-247	90	31	function	function	NOUN
ma-247	90	32	u	u	NOUN
ma-247	90	33	on	on	ADP
ma-247	90	34	ω	ω	PROPN
ma-247	90	35	,	,	PUNCT
ma-247	90	36	that	that	PRON
ma-247	90	37	is	be	AUX
ma-247	90	38	a	a	DET
ma-247	90	39	distribution	distribution	NOUN
ma-247	90	40	such	such	ADJ
ma-247	90	41	that∑	that∑	NOUN
ma-247	90	42	1≤j	1≤j	NOUN
ma-247	90	43	,	,	PUNCT
ma-247	90	44	k≤n	k≤n	PROPN
ma-247	90	45	∂2u	∂2u	PROPN
ma-247	90	46	∂zj∂z̄k	∂zj∂z̄k	PROPN
ma-247	90	47	idzj	idzj	ADJ
ma-247	90	48	∧	∧	PROPN
ma-247	90	49	dzk	dzk	NOUN
ma-247	90	50	is	be	AUX
ma-247	90	51	a	a	DET
ma-247	90	52	positive	positive	ADJ
ma-247	90	53	current	current	NOUN
ma-247	90	54	of	of	ADP
ma-247	90	55	bidimension	bidimension	NOUN
ma-247	90	56	(	(	PUNCT
ma-247	90	57	n	n	CCONJ
ma-247	90	58	−	−	PROPN
ma-247	90	59	1	1	NUM
ma-247	90	60	,	,	PUNCT
ma-247	90	61	n	n	CCONJ
ma-247	90	62	−	−	PROPN
ma-247	90	63	1	1	NUM
ma-247	90	64	)	)	PUNCT
ma-247	90	65	,	,	PUNCT
ma-247	90	66	the	the	DET
ma-247	90	67	derivatives	derivative	NOUN
ma-247	90	68	are	be	AUX
ma-247	90	69	taken	take	VERB
ma-247	90	70	here	here	ADV
ma-247	90	71	in	in	ADP
ma-247	90	72	the	the	DET
ma-247	90	73	sense	sense	NOUN
ma-247	90	74	ofdistributions.we	ofdistributions.we	PRON
ma-247	90	75	denote	denote	NOUN
ma-247	90	76	psh(ω	psh(ω	NOUN
ma-247	90	77	)	)	PUNCT
ma-247	90	78	the	the	DET
ma-247	90	79	set	set	NOUN
ma-247	90	80	of	of	ADP
ma-247	90	81	psh	psh	NOUN
ma-247	90	82	functions	function	NOUN
ma-247	90	83	on	on	ADP
ma-247	90	84	ω	ω	NUM
ma-247	90	85	and	and	CCONJ
ma-247	90	86	l∞loc(ω)∩	l∞loc(ω)∩	ADJ
ma-247	90	87	psh(ω	psh(ω	NOUN
ma-247	90	88	)	)	PUNCT
ma-247	90	89	the	the	DET
ma-247	90	90	subset	subset	NOUN
ma-247	90	91	of	of	ADP
ma-247	90	92	elementsin	elementsin	PROPN
ma-247	90	93	psh(ω	psh(ω	X
ma-247	90	94	)	)	PUNCT
ma-247	90	95	which	which	PRON
ma-247	90	96	are	be	AUX
ma-247	90	97	locally	locally	ADV
ma-247	90	98	bounded	bound	VERB
ma-247	90	99	.	.	PUNCT
ma-247	91	1	the	the	DET
ma-247	91	2	kähler	kähler	PROPN
ma-247	91	3	form	form	NOUN
ma-247	91	4	on	on	ADP
ma-247	91	5	cn	cn	PROPN
ma-247	91	6	is	be	AUX
ma-247	91	7	denoted	denote	VERB
ma-247	91	8	β(t	β(t	PROPN
ma-247	91	9	)	)	PUNCT
ma-247	92	1	=	=	SYM
ma-247	92	2	ddc	ddc	PROPN
ma-247	92	3	|t|2	|t|2	PROPN
ma-247	92	4	.	.	PUNCT
ma-247	93	1	it	it	PRON
ma-247	93	2	canbe	canbe	VERB
ma-247	93	3	decomposed	decompose	VERB
ma-247	93	4	into	into	ADP
ma-247	93	5	the	the	DET
ma-247	93	6	sum	sum	NOUN
ma-247	93	7	β′(t	β′(t	PUNCT
ma-247	94	1	′	′	NUM
ma-247	94	2	)	)	PUNCT
ma-247	95	1	+	+	CCONJ
ma-247	95	2	β′′(t	β′′(t	VERB
ma-247	95	3	′′	′′	NOUN
ma-247	95	4	)	)	PUNCT
ma-247	95	5	where	where	SCONJ
ma-247	95	6	β′	β′	PRON
ma-247	95	7	and	and	CCONJ
ma-247	95	8	β′′	β′′	NOUN
ma-247	95	9	are	be	AUX
ma-247	95	10	kähler	kähler	NOUN
ma-247	95	11	forms	form	NOUN
ma-247	95	12	on	on	ADP
ma-247	95	13	ck	ck	PROPN
ma-247	95	14	and	and	CCONJ
ma-247	95	15	cn−krespectively	cn−krespectively	ADV
ma-247	95	16	.	.	PUNCT
ma-247	96	1	for	for	ADP
ma-247	96	2	a	a	DET
ma-247	96	3	point	point	NOUN
ma-247	96	4	a	a	DET
ma-247	96	5	∈	∈	NOUN
ma-247	96	6	ck	ck	INTJ
ma-247	96	7	,	,	PUNCT
ma-247	96	8	we	we	PRON
ma-247	96	9	denote	denote	VERB
ma-247	96	10	ja	ja	PROPN
ma-247	96	11	the	the	DET
ma-247	96	12	map	map	NOUN
ma-247	96	13	defined	define	VERB
ma-247	96	14	by	by	ADP
ma-247	96	15	ja	ja	PROPN
ma-247	96	16	:	:	PUNCT
ma-247	96	17	cn−k	cn−k	VERB
ma-247	96	18	−→	−→	NOUN
ma-247	96	19	{	{	PUNCT
ma-247	96	20	a	a	DET
ma-247	96	21	}	}	PUNCT
ma-247	96	22	×	×	NOUN
ma-247	96	23	cn−k	cn−k	NOUN
ma-247	96	24	z	z	PROPN
ma-247	97	1	′′	′′	PROPN
ma-247	97	2	7−→	7−→	PROPN
ma-247	97	3	(	(	PUNCT
ma-247	97	4	a	a	DET
ma-247	97	5	,	,	PUNCT
ma-247	97	6	z	z	NOUN
ma-247	97	7	′	′	NOUN
ma-247	97	8	)	)	PUNCT
ma-247	97	9	3	3	NUM
ma-247	97	10	.	.	X
ma-247	97	11	proof	proof	NOUN
ma-247	97	12	of	of	ADP
ma-247	97	13	theorem	theorem	ADJ
ma-247	97	14	1.1	1.1	NUM
ma-247	97	15	:	:	PUNCT
ma-247	97	16	slicing	slicing	NOUN
ma-247	97	17	of	of	ADP
ma-247	97	18	the	the	DET
ma-247	97	19	current	current	ADJ
ma-247	97	20	(	(	PUNCT
ma-247	97	21	log	log	VERB
ma-247	97	22	|f	|f	PRON
ma-247	98	1	|)[x	|)[x	PROPN
ma-247	98	2	]	]	X
ma-247	98	3	first	first	ADV
ma-247	98	4	of	of	ADP
ma-247	98	5	all	all	PRON
ma-247	98	6	,	,	PUNCT
ma-247	98	7	we	we	PRON
ma-247	98	8	require	require	VERB
ma-247	98	9	the	the	DET
ma-247	98	10	following	follow	VERB
ma-247	98	11	well	well	ADV
ma-247	98	12	known	know	VERB
ma-247	98	13	propositions	proposition	NOUN
ma-247	98	14	:	:	PUNCT
ma-247	98	15	proposition	proposition	NOUN
ma-247	98	16	3.1	3.1	NUM
ma-247	98	17	.	.	PUNCT
ma-247	99	1	let	let	VERB
ma-247	99	2	x	x	PRON
ma-247	99	3	be	be	AUX
ma-247	99	4	an	an	DET
ma-247	99	5	analytic	analytic	ADJ
ma-247	99	6	subset	subset	NOUN
ma-247	99	7	of	of	ADP
ma-247	99	8	∆n	∆n	PROPN
ma-247	99	9	and	and	CCONJ
ma-247	99	10	let	let	VERB
ma-247	99	11	m	m	PRON
ma-247	99	12	be	be	AUX
ma-247	99	13	its	its	PRON
ma-247	99	14	complex	complex	ADJ
ma-247	99	15	dimension	dimension	NOUN
ma-247	99	16	.	.	PUNCT
ma-247	100	1	then	then	ADV
ma-247	100	2	,	,	PUNCT
ma-247	100	3	the	the	DET
ma-247	100	4	following	follow	VERB
ma-247	100	5	statements	statement	NOUN
ma-247	100	6	hold	hold	VERB
ma-247	100	7	:	:	PUNCT
ma-247	100	8	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PROPN
ma-247	100	9	eur	eur	PROPN
ma-247	100	10	.	.	PUNCT
ma-247	101	1	j.	j.	PROPN
ma-247	101	2	math	math	PROPN
ma-247	101	3	.	.	PUNCT
ma-247	102	1	anal	anal	PROPN
ma-247	102	2	.	.	PUNCT
ma-247	103	1	10.28924	10.28924	NUM
ma-247	103	2	/	/	SYM
ma-247	103	3	ada	ada	PROPN
ma-247	103	4	/	/	SYM
ma-247	103	5	ma.4.23	ma.4.23	NOUN
ma-247	103	6	5(1	5(1	NUM
ma-247	103	7	)	)	PUNCT
ma-247	103	8	if	if	SCONJ
ma-247	103	9	m	m	VERB
ma-247	103	10	<	<	X
ma-247	103	11	k	k	X
ma-247	103	12	,	,	PUNCT
ma-247	103	13	then	then	ADV
ma-247	103	14	π(x	π(x	NOUN
ma-247	103	15	)	)	PUNCT
ma-247	103	16	is	be	AUX
ma-247	103	17	contained	contain	VERB
ma-247	103	18	in	in	ADP
ma-247	103	19	a	a	DET
ma-247	103	20	countable	countable	ADJ
ma-247	103	21	union	union	NOUN
ma-247	103	22	of	of	ADP
ma-247	103	23	analytic	analytic	ADJ
ma-247	103	24	subsets	subset	NOUN
ma-247	103	25	of	of	ADP
ma-247	103	26	∆k	∆k	PROPN
ma-247	103	27	of	of	ADP
ma-247	103	28	dimensions	dimension	NOUN
ma-247	103	29	≤	≤	PUNCT
ma-247	104	1	m.(2	m.(2	NOUN
ma-247	104	2	)	)	PUNCT
ma-247	104	3	if	if	SCONJ
ma-247	104	4	m	m	PROPN
ma-247	104	5	≥	≥	VERB
ma-247	104	6	k	k	NOUN
ma-247	104	7	,	,	PUNCT
ma-247	104	8	then	then	ADV
ma-247	104	9	the	the	DET
ma-247	104	10	set	set	NOUN
ma-247	104	11	z	z	NOUN
ma-247	104	12	=	=	SYM
ma-247	104	13	{	{	PUNCT
ma-247	104	14	a	a	DET
ma-247	104	15	∈	∈	PROPN
ma-247	104	16	∆k	∆k	NOUN
ma-247	104	17	/	/	SYM
ma-247	104	18	dimc(x	dimc(x	NOUN
ma-247	104	19	∩	∩	ADJ
ma-247	104	20	π−1(x	π−1(x	NOUN
ma-247	104	21	)	)	PUNCT
ma-247	104	22	)	)	PUNCT
ma-247	104	23	≥	≥	NOUN
ma-247	105	1	m	m	VERB
ma-247	105	2	−	−	PROPN
ma-247	106	1	k	k	X
ma-247	106	2	+	+	CCONJ
ma-247	106	3	1	1	NUM
ma-247	106	4	}	}	PUNCT
ma-247	106	5	is	be	AUX
ma-247	106	6	contained	contain	VERB
ma-247	106	7	in	in	ADP
ma-247	106	8	a	a	DET
ma-247	106	9	countable	countable	ADJ
ma-247	106	10	union	union	NOUN
ma-247	106	11	of	of	ADP
ma-247	106	12	analytic	analytic	ADJ
ma-247	106	13	subsets	subset	NOUN
ma-247	106	14	of	of	ADP
ma-247	106	15	∆k	∆k	PROPN
ma-247	106	16	of	of	ADP
ma-247	106	17	dimensions	dimension	NOUN
ma-247	106	18	≤	≤	PUNCT
ma-247	107	1	k	k	NOUN
ma-247	108	1	−	−	PROPN
ma-247	108	2	1	1	X
ma-247	108	3	.	.	PUNCT
ma-247	109	1	as	as	ADP
ma-247	109	2	a	a	DET
ma-247	109	3	consequence	consequence	NOUN
ma-247	109	4	of	of	ADP
ma-247	109	5	the	the	DET
ma-247	109	6	expansion	expansion	NOUN
ma-247	109	7	of	of	ADP
ma-247	109	8	holomorphic	holomorphic	ADJ
ma-247	109	9	functions	function	NOUN
ma-247	109	10	in	in	ADP
ma-247	109	11	power	power	NOUN
ma-247	109	12	series	series	NOUN
ma-247	109	13	we	we	PRON
ma-247	109	14	have	have	VERB
ma-247	109	15	:	:	PUNCT
ma-247	109	16	proposition	proposition	NOUN
ma-247	109	17	3.2	3.2	NUM
ma-247	109	18	.	.	PUNCT
ma-247	110	1	let	let	VERB
ma-247	110	2	f	f	PRON
ma-247	110	3	6≡	6≡	NUM
ma-247	110	4	0	0	NUM
ma-247	110	5	be	be	AUX
ma-247	110	6	a	a	DET
ma-247	110	7	non	non	ADJ
ma-247	110	8	vanishing	vanish	VERB
ma-247	110	9	holomorphic	holomorphic	ADJ
ma-247	110	10	function	function	NOUN
ma-247	110	11	on	on	ADP
ma-247	110	12	∆n	∆n	PROPN
ma-247	110	13	such	such	ADJ
ma-247	110	14	that	that	PRON
ma-247	110	15	for	for	ADP
ma-247	110	16	all	all	DET
ma-247	110	17	z	z	NUM
ma-247	110	18	′′	′′	PROPN
ma-247	110	19	∈	∈	NOUN
ma-247	110	20	∆n−k	∆n−k	INTJ
ma-247	110	21	,	,	PUNCT
ma-247	110	22	f	f	PROPN
ma-247	110	23	(	(	PUNCT
ma-247	110	24	0	0	NUM
ma-247	110	25	,	,	PUNCT
ma-247	110	26	z	z	PROPN
ma-247	110	27	′′	′′	PROPN
ma-247	110	28	)	)	PUNCT
ma-247	110	29	=	=	PUNCT
ma-247	111	1	0	0	X
ma-247	111	2	.	.	PUNCT
ma-247	112	1	then	then	ADV
ma-247	112	2	,	,	PUNCT
ma-247	112	3	there	there	PRON
ma-247	112	4	exist	exist	VERB
ma-247	112	5	a	a	DET
ma-247	112	6	natural	natural	ADJ
ma-247	112	7	number	number	NOUN
ma-247	112	8	m	m	PROPN
ma-247	112	9	≥	≥	NOUN
ma-247	112	10	1	1	NUM
ma-247	112	11	and	and	CCONJ
ma-247	112	12	holomorphic	holomorphic	ADJ
ma-247	112	13	functions	function	NOUN
ma-247	112	14	aµ(z	aµ(z	ADP
ma-247	112	15	′′	′′	PROPN
ma-247	112	16	)	)	PUNCT
ma-247	112	17	on	on	ADP
ma-247	112	18	∆n−k	∆n−k	INTJ
ma-247	112	19	,	,	PUNCT
ma-247	112	20	such	such	ADJ
ma-247	112	21	that	that	SCONJ
ma-247	112	22	,	,	PUNCT
ma-247	112	23	for	for	ADP
ma-247	112	24	all	all	PRON
ma-247	112	25	(	(	PUNCT
ma-247	112	26	z	z	NOUN
ma-247	112	27	′	′	NUM
ma-247	112	28	;	;	PUNCT
ma-247	112	29	z	z	PROPN
ma-247	112	30	′′	′′	PROPN
ma-247	112	31	)	)	PUNCT
ma-247	112	32	∈	∈	PROPN
ma-247	112	33	∆k	∆k	PROPN
ma-247	112	34	×	×	NOUN
ma-247	112	35	∆n−k	∆n−k	INTJ
ma-247	112	36	,	,	PUNCT
ma-247	112	37	we	we	PRON
ma-247	112	38	have	have	AUX
ma-247	112	39	f	f	PROPN
ma-247	112	40	(	(	PUNCT
ma-247	112	41	z	z	NOUN
ma-247	112	42	′	′	NUM
ma-247	112	43	,	,	PUNCT
ma-247	112	44	z	z	PROPN
ma-247	112	45	′′	′′	PROPN
ma-247	112	46	)	)	PUNCT
ma-247	112	47	=	=	PUNCT
ma-247	113	1	∑∞	∑∞	NOUN
ma-247	113	2	j	j	X
ma-247	114	1	=	=	NOUN
ma-247	114	2	mqj(z	mqj(z	PROPN
ma-247	114	3	′	′	NUM
ma-247	114	4	;	;	PUNCT
ma-247	114	5	z	z	PROPN
ma-247	114	6	′′	′′	PROPN
ma-247	114	7	)	)	PUNCT
ma-247	114	8	where	where	SCONJ
ma-247	114	9	qm(z	qm(z	PUNCT
ma-247	114	10	′	′	NOUN
ma-247	114	11	,	,	PUNCT
ma-247	114	12	z	z	PROPN
ma-247	114	13	′′	′′	PROPN
ma-247	114	14	)	)	PUNCT
ma-247	114	15	=	=	PUNCT
ma-247	115	1	∑	∑	PUNCT
ma-247	115	2	|µ|=m	|µ|=m	PROPN
ma-247	115	3	z	z	NOUN
ma-247	115	4	′µaµ(z	′µaµ(z	X
ma-247	115	5	′′	′′	PROPN
ma-247	115	6	)	)	PUNCT
ma-247	115	7	and	and	CCONJ
ma-247	115	8	|µ|	|µ|	PROPN
ma-247	115	9	=	=	PUNCT
ma-247	115	10	µ1	µ1	PROPN
ma-247	115	11	+	+	NOUN
ma-247	115	12	·	·	PUNCT
ma-247	115	13	·	·	PUNCT
ma-247	115	14	·	·	PUNCT
ma-247	115	15	+	+	NUM
ma-247	115	16	µk	µk	INTJ
ma-247	115	17	.	.	PUNCT
ma-247	116	1	now	now	ADV
ma-247	116	2	,	,	PUNCT
ma-247	116	3	given	give	VERB
ma-247	116	4	a	a	DET
ma-247	116	5	psh	psh	NOUN
ma-247	116	6	function	function	NOUN
ma-247	116	7	v	v	NOUN
ma-247	116	8	on	on	ADP
ma-247	116	9	∆n	∆n	PROPN
ma-247	116	10	such	such	ADJ
ma-247	116	11	that	that	DET
ma-247	116	12	v	v	NOUN
ma-247	116	13	is	be	AUX
ma-247	116	14	locally	locally	ADV
ma-247	116	15	integrable	integrable	ADJ
ma-247	116	16	on	on	ADP
ma-247	116	17	the	the	DET
ma-247	116	18	analytic	analytic	ADJ
ma-247	116	19	subset	subset	NOUN
ma-247	116	20	x	x	X
ma-247	116	21	,	,	PUNCT
ma-247	116	22	then	then	ADV
ma-247	116	23	,	,	PUNCT
ma-247	116	24	for	for	ADP
ma-247	116	25	all	all	DET
ma-247	116	26	test	test	NOUN
ma-247	116	27	form	form	NOUN
ma-247	116	28	ψ	ψ	NOUN
ma-247	116	29	,	,	PUNCT
ma-247	116	30	for	for	ADP
ma-247	116	31	all	all	DET
ma-247	116	32	fixed	fix	VERB
ma-247	116	33	ε	ε	PROPN
ma-247	116	34	>	>	PUNCT
ma-247	116	35	0	0	PUNCT
ma-247	116	36	and	and	CCONJ
ma-247	116	37	all	all	DET
ma-247	116	38	a	a	DET
ma-247	116	39	∈	∈	PROPN
ma-247	116	40	sϕ	sϕ	NOUN
ma-247	116	41	,	,	PUNCT
ma-247	116	42	we	we	PRON
ma-247	116	43	shall	shall	AUX
ma-247	116	44	justify	justify	VERB
ma-247	116	45	the	the	DET
ma-247	116	46	well	well	ADJ
ma-247	116	47	definition	definition	NOUN
ma-247	116	48	andthe	andthe	NOUN
ma-247	116	49	finiteness	finiteness	NOUN
ma-247	116	50	of	of	ADP
ma-247	116	51	the	the	DET
ma-247	116	52	quantity	quantity	NOUN
ma-247	116	53	∫	∫	PROPN
ma-247	116	54	bk(a	bk(a	NOUN
ma-247	116	55	,	,	PUNCT
ma-247	116	56	ε)×cn−k	ε)×cn−k	NOUN
ma-247	116	57	v	v	X
ma-247	117	1	[	[	X
ma-247	117	2	x	x	X
ma-247	117	3	]	]	X
ma-247	117	4	∧	∧	PROPN
ma-247	117	5	(	(	PUNCT
ma-247	117	6	ddc	ddc	PROPN
ma-247	117	7	ϕ̃)k	ϕ̃)k	PROPN
ma-247	117	8	∧ψ	∧ψ	PROPN
ma-247	117	9	.	.	PUNCT
ma-247	117	10	proposition	proposition	NOUN
ma-247	117	11	3.3	3.3	NUM
ma-247	117	12	.	.	PUNCT
ma-247	118	1	let	let	VERB
ma-247	118	2	x	x	PRON
ma-247	118	3	⊂	⊂	PROPN
ma-247	118	4	∆n	∆n	PROPN
ma-247	118	5	be	be	AUX
ma-247	118	6	analytic	analytic	ADJ
ma-247	118	7	of	of	ADP
ma-247	118	8	pure	pure	ADJ
ma-247	118	9	dimension	dimension	NOUN
ma-247	118	10	p	p	X
ma-247	118	11	>	>	X
ma-247	118	12	k	k	PROPN
ma-247	118	13	,	,	PUNCT
ma-247	118	14	let	let	VERB
ma-247	118	15	v	v	NUM
ma-247	118	16	∈	∈	PROPN
ma-247	118	17	psh(∆n	psh(∆n	NOUN
ma-247	118	18	)	)	PUNCT
ma-247	118	19	∩	∩	PROPN
ma-247	118	20	l1	l1	PROPN
ma-247	118	21	loc(x	loc(x	PROPN
ma-247	118	22	)	)	PUNCT
ma-247	118	23	be	be	AUX
ma-247	118	24	negative	negative	ADJ
ma-247	118	25	and	and	CCONJ
ma-247	118	26	let	let	VERB
ma-247	118	27	ϕ	ϕ	PROPN
ma-247	118	28	∈	∈	PROPN
ma-247	118	29	psh(∆k)∩l∞loc(∆k	psh(∆k)∩l∞loc(∆k	PROPN
ma-247	118	30	)	)	PUNCT
ma-247	118	31	be	be	AUX
ma-247	118	32	given	give	VERB
ma-247	118	33	such	such	ADJ
ma-247	118	34	that	that	DET
ma-247	118	35	sϕ	sϕ	NOUN
ma-247	118	36	=	=	PUNCT
ma-247	118	37	∆k	∆k	PROPN
ma-247	118	38	.	.	PUNCT
ma-247	119	1	then	then	ADV
ma-247	119	2	,	,	PUNCT
ma-247	119	3	for	for	ADP
ma-247	119	4	any	any	DET
ma-247	119	5	point	point	NOUN
ma-247	119	6	a	a	DET
ma-247	119	7	∈	∈	NOUN
ma-247	119	8	∆k	∆k	NOUN
ma-247	119	9	,	,	PUNCT
ma-247	119	10	for	for	ADP
ma-247	119	11	any	any	DET
ma-247	119	12	fixed	fix	VERB
ma-247	119	13	ε	ε	PROPN
ma-247	119	14	>	>	X
ma-247	119	15	0	0	PUNCT
ma-247	119	16	and	and	CCONJ
ma-247	119	17	for	for	ADP
ma-247	119	18	any	any	DET
ma-247	119	19	positive	positive	ADJ
ma-247	119	20	test	test	NOUN
ma-247	119	21	form	form	NOUN
ma-247	119	22	ψ	ψ	ADP
ma-247	119	23	∈	∈	PROPN
ma-247	119	24	d(p−k	d(p−k	PROPN
ma-247	119	25	,	,	PUNCT
ma-247	119	26	p−k)(∆n	p−k)(∆n	NOUN
ma-247	119	27	)	)	PUNCT
ma-247	119	28	,	,	PUNCT
ma-247	119	29	we	we	PRON
ma-247	119	30	have	have	VERB
ma-247	119	31	1	1	NUM
ma-247	119	32	µϕ(bk(a	µϕ(bk(a	ADJ
ma-247	119	33	,	,	PUNCT
ma-247	119	34	ε	ε	PROPN
ma-247	119	35	)	)	PUNCT
ma-247	119	36	)	)	PUNCT
ma-247	119	37	∫	∫	PROPN
ma-247	119	38	bk(a	bk(a	NOUN
ma-247	119	39	,	,	PUNCT
ma-247	119	40	ε)×cn−k	ε)×cn−k	ADJ
ma-247	119	41	−v	−v	NOUN
ma-247	120	1	[	[	X
ma-247	120	2	x	x	X
ma-247	120	3	]	]	X
ma-247	120	4	∧	∧	PROPN
ma-247	120	5	(	(	PUNCT
ma-247	120	6	ddc	ddc	PROPN
ma-247	120	7	ϕ̃)k	ϕ̃)k	NOUN
ma-247	120	8	∧ψ	∧ψ	PROPN
ma-247	120	9	<	<	X
ma-247	120	10	∞.	∞.	PROPN
ma-247	120	11	proof	proof	NOUN
ma-247	120	12	.	.	PUNCT
ma-247	121	1	the	the	DET
ma-247	121	2	result	result	NOUN
ma-247	121	3	is	be	AUX
ma-247	121	4	local	local	ADJ
ma-247	121	5	.	.	PUNCT
ma-247	122	1	we	we	PRON
ma-247	122	2	may	may	AUX
ma-247	122	3	take	take	VERB
ma-247	122	4	a	a	DET
ma-247	122	5	=	=	NOUN
ma-247	122	6	0	0	NUM
ma-247	122	7	,	,	PUNCT
ma-247	122	8	without	without	ADP
ma-247	122	9	loss	loss	NOUN
ma-247	122	10	of	of	ADP
ma-247	122	11	generality	generality	NOUN
ma-247	122	12	we	we	PRON
ma-247	122	13	may	may	AUX
ma-247	122	14	choose	choose	VERB
ma-247	122	15	the	the	DET
ma-247	122	16	testform	testform	NOUN
ma-247	122	17	ψ	ψ	ADP
ma-247	122	18	such	such	ADJ
ma-247	122	19	that	that	SCONJ
ma-247	122	20	ψ	ψ	NOUN
ma-247	122	21	=	=	PUNCT
ma-247	122	22	ψ(z)β′′n−k	ψ(z)β′′n−k	NOUN
ma-247	122	23	∈	∈	PROPN
ma-247	122	24	d(n−p	d(n−p	NOUN
ma-247	122	25	,	,	PUNCT
ma-247	122	26	n−p)(∆n−k	n−p)(∆n−k	PROPN
ma-247	122	27	)	)	PUNCT
ma-247	122	28	where	where	SCONJ
ma-247	122	29	ψ	ψ	NOUN
ma-247	122	30	is	be	AUX
ma-247	122	31	a	a	DET
ma-247	122	32	positive	positive	ADJ
ma-247	122	33	test	test	NOUN
ma-247	122	34	function	function	NOUN
ma-247	122	35	on	on	ADP
ma-247	122	36	∆n−k	∆n−k	PROPN
ma-247	122	37	.let	.let	PUNCT
ma-247	123	1	denote	denote	VERB
ma-247	123	2	γε	γε	ADJ
ma-247	123	3	:	:	PUNCT
ma-247	123	4	=	=	SYM
ma-247	123	5	1	1	NUM
ma-247	123	6	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	123	7	,	,	PUNCT
ma-247	123	8	ε	ε	PROPN
ma-247	123	9	)	)	PUNCT
ma-247	123	10	)	)	PUNCT
ma-247	123	11	∫	∫	PROPN
ma-247	124	1	bk(a	bk(a	NOUN
ma-247	124	2	,	,	PUNCT
ma-247	124	3	ε)×cn−k	ε)×cn−k	ADJ
ma-247	124	4	−v	−v	NOUN
ma-247	125	1	[	[	X
ma-247	125	2	x	x	X
ma-247	125	3	]	]	X
ma-247	125	4	∧	∧	PROPN
ma-247	125	5	(	(	PUNCT
ma-247	125	6	ddc	ddc	PROPN
ma-247	125	7	ϕ̃)k	ϕ̃)k	PROPN
ma-247	125	8	∧ψ	∧ψ	PROPN
ma-247	125	9	.	.	PUNCT
ma-247	126	1	(	(	PUNCT
ma-247	126	2	3.2	3.2	NUM
ma-247	126	3	)	)	PUNCT
ma-247	126	4	there	there	PRON
ma-247	126	5	exists	exist	VERB
ma-247	126	6	a	a	DET
ma-247	126	7	neighborhood	neighborhood	NOUN
ma-247	126	8	u	u	NOUN
ma-247	126	9	=	=	NOUN
ma-247	126	10	u	u	NOUN
ma-247	126	11	′	′	NOUN
ma-247	126	12	×	×	NOUN
ma-247	126	13	u	u	NOUN
ma-247	127	1	′′	′′	PROPN
ma-247	127	2	⊂	⊂	PROPN
ma-247	127	3	∆k	∆k	PROPN
ma-247	128	1	×	×	PROPN
ma-247	129	1	∆p−k	∆p−k	NUM
ma-247	129	2	×	×	PROPN
ma-247	129	3	∆n−p	∆n−p	NOUN
ma-247	129	4	of	of	ADP
ma-247	129	5	0	0	NUM
ma-247	129	6	and	and	CCONJ
ma-247	129	7	a	a	DET
ma-247	129	8	coordinate	coordinate	NOUN
ma-247	129	9	system	system	NOUN
ma-247	129	10	(	(	PUNCT
ma-247	129	11	t	t	PROPN
ma-247	129	12	,	,	PUNCT
ma-247	129	13	ζ	ζ	NOUN
ma-247	129	14	,	,	PUNCT
ma-247	129	15	z	z	PROPN
ma-247	129	16	′′	′′	PROPN
ma-247	129	17	)	)	PUNCT
ma-247	129	18	such	such	ADJ
ma-247	129	19	that	that	SCONJ
ma-247	129	20	the	the	DET
ma-247	129	21	projection	projection	NOUN
ma-247	129	22	πx	πx	X
ma-247	129	23	:	:	PUNCT
ma-247	129	24	x	x	SYM
ma-247	129	25	∩	∩	ADJ
ma-247	129	26	u	u	NOUN
ma-247	129	27	→	→	SYM
ma-247	129	28	∆p	∆p	PROPN
ma-247	129	29	=	=	PUNCT
ma-247	129	30	∆k	∆k	PROPN
ma-247	129	31	×	×	NOUN
ma-247	129	32	∆p−k	∆p−k	PROPN
ma-247	129	33	(	(	PUNCT
ma-247	129	34	(	(	PUNCT
ma-247	129	35	t	t	PROPN
ma-247	129	36	,	,	PUNCT
ma-247	129	37	ζ	ζ	NOUN
ma-247	129	38	)	)	PUNCT
ma-247	129	39	,	,	PUNCT
ma-247	129	40	z	z	PUNCT
ma-247	129	41	′′	′′	PROPN
ma-247	129	42	)	)	PUNCT
ma-247	129	43	7→	7→	PROPN
ma-247	129	44	(	(	PUNCT
ma-247	129	45	t	t	PROPN
ma-247	129	46	,	,	PUNCT
ma-247	129	47	ζ)is	ζ)is	PROPN
ma-247	129	48	a	a	DET
ma-247	129	49	ramified	ramified	ADJ
ma-247	129	50	covering	covering	NOUN
ma-247	129	51	of	of	ADP
ma-247	129	52	x	x	X
ma-247	129	53	.	.	PUNCT
ma-247	130	1	let	let	VERB
ma-247	130	2	z	z	NOUN
ma-247	130	3	be	be	AUX
ma-247	130	4	the	the	DET
ma-247	130	5	ramification	ramification	NOUN
ma-247	130	6	locus	locus	NOUN
ma-247	130	7	of	of	ADP
ma-247	130	8	πx	πx	PRON
ma-247	130	9	and	and	CCONJ
ma-247	130	10	set	set	VERB
ma-247	130	11	xz	xz	PROPN
ma-247	130	12	=	=	PUNCT
ma-247	131	1	x	x	NOUN
ma-247	131	2	∩	∩	NOUN
ma-247	131	3	(	(	PUNCT
ma-247	131	4	(	(	PUNCT
ma-247	131	5	u	u	NOUN
ma-247	131	6	′	′	NUM
ma-247	131	7	r	r	NOUN
ma-247	131	8	z)×	z)×	NUM
ma-247	131	9	u	u	NOUN
ma-247	131	10	′′	′′	PROPN
ma-247	131	11	)	)	PUNCT
ma-247	131	12	⊂	⊂	PROPN
ma-247	131	13	xreg	xreg	PROPN
ma-247	131	14	.	.	PUNCT
ma-247	132	1	the	the	DET
ma-247	132	2	restriction	restriction	NOUN
ma-247	132	3	of	of	ADP
ma-247	132	4	πxz	πxz	PROPN
ma-247	132	5	:	:	PUNCT
ma-247	132	6	xz	xz	PROPN
ma-247	132	7	→	→	SYM
ma-247	132	8	u	u	NOUN
ma-247	132	9	′	′	NUM
ma-247	132	10	r	r	NOUN
ma-247	132	11	z	z	X
ma-247	132	12	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	NOUN
ma-247	132	13	eur	eur	NOUN
ma-247	132	14	.	.	PUNCT
ma-247	133	1	j.	j.	PROPN
ma-247	133	2	math	math	PROPN
ma-247	133	3	.	.	PUNCT
ma-247	134	1	anal	anal	PROPN
ma-247	134	2	.	.	PUNCT
ma-247	135	1	10.28924	10.28924	NUM
ma-247	135	2	/	/	SYM
ma-247	135	3	ada	ada	PROPN
ma-247	135	4	/	/	SYM
ma-247	135	5	ma.4.23	ma.4.23	PROPN
ma-247	135	6	6is	6is	NOUN
ma-247	135	7	then	then	ADV
ma-247	135	8	a	a	DET
ma-247	135	9	covering	covering	NOUN
ma-247	135	10	with	with	ADP
ma-247	135	11	a	a	DET
ma-247	135	12	finite	finite	ADJ
ma-247	135	13	sheet	sheet	NOUN
ma-247	135	14	number	number	NOUN
ma-247	135	15	.	.	PUNCT
ma-247	136	1	the	the	DET
ma-247	136	2	expression	expression	NOUN
ma-247	136	3	of	of	ADP
ma-247	136	4	γε	γε	PROPN
ma-247	136	5	given	give	VERB
ma-247	136	6	by	by	ADP
ma-247	136	7	(	(	PUNCT
ma-247	136	8	3.2	3.2	NUM
ma-247	136	9	)	)	PUNCT
ma-247	136	10	will	will	AUX
ma-247	136	11	be	be	AUX
ma-247	136	12	transformedas	transformeda	NOUN
ma-247	137	1	γε	γε	PROPN
ma-247	137	2	=	=	SYM
ma-247	137	3	1	1	NUM
ma-247	137	4	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	137	5	,	,	PUNCT
ma-247	137	6	ε	ε	PROPN
ma-247	137	7	)	)	PUNCT
ma-247	137	8	)	)	PUNCT
ma-247	137	9	∫	∫	PROPN
ma-247	138	1	xz∩bk(0,ε)×cn−k	xz∩bk(0,ε)×cn−k	PROPN
ma-247	138	2	πxz	πxz	PROPN
ma-247	138	3	∗(−	∗(−	PROPN
ma-247	138	4	v(ddcϕ)k	v(ddcϕ)k	PROPN
ma-247	138	5	∧ψ	∧ψ	NUM
ma-247	138	6	)	)	PUNCT
ma-247	138	7	.	.	PUNCT
ma-247	139	1	(	(	PUNCT
ma-247	139	2	3.3	3.3	NUM
ma-247	139	3	)	)	PUNCT
ma-247	139	4	we	we	PRON
ma-247	139	5	may	may	AUX
ma-247	139	6	choose	choose	VERB
ma-247	139	7	ε	ε	PROPN
ma-247	139	8	>	>	X
ma-247	139	9	0	0	PROPN
ma-247	139	10	small	small	ADJ
ma-247	139	11	enough	enough	ADV
ma-247	139	12	,	,	PUNCT
ma-247	139	13	so	so	SCONJ
ma-247	139	14	that	that	SCONJ
ma-247	139	15	uε	uε	ADP
ma-247	139	16	:	:	PUNCT
ma-247	139	17	=	=	SYM
ma-247	139	18	xz	xz	PROPN
ma-247	139	19	∩	∩	PROPN
ma-247	139	20	bk(0	bk(0	PROPN
ma-247	139	21	,	,	PUNCT
ma-247	139	22	ε)×	ε)×	X
ma-247	139	23	cn−kbk(0	cn−kbk(0	NOUN
ma-247	139	24	,	,	PUNCT
ma-247	139	25	ε)×	ε)×	PROPN
ma-247	139	26	∆p−k	∆p−k	NUM
ma-247	139	27	×	×	PROPN
ma-247	139	28	{	{	PUNCT
ma-247	139	29	0}cn−p	0}cn−p	NUM
ma-247	139	30	.then	.then	X
ma-247	139	31	,	,	PUNCT
ma-247	139	32	the	the	DET
ma-247	139	33	expression	expression	NOUN
ma-247	139	34	given	give	VERB
ma-247	139	35	by	by	ADP
ma-247	139	36	(	(	PUNCT
ma-247	139	37	3.3	3.3	NUM
ma-247	139	38	)	)	PUNCT
ma-247	139	39	can	can	AUX
ma-247	139	40	be	be	AUX
ma-247	139	41	written	write	VERB
ma-247	139	42	as	as	ADP
ma-247	139	43	γε	γε	NOUN
ma-247	139	44	=	=	PROPN
ma-247	139	45	1	1	NUM
ma-247	139	46	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	139	47	,	,	PUNCT
ma-247	139	48	ε	ε	PROPN
ma-247	139	49	)	)	PUNCT
ma-247	139	50	)	)	PUNCT
ma-247	139	51	∫	∫	PROPN
ma-247	139	52	bk(0,ε	bk(0,ε	NOUN
ma-247	139	53	)	)	PUNCT
ma-247	139	54	w(t)(ddcϕ)k	w(t)(ddcϕ)k	NOUN
ma-247	139	55	,	,	PUNCT
ma-247	139	56	(	(	PUNCT
ma-247	139	57	3.4	3.4	NUM
ma-247	139	58	)	)	PUNCT
ma-247	139	59	where	where	SCONJ
ma-247	139	60	w(t	w(t	X
ma-247	139	61	)	)	PUNCT
ma-247	140	1	=	=	SYM
ma-247	140	2	∫	∫	PROPN
ma-247	140	3	∆p−k	∆p−k	PROPN
ma-247	140	4	π∗xz	π∗xz	PROPN
ma-247	140	5	(	(	PUNCT
ma-247	140	6	−vψ)(t	−vψ)(t	ADJ
ma-247	140	7	,	,	PUNCT
ma-247	140	8	ζ)dλp−k(ζ	ζ)dλp−k(ζ	NOUN
ma-247	140	9	)	)	PUNCT
ma-247	140	10	.	.	PUNCT
ma-247	141	1	furthermore	furthermore	ADV
ma-247	141	2	,	,	PUNCT
ma-247	141	3	following	follow	VERB
ma-247	141	4	[	[	X
ma-247	141	5	6	6	NUM
ma-247	141	6	]	]	PUNCT
ma-247	141	7	we	we	PRON
ma-247	141	8	have	have	VERB
ma-247	141	9	||w(ddcϕ)k	||w(ddcϕ)k	PROPN
ma-247	141	10	||bk(0,ε	||bk(0,ε	NOUN
ma-247	141	11	)	)	PUNCT
ma-247	141	12	≤	≤	NUM
ma-247	141	13	||w	||w	NOUN
ma-247	141	14	||l1(bk(0,2ε))||ϕ||kl∞(bk(0,2ε	||l1(bk(0,2ε))||ϕ||kl∞(bk(0,2ε	NOUN
ma-247	141	15	)	)	PUNCT
ma-247	141	16	)	)	PUNCT
ma-247	141	17	,	,	PUNCT
ma-247	141	18	(	(	PUNCT
ma-247	141	19	3.5	3.5	NUM
ma-247	141	20	)	)	PUNCT
ma-247	141	21	the	the	DET
ma-247	141	22	inequality	inequality	NOUN
ma-247	141	23	(	(	PUNCT
ma-247	141	24	3.5	3.5	NUM
ma-247	141	25	)	)	PUNCT
ma-247	141	26	implies	imply	VERB
ma-247	141	27	that	that	SCONJ
ma-247	141	28	w	w	NOUN
ma-247	141	29	is	be	AUX
ma-247	141	30	a	a	DET
ma-247	141	31	locally	locally	ADV
ma-247	141	32	integrable	integrable	ADJ
ma-247	141	33	function	function	NOUN
ma-247	141	34	with	with	ADP
ma-247	141	35	respect	respect	NOUN
ma-247	141	36	to	to	ADP
ma-247	141	37	the	the	DET
ma-247	141	38	positivemeasure	positivemeasure	NOUN
ma-247	141	39	µϕ	µϕ	ADV
ma-247	141	40	and	and	CCONJ
ma-247	141	41	hence	hence	ADV
ma-247	141	42	the	the	DET
ma-247	141	43	integral	integral	ADJ
ma-247	141	44	given	give	VERB
ma-247	141	45	by	by	ADP
ma-247	141	46	(	(	PUNCT
ma-247	141	47	3.4	3.4	NUM
ma-247	141	48	)	)	PUNCT
ma-247	141	49	is	be	AUX
ma-247	141	50	finite	finite	ADJ
ma-247	141	51	.	.	PUNCT
ma-247	142	1	�	�	PROPN
ma-247	142	2	now	now	ADV
ma-247	142	3	we	we	PRON
ma-247	142	4	are	be	AUX
ma-247	142	5	ready	ready	ADJ
ma-247	142	6	to	to	PART
ma-247	142	7	give	give	VERB
ma-247	142	8	the	the	DET
ma-247	142	9	proof	proof	NOUN
ma-247	142	10	of	of	ADP
ma-247	142	11	theorem	theorem	ADJ
ma-247	142	12	1.1	1.1	NUM
ma-247	142	13	.	.	PUNCT
ma-247	143	1	proof	proof	NOUN
ma-247	143	2	.	.	PUNCT
ma-247	144	1	the	the	DET
ma-247	144	2	result	result	NOUN
ma-247	144	3	is	be	AUX
ma-247	144	4	local	local	ADJ
ma-247	144	5	.	.	PUNCT
ma-247	145	1	without	without	ADP
ma-247	145	2	loss	loss	NOUN
ma-247	145	3	of	of	ADP
ma-247	145	4	generalities	generality	NOUN
ma-247	145	5	we	we	PRON
ma-247	145	6	may	may	AUX
ma-247	145	7	suppose	suppose	VERB
ma-247	145	8	n	n	PROPN
ma-247	145	9	=	=	SYM
ma-247	145	10	1	1	X
ma-247	145	11	.	.	PUNCT
ma-247	146	1	furthermore	furthermore	ADV
ma-247	146	2	,	,	PUNCT
ma-247	146	3	forsimplicity	forsimplicity	NOUN
ma-247	146	4	,	,	PUNCT
ma-247	146	5	we	we	PRON
ma-247	146	6	may	may	AUX
ma-247	146	7	consider	consider	VERB
ma-247	146	8	the	the	DET
ma-247	146	9	slice	slice	NOUN
ma-247	146	10	at	at	ADP
ma-247	146	11	point	point	NOUN
ma-247	146	12	a	a	PRON
ma-247	146	13	=	=	NOUN
ma-247	146	14	0	0	X
ma-247	146	15	.	.	PUNCT
ma-247	147	1	put	put	VERB
ma-247	147	2	t	t	PROPN
ma-247	147	3	=	=	PUNCT
ma-247	147	4	log	log	VERB
ma-247	147	5	|f	|f	PROPN
ma-247	147	6	|[y	|[y	PROPN
ma-247	147	7	]	]	PUNCT
ma-247	147	8	and	and	CCONJ
ma-247	147	9	γε(t	γε(t	ADV
ma-247	147	10	)	)	PUNCT
ma-247	148	1	=	=	SYM
ma-247	148	2	1	1	NUM
ma-247	148	3	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	148	4	,	,	PUNCT
ma-247	148	5	ε	ε	PROPN
ma-247	148	6	)	)	PUNCT
ma-247	148	7	)	)	PUNCT
ma-247	148	8	∫	∫	PROPN
ma-247	149	1	bk(0,ε)×cn−k	bk(0,ε)×cn−k	PROPN
ma-247	149	2	t	t	PROPN
ma-247	149	3	∧	∧	PROPN
ma-247	149	4	(	(	PUNCT
ma-247	149	5	ddc	ddc	PROPN
ma-247	149	6	ϕ̃)k	ϕ̃)k	PROPN
ma-247	149	7	∧ψ	∧ψ	PROPN
ma-247	149	8	,	,	PUNCT
ma-247	149	9	(	(	PUNCT
ma-247	149	10	3.6	3.6	NUM
ma-247	149	11	)	)	PUNCT
ma-247	149	12	where	where	SCONJ
ma-247	149	13	ψ	ψ	NOUN
ma-247	149	14	is	be	AUX
ma-247	149	15	a	a	DET
ma-247	149	16	test	test	NOUN
ma-247	149	17	form	form	NOUN
ma-247	149	18	such	such	ADJ
ma-247	149	19	that	that	SCONJ
ma-247	149	20	ψ	ψ	NOUN
ma-247	149	21	=	=	X
ma-247	149	22	h(z)β′′n−k	h(z)β′′n−k	NOUN
ma-247	149	23	∈	∈	PROPN
ma-247	149	24	d(n−q	d(n−q	PROPN
ma-247	149	25	,	,	PUNCT
ma-247	149	26	n−q)(∆n−k	n−q)(∆n−k	PROPN
ma-247	149	27	)	)	PUNCT
ma-247	149	28	where	where	SCONJ
ma-247	149	29	h	h	NOUN
ma-247	149	30	is	be	AUX
ma-247	149	31	a	a	DET
ma-247	149	32	positive	positive	ADJ
ma-247	149	33	smoothfunction	smoothfunction	NOUN
ma-247	149	34	with	with	ADP
ma-247	149	35	compact	compact	ADJ
ma-247	149	36	support	support	NOUN
ma-247	149	37	.	.	PUNCT
ma-247	150	1	we	we	PRON
ma-247	150	2	may	may	AUX
ma-247	150	3	find	find	VERB
ma-247	150	4	a	a	DET
ma-247	150	5	neighborhood	neighborhood	NOUN
ma-247	150	6	u	u	NOUN
ma-247	150	7	=	=	NOUN
ma-247	150	8	u	u	NOUN
ma-247	150	9	′×u	′×u	NOUN
ma-247	150	10	′′	′′	PROPN
ma-247	150	11	of	of	ADP
ma-247	150	12	0	0	NUM
ma-247	150	13	in	in	ADP
ma-247	150	14	∆k×∆q−k×∆n−qand	∆k×∆q−k×∆n−qand	NOUN
ma-247	150	15	a	a	DET
ma-247	150	16	coordinate	coordinate	NOUN
ma-247	150	17	system	system	NOUN
ma-247	150	18	(	(	PUNCT
ma-247	150	19	t	t	PROPN
ma-247	150	20	,	,	PUNCT
ma-247	150	21	ζ	ζ	NOUN
ma-247	150	22	,	,	PUNCT
ma-247	150	23	z	z	PROPN
ma-247	150	24	′′	′′	PROPN
ma-247	150	25	)	)	PUNCT
ma-247	150	26	such	such	ADJ
ma-247	150	27	that	that	SCONJ
ma-247	150	28	the	the	DET
ma-247	150	29	projection	projection	NOUN
ma-247	150	30	πy	πy	INTJ
ma-247	150	31	:	:	PUNCT
ma-247	150	32	y	y	PROPN
ma-247	150	33	∩	∩	ADJ
ma-247	150	34	u	u	NOUN
ma-247	150	35	→	→	SYM
ma-247	150	36	∆q	∆q	PROPN
ma-247	150	37	(	(	PUNCT
ma-247	150	38	(	(	PUNCT
ma-247	150	39	t	t	PROPN
ma-247	150	40	,	,	PUNCT
ma-247	150	41	ζ	ζ	NOUN
ma-247	150	42	)	)	PUNCT
ma-247	150	43	,	,	PUNCT
ma-247	150	44	z	z	PUNCT
ma-247	150	45	′′	′′	PROPN
ma-247	150	46	)	)	PUNCT
ma-247	150	47	7→	7→	PROPN
ma-247	150	48	(	(	PUNCT
ma-247	150	49	t	t	PROPN
ma-247	150	50	,	,	PUNCT
ma-247	150	51	ζ)defines	ζ)define	VERB
ma-247	150	52	a	a	DET
ma-247	150	53	ramified	ramified	ADJ
ma-247	150	54	covering	covering	NOUN
ma-247	150	55	of	of	ADP
ma-247	150	56	y	y	PROPN
ma-247	150	57	.	.	PUNCT
ma-247	151	1	let	let	VERB
ma-247	151	2	z	z	NOUN
ma-247	151	3	be	be	AUX
ma-247	151	4	the	the	DET
ma-247	151	5	ramification	ramification	NOUN
ma-247	151	6	locus	locus	NOUN
ma-247	151	7	of	of	ADP
ma-247	151	8	π	π	PROPN
ma-247	151	9	and	and	CCONJ
ma-247	151	10	yz	yz	PROPN
ma-247	151	11	=	=	PROPN
ma-247	151	12	y	y	PROPN
ma-247	151	13	∩((u	∩((u	PROPN
ma-247	151	14	′rz)×u	′rz)×u	PROPN
ma-247	151	15	′′	′′	PROPN
ma-247	151	16	)	)	PUNCT
ma-247	151	17	⊂	⊂	PROPN
ma-247	151	18	yreg	yreg	VERB
ma-247	151	19	.	.	PUNCT
ma-247	152	1	the	the	DET
ma-247	152	2	restriction	restriction	NOUN
ma-247	152	3	of	of	ADP
ma-247	152	4	πyz	πyz	NOUN
ma-247	152	5	:	:	PUNCT
ma-247	152	6	yz	yz	PROPN
ma-247	152	7	→	→	SYM
ma-247	152	8	u	u	NOUN
ma-247	152	9	′	′	NUM
ma-247	152	10	r	r	NOUN
ma-247	152	11	z	z	NOUN
ma-247	152	12	is	be	AUX
ma-247	152	13	then	then	ADV
ma-247	152	14	a	a	DET
ma-247	152	15	covering	covering	NOUN
ma-247	152	16	with	with	ADP
ma-247	152	17	a	a	DET
ma-247	152	18	finite	finite	ADJ
ma-247	152	19	sheet	sheet	NOUN
ma-247	152	20	number	number	NOUN
ma-247	152	21	.	.	PUNCT
ma-247	153	1	equality	equality	NOUN
ma-247	153	2	(	(	PUNCT
ma-247	153	3	3.6	3.6	NUM
ma-247	153	4	)	)	PUNCT
ma-247	153	5	can	can	AUX
ma-247	153	6	be	be	AUX
ma-247	153	7	written	write	VERB
ma-247	153	8	as	as	ADP
ma-247	153	9	γε(t	γε(t	X
ma-247	153	10	)	)	PUNCT
ma-247	154	1	=	=	SYM
ma-247	154	2	1	1	NUM
ma-247	154	3	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	154	4	,	,	PUNCT
ma-247	154	5	ε	ε	PROPN
ma-247	154	6	)	)	PUNCT
ma-247	154	7	)	)	PUNCT
ma-247	154	8	∫	∫	PROPN
ma-247	155	1	yz∩bk(0,ε)×cn−k	yz∩bk(0,ε)×cn−k	PRON
ma-247	155	2	π∗	π∗	PROPN
ma-247	155	3	(	(	PUNCT
ma-247	155	4	log	log	VERB
ma-247	155	5	|f	|f	PRON
ma-247	155	6	|(ddcϕ)k	|(ddcϕ)k	PRON
ma-247	155	7	∧ψ	∧ψ	PROPN
ma-247	155	8	)	)	PUNCT
ma-247	155	9	.	.	PUNCT
ma-247	156	1	(	(	PUNCT
ma-247	156	2	3.7	3.7	NUM
ma-247	156	3	)	)	PUNCT
ma-247	156	4	we	we	PRON
ma-247	156	5	have	have	VERB
ma-247	156	6	to	to	PART
ma-247	156	7	find	find	VERB
ma-247	156	8	the	the	DET
ma-247	156	9	limit	limit	NOUN
ma-247	156	10	,	,	PUNCT
ma-247	156	11	as	as	ADP
ma-247	156	12	ε→	ε→	X
ma-247	156	13	0	0	NUM
ma-247	156	14	,	,	PUNCT
ma-247	156	15	of	of	ADP
ma-247	156	16	(	(	PUNCT
ma-247	156	17	6.20	6.20	NUM
ma-247	156	18	)	)	PUNCT
ma-247	156	19	.	.	PUNCT
ma-247	157	1	for	for	ADP
ma-247	157	2	ε	ε	PROPN
ma-247	157	3	>	>	X
ma-247	157	4	0	0	PROPN
ma-247	157	5	small	small	ADJ
ma-247	157	6	enough	enough	ADV
ma-247	157	7	,	,	PUNCT
ma-247	157	8	the	the	DET
ma-247	157	9	set	set	NOUN
ma-247	157	10	uε	uε	INTJ
ma-247	157	11	:	:	PUNCT
ma-247	157	12	=	=	SYM
ma-247	157	13	yz∩bk(0	yz∩bk(0	PROPN
ma-247	157	14	,	,	PUNCT
ma-247	157	15	ε)×	ε)×	PROPN
ma-247	157	16	cn−k	cn−k	NOUN
ma-247	157	17	can	can	AUX
ma-247	157	18	be	be	AUX
ma-247	157	19	viewed	view	VERB
ma-247	157	20	as	as	ADP
ma-247	157	21	bk(0	bk(0	PROPN
ma-247	157	22	,	,	PUNCT
ma-247	157	23	ε	ε	PROPN
ma-247	157	24	)	)	PUNCT
ma-247	157	25	×	×	NOUN
ma-247	157	26	∆q−k	∆q−k	ADJ
ma-247	157	27	×	×	PROPN
ma-247	157	28	{	{	PUNCT
ma-247	157	29	0}cn−q	0}cn−q	NUM
ma-247	157	30	.	.	PUNCT
ma-247	158	1	the	the	DET
ma-247	158	2	integral	integral	ADJ
ma-247	158	3	in	in	ADP
ma-247	158	4	the	the	DET
ma-247	158	5	right	right	ADJ
ma-247	158	6	hand	hand	NOUN
ma-247	158	7	side	side	NOUN
ma-247	158	8	of	of	ADP
ma-247	158	9	(	(	PUNCT
ma-247	158	10	6.20)can	6.20)can	NUM
ma-247	158	11	be	be	AUX
ma-247	158	12	written	write	VERB
ma-247	158	13	such	such	ADJ
ma-247	158	14	that	that	PRON
ma-247	158	15	γε(t	γε(t	PUNCT
ma-247	158	16	)	)	PUNCT
ma-247	159	1	=	=	SYM
ma-247	159	2	1	1	NUM
ma-247	159	3	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	159	4	,	,	PUNCT
ma-247	159	5	ε	ε	PROPN
ma-247	159	6	)	)	PUNCT
ma-247	159	7	)	)	PUNCT
ma-247	159	8	∫	∫	PROPN
ma-247	159	9	bk(0,ε	bk(0,ε	NOUN
ma-247	159	10	)	)	PUNCT
ma-247	159	11	wh(t)(ddcϕ)k	wh(t)(ddcϕ)k	NOUN
ma-247	159	12	,	,	PUNCT
ma-247	159	13	(	(	PUNCT
ma-247	159	14	3.8	3.8	NUM
ma-247	159	15	)	)	PUNCT
ma-247	159	16	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	159	17	eur	eur	PROPN
ma-247	159	18	.	.	PUNCT
ma-247	160	1	j.	j.	PROPN
ma-247	160	2	math	math	PROPN
ma-247	160	3	.	.	PUNCT
ma-247	161	1	anal	anal	PROPN
ma-247	161	2	.	.	PUNCT
ma-247	162	1	10.28924	10.28924	NUM
ma-247	162	2	/	/	SYM
ma-247	162	3	ada	ada	PROPN
ma-247	162	4	/	/	SYM
ma-247	162	5	ma.4.23	ma.4.23	PROPN
ma-247	162	6	7where	7where	NUM
ma-247	162	7	wh(t	wh(t	PUNCT
ma-247	162	8	)	)	PUNCT
ma-247	163	1	=	=	SYM
ma-247	163	2	∫	∫	PROPN
ma-247	164	1	∆q−k	∆q−k	PROPN
ma-247	164	2	log	log	PROPN
ma-247	164	3	|f	|f	PROPN
ma-247	164	4	(	(	PUNCT
ma-247	164	5	t	t	PROPN
ma-247	164	6	,	,	PUNCT
ma-247	164	7	ζ	ζ	NOUN
ma-247	164	8	,	,	PUNCT
ma-247	164	9	0)|h(ζ	0)|h(ζ	NUM
ma-247	164	10	,	,	PUNCT
ma-247	164	11	0)dλq−k(ζ	0)dλq−k(ζ	NUM
ma-247	164	12	)	)	PUNCT
ma-247	165	1	=	=	SYM
ma-247	165	2	∫	∫	PROPN
ma-247	165	3	yz∩π−1(0	yz∩π−1(0	PROPN
ma-247	165	4	)	)	PUNCT
ma-247	165	5	log	log	VERB
ma-247	165	6	|i∗	|i∗	PROPN
ma-247	165	7	yz∩π−1(0	yz∩π−1(0	PROPN
ma-247	165	8	)	)	PUNCT
ma-247	166	1	f	f	PROPN
ma-247	166	2	(	(	PUNCT
ma-247	166	3	t	t	PROPN
ma-247	166	4	,	,	PUNCT
ma-247	166	5	ζ	ζ	NOUN
ma-247	166	6	,	,	PUNCT
ma-247	166	7	z	z	PROPN
ma-247	166	8	′′)|i∗	′′)|i∗	NOUN
ma-247	166	9	yz∩π−1(0	yz∩π−1(0	PROPN
ma-247	166	10	)	)	PUNCT
ma-247	166	11	h(ζ	h(ζ	PROPN
ma-247	166	12	,	,	PUNCT
ma-247	166	13	z	z	PROPN
ma-247	166	14	′′),and	′′),and	X
ma-247	166	15	iyz∩π−1(0	iyz∩π−1(0	PROPN
ma-247	166	16	)	)	PUNCT
ma-247	166	17	:	:	PUNCT
ma-247	167	1	yz	yz	X
ma-247	167	2	∩π−1(0)→	∩π−1(0)→	PROPN
ma-247	167	3	∆n	∆n	PROPN
ma-247	167	4	is	be	AUX
ma-247	167	5	the	the	DET
ma-247	167	6	canonical	canonical	ADJ
ma-247	167	7	injection	injection	NOUN
ma-247	167	8	.	.	PUNCT
ma-247	168	1	let	let	VERB
ma-247	168	2	e	e	PRON
ma-247	168	3	be	be	AUX
ma-247	168	4	the	the	DET
ma-247	168	5	closed	closed	ADJ
ma-247	168	6	pluripolar	pluripolar	ADJ
ma-247	168	7	subsetof	subsetof	NOUN
ma-247	168	8	∆k	∆k	PROPN
ma-247	168	9	defined	define	VERB
ma-247	168	10	by	by	ADP
ma-247	168	11	e	e	X
ma-247	168	12	=	=	PUNCT
ma-247	168	13	{	{	PUNCT
ma-247	168	14	a	a	DET
ma-247	168	15	∈	∈	NOUN
ma-247	168	16	∆k	∆k	NOUN
ma-247	168	17	:	:	PUNCT
ma-247	168	18	w(a	w(a	X
ma-247	168	19	)	)	PUNCT
ma-247	169	1	=	=	SYM
ma-247	169	2	−∞	−∞	PUNCT
ma-247	169	3	or	or	CCONJ
ma-247	169	4	w	w	PROPN
ma-247	169	5	6∈	6∈	PROPN
ma-247	169	6	l1	l1	PROPN
ma-247	169	7	loc(µϕ	loc(µϕ	NUM
ma-247	169	8	)	)	PUNCT
ma-247	169	9	near	near	ADP
ma-247	169	10	a	a	PRON
ma-247	169	11	}	}	PUNCT
ma-247	169	12	.	.	PUNCT
ma-247	170	1	by	by	ADP
ma-247	170	2	theorem	theorem	NOUN
ma-247	170	3	1.1	1.1	NUM
ma-247	170	4	in	in	ADP
ma-247	170	5	[	[	X
ma-247	170	6	11	11	NUM
ma-247	170	7	]	]	PUNCT
ma-247	170	8	,	,	PUNCT
ma-247	170	9	for	for	ADP
ma-247	170	10	all	all	DET
ma-247	170	11	a	a	DET
ma-247	170	12	∈	∈	NOUN
ma-247	170	13	∆k	∆k	PROPN
ma-247	170	14	r	r	NOUN
ma-247	170	15	e	e	NOUN
ma-247	170	16	,	,	PUNCT
ma-247	170	17	we	we	PRON
ma-247	170	18	have:(1	have:(1	PROPN
ma-247	170	19	)	)	PUNCT
ma-247	170	20	if	if	SCONJ
ma-247	170	21	(	(	PUNCT
ma-247	170	22	iyz∩π−1(0))∗f	iyz∩π−1(0))∗f	NOUN
ma-247	170	23	6≡	6≡	NUM
ma-247	170	24	0	0	NUM
ma-247	170	25	,	,	PUNCT
ma-247	170	26	then	then	ADV
ma-247	170	27	,	,	PUNCT
ma-247	170	28	as	as	ADP
ma-247	170	29	ε→	ε→	X
ma-247	170	30	0	0	NUM
ma-247	170	31	,	,	PUNCT
ma-247	170	32	the	the	DET
ma-247	170	33	limit	limit	NOUN
ma-247	170	34	of	of	ADP
ma-247	170	35	the	the	DET
ma-247	170	36	integral	integral	ADJ
ma-247	170	37	given	give	VERB
ma-247	170	38	by	by	ADP
ma-247	170	39	(	(	PUNCT
ma-247	170	40	3.8	3.8	NUM
ma-247	170	41	)	)	PUNCT
ma-247	170	42	will	will	AUX
ma-247	170	43	be	be	AUX
ma-247	170	44	limε→0	limε→0	NOUN
ma-247	170	45	γε	γε	NOUN
ma-247	170	46	=	=	SYM
ma-247	170	47	wh(0	wh(0	NOUN
ma-247	170	48	)	)	PUNCT
ma-247	171	1	=	=	SYM
ma-247	171	2	∫	∫	PROPN
ma-247	172	1	∆q−k	∆q−k	PROPN
ma-247	172	2	log	log	NOUN
ma-247	172	3	|f	|f	PROPN
ma-247	172	4	(	(	PUNCT
ma-247	172	5	0	0	NUM
ma-247	172	6	,	,	PUNCT
ma-247	172	7	ζ	ζ	NOUN
ma-247	172	8	,	,	PUNCT
ma-247	172	9	0)|h(ζ	0)|h(ζ	NUM
ma-247	172	10	,	,	PUNCT
ma-247	172	11	0)dλq−k(ζ	0)dλq−k(ζ	NUM
ma-247	172	12	)	)	PUNCT
ma-247	173	1	=	=	SYM
ma-247	173	2	∫	∫	PROPN
ma-247	174	1	y	y	PROPN
ma-247	174	2	∩π−1(0)(iyz∩π−1(0))∗[log	∩π−1(0)(iyz∩π−1(0))∗[log	PROPN
ma-247	174	3	|f	|f	PROPN
ma-247	174	4	|	|	PROPN
ma-247	174	5	∧ψ	∧ψ	NOUN
ma-247	174	6	]	]	X
ma-247	174	7	=	=	SYM
ma-247	174	8	〈	〈	PROPN
ma-247	174	9	(	(	PUNCT
ma-247	174	10	iyz∩π−1(0	iyz∩π−1(0	PROPN
ma-247	174	11	)	)	PUNCT
ma-247	174	12	)	)	PUNCT
ma-247	174	13	∗	∗	NOUN
ma-247	174	14	(	(	PUNCT
ma-247	174	15	log	log	VERB
ma-247	174	16	|f	|f	PROPN
ma-247	174	17	|	|	NOUN
ma-247	174	18	)	)	PUNCT
ma-247	175	1	[	[	PUNCT
ma-247	175	2	y	y	PROPN
ma-247	175	3	∩	∩	NOUN
ma-247	175	4	π−1(0	π−1(0	PROPN
ma-247	175	5	)	)	PUNCT
ma-247	175	6	]	]	PUNCT
ma-247	175	7	,	,	PUNCT
ma-247	175	8	(	(	PUNCT
ma-247	175	9	iyz∩π−1(0))∗(ψ)〉ϕ.(2	iyz∩π−1(0))∗(ψ)〉ϕ.(2	NOUN
ma-247	175	10	)	)	PUNCT
ma-247	175	11	if	if	SCONJ
ma-247	175	12	(	(	PUNCT
ma-247	175	13	iyz∩π−1(0))∗f	iyz∩π−1(0))∗f	NOUN
ma-247	175	14	≡	≡	PROPN
ma-247	175	15	0	0	NUM
ma-247	175	16	,	,	PUNCT
ma-247	175	17	then	then	ADV
ma-247	175	18	by	by	ADP
ma-247	175	19	proposition	proposition	NOUN
ma-247	175	20	3.2	3.2	NUM
ma-247	175	21	applied	apply	VERB
ma-247	175	22	with	with	ADP
ma-247	175	23	the	the	DET
ma-247	175	24	function	function	NOUN
ma-247	175	25	(	(	PUNCT
ma-247	175	26	iyz∩π−1(0))∗f	iyz∩π−1(0))∗f	NOUN
ma-247	175	27	on	on	ADP
ma-247	175	28	∆q	∆q	PROPN
ma-247	175	29	=	=	PUNCT
ma-247	175	30	∆k	∆k	PROPN
ma-247	175	31	×	×	PROPN
ma-247	175	32	∆q−k	∆q−k	ADV
ma-247	175	33	,	,	PUNCT
ma-247	175	34	there	there	PRON
ma-247	175	35	exist	exist	VERB
ma-247	175	36	an	an	DET
ma-247	175	37	integer	integer	NOUN
ma-247	175	38	m	m	PROPN
ma-247	175	39	∈	∈	PROPN
ma-247	175	40	n	n	CCONJ
ma-247	175	41	,	,	PUNCT
ma-247	175	42	a	a	DET
ma-247	175	43	non	non	AUX
ma-247	175	44	vanishing	vanish	VERB
ma-247	175	45	homogenous	homogenous	ADJ
ma-247	175	46	polynomial	polynomial	ADJ
ma-247	175	47	qm(t	qm(t	NOUN
ma-247	175	48	,	,	PUNCT
ma-247	175	49	ζ	ζ	NOUN
ma-247	175	50	)	)	PUNCT
ma-247	175	51	and	and	CCONJ
ma-247	175	52	smooth	smooth	ADJ
ma-247	175	53	functions	function	NOUN
ma-247	175	54	(	(	PUNCT
ma-247	175	55	t	t	NOUN
ma-247	175	56	,	,	PUNCT
ma-247	175	57	ζ	ζ	NOUN
ma-247	175	58	)	)	PUNCT
ma-247	175	59	7→	7→	NUM
ma-247	175	60	r(t	r(t	NOUN
ma-247	175	61	,	,	PUNCT
ma-247	175	62	ζ	ζ	NOUN
ma-247	175	63	)	)	PUNCT
ma-247	175	64	∈	∈	NOUN
ma-247	175	65	c∞(∆k	c∞(∆k	X
ma-247	175	66	×	×	NOUN
ma-247	175	67	∆q−k	∆q−k	NOUN
ma-247	175	68	)	)	PUNCT
ma-247	175	69	such	such	ADJ
ma-247	175	70	that	that	SCONJ
ma-247	175	71	f	f	PROPN
ma-247	175	72	(	(	PUNCT
ma-247	175	73	t	t	PROPN
ma-247	175	74	,	,	PUNCT
ma-247	175	75	ζ	ζ	NOUN
ma-247	175	76	)	)	PUNCT
ma-247	175	77	=	=	SYM
ma-247	175	78	qm(t	qm(t	NOUN
ma-247	175	79	,	,	PUNCT
ma-247	175	80	ζ	ζ	NOUN
ma-247	175	81	)	)	PUNCT
ma-247	175	82	+	+	NUM
ma-247	175	83	tr(t	tr(t	NUM
ma-247	175	84	,	,	PUNCT
ma-247	175	85	ζ	ζ	NOUN
ma-247	175	86	)	)	PUNCT
ma-247	175	87	on	on	ADP
ma-247	175	88	bk(0	bk(0	PROPN
ma-247	175	89	,	,	PUNCT
ma-247	175	90	ε)×	ε)×	PROPN
ma-247	175	91	∆p−k	∆p−k	PROPN
ma-247	175	92	.	.	PUNCT
ma-247	176	1	thanks	thank	NOUN
ma-247	176	2	to	to	ADP
ma-247	176	3	taylor	taylor	PROPN
ma-247	176	4	’s	’s	PART
ma-247	176	5	formula	formula	NOUN
ma-247	176	6	,	,	PUNCT
ma-247	176	7	the	the	DET
ma-247	176	8	rest	rest	NOUN
ma-247	176	9	r(t	r(t	NOUN
ma-247	176	10	,	,	PUNCT
ma-247	176	11	ζ	ζ	NOUN
ma-247	176	12	)	)	PUNCT
ma-247	176	13	can	can	AUX
ma-247	176	14	be	be	AUX
ma-247	176	15	chosen	choose	VERB
ma-247	176	16	such	such	ADJ
ma-247	176	17	that	that	SCONJ
ma-247	176	18	|r(t	|r(t	PROPN
ma-247	176	19	,	,	PUNCT
ma-247	176	20	ζ)|	ζ)|	NOUN
ma-247	176	21	≤	≤	ADJ
ma-247	176	22	1	1	NUM
ma-247	176	23	.	.	X
ma-247	177	1	thiswith	thiswith	ADP
ma-247	177	2	the	the	DET
ma-247	177	3	triangle	triangle	NOUN
ma-247	177	4	inequality	inequality	NOUN
ma-247	177	5	legitimate	legitimate	ADJ
ma-247	177	6	the	the	DET
ma-247	177	7	following	follow	VERB
ma-247	177	8	inequalities	inequality	NOUN
ma-247	177	9	log	log	VERB
ma-247	177	10	||qm|	||qm|	PUNCT
ma-247	177	11	−	−	PROPN
ma-247	177	12	|tr||	|tr||	X
ma-247	177	13	≤	≤	NUM
ma-247	177	14	log	log	NOUN
ma-247	177	15	||f	||f	PROPN
ma-247	177	16	(	(	PUNCT
ma-247	177	17	t	t	NOUN
ma-247	177	18	,	,	PUNCT
ma-247	177	19	ζ)||	ζ)||	NOUN
ma-247	177	20	=	=	NOUN
ma-247	177	21	log	log	VERB
ma-247	177	22	|(iyz∩π−1(0))∗f	|(iyz∩π−1(0))∗f	NOUN
ma-247	177	23	(	(	PUNCT
ma-247	177	24	t	t	PROPN
ma-247	177	25	,	,	PUNCT
ma-247	177	26	w	w	PROPN
ma-247	177	27	,	,	PUNCT
ma-247	177	28	z	z	NOUN
ma-247	177	29	′′)|	′′)|	VERB
ma-247	177	30	≤	≤	PUNCT
ma-247	177	31	log(|qm|+	log(|qm|+	PROPN
ma-247	177	32	|tr|).since	|tr|).since	NOUN
ma-247	177	33	(	(	PUNCT
ma-247	177	34	t	t	PROPN
ma-247	177	35	,	,	PUNCT
ma-247	177	36	ζ	ζ	NOUN
ma-247	177	37	)	)	PUNCT
ma-247	177	38	∈	∈	PROPN
ma-247	177	39	bk(0	bk(0	PROPN
ma-247	177	40	,	,	PUNCT
ma-247	177	41	ε	ε	PROPN
ma-247	177	42	)	)	PUNCT
ma-247	177	43	×	×	PROPN
ma-247	177	44	∆p−k	∆p−k	PROPN
ma-247	177	45	and	and	CCONJ
ma-247	177	46	|r(t	|r(t	PROPN
ma-247	177	47	,	,	PUNCT
ma-247	177	48	ζ)|	ζ)|	NOUN
ma-247	177	49	≤	≤	ADJ
ma-247	177	50	1	1	NUM
ma-247	177	51	,	,	PUNCT
ma-247	177	52	then	then	ADV
ma-247	177	53	,	,	PUNCT
ma-247	177	54	as	as	ADP
ma-247	177	55	ε	ε	PROPN
ma-247	177	56	→	→	SYM
ma-247	177	57	0	0	NUM
ma-247	177	58	,	,	PUNCT
ma-247	177	59	|t|	|t|	NOUN
ma-247	177	60	→	→	SYM
ma-247	177	61	0	0	NUM
ma-247	177	62	and	and	CCONJ
ma-247	177	63	|tr|	|tr|	NUM
ma-247	177	64	→	→	SYM
ma-247	177	65	0.hence	0.hence	NUM
ma-247	177	66	,	,	PUNCT
ma-247	177	67	we	we	PRON
ma-247	177	68	get	get	VERB
ma-247	177	69	limε→0	limε→0	NOUN
ma-247	177	70	γε	γε	NOUN
ma-247	177	71	=	=	SYM
ma-247	177	72	wh(0	wh(0	NOUN
ma-247	177	73	)	)	PUNCT
ma-247	178	1	=	=	SYM
ma-247	178	2	∫	∫	PROPN
ma-247	178	3	∆q−k	∆q−k	PROPN
ma-247	178	4	log	log	PROPN
ma-247	178	5	|qm(0	|qm(0	NOUN
ma-247	178	6	,	,	PUNCT
ma-247	178	7	ζ)|h(ζ	ζ)|h(ζ	NOUN
ma-247	178	8	,	,	PUNCT
ma-247	178	9	z	z	NOUN
ma-247	178	10	′′)dλq−k(ζ	′′)dλq−k(ζ	NOUN
ma-247	178	11	)	)	PUNCT
ma-247	178	12	=	=	SYM
ma-247	178	13	∫	∫	PROPN
ma-247	179	1	y	y	PROPN
ma-247	179	2	∩π−1(0)(iyz∩π−1(0))∗[log	∩π−1(0)(iyz∩π−1(0))∗[log	PROPN
ma-247	179	3	|qm(0	|qm(0	PROPN
ma-247	179	4	,	,	PUNCT
ma-247	179	5	ζ|	ζ|	PROPN
ma-247	179	6	∧ψ	∧ψ	NUM
ma-247	179	7	]	]	X
ma-247	179	8	=	=	SYM
ma-247	179	9	〈	〈	PROPN
ma-247	179	10	(	(	PUNCT
ma-247	179	11	log	log	NOUN
ma-247	179	12	|(iyz∩π−1(0))∗qm|	|(iyz∩π−1(0))∗qm|	NUM
ma-247	179	13	)	)	PUNCT
ma-247	179	14	[	[	PUNCT
ma-247	179	15	y	y	PROPN
ma-247	179	16	∩	∩	NOUN
ma-247	179	17	π−1(0	π−1(0	PROPN
ma-247	179	18	)	)	PUNCT
ma-247	179	19	]	]	PUNCT
ma-247	179	20	,	,	PUNCT
ma-247	179	21	(	(	PUNCT
ma-247	179	22	iyz∩π−1(0))∗(ψ)〉ϕ.the	iyz∩π−1(0))∗(ψ)〉ϕ.the	DET
ma-247	179	23	proof	proof	NOUN
ma-247	179	24	is	be	AUX
ma-247	179	25	completed	complete	VERB
ma-247	179	26	.	.	PUNCT
ma-247	180	1	�	�	PROPN
ma-247	180	2	the	the	DET
ma-247	180	3	following	follow	VERB
ma-247	180	4	example	example	NOUN
ma-247	180	5	illustrates	illustrate	VERB
ma-247	180	6	theorem	theorem	VERB
ma-247	180	7	1.1	1.1	NUM
ma-247	180	8	.	.	PUNCT
ma-247	180	9	example	example	NOUN
ma-247	180	10	3.4	3.4	NUM
ma-247	180	11	.	.	PUNCT
ma-247	181	1	in	in	ADP
ma-247	181	2	c3	c3	PROPN
ma-247	181	3	we	we	PRON
ma-247	181	4	let	let	VERB
ma-247	181	5	f	f	PROPN
ma-247	181	6	(	(	PUNCT
ma-247	181	7	z	z	NOUN
ma-247	181	8	)	)	PUNCT
ma-247	181	9	=	=	SYM
ma-247	181	10	z2	z2	NOUN
ma-247	181	11	1	1	NUM
ma-247	181	12	+	+	SYM
ma-247	181	13	z2	z2	PROPN
ma-247	181	14	2	2	NUM
ma-247	181	15	−	−	PROPN
ma-247	181	16	z3	z3	PROPN
ma-247	181	17	,	,	PUNCT
ma-247	181	18	y	y	PROPN
ma-247	181	19	=	=	PUNCT
ma-247	181	20	{	{	PUNCT
ma-247	181	21	z1	z1	PROPN
ma-247	181	22	=	=	SYM
ma-247	181	23	−z3	−z3	PROPN
ma-247	181	24	}	}	PUNCT
ma-247	181	25	,	,	PUNCT
ma-247	181	26	k	k	PROPN
ma-247	181	27	=	=	SYM
ma-247	181	28	1	1	NUM
ma-247	181	29	and	and	CCONJ
ma-247	181	30	ϕ(z	ϕ(z	PROPN
ma-247	181	31	′	′	NUM
ma-247	181	32	)	)	PUNCT
ma-247	182	1	=	=	SYM
ma-247	182	2	|z	|z	PROPN
ma-247	182	3	′|2	′|2	PROPN
ma-247	182	4	=	=	NOUN
ma-247	183	1	|z1|2.it	|z1|2.it	NOUN
ma-247	183	2	is	be	AUX
ma-247	183	3	clear	clear	ADJ
ma-247	183	4	that	that	SCONJ
ma-247	183	5	the	the	DET
ma-247	183	6	subsets	subset	NOUN
ma-247	183	7	x	x	PUNCT
ma-247	183	8	=	=	PUNCT
ma-247	183	9	{	{	PUNCT
ma-247	183	10	f	f	NOUN
ma-247	183	11	=	=	SYM
ma-247	183	12	0	0	NUM
ma-247	183	13	}	}	PUNCT
ma-247	183	14	and	and	CCONJ
ma-247	183	15	y	y	PROPN
ma-247	183	16	yield	yield	VERB
ma-247	183	17	a	a	DET
ma-247	183	18	complete	complete	ADJ
ma-247	183	19	intersection	intersection	NOUN
ma-247	183	20	near	near	ADP
ma-247	183	21	0	0	NUM
ma-247	183	22	.	.	PUNCT
ma-247	184	1	hence	hence	ADV
ma-247	184	2	,	,	PUNCT
ma-247	184	3	according	accord	VERB
ma-247	184	4	to	to	ADP
ma-247	184	5	[	[	X
ma-247	184	6	9	9	NUM
ma-247	184	7	]	]	PUNCT
ma-247	184	8	,	,	PUNCT
ma-247	184	9	we	we	PRON
ma-247	184	10	deduce	deduce	VERB
ma-247	184	11	that	that	PRON
ma-247	184	12	log	log	NOUN
ma-247	184	13	|f	|f	PROPN
ma-247	185	1	|	|	ADV
ma-247	185	2	∈	∈	PROPN
ma-247	185	3	l1	l1	PROPN
ma-247	185	4	loc(y	loc(y	PROPN
ma-247	185	5	)	)	PUNCT
ma-247	185	6	.	.	PUNCT
ma-247	186	1	by	by	ADP
ma-247	186	2	theorem	theorem	NOUN
ma-247	186	3	1.1	1.1	NUM
ma-247	186	4	,	,	PUNCT
ma-247	186	5	we	we	PRON
ma-247	186	6	have	have	VERB
ma-247	186	7	〈	〈	PART
ma-247	186	8	log	log	VERB
ma-247	186	9	|f	|f	NOUN
ma-247	186	10	|	|	ADV
ma-247	186	11	.	.	PUNCT
ma-247	187	1	[	[	PUNCT
ma-247	187	2	y	y	X
ma-247	187	3	]	]	PUNCT
ma-247	187	4	,	,	PUNCT
ma-247	187	5	π	π	PROPN
ma-247	187	6	,	,	PUNCT
ma-247	187	7	0	0	NUM
ma-247	187	8	〉	〉	NOUN
ma-247	187	9	=	=	PUNCT
ma-247	187	10	log	log	VERB
ma-247	187	11	|z2	|z2	ADJ
ma-247	187	12	2	2	NUM
ma-247	187	13	−	−	PROPN
ma-247	187	14	z3|	z3|	NUM
ma-247	187	15	.	.	PUNCT
ma-247	188	1	[	[	PUNCT
ma-247	188	2	z1	z1	NOUN
ma-247	188	3	=	=	SYM
ma-247	188	4	z3	z3	PROPN
ma-247	188	5	=	=	SYM
ma-247	188	6	0	0	NUM
ma-247	188	7	]	]	PUNCT
ma-247	188	8	.	.	PUNCT
ma-247	189	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	189	2	eur	eur	PROPN
ma-247	189	3	.	.	PUNCT
ma-247	190	1	j.	j.	PROPN
ma-247	190	2	math	math	PROPN
ma-247	190	3	.	.	PUNCT
ma-247	191	1	anal	anal	PROPN
ma-247	191	2	.	.	PUNCT
ma-247	192	1	10.28924	10.28924	NUM
ma-247	192	2	/	/	SYM
ma-247	192	3	ada	ada	PROPN
ma-247	192	4	/	/	SYM
ma-247	192	5	ma.4.23	ma.4.23	PROPN
ma-247	192	6	84	84	NUM
ma-247	192	7	.	.	PUNCT
ma-247	193	1	proof	proof	NOUN
ma-247	193	2	of	of	ADP
ma-247	193	3	theorem	theorem	ADJ
ma-247	193	4	1.2	1.2	NUM
ma-247	193	5	:	:	PUNCT
ma-247	193	6	slicing	slicing	NOUN
ma-247	193	7	of	of	ADP
ma-247	193	8	the	the	DET
ma-247	193	9	current	current	ADJ
ma-247	193	10	v	v	NOUN
ma-247	193	11	[	[	X
ma-247	193	12	x	x	X
ma-247	193	13	]	]	X
ma-247	193	14	proof	proof	NOUN
ma-247	193	15	.	.	PUNCT
ma-247	194	1	we	we	PRON
ma-247	194	2	use	use	VERB
ma-247	194	3	notations	notation	NOUN
ma-247	194	4	and	and	CCONJ
ma-247	194	5	arguments	argument	NOUN
ma-247	194	6	as	as	ADP
ma-247	194	7	in	in	ADP
ma-247	194	8	the	the	DET
ma-247	194	9	proof	proof	NOUN
ma-247	194	10	of	of	ADP
ma-247	194	11	theorem	theorem	ADJ
ma-247	194	12	1.1	1.1	NUM
ma-247	194	13	and	and	CCONJ
ma-247	194	14	we	we	PRON
ma-247	194	15	may	may	AUX
ma-247	194	16	suppose	suppose	VERB
ma-247	194	17	n	n	PROPN
ma-247	194	18	=	=	NOUN
ma-247	194	19	1.we	1.we	NUM
ma-247	194	20	have	have	VERB
ma-247	194	21	γε(v	γε(v	PUNCT
ma-247	194	22	[	[	X
ma-247	194	23	x	x	X
ma-247	194	24	]	]	X
ma-247	194	25	)	)	PUNCT
ma-247	194	26	=	=	SYM
ma-247	194	27	1	1	NUM
ma-247	194	28	µϕ(bk(0	µϕ(bk(0	NOUN
ma-247	194	29	,	,	PUNCT
ma-247	194	30	ε	ε	PROPN
ma-247	194	31	)	)	PUNCT
ma-247	194	32	)	)	PUNCT
ma-247	195	1	∫	∫	PROPN
ma-247	196	1	bk(0,ε)×cn−k	bk(0,ε)×cn−k	PROPN
ma-247	196	2	v	v	X
ma-247	196	3	[	[	X
ma-247	196	4	x	x	X
ma-247	196	5	]	]	X
ma-247	196	6	∧	∧	PROPN
ma-247	196	7	(	(	PUNCT
ma-247	196	8	ddc	ddc	PROPN
ma-247	196	9	ϕ̃)k	ϕ̃)k	PROPN
ma-247	196	10	∧ψ	∧ψ	PROPN
ma-247	196	11	.	.	PUNCT
ma-247	197	1	(	(	PUNCT
ma-247	197	2	4.9	4.9	NUM
ma-247	197	3	)	)	PUNCT
ma-247	197	4	let	let	VERB
ma-247	197	5	xreg	xreg	NOUN
ma-247	197	6	be	be	AUX
ma-247	197	7	the	the	DET
ma-247	197	8	set	set	NOUN
ma-247	197	9	of	of	ADP
ma-247	197	10	regular	regular	ADJ
ma-247	197	11	points	point	NOUN
ma-247	197	12	of	of	ADP
ma-247	197	13	x.	x.	NOUN
ma-247	197	14	assume	assume	VERB
ma-247	197	15	that	that	SCONJ
ma-247	197	16	0	0	NUM
ma-247	197	17	∈	∈	PROPN
ma-247	197	18	xreg	xreg	NOUN
ma-247	197	19	and	and	CCONJ
ma-247	197	20	put	put	VERB
ma-247	197	21	z1	z1	NOUN
ma-247	197	22	=	=	PUNCT
ma-247	197	23	{	{	PUNCT
ma-247	197	24	a	a	DET
ma-247	197	25	∈	∈	NOUN
ma-247	197	26	∆k	∆k	NOUN
ma-247	197	27	:	:	PUNCT
ma-247	197	28	dimc(x	dimc(x	NOUN
ma-247	197	29	∩	∩	NOUN
ma-247	197	30	π−1(a	π−1(a	PROPN
ma-247	197	31	)	)	PUNCT
ma-247	197	32	)	)	PUNCT
ma-247	197	33	>	>	X
ma-247	198	1	p	p	X
ma-247	198	2	−	−	PROPN
ma-247	198	3	k	k	NOUN
ma-247	198	4	}	}	PUNCT
ma-247	198	5	.	.	PUNCT
ma-247	199	1	then	then	ADV
ma-247	199	2	,	,	PUNCT
ma-247	199	3	by	by	ADP
ma-247	199	4	proposition	proposition	NOUN
ma-247	199	5	3.1	3.1	NUM
ma-247	199	6	,	,	PUNCT
ma-247	199	7	z1	z1	PROPN
ma-247	199	8	is	be	AUX
ma-247	199	9	contained	contain	VERB
ma-247	199	10	in	in	ADP
ma-247	199	11	a	a	DET
ma-247	199	12	countable	countable	ADJ
ma-247	199	13	union	union	NOUN
ma-247	199	14	of	of	ADP
ma-247	199	15	analytic	analytic	ADJ
ma-247	199	16	subsets	subset	NOUN
ma-247	199	17	of	of	ADP
ma-247	199	18	∆k	∆k	PROPN
ma-247	199	19	of	of	ADP
ma-247	199	20	dimen	diman	NOUN
ma-247	199	21	-	-	PUNCT
ma-247	199	22	sions	sion	NOUN
ma-247	199	23	≤	≤	NUM
ma-247	199	24	k	k	NOUN
ma-247	200	1	−	−	PROPN
ma-247	200	2	1	1	X
ma-247	200	3	.	.	PUNCT
ma-247	201	1	as	as	ADP
ma-247	201	2	the	the	DET
ma-247	201	3	dimension	dimension	NOUN
ma-247	201	4	m	m	NOUN
ma-247	201	5	of	of	ADP
ma-247	201	6	the	the	DET
ma-247	201	7	set	set	NOUN
ma-247	201	8	xsing	xsing	PROPN
ma-247	201	9	of	of	ADP
ma-247	201	10	singular	singular	ADJ
ma-247	201	11	points	point	NOUN
ma-247	201	12	,	,	PUNCT
ma-247	201	13	satisfies	satisfie	NOUN
ma-247	201	14	m	m	VERB
ma-247	201	15	≤	≤	NOUN
ma-247	201	16	p	p	PRON
ma-247	201	17	−	−	PROPN
ma-247	201	18	1	1	NUM
ma-247	201	19	,	,	PUNCT
ma-247	201	20	then	then	ADV
ma-247	201	21	,	,	PUNCT
ma-247	201	22	by	by	ADP
ma-247	201	23	proposition	proposition	NOUN
ma-247	201	24	3.1	3.1	NUM
ma-247	201	25	,	,	PUNCT
ma-247	201	26	there	there	PRON
ma-247	201	27	exists	exist	VERB
ma-247	201	28	a	a	DET
ma-247	201	29	set	set	ADJ
ma-247	201	30	z2	z2	NOUN
ma-247	201	31	contained	contain	VERB
ma-247	201	32	in	in	ADP
ma-247	201	33	a	a	DET
ma-247	201	34	countable	countable	ADJ
ma-247	201	35	union	union	NOUN
ma-247	201	36	of	of	ADP
ma-247	201	37	analytic	analytic	ADJ
ma-247	201	38	subsets	subset	NOUN
ma-247	201	39	of	of	ADP
ma-247	201	40	∆kof	∆kof	VERB
ma-247	201	41	dimensions	dimension	NOUN
ma-247	201	42	≤	≤	NUM
ma-247	202	1	k	k	NOUN
ma-247	203	1	−	−	PROPN
ma-247	203	2	1	1	NUM
ma-247	203	3	,	,	PUNCT
ma-247	203	4	such	such	ADJ
ma-247	203	5	that	that	SCONJ
ma-247	203	6	for	for	ADP
ma-247	203	7	all	all	DET
ma-247	203	8	a	a	DET
ma-247	203	9	∈	∈	PROPN
ma-247	203	10	∆k	∆k	NOUN
ma-247	203	11	rz2	rz2	NOUN
ma-247	203	12	,	,	PUNCT
ma-247	203	13	xsing	xsing	NOUN
ma-247	203	14	∩π−1(a	∩π−1(a	NOUN
ma-247	203	15	)	)	PUNCT
ma-247	203	16	is	be	AUX
ma-247	203	17	an	an	DET
ma-247	203	18	analytic	analytic	ADJ
ma-247	203	19	subset	subset	NOUN
ma-247	203	20	of	of	ADP
ma-247	203	21	∆k	∆k	PROPN
ma-247	203	22	ofdimension	ofdimension	NOUN
ma-247	204	1	m	m	NOUN
ma-247	205	1	−	−	PROPN
ma-247	205	2	k	k	NOUN
ma-247	205	3	(	(	PUNCT
ma-247	205	4	otherwise	otherwise	ADV
ma-247	205	5	is	be	AUX
ma-247	205	6	empty	empty	ADJ
ma-247	205	7	)	)	PUNCT
ma-247	205	8	.	.	PUNCT
ma-247	206	1	put	put	VERB
ma-247	206	2	z	z	NOUN
ma-247	206	3	=	=	SYM
ma-247	206	4	z1	z1	X
ma-247	206	5	∪	∪	PROPN
ma-247	206	6	z2	z2	PROPN
ma-247	206	7	and	and	CCONJ
ma-247	206	8	denote	denote	VERB
ma-247	206	9	π̄	π̄	VERB
ma-247	206	10	:	:	PUNCT
ma-247	206	11	=	=	PUNCT
ma-247	206	12	π|xreg	π|xreg	NOUN
ma-247	206	13	.	.	PUNCT
ma-247	207	1	the	the	DET
ma-247	207	2	expression	expression	NOUN
ma-247	207	3	(	(	PUNCT
ma-247	207	4	4.9	4.9	NUM
ma-247	207	5	)	)	PUNCT
ma-247	207	6	can	can	AUX
ma-247	207	7	be	be	AUX
ma-247	207	8	transformed	transform	VERB
ma-247	207	9	to	to	ADP
ma-247	207	10	γε(v	γε(v	PUNCT
ma-247	207	11	[	[	X
ma-247	207	12	x	x	X
ma-247	207	13	]	]	X
ma-247	207	14	)	)	PUNCT
ma-247	207	15	=	=	SYM
ma-247	207	16	1	1	NUM
ma-247	207	17	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	207	18	,	,	PUNCT
ma-247	207	19	ε	ε	PROPN
ma-247	207	20	)	)	PUNCT
ma-247	207	21	)	)	PUNCT
ma-247	208	1	∫	∫	PROPN
ma-247	208	2	xregr(π̄−1(z))∩(bk(a	xregr(π̄−1(z))∩(bk(a	PROPN
ma-247	208	3	,	,	PUNCT
ma-247	208	4	ε)×cn−k	ε)×cn−k	PROPN
ma-247	208	5	)	)	PUNCT
ma-247	208	6	π̄∗[v(ddcϕ)k	π̄∗[v(ddcϕ)k	PUNCT
ma-247	208	7	∧ψ	∧ψ	NOUN
ma-247	208	8	]	]	X
ma-247	208	9	.	.	PUNCT
ma-247	209	1	(	(	PUNCT
ma-247	209	2	4.10	4.10	NUM
ma-247	209	3	)	)	PUNCT
ma-247	209	4	in	in	ADP
ma-247	209	5	a	a	DET
ma-247	209	6	local	local	ADJ
ma-247	209	7	chart	chart	NOUN
ma-247	209	8	of	of	ADP
ma-247	209	9	coordinates	coordinate	NOUN
ma-247	209	10	(	(	PUNCT
ma-247	209	11	z1	z1	PROPN
ma-247	209	12	,	,	PUNCT
ma-247	209	13	.	.	PUNCT
ma-247	209	14	.	.	PUNCT
ma-247	209	15	.	.	PUNCT
ma-247	210	1	,	,	PUNCT
ma-247	210	2	zk	zk	PROPN
ma-247	210	3	,	,	PUNCT
ma-247	210	4	w1	w1	PROPN
ma-247	210	5	,	,	PUNCT
ma-247	210	6	.	.	PUNCT
ma-247	210	7	.	.	PUNCT
ma-247	210	8	.	.	PUNCT
ma-247	211	1	,	,	PUNCT
ma-247	211	2	wn−k	wn−k	PROPN
ma-247	211	3	)	)	PUNCT
ma-247	211	4	,	,	PUNCT
ma-247	211	5	such	such	ADJ
ma-247	211	6	that	that	SCONJ
ma-247	211	7	xreg	xreg	PROPN
ma-247	211	8	r	r	NOUN
ma-247	211	9	(	(	PUNCT
ma-247	211	10	π̄−1(z	π̄−1(z	PROPN
ma-247	211	11	)	)	PUNCT
ma-247	211	12	)	)	PUNCT
ma-247	211	13	∩	∩	NOUN
ma-247	211	14	(	(	PUNCT
ma-247	211	15	bk(a	bk(a	X
ma-247	211	16	,	,	PUNCT
ma-247	211	17	ε)×	ε)×	PROPN
ma-247	211	18	cn−k	cn−k	PROPN
ma-247	211	19	)	)	PUNCT
ma-247	211	20	=	=	PUNCT
ma-247	211	21	bk(a	bk(a	NOUN
ma-247	211	22	,	,	PUNCT
ma-247	211	23	ε)×	ε)×	PROPN
ma-247	211	24	cp−k	cp−k	PROPN
ma-247	211	25	×	×	PROPN
ma-247	211	26	{	{	PUNCT
ma-247	211	27	0}cn−p	0}cn−p	PROPN
ma-247	211	28	,	,	PUNCT
ma-247	211	29	when	when	SCONJ
ma-247	211	30	ε	ε	PROPN
ma-247	211	31	>	>	X
ma-247	211	32	0	0	NUM
ma-247	211	33	is	be	AUX
ma-247	211	34	small	small	ADJ
ma-247	211	35	enough	enough	ADV
ma-247	211	36	,	,	PUNCT
ma-247	211	37	the	the	DET
ma-247	211	38	integral	integral	ADJ
ma-247	211	39	(	(	PUNCT
ma-247	211	40	4.10	4.10	NUM
ma-247	211	41	)	)	PUNCT
ma-247	211	42	,	,	PUNCT
ma-247	211	43	can	can	AUX
ma-247	211	44	be	be	AUX
ma-247	211	45	written	write	VERB
ma-247	211	46	as	as	ADP
ma-247	211	47	γε	γε	NOUN
ma-247	211	48	=	=	SYM
ma-247	211	49	1	1	NUM
ma-247	211	50	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	211	51	,	,	PUNCT
ma-247	211	52	ε	ε	PROPN
ma-247	211	53	)	)	PUNCT
ma-247	211	54	)	)	PUNCT
ma-247	212	1	∫	∫	PROPN
ma-247	212	2	∆pr(π̄−1(z))∩(bk(a	∆pr(π̄−1(z))∩(bk(a	PROPN
ma-247	212	3	,	,	PUNCT
ma-247	212	4	ε)×cn−k	ε)×cn−k	PROPN
ma-247	212	5	)	)	PUNCT
ma-247	212	6	v(z	v(z	PROPN
ma-247	212	7	′	′	NUM
ma-247	212	8	,	,	PUNCT
ma-247	212	9	w)(ddcϕ)k	w)(ddcϕ)k	PROPN
ma-247	212	10	∧ψ(z	∧ψ(z	NUM
ma-247	212	11	′	′	NUM
ma-247	212	12	,	,	PUNCT
ma-247	212	13	w	w	NOUN
ma-247	212	14	)	)	PUNCT
ma-247	212	15	.	.	PUNCT
ma-247	213	1	(	(	PUNCT
ma-247	213	2	4.11	4.11	NUM
ma-247	213	3	)	)	PUNCT
ma-247	213	4	according	accord	VERB
ma-247	213	5	to	to	ADP
ma-247	213	6	[	[	X
ma-247	213	7	11	11	NUM
ma-247	213	8	]	]	PUNCT
ma-247	213	9	,	,	PUNCT
ma-247	213	10	since	since	SCONJ
ma-247	213	11	z	z	NOUN
ma-247	213	12	′	′	NOUN
ma-247	213	13	7→	7→	NUM
ma-247	213	14	v(z	v(z	NOUN
ma-247	213	15	′	′	NOUN
ma-247	213	16	,	,	PUNCT
ma-247	213	17	w	w	NOUN
ma-247	213	18	)	)	PUNCT
ma-247	213	19	is	be	AUX
ma-247	213	20	locally	locally	ADV
ma-247	213	21	integrable	integrable	ADJ
ma-247	213	22	on	on	ADP
ma-247	213	23	xreg	xreg	PROPN
ma-247	213	24	r	r	PROPN
ma-247	213	25	(	(	PUNCT
ma-247	213	26	π̄−1(z	π̄−1(z	PROPN
ma-247	213	27	)	)	PUNCT
ma-247	213	28	,	,	PUNCT
ma-247	213	29	then	then	ADV
ma-247	213	30	when	when	SCONJ
ma-247	213	31	ε→	ε→	PUNCT
ma-247	213	32	0,we	0,we	NUM
ma-247	213	33	get	get	VERB
ma-247	213	34	the	the	DET
ma-247	213	35	limit	limit	NOUN
ma-247	213	36	of	of	ADP
ma-247	213	37	(	(	PUNCT
ma-247	213	38	4.11	4.11	NUM
ma-247	213	39	)	)	PUNCT
ma-247	213	40	,	,	PUNCT
ma-247	213	41	as	as	SCONJ
ma-247	213	42	follows	follow	VERB
ma-247	213	43	limε→0	limε→0	NOUN
ma-247	213	44	γε(v	γε(v	PUNCT
ma-247	214	1	[	[	X
ma-247	214	2	x	x	X
ma-247	214	3	]	]	X
ma-247	214	4	)	)	PUNCT
ma-247	214	5	=	=	SYM
ma-247	214	6	∫	∫	PROPN
ma-247	215	1	∆p−k	∆p−k	PROPN
ma-247	215	2	v(a	v(a	PROPN
ma-247	215	3	,	,	PUNCT
ma-247	215	4	.)ψ(a	.)ψ(a	PROPN
ma-247	215	5	,	,	PUNCT
ma-247	215	6	w)β′′p−k	w)β′′p−k	PROPN
ma-247	215	7	=	=	SYM
ma-247	215	8	∫	∫	PROPN
ma-247	215	9	xreg\(π̄−1(z))∩π−1{a	xreg\(π̄−1(z))∩π−1{a	NUM
ma-247	215	10	}	}	PUNCT
ma-247	215	11	v(a	v(a	PROPN
ma-247	215	12	,	,	PUNCT
ma-247	215	13	.)ψ(a	.)ψ(a	PROPN
ma-247	215	14	,	,	PUNCT
ma-247	215	15	w)β′′p−k	w)β′′p−k	PROPN
ma-247	215	16	=	=	SYM
ma-247	215	17	〈	〈	PROPN
ma-247	215	18	v|x∩π−1(a	v|x∩π−1(a	NOUN
ma-247	215	19	)	)	PUNCT
ma-247	215	20	.	.	PUNCT
ma-247	216	1	[	[	PUNCT
ma-247	216	2	x	x	SYM
ma-247	216	3	∩	∩	X
ma-247	216	4	π−1(a	π−1(a	PROPN
ma-247	216	5	)	)	PUNCT
ma-247	216	6	]	]	PUNCT
ma-247	216	7	,	,	PUNCT
ma-247	216	8	(	(	PUNCT
ma-247	216	9	ix∩π−1(a))∗ψ〉.this	ix∩π−1(a))∗ψ〉.this	PRON
ma-247	216	10	finishes	finish	VERB
ma-247	216	11	the	the	DET
ma-247	216	12	proof	proof	NOUN
ma-247	216	13	.	.	PUNCT
ma-247	217	1	�	�	PROPN
ma-247	217	2	we	we	PRON
ma-247	217	3	illustrate	illustrate	VERB
ma-247	217	4	theorem	theorem	ADJ
ma-247	217	5	1.2	1.2	NUM
ma-247	217	6	by	by	ADP
ma-247	217	7	the	the	DET
ma-247	217	8	following	follow	VERB
ma-247	217	9	example	example	NOUN
ma-247	217	10	.	.	PUNCT
ma-247	218	1	example	example	NOUN
ma-247	218	2	4.1	4.1	NUM
ma-247	218	3	.	.	PUNCT
ma-247	219	1	in	in	ADP
ma-247	219	2	c4	c4	NOUN
ma-247	219	3	=	=	SYM
ma-247	219	4	c×	c×	PROPN
ma-247	219	5	c3	c3	PROPN
ma-247	219	6	,	,	PUNCT
ma-247	219	7	we	we	PRON
ma-247	219	8	take	take	VERB
ma-247	219	9	v(z	v(z	NOUN
ma-247	219	10	)	)	PUNCT
ma-247	220	1	=	=	PUNCT
ma-247	220	2	log(|z1|3	log(|z1|3	X
ma-247	220	3	+	+	NUM
ma-247	220	4	|z3|5	|z3|5	PROPN
ma-247	220	5	)	)	PUNCT
ma-247	220	6	,	,	PUNCT
ma-247	220	7	y	y	PROPN
ma-247	220	8	=	=	PUNCT
ma-247	220	9	{	{	PUNCT
ma-247	220	10	z4	z4	PROPN
ma-247	220	11	=	=	SYM
ma-247	220	12	z2	z2	PROPN
ma-247	220	13	1	1	NUM
ma-247	220	14	z	z	NOUN
ma-247	220	15	4	4	NUM
ma-247	220	16	2	2	NUM
ma-247	220	17	}	}	PUNCT
ma-247	220	18	,	,	PUNCT
ma-247	220	19	k	k	PROPN
ma-247	220	20	=	=	SYM
ma-247	220	21	1	1	NUM
ma-247	220	22	and	and	CCONJ
ma-247	220	23	ϕ(z	ϕ(z	PROPN
ma-247	220	24	′	′	NUM
ma-247	220	25	)	)	PUNCT
ma-247	221	1	=	=	SYM
ma-247	221	2	|z	|z	PROPN
ma-247	221	3	′|2	′|2	PROPN
ma-247	221	4	=	=	PUNCT
ma-247	221	5	|z1|2	|z1|2	PROPN
ma-247	221	6	.	.	PUNCT
ma-247	222	1	the	the	DET
ma-247	222	2	subset	subset	NOUN
ma-247	222	3	x	x	X
ma-247	222	4	=	=	SYM
ma-247	222	5	{	{	PUNCT
ma-247	222	6	z1	z1	NOUN
ma-247	222	7	=	=	SYM
ma-247	222	8	z3	z3	PROPN
ma-247	222	9	=	=	SYM
ma-247	222	10	0	0	NUM
ma-247	222	11	}	}	PUNCT
ma-247	222	12	that	that	PRON
ma-247	222	13	contains	contain	VERB
ma-247	222	14	{	{	PUNCT
ma-247	222	15	v	v	NOUN
ma-247	222	16	=	=	SYM
ma-247	222	17	−∞	−∞	NOUN
ma-247	222	18	}	}	PUNCT
ma-247	222	19	,	,	PUNCT
ma-247	222	20	yields	yield	VERB
ma-247	222	21	a	a	DET
ma-247	222	22	complete	complete	ADJ
ma-247	222	23	intersection	intersection	NOUN
ma-247	222	24	with	with	ADP
ma-247	222	25	thesubset	thesubset	ADJ
ma-247	222	26	y	y	PROPN
ma-247	222	27	near	near	ADP
ma-247	222	28	0	0	NUM
ma-247	222	29	.	.	PUNCT
ma-247	223	1	by	by	ADP
ma-247	223	2	theorem	theorem	NOUN
ma-247	223	3	1.2	1.2	NUM
ma-247	223	4	,	,	PUNCT
ma-247	223	5	up	up	ADP
ma-247	223	6	to	to	ADP
ma-247	223	7	a	a	DET
ma-247	223	8	constant	constant	ADJ
ma-247	223	9	,	,	PUNCT
ma-247	223	10	we	we	PRON
ma-247	223	11	have	have	VERB
ma-247	223	12	〈	〈	PROPN
ma-247	223	13	v	v	ADP
ma-247	223	14	[	[	X
ma-247	223	15	y	y	X
ma-247	223	16	]	]	PUNCT
ma-247	223	17	,	,	PUNCT
ma-247	223	18	π	π	PROPN
ma-247	223	19	,	,	PUNCT
ma-247	223	20	0	0	NUM
ma-247	223	21	〉	〉	NOUN
ma-247	223	22	=	=	PUNCT
ma-247	223	23	log	log	PROPN
ma-247	223	24	|z3|	|z3|	NOUN
ma-247	223	25	[	[	PUNCT
ma-247	223	26	z1	z1	PROPN
ma-247	223	27	=	=	SYM
ma-247	223	28	z2	z2	PROPN
ma-247	223	29	=	=	SYM
ma-247	223	30	z4	z4	PROPN
ma-247	223	31	=	=	SYM
ma-247	223	32	0	0	NUM
ma-247	223	33	]	]	PUNCT
ma-247	223	34	.	.	PUNCT
ma-247	224	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	224	2	eur	eur	PROPN
ma-247	224	3	.	.	PUNCT
ma-247	225	1	j.	j.	PROPN
ma-247	225	2	math	math	PROPN
ma-247	225	3	.	.	PUNCT
ma-247	226	1	anal	anal	PROPN
ma-247	226	2	.	.	PUNCT
ma-247	227	1	10.28924	10.28924	NUM
ma-247	227	2	/	/	SYM
ma-247	227	3	ada	ada	PROPN
ma-247	227	4	/	/	SYM
ma-247	227	5	ma.4.23	ma.4.23	PROPN
ma-247	228	1	9for	9for	NUM
ma-247	228	2	the	the	DET
ma-247	228	3	next	next	ADJ
ma-247	228	4	section	section	NOUN
ma-247	228	5	,	,	PUNCT
ma-247	228	6	we	we	PRON
ma-247	228	7	will	will	AUX
ma-247	228	8	be	be	AUX
ma-247	228	9	concerned	concern	VERB
ma-247	228	10	with	with	ADP
ma-247	228	11	points	point	NOUN
ma-247	228	12	a	a	DET
ma-247	228	13	∈	∈	NOUN
ma-247	228	14	∆k	∆k	VERB
ma-247	228	15	at	at	ADP
ma-247	228	16	which	which	PRON
ma-247	228	17	the	the	DET
ma-247	228	18	slice	slice	NOUN
ma-247	228	19	〈	〈	PROPN
ma-247	228	20	u[x]∧	u[x]∧	PROPN
ma-247	229	1	[	[	X
ma-247	229	2	y	y	X
ma-247	229	3	]	]	PUNCT
ma-247	229	4	,	,	PUNCT
ma-247	229	5	π	π	PROPN
ma-247	229	6	,	,	PUNCT
ma-247	229	7	a〉of	a〉of	PROPN
ma-247	229	8	the	the	DET
ma-247	229	9	current	current	ADJ
ma-247	229	10	u[x	u[x	X
ma-247	229	11	]	]	X
ma-247	229	12	∧	∧	PROPN
ma-247	229	13	[	[	X
ma-247	229	14	y	y	X
ma-247	229	15	]	]	PUNCT
ma-247	229	16	,	,	PUNCT
ma-247	229	17	is	be	AUX
ma-247	229	18	well	well	ADV
ma-247	229	19	defined	define	VERB
ma-247	229	20	and	and	CCONJ
ma-247	229	21	how	how	SCONJ
ma-247	229	22	it	it	PRON
ma-247	229	23	can	can	AUX
ma-247	229	24	be	be	AUX
ma-247	229	25	expressed.remember	expressed.remember	NOUN
ma-247	229	26	that	that	SCONJ
ma-247	229	27	the	the	DET
ma-247	229	28	wedge	wedge	NOUN
ma-247	229	29	product	product	NOUN
ma-247	229	30	u[x	u[x	X
ma-247	229	31	]	]	X
ma-247	229	32	∧	∧	PROPN
ma-247	229	33	[	[	X
ma-247	229	34	y	y	X
ma-247	229	35	]	]	PUNCT
ma-247	229	36	was	be	AUX
ma-247	229	37	defined	define	VERB
ma-247	229	38	in	in	ADP
ma-247	229	39	[	[	X
ma-247	229	40	10	10	NUM
ma-247	229	41	]	]	PUNCT
ma-247	229	42	as	as	ADP
ma-247	229	43	a	a	DET
ma-247	229	44	current	current	NOUN
ma-247	229	45	on	on	ADP
ma-247	229	46	y	y	PROPN
ma-247	229	47	,	,	PUNCT
ma-247	229	48	as	as	SCONJ
ma-247	229	49	wellas	wella	NOUN
ma-247	229	50	,	,	PUNCT
ma-247	229	51	the	the	DET
ma-247	229	52	analytic	analytic	ADJ
ma-247	229	53	subsets	subset	NOUN
ma-247	229	54	x	x	PUNCT
ma-247	229	55	and	and	CCONJ
ma-247	229	56	y	y	PROPN
ma-247	229	57	yield	yield	VERB
ma-247	229	58	a	a	DET
ma-247	229	59	complete	complete	ADJ
ma-247	229	60	intersection	intersection	NOUN
ma-247	229	61	in	in	ADP
ma-247	229	62	ω	ω	PROPN
ma-247	229	63	.	.	PUNCT
ma-247	230	1	actually	actually	ADV
ma-247	230	2	a	a	DET
ma-247	230	3	necessary	necessary	ADJ
ma-247	230	4	andsufficient	andsufficient	NOUN
ma-247	230	5	condition	condition	NOUN
ma-247	230	6	was	be	AUX
ma-247	230	7	established	establish	VERB
ma-247	230	8	,	,	PUNCT
ma-247	230	9	by	by	ADP
ma-247	230	10	expressing	express	VERB
ma-247	230	11	that	that	SCONJ
ma-247	230	12	the	the	DET
ma-247	230	13	current	current	ADJ
ma-247	230	14	u[x	u[x	X
ma-247	230	15	]	]	PUNCT
ma-247	230	16	has	have	VERB
ma-247	230	17	integrable	integrable	ADJ
ma-247	230	18	coefficientswith	coefficientswith	ADJ
ma-247	230	19	respect	respect	NOUN
ma-247	230	20	to	to	ADP
ma-247	230	21	the	the	DET
ma-247	230	22	trace	trace	NOUN
ma-247	230	23	measure	measure	NOUN
ma-247	230	24	of	of	ADP
ma-247	230	25	the	the	DET
ma-247	230	26	current	current	ADJ
ma-247	230	27	[	[	X
ma-247	230	28	y	y	X
ma-247	230	29	]	]	X
ma-247	230	30	,	,	PUNCT
ma-247	230	31	if	if	SCONJ
ma-247	230	32	and	and	CCONJ
ma-247	230	33	only	only	ADV
ma-247	230	34	if	if	SCONJ
ma-247	230	35	x	x	PRON
ma-247	230	36	and	and	CCONJ
ma-247	230	37	y	y	PROPN
ma-247	230	38	yield	yield	VERB
ma-247	230	39	a	a	DET
ma-247	230	40	completeintersection	completeintersection	NOUN
ma-247	230	41	in	in	ADP
ma-247	230	42	ω	ω	PROPN
ma-247	230	43	.	.	PUNCT
ma-247	231	1	the	the	DET
ma-247	231	2	condition	condition	NOUN
ma-247	231	3	of	of	ADP
ma-247	231	4	complete	complete	ADJ
ma-247	231	5	intersection	intersection	NOUN
ma-247	231	6	is	be	AUX
ma-247	231	7	optimal	optimal	ADJ
ma-247	231	8	for	for	ADP
ma-247	231	9	the	the	DET
ma-247	231	10	definition	definition	NOUN
ma-247	231	11	of	of	ADP
ma-247	231	12	this	this	DET
ma-247	231	13	wedgeproduct	wedgeproduct	NOUN
ma-247	231	14	in	in	ADP
ma-247	231	15	the	the	DET
ma-247	231	16	weak	weak	ADJ
ma-247	231	17	sense	sense	NOUN
ma-247	231	18	of	of	ADP
ma-247	231	19	currents	current	NOUN
ma-247	231	20	.	.	PUNCT
ma-247	232	1	recall	recall	VERB
ma-247	232	2	that	that	PRON
ma-247	232	3	,	,	PUNCT
ma-247	232	4	x	x	PUNCT
ma-247	232	5	and	and	CCONJ
ma-247	232	6	y	y	PROPN
ma-247	232	7	yield	yield	VERB
ma-247	232	8	a	a	DET
ma-247	232	9	complete	complete	ADJ
ma-247	232	10	intersection	intersection	NOUN
ma-247	232	11	,	,	PUNCT
ma-247	232	12	if	if	SCONJ
ma-247	232	13	for	for	ADP
ma-247	232	14	anyirreducible	anyirreducible	ADJ
ma-247	232	15	components	component	NOUN
ma-247	232	16	xj	xj	PROPN
ma-247	232	17	of	of	ADP
ma-247	232	18	x	x	PROPN
ma-247	232	19	and	and	CCONJ
ma-247	232	20	yk	yk	PROPN
ma-247	232	21	of	of	ADP
ma-247	232	22	y	y	PROPN
ma-247	232	23	,	,	PUNCT
ma-247	232	24	we	we	PRON
ma-247	232	25	have	have	VERB
ma-247	232	26	codim(xj	codim(xj	ADJ
ma-247	232	27	∩	∩	ADJ
ma-247	232	28	yk	yk	NOUN
ma-247	232	29	)	)	PUNCT
ma-247	233	1	=	=	SYM
ma-247	233	2	codimxj	codimxj	NOUN
ma-247	233	3	+	+	CCONJ
ma-247	233	4	codimyk	codimyk	NOUN
ma-247	233	5	.	.	PUNCT
ma-247	234	1	5	5	X
ma-247	234	2	.	.	X
ma-247	234	3	proof	proof	NOUN
ma-247	234	4	of	of	ADP
ma-247	234	5	theorem	theorem	ADJ
ma-247	234	6	1.3	1.3	NUM
ma-247	234	7	:	:	PUNCT
ma-247	234	8	slicing	slicing	NOUN
ma-247	234	9	of	of	ADP
ma-247	234	10	the	the	DET
ma-247	234	11	current	current	ADJ
ma-247	234	12	u[x	u[x	X
ma-247	234	13	]	]	X
ma-247	234	14	∧	∧	PROPN
ma-247	235	1	[	[	X
ma-247	235	2	y	y	X
ma-247	235	3	]	]	PUNCT
ma-247	235	4	in	in	ADP
ma-247	235	5	order	order	NOUN
ma-247	235	6	to	to	PART
ma-247	235	7	avoid	avoid	VERB
ma-247	235	8	complications	complication	NOUN
ma-247	235	9	,	,	PUNCT
ma-247	235	10	we	we	PRON
ma-247	235	11	assume	assume	VERB
ma-247	235	12	here	here	ADV
ma-247	235	13	that	that	SCONJ
ma-247	235	14	the	the	DET
ma-247	235	15	function	function	NOUN
ma-247	235	16	ϕ	ϕ	NOUN
ma-247	235	17	is	be	AUX
ma-247	235	18	smooth	smooth	ADJ
ma-247	235	19	and	and	CCONJ
ma-247	235	20	hence	hence	ADV
ma-247	235	21	themeasure	themeasure	NOUN
ma-247	235	22	µϕ	µϕ	ADV
ma-247	235	23	can	can	AUX
ma-247	235	24	be	be	AUX
ma-247	235	25	considered	consider	VERB
ma-247	235	26	as	as	ADP
ma-247	235	27	the	the	DET
ma-247	235	28	lebesgue	lebesgue	ADJ
ma-247	235	29	measure	measure	NOUN
ma-247	235	30	on	on	ADP
ma-247	235	31	ck	ck	INTJ
ma-247	235	32	,	,	PUNCT
ma-247	235	33	with	with	ADP
ma-247	235	34	density	density	NOUN
ma-247	235	35	a	a	DET
ma-247	235	36	smooth	smooth	ADJ
ma-247	235	37	functiondenoted	functiondenoted	ADJ
ma-247	235	38	µϕ(z	µϕ(z	PUNCT
ma-247	235	39	′	′	NUM
ma-247	235	40	)	)	PUNCT
ma-247	235	41	.	.	PUNCT
ma-247	236	1	proof	proof	NOUN
ma-247	236	2	.	.	PUNCT
ma-247	237	1	let	let	VERB
ma-247	237	2	ψ	ψ	PRON
ma-247	237	3	∈	∈	PROPN
ma-247	237	4	d(p+q−n+1−k	d(p+q−n+1−k	NOUN
ma-247	237	5	,	,	PUNCT
ma-247	237	6	p+q−n+1−k)(∆n	p+q−n+1−k)(∆n	NOUN
ma-247	237	7	)	)	PUNCT
ma-247	237	8	be	be	AUX
ma-247	237	9	a	a	DET
ma-247	237	10	test	test	NOUN
ma-247	237	11	form	form	NOUN
ma-247	237	12	.	.	PUNCT
ma-247	238	1	we	we	PRON
ma-247	238	2	have	have	VERB
ma-247	238	3	to	to	PART
ma-247	238	4	prove	prove	VERB
ma-247	238	5	the	the	DET
ma-247	238	6	existence	existence	NOUN
ma-247	238	7	of	of	ADP
ma-247	238	8	apluripolar	apluripolar	ADJ
ma-247	238	9	subset	subset	NOUN
ma-247	238	10	e	e	NOUN
ma-247	238	11	of	of	ADP
ma-247	238	12	∆k	∆k	PROPN
ma-247	238	13	such	such	ADJ
ma-247	238	14	that	that	PRON
ma-247	238	15	for	for	ADP
ma-247	238	16	all	all	DET
ma-247	238	17	a	a	DET
ma-247	238	18	6∈	6∈	NOUN
ma-247	238	19	e	e	NOUN
ma-247	238	20	,	,	PUNCT
ma-247	238	21	we	we	PRON
ma-247	238	22	have	have	VERB
ma-247	238	23	lim	lim	NOUN
ma-247	238	24	ε→0	ε→0	NOUN
ma-247	238	25	γε	γε	PROPN
ma-247	238	26	=	=	SYM
ma-247	238	27	∫	∫	PROPN
ma-247	239	1	∆n	∆n	PROPN
ma-247	239	2	j∗au[x	j∗au[x	PROPN
ma-247	239	3	]	]	X
ma-247	240	1	∧	∧	PROPN
ma-247	240	2	[	[	PUNCT
ma-247	240	3	y	y	PROPN
ma-247	240	4	∩	∩	PROPN
ma-247	240	5	π−1(a	π−1(a	PROPN
ma-247	240	6	)	)	PUNCT
ma-247	240	7	]	]	PUNCT
ma-247	241	1	∧	∧	NOUN
ma-247	241	2	j∗a	j∗a	NUM
ma-247	241	3	(	(	PUNCT
ma-247	241	4	ψ	ψ	NOUN
ma-247	241	5	)	)	PUNCT
ma-247	241	6	where	where	SCONJ
ma-247	241	7	γε	γε	ADJ
ma-247	241	8	=	=	SYM
ma-247	241	9	1	1	NUM
ma-247	241	10	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	241	11	,	,	PUNCT
ma-247	241	12	ε	ε	PROPN
ma-247	241	13	)	)	PUNCT
ma-247	241	14	)	)	PUNCT
ma-247	241	15	∫	∫	PROPN
ma-247	241	16	bk(a	bk(a	NOUN
ma-247	241	17	,	,	PUNCT
ma-247	241	18	ε)×cn−k	ε)×cn−k	X
ma-247	241	19	u[x	u[x	X
ma-247	241	20	]	]	X
ma-247	241	21	∧	∧	PROPN
ma-247	241	22	[	[	X
ma-247	241	23	y	y	X
ma-247	241	24	]	]	X
ma-247	241	25	∧	∧	PROPN
ma-247	241	26	(	(	PUNCT
ma-247	241	27	ddc	ddc	PROPN
ma-247	241	28	ϕ̃)k	ϕ̃)k	PROPN
ma-247	241	29	∧ψ	∧ψ	PROPN
ma-247	241	30	.	.	PUNCT
ma-247	242	1	(	(	PUNCT
ma-247	242	2	5.12	5.12	NUM
ma-247	242	3	)	)	PUNCT
ma-247	242	4	since	since	SCONJ
ma-247	242	5	the	the	DET
ma-247	242	6	potential	potential	ADJ
ma-247	242	7	u[x	u[x	X
ma-247	242	8	]	]	PUNCT
ma-247	242	9	involved	involve	VERB
ma-247	242	10	in	in	ADP
ma-247	242	11	(	(	PUNCT
ma-247	242	12	5.12	5.12	NUM
ma-247	242	13	)	)	PUNCT
ma-247	242	14	,	,	PUNCT
ma-247	242	15	is	be	AUX
ma-247	242	16	a	a	DET
ma-247	242	17	finite	finite	ADJ
ma-247	242	18	linear	linear	ADJ
ma-247	242	19	combination	combination	NOUN
ma-247	242	20	of	of	ADP
ma-247	242	21	forms	form	NOUN
ma-247	242	22	with	with	ADP
ma-247	242	23	coefficients	coefficient	NOUN
ma-247	242	24	bi	bi	PROPN
ma-247	242	25	,	,	PUNCT
ma-247	242	26	j(s	j(s	PROPN
ma-247	242	27	,	,	PUNCT
ma-247	242	28	z	z	NOUN
ma-247	242	29	)	)	PUNCT
ma-247	242	30	given	give	VERB
ma-247	242	31	by	by	ADP
ma-247	242	32	the	the	DET
ma-247	242	33	following	following	ADJ
ma-247	242	34	expression	expression	NOUN
ma-247	242	35	bi	bi	NOUN
ma-247	242	36	,	,	PUNCT
ma-247	242	37	j(s	j(s	PROPN
ma-247	242	38	,	,	PUNCT
ma-247	242	39	z	z	NOUN
ma-247	242	40	)	)	PUNCT
ma-247	243	1	=	=	SYM
ma-247	243	2	∫	∫	PROPN
ma-247	244	1	ξ∈cn	ξ∈cn	PROPN
ma-247	244	2	η(ξ)n(z	η(ξ)n(z	VERB
ma-247	244	3	−	−	PROPN
ma-247	244	4	ξ)[x](ξ	ξ)[x](ξ	NOUN
ma-247	244	5	)	)	PUNCT
ma-247	244	6	∧	∧	NOUN
ma-247	244	7	βp−s(ξ	βp−s(ξ	NOUN
ma-247	244	8	)	)	PUNCT
ma-247	245	1	∧	∧	NOUN
ma-247	246	1	2s	2s	NUM
ma-247	246	2	i	i	PRON
ma-247	246	3	s	s	PROPN
ma-247	246	4	2	2	NUM
ma-247	246	5	dξi	dξi	NOUN
ma-247	246	6	∧	∧	NOUN
ma-247	246	7	dξj	dξj	NOUN
ma-247	246	8	(	(	PUNCT
ma-247	246	9	|i|	|i|	NOUN
ma-247	246	10	=	=	SYM
ma-247	246	11	|j|	|j|	X
ma-247	246	12	=	=	SYM
ma-247	246	13	s	s	NOUN
ma-247	246	14	)	)	PUNCT
ma-247	246	15	.	.	PUNCT
ma-247	247	1	(	(	PUNCT
ma-247	247	2	5.13	5.13	NUM
ma-247	247	3	)	)	PUNCT
ma-247	247	4	we	we	PRON
ma-247	247	5	need	need	VERB
ma-247	247	6	to	to	PART
ma-247	247	7	find	find	VERB
ma-247	247	8	limε→0	limε→0	PROPN
ma-247	247	9	γε(bi	γε(bi	PROPN
ma-247	247	10	,	,	PUNCT
ma-247	247	11	j(s	j(s	PROPN
ma-247	247	12	,	,	PUNCT
ma-247	247	13	z	z	NOUN
ma-247	247	14	)	)	PUNCT
ma-247	247	15	)	)	PUNCT
ma-247	247	16	,	,	PUNCT
ma-247	247	17	where	where	SCONJ
ma-247	247	18	γε(bi	γε(bi	PROPN
ma-247	247	19	,	,	PUNCT
ma-247	247	20	j(s	j(s	PROPN
ma-247	247	21	,	,	PUNCT
ma-247	247	22	z	z	NOUN
ma-247	247	23	)	)	PUNCT
ma-247	247	24	)	)	PUNCT
ma-247	247	25	is	be	AUX
ma-247	247	26	given	give	VERB
ma-247	247	27	by	by	ADP
ma-247	247	28	γε(bi	γε(bi	PROPN
ma-247	247	29	,	,	PUNCT
ma-247	247	30	j(s	j(s	PROPN
ma-247	247	31	,	,	PUNCT
ma-247	247	32	z	z	NOUN
ma-247	247	33	)	)	PUNCT
ma-247	247	34	)	)	PUNCT
ma-247	248	1	:	:	PUNCT
ma-247	249	1	=	=	SYM
ma-247	249	2	1	1	NUM
ma-247	249	3	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	249	4	,	,	PUNCT
ma-247	249	5	ε	ε	PROPN
ma-247	249	6	)	)	PUNCT
ma-247	249	7	)	)	PUNCT
ma-247	249	8	∫	∫	PROPN
ma-247	249	9	bk(a	bk(a	NOUN
ma-247	249	10	,	,	PUNCT
ma-247	249	11	ε)×cn−k	ε)×cn−k	ADJ
ma-247	249	12	bi	bi	PROPN
ma-247	249	13	,	,	PUNCT
ma-247	249	14	j(s	j(s	PROPN
ma-247	249	15	,	,	PUNCT
ma-247	249	16	z	z	NOUN
ma-247	249	17	)	)	PUNCT
ma-247	249	18	∧	∧	PROPN
ma-247	249	19	[	[	X
ma-247	249	20	y	y	X
ma-247	249	21	]	]	X
ma-247	249	22	∧	∧	PROPN
ma-247	249	23	(	(	PUNCT
ma-247	249	24	ddc	ddc	PROPN
ma-247	249	25	ϕ̃)k	ϕ̃)k	PROPN
ma-247	249	26	∧ψ	∧ψ	PROPN
ma-247	249	27	.	.	PUNCT
ma-247	249	28	(	(	PUNCT
ma-247	249	29	5.14	5.14	NUM
ma-247	249	30	)	)	PUNCT
ma-247	249	31	since	since	SCONJ
ma-247	249	32	the	the	DET
ma-247	249	33	current	current	ADJ
ma-247	249	34	u[x]∧	u[x]∧	PROPN
ma-247	250	1	[	[	X
ma-247	250	2	y	y	X
ma-247	250	3	]	]	PUNCT
ma-247	250	4	is	be	AUX
ma-247	250	5	well	well	ADV
ma-247	250	6	defined	define	VERB
ma-247	250	7	then	then	ADV
ma-247	250	8	,	,	PUNCT
ma-247	250	9	the	the	DET
ma-247	250	10	functions	function	NOUN
ma-247	250	11	z	z	PROPN
ma-247	250	12	7→	7→	NUM
ma-247	250	13	bi	bi	NOUN
ma-247	250	14	,	,	PUNCT
ma-247	250	15	j(s	j(s	PROPN
ma-247	250	16	,	,	PUNCT
ma-247	250	17	z	z	NOUN
ma-247	250	18	)	)	PUNCT
ma-247	250	19	are	be	AUX
ma-247	250	20	locally	locally	ADV
ma-247	250	21	integrableon	integrableon	NOUN
ma-247	250	22	y	y	PROPN
ma-247	250	23	,	,	PUNCT
ma-247	250	24	and	and	CCONJ
ma-247	250	25	hence	hence	ADV
ma-247	250	26	γε(bi	γε(bi	PROPN
ma-247	250	27	,	,	PUNCT
ma-247	250	28	j(s	j(s	PROPN
ma-247	250	29	,	,	PUNCT
ma-247	250	30	z	z	NOUN
ma-247	250	31	)	)	PUNCT
ma-247	250	32	)	)	PUNCT
ma-247	250	33	given	give	VERB
ma-247	250	34	by	by	ADP
ma-247	250	35	(	(	PUNCT
ma-247	250	36	5.14	5.14	NUM
ma-247	250	37	)	)	PUNCT
ma-247	250	38	,	,	PUNCT
ma-247	250	39	can	can	AUX
ma-247	250	40	be	be	AUX
ma-247	250	41	written	write	VERB
ma-247	250	42	as	as	ADP
ma-247	250	43	γε	γε	ADJ
ma-247	250	44	:	:	PUNCT
ma-247	251	1	=	=	SYM
ma-247	251	2	γε(bi	γε(bi	PROPN
ma-247	251	3	,	,	PUNCT
ma-247	251	4	j(s	j(s	PROPN
ma-247	251	5	,	,	PUNCT
ma-247	251	6	z	z	NOUN
ma-247	251	7	)	)	PUNCT
ma-247	251	8	)	)	PUNCT
ma-247	252	1	=	=	SYM
ma-247	252	2	1	1	NUM
ma-247	253	1	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	253	2	,	,	PUNCT
ma-247	253	3	ε	ε	PROPN
ma-247	253	4	)	)	PUNCT
ma-247	253	5	)	)	PUNCT
ma-247	253	6	∫	∫	PROPN
ma-247	253	7	(	(	PUNCT
ma-247	253	8	ξ	ξ	PROPN
ma-247	253	9	,	,	PUNCT
ma-247	253	10	z)∈vε	z)∈vε	PROPN
ma-247	253	11	η(ξ)n(z	η(ξ)n(z	VERB
ma-247	253	12	−	−	PROPN
ma-247	253	13	ξ	ξ	NOUN
ma-247	253	14	)	)	PUNCT
ma-247	254	1	[	[	PUNCT
ma-247	254	2	x	x	SYM
ma-247	254	3	×	×	NOUN
ma-247	254	4	y	y	PROPN
ma-247	254	5	]	]	PUNCT
ma-247	254	6	(	(	PUNCT
ma-247	254	7	ξ	ξ	X
ma-247	254	8	,	,	PUNCT
ma-247	254	9	z	z	NOUN
ma-247	254	10	)	)	PUNCT
ma-247	254	11	∧	∧	PROPN
ma-247	254	12	(	(	PUNCT
ma-247	254	13	ddcϕ)k	ddcϕ)k	PROPN
ma-247	254	14	∧	∧	PROPN
ma-247	254	15	f	f	X
ma-247	254	16	(	(	PUNCT
ma-247	254	17	ξ	ξ	PROPN
ma-247	254	18	,	,	PUNCT
ma-247	254	19	z	z	NOUN
ma-247	254	20	)	)	PUNCT
ma-247	254	21	,	,	PUNCT
ma-247	254	22	where	where	SCONJ
ma-247	254	23	f	f	PROPN
ma-247	254	24	(	(	PUNCT
ma-247	254	25	ξ	ξ	PROPN
ma-247	254	26	,	,	PUNCT
ma-247	254	27	z	z	NOUN
ma-247	254	28	)	)	PUNCT
ma-247	254	29	=	=	SYM
ma-247	254	30	βp−s(ξ	βp−s(ξ	PROPN
ma-247	254	31	)	)	PUNCT
ma-247	254	32	∧	∧	NOUN
ma-247	255	1	2s	2s	NUM
ma-247	256	1	i	i	PRON
ma-247	256	2	s	s	PROPN
ma-247	256	3	2	2	NUM
ma-247	256	4	dξi	dξi	NOUN
ma-247	256	5	∧	∧	PROPN
ma-247	256	6	dξj	dξj	NOUN
ma-247	256	7	∧ψ(z	∧ψ(z	NUM
ma-247	256	8	)	)	PUNCT
ma-247	256	9	and	and	CCONJ
ma-247	256	10	vε	vε	VERB
ma-247	256	11	=	=	SYM
ma-247	256	12	cn	cn	PROPN
ma-247	256	13	×	×	NOUN
ma-247	256	14	[	[	PUNCT
ma-247	256	15	bk(a	bk(a	X
ma-247	256	16	,	,	PUNCT
ma-247	256	17	ε)×	ε)×	X
ma-247	256	18	cn−k	cn−k	PROPN
ma-247	256	19	]	]	PUNCT
ma-247	256	20	.	.	PUNCT
ma-247	257	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	257	2	eur	eur	PROPN
ma-247	257	3	.	.	PUNCT
ma-247	258	1	j.	j.	PROPN
ma-247	258	2	math	math	PROPN
ma-247	258	3	.	.	PUNCT
ma-247	259	1	anal	anal	PROPN
ma-247	259	2	.	.	PUNCT
ma-247	260	1	10.28924	10.28924	NUM
ma-247	260	2	/	/	SYM
ma-247	260	3	ada	ada	PROPN
ma-247	260	4	/	/	SYM
ma-247	260	5	ma.4.23	ma.4.23	PROPN
ma-247	260	6	10let	10let	NOUN
ma-247	261	1	[	[	X
ma-247	261	2	x	x	X
ma-247	261	3	×	×	VERB
ma-247	261	4	y	y	PROPN
ma-247	261	5	]	]	PUNCT
ma-247	261	6	reg	reg	NUM
ma-247	261	7	be	be	VERB
ma-247	261	8	the	the	DET
ma-247	261	9	set	set	NOUN
ma-247	261	10	of	of	ADP
ma-247	261	11	regular	regular	ADJ
ma-247	261	12	points	point	NOUN
ma-247	261	13	of	of	ADP
ma-247	261	14	x	x	X
ma-247	261	15	×	×	PROPN
ma-247	261	16	y.	y.	NOUN
ma-247	261	17	we	we	PRON
ma-247	261	18	may	may	AUX
ma-247	261	19	assume	assume	VERB
ma-247	261	20	that	that	SCONJ
ma-247	261	21	0	0	NUM
ma-247	261	22	∈	∈	PROPN
ma-247	262	1	[	[	X
ma-247	262	2	x	x	X
ma-247	262	3	×	×	VERB
ma-247	262	4	y	y	PROPN
ma-247	262	5	]	]	X
ma-247	262	6	reg	reg	X
ma-247	262	7	.	.	PUNCT
ma-247	263	1	put	put	VERB
ma-247	263	2	z1	z1	NOUN
ma-247	263	3	=	=	PUNCT
ma-247	263	4	{	{	PUNCT
ma-247	263	5	a	a	DET
ma-247	263	6	∈	∈	NOUN
ma-247	263	7	∆k	∆k	NOUN
ma-247	263	8	:	:	PUNCT
ma-247	263	9	dimc((x	dimc((x	NOUN
ma-247	263	10	×	×	PROPN
ma-247	263	11	y	y	PROPN
ma-247	263	12	)	)	PUNCT
ma-247	263	13	∩	∩	NOUN
ma-247	263	14	(	(	PUNCT
ma-247	263	15	π−1(a)×	π−1(a)×	PROPN
ma-247	263	16	π−1(a	π−1(a	PROPN
ma-247	263	17	)	)	PUNCT
ma-247	263	18	)	)	PUNCT
ma-247	264	1	>	>	X
ma-247	265	1	p	p	X
ma-247	266	1	+	+	NOUN
ma-247	266	2	q	q	PROPN
ma-247	267	1	−	−	PROPN
ma-247	267	2	k	k	NOUN
ma-247	267	3	}	}	PUNCT
ma-247	267	4	=	=	SYM
ma-247	267	5	{	{	PUNCT
ma-247	267	6	a	a	DET
ma-247	267	7	∈	∈	NOUN
ma-247	267	8	∆k	∆k	NOUN
ma-247	267	9	:	:	PUNCT
ma-247	267	10	dimc(x	dimc(x	VERB
ma-247	267	11	∩	∩	NOUN
ma-247	267	12	π−1(a))×	π−1(a))×	PROPN
ma-247	267	13	(	(	PUNCT
ma-247	267	14	y	y	PROPN
ma-247	267	15	∩	∩	PROPN
ma-247	267	16	π−1(a	π−1(a	PROPN
ma-247	267	17	)	)	PUNCT
ma-247	267	18	)	)	PUNCT
ma-247	268	1	>	>	X
ma-247	269	1	p	p	X
ma-247	270	1	+	+	NOUN
ma-247	270	2	q	q	NOUN
ma-247	270	3	−	−	PROPN
ma-247	270	4	k	k	NOUN
ma-247	270	5	}	}	PUNCT
ma-247	270	6	.	.	PUNCT
ma-247	271	1	following	follow	VERB
ma-247	271	2	proposition	proposition	NOUN
ma-247	271	3	3.1	3.1	NUM
ma-247	271	4	,	,	PUNCT
ma-247	271	5	z1	z1	PROPN
ma-247	271	6	is	be	AUX
ma-247	271	7	contained	contain	VERB
ma-247	271	8	in	in	ADP
ma-247	271	9	a	a	DET
ma-247	271	10	countable	countable	ADJ
ma-247	271	11	union	union	NOUN
ma-247	271	12	of	of	ADP
ma-247	271	13	analytic	analytic	ADJ
ma-247	271	14	subsets	subset	NOUN
ma-247	271	15	of	of	ADP
ma-247	271	16	∆k	∆k	PROPN
ma-247	271	17	ofdimensions	ofdimension	NOUN
ma-247	271	18	≤	≤	NOUN
ma-247	272	1	k	k	NOUN
ma-247	273	1	−	−	PROPN
ma-247	273	2	1	1	X
ma-247	273	3	.	.	PUNCT
ma-247	274	1	as	as	ADP
ma-247	274	2	the	the	DET
ma-247	274	3	dimension	dimension	NOUN
ma-247	274	4	m	m	NOUN
ma-247	274	5	of	of	ADP
ma-247	274	6	the	the	DET
ma-247	274	7	subset	subset	NOUN
ma-247	274	8	(	(	PUNCT
ma-247	274	9	x	x	SYM
ma-247	274	10	×	×	PROPN
ma-247	274	11	y	y	PROPN
ma-247	274	12	)	)	PUNCT
ma-247	274	13	sing	sing	NOUN
ma-247	274	14	of	of	ADP
ma-247	274	15	singular	singular	ADJ
ma-247	274	16	points	point	NOUN
ma-247	274	17	,	,	PUNCT
ma-247	274	18	satisfies	satisfie	NOUN
ma-247	274	19	m	m	VERB
ma-247	274	20	≤	≤	NOUN
ma-247	274	21	p	p	NOUN
ma-247	275	1	+	+	NOUN
ma-247	275	2	q	q	NOUN
ma-247	275	3	−	−	PROPN
ma-247	275	4	1	1	NUM
ma-247	275	5	,	,	PUNCT
ma-247	275	6	then	then	ADV
ma-247	275	7	,	,	PUNCT
ma-247	276	1	again	again	ADV
ma-247	276	2	by	by	ADP
ma-247	276	3	proposition	proposition	NOUN
ma-247	276	4	3.1	3.1	NUM
ma-247	276	5	,	,	PUNCT
ma-247	276	6	there	there	PRON
ma-247	276	7	exists	exist	VERB
ma-247	276	8	a	a	DET
ma-247	276	9	subset	subset	ADJ
ma-247	276	10	z2	z2	NOUN
ma-247	276	11	contained	contain	VERB
ma-247	276	12	in	in	ADP
ma-247	276	13	a	a	DET
ma-247	276	14	countableunion	countableunion	NOUN
ma-247	276	15	of	of	ADP
ma-247	276	16	analytic	analytic	ADJ
ma-247	276	17	subsets	subset	NOUN
ma-247	276	18	of	of	ADP
ma-247	276	19	∆k	∆k	PROPN
ma-247	276	20	of	of	ADP
ma-247	276	21	dimensions	dimension	NOUN
ma-247	276	22	≤	≤	PUNCT
ma-247	277	1	k	k	CCONJ
ma-247	278	1	−	−	PROPN
ma-247	278	2	1	1	NUM
ma-247	278	3	,	,	PUNCT
ma-247	278	4	such	such	ADJ
ma-247	278	5	that	that	PRON
ma-247	278	6	for	for	ADP
ma-247	278	7	any	any	DET
ma-247	278	8	point	point	NOUN
ma-247	278	9	a	a	DET
ma-247	278	10	∈	∈	PROPN
ma-247	278	11	∆k	∆k	PROPN
ma-247	278	12	rz2	rz2	NOUN
ma-247	278	13	,	,	PUNCT
ma-247	278	14	the	the	DET
ma-247	278	15	set	set	NOUN
ma-247	278	16	(	(	PUNCT
ma-247	278	17	x	x	SYM
ma-247	278	18	×	×	PROPN
ma-247	278	19	y	y	PROPN
ma-247	278	20	)	)	PUNCT
ma-247	278	21	sing	sing	NOUN
ma-247	278	22	∩	∩	NOUN
ma-247	278	23	(	(	PUNCT
ma-247	278	24	π−1(a	π−1(a	PROPN
ma-247	278	25	)	)	PUNCT
ma-247	278	26	×	×	NOUN
ma-247	278	27	π−1(a	π−1(a	PROPN
ma-247	278	28	)	)	PUNCT
ma-247	278	29	is	be	AUX
ma-247	278	30	analytic	analytic	ADJ
ma-247	278	31	in	in	ADP
ma-247	278	32	∆k	∆k	PROPN
ma-247	278	33	of	of	ADP
ma-247	278	34	dimension	dimension	NOUN
ma-247	278	35	m	m	VERB
ma-247	279	1	−	−	PROPN
ma-247	279	2	k	k	NOUN
ma-247	279	3	(	(	PUNCT
ma-247	279	4	otherwise	otherwise	ADV
ma-247	279	5	is	be	AUX
ma-247	279	6	empty	empty	ADJ
ma-247	279	7	)	)	PUNCT
ma-247	279	8	.	.	PUNCT
ma-247	280	1	put	put	VERB
ma-247	280	2	z	z	NOUN
ma-247	280	3	=	=	SYM
ma-247	280	4	z1	z1	X
ma-247	280	5	∪	∪	PROPN
ma-247	280	6	z2	z2	PROPN
ma-247	280	7	and	and	CCONJ
ma-247	280	8	denote	denote	VERB
ma-247	280	9	π̄	π̄	VERB
ma-247	280	10	:	:	PUNCT
ma-247	280	11	=	=	SYM
ma-247	280	12	π|(x×y	π|(x×y	NOUN
ma-247	280	13	)	)	PUNCT
ma-247	280	14	reg	reg	NOUN
ma-247	280	15	,	,	PUNCT
ma-247	280	16	we	we	PRON
ma-247	280	17	have	have	VERB
ma-247	280	18	γε	γε	PROPN
ma-247	280	19	=	=	SYM
ma-247	280	20	∫	∫	PROPN
ma-247	280	21	(	(	PUNCT
ma-247	280	22	x×y	x×y	PROPN
ma-247	280	23	)	)	PUNCT
ma-247	281	1	regr(π̄−1(z))∩vε	regr(π̄−1(z))∩vε	PROPN
ma-247	281	2	π̄∗	π̄∗	PROPN
ma-247	281	3	[	[	PUNCT
ma-247	281	4	η(ξ)n(z	η(ξ)n(z	VERB
ma-247	281	5	−	−	PROPN
ma-247	281	6	ξ)(ddcϕ)k	ξ)(ddcϕ)k	SYM
ma-247	282	1	∧	∧	PROPN
ma-247	282	2	f	f	X
ma-247	282	3	(	(	PUNCT
ma-247	282	4	ξ	ξ	PROPN
ma-247	282	5	,	,	PUNCT
ma-247	282	6	z	z	NOUN
ma-247	282	7	)	)	PUNCT
ma-247	282	8	]	]	PUNCT
ma-247	282	9	.	.	PUNCT
ma-247	283	1	(	(	PUNCT
ma-247	283	2	5.15	5.15	NUM
ma-247	283	3	)	)	PUNCT
ma-247	283	4	we	we	PRON
ma-247	283	5	can	can	AUX
ma-247	283	6	find	find	VERB
ma-247	283	7	local	local	ADJ
ma-247	283	8	coordinates	coordinate	NOUN
ma-247	283	9	(	(	PUNCT
ma-247	283	10	ξ	ξ	PROPN
ma-247	283	11	,	,	PUNCT
ma-247	283	12	z	z	NOUN
ma-247	283	13	)	)	PUNCT
ma-247	283	14	=	=	SYM
ma-247	283	15	(	(	PUNCT
ma-247	283	16	ξ	ξ	PROPN
ma-247	283	17	,	,	PUNCT
ma-247	283	18	(	(	PUNCT
ma-247	283	19	z	z	NOUN
ma-247	283	20	′	′	NUM
ma-247	283	21	,	,	PUNCT
ma-247	283	22	w	w	NOUN
ma-247	283	23	)	)	PUNCT
ma-247	283	24	)	)	PUNCT
ma-247	284	1	=	=	PRON
ma-247	284	2	(	(	PUNCT
ma-247	284	3	ξ	ξ	PROPN
ma-247	284	4	,	,	PUNCT
ma-247	284	5	(	(	PUNCT
ma-247	284	6	z1	z1	NOUN
ma-247	284	7	,	,	PUNCT
ma-247	284	8	.	.	PUNCT
ma-247	284	9	.	.	PUNCT
ma-247	285	1	.	.	PUNCT
ma-247	286	1	,	,	PUNCT
ma-247	286	2	zk	zk	PROPN
ma-247	286	3	,	,	PUNCT
ma-247	286	4	w1	w1	PROPN
ma-247	286	5	,	,	PUNCT
ma-247	286	6	.	.	PUNCT
ma-247	286	7	.	.	PUNCT
ma-247	286	8	.	.	PUNCT
ma-247	287	1	,	,	PUNCT
ma-247	287	2	wn−k	wn−k	PROPN
ma-247	287	3	)	)	PUNCT
ma-247	287	4	)	)	PUNCT
ma-247	287	5	,	,	PUNCT
ma-247	287	6	so	so	SCONJ
ma-247	287	7	that	that	SCONJ
ma-247	287	8	,	,	PUNCT
ma-247	287	9	for	for	ADP
ma-247	287	10	all	all	DET
ma-247	287	11	ε	ε	PROPN
ma-247	287	12	>	>	X
ma-247	287	13	0	0	PROPN
ma-247	287	14	small	small	ADJ
ma-247	287	15	enough	enough	ADV
ma-247	287	16	,	,	PUNCT
ma-247	287	17	(	(	PUNCT
ma-247	287	18	x	x	SYM
ma-247	287	19	×	×	PROPN
ma-247	287	20	y	y	PROPN
ma-247	287	21	)	)	PUNCT
ma-247	287	22	reg	reg	PROPN
ma-247	287	23	r	r	NOUN
ma-247	287	24	(	(	PUNCT
ma-247	287	25	π̄−1(z	π̄−1(z	PROPN
ma-247	287	26	)	)	PUNCT
ma-247	287	27	)	)	PUNCT
ma-247	287	28	∩	∩	NOUN
ma-247	287	29	vε	vε	NOUN
ma-247	287	30	=	=	SYM
ma-247	287	31	cp	cp	INTJ
ma-247	287	32	×	×	PROPN
ma-247	287	33	{	{	PUNCT
ma-247	287	34	0}cn−p	0}cn−p	PROPN
ma-247	287	35	×	×	NOUN
ma-247	287	36	bk(a	bk(a	NOUN
ma-247	287	37	,	,	PUNCT
ma-247	287	38	ε)×	ε)×	PROPN
ma-247	287	39	cq−k	cq−k	PROPN
ma-247	287	40	×	×	PROPN
ma-247	287	41	{	{	PUNCT
ma-247	287	42	0}cn−q	0}cn−q	PROPN
ma-247	287	43	.	.	PUNCT
ma-247	288	1	since	since	SCONJ
ma-247	288	2	µϕ(bk(a	µϕ(bk(a	PUNCT
ma-247	288	3	,	,	PUNCT
ma-247	288	4	ε	ε	PROPN
ma-247	288	5	)	)	PUNCT
ma-247	288	6	)	)	PUNCT
ma-247	288	7	∼	∼	NOUN
ma-247	288	8	ω2kε	ω2kε	PUNCT
ma-247	288	9	2kµϕ(a	2kµϕ(a	NUM
ma-247	288	10	)	)	PUNCT
ma-247	288	11	as	as	ADP
ma-247	288	12	ε	ε	PROPN
ma-247	288	13	→	→	SYM
ma-247	288	14	0	0	NUM
ma-247	288	15	.	.	PUNCT
ma-247	289	1	then	then	ADV
ma-247	289	2	,	,	PUNCT
ma-247	289	3	by	by	ADP
ma-247	289	4	an	an	DET
ma-247	289	5	application	application	NOUN
ma-247	289	6	of	of	ADP
ma-247	289	7	fubini	fubini	NOUN
ma-247	289	8	’s	’s	PART
ma-247	289	9	theorem	theorem	ADJ
ma-247	289	10	andby	andby	PROPN
ma-247	289	11	the	the	DET
ma-247	289	12	change	change	NOUN
ma-247	289	13	of	of	ADP
ma-247	289	14	variables	variable	NOUN
ma-247	289	15	(	(	PUNCT
ma-247	289	16	ξ	ξ	X
ma-247	289	17	,	,	PUNCT
ma-247	289	18	z	z	NOUN
ma-247	289	19	′	′	NOUN
ma-247	289	20	)	)	PUNCT
ma-247	289	21	↔	↔	PROPN
ma-247	289	22	(	(	PUNCT
ma-247	289	23	ξ	ξ	PROPN
ma-247	289	24	,	,	PUNCT
ma-247	289	25	z	z	NOUN
ma-247	289	26	′−a	′−a	PROPN
ma-247	289	27	ε	ε	PROPN
ma-247	289	28	)	)	PUNCT
ma-247	289	29	,	,	PUNCT
ma-247	289	30	when	when	SCONJ
ma-247	289	31	ε	ε	PROPN
ma-247	289	32	>	>	X
ma-247	289	33	0	0	NUM
ma-247	289	34	is	be	AUX
ma-247	289	35	small	small	ADJ
ma-247	289	36	enough	enough	ADV
ma-247	289	37	,	,	PUNCT
ma-247	289	38	the	the	DET
ma-247	289	39	equality	equality	NOUN
ma-247	289	40	(	(	PUNCT
ma-247	289	41	5.15)can	5.15)can	NUM
ma-247	289	42	be	be	AUX
ma-247	289	43	transformed	transform	VERB
ma-247	289	44	to	to	ADP
ma-247	289	45	the	the	DET
ma-247	289	46	following	follow	VERB
ma-247	289	47	γε	γε	X
ma-247	289	48	=	=	NOUN
ma-247	289	49	1	1	NUM
ma-247	289	50	ω2kµϕ(a	ω2kµϕ(a	NUM
ma-247	289	51	)	)	PUNCT
ma-247	289	52	∫	∫	PROPN
ma-247	289	53	v	v	ADP
ma-247	289	54	µϕ(a	µϕ(a	PUNCT
ma-247	289	55	+	+	CCONJ
ma-247	289	56	εt)η(ξ)n((a	εt)η(ξ)n((a	X
ma-247	290	1	+	+	CCONJ
ma-247	290	2	εt	εt	PROPN
ma-247	290	3	,	,	PUNCT
ma-247	290	4	w)−	w)−	PROPN
ma-247	290	5	ξ)f	ξ)f	NOUN
ma-247	290	6	(	(	PUNCT
ma-247	290	7	ξ	ξ	PROPN
ma-247	290	8	,	,	PUNCT
ma-247	290	9	(	(	PUNCT
ma-247	290	10	a	a	DET
ma-247	290	11	+	+	X
ma-247	290	12	εt	εt	PROPN
ma-247	290	13	,	,	PUNCT
ma-247	290	14	w))dν(t	w))dν(t	X
ma-247	290	15	,	,	PUNCT
ma-247	290	16	w	w	NOUN
ma-247	290	17	)	)	PUNCT
ma-247	290	18	,	,	PUNCT
ma-247	290	19	(	(	PUNCT
ma-247	290	20	5.16	5.16	NUM
ma-247	290	21	)	)	PUNCT
ma-247	290	22	where	where	SCONJ
ma-247	290	23	v	v	NOUN
ma-247	290	24	=	=	SYM
ma-247	290	25	cp+q−k	cp+q−k	PROPN
ma-247	290	26	×bk(0	×bk(0	PROPN
ma-247	290	27	,	,	PUNCT
ma-247	290	28	1	1	NUM
ma-247	290	29	)	)	PUNCT
ma-247	290	30	and	and	CCONJ
ma-247	290	31	dν(t	dν(t	ADP
ma-247	290	32	,	,	PUNCT
ma-247	290	33	w	w	NOUN
ma-247	290	34	)	)	PUNCT
ma-247	290	35	=	=	NOUN
ma-247	290	36	dλk(t)⊗	dλk(t)⊗	VERB
ma-247	290	37	dλp+q−k(w	dλp+q−k(w	NOUN
ma-247	290	38	)	)	PUNCT
ma-247	290	39	.	.	PUNCT
ma-247	291	1	by	by	ADP
ma-247	291	2	letting	let	VERB
ma-247	291	3	ε→	ε→	INTJ
ma-247	291	4	0	0	NUM
ma-247	292	1	in	in	ADP
ma-247	292	2	(	(	PUNCT
ma-247	292	3	5.16),we	5.16),we	NUM
ma-247	292	4	get	get	VERB
ma-247	292	5	limε→0	limε→0	NOUN
ma-247	292	6	γε	γε	NOUN
ma-247	292	7	=	=	SYM
ma-247	292	8	∫	∫	PROPN
ma-247	292	9	cp+q−k	cp+q−k	PROPN
ma-247	292	10	η(ξ)n((a	η(ξ)n((a	PROPN
ma-247	292	11	,	,	PUNCT
ma-247	292	12	w)−	w)−	PROPN
ma-247	292	13	ξ)f	ξ)f	NOUN
ma-247	292	14	(	(	PUNCT
ma-247	292	15	ξ	ξ	PROPN
ma-247	292	16	,	,	PUNCT
ma-247	292	17	(	(	PUNCT
ma-247	292	18	a	a	PRON
ma-247	292	19	,	,	PUNCT
ma-247	292	20	w))dλp+q−k(w	w))dλp+q−k(w	NOUN
ma-247	292	21	)	)	PUNCT
ma-247	292	22	=	=	SYM
ma-247	292	23	∫	∫	PROPN
ma-247	292	24	(	(	PUNCT
ma-247	292	25	x×y	x×y	PROPN
ma-247	292	26	)	)	PUNCT
ma-247	292	27	reg∩(π−1(a)×π−1(a	reg∩(π−1(a)×π−1(a	NUM
ma-247	292	28	)	)	PUNCT
ma-247	292	29	)	)	PUNCT
ma-247	292	30	η(ξ)n((a	η(ξ)n((a	NUM
ma-247	292	31	,	,	PUNCT
ma-247	292	32	w)−	w)−	PROPN
ma-247	292	33	ξ	ξ	PROPN
ma-247	292	34	)	)	PUNCT
ma-247	292	35	∧	∧	PROPN
ma-247	292	36	f	f	X
ma-247	292	37	(	(	PUNCT
ma-247	292	38	ξ	ξ	PROPN
ma-247	292	39	,	,	PUNCT
ma-247	292	40	(	(	PUNCT
ma-247	292	41	a	a	DET
ma-247	292	42	,	,	PUNCT
ma-247	292	43	w	w	NOUN
ma-247	292	44	)	)	PUNCT
ma-247	292	45	)	)	PUNCT
ma-247	292	46	=	=	SYM
ma-247	293	1	∫	∫	PROPN
ma-247	293	2	(	(	PUNCT
ma-247	293	3	x∩π−1(a))reg×(y	x∩π−1(a))reg×(y	PROPN
ma-247	293	4	∩π−1(a))reg	∩π−1(a))reg	PROPN
ma-247	293	5	η(ξ)n((a	η(ξ)n((a	NUM
ma-247	293	6	,	,	PUNCT
ma-247	293	7	w)−	w)−	PROPN
ma-247	293	8	ξ	ξ	PROPN
ma-247	293	9	)	)	PUNCT
ma-247	293	10	∧	∧	PROPN
ma-247	293	11	f	f	X
ma-247	293	12	(	(	PUNCT
ma-247	293	13	ξ	ξ	PROPN
ma-247	293	14	,	,	PUNCT
ma-247	293	15	(	(	PUNCT
ma-247	293	16	a	a	DET
ma-247	293	17	,	,	PUNCT
ma-247	293	18	w	w	NOUN
ma-247	293	19	)	)	PUNCT
ma-247	293	20	)	)	PUNCT
ma-247	294	1	=	=	SYM
ma-247	294	2	∫	∫	PROPN
ma-247	294	3	cn−k	cn−k	PROPN
ma-247	294	4	bi	bi	PROPN
ma-247	294	5	,	,	PUNCT
ma-247	294	6	j(s	j(s	PROPN
ma-247	294	7	,	,	PUNCT
ma-247	294	8	(	(	PUNCT
ma-247	294	9	a	a	PRON
ma-247	294	10	,	,	PUNCT
ma-247	294	11	w	w	NOUN
ma-247	294	12	)	)	PUNCT
ma-247	294	13	)	)	PUNCT
ma-247	295	1	∧	∧	PROPN
ma-247	296	1	[	[	X
ma-247	296	2	y	y	PROPN
ma-247	296	3	∩	∩	X
ma-247	296	4	π−1(a	π−1(a	PROPN
ma-247	296	5	)	)	PUNCT
ma-247	296	6	]	]	PUNCT
ma-247	296	7	∧ψ(a	∧ψ(a	PROPN
ma-247	296	8	,	,	PUNCT
ma-247	296	9	z	z	PROPN
ma-247	296	10	′′	′′	PROPN
ma-247	296	11	)	)	PUNCT
ma-247	296	12	=	=	PUNCT
ma-247	297	1	〈	〈	PROPN
ma-247	297	2	j∗a(bi	j∗a(bi	PROPN
ma-247	297	3	,	,	PUNCT
ma-247	297	4	j(s	j(s	PROPN
ma-247	297	5	,	,	PUNCT
ma-247	297	6	z	z	NOUN
ma-247	297	7	)	)	PUNCT
ma-247	297	8	)	)	PUNCT
ma-247	298	1	∧	∧	PROPN
ma-247	299	1	[	[	X
ma-247	299	2	y	y	PROPN
ma-247	299	3	∩	∩	X
ma-247	299	4	π−1(a	π−1(a	PROPN
ma-247	299	5	)	)	PUNCT
ma-247	299	6	]	]	PUNCT
ma-247	299	7	,	,	PUNCT
ma-247	299	8	j∗a	j∗a	NUM
ma-247	299	9	(	(	PUNCT
ma-247	299	10	ψ)〉.the	ψ)〉.the	DET
ma-247	299	11	last	last	ADJ
ma-247	299	12	equality	equality	NOUN
ma-247	299	13	holds	hold	VERB
ma-247	299	14	true	true	ADJ
ma-247	299	15	since	since	SCONJ
ma-247	299	16	we	we	PRON
ma-247	299	17	have	have	AUX
ma-247	299	18	dim	dim	VERB
ma-247	299	19	(	(	PUNCT
ma-247	299	20	(	(	PUNCT
ma-247	299	21	x	x	SYM
ma-247	299	22	×	×	PROPN
ma-247	299	23	y	y	PROPN
ma-247	299	24	)	)	PUNCT
ma-247	299	25	sing	sing	NOUN
ma-247	299	26	∩	∩	NOUN
ma-247	299	27	(	(	PUNCT
ma-247	299	28	π−1(a)×	π−1(a)×	PROPN
ma-247	299	29	π−1(a	π−1(a	PROPN
ma-247	299	30	)	)	PUNCT
ma-247	299	31	)	)	PUNCT
ma-247	299	32	)	)	PUNCT
ma-247	300	1	<	<	X
ma-247	301	1	p	p	X
ma-247	302	1	+	+	NOUN
ma-247	302	2	q	q	PROPN
ma-247	303	1	−	−	PROPN
ma-247	303	2	k.	k.	PROPN
ma-247	303	3	this	this	PRON
ma-247	303	4	achieves	achieve	VERB
ma-247	303	5	the	the	DET
ma-247	303	6	proof	proof	NOUN
ma-247	303	7	of	of	ADP
ma-247	303	8	theorem	theorem	ADJ
ma-247	303	9	1.3	1.3	NUM
ma-247	303	10	.	.	PUNCT
ma-247	304	1	�	�	PROPN
ma-247	304	2	let	let	VERB
ma-247	304	3	illustrate	illustrate	VERB
ma-247	304	4	theorem	theorem	VERB
ma-247	304	5	1.3	1.3	NUM
ma-247	304	6	with	with	ADP
ma-247	304	7	the	the	DET
ma-247	304	8	following	follow	VERB
ma-247	304	9	example	example	NOUN
ma-247	304	10	.	.	PUNCT
ma-247	305	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	305	2	eur	eur	PROPN
ma-247	305	3	.	.	PUNCT
ma-247	306	1	j.	j.	PROPN
ma-247	306	2	math	math	PROPN
ma-247	306	3	.	.	PUNCT
ma-247	307	1	anal	anal	PROPN
ma-247	307	2	.	.	PUNCT
ma-247	308	1	10.28924	10.28924	NUM
ma-247	308	2	/	/	SYM
ma-247	308	3	ada	ada	PROPN
ma-247	308	4	/	/	SYM
ma-247	308	5	ma.4.23	ma.4.23	PROPN
ma-247	308	6	11	11	NUM
ma-247	308	7	example	example	NOUN
ma-247	308	8	5.1	5.1	NUM
ma-247	308	9	.	.	PUNCT
ma-247	309	1	consider	consider	VERB
ma-247	309	2	in	in	ADP
ma-247	309	3	c5	c5	PROPN
ma-247	309	4	=	=	PUNCT
ma-247	309	5	c	c	PROPN
ma-247	309	6	×	×	PROPN
ma-247	309	7	c4	c4	NOUN
ma-247	309	8	,	,	PUNCT
ma-247	309	9	x	x	X
ma-247	309	10	=	=	PRON
ma-247	309	11	{	{	PUNCT
ma-247	309	12	z2	z2	NOUN
ma-247	309	13	=	=	SYM
ma-247	309	14	z3	z3	PROPN
ma-247	309	15	=	=	PUNCT
ma-247	309	16	z4	z4	PROPN
ma-247	309	17	=	=	SYM
ma-247	309	18	0	0	NUM
ma-247	309	19	}	}	PUNCT
ma-247	309	20	and	and	CCONJ
ma-247	309	21	y	y	PROPN
ma-247	309	22	=	=	SYM
ma-247	309	23	{	{	PUNCT
ma-247	309	24	z4	z4	PROPN
ma-247	309	25	=	=	SYM
ma-247	309	26	z2	z2	PROPN
ma-247	309	27	2	2	NUM
ma-247	309	28	z	z	NOUN
ma-247	309	29	2	2	NUM
ma-247	309	30	3	3	NUM
ma-247	309	31	}	}	PUNCT
ma-247	309	32	take	take	VERB
ma-247	309	33	k	k	NOUN
ma-247	309	34	=	=	SYM
ma-247	309	35	1	1	NUM
ma-247	309	36	and	and	CCONJ
ma-247	309	37	ϕ(z	ϕ(z	PROPN
ma-247	309	38	′	′	NUM
ma-247	309	39	)	)	PUNCT
ma-247	310	1	=	=	SYM
ma-247	310	2	|z	|z	PROPN
ma-247	310	3	′|2	′|2	PROPN
ma-247	310	4	=	=	PUNCT
ma-247	310	5	|z1|2	|z1|2	PROPN
ma-247	310	6	.	.	PUNCT
ma-247	311	1	since	since	SCONJ
ma-247	311	2	x	x	PRON
ma-247	311	3	is	be	AUX
ma-247	311	4	smooth	smooth	ADJ
ma-247	311	5	,	,	PUNCT
ma-247	311	6	then	then	ADV
ma-247	311	7	a	a	DET
ma-247	311	8	potential	potential	NOUN
ma-247	311	9	of	of	ADP
ma-247	311	10	[	[	X
ma-247	311	11	x	x	X
ma-247	311	12	]	]	X
ma-247	311	13	is	be	AUX
ma-247	311	14	u	u	NOUN
ma-247	311	15	=	=	PUNCT
ma-247	311	16	log(|z2|2	log(|z2|2	X
ma-247	311	17	+	+	PUNCT
ma-247	311	18	|z3|2	|z3|2	PUNCT
ma-247	311	19	+	+	X
ma-247	311	20	|z4|2)ddc	|z4|2)ddc	NOUN
ma-247	311	21	log(|z2|2	log(|z2|2	NOUN
ma-247	311	22	+	+	PUNCT
ma-247	311	23	|z3|2	|z3|2	PUNCT
ma-247	311	24	+	+	X
ma-247	311	25	|z4|2	|z4|2	NOUN
ma-247	311	26	)	)	PUNCT
ma-247	311	27	|z2|2	|z2|2	PUNCT
ma-247	312	1	+	+	PUNCT
ma-247	312	2	|z3|2	|z3|2	PUNCT
ma-247	312	3	+	+	CCONJ
ma-247	312	4	|z4|2	|z4|2	X
ma-247	312	5	.	.	PUNCT
ma-247	313	1	as	as	SCONJ
ma-247	313	2	x	x	PROPN
ma-247	313	3	and	and	CCONJ
ma-247	313	4	y	y	PROPN
ma-247	313	5	yield	yield	VERB
ma-247	313	6	a	a	DET
ma-247	313	7	complete	complete	ADJ
ma-247	313	8	intersection	intersection	NOUN
ma-247	313	9	,	,	PUNCT
ma-247	313	10	then	then	ADV
ma-247	313	11	by	by	ADP
ma-247	313	12	[	[	PUNCT
ma-247	313	13	9	9	NUM
ma-247	313	14	]	]	PUNCT
ma-247	313	15	,	,	PUNCT
ma-247	314	1	the	the	DET
ma-247	314	2	current	current	ADJ
ma-247	314	3	u	u	NOUN
ma-247	314	4	∧	∧	PROPN
ma-247	314	5	[	[	X
ma-247	314	6	y	y	X
ma-247	314	7	]	]	PUNCT
ma-247	314	8	is	be	AUX
ma-247	314	9	well	well	ADV
ma-247	314	10	defined	define	VERB
ma-247	314	11	.	.	PUNCT
ma-247	315	1	thecurrent	thecurrent	NOUN
ma-247	315	2	u	u	PROPN
ma-247	315	3	∧	∧	PROPN
ma-247	315	4	[	[	X
ma-247	315	5	y	y	X
ma-247	315	6	]	]	PUNCT
ma-247	315	7	is	be	AUX
ma-247	315	8	given	give	VERB
ma-247	315	9	by	by	ADP
ma-247	315	10	the	the	DET
ma-247	315	11	following	follow	VERB
ma-247	315	12	u	u	NOUN
ma-247	315	13	∧	∧	PROPN
ma-247	316	1	[	[	X
ma-247	316	2	y	y	X
ma-247	316	3	]	]	X
ma-247	316	4	=	=	PUNCT
ma-247	316	5	i∗y	i∗y	PUNCT
ma-247	316	6	[	[	PUNCT
ma-247	316	7	log(|z2|2	log(|z2|2	X
ma-247	316	8	+	+	PUNCT
ma-247	316	9	|z3|2	|z3|2	PUNCT
ma-247	316	10	+	+	PUNCT
ma-247	316	11	|z2z3|4)ddc	|z2z3|4)ddc	NOUN
ma-247	316	12	log(|z2|2	log(|z2|2	NOUN
ma-247	316	13	+	+	PUNCT
ma-247	316	14	|z3|2	|z3|2	PUNCT
ma-247	316	15	+	+	SYM
ma-247	316	16	|z2z3|4	|z2z3|4	NOUN
ma-247	316	17	)	)	PUNCT
ma-247	316	18	|z2|2	|z2|2	PUNCT
ma-247	316	19	+	+	PUNCT
ma-247	316	20	|z3|2	|z3|2	PUNCT
ma-247	316	21	+	+	SYM
ma-247	316	22	|z2z3|4	|z2z3|4	NOUN
ma-247	316	23	]	]	PUNCT
ma-247	316	24	.	.	PUNCT
ma-247	317	1	we	we	PRON
ma-247	317	2	have	have	VERB
ma-247	317	3	x	x	NOUN
ma-247	317	4	∩	∩	ADJ
ma-247	317	5	π−1(0	π−1(0	NOUN
ma-247	317	6	)	)	PUNCT
ma-247	317	7	=	=	SYM
ma-247	317	8	{	{	PUNCT
ma-247	317	9	z1	z1	PROPN
ma-247	317	10	=	=	SYM
ma-247	317	11	z2	z2	PROPN
ma-247	317	12	=	=	SYM
ma-247	317	13	z3	z3	PROPN
ma-247	317	14	=	=	PUNCT
ma-247	317	15	z4	z4	PROPN
ma-247	317	16	=	=	SYM
ma-247	317	17	0	0	NUM
ma-247	317	18	}	}	PUNCT
ma-247	317	19	and	and	CCONJ
ma-247	317	20	y	y	PROPN
ma-247	317	21	∩	∩	ADJ
ma-247	317	22	π−1(0	π−1(0	PROPN
ma-247	317	23	)	)	PUNCT
ma-247	317	24	=	=	SYM
ma-247	317	25	{	{	PUNCT
ma-247	317	26	z1	z1	NOUN
ma-247	317	27	=	=	SYM
ma-247	317	28	z2	z2	PROPN
ma-247	317	29	4	4	NUM
ma-247	317	30	−	−	PROPN
ma-247	317	31	z2z3	z2z3	NOUN
ma-247	317	32	=	=	NOUN
ma-247	317	33	0	0	NUM
ma-247	317	34	}	}	PUNCT
ma-247	317	35	.	.	PUNCT
ma-247	318	1	it	it	PRON
ma-247	318	2	is	be	AUX
ma-247	318	3	clear	clear	ADJ
ma-247	318	4	that	that	SCONJ
ma-247	318	5	x∩π−1(0	x∩π−1(0	PROPN
ma-247	318	6	)	)	PUNCT
ma-247	318	7	and	and	CCONJ
ma-247	318	8	y	y	PROPN
ma-247	318	9	∩π−1(0	∩π−1(0	NOUN
ma-247	318	10	)	)	PUNCT
ma-247	318	11	yield	yield	VERB
ma-247	318	12	a	a	DET
ma-247	318	13	complete	complete	ADJ
ma-247	318	14	intersection	intersection	NOUN
ma-247	318	15	in	in	ADP
ma-247	318	16	{	{	PUNCT
ma-247	318	17	0}×c4	0}×c4	NUM
ma-247	318	18	.	.	PUNCT
ma-247	319	1	if	if	SCONJ
ma-247	319	2	σ	σ	PROPN
ma-247	319	3	=	=	PUNCT
ma-247	319	4	iy	iy	PROPN
ma-247	319	5	∩π−1(0),then	∩π−1(0),then	PROPN
ma-247	319	6	,	,	PUNCT
ma-247	319	7	〈	〈	PROPN
ma-247	319	8	u	u	NOUN
ma-247	319	9	∧	∧	PROPN
ma-247	319	10	[	[	X
ma-247	319	11	y	y	X
ma-247	319	12	]	]	X
ma-247	319	13	,	,	PUNCT
ma-247	319	14	π	π	PROPN
ma-247	319	15	,	,	PUNCT
ma-247	319	16	0	0	NUM
ma-247	319	17	〉	〉	NOUN
ma-247	319	18	=	=	NOUN
ma-247	319	19	σ∗	σ∗	X
ma-247	319	20	[	[	PUNCT
ma-247	319	21	log(|z2|2	log(|z2|2	X
ma-247	319	22	+	+	PUNCT
ma-247	319	23	|z3|2	|z3|2	PUNCT
ma-247	319	24	+	+	PUNCT
ma-247	319	25	|z2z3|4)ddc	|z2z3|4)ddc	NOUN
ma-247	319	26	log(|z2|2	log(|z2|2	NOUN
ma-247	319	27	+	+	PUNCT
ma-247	319	28	|z3|2	|z3|2	PUNCT
ma-247	319	29	+	+	SYM
ma-247	319	30	|z2z3|4	|z2z3|4	NOUN
ma-247	319	31	)	)	PUNCT
ma-247	319	32	|z2|2	|z2|2	PUNCT
ma-247	319	33	+	+	PUNCT
ma-247	319	34	|z3|2	|z3|2	PUNCT
ma-247	319	35	+	+	SYM
ma-247	319	36	|z2z3|4	|z2z3|4	NOUN
ma-247	319	37	]	]	PUNCT
ma-247	319	38	.	.	PUNCT
ma-247	320	1	6	6	X
ma-247	320	2	.	.	X
ma-247	320	3	proofs	proof	NOUN
ma-247	320	4	of	of	ADP
ma-247	320	5	theorem	theorem	ADJ
ma-247	320	6	1.4	1.4	NUM
ma-247	320	7	and	and	CCONJ
ma-247	320	8	theorem	theorem	VERB
ma-247	320	9	1.5	1.5	NUM
ma-247	320	10	with	with	ADP
ma-247	320	11	applications	application	NOUN
ma-247	320	12	this	this	DET
ma-247	320	13	section	section	NOUN
ma-247	320	14	aims	aim	VERB
ma-247	320	15	to	to	PART
ma-247	320	16	give	give	VERB
ma-247	320	17	the	the	DET
ma-247	320	18	proof	proof	NOUN
ma-247	320	19	of	of	ADP
ma-247	320	20	theorem	theorem	ADJ
ma-247	320	21	1.4	1.4	NUM
ma-247	320	22	and	and	CCONJ
ma-247	320	23	the	the	DET
ma-247	320	24	proof	proof	NOUN
ma-247	320	25	of	of	ADP
ma-247	320	26	theorem	theorem	ADJ
ma-247	320	27	1.5	1.5	NUM
ma-247	320	28	.	.	PUNCT
ma-247	321	1	we	we	PRON
ma-247	321	2	may	may	AUX
ma-247	321	3	suppose	suppose	VERB
ma-247	321	4	n	n	PROPN
ma-247	321	5	=	=	SYM
ma-247	321	6	1	1	X
ma-247	321	7	.	.	PUNCT
ma-247	321	8	indeed	indeed	ADV
ma-247	321	9	,	,	PUNCT
ma-247	321	10	for	for	ADP
ma-247	321	11	the	the	DET
ma-247	321	12	proof	proof	NOUN
ma-247	321	13	of	of	ADP
ma-247	321	14	theorem	theorem	ADJ
ma-247	321	15	1.5	1.5	NUM
ma-247	321	16	,	,	PUNCT
ma-247	321	17	we	we	PRON
ma-247	321	18	use	use	VERB
ma-247	321	19	the	the	DET
ma-247	321	20	fact	fact	NOUN
ma-247	321	21	that	that	SCONJ
ma-247	321	22	a	a	DET
ma-247	321	23	finite	finite	ADJ
ma-247	321	24	union	union	NOUN
ma-247	321	25	of	of	ADP
ma-247	321	26	pluripolar	pluripolar	ADJ
ma-247	321	27	subsetsis	subsetsis	NOUN
ma-247	321	28	pluripolar	pluripolar	ADJ
ma-247	321	29	.	.	PUNCT
ma-247	322	1	in	in	ADP
ma-247	322	2	addition	addition	NOUN
ma-247	322	3	,	,	PUNCT
ma-247	322	4	theorem	theorem	VERB
ma-247	322	5	1.4	1.4	NUM
ma-247	322	6	is	be	AUX
ma-247	322	7	an	an	DET
ma-247	322	8	immediate	immediate	ADJ
ma-247	322	9	application	application	NOUN
ma-247	322	10	of	of	ADP
ma-247	322	11	the	the	DET
ma-247	322	12	following	following	NOUN
ma-247	322	13	theorem	theorem	NOUN
ma-247	322	14	6.1.in	6.1.in	NUM
ma-247	322	15	fact	fact	NOUN
ma-247	322	16	,	,	PUNCT
ma-247	322	17	if	if	SCONJ
ma-247	322	18	for	for	ADP
ma-247	322	19	all	all	DET
ma-247	322	20	1	1	NUM
ma-247	322	21	≤	≤	NUM
ma-247	322	22	j	j	PROPN
ma-247	322	23	≤	≤	PROPN
ma-247	322	24	n	n	CCONJ
ma-247	322	25	,	,	PUNCT
ma-247	322	26	there	there	PRON
ma-247	322	27	exists	exist	VERB
ma-247	322	28	αj	αj	X
ma-247	322	29	>	>	X
ma-247	322	30	0	0	PUNCT
ma-247	323	1	for	for	ADP
ma-247	323	2	which	which	PRON
ma-247	323	3	theorem	theorem	VERB
ma-247	323	4	6.1	6.1	NUM
ma-247	323	5	holds	hold	NOUN
ma-247	323	6	for	for	ADP
ma-247	323	7	the	the	DET
ma-247	323	8	psh	psh	NOUN
ma-247	323	9	function	function	NOUN
ma-247	323	10	vj	vj	INTJ
ma-247	323	11	,	,	PUNCT
ma-247	323	12	then	then	ADV
ma-247	323	13	for	for	ADP
ma-247	323	14	α	α	PROPN
ma-247	323	15	=	=	SYM
ma-247	323	16	min1≤j≤n	min1≤j≤n	PROPN
ma-247	323	17	(	(	PUNCT
ma-247	323	18	αj	αj	NOUN
ma-247	323	19	)	)	PUNCT
ma-247	323	20	,	,	PUNCT
ma-247	323	21	theorem	theorem	VERB
ma-247	323	22	1.4	1.4	NUM
ma-247	323	23	can	can	AUX
ma-247	323	24	be	be	AUX
ma-247	323	25	deduced	deduce	VERB
ma-247	323	26	from	from	ADP
ma-247	323	27	theorem	theorem	ADJ
ma-247	323	28	6.1	6.1	NUM
ma-247	323	29	in	in	ADP
ma-247	323	30	terms	term	NOUN
ma-247	323	31	of	of	ADP
ma-247	323	32	slicesand	slicesand	NOUN
ma-247	323	33	it	it	PRON
ma-247	323	34	works	work	VERB
ma-247	323	35	with	with	ADP
ma-247	323	36	the	the	DET
ma-247	323	37	psh	psh	NOUN
ma-247	323	38	function	function	NOUN
ma-247	323	39	v	v	ADP
ma-247	323	40	=	=	PUNCT
ma-247	323	41	∑	∑	PROPN
ma-247	323	42	1≤j≤n	1≤j≤n	NUM
ma-247	323	43	exp(−αvj	exp(−αvj	NOUN
ma-247	323	44	)	)	PUNCT
ma-247	323	45	.	.	PUNCT
ma-247	324	1	theorem	theorem	NOUN
ma-247	324	2	6.1	6.1	NUM
ma-247	324	3	.	.	PUNCT
ma-247	325	1	assume	assume	VERB
ma-247	325	2	we	we	PRON
ma-247	325	3	have	have	VERB
ma-247	325	4	a	a	DET
ma-247	325	5	psh	psh	NOUN
ma-247	325	6	function	function	NOUN
ma-247	325	7	v	v	NOUN
ma-247	325	8	on	on	ADP
ma-247	325	9	∆n	∆n	PROPN
ma-247	325	10	such	such	ADJ
ma-247	325	11	that	that	SCONJ
ma-247	325	12	its	its	PRON
ma-247	325	13	set	set	NOUN
ma-247	325	14	of	of	ADP
ma-247	325	15	singular	singular	ADJ
ma-247	325	16	points	point	NOUN
ma-247	325	17	is	be	AUX
ma-247	325	18	contained	contain	VERB
ma-247	325	19	in	in	ADP
ma-247	325	20	a	a	DET
ma-247	325	21	hypersurface	hypersurface	NOUN
ma-247	325	22	x	x	PUNCT
ma-247	325	23	forming	form	VERB
ma-247	325	24	a	a	DET
ma-247	325	25	complete	complete	ADJ
ma-247	325	26	intersection	intersection	NOUN
ma-247	325	27	with	with	ADP
ma-247	325	28	the	the	DET
ma-247	325	29	analytic	analytic	ADJ
ma-247	325	30	subset	subset	NOUN
ma-247	325	31	y	y	PROPN
ma-247	325	32	of	of	ADP
ma-247	325	33	∆n	∆n	PROPN
ma-247	325	34	.	.	PUNCT
ma-247	326	1	then	then	ADV
ma-247	326	2	,	,	PUNCT
ma-247	326	3	there	there	PRON
ma-247	326	4	exists	exist	VERB
ma-247	326	5	α	α	PROPN
ma-247	326	6	>	>	X
ma-247	326	7	0	0	NUM
ma-247	326	8	,	,	PUNCT
ma-247	326	9	such	such	ADJ
ma-247	326	10	that	that	SCONJ
ma-247	326	11	the	the	DET
ma-247	326	12	function	function	NOUN
ma-247	326	13	exp	exp	NOUN
ma-247	326	14	(	(	PUNCT
ma-247	326	15	−αv)|y	−αv)|y	NOUN
ma-247	326	16	lies	lie	VERB
ma-247	326	17	in	in	ADP
ma-247	326	18	l1	l1	PROPN
ma-247	326	19	loc(y	loc(y	PROPN
ma-247	326	20	)	)	PUNCT
ma-247	326	21	.	.	PUNCT
ma-247	327	1	proof	proof	NOUN
ma-247	327	2	.	.	PUNCT
ma-247	328	1	the	the	DET
ma-247	328	2	result	result	NOUN
ma-247	328	3	is	be	AUX
ma-247	328	4	local	local	ADJ
ma-247	328	5	,	,	PUNCT
ma-247	328	6	so	so	SCONJ
ma-247	328	7	it	it	PRON
ma-247	328	8	’s	’	VERB
ma-247	328	9	enough	enough	ADV
ma-247	328	10	to	to	PART
ma-247	328	11	prove	prove	VERB
ma-247	328	12	it	it	PRON
ma-247	328	13	near	near	ADP
ma-247	328	14	a	a	DET
ma-247	328	15	singular	singular	ADJ
ma-247	328	16	point	point	NOUN
ma-247	328	17	z0	z0	NOUN
ma-247	328	18	=	=	SYM
ma-247	328	19	0	0	NUM
ma-247	328	20	∈	∈	PROPN
ma-247	328	21	{	{	PUNCT
ma-247	328	22	v	v	NOUN
ma-247	328	23	=	=	SYM
ma-247	328	24	−∞	−∞	NOUN
ma-247	328	25	}	}	PUNCT
ma-247	328	26	⊂	⊂	PROPN
ma-247	328	27	x	x	PUNCT
ma-247	328	28	.we	.we	PUNCT
ma-247	328	29	may	may	AUX
ma-247	328	30	suppose	suppose	VERB
ma-247	328	31	z0	z0	PROPN
ma-247	328	32	is	be	AUX
ma-247	328	33	a	a	DET
ma-247	328	34	regular	regular	ADJ
ma-247	328	35	point	point	NOUN
ma-247	328	36	of	of	ADP
ma-247	328	37	x	x	NOUN
ma-247	328	38	∩y	∩y	NOUN
ma-247	328	39	.	.	PUNCT
ma-247	329	1	let	let	VERB
ma-247	329	2	q	q	PRON
ma-247	329	3	be	be	AUX
ma-247	329	4	the	the	DET
ma-247	329	5	dimension	dimension	NOUN
ma-247	329	6	of	of	ADP
ma-247	329	7	y	y	PROPN
ma-247	329	8	at	at	ADP
ma-247	329	9	z0	z0	PROPN
ma-247	329	10	(	(	PUNCT
ma-247	329	11	1	1	NUM
ma-247	329	12	≤	≤	PROPN
ma-247	329	13	q	q	ADJ
ma-247	329	14	≤	≤	NUM
ma-247	329	15	n−1),since	n−1),since	NOUN
ma-247	329	16	x	x	PUNCT
ma-247	329	17	and	and	CCONJ
ma-247	329	18	y	y	PROPN
ma-247	329	19	yield	yield	VERB
ma-247	329	20	a	a	DET
ma-247	329	21	complete	complete	ADJ
ma-247	329	22	intersection	intersection	NOUN
ma-247	329	23	near	near	ADP
ma-247	329	24	z0	z0	PROPN
ma-247	329	25	,	,	PUNCT
ma-247	329	26	then	then	ADV
ma-247	329	27	we	we	PRON
ma-247	329	28	may	may	AUX
ma-247	329	29	find	find	VERB
ma-247	329	30	a	a	DET
ma-247	329	31	neighborhood	neighborhood	NOUN
ma-247	329	32	v	v	NOUN
ma-247	329	33	(	(	PUNCT
ma-247	329	34	z0	z0	PROPN
ma-247	329	35	)	)	PUNCT
ma-247	329	36	of	of	ADP
ma-247	329	37	z0	z0	PROPN
ma-247	329	38	and	and	CCONJ
ma-247	329	39	local	local	ADJ
ma-247	329	40	coordinates	coordinate	NOUN
ma-247	329	41	(	(	PUNCT
ma-247	329	42	z1	z1	PROPN
ma-247	329	43	,	,	PUNCT
ma-247	329	44	.	.	PUNCT
ma-247	329	45	.	.	PUNCT
ma-247	330	1	.	.	PUNCT
ma-247	331	1	,	,	PUNCT
ma-247	331	2	zn	zn	X
ma-247	331	3	)	)	PUNCT
ma-247	331	4	such	such	ADJ
ma-247	331	5	that	that	SCONJ
ma-247	331	6	k	k	PROPN
ma-247	331	7	∩x	∩x	ADJ
ma-247	331	8	∩	∩	PROPN
ma-247	331	9	y	y	PROPN
ma-247	331	10	⊂	⊂	PROPN
ma-247	331	11	k	k	PROPN
ma-247	331	12	∩	∩	PROPN
ma-247	331	13	y	y	PROPN
ma-247	331	14	∩	∩	X
ma-247	331	15	{	{	PUNCT
ma-247	331	16	zq	zq	PROPN
ma-247	331	17	=	=	SYM
ma-247	331	18	zq+1	zq+1	PROPN
ma-247	331	19	=	=	SYM
ma-247	331	20	·	·	PUNCT
ma-247	331	21	·	·	PUNCT
ma-247	331	22	·	·	PUNCT
ma-247	332	1	=	=	PUNCT
ma-247	332	2	zn	zn	X
ma-247	332	3	=	=	SYM
ma-247	332	4	0	0	NUM
ma-247	332	5	}	}	PUNCT
ma-247	332	6	,	,	PUNCT
ma-247	332	7	k	k	PROPN
ma-247	332	8	=	=	SYM
ma-247	332	9	v	v	PROPN
ma-247	332	10	(	(	PUNCT
ma-247	332	11	z0	z0	PROPN
ma-247	332	12	)	)	PUNCT
ma-247	332	13	.	.	PUNCT
ma-247	333	1	(	(	PUNCT
ma-247	333	2	6.17	6.17	NUM
ma-247	333	3	)	)	PUNCT
ma-247	333	4	let	let	VERB
ma-247	333	5	(	(	PUNCT
ma-247	333	6	vδ)δ>0	vδ)δ>0	ADP
ma-247	333	7	be	be	AUX
ma-247	333	8	a	a	DET
ma-247	333	9	decreasing	decrease	VERB
ma-247	333	10	sequence	sequence	NOUN
ma-247	333	11	of	of	ADP
ma-247	333	12	continuous	continuous	ADJ
ma-247	333	13	psh	psh	NOUN
ma-247	333	14	functions	function	NOUN
ma-247	333	15	such	such	ADJ
ma-247	333	16	that	that	SCONJ
ma-247	333	17	limδ→0	limδ→0	PROPN
ma-247	333	18	vδ	vδ	NOUN
ma-247	333	19	=	=	X
ma-247	333	20	v	v	PROPN
ma-247	333	21	pointwise.we	pointwise.we	NOUN
ma-247	333	22	choose	choose	VERB
ma-247	333	23	a	a	DET
ma-247	333	24	fixed	fixed	ADJ
ma-247	333	25	δ0	δ0	NOUN
ma-247	333	26	>	>	X
ma-247	333	27	0	0	PUNCT
ma-247	334	1	small	small	ADJ
ma-247	334	2	enough	enough	ADV
ma-247	334	3	,	,	PUNCT
ma-247	334	4	such	such	ADJ
ma-247	334	5	that	that	DET
ma-247	334	6	vδ0	vδ0	NOUN
ma-247	334	7	(	(	PUNCT
ma-247	334	8	z)−	z)−	PROPN
ma-247	334	9	1	1	NUM
ma-247	334	10	≤	≤	NUM
ma-247	334	11	v(z	v(z	NOUN
ma-247	334	12	)	)	PUNCT
ma-247	334	13	≤	≤	NUM
ma-247	334	14	vδ0	vδ0	NOUN
ma-247	334	15	(	(	PUNCT
ma-247	334	16	z	z	NOUN
ma-247	334	17	)	)	PUNCT
ma-247	334	18	,	,	PUNCT
ma-247	334	19	∀	∀	PUNCT
ma-247	334	20	z	z	NOUN
ma-247	334	21	∈	∈	PROPN
ma-247	335	1	k	k	X
ma-247	335	2	rx	rx	X
ma-247	335	3	∩	∩	PROPN
ma-247	335	4	y.	y.	NOUN
ma-247	335	5	(	(	PUNCT
ma-247	335	6	6.18	6.18	NUM
ma-247	335	7	)	)	PUNCT
ma-247	335	8	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	335	9	eur	eur	PROPN
ma-247	335	10	.	.	PUNCT
ma-247	336	1	j.	j.	PROPN
ma-247	336	2	math	math	PROPN
ma-247	336	3	.	.	PUNCT
ma-247	337	1	anal	anal	PROPN
ma-247	337	2	.	.	PUNCT
ma-247	338	1	10.28924	10.28924	NUM
ma-247	338	2	/	/	SYM
ma-247	338	3	ada	ada	PROPN
ma-247	338	4	/	/	SYM
ma-247	338	5	ma.4.23	ma.4.23	NOUN
ma-247	338	6	12according	12accorde	VERB
ma-247	338	7	to	to	ADP
ma-247	338	8	a	a	DET
ma-247	338	9	classical	classical	ADJ
ma-247	338	10	result	result	NOUN
ma-247	338	11	due	due	ADP
ma-247	338	12	to	to	ADP
ma-247	338	13	h.	h.	PROPN
ma-247	338	14	j.	j.	PROPN
ma-247	338	15	bremermann	bremermann	PROPN
ma-247	338	16	and	and	CCONJ
ma-247	338	17	p.	p.	PROPN
ma-247	338	18	lelong	lelong	PROPN
ma-247	339	1	[	[	X
ma-247	339	2	4	4	NUM
ma-247	339	3	,	,	PUNCT
ma-247	339	4	14	14	NUM
ma-247	339	5	]	]	PUNCT
ma-247	339	6	,	,	PUNCT
ma-247	339	7	there	there	PRON
ma-247	339	8	exists	exist	VERB
ma-247	339	9	(	(	PUNCT
ma-247	339	10	fj)j	fj)j	NOUN
ma-247	339	11	asequence	asequence	NOUN
ma-247	339	12	of	of	ADP
ma-247	339	13	holomorphic	holomorphic	ADJ
ma-247	339	14	functions	function	NOUN
ma-247	339	15	on	on	ADP
ma-247	339	16	∆n	∆n	PROPN
ma-247	339	17	such	such	ADJ
ma-247	339	18	that	that	SCONJ
ma-247	339	19	the	the	DET
ma-247	339	20	function	function	NOUN
ma-247	339	21	vδ0	vδ0	NOUN
ma-247	339	22	is	be	AUX
ma-247	339	23	the	the	DET
ma-247	339	24	regularized	regularize	VERB
ma-247	339	25	supremumlimit	supremumlimit	NOUN
ma-247	339	26	on	on	ADP
ma-247	339	27	k	k	PROPN
ma-247	339	28	of	of	ADP
ma-247	339	29	(	(	PUNCT
ma-247	339	30	1	1	NUM
ma-247	339	31	j	j	NOUN
ma-247	339	32	log	log	NOUN
ma-247	339	33	|fj	|fj	X
ma-247	340	1	|	|	ADV
ma-247	340	2	)	)	PUNCT
ma-247	340	3	j	j	PROPN
ma-247	340	4	.	.	PUNCT
ma-247	341	1	which	which	PRON
ma-247	341	2	means	mean	VERB
ma-247	341	3	that	that	SCONJ
ma-247	341	4	vδ0	vδ0	NOUN
ma-247	341	5	(	(	PUNCT
ma-247	341	6	z	z	NOUN
ma-247	341	7	)	)	PUNCT
ma-247	342	1	=	=	PUNCT
ma-247	342	2	[	[	PUNCT
ma-247	342	3	limj→∞	limj→∞	PROPN
ma-247	342	4	sup	sup	NOUN
ma-247	342	5	1	1	NUM
ma-247	342	6	j	j	NOUN
ma-247	342	7	log	log	NOUN
ma-247	342	8	|fj(z)|	|fj(z)|	PROPN
ma-247	342	9	]	]	PUNCT
ma-247	342	10	∗	∗	NOUN
ma-247	342	11	=	=	SYM
ma-247	342	12	limε→0	limε→0	PROPN
ma-247	342	13	supζ∈b(z	supζ∈b(z	NOUN
ma-247	342	14	,	,	PUNCT
ma-247	342	15	ε)[limj→∞	ε)[limj→∞	PROPN
ma-247	342	16	1	1	NUM
ma-247	342	17	j	j	PROPN
ma-247	342	18	log	log	NOUN
ma-247	342	19	|fj(ζ	|fj(ζ	NOUN
ma-247	342	20	)	)	PUNCT
ma-247	342	21	]	]	PUNCT
ma-247	342	22	,	,	PUNCT
ma-247	342	23	z	z	PROPN
ma-247	342	24	∈	∈	PROPN
ma-247	342	25	k.	k.	PROPN
ma-247	342	26	(	(	PUNCT
ma-247	342	27	6.19	6.19	NUM
ma-247	342	28	)	)	PUNCT
ma-247	342	29	for	for	ADP
ma-247	342	30	the	the	DET
ma-247	342	31	compact	compact	ADJ
ma-247	342	32	k	k	NOUN
ma-247	342	33	,	,	PUNCT
ma-247	342	34	in	in	ADP
ma-247	342	35	view	view	NOUN
ma-247	342	36	of	of	ADP
ma-247	342	37	(	(	PUNCT
ma-247	342	38	6.18	6.18	NUM
ma-247	342	39	)	)	PUNCT
ma-247	342	40	and	and	CCONJ
ma-247	342	41	(	(	PUNCT
ma-247	342	42	6.19	6.19	NUM
ma-247	342	43	)	)	PUNCT
ma-247	342	44	,	,	PUNCT
ma-247	342	45	we	we	PRON
ma-247	342	46	can	can	AUX
ma-247	342	47	find	find	VERB
ma-247	342	48	a	a	DET
ma-247	342	49	natural	natural	ADJ
ma-247	342	50	number	number	NOUN
ma-247	342	51	n	n	PRON
ma-247	342	52	that	that	PRON
ma-247	342	53	depends	depend	VERB
ma-247	342	54	on	on	ADP
ma-247	342	55	k	k	PROPN
ma-247	342	56	,	,	PUNCT
ma-247	342	57	and	and	CCONJ
ma-247	342	58	analytic	analytic	ADJ
ma-247	342	59	functions	function	NOUN
ma-247	342	60	on	on	ADP
ma-247	342	61	∆n	∆n	PROPN
ma-247	342	62	,	,	PUNCT
ma-247	342	63	f1	f1	NOUN
ma-247	342	64	,	,	PUNCT
ma-247	342	65	.	.	PUNCT
ma-247	342	66	.	.	PUNCT
ma-247	342	67	.	.	PUNCT
ma-247	343	1	,	,	PUNCT
ma-247	343	2	fn	fn	INTJ
ma-247	343	3	,	,	PUNCT
ma-247	343	4	such	such	ADJ
ma-247	343	5	that	that	SCONJ
ma-247	343	6	the	the	DET
ma-247	343	7	function	function	NOUN
ma-247	343	8	v	v	NOUN
ma-247	343	9	and	and	CCONJ
ma-247	343	10	the	the	DET
ma-247	343	11	function	function	NOUN
ma-247	343	12	vn	vn	VERB
ma-247	343	13	definedby	definedby	ADJ
ma-247	343	14	vn(z	vn(z	NOUN
ma-247	343	15	)	)	PUNCT
ma-247	344	1	=	=	SYM
ma-247	344	2	max	max	PROPN
ma-247	344	3	1≤j≤n	1≤j≤n	NUM
ma-247	344	4	1	1	NUM
ma-247	344	5	j	j	PROPN
ma-247	344	6	log	log	NOUN
ma-247	344	7	|fj(z)|	|fj(z)|	PROPN
ma-247	344	8	,	,	PUNCT
ma-247	344	9	(	(	PUNCT
ma-247	344	10	6.20	6.20	NUM
ma-247	344	11	)	)	PUNCT
ma-247	344	12	satisfy	satisfy	VERB
ma-247	344	13	the	the	DET
ma-247	344	14	following	follow	VERB
ma-247	344	15	inequalities	inequality	NOUN
ma-247	344	16	vn	vn	VERB
ma-247	344	17	−	−	PROPN
ma-247	344	18	1	1	NUM
ma-247	344	19	≤	≤	NOUN
ma-247	344	20	v	v	PRON
ma-247	344	21	≤	≤	NUM
ma-247	344	22	vn	vn	VERB
ma-247	344	23	≤	≤	NOUN
ma-247	344	24	0	0	NUM
ma-247	345	1	on	on	ADP
ma-247	345	2	k	k	X
ma-247	345	3	rx	rx	X
ma-247	345	4	∩	∩	PROPN
ma-247	345	5	y	y	PROPN
ma-247	345	6	.	.	PUNCT
ma-247	346	1	(	(	PUNCT
ma-247	346	2	6.21	6.21	NUM
ma-247	346	3	)	)	PUNCT
ma-247	346	4	we	we	PRON
ma-247	346	5	may	may	AUX
ma-247	346	6	suppose	suppose	VERB
ma-247	346	7	the	the	DET
ma-247	346	8	compact	compact	ADJ
ma-247	346	9	k	k	PROPN
ma-247	346	10	is	be	AUX
ma-247	346	11	sufficiently	sufficiently	ADV
ma-247	346	12	small	small	ADJ
ma-247	346	13	so	so	SCONJ
ma-247	346	14	that	that	SCONJ
ma-247	346	15	k	k	PROPN
ma-247	346	16	∩x	∩x	ADJ
ma-247	346	17	∩	∩	PROPN
ma-247	346	18	y	y	PROPN
ma-247	346	19	⊂	⊂	PROPN
ma-247	346	20	k	k	PROPN
ma-247	346	21	∩	∩	PROPN
ma-247	346	22	y	y	PROPN
ma-247	346	23	∩	∩	X
ma-247	346	24	{	{	PUNCT
ma-247	346	25	fj	fj	PROPN
ma-247	346	26	=	=	SYM
ma-247	346	27	0	0	PROPN
ma-247	346	28	,	,	PUNCT
ma-247	346	29	j	j	PROPN
ma-247	346	30	=	=	SYM
ma-247	346	31	1	1	NUM
ma-247	346	32	,	,	PUNCT
ma-247	346	33	.	.	PUNCT
ma-247	346	34	.	.	PUNCT
ma-247	347	1	.	.	PUNCT
ma-247	347	2	,	,	PUNCT
ma-247	347	3	n	n	CCONJ
ma-247	347	4	}	}	PUNCT
ma-247	347	5	.	.	PUNCT
ma-247	348	1	by	by	ADP
ma-247	348	2	the	the	DET
ma-247	348	3	weierstrass	weierstrass	NOUN
ma-247	348	4	preparation	preparation	NOUN
ma-247	348	5	theorem	theorem	NOUN
ma-247	348	6	(	(	PUNCT
ma-247	348	7	see	see	VERB
ma-247	348	8	[	[	X
ma-247	348	9	6	6	NUM
ma-247	348	10	]	]	NUM
ma-247	348	11	)	)	PUNCT
ma-247	348	12	,	,	PUNCT
ma-247	348	13	for	for	ADP
ma-247	348	14	each	each	DET
ma-247	348	15	indice	indice	NOUN
ma-247	348	16	1	1	NUM
ma-247	348	17	≤	≤	NUM
ma-247	348	18	j	j	PROPN
ma-247	348	19	≤	≤	PROPN
ma-247	348	20	n	n	PRON
ma-247	348	21	,	,	PUNCT
ma-247	348	22	one	one	PRON
ma-247	348	23	can	can	AUX
ma-247	348	24	write	write	VERB
ma-247	348	25	fj	fj	ADP
ma-247	348	26	suchthat	suchthat	PROPN
ma-247	348	27	fj(z	fj(z	NOUN
ma-247	348	28	)	)	PUNCT
ma-247	348	29	=	=	SYM
ma-247	349	1	hj(z)pj(z	hj(z)pj(z	NOUN
ma-247	349	2	′	′	NUM
ma-247	349	3	,	,	PUNCT
ma-247	349	4	zn	zn	PROPN
ma-247	349	5	)	)	PUNCT
ma-247	349	6	,	,	PUNCT
ma-247	349	7	(	(	PUNCT
ma-247	349	8	6.22	6.22	NUM
ma-247	349	9	)	)	PUNCT
ma-247	349	10	where	where	SCONJ
ma-247	349	11	hj	hj	PROPN
ma-247	349	12	is	be	AUX
ma-247	349	13	an	an	DET
ma-247	349	14	invertible	invertible	ADJ
ma-247	349	15	holomorphic	holomorphic	ADJ
ma-247	349	16	function	function	NOUN
ma-247	349	17	on	on	ADP
ma-247	349	18	∆n	∆n	PROPN
ma-247	349	19	and	and	CCONJ
ma-247	349	20	pj(z	pj(z	NOUN
ma-247	349	21	′	′	NUM
ma-247	349	22	,	,	PUNCT
ma-247	349	23	zn	zn	X
ma-247	349	24	)	)	PUNCT
ma-247	349	25	is	be	AUX
ma-247	349	26	a	a	DET
ma-247	349	27	weierstrass	weierstrass	NOUN
ma-247	349	28	polynomial	polynomial	NOUN
ma-247	349	29	in	in	ADP
ma-247	349	30	zn	zn	PROPN
ma-247	349	31	,	,	PUNCT
ma-247	349	32	of	of	ADP
ma-247	349	33	the	the	DET
ma-247	349	34	form	form	NOUN
ma-247	349	35	pj(z	pj(z	NOUN
ma-247	349	36	′	′	NUM
ma-247	349	37	,	,	PUNCT
ma-247	349	38	zn	zn	NOUN
ma-247	349	39	)	)	PUNCT
ma-247	349	40	=	=	PUNCT
ma-247	349	41	z	z	NOUN
ma-247	349	42	mj	mj	NOUN
ma-247	349	43	n	n	PROPN
ma-247	349	44	+	+	CCONJ
ma-247	349	45	a1,j(z	a1,j(z	X
ma-247	349	46	′)z	′)z	PROPN
ma-247	349	47	mj−1	mj−1	NOUN
ma-247	349	48	n	n	PROPN
ma-247	349	49	+	+	CCONJ
ma-247	349	50	·	·	PUNCT
ma-247	349	51	·	·	PUNCT
ma-247	349	52	·	·	PUNCT
ma-247	349	53	+	+	NUM
ma-247	349	54	aν	aν	NOUN
ma-247	349	55	,	,	PUNCT
ma-247	349	56	j(z	j(z	PROPN
ma-247	349	57	′)z	′)z	PROPN
ma-247	349	58	mj−ν	mj−ν	PROPN
ma-247	349	59	n	n	PROPN
ma-247	349	60	+	+	CCONJ
ma-247	349	61	·	·	PUNCT
ma-247	349	62	·	·	PUNCT
ma-247	349	63	·	·	PUNCT
ma-247	350	1	+	+	CCONJ
ma-247	350	2	amj	amj	NOUN
ma-247	350	3	,	,	PUNCT
ma-247	350	4	j,(z	j,(z	NOUN
ma-247	350	5	′	′	NOUN
ma-247	350	6	)	)	PUNCT
ma-247	350	7	,	,	PUNCT
ma-247	350	8	aν	aν	NOUN
ma-247	350	9	,	,	PUNCT
ma-247	350	10	j(0	j(0	PROPN
ma-247	350	11	)	)	PUNCT
ma-247	350	12	=	=	PUNCT
ma-247	350	13	0	0	NUM
ma-247	350	14	,	,	PUNCT
ma-247	350	15	with	with	SCONJ
ma-247	350	16	mj	mj	PROPN
ma-247	350	17	≥	≥	PROPN
ma-247	350	18	1	1	NUM
ma-247	350	19	is	be	AUX
ma-247	350	20	the	the	DET
ma-247	350	21	vanishing	vanish	VERB
ma-247	350	22	order	order	NOUN
ma-247	350	23	of	of	ADP
ma-247	350	24	fj	fj	PROPN
ma-247	350	25	at	at	ADP
ma-247	350	26	z0	z0	PROPN
ma-247	350	27	=	=	PROPN
ma-247	350	28	0	0	PROPN
ma-247	350	29	and	and	CCONJ
ma-247	350	30	(	(	PUNCT
ma-247	350	31	aν	aν	NOUN
ma-247	350	32	,	,	PUNCT
ma-247	350	33	j,(z	j,(z	NOUN
ma-247	350	34	′))1≤µ≤mj	′))1≤µ≤mj	NOUN
ma-247	350	35	are	be	AUX
ma-247	350	36	holomorphic	holomorphic	ADJ
ma-247	350	37	coefficientson	coefficientson	NOUN
ma-247	350	38	the	the	DET
ma-247	350	39	polydisc	polydisc	NOUN
ma-247	350	40	∆n−1	∆n−1	PROPN
ma-247	350	41	in	in	ADP
ma-247	350	42	cn−1	cn−1	PROPN
ma-247	350	43	.	.	PUNCT
ma-247	351	1	furthermore	furthermore	ADV
ma-247	351	2	,	,	PUNCT
ma-247	351	3	for	for	ADP
ma-247	351	4	all	all	DET
ma-247	351	5	1	1	NUM
ma-247	351	6	≤	≤	NUM
ma-247	351	7	j	j	PROPN
ma-247	351	8	≤	≤	PROPN
ma-247	351	9	n	n	CCONJ
ma-247	351	10	and	and	CCONJ
ma-247	351	11	all	all	DET
ma-247	351	12	1	1	NUM
ma-247	351	13	≤	≤	NUM
ma-247	351	14	ν	ν	NOUN
ma-247	351	15	≤	≤	PROPN
ma-247	351	16	mj	mj	NOUN
ma-247	351	17	,	,	PUNCT
ma-247	351	18	there	there	PRON
ma-247	351	19	is	be	VERB
ma-247	351	20	apositive	apositive	ADJ
ma-247	351	21	constant	constant	ADJ
ma-247	351	22	c1	c1	NOUN
ma-247	351	23	such	such	ADJ
ma-247	351	24	that	that	SCONJ
ma-247	351	25	for	for	ADP
ma-247	351	26	all	all	DET
ma-247	351	27	z	z	NOUN
ma-247	351	28	′	′	NUM
ma-247	352	1	∈	∈	PROPN
ma-247	352	2	∆n−1	∆n−1	PROPN
ma-247	352	3	,	,	PUNCT
ma-247	352	4	the	the	DET
ma-247	352	5	following	follow	VERB
ma-247	352	6	inequality	inequality	NOUN
ma-247	352	7	holds	hold	VERB
ma-247	352	8	|aν	|aν	ADP
ma-247	352	9	,	,	PUNCT
ma-247	352	10	j(z	j(z	PROPN
ma-247	352	11	′)|	′)|	PUNCT
ma-247	352	12	≤	≤	NUM
ma-247	352	13	c1||z	c1||z	NOUN
ma-247	352	14	′||ν	′||ν	PROPN
ma-247	352	15	.	.	PUNCT
ma-247	353	1	(	(	PUNCT
ma-247	353	2	6.23	6.23	NUM
ma-247	353	3	)	)	PUNCT
ma-247	353	4	therefore	therefore	ADV
ma-247	353	5	,	,	PUNCT
ma-247	353	6	by	by	ADP
ma-247	353	7	the	the	DET
ma-247	353	8	triangle	triangle	NOUN
ma-247	353	9	inequality	inequality	NOUN
ma-247	353	10	,	,	PUNCT
ma-247	353	11	(	(	PUNCT
ma-247	353	12	6.23	6.23	NUM
ma-247	353	13	)	)	PUNCT
ma-247	353	14	provides	provide	VERB
ma-247	353	15	the	the	DET
ma-247	353	16	existence	existence	NOUN
ma-247	353	17	of	of	ADP
ma-247	353	18	a	a	DET
ma-247	353	19	constant	constant	ADJ
ma-247	353	20	c2	c2	PROPN
ma-247	353	21	=	=	SYM
ma-247	353	22	c3(k	c3(k	PROPN
ma-247	353	23	)	)	PUNCT
ma-247	353	24	>	>	X
ma-247	354	1	0such	0such	PUNCT
ma-247	355	1	that	that	SCONJ
ma-247	355	2	for	for	ADP
ma-247	355	3	all	all	DET
ma-247	355	4	1	1	NUM
ma-247	355	5	≤	≤	NUM
ma-247	355	6	j	j	PROPN
ma-247	355	7	≤	≤	PROPN
ma-247	355	8	n	n	CCONJ
ma-247	355	9	and	and	CCONJ
ma-247	355	10	for	for	ADP
ma-247	355	11	all	all	DET
ma-247	355	12	z	z	NOUN
ma-247	355	13	=	=	SYM
ma-247	355	14	(	(	PUNCT
ma-247	355	15	z	z	NOUN
ma-247	355	16	′	′	NUM
ma-247	355	17	,	,	PUNCT
ma-247	355	18	zn	zn	X
ma-247	355	19	)	)	PUNCT
ma-247	355	20	∈	∈	PROPN
ma-247	356	1	k	k	NOUN
ma-247	356	2	,	,	PUNCT
ma-247	356	3	the	the	DET
ma-247	356	4	following	follow	VERB
ma-247	356	5	inequality	inequality	NOUN
ma-247	356	6	holds	hold	VERB
ma-247	356	7	|pj(z	|pj(z	PROPN
ma-247	356	8	′	′	PROPN
ma-247	356	9	,	,	PUNCT
ma-247	356	10	zn)|	zn)|	PROPN
ma-247	356	11	≤	≤	NUM
ma-247	356	12	c2||z	c2||z	ADJ
ma-247	356	13	′||mj	′||mj	NOUN
ma-247	356	14	.	.	PUNCT
ma-247	357	1	(	(	PUNCT
ma-247	357	2	6.24	6.24	NUM
ma-247	357	3	)	)	PUNCT
ma-247	357	4	on	on	ADP
ma-247	357	5	the	the	DET
ma-247	357	6	other	other	ADJ
ma-247	357	7	hand	hand	NOUN
ma-247	357	8	,	,	PUNCT
ma-247	357	9	since	since	SCONJ
ma-247	357	10	{	{	PUNCT
ma-247	357	11	fj	fj	PROPN
ma-247	357	12	=	=	SYM
ma-247	357	13	0	0	PROPN
ma-247	357	14	,	,	PUNCT
ma-247	357	15	j	j	PROPN
ma-247	357	16	=	=	SYM
ma-247	357	17	1	1	NUM
ma-247	357	18	,	,	PUNCT
ma-247	357	19	.	.	PUNCT
ma-247	357	20	.	.	PUNCT
ma-247	358	1	.	.	PUNCT
ma-247	359	1	,	,	PUNCT
ma-247	359	2	n	n	CCONJ
ma-247	359	3	}	}	PUNCT
ma-247	359	4	=	=	SYM
ma-247	359	5	{	{	PUNCT
ma-247	359	6	pj	pj	PROPN
ma-247	359	7	=	=	SYM
ma-247	359	8	0	0	PROPN
ma-247	359	9	,	,	PUNCT
ma-247	359	10	j	j	PROPN
ma-247	359	11	=	=	SYM
ma-247	359	12	1	1	NUM
ma-247	359	13	,	,	PUNCT
ma-247	359	14	.	.	PUNCT
ma-247	359	15	.	.	PUNCT
ma-247	360	1	.	.	PUNCT
ma-247	361	1	,	,	PUNCT
ma-247	362	1	n	n	CCONJ
ma-247	362	2	}	}	PUNCT
ma-247	362	3	and	and	CCONJ
ma-247	362	4	since	since	SCONJ
ma-247	362	5	the	the	DET
ma-247	362	6	polynomial	polynomial	ADJ
ma-247	362	7	pj	pj	PROPN
ma-247	362	8	vanishes	vanish	VERB
ma-247	362	9	at	at	ADP
ma-247	362	10	z0	z0	PROPN
ma-247	362	11	=	=	SYM
ma-247	362	12	0	0	PROPN
ma-247	362	13	,	,	PUNCT
ma-247	362	14	then	then	ADV
ma-247	362	15	there	there	PRON
ma-247	362	16	is	be	VERB
ma-247	362	17	a	a	DET
ma-247	362	18	constant	constant	ADJ
ma-247	362	19	c3	c3	NOUN
ma-247	362	20	>	>	X
ma-247	362	21	0	0	PROPN
ma-247	362	22	,	,	PUNCT
ma-247	362	23	such	such	ADJ
ma-247	362	24	that	that	PRON
ma-247	362	25	for	for	ADP
ma-247	362	26	any	any	DET
ma-247	362	27	point	point	NOUN
ma-247	362	28	z	z	NOUN
ma-247	363	1	=	=	SYM
ma-247	364	1	(	(	PUNCT
ma-247	364	2	z	z	NOUN
ma-247	364	3	′	′	NUM
ma-247	364	4	,	,	PUNCT
ma-247	364	5	zn	zn	X
ma-247	364	6	)	)	PUNCT
ma-247	364	7	∈	∈	PROPN
ma-247	364	8	ksufficiently	ksufficiently	ADV
ma-247	364	9	close	close	ADJ
ma-247	364	10	to	to	ADP
ma-247	364	11	0	0	NUM
ma-247	364	12	,	,	PUNCT
ma-247	364	13	the	the	DET
ma-247	364	14	following	follow	VERB
ma-247	364	15	inequality	inequality	NOUN
ma-247	364	16	holds	hold	VERB
ma-247	364	17	|zn|	|zn|	NOUN
ma-247	364	18	≤	≤	NOUN
ma-247	364	19	c3||z	c3||z	NOUN
ma-247	364	20	′||	′||	PROPN
ma-247	364	21	.	.	PUNCT
ma-247	365	1	(	(	PUNCT
ma-247	365	2	6.25	6.25	NUM
ma-247	365	3	)	)	PUNCT
ma-247	365	4	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	365	5	eur	eur	PROPN
ma-247	365	6	.	.	PUNCT
ma-247	366	1	j.	j.	PROPN
ma-247	366	2	math	math	PROPN
ma-247	366	3	.	.	PUNCT
ma-247	367	1	anal	anal	PROPN
ma-247	367	2	.	.	PUNCT
ma-247	368	1	10.28924	10.28924	NUM
ma-247	368	2	/	/	SYM
ma-247	368	3	ada	ada	PROPN
ma-247	368	4	/	/	SYM
ma-247	368	5	ma.4.23	ma.4.23	PROPN
ma-247	368	6	13hence	13hence	NOUN
ma-247	368	7	,	,	PUNCT
ma-247	368	8	using	use	VERB
ma-247	368	9	the	the	DET
ma-247	368	10	triangle	triangle	NOUN
ma-247	368	11	inequality	inequality	NOUN
ma-247	368	12	with	with	ADP
ma-247	368	13	(	(	PUNCT
ma-247	368	14	6.25	6.25	NUM
ma-247	368	15	)	)	PUNCT
ma-247	368	16	and	and	CCONJ
ma-247	368	17	reasoning	reason	VERB
ma-247	368	18	by	by	ADP
ma-247	368	19	induction	induction	NOUN
ma-247	368	20	on	on	ADP
ma-247	368	21	the	the	DET
ma-247	368	22	degree	degree	NOUN
ma-247	368	23	mj	mj	PROPN
ma-247	368	24	of	of	ADP
ma-247	368	25	pj	pj	PROPN
ma-247	368	26	,	,	PUNCT
ma-247	368	27	1	1	NUM
ma-247	368	28	≤	≤	NUM
ma-247	368	29	j	j	PROPN
ma-247	368	30	≤	≤	PROPN
ma-247	368	31	n	n	CCONJ
ma-247	368	32	,	,	PUNCT
ma-247	368	33	we	we	PRON
ma-247	368	34	can	can	AUX
ma-247	368	35	find	find	VERB
ma-247	368	36	a	a	DET
ma-247	368	37	constant	constant	ADJ
ma-247	368	38	c4	c4	NOUN
ma-247	368	39	=	=	SYM
ma-247	368	40	c4(k	c4(k	PROPN
ma-247	368	41	)	)	PUNCT
ma-247	368	42	>	>	X
ma-247	368	43	0	0	NUM
ma-247	368	44	,	,	PUNCT
ma-247	368	45	such	such	ADJ
ma-247	368	46	that	that	SCONJ
ma-247	368	47	for	for	ADP
ma-247	368	48	all	all	DET
ma-247	368	49	1	1	NUM
ma-247	368	50	≤	≤	NUM
ma-247	368	51	j	j	PROPN
ma-247	368	52	≤	≤	PROPN
ma-247	368	53	n	n	CCONJ
ma-247	368	54	and	and	CCONJ
ma-247	368	55	for	for	ADP
ma-247	368	56	all	all	DET
ma-247	368	57	z	z	NOUN
ma-247	368	58	=	=	SYM
ma-247	368	59	(	(	PUNCT
ma-247	368	60	z	z	NOUN
ma-247	368	61	′	′	NUM
ma-247	368	62	,	,	PUNCT
ma-247	368	63	zn	zn	X
ma-247	368	64	)	)	PUNCT
ma-247	368	65	∈	∈	PROPN
ma-247	369	1	k	k	NOUN
ma-247	369	2	,	,	PUNCT
ma-247	369	3	the	the	DET
ma-247	369	4	polynomial	polynomial	ADJ
ma-247	369	5	pj(z	pj(z	NOUN
ma-247	369	6	′	′	NOUN
ma-247	369	7	,	,	PUNCT
ma-247	369	8	zn	zn	NOUN
ma-247	369	9	)	)	PUNCT
ma-247	369	10	satisfies	satisfy	VERB
ma-247	369	11	the	the	DET
ma-247	369	12	following	follow	VERB
ma-247	369	13	inequality	inequality	NOUN
ma-247	369	14	c4||z	c4||z	NOUN
ma-247	369	15	′||mj	′||mj	VERB
ma-247	369	16	≤	≤	NUM
ma-247	369	17	|pj(z	|pj(z	PROPN
ma-247	369	18	′	′	NOUN
ma-247	369	19	,	,	PUNCT
ma-247	369	20	zn)|	zn)|	X
ma-247	369	21	.	.	PUNCT
ma-247	370	1	(	(	PUNCT
ma-247	370	2	6.26	6.26	NUM
ma-247	370	3	)	)	PUNCT
ma-247	370	4	in	in	ADP
ma-247	370	5	addition	addition	NOUN
ma-247	370	6	,	,	PUNCT
ma-247	370	7	since	since	SCONJ
ma-247	370	8	for	for	ADP
ma-247	370	9	all	all	DET
ma-247	370	10	1	1	NUM
ma-247	370	11	≤	≤	NUM
ma-247	370	12	j	j	PROPN
ma-247	370	13	≤	≤	ADJ
ma-247	370	14	n	n	CCONJ
ma-247	370	15	,	,	PUNCT
ma-247	370	16	the	the	DET
ma-247	370	17	holomorphic	holomorphic	ADJ
ma-247	370	18	function	function	NOUN
ma-247	370	19	hj	hj	PROPN
ma-247	370	20	is	be	AUX
ma-247	370	21	invertible	invertible	ADJ
ma-247	370	22	and	and	CCONJ
ma-247	370	23	n	n	ADV
ma-247	370	24	depends	depend	VERB
ma-247	370	25	onlyon	onlyon	NOUN
ma-247	371	1	k	k	NOUN
ma-247	371	2	,	,	PUNCT
ma-247	371	3	then	then	ADV
ma-247	371	4	we	we	PRON
ma-247	371	5	can	can	AUX
ma-247	371	6	find	find	VERB
ma-247	371	7	a	a	DET
ma-247	371	8	constant	constant	ADJ
ma-247	371	9	c5	c5	PROPN
ma-247	371	10	=	=	SYM
ma-247	371	11	c5(k	c5(k	PROPN
ma-247	371	12	)	)	PUNCT
ma-247	371	13	>	>	X
ma-247	371	14	0	0	NUM
ma-247	372	1	such	such	ADJ
ma-247	372	2	that	that	PRON
ma-247	372	3	for	for	ADP
ma-247	372	4	all	all	DET
ma-247	372	5	z	z	NOUN
ma-247	372	6	∈	∈	PROPN
ma-247	372	7	k	k	NOUN
ma-247	372	8	,	,	PUNCT
ma-247	372	9	the	the	DET
ma-247	372	10	function	function	NOUN
ma-247	372	11	defined	define	VERB
ma-247	372	12	by	by	ADP
ma-247	372	13	v	v	PRON
ma-247	372	14	′n(z	′n(z	PROPN
ma-247	372	15	)	)	PUNCT
ma-247	372	16	=	=	SYM
ma-247	372	17	max	max	PROPN
ma-247	372	18	1≤j≤n	1≤j≤n	NUM
ma-247	372	19	1	1	NUM
ma-247	372	20	j	j	NOUN
ma-247	372	21	log	log	NOUN
ma-247	372	22	|hj(z)|	|hj(z)|	VERB
ma-247	372	23	satisfies	satisfy	VERB
ma-247	372	24	the	the	DET
ma-247	372	25	following	follow	VERB
ma-247	372	26	inequality	inequality	NOUN
ma-247	372	27	−	−	PROPN
ma-247	372	28	c5	c5	PROPN
ma-247	372	29	≤	≤	PROPN
ma-247	372	30	v	v	ADP
ma-247	372	31	′n(z	′n(z	PROPN
ma-247	372	32	)	)	PUNCT
ma-247	372	33	≤	≤	NOUN
ma-247	372	34	0	0	NUM
ma-247	372	35	.	.	PUNCT
ma-247	373	1	(	(	PUNCT
ma-247	373	2	6.27)if	6.27)if	NOUN
ma-247	373	3	we	we	PRON
ma-247	373	4	denote	denote	VERB
ma-247	373	5	βn	βn	ADV
ma-247	373	6	=	=	VERB
ma-247	373	7	min1≤j≤n	min1≤j≤n	PROPN
ma-247	373	8	mj	mj	PROPN
ma-247	373	9	j	j	PROPN
ma-247	373	10	and	and	CCONJ
ma-247	373	11	wn(z	wn(z	PUNCT
ma-247	373	12	)	)	PUNCT
ma-247	373	13	=	=	PRON
ma-247	373	14	wn(z	wn(z	PUNCT
ma-247	373	15	′	′	PROPN
ma-247	373	16	,	,	PUNCT
ma-247	373	17	zn	zn	NOUN
ma-247	373	18	)	)	PUNCT
ma-247	373	19	=	=	SYM
ma-247	373	20	βn	βn	NOUN
ma-247	373	21	log	log	NOUN
ma-247	373	22	||z	||z	NOUN
ma-247	373	23	′||	′||	PROPN
ma-247	373	24	,	,	PUNCT
ma-247	373	25	then	then	ADV
ma-247	373	26	following	follow	VERB
ma-247	373	27	(	(	PUNCT
ma-247	373	28	6.22	6.22	NUM
ma-247	373	29	)	)	PUNCT
ma-247	373	30	thefunction	thefunction	NOUN
ma-247	373	31	vn	vn	PROPN
ma-247	373	32	defined	define	VERB
ma-247	373	33	by	by	ADP
ma-247	373	34	(	(	PUNCT
ma-247	373	35	6.20	6.20	NUM
ma-247	373	36	)	)	PUNCT
ma-247	373	37	is	be	AUX
ma-247	373	38	such	such	ADJ
ma-247	373	39	that	that	SCONJ
ma-247	373	40	vn(z	vn(z	PUNCT
ma-247	373	41	)	)	PUNCT
ma-247	373	42	=	=	PRON
ma-247	373	43	max1≤j≤n	max1≤j≤n	NOUN
ma-247	373	44	1	1	NUM
ma-247	373	45	j	j	NOUN
ma-247	373	46	log	log	NOUN
ma-247	373	47	|pj(z	|pj(z	PROPN
ma-247	373	48	′	′	PROPN
ma-247	373	49	,	,	PUNCT
ma-247	373	50	z	z	PROPN
ma-247	373	51	,	,	PUNCT
ma-247	373	52	n	n	NOUN
ma-247	373	53	)	)	PUNCT
ma-247	373	54	|+	|+	NOUN
ma-247	374	1	max1≤j≤n	max1≤j≤n	PROPN
ma-247	374	2	1	1	NUM
ma-247	374	3	j	j	PROPN
ma-247	374	4	log	log	VERB
ma-247	374	5	|hj	|hj	PROPN
ma-247	374	6	,	,	PUNCT
ma-247	374	7	δ>0(z)|	δ>0(z)|	ADJ
ma-247	374	8	≤	≤	NUM
ma-247	374	9	βn	βn	NOUN
ma-247	374	10	log	log	NOUN
ma-247	374	11	||z	||z	NOUN
ma-247	374	12	′||+	′||+	PROPN
ma-247	374	13	max1≤j≤n	max1≤j≤n	NOUN
ma-247	374	14	1	1	NUM
ma-247	374	15	j	j	NOUN
ma-247	374	16	log	log	NOUN
ma-247	374	17	|hj(z)|	|hj(z)|	NOUN
ma-247	374	18	=	=	PUNCT
ma-247	374	19	wn(z	wn(z	PRON
ma-247	374	20	)	)	PUNCT
ma-247	374	21	+	+	CCONJ
ma-247	374	22	v	v	ADP
ma-247	374	23	′n(z	′n(z	NOUN
ma-247	374	24	)	)	PUNCT
ma-247	374	25	≤	≤	NOUN
ma-247	374	26	0	0	NUM
ma-247	374	27	.	.	PUNCT
ma-247	375	1	(	(	PUNCT
ma-247	375	2	6.28	6.28	NUM
ma-247	375	3	)	)	PUNCT
ma-247	375	4	hence	hence	ADV
ma-247	375	5	,	,	PUNCT
ma-247	375	6	in	in	ADP
ma-247	375	7	view	view	NOUN
ma-247	375	8	of	of	ADP
ma-247	375	9	(	(	PUNCT
ma-247	375	10	6.21	6.21	NUM
ma-247	375	11	)	)	PUNCT
ma-247	375	12	,	,	PUNCT
ma-247	375	13	(	(	PUNCT
ma-247	375	14	6.24	6.24	NUM
ma-247	375	15	)	)	PUNCT
ma-247	375	16	,	,	PUNCT
ma-247	375	17	(	(	PUNCT
ma-247	375	18	6.26	6.26	NUM
ma-247	375	19	)	)	PUNCT
ma-247	375	20	and	and	CCONJ
ma-247	375	21	(	(	PUNCT
ma-247	375	22	6.28	6.28	NUM
ma-247	375	23	)	)	PUNCT
ma-247	375	24	,	,	PUNCT
ma-247	375	25	we	we	PRON
ma-247	375	26	can	can	AUX
ma-247	375	27	find	find	VERB
ma-247	375	28	constants	constant	NOUN
ma-247	375	29	c6	c6	PROPN
ma-247	375	30	=	=	SYM
ma-247	375	31	c6(k	c6(k	PROPN
ma-247	375	32	)	)	PUNCT
ma-247	375	33	>	>	X
ma-247	375	34	0	0	PUNCT
ma-247	376	1	and	and	CCONJ
ma-247	376	2	c7	c7	PROPN
ma-247	376	3	=	=	SYM
ma-247	376	4	c7(k	c7(k	PROPN
ma-247	376	5	)	)	PUNCT
ma-247	376	6	>	>	X
ma-247	376	7	0	0	PUNCT
ma-247	377	1	such	such	ADJ
ma-247	377	2	that	that	PRON
ma-247	377	3	for	for	ADP
ma-247	377	4	all	all	DET
ma-247	377	5	z	z	NOUN
ma-247	377	6	∈	∈	PROPN
ma-247	377	7	k	k	NOUN
ma-247	377	8	r	r	NOUN
ma-247	377	9	y	y	PROPN
ma-247	377	10	∩	∩	PROPN
ma-247	377	11	x	x	SYM
ma-247	377	12	,	,	PUNCT
ma-247	377	13	the	the	DET
ma-247	377	14	functions	function	NOUN
ma-247	377	15	v	v	VERB
ma-247	377	16	and	and	CCONJ
ma-247	377	17	wn	wn	PROPN
ma-247	377	18	satisfy	satisfy	VERB
ma-247	377	19	the	the	DET
ma-247	377	20	followinginequalities	followinginequalitie	NOUN
ma-247	377	21	c6wn(z)−	c6wn(z)−	PROPN
ma-247	377	22	c7	c7	PROPN
ma-247	377	23	≤	≤	PROPN
ma-247	377	24	v(z	v(z	NOUN
ma-247	377	25	)	)	PUNCT
ma-247	377	26	≤	≤	NOUN
ma-247	377	27	wn(z	wn(z	PUNCT
ma-247	377	28	)	)	PUNCT
ma-247	377	29	≤	≤	NOUN
ma-247	377	30	0	0	NUM
ma-247	377	31	.	.	PUNCT
ma-247	378	1	(	(	PUNCT
ma-247	378	2	6.29)it	6.29)it	NOUN
ma-247	378	3	is	be	AUX
ma-247	378	4	clear	clear	ADJ
ma-247	378	5	that	that	SCONJ
ma-247	378	6	(	(	PUNCT
ma-247	378	7	βn)n	βn)n	PROPN
ma-247	378	8	is	be	AUX
ma-247	378	9	a	a	DET
ma-247	378	10	positive	positive	ADJ
ma-247	378	11	and	and	CCONJ
ma-247	378	12	decreasing	decrease	VERB
ma-247	378	13	sequence	sequence	NOUN
ma-247	378	14	.	.	PUNCT
ma-247	379	1	hence	hence	ADV
ma-247	379	2	it	it	PRON
ma-247	379	3	has	have	VERB
ma-247	379	4	a	a	DET
ma-247	379	5	limit	limit	NOUN
ma-247	379	6	β	β	X
ma-247	379	7	≥	≥	NOUN
ma-247	379	8	0	0	PUNCT
ma-247	379	9	as	as	ADP
ma-247	379	10	n	n	PRON
ma-247	379	11	→	→	PUNCT
ma-247	379	12	+	+	ADJ
ma-247	379	13	∞.in	∞.in	ADJ
ma-247	379	14	view	view	NOUN
ma-247	379	15	of	of	ADP
ma-247	379	16	(	(	PUNCT
ma-247	379	17	6.17	6.17	NUM
ma-247	379	18	)	)	PUNCT
ma-247	379	19	and	and	CCONJ
ma-247	379	20	(	(	PUNCT
ma-247	379	21	6.29	6.29	NUM
ma-247	379	22	)	)	PUNCT
ma-247	379	23	we	we	PRON
ma-247	379	24	have	have	VERB
ma-247	379	25	β	β	X
ma-247	379	26	>	>	X
ma-247	380	1	0	0	X
ma-247	380	2	.	.	PUNCT
ma-247	381	1	indeed	indeed	ADV
ma-247	381	2	,	,	PUNCT
ma-247	381	3	if	if	SCONJ
ma-247	381	4	not	not	PART
ma-247	381	5	,	,	PUNCT
ma-247	381	6	the	the	DET
ma-247	381	7	function	function	NOUN
ma-247	381	8	v	v	NOUN
ma-247	381	9	will	will	AUX
ma-247	381	10	be	be	AUX
ma-247	381	11	bounded	bound	VERB
ma-247	381	12	near	near	ADP
ma-247	381	13	0and	0and	PROPN
ma-247	381	14	then	then	ADV
ma-247	381	15	the	the	DET
ma-247	381	16	point	point	NOUN
ma-247	381	17	z0	z0	PROPN
ma-247	381	18	is	be	AUX
ma-247	381	19	not	not	PART
ma-247	381	20	a	a	DET
ma-247	381	21	singular	singular	ADJ
ma-247	381	22	point	point	NOUN
ma-247	381	23	of	of	ADP
ma-247	381	24	v	v	NOUN
ma-247	381	25	.	.	PUNCT
ma-247	382	1	therefore	therefore	ADV
ma-247	382	2	,	,	PUNCT
ma-247	382	3	by	by	ADP
ma-247	382	4	letting	let	VERB
ma-247	382	5	n	n	X
ma-247	382	6	→	→	PUNCT
ma-247	382	7	+	+	NUM
ma-247	382	8	∞	∞	PROPN
ma-247	382	9	in	in	ADP
ma-247	382	10	(	(	PUNCT
ma-247	382	11	6.29	6.29	NUM
ma-247	382	12	)	)	PUNCT
ma-247	382	13	,	,	PUNCT
ma-247	382	14	we	we	PRON
ma-247	382	15	canfind	canfind	VERB
ma-247	382	16	constants	constant	NOUN
ma-247	382	17	β	β	X
ma-247	382	18	>	>	X
ma-247	382	19	0	0	PUNCT
ma-247	382	20	and	and	CCONJ
ma-247	382	21	c8	c8	PROPN
ma-247	382	22	=	=	SYM
ma-247	382	23	c8(k	c8(k	PROPN
ma-247	382	24	)	)	PUNCT
ma-247	382	25	>	>	X
ma-247	382	26	0	0	NUM
ma-247	382	27	such	such	ADJ
ma-247	382	28	that	that	PRON
ma-247	382	29	for	for	ADP
ma-247	382	30	all	all	DET
ma-247	382	31	z	z	NOUN
ma-247	382	32	=	=	SYM
ma-247	382	33	(	(	PUNCT
ma-247	382	34	z	z	NOUN
ma-247	382	35	′	′	NUM
ma-247	382	36	,	,	PUNCT
ma-247	382	37	zn	zn	X
ma-247	382	38	)	)	PUNCT
ma-247	382	39	∈	∈	PROPN
ma-247	383	1	k	k	NOUN
ma-247	383	2	r	r	NOUN
ma-247	383	3	y	y	PROPN
ma-247	383	4	∩x	∩x	NOUN
ma-247	383	5	,	,	PUNCT
ma-247	383	6	the	the	DET
ma-247	383	7	function	function	NOUN
ma-247	383	8	v	v	NOUN
ma-247	383	9	satisfies	satisfy	VERB
ma-247	383	10	the	the	DET
ma-247	383	11	following	follow	VERB
ma-247	383	12	inequality	inequality	NOUN
ma-247	383	13	exp	exp	NOUN
ma-247	383	14	(	(	PUNCT
ma-247	383	15	−v(z	−v(z	PROPN
ma-247	383	16	)	)	PUNCT
ma-247	383	17	)	)	PUNCT
ma-247	383	18	≤	≤	NUM
ma-247	383	19	c8||z	c8||z	NOUN
ma-247	383	20	′||−β	′||−β	NUM
ma-247	383	21	.	.	PUNCT
ma-247	384	1	(	(	PUNCT
ma-247	384	2	6.30	6.30	NUM
ma-247	384	3	)	)	PUNCT
ma-247	384	4	taking	take	VERB
ma-247	384	5	α	α	PRON
ma-247	384	6	>	>	X
ma-247	384	7	0	0	PUNCT
ma-247	385	1	so	so	SCONJ
ma-247	385	2	that	that	SCONJ
ma-247	385	3	αβ	αβ	INTJ
ma-247	385	4	∈	∈	NOUN
ma-247	385	5	(	(	PUNCT
ma-247	385	6	0	0	NUM
ma-247	385	7	,	,	PUNCT
ma-247	385	8	2q	2q	NUM
ma-247	385	9	−	−	PROPN
ma-247	385	10	2	2	NUM
ma-247	385	11	)	)	PUNCT
ma-247	385	12	,	,	PUNCT
ma-247	385	13	hence	hence	ADV
ma-247	385	14	z	z	NOUN
ma-247	385	15	7→	7→	NUM
ma-247	385	16	1	1	NUM
ma-247	385	17	||z	||z	NOUN
ma-247	385	18	||αβ	||αβ	ADV
ma-247	385	19	∈	∈	PROPN
ma-247	385	20	l	l	NOUN
ma-247	385	21	1	1	NUM
ma-247	385	22	loc(cq−1	loc(cq−1	NOUN
ma-247	385	23	)	)	PUNCT
ma-247	385	24	,	,	PUNCT
ma-247	385	25	and	and	CCONJ
ma-247	385	26	investigating	investigate	VERB
ma-247	385	27	(	(	PUNCT
ma-247	385	28	6.17)with	6.17)with	NUM
ma-247	385	29	(	(	PUNCT
ma-247	385	30	6.30	6.30	NUM
ma-247	385	31	)	)	PUNCT
ma-247	385	32	,	,	PUNCT
ma-247	385	33	we	we	PRON
ma-247	385	34	get	get	VERB
ma-247	385	35	the	the	DET
ma-247	385	36	following∫	following∫	NOUN
ma-247	385	37	z∈k\x∩y	z∈k\x∩y	PROPN
ma-247	385	38	exp(−αv(z	exp(−αv(z	NOUN
ma-247	385	39	)	)	PUNCT
ma-247	385	40	)	)	PUNCT
ma-247	386	1	≤	≤	PROPN
ma-247	386	2	c8	c8	PROPN
ma-247	386	3	∫	∫	PROPN
ma-247	387	1	z∈∆q−1	z∈∆q−1	X
ma-247	387	2	dλq−1(z	dλq−1(z	PROPN
ma-247	387	3	)	)	PUNCT
ma-247	387	4	||z	||z	NOUN
ma-247	387	5	||αβ	||αβ	ADJ
ma-247	387	6	<	<	X
ma-247	387	7	∞.	∞.	PROPN
ma-247	387	8	(	(	PUNCT
ma-247	387	9	6.31	6.31	NUM
ma-247	387	10	)	)	PUNCT
ma-247	387	11	consequently	consequently	ADV
ma-247	387	12	,	,	PUNCT
ma-247	387	13	our	our	PRON
ma-247	387	14	proof	proof	NOUN
ma-247	387	15	is	be	AUX
ma-247	387	16	achieved	achieve	VERB
ma-247	387	17	thanks	thank	NOUN
ma-247	387	18	to	to	ADP
ma-247	387	19	(	(	PUNCT
ma-247	387	20	6.31	6.31	NUM
ma-247	387	21	)	)	PUNCT
ma-247	387	22	.	.	PUNCT
ma-247	388	1	�	�	PROPN
ma-247	388	2	remark	remark	VERB
ma-247	388	3	6.2	6.2	NUM
ma-247	388	4	.	.	PUNCT
ma-247	389	1	theorem	theorem	VERB
ma-247	389	2	6.1	6.1	NUM
ma-247	389	3	generalizes	generalize	VERB
ma-247	389	4	a	a	DET
ma-247	389	5	result	result	NOUN
ma-247	389	6	in	in	ADP
ma-247	389	7	[	[	X
ma-247	389	8	9	9	NUM
ma-247	389	9	]	]	PUNCT
ma-247	389	10	showing	show	VERB
ma-247	389	11	that	that	SCONJ
ma-247	389	12	,	,	PUNCT
ma-247	389	13	if	if	SCONJ
ma-247	389	14	y	y	PROPN
ma-247	389	15	=	=	PRON
ma-247	389	16	{	{	PUNCT
ma-247	389	17	f	f	NOUN
ma-247	389	18	=	=	SYM
ma-247	389	19	0	0	NUM
ma-247	389	20	}	}	PUNCT
ma-247	389	21	is	be	AUX
ma-247	389	22	a	a	DET
ma-247	389	23	hyper	hyper	NOUN
ma-247	389	24	-	-	NOUN
ma-247	389	25	surface	surface	NOUN
ma-247	389	26	in	in	ADP
ma-247	389	27	ω	ω	PROPN
ma-247	389	28	given	give	VERB
ma-247	389	29	by	by	ADP
ma-247	389	30	a	a	DET
ma-247	389	31	holomorphic	holomorphic	ADJ
ma-247	389	32	function	function	NOUN
ma-247	389	33	not	not	PART
ma-247	389	34	identically	identically	ADV
ma-247	389	35	vanishing	vanish	VERB
ma-247	389	36	and	and	CCONJ
ma-247	389	37	if	if	SCONJ
ma-247	389	38	y	y	PROPN
ma-247	389	39	yields	yield	VERB
ma-247	389	40	a	a	DET
ma-247	389	41	completeintersection	completeintersection	NOUN
ma-247	389	42	with	with	ADP
ma-247	389	43	another	another	DET
ma-247	389	44	analytic	analytic	ADJ
ma-247	389	45	subset	subset	NOUN
ma-247	389	46	x	x	X
ma-247	389	47	⊂	⊂	PROPN
ma-247	389	48	ω	ω	PROPN
ma-247	389	49	,	,	PUNCT
ma-247	389	50	then	then	ADV
ma-247	389	51	there	there	PRON
ma-247	389	52	exists	exist	VERB
ma-247	389	53	δ	δ	PROPN
ma-247	389	54	>	>	X
ma-247	389	55	0	0	NUM
ma-247	390	1	such	such	ADJ
ma-247	390	2	that	that	SCONJ
ma-247	390	3	the	the	DET
ma-247	390	4	coefficientsof	coefficientsof	NOUN
ma-247	390	5	the	the	DET
ma-247	390	6	current	current	ADJ
ma-247	390	7	log	log	NOUN
ma-247	390	8	|f	|f	PROPN
ma-247	390	9	|[x	|[x	PROPN
ma-247	390	10	]	]	PUNCT
ma-247	390	11	lies	lie	VERB
ma-247	390	12	in	in	ADP
ma-247	390	13	l1+δ	l1+δ	PROPN
ma-247	390	14	loc	loc	PROPN
ma-247	390	15	(	(	PUNCT
ma-247	390	16	x	x	NOUN
ma-247	390	17	)	)	PUNCT
ma-247	390	18	.	.	PUNCT
ma-247	391	1	moreover	moreover	ADV
ma-247	391	2	,	,	PUNCT
ma-247	391	3	there	there	PRON
ma-247	391	4	exists	exist	VERB
ma-247	391	5	α	α	PROPN
ma-247	391	6	>	>	X
ma-247	391	7	0	0	NUM
ma-247	391	8	,	,	PUNCT
ma-247	391	9	such	such	ADJ
ma-247	391	10	that	that	SCONJ
ma-247	391	11	the	the	DET
ma-247	391	12	coefficients	coefficient	NOUN
ma-247	391	13	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	391	14	eur	eur	PROPN
ma-247	391	15	.	.	PUNCT
ma-247	392	1	j.	j.	PROPN
ma-247	392	2	math	math	PROPN
ma-247	392	3	.	.	PUNCT
ma-247	393	1	anal	anal	PROPN
ma-247	393	2	.	.	PUNCT
ma-247	394	1	10.28924	10.28924	NUM
ma-247	394	2	/	/	SYM
ma-247	394	3	ada	ada	PROPN
ma-247	394	4	/	/	SYM
ma-247	394	5	ma.4.23	ma.4.23	NOUN
ma-247	394	6	14of	14of	ADJ
ma-247	394	7	the	the	DET
ma-247	394	8	current	current	NOUN
ma-247	394	9	|f	|f	PROPN
ma-247	395	1	|−α[x	|−α[x	NOUN
ma-247	395	2	]	]	PUNCT
ma-247	395	3	lie	lie	NOUN
ma-247	395	4	in	in	ADP
ma-247	395	5	l1	l1	PROPN
ma-247	395	6	loc(x	loc(x	PROPN
ma-247	395	7	)	)	PUNCT
ma-247	395	8	.	.	PUNCT
ma-247	396	1	if	if	SCONJ
ma-247	396	2	we	we	PRON
ma-247	396	3	replace	replace	VERB
ma-247	396	4	the	the	DET
ma-247	396	5	hypersurface	hypersurface	NOUN
ma-247	396	6	y	y	PROPN
ma-247	396	7	=	=	PUNCT
ma-247	396	8	{	{	PUNCT
ma-247	396	9	f	f	NOUN
ma-247	396	10	=	=	SYM
ma-247	396	11	0	0	NUM
ma-247	396	12	}	}	PUNCT
ma-247	396	13	by	by	ADP
ma-247	396	14	a	a	DET
ma-247	396	15	positive	positive	ADJ
ma-247	396	16	(	(	PUNCT
ma-247	396	17	1	1	NUM
ma-247	396	18	,	,	PUNCT
ma-247	396	19	1)-closed	1)-closed	NUM
ma-247	396	20	current	current	NOUN
ma-247	396	21	of	of	ADP
ma-247	396	22	the	the	DET
ma-247	396	23	form	form	NOUN
ma-247	396	24	t	t	NOUN
ma-247	396	25	=	=	SYM
ma-247	396	26	ddcv	ddcv	PROPN
ma-247	396	27	for	for	ADP
ma-247	396	28	some	some	DET
ma-247	396	29	psh	psh	NOUN
ma-247	396	30	function	function	NOUN
ma-247	396	31	v	v	NOUN
ma-247	396	32	,	,	PUNCT
ma-247	396	33	then	then	ADV
ma-247	396	34	we	we	PRON
ma-247	396	35	provide	provide	VERB
ma-247	396	36	theorem	theorem	VERB
ma-247	396	37	1.4.note	1.4.note	NUM
ma-247	396	38	that	that	SCONJ
ma-247	396	39	we	we	PRON
ma-247	396	40	invite	invite	VERB
ma-247	396	41	interested	interested	ADJ
ma-247	396	42	readers	reader	NOUN
ma-247	396	43	to	to	PART
ma-247	396	44	proceed	proceed	VERB
ma-247	396	45	defining	define	VERB
ma-247	396	46	and	and	CCONJ
ma-247	396	47	investigating	investigate	VERB
ma-247	396	48	quaternionic	quaternionic	ADJ
ma-247	396	49	hyper	hyper	NOUN
ma-247	396	50	-	-	NOUN
ma-247	396	51	surfaces	surface	NOUN
ma-247	396	52	and	and	CCONJ
ma-247	396	53	quaternionic	quaternionic	ADJ
ma-247	396	54	analytic	analytic	ADJ
ma-247	396	55	sets	set	NOUN
ma-247	396	56	in	in	ADP
ma-247	396	57	the	the	DET
ma-247	396	58	space	space	NOUN
ma-247	396	59	hc	hc	PROPN
ma-247	396	60	recently	recently	ADV
ma-247	396	61	studied	study	VERB
ma-247	396	62	in	in	ADP
ma-247	396	63	[	[	X
ma-247	396	64	13	13	NUM
ma-247	396	65	]	]	PUNCT
ma-247	396	66	and	and	CCONJ
ma-247	396	67	find	find	VERB
ma-247	396	68	similarresults	similarresult	NOUN
ma-247	396	69	as	as	SCONJ
ma-247	396	70	given	give	VERB
ma-247	396	71	in	in	ADP
ma-247	396	72	this	this	DET
ma-247	396	73	paper	paper	NOUN
ma-247	396	74	.	.	PUNCT
ma-247	397	1	in	in	ADP
ma-247	397	2	the	the	DET
ma-247	397	3	direction	direction	NOUN
ma-247	397	4	of	of	ADP
ma-247	397	5	theorem	theorem	ADJ
ma-247	397	6	1.3	1.3	NUM
ma-247	397	7	and	and	CCONJ
ma-247	397	8	proceeding	proceeding	NOUN
ma-247	397	9	as	as	ADP
ma-247	397	10	in	in	ADP
ma-247	397	11	the	the	DET
ma-247	397	12	proof	proof	NOUN
ma-247	397	13	of	of	ADP
ma-247	397	14	theorem	theorem	ADJ
ma-247	397	15	3.3	3.3	NUM
ma-247	397	16	in	in	ADP
ma-247	397	17	[	[	PUNCT
ma-247	397	18	9	9	NUM
ma-247	397	19	]	]	PUNCT
ma-247	397	20	,	,	PUNCT
ma-247	397	21	we	we	PRON
ma-247	397	22	finallygive	finallygive	VERB
ma-247	397	23	the	the	DET
ma-247	397	24	proof	proof	NOUN
ma-247	397	25	of	of	ADP
ma-247	397	26	theorem	theorem	ADJ
ma-247	397	27	1.5	1.5	NUM
ma-247	397	28	.	.	PUNCT
ma-247	398	1	proof	proof	NOUN
ma-247	398	2	.	.	PUNCT
ma-247	399	1	we	we	PRON
ma-247	399	2	consider	consider	VERB
ma-247	399	3	the	the	DET
ma-247	399	4	analytic	analytic	ADJ
ma-247	399	5	subsets	subset	NOUN
ma-247	399	6	x	x	SYM
ma-247	399	7	∩	∩	PROPN
ma-247	399	8	π−1(a	π−1(a	PROPN
ma-247	399	9	)	)	PUNCT
ma-247	399	10	,	,	PUNCT
ma-247	399	11	y	y	PROPN
ma-247	399	12	∩	∩	PROPN
ma-247	399	13	π−1(a	π−1(a	PROPN
ma-247	399	14	)	)	PUNCT
ma-247	399	15	and	and	CCONJ
ma-247	399	16	e	e	X
ma-247	399	17	=	=	PRON
ma-247	399	18	{	{	PUNCT
ma-247	399	19	a	a	DET
ma-247	399	20	∈	∈	NOUN
ma-247	399	21	∆k	∆k	NOUN
ma-247	399	22	:	:	PUNCT
ma-247	399	23	dimc(x	dimc(x	VERB
ma-247	399	24	∩	∩	NOUN
ma-247	399	25	π−1(a))×	π−1(a))×	PROPN
ma-247	399	26	(	(	PUNCT
ma-247	399	27	y	y	PROPN
ma-247	399	28	∩	∩	PROPN
ma-247	399	29	π−1(a	π−1(a	PROPN
ma-247	399	30	)	)	PUNCT
ma-247	399	31	)	)	PUNCT
ma-247	399	32	>	>	X
ma-247	400	1	p	p	X
ma-247	401	1	+	+	NOUN
ma-247	401	2	q	q	NOUN
ma-247	401	3	−	−	PROPN
ma-247	401	4	k	k	NOUN
ma-247	401	5	}	}	PUNCT
ma-247	401	6	.	.	PUNCT
ma-247	402	1	by	by	ADP
ma-247	402	2	proposition	proposition	NOUN
ma-247	402	3	3.1	3.1	NUM
ma-247	402	4	,	,	PUNCT
ma-247	402	5	e	e	X
ma-247	402	6	is	be	AUX
ma-247	402	7	contained	contain	VERB
ma-247	402	8	in	in	ADP
ma-247	402	9	a	a	DET
ma-247	402	10	countable	countable	ADJ
ma-247	402	11	union	union	NOUN
ma-247	402	12	of	of	ADP
ma-247	402	13	analytic	analytic	ADJ
ma-247	402	14	subsets	subset	NOUN
ma-247	402	15	of	of	ADP
ma-247	402	16	∆k	∆k	PROPN
ma-247	402	17	of	of	ADP
ma-247	402	18	dimension	dimension	NOUN
ma-247	402	19	≤	≤	PUNCT
ma-247	402	20	k−1	k−1	PROPN
ma-247	402	21	.	.	PUNCT
ma-247	403	1	therefore	therefore	ADV
ma-247	403	2	the	the	DET
ma-247	403	3	set	set	NOUN
ma-247	403	4	e	e	NOUN
ma-247	403	5	is	be	AUX
ma-247	403	6	pluripolar	pluripolar	ADJ
ma-247	403	7	.	.	PUNCT
ma-247	404	1	we	we	PRON
ma-247	404	2	work	work	VERB
ma-247	404	3	as	as	ADP
ma-247	404	4	in	in	ADP
ma-247	404	5	[	[	X
ma-247	404	6	9	9	NUM
ma-247	404	7	]	]	X
ma-247	404	8	(	(	PUNCT
ma-247	404	9	proof	proof	NOUN
ma-247	404	10	of	of	ADP
ma-247	404	11	theorem	theorem	ADJ
ma-247	404	12	3.3	3.3	NUM
ma-247	404	13	)	)	PUNCT
ma-247	404	14	around	around	ADP
ma-247	404	15	a	a	DET
ma-247	404	16	point	point	NOUN
ma-247	404	17	a	a	DET
ma-247	404	18	6∈	6∈	PROPN
ma-247	404	19	e.	e.	PROPN
ma-247	404	20	�	�	PROPN
ma-247	404	21	competing	compete	VERB
ma-247	404	22	interests	interest	NOUN
ma-247	404	23	the	the	DET
ma-247	404	24	author(s	author(s	PROPN
ma-247	404	25	)	)	PUNCT
ma-247	404	26	declare(s	declare(s	NOUN
ma-247	404	27	)	)	PUNCT
ma-247	404	28	that	that	SCONJ
ma-247	404	29	there	there	PRON
ma-247	404	30	is	be	VERB
ma-247	404	31	no	no	DET
ma-247	404	32	conflict	conflict	NOUN
ma-247	404	33	of	of	ADP
ma-247	404	34	interest	interest	NOUN
ma-247	404	35	regarding	regard	VERB
ma-247	404	36	the	the	DET
ma-247	404	37	publication	publication	NOUN
ma-247	404	38	of	of	ADP
ma-247	404	39	thispaper	thispaper	NOUN
ma-247	404	40	.	.	PUNCT
ma-247	405	1	references	reference	NOUN
ma-247	405	2	[	[	X
ma-247	405	3	1	1	NUM
ma-247	405	4	]	]	PUNCT
ma-247	405	5	e.	e.	PROPN
ma-247	405	6	bedford	bedford	PROPN
ma-247	405	7	,	,	PUNCT
ma-247	405	8	b.a	b.a	PROPN
ma-247	405	9	.	.	PROPN
ma-247	405	10	taylor	taylor	PROPN
ma-247	405	11	,	,	PUNCT
ma-247	405	12	a	a	DET
ma-247	405	13	new	new	ADJ
ma-247	405	14	capacity	capacity	NOUN
ma-247	405	15	for	for	ADP
ma-247	405	16	plurisubharmonic	plurisubharmonic	ADJ
ma-247	405	17	functions	function	NOUN
ma-247	405	18	,	,	PUNCT
ma-247	405	19	acta	acta	PROPN
ma-247	405	20	math	math	PROPN
ma-247	405	21	.	.	PUNCT
ma-247	406	1	149	149	NUM
ma-247	406	2	(	(	PUNCT
ma-247	406	3	1982	1982	NUM
ma-247	406	4	)	)	PUNCT
ma-247	407	1	1–41.[2	1–41.[2	NUM
ma-247	407	2	]	]	X
ma-247	407	3	h.	h.	PROPN
ma-247	407	4	ben	ben	PROPN
ma-247	407	5	messaoud	messaoud	PROPN
ma-247	407	6	,	,	PUNCT
ma-247	407	7	h.	h.	PROPN
ma-247	407	8	el	el	PROPN
ma-247	407	9	mir	mir	PROPN
ma-247	407	10	,	,	PUNCT
ma-247	407	11	operateur	operateur	ADJ
ma-247	407	12	de	de	X
ma-247	407	13	monge	monge	PROPN
ma-247	407	14	-	-	PUNCT
ma-247	407	15	ampre	ampre	PROPN
ma-247	407	16	et	et	PROPN
ma-247	407	17	formule	formule	PROPN
ma-247	407	18	de	de	PROPN
ma-247	407	19	tranchage	tranchage	PROPN
ma-247	407	20	pour	pour	PROPN
ma-247	407	21	un	un	PROPN
ma-247	407	22	courant	courant	PROPN
ma-247	407	23	positifs	positifs	PROPN
ma-247	407	24	fermé	fermé	PROPN
ma-247	407	25	,	,	PUNCT
ma-247	407	26	c.r.a.s	c.r.a.s	PROPN
ma-247	407	27	.	.	PROPN
ma-247	407	28	paris	paris	PROPN
ma-247	407	29	,	,	PUNCT
ma-247	407	30	t316	t316	PROPN
ma-247	407	31	i	i	PRON
ma-247	407	32	(	(	PUNCT
ma-247	407	33	1993	1993	NUM
ma-247	407	34	)	)	PUNCT
ma-247	407	35	,	,	PUNCT
ma-247	407	36	1173–1176.[3	1173–1176.[3	NUM
ma-247	407	37	]	]	X
ma-247	407	38	h.	h.	PROPN
ma-247	407	39	ben	ben	PROPN
ma-247	407	40	messaoud	messaoud	PROPN
ma-247	407	41	,	,	PUNCT
ma-247	407	42	h.	h.	PROPN
ma-247	407	43	el	el	PROPN
ma-247	407	44	mir	mir	PROPN
ma-247	407	45	,	,	PUNCT
ma-247	407	46	tranchage	tranchage	NOUN
ma-247	407	47	et	et	PROPN
ma-247	407	48	prolongement	prolongement	PROPN
ma-247	407	49	des	des	PROPN
ma-247	407	50	courants	courants	PROPN
ma-247	407	51	positifs	positifs	PROPN
ma-247	407	52	fermés	fermés	PROPN
ma-247	407	53	,	,	PUNCT
ma-247	407	54	math	math	NOUN
ma-247	407	55	.	.	PUNCT
ma-247	408	1	ann	ann	PROPN
ma-247	408	2	.	.	PROPN
ma-247	409	1	307	307	NUM
ma-247	409	2	(	(	PUNCT
ma-247	409	3	1997),473–487.[4	1997),473–487.[4	NUM
ma-247	409	4	]	]	X
ma-247	409	5	h.-j	h.-j	PROPN
ma-247	409	6	.	.	PUNCT
ma-247	409	7	bremermann	bremermann	PROPN
ma-247	409	8	,	,	PUNCT
ma-247	409	9	on	on	ADP
ma-247	409	10	the	the	DET
ma-247	409	11	conjecture	conjecture	NOUN
ma-247	409	12	of	of	ADP
ma-247	409	13	the	the	DET
ma-247	409	14	equivalence	equivalence	NOUN
ma-247	409	15	of	of	ADP
ma-247	409	16	the	the	DET
ma-247	409	17	plurisubharmonic	plurisubharmonic	ADJ
ma-247	409	18	functions	function	NOUN
ma-247	409	19	and	and	CCONJ
ma-247	409	20	the	the	DET
ma-247	409	21	hartogs	hartog	NOUN
ma-247	409	22	functions	function	NOUN
ma-247	409	23	,	,	PUNCT
ma-247	409	24	math	math	NOUN
ma-247	409	25	.	.	PUNCT
ma-247	410	1	ann	ann	PROPN
ma-247	410	2	.	.	PROPN
ma-247	411	1	131	131	NUM
ma-247	411	2	(	(	PUNCT
ma-247	411	3	1956	1956	NUM
ma-247	411	4	)	)	PUNCT
ma-247	411	5	,	,	PUNCT
ma-247	411	6	76–86.[5	76–86.[5	NUM
ma-247	411	7	]	]	X
ma-247	411	8	c.o	c.o	PROPN
ma-247	411	9	.	.	PROPN
ma-247	411	10	kiselman	kiselman	PROPN
ma-247	411	11	,	,	PUNCT
ma-247	411	12	sur	sur	PROPN
ma-247	411	13	la	la	PROPN
ma-247	411	14	définition	définition	PROPN
ma-247	411	15	de	de	X
ma-247	411	16	l’opérateur	l’opérateur	X
ma-247	411	17	de	de	X
ma-247	411	18	monge	monge	VERB
ma-247	411	19	-	-	PUNCT
ma-247	411	20	ampère	ampère	NOUN
ma-247	411	21	complexe	complexe	PROPN
ma-247	411	22	,	,	PUNCT
ma-247	411	23	analyse	analyse	PROPN
ma-247	411	24	complexe	complexe	PROPN
ma-247	411	25	,	,	PUNCT
ma-247	411	26	proceedings	proceeding	NOUN
ma-247	411	27	ofjournées	ofjournées	VERB
ma-247	411	28	fermat	fermat	PROPN
ma-247	411	29	(	(	PUNCT
ma-247	411	30	smf	smf	PROPN
ma-247	411	31	)	)	PUNCT
ma-247	411	32	,	,	PUNCT
ma-247	411	33	toulouse	toulouse	NOUN
ma-247	411	34	,	,	PUNCT
ma-247	411	35	may	may	AUX
ma-247	411	36	,	,	PUNCT
ma-247	411	37	1983.[6	1983.[6	NUM
ma-247	411	38	]	]	X
ma-247	411	39	j.p	j.p	PROPN
ma-247	411	40	.	.	PROPN
ma-247	411	41	demailly	demailly	ADV
ma-247	411	42	,	,	PUNCT
ma-247	411	43	complex	complex	ADJ
ma-247	411	44	analytic	analytic	ADJ
ma-247	411	45	and	and	CCONJ
ma-247	411	46	differential	differential	ADJ
ma-247	411	47	geometry	geometry	NOUN
ma-247	411	48	,	,	PUNCT
ma-247	411	49	e	e	NOUN
ma-247	411	50	-	-	NOUN
ma-247	411	51	book	book	NOUN
ma-247	411	52	,	,	PUNCT
ma-247	411	53	2007.[7	2007.[7	NUM
ma-247	411	54	]	]	X
ma-247	411	55	h.	h.	PROPN
ma-247	411	56	federer	federer	PROPN
ma-247	411	57	,	,	PUNCT
ma-247	411	58	geometric	geometric	ADJ
ma-247	411	59	measure	measure	NOUN
ma-247	411	60	theory	theory	NOUN
ma-247	411	61	,	,	PUNCT
ma-247	411	62	springer	springer	NOUN
ma-247	411	63	,	,	PUNCT
ma-247	411	64	1969.[8	1969.[8	NUM
ma-247	411	65	]	]	PUNCT
ma-247	411	66	r.	r.	PROPN
ma-247	411	67	harvey	harvey	PROPN
ma-247	411	68	,	,	PUNCT
ma-247	411	69	b.	b.	PROPN
ma-247	411	70	shiffman	shiffman	PROPN
ma-247	411	71	,	,	PUNCT
ma-247	411	72	a	a	DET
ma-247	411	73	characterization	characterization	NOUN
ma-247	411	74	of	of	ADP
ma-247	411	75	holomorphic	holomorphic	ADJ
ma-247	411	76	chains	chain	NOUN
ma-247	411	77	,	,	PUNCT
ma-247	411	78	ann	ann	PROPN
ma-247	411	79	.	.	PROPN
ma-247	411	80	math	math	PROPN
ma-247	411	81	.	.	PUNCT
ma-247	412	1	28	28	NUM
ma-247	412	2	(	(	PUNCT
ma-247	412	3	1974	1974	NUM
ma-247	412	4	)	)	PUNCT
ma-247	412	5	,	,	PUNCT
ma-247	412	6	553–587.[9	553–587.[9	NUM
ma-247	412	7	]	]	X
ma-247	412	8	h.	h.	PROPN
ma-247	412	9	khedhiri	khedhiri	PROPN
ma-247	412	10	,	,	PUNCT
ma-247	412	11	wedge	wedge	NOUN
ma-247	412	12	product	product	NOUN
ma-247	412	13	of	of	ADP
ma-247	412	14	currents	current	NOUN
ma-247	412	15	,	,	PUNCT
ma-247	412	16	lobachevskii	lobachevskii	VERB
ma-247	412	17	j.	j.	PROPN
ma-247	412	18	math	math	PROPN
ma-247	412	19	.	.	PUNCT
ma-247	413	1	31	31	NUM
ma-247	413	2	(	(	PUNCT
ma-247	413	3	2010	2010	NUM
ma-247	413	4	)	)	PUNCT
ma-247	413	5	,	,	PUNCT
ma-247	414	1	224–231.[10	224–231.[10	NUM
ma-247	414	2	]	]	X
ma-247	414	3	h.	h.	PROPN
ma-247	414	4	khedhiri	khedhiri	PROPN
ma-247	414	5	,	,	PUNCT
ma-247	414	6	slicing	slicing	NOUN
ma-247	414	7	of	of	ADP
ma-247	414	8	currents	current	NOUN
ma-247	414	9	associated	associate	VERB
ma-247	414	10	to	to	ADP
ma-247	414	11	a	a	DET
ma-247	414	12	plurisubharmonic	plurisubharmonic	ADJ
ma-247	414	13	function	function	NOUN
ma-247	414	14	,	,	PUNCT
ma-247	414	15	punj	punj	PROPN
ma-247	414	16	univ	univ	PROPN
ma-247	414	17	.	.	PUNCT
ma-247	415	1	j.	j.	PROPN
ma-247	415	2	math	math	PROPN
ma-247	415	3	.	.	PUNCT
ma-247	416	1	47	47	NUM
ma-247	416	2	(	(	PUNCT
ma-247	416	3	2015	2015	NUM
ma-247	416	4	)	)	PUNCT
ma-247	416	5	,	,	PUNCT
ma-247	416	6	21–34.[11	21–34.[11	NUM
ma-247	416	7	]	]	X
ma-247	416	8	h.	h.	PROPN
ma-247	416	9	khedhiri	khedhiri	PROPN
ma-247	416	10	,	,	PUNCT
ma-247	416	11	ϕ-slicing	ϕ-slice	VERB
ma-247	416	12	results	result	NOUN
ma-247	416	13	for	for	ADP
ma-247	416	14	negative	negative	ADJ
ma-247	416	15	plurisubharmonic	plurisubharmonic	ADJ
ma-247	416	16	currents	current	NOUN
ma-247	416	17	,	,	PUNCT
ma-247	416	18	uzbek	uzbek	PROPN
ma-247	416	19	j.	j.	PROPN
ma-247	416	20	math	math	PROPN
ma-247	416	21	.	.	PUNCT
ma-247	417	1	68	68	NUM
ma-247	417	2	(	(	PUNCT
ma-247	417	3	2024	2024	NUM
ma-247	417	4	)	)	PUNCT
ma-247	417	5	,	,	PUNCT
ma-247	417	6	102	102	NUM
ma-247	417	7	-	-	SYM
ma-247	417	8	112.[12	112.[12	NUM
ma-247	417	9	]	]	PUNCT
ma-247	417	10	h.	h.	PROPN
ma-247	417	11	khedhiri	khedhiri	PROPN
ma-247	417	12	,	,	PUNCT
ma-247	417	13	on	on	ADP
ma-247	417	14	construction	construction	NOUN
ma-247	417	15	of	of	ADP
ma-247	417	16	positive	positive	ADJ
ma-247	417	17	closed	closed	ADJ
ma-247	417	18	currents	current	NOUN
ma-247	417	19	with	with	ADP
ma-247	417	20	prescribed	prescribed	ADJ
ma-247	417	21	lelong	lelong	PROPN
ma-247	417	22	numbers	number	NOUN
ma-247	417	23	,	,	PUNCT
ma-247	417	24	j.	j.	PROPN
ma-247	417	25	sib	sib	PROPN
ma-247	417	26	.	.	PUNCT
ma-247	418	1	fed	fed	PROPN
ma-247	418	2	.	.	PUNCT
ma-247	419	1	univ	univ	PROPN
ma-247	419	2	.	.	PUNCT
ma-247	420	1	math.phys	math.phy	NOUN
ma-247	420	2	.	.	PROPN
ma-247	420	3	13	13	NUM
ma-247	420	4	(	(	PUNCT
ma-247	420	5	2020	2020	NUM
ma-247	420	6	)	)	PUNCT
ma-247	420	7	,	,	PUNCT
ma-247	420	8	331	331	NUM
ma-247	420	9	-	-	SYM
ma-247	420	10	341.[13	341.[13	NUM
ma-247	420	11	]	]	PUNCT
ma-247	420	12	h.	h.	PROPN
ma-247	420	13	khedhiri	khedhiri	PROPN
ma-247	420	14	,	,	PUNCT
ma-247	420	15	t.	t.	PROPN
ma-247	420	16	mkademi	mkademi	PROPN
ma-247	420	17	,	,	PUNCT
ma-247	420	18	foundational	foundational	ADJ
ma-247	420	19	aspects	aspect	NOUN
ma-247	420	20	of	of	ADP
ma-247	420	21	a	a	DET
ma-247	420	22	new	new	ADJ
ma-247	420	23	matrix	matrix	NOUN
ma-247	420	24	holomorphic	holomorphic	ADJ
ma-247	420	25	structure	structure	NOUN
ma-247	420	26	,	,	PUNCT
ma-247	420	27	arab	arab	PROPN
ma-247	420	28	j.	j.	PROPN
ma-247	420	29	math	math	PROPN
ma-247	420	30	.	.	PUNCT
ma-247	421	1	sci	sci	PROPN
ma-247	421	2	.	.	PUNCT
ma-247	422	1	(	(	PUNCT
ma-247	422	2	2024).[14	2024).[14	NUM
ma-247	422	3	]	]	X
ma-247	422	4	p.	p.	PROPN
ma-247	422	5	lelong	lelong	PROPN
ma-247	422	6	,	,	PUNCT
ma-247	422	7	intégration	intégration	PROPN
ma-247	422	8	sur	sur	PROPN
ma-247	422	9	un	un	PROPN
ma-247	422	10	ensemble	ensemble	ADJ
ma-247	422	11	analytique	analytique	ADJ
ma-247	422	12	complexe	complexe	PROPN
ma-247	422	13	,	,	PUNCT
ma-247	422	14	bull	bull	NOUN
ma-247	422	15	.	.	PUNCT
ma-247	423	1	soc	soc	PROPN
ma-247	423	2	.	.	PUNCT
ma-247	424	1	math	math	PROPN
ma-247	424	2	france	france	PROPN
ma-247	424	3	.	.	PUNCT
ma-247	425	1	853	853	NUM
ma-247	425	2	(	(	PUNCT
ma-247	425	3	1957	1957	NUM
ma-247	425	4	)	)	PUNCT
ma-247	425	5	,	,	PUNCT
ma-247	425	6	239–262	239–262	NUM
ma-247	425	7	.	.	PUNCT
ma-247	426	1	https://doi.org/10.28924/ada/ma.4.23	https://doi.org/10.28924/ada/ma.4.23	PRON
ma-247	426	2	1	1	NUM
ma-247	426	3	.	.	PUNCT
ma-247	426	4	introduction	introduction	NOUN
ma-247	426	5	2	2	NUM
ma-247	426	6	.	.	PUNCT
ma-247	426	7	preliminaries	preliminary	NOUN
ma-247	426	8	3	3	NUM
ma-247	426	9	.	.	PUNCT
ma-247	426	10	proof	proof	NOUN
ma-247	426	11	of	of	ADP
ma-247	426	12	theorem	theorem	ADJ
ma-247	426	13	1.1	1.1	NUM
ma-247	426	14	:	:	PUNCT
ma-247	426	15	slicing	slicing	NOUN
ma-247	426	16	of	of	ADP
ma-247	426	17	the	the	DET
ma-247	426	18	current	current	ADJ
ma-247	426	19	(	(	PUNCT
ma-247	426	20	log|f|)[x	log|f|)[x	PROPN
ma-247	426	21	]	]	X
ma-247	426	22	4	4	NUM
ma-247	426	23	.	.	X
ma-247	426	24	proof	proof	NOUN
ma-247	426	25	of	of	ADP
ma-247	426	26	theorem	theorem	ADJ
ma-247	426	27	1.2	1.2	NUM
ma-247	426	28	:	:	PUNCT
ma-247	426	29	slicing	slicing	NOUN
ma-247	426	30	of	of	ADP
ma-247	426	31	the	the	DET
ma-247	426	32	current	current	ADJ
ma-247	426	33	v[x	v[x	NOUN
ma-247	426	34	]	]	X
ma-247	426	35	5	5	NUM
ma-247	426	36	.	.	X
ma-247	426	37	proof	proof	NOUN
ma-247	426	38	of	of	ADP
ma-247	426	39	theorem	theorem	ADJ
ma-247	426	40	1.3	1.3	NUM
ma-247	426	41	:	:	PUNCT
ma-247	426	42	slicing	slicing	NOUN
ma-247	426	43	of	of	ADP
ma-247	426	44	the	the	DET
ma-247	426	45	current	current	ADJ
ma-247	426	46	u[x][y	u[x][y	NOUN
ma-247	426	47	]	]	X
ma-247	426	48	6	6	NUM
ma-247	426	49	.	.	PUNCT
ma-247	427	1	proofs	proof	NOUN
ma-247	427	2	of	of	ADP
ma-247	427	3	theorem	theorem	ADJ
ma-247	427	4	1.4	1.4	NUM
ma-247	427	5	and	and	CCONJ
ma-247	427	6	theorem	theorem	VERB
ma-247	427	7	1.5	1.5	NUM
ma-247	427	8	with	with	ADP
ma-247	427	9	applications	application	NOUN
ma-247	427	10	competing	compete	VERB
ma-247	427	11	interests	interest	NOUN
ma-247	427	12	references	reference	NOUN