id	sid	tid	token	lemma	pos
ap-6125	1	1	acta	acta	PROPN
ap-6125	1	2	polytechnica	polytechnica	PROPN
ap-6125	1	3	https://doi.org/10.14311/ap.2021.61.0148	https://doi.org/10.14311/ap.2021.61.0148	PROPN
ap-6125	1	4	acta	acta	PROPN
ap-6125	1	5	polytechnica	polytechnica	PROPN
ap-6125	1	6	61(si):148–154	61(si):148–154	PROPN
ap-6125	1	7	,	,	PUNCT
ap-6125	1	8	2021	2021	NUM
ap-6125	1	9	©	©	ADP
ap-6125	1	10	2021	2021	NUM
ap-6125	1	11	the	the	DET
ap-6125	1	12	author(s	author(s	NOUN
ap-6125	1	13	)	)	PUNCT
ap-6125	1	14	.	.	PUNCT
ap-6125	2	1	licensed	license	VERB
ap-6125	2	2	under	under	ADP
ap-6125	2	3	a	a	DET
ap-6125	2	4	cc	cc	NOUN
ap-6125	2	5	-	-	PUNCT
ap-6125	2	6	by	by	ADP
ap-6125	2	7	4.0	4.0	NUM
ap-6125	2	8	licence	licence	NOUN
ap-6125	2	9	published	publish	VERB
ap-6125	2	10	by	by	ADP
ap-6125	2	11	the	the	DET
ap-6125	2	12	czech	czech	PROPN
ap-6125	2	13	technical	technical	PROPN
ap-6125	2	14	university	university	PROPN
ap-6125	2	15	in	in	ADP
ap-6125	2	16	prague	prague	PROPN
ap-6125	2	17	multivariate	multivariate	NOUN
ap-6125	2	18	interpolation	interpolation	NOUN
ap-6125	2	19	using	use	VERB
ap-6125	2	20	polyharmonic	polyharmonic	ADJ
ap-6125	2	21	splines	spline	NOUN
ap-6125	2	22	karel	karel	PROPN
ap-6125	2	23	segeth	segeth	PROPN
ap-6125	2	24	czech	czech	PROPN
ap-6125	2	25	academy	academy	PROPN
ap-6125	2	26	of	of	ADP
ap-6125	2	27	sciences	sciences	PROPN
ap-6125	2	28	,	,	PUNCT
ap-6125	2	29	institute	institute	NOUN
ap-6125	2	30	of	of	ADP
ap-6125	2	31	mathematics	mathematics	PROPN
ap-6125	2	32	,	,	PUNCT
ap-6125	2	33	žitná	žitná	NOUN
ap-6125	2	34	25	25	NUM
ap-6125	2	35	,	,	PUNCT
ap-6125	2	36	115	115	NUM
ap-6125	2	37	67	67	NUM
ap-6125	2	38	praha	praha	NOUN
ap-6125	2	39	1	1	NUM
ap-6125	2	40	,	,	PUNCT
ap-6125	2	41	czech	czech	PROPN
ap-6125	2	42	republic	republic	NOUN
ap-6125	2	43	correspondence	correspondence	NOUN
ap-6125	2	44	:	:	PUNCT
ap-6125	2	45	segeth@math.cas.cz	segeth@math.cas.cz	NOUN
ap-6125	2	46	abstract	abstract	NOUN
ap-6125	2	47	.	.	PUNCT
ap-6125	3	1	data	datum	NOUN
ap-6125	3	2	measuring	measure	VERB
ap-6125	3	3	and	and	CCONJ
ap-6125	3	4	further	further	ADJ
ap-6125	3	5	processing	processing	NOUN
ap-6125	3	6	is	be	AUX
ap-6125	3	7	the	the	DET
ap-6125	3	8	fundamental	fundamental	ADJ
ap-6125	3	9	activity	activity	NOUN
ap-6125	3	10	in	in	ADP
ap-6125	3	11	all	all	DET
ap-6125	3	12	branches	branch	NOUN
ap-6125	3	13	of	of	ADP
ap-6125	3	14	science	science	NOUN
ap-6125	3	15	and	and	CCONJ
ap-6125	3	16	technology	technology	NOUN
ap-6125	3	17	.	.	PUNCT
ap-6125	4	1	data	datum	NOUN
ap-6125	4	2	interpolation	interpolation	NOUN
ap-6125	4	3	has	have	AUX
ap-6125	4	4	been	be	AUX
ap-6125	4	5	an	an	DET
ap-6125	4	6	important	important	ADJ
ap-6125	4	7	part	part	NOUN
ap-6125	4	8	of	of	ADP
ap-6125	4	9	computational	computational	ADJ
ap-6125	4	10	mathematics	mathematic	NOUN
ap-6125	4	11	for	for	ADP
ap-6125	4	12	a	a	DET
ap-6125	4	13	long	long	ADJ
ap-6125	4	14	time	time	NOUN
ap-6125	4	15	.	.	PUNCT
ap-6125	5	1	in	in	ADP
ap-6125	5	2	the	the	DET
ap-6125	5	3	paper	paper	NOUN
ap-6125	5	4	,	,	PUNCT
ap-6125	5	5	we	we	PRON
ap-6125	5	6	are	be	AUX
ap-6125	5	7	concerned	concerned	ADJ
ap-6125	5	8	with	with	ADP
ap-6125	5	9	the	the	DET
ap-6125	5	10	interpolation	interpolation	NOUN
ap-6125	5	11	by	by	ADP
ap-6125	5	12	polyharmonic	polyharmonic	ADJ
ap-6125	5	13	splines	spline	NOUN
ap-6125	5	14	in	in	ADP
ap-6125	5	15	an	an	DET
ap-6125	5	16	arbitrary	arbitrary	ADJ
ap-6125	5	17	dimension	dimension	NOUN
ap-6125	5	18	.	.	PUNCT
ap-6125	6	1	we	we	PRON
ap-6125	6	2	show	show	VERB
ap-6125	6	3	the	the	DET
ap-6125	6	4	connection	connection	NOUN
ap-6125	6	5	of	of	ADP
ap-6125	6	6	this	this	DET
ap-6125	6	7	interpolation	interpolation	NOUN
ap-6125	6	8	with	with	ADP
ap-6125	6	9	the	the	DET
ap-6125	6	10	interpolation	interpolation	NOUN
ap-6125	6	11	by	by	ADP
ap-6125	6	12	radial	radial	ADJ
ap-6125	6	13	basis	basis	NOUN
ap-6125	6	14	functions	function	NOUN
ap-6125	6	15	and	and	CCONJ
ap-6125	6	16	the	the	DET
ap-6125	6	17	smooth	smooth	ADJ
ap-6125	6	18	interpolation	interpolation	NOUN
ap-6125	6	19	by	by	ADP
ap-6125	6	20	generating	generating	NOUN
ap-6125	6	21	functions	function	NOUN
ap-6125	6	22	,	,	PUNCT
ap-6125	6	23	which	which	PRON
ap-6125	6	24	provide	provide	VERB
ap-6125	6	25	means	mean	NOUN
ap-6125	6	26	for	for	ADP
ap-6125	6	27	minimizing	minimize	VERB
ap-6125	6	28	the	the	DET
ap-6125	6	29	l2	l2	NOUN
ap-6125	6	30	norm	norm	NOUN
ap-6125	6	31	of	of	ADP
ap-6125	6	32	chosen	choose	VERB
ap-6125	6	33	derivatives	derivative	NOUN
ap-6125	6	34	of	of	ADP
ap-6125	6	35	the	the	DET
ap-6125	6	36	interpolant	interpolant	NOUN
ap-6125	6	37	.	.	PUNCT
ap-6125	7	1	this	this	PRON
ap-6125	7	2	can	can	AUX
ap-6125	7	3	be	be	AUX
ap-6125	7	4	useful	useful	ADJ
ap-6125	7	5	in	in	ADP
ap-6125	7	6	2d	2d	NUM
ap-6125	7	7	and	and	CCONJ
ap-6125	7	8	3d	3d	NUM
ap-6125	7	9	,	,	PUNCT
ap-6125	7	10	e.g.	e.g.	ADV
ap-6125	7	11	,	,	PUNCT
ap-6125	7	12	in	in	ADP
ap-6125	7	13	the	the	DET
ap-6125	7	14	construction	construction	NOUN
ap-6125	7	15	of	of	ADP
ap-6125	7	16	geographic	geographic	ADJ
ap-6125	7	17	information	information	NOUN
ap-6125	7	18	systems	system	NOUN
ap-6125	7	19	or	or	CCONJ
ap-6125	7	20	computer	computer	NOUN
ap-6125	7	21	aided	aid	VERB
ap-6125	7	22	geometric	geometric	ADJ
ap-6125	7	23	design	design	NOUN
ap-6125	7	24	.	.	PUNCT
ap-6125	8	1	we	we	PRON
ap-6125	8	2	prove	prove	VERB
ap-6125	8	3	the	the	DET
ap-6125	8	4	properties	property	NOUN
ap-6125	8	5	of	of	ADP
ap-6125	8	6	the	the	DET
ap-6125	8	7	piecewise	piecewise	NOUN
ap-6125	8	8	polyharmonic	polyharmonic	ADJ
ap-6125	8	9	spline	spline	NOUN
ap-6125	8	10	interpolant	interpolant	NOUN
ap-6125	8	11	and	and	CCONJ
ap-6125	8	12	present	present	VERB
ap-6125	8	13	a	a	DET
ap-6125	8	14	simple	simple	ADJ
ap-6125	8	15	1d	1d	NUM
ap-6125	8	16	example	example	NOUN
ap-6125	8	17	to	to	PART
ap-6125	8	18	illustrate	illustrate	VERB
ap-6125	8	19	them	they	PRON
ap-6125	8	20	.	.	PUNCT
ap-6125	9	1	keywords	keyword	NOUN
ap-6125	9	2	:	:	PUNCT
ap-6125	9	3	data	datum	NOUN
ap-6125	9	4	interpolation	interpolation	NOUN
ap-6125	9	5	,	,	PUNCT
ap-6125	9	6	smooth	smooth	ADJ
ap-6125	9	7	interpolation	interpolation	NOUN
ap-6125	9	8	,	,	PUNCT
ap-6125	9	9	polyharmonic	polyharmonic	ADJ
ap-6125	9	10	spline	spline	NOUN
ap-6125	9	11	,	,	PUNCT
ap-6125	9	12	fourier	fourier	NOUN
ap-6125	9	13	transform	transform	NOUN
ap-6125	9	14	.	.	PUNCT
ap-6125	10	1	1	1	X
ap-6125	10	2	.	.	X
ap-6125	10	3	introduction	introduction	NOUN
ap-6125	10	4	measuring	measure	VERB
ap-6125	10	5	data	datum	NOUN
ap-6125	10	6	of	of	ADP
ap-6125	10	7	all	all	DET
ap-6125	10	8	different	different	ADJ
ap-6125	10	9	types	type	NOUN
ap-6125	10	10	and	and	CCONJ
ap-6125	10	11	formats	format	NOUN
ap-6125	10	12	is	be	AUX
ap-6125	10	13	the	the	DET
ap-6125	10	14	basic	basic	ADJ
ap-6125	10	15	means	mean	NOUN
ap-6125	10	16	of	of	ADP
ap-6125	10	17	research	research	NOUN
ap-6125	10	18	in	in	ADP
ap-6125	10	19	all	all	DET
ap-6125	10	20	branches	branch	NOUN
ap-6125	10	21	of	of	ADP
ap-6125	10	22	science	science	NOUN
ap-6125	10	23	and	and	CCONJ
ap-6125	10	24	technology	technology	NOUN
ap-6125	10	25	.	.	PUNCT
ap-6125	11	1	it	it	PRON
ap-6125	11	2	is	be	AUX
ap-6125	11	3	a	a	DET
ap-6125	11	4	discrete	discrete	ADJ
ap-6125	11	5	process	process	NOUN
ap-6125	11	6	providing	provide	VERB
ap-6125	11	7	a	a	DET
ap-6125	11	8	finite	finite	ADJ
ap-6125	11	9	number	number	NOUN
ap-6125	11	10	of	of	ADP
ap-6125	11	11	numerical	numerical	ADJ
ap-6125	11	12	values	value	NOUN
ap-6125	11	13	over	over	ADP
ap-6125	11	14	some	some	DET
ap-6125	11	15	domain	domain	NOUN
ap-6125	11	16	.	.	PUNCT
ap-6125	12	1	the	the	DET
ap-6125	12	2	first	first	ADJ
ap-6125	12	3	stage	stage	NOUN
ap-6125	12	4	of	of	ADP
ap-6125	12	5	data	datum	NOUN
ap-6125	12	6	processing	processing	NOUN
ap-6125	12	7	usually	usually	ADV
ap-6125	12	8	consists	consist	VERB
ap-6125	12	9	in	in	ADP
ap-6125	12	10	its	its	PRON
ap-6125	12	11	approximation	approximation	NOUN
ap-6125	12	12	,	,	PUNCT
ap-6125	12	13	i.e.	i.e.	X
ap-6125	12	14	,	,	PUNCT
ap-6125	12	15	computing	compute	VERB
ap-6125	12	16	reliable	reliable	ADJ
ap-6125	12	17	data	data	NOUN
ap-6125	12	18	values	value	NOUN
ap-6125	12	19	at	at	ADP
ap-6125	12	20	an	an	DET
ap-6125	12	21	arbitrary	arbitrary	ADJ
ap-6125	12	22	point	point	NOUN
ap-6125	12	23	in	in	ADP
ap-6125	12	24	the	the	DET
ap-6125	12	25	domain	domain	NOUN
ap-6125	12	26	of	of	ADP
ap-6125	12	27	interest	interest	NOUN
ap-6125	12	28	.	.	PUNCT
ap-6125	13	1	in	in	ADP
ap-6125	13	2	the	the	DET
ap-6125	13	3	paper	paper	NOUN
ap-6125	13	4	,	,	PUNCT
ap-6125	13	5	we	we	PRON
ap-6125	13	6	are	be	AUX
ap-6125	13	7	concerned	concerned	ADJ
ap-6125	13	8	with	with	ADP
ap-6125	13	9	the	the	DET
ap-6125	13	10	problem	problem	NOUN
ap-6125	13	11	of	of	ADP
ap-6125	13	12	data	datum	NOUN
ap-6125	13	13	interpolation	interpolation	NOUN
ap-6125	13	14	in	in	ADP
ap-6125	13	15	an	an	DET
ap-6125	13	16	arbitrary	arbitrary	ADJ
ap-6125	13	17	dimension	dimension	NOUN
ap-6125	13	18	.	.	PUNCT
ap-6125	14	1	in	in	ADP
ap-6125	14	2	particular	particular	ADJ
ap-6125	14	3	,	,	PUNCT
ap-6125	14	4	we	we	PRON
ap-6125	14	5	consider	consider	VERB
ap-6125	14	6	the	the	DET
ap-6125	14	7	interpolation	interpolation	NOUN
ap-6125	14	8	with	with	ADP
ap-6125	14	9	radial	radial	ADJ
ap-6125	14	10	basis	basis	NOUN
ap-6125	14	11	functions	function	NOUN
ap-6125	14	12	that	that	PRON
ap-6125	14	13	is	be	AUX
ap-6125	14	14	reasonable	reasonable	ADJ
ap-6125	14	15	if	if	SCONJ
ap-6125	14	16	we	we	PRON
ap-6125	14	17	assume	assume	VERB
ap-6125	14	18	that	that	SCONJ
ap-6125	14	19	the	the	DET
ap-6125	14	20	value	value	NOUN
ap-6125	14	21	at	at	ADP
ap-6125	14	22	a	a	DET
ap-6125	14	23	point	point	NOUN
ap-6125	14	24	x	x	PUNCT
ap-6125	14	25	depends	depend	VERB
ap-6125	14	26	in	in	ADP
ap-6125	14	27	some	some	DET
ap-6125	14	28	way	way	NOUN
ap-6125	14	29	on	on	ADP
ap-6125	14	30	the	the	DET
ap-6125	14	31	euclidean	euclidean	ADJ
ap-6125	14	32	distance	distance	NOUN
ap-6125	14	33	r(x	r(x	PROPN
ap-6125	14	34	,	,	PUNCT
ap-6125	14	35	xj	xj	PROPN
ap-6125	14	36	)	)	PUNCT
ap-6125	14	37	between	between	ADP
ap-6125	14	38	x	x	SYM
ap-6125	14	39	and	and	CCONJ
ap-6125	14	40	the	the	DET
ap-6125	14	41	nodes	node	NOUN
ap-6125	14	42	xj	xj	PROPN
ap-6125	14	43	where	where	SCONJ
ap-6125	14	44	the	the	DET
ap-6125	14	45	values	value	NOUN
ap-6125	14	46	have	have	AUX
ap-6125	14	47	been	be	AUX
ap-6125	14	48	measured	measure	VERB
ap-6125	14	49	.	.	PUNCT
ap-6125	15	1	the	the	DET
ap-6125	15	2	background	background	NOUN
ap-6125	15	3	of	of	ADP
ap-6125	15	4	the	the	DET
ap-6125	15	5	paper	paper	NOUN
ap-6125	15	6	is	be	AUX
ap-6125	15	7	the	the	DET
ap-6125	15	8	so	so	ADV
ap-6125	15	9	-	-	PUNCT
ap-6125	15	10	called	call	VERB
ap-6125	15	11	smooth	smooth	ADJ
ap-6125	15	12	interpolation	interpolation	NOUN
ap-6125	15	13	[	[	X
ap-6125	15	14	1	1	NUM
ap-6125	15	15	]	]	PUNCT
ap-6125	15	16	,	,	PUNCT
ap-6125	15	17	[	[	X
ap-6125	15	18	2	2	NUM
ap-6125	15	19	]	]	PUNCT
ap-6125	15	20	allowing	allow	VERB
ap-6125	15	21	for	for	ADP
ap-6125	15	22	the	the	DET
ap-6125	15	23	minimization	minimization	NOUN
ap-6125	15	24	of	of	ADP
ap-6125	15	25	some	some	DET
ap-6125	15	26	functionals	functional	NOUN
ap-6125	15	27	applied	apply	VERB
ap-6125	15	28	to	to	ADP
ap-6125	15	29	the	the	DET
ap-6125	15	30	interpolation	interpolation	NOUN
ap-6125	15	31	formula	formula	NOUN
ap-6125	15	32	.	.	PUNCT
ap-6125	16	1	choosing	choose	VERB
ap-6125	16	2	particular	particular	ADJ
ap-6125	16	3	basis	basis	NOUN
ap-6125	16	4	functions	function	NOUN
ap-6125	16	5	in	in	ADP
ap-6125	16	6	the	the	DET
ap-6125	16	7	minimization	minimization	NOUN
ap-6125	16	8	space	space	NOUN
ap-6125	16	9	,	,	PUNCT
ap-6125	16	10	we	we	PRON
ap-6125	16	11	can	can	AUX
ap-6125	16	12	get	get	VERB
ap-6125	16	13	an	an	DET
ap-6125	16	14	interpolation	interpolation	NOUN
ap-6125	16	15	formula	formula	NOUN
ap-6125	16	16	whose	whose	DET
ap-6125	16	17	principal	principal	ADJ
ap-6125	16	18	part	part	NOUN
ap-6125	16	19	is	be	AUX
ap-6125	16	20	a	a	DET
ap-6125	16	21	linear	linear	ADJ
ap-6125	16	22	combination	combination	NOUN
ap-6125	16	23	of	of	ADP
ap-6125	16	24	polyharmonic	polyharmonic	ADJ
ap-6125	16	25	splines	spline	NOUN
ap-6125	16	26	of	of	ADP
ap-6125	16	27	fixed	fix	VERB
ap-6125	16	28	order	order	NOUN
ap-6125	16	29	that	that	PRON
ap-6125	16	30	are	be	AUX
ap-6125	16	31	,	,	PUNCT
ap-6125	16	32	at	at	ADP
ap-6125	16	33	the	the	DET
ap-6125	16	34	same	same	ADJ
ap-6125	16	35	time	time	NOUN
ap-6125	16	36	,	,	PUNCT
ap-6125	16	37	radial	radial	ADJ
ap-6125	16	38	functions	function	NOUN
ap-6125	16	39	.	.	PUNCT
ap-6125	17	1	we	we	PRON
ap-6125	17	2	construct	construct	VERB
ap-6125	17	3	such	such	DET
ap-6125	17	4	a	a	DET
ap-6125	17	5	radial	radial	ADJ
ap-6125	17	6	basis	basis	NOUN
ap-6125	17	7	,	,	PUNCT
ap-6125	17	8	i.e.	i.e.	X
ap-6125	17	9	polyharmonic	polyharmonic	ADJ
ap-6125	17	10	splines	spline	NOUN
ap-6125	17	11	,	,	PUNCT
ap-6125	17	12	and	and	CCONJ
ap-6125	17	13	show	show	VERB
ap-6125	17	14	its	its	PRON
ap-6125	17	15	properties	property	NOUN
ap-6125	17	16	.	.	PUNCT
ap-6125	18	1	among	among	ADP
ap-6125	18	2	other	other	ADJ
ap-6125	18	3	things	thing	NOUN
ap-6125	18	4	,	,	PUNCT
ap-6125	18	5	we	we	PRON
ap-6125	18	6	prove	prove	VERB
ap-6125	18	7	that	that	SCONJ
ap-6125	18	8	the	the	DET
ap-6125	18	9	interpolant	interpolant	NOUN
ap-6125	18	10	is	be	AUX
ap-6125	18	11	piecewise	piecewise	NOUN
ap-6125	18	12	polyharmonic	polyharmonic	NOUN
ap-6125	18	13	.	.	PUNCT
ap-6125	19	1	we	we	PRON
ap-6125	19	2	present	present	VERB
ap-6125	19	3	a	a	DET
ap-6125	19	4	1d	1d	NUM
ap-6125	19	5	example	example	NOUN
ap-6125	19	6	that	that	PRON
ap-6125	19	7	shows	show	VERB
ap-6125	19	8	the	the	DET
ap-6125	19	9	result	result	NOUN
ap-6125	19	10	of	of	ADP
ap-6125	19	11	interpolation	interpolation	NOUN
ap-6125	19	12	if	if	SCONJ
ap-6125	19	13	different	different	ADJ
ap-6125	19	14	derivatives	derivative	NOUN
ap-6125	19	15	of	of	ADP
ap-6125	19	16	the	the	DET
ap-6125	19	17	interpolant	interpolant	NOUN
ap-6125	19	18	are	be	AUX
ap-6125	19	19	minimized	minimize	VERB
ap-6125	19	20	in	in	ADP
ap-6125	19	21	the	the	DET
ap-6125	19	22	l2	l2	NOUN
ap-6125	19	23	norm	norm	NOUN
ap-6125	19	24	.	.	PUNCT
ap-6125	20	1	the	the	DET
ap-6125	20	2	example	example	NOUN
ap-6125	20	3	shows	show	VERB
ap-6125	20	4	that	that	SCONJ
ap-6125	20	5	the	the	DET
ap-6125	20	6	respective	respective	ADJ
ap-6125	20	7	interpolations	interpolation	NOUN
ap-6125	20	8	give	give	VERB
ap-6125	20	9	expected	expect	VERB
ap-6125	20	10	results	result	NOUN
ap-6125	20	11	.	.	PUNCT
ap-6125	21	1	interpolation	interpolation	NOUN
ap-6125	21	2	of	of	ADP
ap-6125	21	3	this	this	DET
ap-6125	21	4	nature	nature	NOUN
ap-6125	21	5	is	be	AUX
ap-6125	21	6	often	often	ADV
ap-6125	21	7	employed	employ	VERB
ap-6125	21	8	in	in	ADP
ap-6125	21	9	signal	signal	ADJ
ap-6125	21	10	processing	processing	NOUN
ap-6125	21	11	,	,	PUNCT
ap-6125	21	12	construction	construction	NOUN
ap-6125	21	13	of	of	ADP
ap-6125	21	14	geographic	geographic	ADJ
ap-6125	21	15	information	information	NOUN
ap-6125	21	16	systems	system	NOUN
ap-6125	21	17	,	,	PUNCT
ap-6125	21	18	or	or	CCONJ
ap-6125	21	19	computer	computer	NOUN
ap-6125	21	20	aided	aid	VERB
ap-6125	21	21	geometric	geometric	ADJ
ap-6125	21	22	design	design	NOUN
ap-6125	21	23	.	.	PUNCT
ap-6125	22	1	moreover	moreover	ADV
ap-6125	22	2	,	,	PUNCT
ap-6125	22	3	if	if	SCONJ
ap-6125	22	4	the	the	DET
ap-6125	22	5	field	field	NOUN
ap-6125	22	6	measured	measure	VERB
ap-6125	22	7	is	be	AUX
ap-6125	22	8	known	know	VERB
ap-6125	22	9	to	to	PART
ap-6125	22	10	be	be	AUX
ap-6125	22	11	polyharmonic	polyharmonic	ADJ
ap-6125	22	12	,	,	PUNCT
ap-6125	22	13	it	it	PRON
ap-6125	22	14	is	be	AUX
ap-6125	22	15	worth	worth	ADJ
ap-6125	22	16	to	to	PART
ap-6125	22	17	interpolate	interpolate	VERB
ap-6125	22	18	it	it	PRON
ap-6125	22	19	by	by	ADP
ap-6125	22	20	a	a	DET
ap-6125	22	21	formula	formula	NOUN
ap-6125	22	22	preserving	preserve	VERB
ap-6125	22	23	the	the	DET
ap-6125	22	24	polyharmonicity	polyharmonicity	NOUN
ap-6125	22	25	.	.	PUNCT
ap-6125	23	1	frequent	frequent	ADJ
ap-6125	23	2	citations	citation	NOUN
ap-6125	23	3	of	of	ADP
ap-6125	23	4	the	the	DET
ap-6125	23	5	author	author	NOUN
ap-6125	23	6	’s	’s	PART
ap-6125	23	7	paper	paper	NOUN
ap-6125	23	8	[	[	X
ap-6125	23	9	2	2	X
ap-6125	23	10	]	]	PUNCT
ap-6125	23	11	have	have	AUX
ap-6125	23	12	been	be	AUX
ap-6125	23	13	used	use	VERB
ap-6125	23	14	to	to	PART
ap-6125	23	15	introduce	introduce	VERB
ap-6125	23	16	the	the	DET
ap-6125	23	17	notation	notation	NOUN
ap-6125	23	18	and	and	CCONJ
ap-6125	23	19	basic	basic	ADJ
ap-6125	23	20	properties	property	NOUN
ap-6125	23	21	of	of	ADP
ap-6125	23	22	notions	notion	NOUN
ap-6125	23	23	used	use	VERB
ap-6125	23	24	.	.	PUNCT
ap-6125	24	1	the	the	DET
ap-6125	24	2	conclusion	conclusion	NOUN
ap-6125	24	3	of	of	ADP
ap-6125	24	4	the	the	DET
ap-6125	24	5	present	present	ADJ
ap-6125	24	6	paper	paper	NOUN
ap-6125	24	7	is	be	AUX
ap-6125	24	8	more	more	ADV
ap-6125	24	9	advanced	advanced	ADJ
ap-6125	24	10	,	,	PUNCT
ap-6125	24	11	it	it	PRON
ap-6125	24	12	shows	show	VERB
ap-6125	24	13	that	that	SCONJ
ap-6125	24	14	for	for	ADP
ap-6125	24	15	interpolation	interpolation	NOUN
ap-6125	24	16	it	it	PRON
ap-6125	24	17	is	be	AUX
ap-6125	24	18	possible	possible	ADJ
ap-6125	24	19	to	to	PART
ap-6125	24	20	use	use	VERB
ap-6125	24	21	polyharmonic	polyharmonic	ADJ
ap-6125	24	22	functions	function	NOUN
ap-6125	24	23	of	of	ADP
ap-6125	24	24	different	different	ADJ
ap-6125	24	25	orders	order	NOUN
ap-6125	24	26	m	m	VERB
ap-6125	24	27	that	that	PRON
ap-6125	24	28	minimize	minimize	VERB
ap-6125	24	29	different	different	ADJ
ap-6125	24	30	norms	norm	NOUN
ap-6125	24	31	of	of	ADP
ap-6125	24	32	the	the	DET
ap-6125	24	33	interpolant	interpolant	PROPN
ap-6125	24	34	u	u	NOUN
ap-6125	24	35	,	,	PUNCT
ap-6125	24	36	i.e.	i.e.	X
ap-6125	24	37	the	the	DET
ap-6125	24	38	l2	l2	NOUN
ap-6125	24	39	norm	norm	NOUN
ap-6125	24	40	of	of	ADP
ap-6125	24	41	the	the	PRON
ap-6125	24	42	(	(	PUNCT
ap-6125	24	43	m	m	PROPN
ap-6125	24	44	+	+	NOUN
ap-6125	25	1	l)th	l)th	PROPN
ap-6125	25	2	derivative	derivative	NOUN
ap-6125	25	3	of	of	ADP
ap-6125	25	4	u	u	NOUN
ap-6125	25	5	for	for	ADP
ap-6125	25	6	any	any	DET
ap-6125	25	7	l	l	NOUN
ap-6125	25	8	positive	positive	ADJ
ap-6125	25	9	.	.	PUNCT
ap-6125	26	1	we	we	PRON
ap-6125	26	2	state	state	VERB
ap-6125	26	3	the	the	DET
ap-6125	26	4	problem	problem	NOUN
ap-6125	26	5	of	of	ADP
ap-6125	26	6	data	datum	NOUN
ap-6125	26	7	interpolation	interpolation	NOUN
ap-6125	26	8	in	in	ADP
ap-6125	26	9	sec	sec	PROPN
ap-6125	26	10	.	.	PROPN
ap-6125	26	11	2	2	NUM
ap-6125	26	12	,	,	PUNCT
ap-6125	26	13	introduce	introduce	VERB
ap-6125	26	14	radial	radial	ADJ
ap-6125	26	15	functions	function	NOUN
ap-6125	26	16	in	in	ADP
ap-6125	26	17	sec	sec	PROPN
ap-6125	26	18	.	.	PROPN
ap-6125	26	19	3	3	NUM
ap-6125	26	20	,	,	PUNCT
ap-6125	26	21	and	and	CCONJ
ap-6125	26	22	polyharmonic	polyharmonic	ADJ
ap-6125	26	23	splines	spline	NOUN
ap-6125	26	24	in	in	ADP
ap-6125	26	25	sec	sec	PROPN
ap-6125	26	26	.	.	PROPN
ap-6125	26	27	4	4	NUM
ap-6125	26	28	.	.	X
ap-6125	27	1	further	far	ADV
ap-6125	27	2	,	,	PUNCT
ap-6125	27	3	we	we	PRON
ap-6125	27	4	define	define	VERB
ap-6125	27	5	the	the	DET
ap-6125	27	6	spaces	space	NOUN
ap-6125	27	7	wl	wl	X
ap-6125	27	8	where	where	SCONJ
ap-6125	27	9	we	we	PRON
ap-6125	27	10	are	be	AUX
ap-6125	27	11	going	go	VERB
ap-6125	27	12	to	to	PART
ap-6125	27	13	carry	carry	VERB
ap-6125	27	14	out	out	ADP
ap-6125	27	15	the	the	DET
ap-6125	27	16	minimization	minimization	NOUN
ap-6125	27	17	and	and	CCONJ
ap-6125	27	18	present	present	VERB
ap-6125	27	19	a	a	DET
ap-6125	27	20	general	general	ADJ
ap-6125	27	21	form	form	NOUN
ap-6125	27	22	of	of	ADP
ap-6125	27	23	the	the	DET
ap-6125	27	24	interpolation	interpolation	NOUN
ap-6125	27	25	formula	formula	NOUN
ap-6125	27	26	in	in	ADP
ap-6125	27	27	sec	sec	PROPN
ap-6125	27	28	.	.	PROPN
ap-6125	28	1	5	5	NUM
ap-6125	28	2	.	.	PUNCT
ap-6125	29	1	moreover	moreover	ADV
ap-6125	29	2	,	,	PUNCT
ap-6125	29	3	we	we	PRON
ap-6125	29	4	quote	quote	VERB
ap-6125	29	5	a	a	DET
ap-6125	29	6	theorem	theorem	NOUN
ap-6125	29	7	from	from	ADP
ap-6125	29	8	[	[	X
ap-6125	29	9	2	2	X
ap-6125	29	10	]	]	PUNCT
ap-6125	29	11	that	that	PRON
ap-6125	29	12	states	state	VERB
ap-6125	29	13	the	the	DET
ap-6125	29	14	existence	existence	NOUN
ap-6125	29	15	and	and	CCONJ
ap-6125	29	16	unicity	unicity	NOUN
ap-6125	29	17	of	of	ADP
ap-6125	29	18	the	the	DET
ap-6125	29	19	solution	solution	NOUN
ap-6125	29	20	of	of	ADP
ap-6125	29	21	the	the	DET
ap-6125	29	22	interpolation	interpolation	NOUN
ap-6125	29	23	problem	problem	NOUN
ap-6125	29	24	.	.	PUNCT
ap-6125	30	1	in	in	ADP
ap-6125	30	2	sec	sec	PROPN
ap-6125	30	3	.	.	PROPN
ap-6125	30	4	6	6	NUM
ap-6125	30	5	,	,	PUNCT
ap-6125	30	6	we	we	PRON
ap-6125	30	7	choose	choose	VERB
ap-6125	30	8	exponential	exponential	ADJ
ap-6125	30	9	functions	function	NOUN
ap-6125	30	10	of	of	ADP
ap-6125	30	11	a	a	DET
ap-6125	30	12	pure	pure	ADJ
ap-6125	30	13	imaginary	imaginary	ADJ
ap-6125	30	14	argument	argument	NOUN
ap-6125	30	15	for	for	ADP
ap-6125	30	16	the	the	DET
ap-6125	30	17	basis	basis	NOUN
ap-6125	30	18	functions	function	NOUN
ap-6125	30	19	in	in	ADP
ap-6125	30	20	wl	wl	PROPN
ap-6125	30	21	.	.	PUNCT
ap-6125	31	1	for	for	ADP
ap-6125	31	2	this	this	DET
ap-6125	31	3	choice	choice	NOUN
ap-6125	31	4	of	of	ADP
ap-6125	31	5	basis	basis	NOUN
ap-6125	31	6	functions	function	NOUN
ap-6125	31	7	,	,	PUNCT
ap-6125	31	8	we	we	PRON
ap-6125	31	9	obtain	obtain	VERB
ap-6125	31	10	a	a	DET
ap-6125	31	11	radial	radial	ADJ
ap-6125	31	12	basis	basis	NOUN
ap-6125	31	13	function	function	NOUN
ap-6125	31	14	interpolation	interpolation	NOUN
ap-6125	31	15	formula	formula	NOUN
ap-6125	31	16	,	,	PUNCT
ap-6125	31	17	where	where	SCONJ
ap-6125	31	18	the	the	DET
ap-6125	31	19	radial	radial	ADJ
ap-6125	31	20	functions	function	NOUN
ap-6125	31	21	are	be	AUX
ap-6125	31	22	polyharmonic	polyharmonic	ADJ
ap-6125	31	23	functions	function	NOUN
ap-6125	31	24	,	,	PUNCT
ap-6125	31	25	which	which	PRON
ap-6125	31	26	can	can	AUX
ap-6125	31	27	be	be	AUX
ap-6125	31	28	seen	see	VERB
ap-6125	31	29	in	in	ADP
ap-6125	31	30	sec	sec	PROPN
ap-6125	31	31	.	.	PROPN
ap-6125	31	32	7	7	NUM
ap-6125	31	33	,	,	PUNCT
ap-6125	31	34	present	present	VERB
ap-6125	31	35	some	some	DET
ap-6125	31	36	properties	property	NOUN
ap-6125	31	37	(	(	PUNCT
ap-6125	31	38	polyharmonicity	polyharmonicity	NOUN
ap-6125	31	39	)	)	PUNCT
ap-6125	31	40	of	of	ADP
ap-6125	31	41	such	such	DET
ap-6125	31	42	an	an	DET
ap-6125	31	43	interpolant	interpolant	NOUN
ap-6125	31	44	in	in	ADP
ap-6125	31	45	sec	sec	PROPN
ap-6125	31	46	.	.	PROPN
ap-6125	31	47	8	8	NUM
ap-6125	31	48	.	.	PUNCT
ap-6125	31	49	and	and	CCONJ
ap-6125	31	50	show	show	VERB
ap-6125	31	51	a	a	DET
ap-6125	31	52	simple	simple	ADJ
ap-6125	31	53	computional	computional	ADJ
ap-6125	31	54	example	example	NOUN
ap-6125	31	55	in	in	ADP
ap-6125	31	56	sec	sec	PROPN
ap-6125	31	57	.	.	PROPN
ap-6125	31	58	9	9	NUM
ap-6125	31	59	.	.	NOUN
ap-6125	31	60	2	2	NUM
ap-6125	31	61	.	.	X
ap-6125	31	62	problem	problem	NOUN
ap-6125	31	63	of	of	ADP
ap-6125	31	64	data	datum	NOUN
ap-6125	31	65	interpolation	interpolation	NOUN
ap-6125	31	66	fundamental	fundamental	ADJ
ap-6125	31	67	notation	notation	NOUN
ap-6125	31	68	and	and	CCONJ
ap-6125	31	69	basic	basic	ADJ
ap-6125	31	70	statements	statement	NOUN
ap-6125	31	71	are	be	AUX
ap-6125	31	72	taken	take	VERB
ap-6125	31	73	mostly	mostly	ADV
ap-6125	31	74	from	from	ADP
ap-6125	31	75	[	[	X
ap-6125	31	76	2	2	NUM
ap-6125	31	77	]	]	PUNCT
ap-6125	31	78	.	.	PUNCT
ap-6125	32	1	consider	consider	VERB
ap-6125	32	2	a	a	DET
ap-6125	32	3	finite	finite	ADJ
ap-6125	32	4	number	number	NOUN
ap-6125	32	5	n	n	PROPN
ap-6125	32	6	of	of	ADP
ap-6125	32	7	(	(	PUNCT
ap-6125	32	8	complex	complex	ADJ
ap-6125	32	9	,	,	PUNCT
ap-6125	32	10	in	in	ADP
ap-6125	32	11	general	general	ADJ
ap-6125	32	12	)	)	PUNCT
ap-6125	32	13	measured	measure	VERB
ap-6125	32	14	(	(	PUNCT
ap-6125	32	15	sampled	sample	VERB
ap-6125	32	16	)	)	PUNCT
ap-6125	32	17	values	value	NOUN
ap-6125	32	18	f1	f1	NOUN
ap-6125	32	19	,	,	PUNCT
ap-6125	32	20	.	.	PUNCT
ap-6125	32	21	.	.	PUNCT
ap-6125	33	1	.	.	PUNCT
ap-6125	34	1	,	,	PUNCT
ap-6125	34	2	fn	fn	PROPN
ap-6125	34	3	∈	∈	PROPN
ap-6125	34	4	c	c	NOUN
ap-6125	34	5	obtained	obtain	VERB
ap-6125	34	6	at	at	ADP
ap-6125	34	7	n	n	ADV
ap-6125	34	8	given	give	VERB
ap-6125	34	9	nodes	node	NOUN
ap-6125	34	10	x1	x1	PROPN
ap-6125	34	11	,	,	PUNCT
ap-6125	34	12	.	.	PUNCT
ap-6125	34	13	.	.	PUNCT
ap-6125	35	1	.	.	PUNCT
ap-6125	36	1	,	,	PUNCT
ap-6125	36	2	xn	xn	PROPN
ap-6125	36	3	∈	∈	PROPN
ap-6125	36	4	ω	ω	PROPN
ap-6125	36	5	,	,	PUNCT
ap-6125	36	6	xj	xj	PROPN
ap-6125	36	7	=	=	SYM
ap-6125	36	8	(	(	PUNCT
ap-6125	36	9	xj1	xj1	PROPN
ap-6125	36	10	,	,	PUNCT
ap-6125	36	11	.	.	PUNCT
ap-6125	36	12	.	.	PUNCT
ap-6125	36	13	.	.	PUNCT
ap-6125	37	1	,	,	PUNCT
ap-6125	37	2	xjn	xjn	PROPN
ap-6125	37	3	)	)	PUNCT
ap-6125	37	4	,	,	PUNCT
ap-6125	37	5	that	that	PRON
ap-6125	37	6	are	be	AUX
ap-6125	37	7	mutually	mutually	ADV
ap-6125	37	8	distinct	distinct	ADJ
ap-6125	37	9	,	,	PUNCT
ap-6125	37	10	where	where	SCONJ
ap-6125	37	11	n	n	PRON
ap-6125	37	12	is	be	AUX
ap-6125	37	13	a	a	DET
ap-6125	37	14	positive	positive	ADJ
ap-6125	37	15	integer	integer	NOUN
ap-6125	37	16	and	and	CCONJ
ap-6125	37	17	ω	ω	NUM
ap-6125	37	18	∈	∈	PROPN
ap-6125	37	19	rn	rn	PROPN
ap-6125	37	20	is	be	AUX
ap-6125	37	21	a	a	DET
ap-6125	37	22	cube	cube	NOUN
ap-6125	37	23	.	.	PUNCT
ap-6125	38	1	usually	usually	ADV
ap-6125	38	2	,	,	PUNCT
ap-6125	38	3	we	we	PRON
ap-6125	38	4	need	need	VERB
ap-6125	38	5	also	also	ADV
ap-6125	38	6	the	the	DET
ap-6125	38	7	values	value	NOUN
ap-6125	38	8	corresponding	correspond	VERB
ap-6125	38	9	to	to	ADP
ap-6125	38	10	other	other	ADJ
ap-6125	38	11	points	point	NOUN
ap-6125	38	12	in	in	ADP
ap-6125	38	13	ω	ω	NUM
ap-6125	38	14	that	that	PRON
ap-6125	38	15	are	be	AUX
ap-6125	38	16	not	not	PART
ap-6125	38	17	known	know	VERB
ap-6125	38	18	.	.	PUNCT
ap-6125	39	1	let	let	VERB
ap-6125	39	2	fj	fj	PROPN
ap-6125	39	3	=	=	PUNCT
ap-6125	39	4	f(xj	f(xj	PROPN
ap-6125	39	5	)	)	PUNCT
ap-6125	39	6	be	be	AUX
ap-6125	39	7	measured	measure	VERB
ap-6125	39	8	values	value	NOUN
ap-6125	39	9	of	of	ADP
ap-6125	39	10	a	a	DET
ap-6125	39	11	148	148	NUM
ap-6125	39	12	https://doi.org/10.14311/ap.2021.61.0148	https://doi.org/10.14311/ap.2021.61.0148	NOUN
ap-6125	39	13	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-6125	39	14	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-6125	39	15	vol	vol	NOUN
ap-6125	39	16	.	.	PUNCT
ap-6125	40	1	61	61	NUM
ap-6125	40	2	special	special	ADJ
ap-6125	40	3	issue/2021	issue/2021	NOUN
ap-6125	40	4	multivariate	multivariate	NOUN
ap-6125	40	5	interpolation	interpolation	NOUN
ap-6125	40	6	using	use	VERB
ap-6125	40	7	polyharmonic	polyharmonic	ADJ
ap-6125	40	8	splines	spline	NOUN
ap-6125	40	9	complex	complex	ADV
ap-6125	40	10	-	-	PUNCT
ap-6125	40	11	valued	value	VERB
ap-6125	40	12	function	function	NOUN
ap-6125	40	13	f	f	X
ap-6125	40	14	continuous	continuous	ADJ
ap-6125	40	15	in	in	ADP
ap-6125	40	16	ω	ω	PROPN
ap-6125	40	17	and	and	CCONJ
ap-6125	40	18	z	z	NOUN
ap-6125	40	19	is	be	AUX
ap-6125	40	20	an	an	DET
ap-6125	40	21	approximating	approximate	VERB
ap-6125	40	22	function	function	NOUN
ap-6125	40	23	to	to	PART
ap-6125	40	24	be	be	AUX
ap-6125	40	25	constructed	construct	VERB
ap-6125	40	26	.	.	PUNCT
ap-6125	41	1	definition	definition	NOUN
ap-6125	41	2	1	1	NUM
ap-6125	41	3	.	.	PUNCT
ap-6125	42	1	the	the	DET
ap-6125	42	2	interpolating	interpolate	VERB
ap-6125	42	3	function	function	NOUN
ap-6125	42	4	(	(	PUNCT
ap-6125	42	5	interpolant	interpolant	NOUN
ap-6125	42	6	)	)	PUNCT
ap-6125	42	7	z	z	NOUN
ap-6125	42	8	is	be	AUX
ap-6125	42	9	constructed	construct	VERB
ap-6125	42	10	to	to	PART
ap-6125	42	11	fulfill	fulfill	VERB
ap-6125	42	12	the	the	DET
ap-6125	42	13	interpolation	interpolation	NOUN
ap-6125	42	14	conditions	condition	NOUN
ap-6125	42	15	z(xj	z(xj	ADV
ap-6125	42	16	)	)	PUNCT
ap-6125	43	1	=	=	SYM
ap-6125	43	2	fj	fj	PROPN
ap-6125	43	3	,	,	PUNCT
ap-6125	43	4	j	j	PROPN
ap-6125	43	5	=	=	SYM
ap-6125	43	6	1	1	NUM
ap-6125	43	7	,	,	PUNCT
ap-6125	43	8	.	.	PUNCT
ap-6125	43	9	.	.	PUNCT
ap-6125	43	10	.	.	PUNCT
ap-6125	44	1	,	,	PUNCT
ap-6125	44	2	n	n	CCONJ
ap-6125	44	3	,	,	PUNCT
ap-6125	44	4	(	(	PUNCT
ap-6125	44	5	1	1	X
ap-6125	44	6	)	)	PUNCT
ap-6125	44	7	cf	cf	NOUN
ap-6125	44	8	.	.	PUNCT
ap-6125	45	1	definition	definition	NOUN
ap-6125	45	2	1	1	NUM
ap-6125	45	3	of	of	ADP
ap-6125	45	4	[	[	X
ap-6125	45	5	2	2	NUM
ap-6125	45	6	]	]	PUNCT
ap-6125	45	7	.	.	PUNCT
ap-6125	46	1	various	various	ADJ
ap-6125	46	2	additional	additional	ADJ
ap-6125	46	3	conditions	condition	NOUN
ap-6125	46	4	can	can	AUX
ap-6125	46	5	be	be	AUX
ap-6125	46	6	considered	consider	VERB
ap-6125	46	7	,	,	PUNCT
ap-6125	46	8	e.g.	e.g.	ADV
ap-6125	46	9	,	,	PUNCT
ap-6125	46	10	minimization	minimization	NOUN
ap-6125	46	11	of	of	ADP
ap-6125	46	12	some	some	DET
ap-6125	46	13	functionals	functional	NOUN
ap-6125	46	14	applied	apply	VERB
ap-6125	46	15	to	to	ADP
ap-6125	46	16	z.	z.	PROPN
ap-6125	46	17	the	the	DET
ap-6125	46	18	problem	problem	NOUN
ap-6125	46	19	of	of	ADP
ap-6125	46	20	data	datum	NOUN
ap-6125	46	21	interpolation	interpolation	NOUN
ap-6125	46	22	does	do	AUX
ap-6125	46	23	not	not	PART
ap-6125	46	24	have	have	VERB
ap-6125	46	25	a	a	DET
ap-6125	46	26	unique	unique	ADJ
ap-6125	46	27	solution	solution	NOUN
ap-6125	46	28	.	.	PUNCT
ap-6125	47	1	the	the	DET
ap-6125	47	2	property	property	NOUN
ap-6125	47	3	(	(	PUNCT
ap-6125	47	4	1	1	NUM
ap-6125	47	5	)	)	PUNCT
ap-6125	47	6	of	of	ADP
ap-6125	47	7	the	the	DET
ap-6125	47	8	interpolant	interpolant	NOUN
ap-6125	47	9	is	be	AUX
ap-6125	47	10	clearly	clearly	ADV
ap-6125	47	11	formulated	formulate	VERB
ap-6125	47	12	by	by	ADP
ap-6125	47	13	mathematical	mathematical	ADJ
ap-6125	47	14	means	mean	NOUN
ap-6125	47	15	but	but	CCONJ
ap-6125	47	16	the	the	DET
ap-6125	47	17	behavior	behavior	NOUN
ap-6125	47	18	of	of	ADP
ap-6125	47	19	the	the	DET
ap-6125	47	20	interpolating	interpolate	VERB
ap-6125	47	21	curve	curve	NOUN
ap-6125	47	22	or	or	CCONJ
ap-6125	47	23	the	the	DET
ap-6125	47	24	surface	surface	NOUN
ap-6125	47	25	between	between	ADP
ap-6125	47	26	nodes	node	NOUN
ap-6125	47	27	can	can	AUX
ap-6125	47	28	differ	differ	VERB
ap-6125	47	29	case	case	NOUN
ap-6125	47	30	from	from	ADP
ap-6125	47	31	case	case	NOUN
ap-6125	47	32	and	and	CCONJ
ap-6125	47	33	can	can	AUX
ap-6125	47	34	not	not	PART
ap-6125	47	35	be	be	AUX
ap-6125	47	36	formalized	formalize	VERB
ap-6125	47	37	easily	easily	ADV
ap-6125	47	38	.	.	PUNCT
ap-6125	48	1	the	the	DET
ap-6125	48	2	problem	problem	NOUN
ap-6125	48	3	of	of	ADP
ap-6125	48	4	least	least	ADJ
ap-6125	48	5	squares	square	NOUN
ap-6125	48	6	data	datum	NOUN
ap-6125	48	7	approximation	approximation	NOUN
ap-6125	48	8	is	be	AUX
ap-6125	48	9	more	more	ADV
ap-6125	48	10	general	general	ADJ
ap-6125	48	11	than	than	ADP
ap-6125	48	12	the	the	DET
ap-6125	48	13	problem	problem	NOUN
ap-6125	48	14	of	of	ADP
ap-6125	48	15	data	datum	NOUN
ap-6125	48	16	interpolation	interpolation	NOUN
ap-6125	48	17	.	.	PUNCT
ap-6125	49	1	no	no	DET
ap-6125	49	2	explicit	explicit	ADJ
ap-6125	49	3	interpolation	interpolation	NOUN
ap-6125	49	4	conditions	condition	NOUN
ap-6125	49	5	of	of	ADP
ap-6125	49	6	the	the	DET
ap-6125	49	7	form	form	NOUN
ap-6125	49	8	(	(	PUNCT
ap-6125	49	9	1	1	X
ap-6125	49	10	)	)	PUNCT
ap-6125	49	11	are	be	AUX
ap-6125	49	12	to	to	PART
ap-6125	49	13	be	be	AUX
ap-6125	49	14	satisfied	satisfied	ADJ
ap-6125	49	15	,	,	PUNCT
ap-6125	49	16	but	but	CCONJ
ap-6125	49	17	the	the	DET
ap-6125	49	18	approximating	approximate	VERB
ap-6125	49	19	function	function	NOUN
ap-6125	49	20	z	z	NOUN
ap-6125	49	21	is	be	AUX
ap-6125	49	22	constructed	construct	VERB
ap-6125	49	23	to	to	PART
ap-6125	49	24	minimize	minimize	VERB
ap-6125	49	25	the	the	DET
ap-6125	49	26	least	least	ADJ
ap-6125	49	27	squares	square	NOUN
ap-6125	49	28	functional	functional	ADJ
ap-6125	49	29	n∑	n∑	NOUN
ap-6125	49	30	j=1	j=1	PROPN
ap-6125	49	31	wj(z(xj)−	wj(z(xj)−	PROPN
ap-6125	49	32	fj)(z(xj)−	fj)(z(xj)−	PROPN
ap-6125	50	1	fj)∗	fj)∗	PROPN
ap-6125	50	2	,	,	PUNCT
ap-6125	50	3	where	where	SCONJ
ap-6125	50	4	wj	wj	PROPN
ap-6125	50	5	,	,	PUNCT
ap-6125	50	6	j	j	PROPN
ap-6125	50	7	=	=	SYM
ap-6125	50	8	1	1	NUM
ap-6125	50	9	,	,	PUNCT
ap-6125	50	10	.	.	PUNCT
ap-6125	50	11	.	.	PUNCT
ap-6125	50	12	.	.	PUNCT
ap-6125	51	1	,	,	PUNCT
ap-6125	51	2	n	n	CCONJ
ap-6125	51	3	,	,	PUNCT
ap-6125	51	4	are	be	AUX
ap-6125	51	5	positive	positive	ADJ
ap-6125	51	6	weights	weight	NOUN
ap-6125	51	7	and	and	CCONJ
ap-6125	51	8	∗	∗	NOUN
ap-6125	51	9	denotes	denote	VERB
ap-6125	51	10	the	the	DET
ap-6125	51	11	complex	complex	ADJ
ap-6125	51	12	conjugate	conjugate	NOUN
ap-6125	51	13	,	,	PUNCT
ap-6125	51	14	cf	cf	NOUN
ap-6125	51	15	.	.	PUNCT
ap-6125	51	16	,	,	PUNCT
ap-6125	51	17	e.g.	e.g.	ADV
ap-6125	51	18	,	,	PUNCT
ap-6125	51	19	[	[	X
ap-6125	51	20	1	1	NUM
ap-6125	51	21	]	]	PUNCT
ap-6125	51	22	.	.	PUNCT
ap-6125	52	1	various	various	ADJ
ap-6125	52	2	additional	additional	ADJ
ap-6125	52	3	conditions	condition	NOUN
ap-6125	52	4	can	can	AUX
ap-6125	52	5	be	be	AUX
ap-6125	52	6	considered	consider	VERB
ap-6125	52	7	,	,	PUNCT
ap-6125	52	8	for	for	ADP
ap-6125	52	9	example	example	NOUN
ap-6125	52	10	,	,	PUNCT
ap-6125	52	11	the	the	DET
ap-6125	52	12	minimization	minimization	NOUN
ap-6125	52	13	of	of	ADP
ap-6125	52	14	some	some	DET
ap-6125	52	15	further	further	ADJ
ap-6125	52	16	functionals	functional	NOUN
ap-6125	52	17	applied	apply	VERB
ap-6125	52	18	to	to	ADP
ap-6125	52	19	z.	z.	PROPN
ap-6125	52	20	in	in	ADP
ap-6125	52	21	some	some	DET
ap-6125	52	22	branches	branch	NOUN
ap-6125	52	23	of	of	ADP
ap-6125	52	24	science	science	NOUN
ap-6125	52	25	,	,	PUNCT
ap-6125	52	26	the	the	DET
ap-6125	52	27	terminology	terminology	NOUN
ap-6125	52	28	may	may	AUX
ap-6125	52	29	differ	differ	VERB
ap-6125	52	30	.	.	PUNCT
ap-6125	53	1	the	the	DET
ap-6125	53	2	terms	term	NOUN
ap-6125	53	3	exact	exact	ADJ
ap-6125	53	4	and	and	CCONJ
ap-6125	53	5	inexact	inexact	ADJ
ap-6125	53	6	interpolation	interpolation	NOUN
ap-6125	53	7	are	be	AUX
ap-6125	53	8	also	also	ADV
ap-6125	53	9	used	use	VERB
ap-6125	53	10	if	if	SCONJ
ap-6125	53	11	the	the	DET
ap-6125	53	12	interpolant	interpolant	NOUN
ap-6125	53	13	or	or	CCONJ
ap-6125	53	14	approximant	approximant	ADJ
ap-6125	53	15	satisfies	satisfie	NOUN
ap-6125	53	16	the	the	DET
ap-6125	53	17	conditions	condition	NOUN
ap-6125	53	18	(	(	PUNCT
ap-6125	53	19	1	1	NUM
ap-6125	53	20	)	)	PUNCT
ap-6125	53	21	or	or	CCONJ
ap-6125	53	22	not	not	PART
ap-6125	53	23	.	.	PUNCT
ap-6125	54	1	we	we	PRON
ap-6125	54	2	are	be	AUX
ap-6125	54	3	not	not	PART
ap-6125	54	4	concerned	concern	VERB
ap-6125	54	5	with	with	ADP
ap-6125	54	6	the	the	DET
ap-6125	54	7	general	general	ADJ
ap-6125	54	8	data	datum	NOUN
ap-6125	54	9	approximation	approximation	NOUN
ap-6125	54	10	in	in	ADP
ap-6125	54	11	the	the	DET
ap-6125	54	12	paper	paper	NOUN
ap-6125	54	13	.	.	PUNCT
ap-6125	55	1	3	3	X
ap-6125	55	2	.	.	X
ap-6125	55	3	interpolation	interpolation	NOUN
ap-6125	55	4	with	with	ADP
ap-6125	55	5	radial	radial	ADJ
ap-6125	55	6	basis	basis	NOUN
ap-6125	55	7	functions	function	NOUN
ap-6125	55	8	let	let	VERB
ap-6125	55	9	x	x	PRON
ap-6125	55	10	,	,	PUNCT
ap-6125	55	11	y	y	PROPN
ap-6125	55	12	∈	∈	PROPN
ap-6125	55	13	rn	rn	PROPN
ap-6125	55	14	and	and	CCONJ
ap-6125	55	15	r(x	r(x	PROPN
ap-6125	55	16	,	,	PUNCT
ap-6125	55	17	y	y	NOUN
ap-6125	55	18	)	)	PUNCT
ap-6125	55	19	=	=	SYM
ap-6125	56	1	‖x−	‖x−	PROPN
ap-6125	56	2	y‖e	y‖e	NOUN
ap-6125	56	3	=	=	PUNCT
ap-6125	56	4	√√√√	√√√√	PRON
ap-6125	56	5	n∑	n∑	NOUN
ap-6125	56	6	s=1	s=1	X
ap-6125	56	7	(	(	PUNCT
ap-6125	56	8	xs	xs	PROPN
ap-6125	56	9	−	−	PROPN
ap-6125	56	10	ys)2	ys)2	PROPN
ap-6125	56	11	(	(	PUNCT
ap-6125	56	12	2	2	X
ap-6125	56	13	)	)	PUNCT
ap-6125	56	14	be	be	AUX
ap-6125	56	15	the	the	DET
ap-6125	56	16	euclidean	euclidean	ADJ
ap-6125	56	17	norm	norm	NOUN
ap-6125	56	18	of	of	ADP
ap-6125	56	19	the	the	DET
ap-6125	56	20	vector	vector	NOUN
ap-6125	56	21	x	x	PUNCT
ap-6125	56	22	−	−	PROPN
ap-6125	56	23	y	y	PROPN
ap-6125	56	24	∈	∈	PROPN
ap-6125	56	25	rn	rn	PROPN
ap-6125	56	26	.	.	PUNCT
ap-6125	57	1	the	the	DET
ap-6125	57	2	dimension	dimension	NOUN
ap-6125	57	3	n	n	PROPN
ap-6125	57	4	of	of	ADP
ap-6125	57	5	the	the	DET
ap-6125	57	6	independent	independent	ADJ
ap-6125	57	7	variable	variable	NOUN
ap-6125	57	8	can	can	AUX
ap-6125	57	9	be	be	AUX
ap-6125	57	10	arbitrary	arbitrary	ADJ
ap-6125	57	11	.	.	PUNCT
ap-6125	58	1	definition	definition	NOUN
ap-6125	58	2	2	2	NUM
ap-6125	58	3	(	(	PUNCT
ap-6125	58	4	radial	radial	ADJ
ap-6125	58	5	function	function	NOUN
ap-6125	58	6	)	)	PUNCT
ap-6125	58	7	.	.	PUNCT
ap-6125	59	1	we	we	PRON
ap-6125	59	2	say	say	VERB
ap-6125	59	3	that	that	SCONJ
ap-6125	59	4	the	the	DET
ap-6125	59	5	function	function	NOUN
ap-6125	59	6	f	f	X
ap-6125	59	7	(	(	PUNCT
ap-6125	59	8	x	x	PROPN
ap-6125	59	9	,	,	PUNCT
ap-6125	59	10	y	y	NOUN
ap-6125	59	11	)	)	PUNCT
ap-6125	59	12	=	=	SYM
ap-6125	59	13	f̂	f̂	X
ap-6125	59	14	(	(	PUNCT
ap-6125	59	15	r(x	r(x	PROPN
ap-6125	59	16	,	,	PUNCT
ap-6125	59	17	y	y	NOUN
ap-6125	59	18	)	)	PUNCT
ap-6125	59	19	)	)	PUNCT
ap-6125	59	20	depending	depend	VERB
ap-6125	59	21	only	only	ADV
ap-6125	59	22	on	on	ADP
ap-6125	59	23	r	r	NOUN
ap-6125	59	24	from	from	ADP
ap-6125	59	25	(	(	PUNCT
ap-6125	59	26	2	2	NUM
ap-6125	59	27	)	)	PUNCT
ap-6125	59	28	is	be	AUX
ap-6125	59	29	a	a	DET
ap-6125	59	30	radial	radial	ADJ
ap-6125	59	31	function	function	NOUN
ap-6125	59	32	.	.	PUNCT
ap-6125	60	1	radial	radial	ADJ
ap-6125	60	2	functions	function	NOUN
ap-6125	60	3	are	be	AUX
ap-6125	60	4	radially	radially	ADV
ap-6125	60	5	symmetric	symmetric	ADJ
ap-6125	60	6	.	.	PUNCT
ap-6125	61	1	they	they	PRON
ap-6125	61	2	are	be	AUX
ap-6125	61	3	often	often	ADV
ap-6125	61	4	used	use	VERB
ap-6125	61	5	as	as	ADP
ap-6125	61	6	basis	basis	NOUN
ap-6125	61	7	functions	function	NOUN
ap-6125	61	8	for	for	ADP
ap-6125	61	9	interpolation	interpolation	NOUN
ap-6125	61	10	as	as	ADV
ap-6125	61	11	well	well	ADV
ap-6125	61	12	as	as	ADP
ap-6125	61	13	approximation	approximation	NOUN
ap-6125	61	14	.	.	PUNCT
ap-6125	62	1	it	it	PRON
ap-6125	62	2	is	be	AUX
ap-6125	62	3	assumed	assume	VERB
ap-6125	62	4	that	that	SCONJ
ap-6125	62	5	every	every	DET
ap-6125	62	6	item	item	NOUN
ap-6125	62	7	fj	fj	PROPN
ap-6125	62	8	of	of	ADP
ap-6125	62	9	the	the	DET
ap-6125	62	10	measured	measure	VERB
ap-6125	62	11	data	datum	NOUN
ap-6125	62	12	at	at	ADP
ap-6125	62	13	the	the	DET
ap-6125	62	14	node	node	PROPN
ap-6125	62	15	xj	xj	PROPN
ap-6125	62	16	influences	influence	VERB
ap-6125	62	17	the	the	DET
ap-6125	62	18	result	result	NOUN
ap-6125	62	19	of	of	ADP
ap-6125	62	20	interpolation	interpolation	NOUN
ap-6125	62	21	or	or	CCONJ
ap-6125	62	22	approximation	approximation	NOUN
ap-6125	62	23	at	at	ADP
ap-6125	62	24	a	a	DET
ap-6125	62	25	point	point	NOUN
ap-6125	62	26	x	x	X
ap-6125	62	27	in	in	ADP
ap-6125	62	28	the	the	DET
ap-6125	62	29	vicinity	vicinity	NOUN
ap-6125	62	30	of	of	ADP
ap-6125	62	31	xj	xj	PROPN
ap-6125	62	32	proportionally	proportionally	ADV
ap-6125	62	33	,	,	PUNCT
ap-6125	62	34	in	in	ADP
ap-6125	62	35	some	some	DET
ap-6125	62	36	sense	sense	NOUN
ap-6125	62	37	,	,	PUNCT
ap-6125	62	38	to	to	ADP
ap-6125	62	39	its	its	PRON
ap-6125	62	40	distance	distance	NOUN
ap-6125	62	41	r(x	r(x	PROPN
ap-6125	62	42	,	,	PUNCT
ap-6125	62	43	xj	xj	PROPN
ap-6125	62	44	)	)	PUNCT
ap-6125	62	45	from	from	ADP
ap-6125	62	46	xj	xj	PROPN
ap-6125	62	47	if	if	SCONJ
ap-6125	62	48	this	this	DET
ap-6125	62	49	vicinity	vicinity	NOUN
ap-6125	62	50	can	can	AUX
ap-6125	62	51	be	be	AUX
ap-6125	62	52	considered	consider	VERB
ap-6125	62	53	“	"	PUNCT
ap-6125	62	54	homogeneous	homogeneous	ADJ
ap-6125	62	55	”	"	PUNCT
ap-6125	62	56	.	.	PUNCT
ap-6125	63	1	the	the	DET
ap-6125	63	2	vector	vector	NOUN
ap-6125	63	3	α	α	NOUN
ap-6125	63	4	=	=	SYM
ap-6125	63	5	(	(	PUNCT
ap-6125	63	6	α1	α1	PROPN
ap-6125	63	7	,	,	PUNCT
ap-6125	63	8	.	.	PUNCT
ap-6125	63	9	.	.	PUNCT
ap-6125	63	10	.	.	PUNCT
ap-6125	64	1	,	,	PUNCT
ap-6125	64	2	αn	αn	NOUN
ap-6125	64	3	)	)	PUNCT
ap-6125	64	4	,	,	PUNCT
ap-6125	64	5	where	where	SCONJ
ap-6125	64	6	αs	αs	ADJ
ap-6125	64	7	,	,	PUNCT
ap-6125	64	8	s	s	PART
ap-6125	64	9	=	=	NOUN
ap-6125	64	10	1	1	NUM
ap-6125	64	11	,	,	PUNCT
ap-6125	64	12	.	.	PUNCT
ap-6125	64	13	.	.	PUNCT
ap-6125	65	1	.	.	PUNCT
ap-6125	66	1	,	,	PUNCT
ap-6125	66	2	n	n	CCONJ
ap-6125	66	3	,	,	PUNCT
ap-6125	66	4	are	be	AUX
ap-6125	66	5	integers	integer	NOUN
ap-6125	66	6	,	,	PUNCT
ap-6125	66	7	is	be	AUX
ap-6125	66	8	called	call	VERB
ap-6125	66	9	a	a	DET
ap-6125	66	10	multiindex	multiindex	NOUN
ap-6125	66	11	.	.	PUNCT
ap-6125	67	1	denote	denote	VERB
ap-6125	67	2	the	the	DET
ap-6125	67	3	length	length	NOUN
ap-6125	67	4	of	of	ADP
ap-6125	67	5	a	a	DET
ap-6125	67	6	multiindex	multiindex	NOUN
ap-6125	67	7	α	α	NOUN
ap-6125	67	8	by	by	ADP
ap-6125	67	9	|α|	|α|	PROPN
ap-6125	67	10	=	=	SYM
ap-6125	67	11	n∑	n∑	NOUN
ap-6125	68	1	s=1	s=1	X
ap-6125	68	2	|αs|	|αs|	NOUN
ap-6125	68	3	,	,	PUNCT
ap-6125	68	4	(	(	PUNCT
ap-6125	68	5	3	3	X
ap-6125	68	6	)	)	PUNCT
ap-6125	68	7	where	where	SCONJ
ap-6125	68	8	|αs|	|αs|	NOUN
ap-6125	68	9	means	mean	VERB
ap-6125	68	10	the	the	DET
ap-6125	68	11	absolute	absolute	ADJ
ap-6125	68	12	value	value	NOUN
ap-6125	68	13	of	of	ADP
ap-6125	68	14	the	the	DET
ap-6125	68	15	component	component	NOUN
ap-6125	69	1	αs	αs	INTJ
ap-6125	69	2	.	.	PUNCT
ap-6125	70	1	we	we	PRON
ap-6125	70	2	say	say	VERB
ap-6125	70	3	that	that	SCONJ
ap-6125	70	4	α	α	PRON
ap-6125	70	5	is	be	AUX
ap-6125	70	6	a	a	DET
ap-6125	70	7	nonnegative	nonnegative	ADJ
ap-6125	70	8	multiindex	multiindex	NOUN
ap-6125	70	9	if	if	SCONJ
ap-6125	70	10	αs	αs	PROPN
ap-6125	70	11	≥	≥	X
ap-6125	70	12	0	0	NUM
ap-6125	70	13	holds	hold	VERB
ap-6125	70	14	for	for	ADP
ap-6125	70	15	all	all	PRON
ap-6125	70	16	s	s	PART
ap-6125	70	17	=	=	NOUN
ap-6125	70	18	1	1	NUM
ap-6125	70	19	,	,	PUNCT
ap-6125	70	20	.	.	PUNCT
ap-6125	70	21	.	.	PUNCT
ap-6125	70	22	.	.	PUNCT
ap-6125	71	1	,	,	PUNCT
ap-6125	71	2	n.	n.	PROPN
ap-6125	71	3	choose	choose	VERB
ap-6125	71	4	a	a	DET
ap-6125	71	5	nonnegative	nonnegative	ADJ
ap-6125	71	6	integer	integer	NOUN
ap-6125	71	7	l	l	NOUN
ap-6125	71	8	and	and	CCONJ
ap-6125	71	9	consider	consider	VERB
ap-6125	71	10	the	the	DET
ap-6125	71	11	interpolant	interpolant	NOUN
ap-6125	71	12	z(x	z(x	NOUN
ap-6125	71	13	)	)	PUNCT
ap-6125	71	14	=	=	SYM
ap-6125	71	15	n∑	n∑	PROPN
ap-6125	71	16	j=1	j=1	PROPN
ap-6125	71	17	λjf	λjf	PROPN
ap-6125	71	18	(	(	PUNCT
ap-6125	71	19	x	x	X
ap-6125	71	20	,	,	PUNCT
ap-6125	71	21	xj	xj	PROPN
ap-6125	71	22	)	)	PUNCT
ap-6125	72	1	+	+	CCONJ
ap-6125	72	2	∑	∑	PROPN
ap-6125	72	3	|α|≤l−1	|α|≤l−1	PROPN
ap-6125	72	4	aαϕα(x	aαϕα(x	PROPN
ap-6125	72	5	)	)	PUNCT
ap-6125	72	6	,	,	PUNCT
ap-6125	72	7	(	(	PUNCT
ap-6125	72	8	4	4	X
ap-6125	72	9	)	)	PUNCT
ap-6125	72	10	where	where	SCONJ
ap-6125	72	11	α	α	NOUN
ap-6125	72	12	is	be	AUX
ap-6125	72	13	a	a	DET
ap-6125	72	14	nonnegative	nonnegative	ADJ
ap-6125	72	15	multiindex	multiindex	NOUN
ap-6125	72	16	,	,	PUNCT
ap-6125	72	17	f	f	PROPN
ap-6125	72	18	(	(	PUNCT
ap-6125	72	19	x	x	PROPN
ap-6125	72	20	,	,	PUNCT
ap-6125	72	21	y	y	NOUN
ap-6125	72	22	)	)	PUNCT
ap-6125	72	23	=	=	SYM
ap-6125	72	24	f̂	f̂	X
ap-6125	72	25	(	(	PUNCT
ap-6125	72	26	r(x	r(x	PROPN
ap-6125	72	27	,	,	PUNCT
ap-6125	72	28	y	y	NOUN
ap-6125	72	29	)	)	PUNCT
ap-6125	72	30	)	)	PUNCT
ap-6125	72	31	is	be	AUX
ap-6125	72	32	a	a	DET
ap-6125	72	33	radial	radial	ADJ
ap-6125	72	34	basis	basis	NOUN
ap-6125	72	35	function	function	NOUN
ap-6125	72	36	(	(	PUNCT
ap-6125	72	37	e.g.	e.g.	ADV
ap-6125	72	38	a	a	DET
ap-6125	72	39	proper	proper	ADJ
ap-6125	72	40	polyharmonic	polyharmonic	ADJ
ap-6125	72	41	spline	spline	NOUN
ap-6125	72	42	,	,	PUNCT
ap-6125	72	43	see	see	VERB
ap-6125	72	44	sec	sec	PROPN
ap-6125	72	45	.	.	PROPN
ap-6125	72	46	4	4	NUM
ap-6125	72	47	)	)	PUNCT
ap-6125	72	48	,	,	PUNCT
ap-6125	72	49	ϕα	ϕα	ADV
ap-6125	72	50	are	be	AUX
ap-6125	72	51	all	all	DET
ap-6125	72	52	the	the	DET
ap-6125	72	53	monomials	monomial	NOUN
ap-6125	72	54	of	of	ADP
ap-6125	72	55	the	the	DET
ap-6125	72	56	form	form	NOUN
ap-6125	72	57	ϕα(x	ϕα(x	PUNCT
ap-6125	72	58	)	)	PUNCT
ap-6125	72	59	=	=	PUNCT
ap-6125	73	1	xα1	xα1	NOUN
ap-6125	73	2	1	1	X
ap-6125	73	3	.	.	PUNCT
ap-6125	73	4	.	.	PUNCT
ap-6125	73	5	.	.	PUNCT
ap-6125	74	1	xαn	xαn	PROPN
ap-6125	74	2	n	n	PROPN
ap-6125	74	3	(	(	PUNCT
ap-6125	74	4	5	5	NUM
ap-6125	74	5	)	)	PUNCT
ap-6125	74	6	of	of	ADP
ap-6125	74	7	degree	degree	NOUN
ap-6125	74	8	|α|	|α|	PROPN
ap-6125	74	9	≤	≤	NOUN
ap-6125	75	1	l	l	NOUN
ap-6125	75	2	−	−	PROPN
ap-6125	75	3	1	1	NUM
ap-6125	75	4	called	call	VERB
ap-6125	75	5	trend	trend	NOUN
ap-6125	75	6	functions	function	NOUN
ap-6125	75	7	,	,	PUNCT
ap-6125	75	8	and	and	CCONJ
ap-6125	75	9	λj	λj	INTJ
ap-6125	75	10	,	,	PUNCT
ap-6125	75	11	j	j	PROPN
ap-6125	75	12	=	=	SYM
ap-6125	75	13	1	1	NUM
ap-6125	75	14	,	,	PUNCT
ap-6125	75	15	.	.	PUNCT
ap-6125	75	16	.	.	PUNCT
ap-6125	76	1	.	.	PUNCT
ap-6125	77	1	,	,	PUNCT
ap-6125	77	2	n	n	CCONJ
ap-6125	77	3	,	,	PUNCT
ap-6125	77	4	and	and	CCONJ
ap-6125	77	5	aα	aα	NOUN
ap-6125	77	6	,	,	PUNCT
ap-6125	77	7	|α|	|α|	PROPN
ap-6125	77	8	≤	≤	NUM
ap-6125	77	9	l−1	l−1	PROPN
ap-6125	77	10	,	,	PUNCT
ap-6125	77	11	are	be	AUX
ap-6125	77	12	coefficients	coefficient	NOUN
ap-6125	77	13	to	to	PART
ap-6125	77	14	be	be	AUX
ap-6125	77	15	found	find	VERB
ap-6125	77	16	;	;	PUNCT
ap-6125	77	17	the	the	DET
ap-6125	77	18	second	second	ADJ
ap-6125	77	19	sum	sum	NOUN
ap-6125	77	20	in	in	ADP
ap-6125	77	21	the	the	DET
ap-6125	77	22	formula	formula	NOUN
ap-6125	77	23	(	(	PUNCT
ap-6125	77	24	4	4	X
ap-6125	77	25	)	)	PUNCT
ap-6125	77	26	is	be	AUX
ap-6125	77	27	empty	empty	ADJ
ap-6125	77	28	if	if	SCONJ
ap-6125	77	29	l	l	NOUN
ap-6125	77	30	=	=	SYM
ap-6125	77	31	0	0	PUNCT
ap-6125	77	32	(	(	PUNCT
ap-6125	77	33	cf	cf	NOUN
ap-6125	77	34	.	.	PUNCT
ap-6125	78	1	[	[	X
ap-6125	78	2	2	2	NUM
ap-6125	78	3	]	]	PUNCT
ap-6125	78	4	)	)	PUNCT
ap-6125	78	5	.	.	PUNCT
ap-6125	79	1	the	the	DET
ap-6125	79	2	interpolation	interpolation	NOUN
ap-6125	79	3	with	with	ADP
ap-6125	79	4	radial	radial	ADJ
ap-6125	79	5	basis	basis	NOUN
ap-6125	79	6	functions	function	NOUN
ap-6125	79	7	is	be	AUX
ap-6125	79	8	widely	widely	ADV
ap-6125	79	9	used	use	VERB
ap-6125	79	10	in	in	ADP
ap-6125	79	11	computational	computational	ADJ
ap-6125	79	12	practice	practice	NOUN
ap-6125	79	13	.	.	PUNCT
ap-6125	80	1	for	for	ADP
ap-6125	80	2	practical	practical	ADJ
ap-6125	80	3	reasons	reason	NOUN
ap-6125	80	4	,	,	PUNCT
ap-6125	80	5	the	the	DET
ap-6125	80	6	basis	basis	NOUN
ap-6125	80	7	functions	function	NOUN
ap-6125	80	8	are	be	AUX
ap-6125	80	9	usually	usually	ADV
ap-6125	80	10	taken	take	VERB
ap-6125	80	11	only	only	ADV
ap-6125	80	12	from	from	ADP
ap-6125	80	13	a	a	DET
ap-6125	80	14	very	very	ADV
ap-6125	80	15	small	small	ADJ
ap-6125	80	16	set	set	NOUN
ap-6125	80	17	of	of	ADP
ap-6125	80	18	functions	function	NOUN
ap-6125	80	19	,	,	PUNCT
ap-6125	80	20	e.g.	e.g.	ADV
ap-6125	80	21	,	,	PUNCT
ap-6125	80	22	√	√	ADP
ap-6125	80	23	r2	r2	PROPN
ap-6125	80	24	+	+	CCONJ
ap-6125	80	25	s2	s2	PROPN
ap-6125	80	26	,	,	PUNCT
ap-6125	80	27	1/	1/	NUM
ap-6125	80	28	√	√	NUM
ap-6125	80	29	r2	r2	PROPN
ap-6125	80	30	+	+	CCONJ
ap-6125	80	31	s2	s2	PROPN
ap-6125	80	32	,	,	PUNCT
ap-6125	80	33	exp(−sr2	exp(−sr2	NOUN
ap-6125	80	34	)	)	PUNCT
ap-6125	80	35	,	,	PUNCT
ap-6125	80	36	or	or	CCONJ
ap-6125	80	37	r2	r2	PROPN
ap-6125	80	38	ln(r	ln(r	NOUN
ap-6125	80	39	/	/	SYM
ap-6125	80	40	s	s	NOUN
ap-6125	80	41	)	)	PUNCT
ap-6125	80	42	,	,	PUNCT
ap-6125	80	43	where	where	SCONJ
ap-6125	80	44	s	s	VERB
ap-6125	80	45	>	>	X
ap-6125	80	46	0	0	NUM
ap-6125	80	47	is	be	AUX
ap-6125	80	48	a	a	DET
ap-6125	80	49	constant	constant	ADJ
ap-6125	80	50	.	.	PUNCT
ap-6125	81	1	4	4	X
ap-6125	81	2	.	.	NOUN
ap-6125	81	3	polyharmonic	polyharmonic	ADJ
ap-6125	81	4	splines	spline	NOUN
ap-6125	81	5	definition	definition	NOUN
ap-6125	81	6	3	3	NUM
ap-6125	81	7	(	(	PUNCT
ap-6125	81	8	polyharmonic	polyharmonic	ADJ
ap-6125	81	9	spline	spline	NOUN
ap-6125	81	10	)	)	PUNCT
ap-6125	81	11	.	.	PUNCT
ap-6125	82	1	let	let	VERB
ap-6125	82	2	r(x	r(x	PROPN
ap-6125	82	3	,	,	PUNCT
ap-6125	82	4	y	y	PROPN
ap-6125	82	5	)	)	PUNCT
ap-6125	82	6	be	be	VERB
ap-6125	82	7	the	the	DET
ap-6125	82	8	euclidean	euclidean	ADJ
ap-6125	82	9	norm	norm	NOUN
ap-6125	82	10	(	(	PUNCT
ap-6125	82	11	2	2	NUM
ap-6125	82	12	)	)	PUNCT
ap-6125	82	13	of	of	ADP
ap-6125	82	14	the	the	DET
ap-6125	82	15	vector	vector	NOUN
ap-6125	82	16	x−	x−	PROPN
ap-6125	82	17	y	y	PROPN
ap-6125	82	18	∈	∈	PROPN
ap-6125	82	19	rn	rn	PROPN
ap-6125	82	20	.	.	PUNCT
ap-6125	83	1	the	the	DET
ap-6125	83	2	functions	function	NOUN
ap-6125	83	3	rq	rq	VERB
ap-6125	83	4	,	,	PUNCT
ap-6125	83	5	q	q	NOUN
ap-6125	83	6	=	=	NOUN
ap-6125	83	7	1	1	NUM
ap-6125	83	8	,	,	PUNCT
ap-6125	83	9	3	3	NUM
ap-6125	83	10	,	,	PUNCT
ap-6125	83	11	.	.	PUNCT
ap-6125	83	12	.	.	PUNCT
ap-6125	83	13	.	.	PUNCT
ap-6125	84	1	,	,	PUNCT
ap-6125	84	2	(	(	PUNCT
ap-6125	84	3	6	6	X
ap-6125	84	4	)	)	PUNCT
ap-6125	84	5	rq	rq	X
ap-6125	84	6	ln	ln	ADJ
ap-6125	84	7	r	r	NOUN
ap-6125	84	8	,	,	PUNCT
ap-6125	84	9	q	q	NOUN
ap-6125	84	10	=	=	SYM
ap-6125	84	11	2	2	NUM
ap-6125	84	12	,	,	PUNCT
ap-6125	84	13	4	4	NUM
ap-6125	84	14	,	,	PUNCT
ap-6125	84	15	.	.	PUNCT
ap-6125	84	16	.	.	PUNCT
ap-6125	84	17	.	.	PUNCT
ap-6125	85	1	,	,	PUNCT
ap-6125	85	2	(	(	PUNCT
ap-6125	85	3	7	7	X
ap-6125	85	4	)	)	PUNCT
ap-6125	85	5	are	be	AUX
ap-6125	85	6	called	call	VERB
ap-6125	85	7	polyharmonic	polyharmonic	ADJ
ap-6125	85	8	splines	spline	NOUN
ap-6125	85	9	.	.	PUNCT
ap-6125	86	1	the	the	DET
ap-6125	86	2	equation	equation	NOUN
ap-6125	86	3	∆m	∆m	PROPN
ap-6125	86	4	u(x1	u(x1	PROPN
ap-6125	86	5	,	,	PUNCT
ap-6125	86	6	.	.	PUNCT
ap-6125	86	7	.	.	PUNCT
ap-6125	86	8	.	.	PUNCT
ap-6125	87	1	,	,	PUNCT
ap-6125	87	2	xn	xn	X
ap-6125	87	3	)	)	PUNCT
ap-6125	87	4	=	=	SYM
ap-6125	87	5	0	0	NUM
ap-6125	87	6	,	,	PUNCT
ap-6125	87	7	(	(	PUNCT
ap-6125	87	8	8)	8)	NUM
ap-6125	87	9	where	where	SCONJ
ap-6125	87	10	∆	∆	VERB
ap-6125	87	11	=	=	SYM
ap-6125	87	12	∂2/∂x2	∂2/∂x2	NOUN
ap-6125	87	13	1	1	NUM
ap-6125	87	14	+	+	CCONJ
ap-6125	87	15	·	·	PUNCT
ap-6125	87	16	·	·	PUNCT
ap-6125	87	17	·	·	PUNCT
ap-6125	88	1	+	+	NOUN
ap-6125	88	2	∂2/∂x2	∂2/∂x2	NOUN
ap-6125	88	3	n	n	PRON
ap-6125	88	4	is	be	AUX
ap-6125	88	5	the	the	DET
ap-6125	88	6	laplace	laplace	NOUN
ap-6125	88	7	operator	operator	NOUN
ap-6125	88	8	,	,	PUNCT
ap-6125	88	9	is	be	AUX
ap-6125	88	10	called	call	VERB
ap-6125	88	11	the	the	DET
ap-6125	88	12	polyharmonic	polyharmonic	ADJ
ap-6125	88	13	equation	equation	NOUN
ap-6125	88	14	of	of	ADP
ap-6125	88	15	order	order	NOUN
ap-6125	88	16	m	m	NOUN
ap-6125	88	17	,	,	PUNCT
ap-6125	88	18	cf	cf	INTJ
ap-6125	88	19	.	.	PUNCT
ap-6125	89	1	[	[	X
ap-6125	89	2	2	2	NUM
ap-6125	89	3	]	]	PUNCT
ap-6125	89	4	.	.	PUNCT
ap-6125	90	1	apparently	apparently	ADV
ap-6125	90	2	,	,	PUNCT
ap-6125	90	3	all	all	DET
ap-6125	90	4	the	the	DET
ap-6125	90	5	derivatives	derivative	NOUN
ap-6125	90	6	in	in	ADP
ap-6125	90	7	the	the	DET
ap-6125	90	8	equation	equation	NOUN
ap-6125	90	9	(	(	PUNCT
ap-6125	90	10	8)	8)	NUM
ap-6125	90	11	are	be	AUX
ap-6125	90	12	of	of	ADP
ap-6125	90	13	order	order	NOUN
ap-6125	90	14	2	2	NUM
ap-6125	90	15	m.	m.	NOUN
ap-6125	90	16	polyharmonic	polyharmonic	ADJ
ap-6125	90	17	splines	spline	NOUN
ap-6125	90	18	solve	solve	VERB
ap-6125	90	19	the	the	DET
ap-6125	90	20	respective	respective	ADJ
ap-6125	90	21	polyharmonic	polyharmonic	ADJ
ap-6125	90	22	equation	equation	NOUN
ap-6125	90	23	.	.	PUNCT
ap-6125	91	1	the	the	DET
ap-6125	91	2	next	next	ADJ
ap-6125	91	3	theorem	theorem	NOUN
ap-6125	91	4	presents	present	VERB
ap-6125	91	5	the	the	DET
ap-6125	91	6	exact	exact	ADJ
ap-6125	91	7	statement	statement	NOUN
ap-6125	91	8	.	.	PUNCT
ap-6125	92	1	149	149	NUM
ap-6125	92	2	karel	karel	PROPN
ap-6125	92	3	segeth	segeth	PROPN
ap-6125	92	4	acta	acta	PROPN
ap-6125	92	5	polytechnica	polytechnica	PROPN
ap-6125	92	6	theorem	theorem	VERB
ap-6125	92	7	1	1	X
ap-6125	92	8	.	.	PUNCT
ap-6125	93	1	fix	fix	VERB
ap-6125	93	2	the	the	DET
ap-6125	93	3	vector	vector	NOUN
ap-6125	93	4	y	y	PROPN
ap-6125	93	5	∈	∈	PROPN
ap-6125	93	6	rn	rn	PROPN
ap-6125	93	7	.	.	PROPN
ap-6125	94	1	then	then	ADV
ap-6125	94	2	the	the	DET
ap-6125	94	3	polyharmonic	polyharmonic	ADJ
ap-6125	94	4	spline	spline	NOUN
ap-6125	94	5	rq	rq	NOUN
ap-6125	94	6	(	(	PUNCT
ap-6125	94	7	or	or	CCONJ
ap-6125	94	8	rq	rq	VERB
ap-6125	94	9	ln	ln	ADJ
ap-6125	94	10	r	r	NOUN
ap-6125	94	11	)	)	PUNCT
ap-6125	94	12	solves	solve	VERB
ap-6125	94	13	the	the	DET
ap-6125	94	14	polyharmonic	polyharmonic	ADJ
ap-6125	94	15	equation	equation	NOUN
ap-6125	94	16	in	in	ADP
ap-6125	94	17	variable	variable	NOUN
ap-6125	94	18	x	x	X
ap-6125	94	19	of	of	ADP
ap-6125	94	20	order	order	NOUN
ap-6125	94	21	m	m	VERB
ap-6125	94	22	=	=	SYM
ap-6125	94	23	1	1	NUM
ap-6125	94	24	2	2	NUM
ap-6125	94	25	(	(	PUNCT
ap-6125	94	26	q+n	q+n	NUM
ap-6125	94	27	)	)	PUNCT
ap-6125	94	28	in	in	ADP
ap-6125	94	29	rny	rny	NOUN
ap-6125	94	30	=	=	SYM
ap-6125	94	31	rn	rn	PROPN
ap-6125	94	32	\	\	PROPN
ap-6125	94	33	{	{	PUNCT
ap-6125	94	34	x	x	SYM
ap-6125	94	35	=	=	SYM
ap-6125	94	36	y	y	NOUN
ap-6125	94	37	}	}	PUNCT
ap-6125	94	38	for	for	ADP
ap-6125	94	39	n	n	PRON
ap-6125	94	40	odd	odd	ADJ
ap-6125	94	41	(	(	PUNCT
ap-6125	94	42	or	or	CCONJ
ap-6125	94	43	for	for	ADP
ap-6125	94	44	n	n	PRON
ap-6125	94	45	even	even	ADV
ap-6125	94	46	)	)	PUNCT
ap-6125	94	47	.	.	PUNCT
ap-6125	95	1	proof	proof	NOUN
ap-6125	95	2	.	.	PUNCT
ap-6125	96	1	it	it	PRON
ap-6125	96	2	is	be	AUX
ap-6125	96	3	easy	easy	ADJ
ap-6125	96	4	to	to	PART
ap-6125	96	5	prove	prove	VERB
ap-6125	96	6	the	the	DET
ap-6125	96	7	statement	statement	NOUN
ap-6125	96	8	by	by	ADP
ap-6125	96	9	direct	direct	ADJ
ap-6125	96	10	computation	computation	NOUN
ap-6125	96	11	.	.	PUNCT
ap-6125	97	1	note	note	VERB
ap-6125	97	2	that	that	SCONJ
ap-6125	97	3	the	the	DET
ap-6125	97	4	term	term	NOUN
ap-6125	97	5	spline	spline	NOUN
ap-6125	97	6	is	be	AUX
ap-6125	97	7	used	use	VERB
ap-6125	97	8	here	here	ADV
ap-6125	97	9	also	also	ADV
ap-6125	97	10	for	for	ADP
ap-6125	97	11	a	a	DET
ap-6125	97	12	nonpolynomial	nonpolynomial	ADJ
ap-6125	97	13	function	function	NOUN
ap-6125	97	14	.	.	PUNCT
ap-6125	98	1	another	another	DET
ap-6125	98	2	(	(	PUNCT
ap-6125	98	3	weak	weak	ADJ
ap-6125	98	4	)	)	PUNCT
ap-6125	98	5	definition	definition	NOUN
ap-6125	98	6	of	of	ADP
ap-6125	98	7	the	the	DET
ap-6125	98	8	polyharmonic	polyharmonic	ADJ
ap-6125	98	9	spline	spline	NOUN
ap-6125	98	10	can	can	AUX
ap-6125	98	11	be	be	AUX
ap-6125	98	12	given	give	VERB
ap-6125	98	13	with	with	ADP
ap-6125	98	14	the	the	DET
ap-6125	98	15	help	help	NOUN
ap-6125	98	16	of	of	ADP
ap-6125	98	17	the	the	DET
ap-6125	98	18	dirac	dirac	NOUN
ap-6125	98	19	function	function	NOUN
ap-6125	98	20	.	.	PUNCT
ap-6125	99	1	apparently	apparently	ADV
ap-6125	99	2	,	,	PUNCT
ap-6125	99	3	r(x	r(x	PROPN
ap-6125	99	4	,	,	PUNCT
ap-6125	99	5	y	y	PROPN
ap-6125	99	6	)	)	PUNCT
ap-6125	99	7	is	be	AUX
ap-6125	99	8	a	a	DET
ap-6125	99	9	real	real	ADJ
ap-6125	99	10	radial	radial	ADJ
ap-6125	99	11	basis	basis	NOUN
ap-6125	99	12	function	function	NOUN
ap-6125	99	13	.	.	PUNCT
ap-6125	100	1	all	all	DET
ap-6125	100	2	polyharmonic	polyharmonic	ADJ
ap-6125	100	3	splines	spline	NOUN
ap-6125	100	4	(	(	PUNCT
ap-6125	100	5	6	6	NUM
ap-6125	100	6	)	)	PUNCT
ap-6125	100	7	,	,	PUNCT
ap-6125	100	8	(	(	PUNCT
ap-6125	100	9	7	7	X
ap-6125	100	10	)	)	PUNCT
ap-6125	100	11	also	also	ADV
ap-6125	100	12	possess	possess	VERB
ap-6125	100	13	the	the	DET
ap-6125	100	14	same	same	ADJ
ap-6125	100	15	property	property	NOUN
ap-6125	100	16	.	.	PUNCT
ap-6125	101	1	for	for	ADP
ap-6125	101	2	a	a	DET
ap-6125	101	3	practical	practical	ADJ
ap-6125	101	4	use	use	NOUN
ap-6125	101	5	in	in	ADP
ap-6125	101	6	approximation	approximation	NOUN
ap-6125	101	7	,	,	PUNCT
ap-6125	101	8	the	the	DET
ap-6125	101	9	polyharmonic	polyharmonic	ADJ
ap-6125	101	10	splines	spline	NOUN
ap-6125	101	11	are	be	AUX
ap-6125	101	12	combined	combine	VERB
ap-6125	101	13	with	with	ADP
ap-6125	101	14	lower	low	ADJ
ap-6125	101	15	order	order	NOUN
ap-6125	101	16	polynomial	polynomial	ADJ
ap-6125	101	17	terms	term	NOUN
ap-6125	101	18	(	(	PUNCT
ap-6125	101	19	trends	trend	NOUN
ap-6125	101	20	)	)	PUNCT
ap-6125	101	21	to	to	PART
ap-6125	101	22	form	form	VERB
ap-6125	101	23	an	an	DET
ap-6125	101	24	interpolation	interpolation	NOUN
ap-6125	101	25	or	or	CCONJ
ap-6125	101	26	approximation	approximation	NOUN
ap-6125	101	27	formula	formula	NOUN
ap-6125	101	28	as	as	ADP
ap-6125	101	29	in	in	ADP
ap-6125	101	30	(	(	PUNCT
ap-6125	101	31	4	4	NUM
ap-6125	101	32	)	)	PUNCT
ap-6125	101	33	.	.	PUNCT
ap-6125	102	1	5	5	X
ap-6125	102	2	.	.	X
ap-6125	102	3	smooth	smooth	ADJ
ap-6125	102	4	interpolation	interpolation	NOUN
ap-6125	102	5	let	let	VERB
ap-6125	102	6	us	we	PRON
ap-6125	102	7	briefly	briefly	ADV
ap-6125	102	8	present	present	VERB
ap-6125	102	9	some	some	DET
ap-6125	102	10	properties	property	NOUN
ap-6125	102	11	of	of	ADP
ap-6125	102	12	polyharmonic	polyharmonic	ADJ
ap-6125	102	13	splines	spline	NOUN
ap-6125	102	14	in	in	ADP
ap-6125	102	15	the	the	DET
ap-6125	102	16	smooth	smooth	ADJ
ap-6125	102	17	approximation	approximation	NOUN
ap-6125	102	18	or	or	CCONJ
ap-6125	102	19	variational	variational	ADJ
ap-6125	102	20	spline	spline	NOUN
ap-6125	102	21	theory	theory	NOUN
ap-6125	102	22	.	.	PUNCT
ap-6125	103	1	these	these	DET
ap-6125	103	2	properties	property	NOUN
ap-6125	103	3	show	show	VERB
ap-6125	103	4	the	the	DET
ap-6125	103	5	place	place	NOUN
ap-6125	103	6	of	of	ADP
ap-6125	103	7	splines	spline	NOUN
ap-6125	103	8	in	in	ADP
ap-6125	103	9	the	the	DET
ap-6125	103	10	context	context	NOUN
ap-6125	103	11	of	of	ADP
ap-6125	103	12	radial	radial	ADJ
ap-6125	103	13	basis	basis	NOUN
ap-6125	103	14	function	function	NOUN
ap-6125	103	15	interpolation	interpolation	NOUN
ap-6125	103	16	.	.	PUNCT
ap-6125	104	1	alternatively	alternatively	ADV
ap-6125	104	2	,	,	PUNCT
ap-6125	104	3	the	the	DET
ap-6125	104	4	splines	spline	NOUN
ap-6125	104	5	can	can	AUX
ap-6125	104	6	be	be	AUX
ap-6125	104	7	derived	derive	VERB
ap-6125	104	8	with	with	ADP
ap-6125	104	9	the	the	DET
ap-6125	104	10	help	help	NOUN
ap-6125	104	11	of	of	ADP
ap-6125	104	12	the	the	DET
ap-6125	104	13	algebraic	algebraic	PROPN
ap-6125	104	14	spline	spline	PROPN
ap-6125	104	15	theory	theory	NOUN
ap-6125	104	16	,	,	PUNCT
ap-6125	104	17	cf	cf	NOUN
ap-6125	104	18	.	.	PUNCT
ap-6125	105	1	an	an	DET
ap-6125	105	2	example	example	NOUN
ap-6125	105	3	of	of	ADP
ap-6125	105	4	the	the	DET
ap-6125	105	5	1d	1d	NUM
ap-6125	105	6	cubic	cubic	ADJ
ap-6125	105	7	spline	spline	NOUN
ap-6125	105	8	in	in	ADP
ap-6125	105	9	[	[	X
ap-6125	105	10	3	3	NUM
ap-6125	105	11	]	]	PUNCT
ap-6125	105	12	.	.	PUNCT
ap-6125	106	1	we	we	PRON
ap-6125	106	2	employ	employ	VERB
ap-6125	106	3	the	the	DET
ap-6125	106	4	usual	usual	ADJ
ap-6125	106	5	lebesgue	lebesgue	NOUN
ap-6125	106	6	space	space	NOUN
ap-6125	106	7	l2(ω	l2(ω	NOUN
ap-6125	106	8	)	)	PUNCT
ap-6125	106	9	of	of	ADP
ap-6125	106	10	generalized	generalized	ADJ
ap-6125	106	11	complex	complex	ADJ
ap-6125	106	12	-	-	PUNCT
ap-6125	106	13	valued	value	VERB
ap-6125	106	14	functions	function	NOUN
ap-6125	106	15	with	with	ADP
ap-6125	106	16	the	the	DET
ap-6125	106	17	norm	norm	NOUN
ap-6125	106	18	‖g‖2	‖g‖2	NOUN
ap-6125	106	19	l2	l2	NOUN
ap-6125	106	20	=	=	SYM
ap-6125	106	21	∫	∫	PROPN
ap-6125	106	22	ω	ω	PROPN
ap-6125	106	23	|g(x)|2	|g(x)|2	NUM
ap-6125	106	24	dx	dx	PROPN
ap-6125	106	25	.	.	PUNCT
ap-6125	107	1	we	we	PRON
ap-6125	107	2	follow	follow	VERB
ap-6125	107	3	[	[	X
ap-6125	107	4	1	1	NUM
ap-6125	107	5	]	]	PUNCT
ap-6125	107	6	and	and	CCONJ
ap-6125	107	7	[	[	X
ap-6125	107	8	4	4	NUM
ap-6125	107	9	]	]	PUNCT
ap-6125	107	10	,	,	PUNCT
ap-6125	107	11	and	and	CCONJ
ap-6125	107	12	formulate	formulate	VERB
ap-6125	107	13	and	and	CCONJ
ap-6125	107	14	solve	solve	VERB
ap-6125	107	15	the	the	DET
ap-6125	107	16	problem	problem	NOUN
ap-6125	107	17	of	of	ADP
ap-6125	107	18	smooth	smooth	ADJ
ap-6125	107	19	interpolation	interpolation	NOUN
ap-6125	107	20	[	[	X
ap-6125	107	21	2	2	NUM
ap-6125	107	22	]	]	PUNCT
ap-6125	107	23	.	.	PUNCT
ap-6125	108	1	choose	choose	VERB
ap-6125	108	2	a	a	DET
ap-6125	108	3	set	set	NOUN
ap-6125	108	4	{	{	PUNCT
ap-6125	108	5	bα	bα	NOUN
ap-6125	108	6	}	}	PUNCT
ap-6125	108	7	of	of	ADP
ap-6125	108	8	nonnegative	nonnegative	ADJ
ap-6125	108	9	numbers	number	NOUN
ap-6125	108	10	,	,	PUNCT
ap-6125	108	11	where	where	SCONJ
ap-6125	108	12	α	α	PROPN
ap-6125	108	13	is	be	AUX
ap-6125	108	14	a	a	DET
ap-6125	108	15	nonnegative	nonnegative	ADJ
ap-6125	108	16	multiindex	multiindex	NOUN
ap-6125	108	17	.	.	PUNCT
ap-6125	109	1	let	let	VERB
ap-6125	109	2	l	l	NOUN
ap-6125	109	3	be	be	AUX
ap-6125	109	4	the	the	DET
ap-6125	109	5	smallest	small	ADJ
ap-6125	109	6	nonnegative	nonnegative	ADJ
ap-6125	109	7	integer	integer	NOUN
ap-6125	109	8	such	such	DET
ap-6125	109	9	that	that	SCONJ
ap-6125	109	10	bα	bα	PROPN
ap-6125	109	11	>	>	X
ap-6125	109	12	0	0	PUNCT
ap-6125	110	1	for	for	ADP
ap-6125	110	2	at	at	ADV
ap-6125	110	3	least	least	ADV
ap-6125	110	4	one	one	NUM
ap-6125	110	5	α	α	NOUN
ap-6125	110	6	,	,	PUNCT
ap-6125	110	7	|α|	|α|	PROPN
ap-6125	110	8	=	=	SYM
ap-6125	110	9	l	l	NOUN
ap-6125	110	10	,	,	PUNCT
ap-6125	110	11	while	while	SCONJ
ap-6125	110	12	bα	bα	PROPN
ap-6125	110	13	=	=	NOUN
ap-6125	110	14	0	0	NUM
ap-6125	110	15	for	for	ADP
ap-6125	110	16	all	all	DET
ap-6125	110	17	α	α	NOUN
ap-6125	110	18	,	,	PUNCT
ap-6125	110	19	|α|	|α|	PROPN
ap-6125	110	20	<	<	X
ap-6125	110	21	l.	l.	PROPN
ap-6125	110	22	recall	recall	PROPN
ap-6125	110	23	that	that	SCONJ
ap-6125	110	24	the	the	DET
ap-6125	110	25	nodes	node	NOUN
ap-6125	110	26	x1	x1	PROPN
ap-6125	110	27	,	,	PUNCT
ap-6125	110	28	.	.	PUNCT
ap-6125	110	29	.	.	PUNCT
ap-6125	111	1	.	.	PUNCT
ap-6125	112	1	,	,	PUNCT
ap-6125	112	2	xn	xn	PROPN
ap-6125	112	3	∈	∈	PROPN
ap-6125	112	4	ω	ω	NOUN
ap-6125	112	5	are	be	AUX
ap-6125	112	6	supposed	suppose	VERB
ap-6125	112	7	to	to	PART
ap-6125	112	8	be	be	AUX
ap-6125	112	9	mutually	mutually	ADV
ap-6125	112	10	distinct	distinct	ADJ
ap-6125	112	11	.	.	PUNCT
ap-6125	113	1	let	let	VERB
ap-6125	113	2	w̃	w̃	PROPN
ap-6125	113	3	be	be	AUX
ap-6125	113	4	a	a	DET
ap-6125	113	5	linear	linear	ADJ
ap-6125	113	6	vector	vector	NOUN
ap-6125	113	7	space	space	NOUN
ap-6125	113	8	of	of	ADP
ap-6125	113	9	complex	complex	ADJ
ap-6125	113	10	valued	value	VERB
ap-6125	113	11	functions	function	NOUN
ap-6125	113	12	g	g	PRON
ap-6125	113	13	continuous	continuous	ADJ
ap-6125	113	14	together	together	ADV
ap-6125	113	15	with	with	ADP
ap-6125	113	16	all	all	DET
ap-6125	113	17	their	their	PRON
ap-6125	113	18	partial	partial	ADJ
ap-6125	113	19	derivatives	derivative	NOUN
ap-6125	113	20	of	of	ADP
ap-6125	113	21	all	all	DET
ap-6125	113	22	orders	order	NOUN
ap-6125	113	23	in	in	ADP
ap-6125	113	24	ω	ω	PROPN
ap-6125	113	25	.	.	PUNCT
ap-6125	114	1	for	for	ADP
ap-6125	114	2	g	g	PROPN
ap-6125	114	3	,	,	PUNCT
ap-6125	114	4	h	h	NOUN
ap-6125	114	5	∈	∈	PROPN
ap-6125	115	1	w̃	w̃	PROPN
ap-6125	115	2	we	we	PRON
ap-6125	115	3	put	put	VERB
ap-6125	115	4	(	(	PUNCT
ap-6125	115	5	g	g	NOUN
ap-6125	115	6	,	,	PUNCT
ap-6125	115	7	h)l	h)l	ADJ
ap-6125	115	8	=	=	PUNCT
ap-6125	115	9	∑	∑	PUNCT
ap-6125	116	1	l≤|α|	l≤|α|	PROPN
ap-6125	116	2	bα	bα	PROPN
ap-6125	116	3	∫	∫	PROPN
ap-6125	116	4	ω	ω	NUM
ap-6125	116	5	∂|α|g(x	∂|α|g(x	NOUN
ap-6125	116	6	)	)	PUNCT
ap-6125	116	7	∂xα1	∂xα1	NOUN
ap-6125	116	8	1	1	NUM
ap-6125	116	9	.	.	PUNCT
ap-6125	116	10	.	.	PUNCT
ap-6125	116	11	.	.	PUNCT
ap-6125	117	1	∂xαn	∂xαn	NUM
ap-6125	117	2	n	n	CCONJ
ap-6125	117	3	(	(	PUNCT
ap-6125	117	4	∂|α|h(x	∂|α|h(x	NOUN
ap-6125	117	5	)	)	PUNCT
ap-6125	117	6	∂xα1	∂xα1	NOUN
ap-6125	117	7	1	1	NUM
ap-6125	117	8	.	.	PUNCT
ap-6125	117	9	.	.	PUNCT
ap-6125	118	1	.	.	PUNCT
ap-6125	119	1	∂xαn	∂xαn	NUM
ap-6125	119	2	n	n	CCONJ
ap-6125	119	3	)	)	PUNCT
ap-6125	119	4	∗	∗	NOUN
ap-6125	119	5	dx	dx	PROPN
ap-6125	119	6	(	(	PUNCT
ap-6125	119	7	9	9	NUM
ap-6125	119	8	)	)	PUNCT
ap-6125	119	9	and	and	CCONJ
ap-6125	119	10	similarly	similarly	ADV
ap-6125	119	11	|g|2l	|g|2l	NOUN
ap-6125	119	12	=	=	SYM
ap-6125	119	13	∑	∑	PUNCT
ap-6125	120	1	l≤|α|	l≤|α|	PROPN
ap-6125	120	2	bα	bα	PROPN
ap-6125	120	3	∫	∫	PROPN
ap-6125	120	4	ω	ω	PROPN
ap-6125	120	5	∣∣∣∣	∣∣∣∣	PROPN
ap-6125	120	6	∂|α|g(x	∂|α|g(x	NOUN
ap-6125	120	7	)	)	PUNCT
ap-6125	120	8	∂xα1	∂xα1	NOUN
ap-6125	120	9	1	1	NUM
ap-6125	120	10	.	.	PUNCT
ap-6125	120	11	.	.	PUNCT
ap-6125	121	1	.	.	PUNCT
ap-6125	122	1	∂xαn	∂xαn	NUM
ap-6125	122	2	n	n	PRON
ap-6125	122	3	∣∣∣∣2	∣∣∣∣2	NOUN
ap-6125	122	4	dx	dx	PROPN
ap-6125	122	5	(	(	PUNCT
ap-6125	122	6	10	10	NUM
ap-6125	122	7	)	)	PUNCT
ap-6125	122	8	if	if	SCONJ
ap-6125	122	9	the	the	DET
ap-6125	122	10	values	value	NOUN
ap-6125	122	11	of	of	ADP
ap-6125	122	12	|g|l	|g|l	PROPN
ap-6125	122	13	and	and	CCONJ
ap-6125	122	14	|h|l	|h|l	NOUN
ap-6125	122	15	exist	exist	VERB
ap-6125	122	16	and	and	CCONJ
ap-6125	122	17	are	be	AUX
ap-6125	122	18	finite	finite	ADJ
ap-6125	122	19	.	.	PUNCT
ap-6125	123	1	if	if	SCONJ
ap-6125	123	2	l	l	NOUN
ap-6125	123	3	=	=	SYM
ap-6125	123	4	0	0	PUNCT
ap-6125	123	5	(	(	PUNCT
ap-6125	123	6	i.e.	i.e.	X
ap-6125	123	7	bα	bα	NOUN
ap-6125	123	8	>	>	X
ap-6125	123	9	0	0	NUM
ap-6125	123	10	for	for	ADP
ap-6125	123	11	|α|	|α|	PROPN
ap-6125	123	12	=	=	SYM
ap-6125	123	13	0	0	NUM
ap-6125	123	14	)	)	PUNCT
ap-6125	123	15	,	,	PUNCT
ap-6125	123	16	consider	consider	VERB
ap-6125	123	17	functions	function	NOUN
ap-6125	123	18	g	g	NOUN
ap-6125	123	19	,	,	PUNCT
ap-6125	123	20	h	h	NOUN
ap-6125	123	21	∈	∈	PROPN
ap-6125	124	1	w̃	w̃	PROPN
ap-6125	124	2	such	such	ADJ
ap-6125	124	3	that	that	SCONJ
ap-6125	124	4	the	the	DET
ap-6125	124	5	values	value	NOUN
ap-6125	124	6	of	of	ADP
ap-6125	124	7	|g|0	|g|0	NOUN
ap-6125	124	8	and	and	CCONJ
ap-6125	124	9	|h|0	|h|0	NOUN
ap-6125	124	10	exist	exist	VERB
ap-6125	124	11	and	and	CCONJ
ap-6125	124	12	are	be	AUX
ap-6125	124	13	finite	finite	ADJ
ap-6125	124	14	.	.	PUNCT
ap-6125	125	1	then	then	ADV
ap-6125	125	2	(	(	PUNCT
ap-6125	125	3	g	g	NOUN
ap-6125	125	4	,	,	PUNCT
ap-6125	125	5	h)0	h)0	PROPN
ap-6125	125	6	has	have	VERB
ap-6125	125	7	the	the	DET
ap-6125	125	8	properties	property	NOUN
ap-6125	125	9	of	of	ADP
ap-6125	125	10	inner	inner	ADJ
ap-6125	125	11	product	product	NOUN
ap-6125	125	12	and	and	CCONJ
ap-6125	125	13	the	the	DET
ap-6125	125	14	expression	expression	NOUN
ap-6125	125	15	‖g‖0	‖g‖0	NOUN
ap-6125	125	16	=	=	PUNCT
ap-6125	125	17	|g|0	|g|0	NOUN
ap-6125	125	18	is	be	AUX
ap-6125	125	19	norm	norm	NOUN
ap-6125	125	20	in	in	ADP
ap-6125	125	21	a	a	DET
ap-6125	125	22	normed	normed	ADJ
ap-6125	125	23	space	space	NOUN
ap-6125	125	24	w0	w0	PROPN
ap-6125	125	25	=	=	PUNCT
ap-6125	126	1	w̃.	w̃.	PROPN
ap-6125	126	2	let	let	VERB
ap-6125	126	3	l	l	PROPN
ap-6125	126	4	>	>	X
ap-6125	126	5	0	0	X
ap-6125	126	6	.	.	PUNCT
ap-6125	127	1	consider	consider	VERB
ap-6125	127	2	again	again	ADV
ap-6125	127	3	functions	function	VERB
ap-6125	127	4	g	g	NOUN
ap-6125	127	5	,	,	PUNCT
ap-6125	127	6	h	h	NOUN
ap-6125	127	7	∈	∈	PROPN
ap-6125	128	1	w̃	w̃	PROPN
ap-6125	128	2	such	such	ADJ
ap-6125	128	3	that	that	SCONJ
ap-6125	128	4	the	the	DET
ap-6125	128	5	values	value	NOUN
ap-6125	128	6	of	of	ADP
ap-6125	128	7	|g|l	|g|l	PROPN
ap-6125	128	8	and	and	CCONJ
ap-6125	128	9	|h|l	|h|l	NOUN
ap-6125	128	10	exist	exist	VERB
ap-6125	128	11	and	and	CCONJ
ap-6125	128	12	are	be	AUX
ap-6125	128	13	finite	finite	ADJ
ap-6125	128	14	.	.	PUNCT
ap-6125	129	1	let	let	VERB
ap-6125	129	2	pl−1	pl−1	PROPN
ap-6125	129	3	⊂	⊂	PRON
ap-6125	129	4	w̃	w̃	PROPN
ap-6125	129	5	be	be	AUX
ap-6125	129	6	the	the	DET
ap-6125	129	7	subspace	subspace	NOUN
ap-6125	129	8	whose	whose	DET
ap-6125	129	9	basis	basis	NOUN
ap-6125	129	10	{	{	PUNCT
ap-6125	129	11	ϕα	ϕα	NOUN
ap-6125	129	12	}	}	PUNCT
ap-6125	129	13	,	,	PUNCT
ap-6125	129	14	where	where	SCONJ
ap-6125	129	15	α	α	NOUN
ap-6125	129	16	is	be	AUX
ap-6125	129	17	a	a	DET
ap-6125	129	18	nonnegative	nonnegative	ADJ
ap-6125	129	19	multiindex	multiindex	NOUN
ap-6125	129	20	,	,	PUNCT
ap-6125	129	21	|α|	|α|	NOUN
ap-6125	129	22	≤	≤	NUM
ap-6125	129	23	l	l	NOUN
ap-6125	129	24	−	−	NOUN
ap-6125	129	25	1	1	NUM
ap-6125	129	26	,	,	PUNCT
ap-6125	129	27	consists	consist	VERB
ap-6125	129	28	of	of	ADP
ap-6125	129	29	all	all	DET
ap-6125	129	30	the	the	DET
ap-6125	129	31	trend	trend	NOUN
ap-6125	129	32	functions	function	NOUN
ap-6125	129	33	(	(	PUNCT
ap-6125	129	34	5	5	NUM
ap-6125	129	35	)	)	PUNCT
ap-6125	129	36	of	of	ADP
ap-6125	129	37	degree	degree	NOUN
ap-6125	129	38	l−	l−	NOUN
ap-6125	129	39	1	1	NUM
ap-6125	129	40	,	,	PUNCT
ap-6125	129	41	at	at	ADV
ap-6125	129	42	most	most	ADJ
ap-6125	129	43	.	.	PUNCT
ap-6125	130	1	then	then	ADV
ap-6125	130	2	,	,	PUNCT
ap-6125	130	3	for	for	ADP
ap-6125	130	4	a	a	DET
ap-6125	130	5	nonnegative	nonnegative	ADJ
ap-6125	130	6	multiindex	multiindex	NOUN
ap-6125	130	7	β	β	NOUN
ap-6125	130	8	,	,	PUNCT
ap-6125	130	9	(	(	PUNCT
ap-6125	130	10	ϕα	ϕα	ADV
ap-6125	130	11	,	,	PUNCT
ap-6125	130	12	ϕβ)l	ϕβ)l	PROPN
ap-6125	130	13	=	=	SYM
ap-6125	130	14	0	0	NUM
ap-6125	130	15	and	and	CCONJ
ap-6125	130	16	|ϕα|l	|ϕα|l	PRON
ap-6125	130	17	=	=	SYM
ap-6125	130	18	0	0	NUM
ap-6125	130	19	for	for	ADP
ap-6125	130	20	|α|	|α|	PROPN
ap-6125	130	21	≤	≤	NUM
ap-6125	130	22	l−	l−	NOUN
ap-6125	130	23	1	1	NUM
ap-6125	130	24	and	and	CCONJ
ap-6125	130	25	|β|	|β|	PRON
ap-6125	130	26	≤	≤	X
ap-6125	130	27	l−	l−	NOUN
ap-6125	130	28	1	1	NUM
ap-6125	130	29	.	.	PUNCT
ap-6125	131	1	(	(	PUNCT
ap-6125	131	2	11	11	NUM
ap-6125	131	3	)	)	PUNCT
ap-6125	131	4	using	use	VERB
ap-6125	131	5	(	(	PUNCT
ap-6125	131	6	9	9	NUM
ap-6125	131	7	)	)	PUNCT
ap-6125	131	8	and	and	CCONJ
ap-6125	131	9	(	(	PUNCT
ap-6125	131	10	10	10	NUM
ap-6125	131	11	)	)	PUNCT
ap-6125	131	12	,	,	PUNCT
ap-6125	131	13	we	we	PRON
ap-6125	131	14	construct	construct	VERB
ap-6125	131	15	the	the	DET
ap-6125	131	16	quotient	quotient	NOUN
ap-6125	131	17	space	space	NOUN
ap-6125	131	18	w̃/pl−1	w̃/pl−1	NOUN
ap-6125	131	19	whose	whose	DET
ap-6125	131	20	zero	zero	NUM
ap-6125	131	21	class	class	NOUN
ap-6125	131	22	is	be	AUX
ap-6125	131	23	the	the	DET
ap-6125	131	24	subspace	subspace	NOUN
ap-6125	131	25	pl−1	pl−1	NOUN
ap-6125	131	26	.	.	PUNCT
ap-6125	132	1	finally	finally	ADV
ap-6125	132	2	,	,	PUNCT
ap-6125	132	3	considering	consider	VERB
ap-6125	132	4	(	(	PUNCT
ap-6125	132	5	·	·	PUNCT
ap-6125	132	6	,	,	PUNCT
ap-6125	132	7	·	·	PUNCT
ap-6125	132	8	)	)	PUNCT
ap-6125	132	9	l	l	NOUN
ap-6125	132	10	and	and	CCONJ
ap-6125	132	11	|	|	ADV
ap-6125	132	12	·	·	PUNCT
ap-6125	132	13	|l	|l	PROPN
ap-6125	132	14	in	in	ADP
ap-6125	132	15	every	every	DET
ap-6125	132	16	equivalence	equivalence	NOUN
ap-6125	132	17	class	class	NOUN
ap-6125	132	18	,	,	PUNCT
ap-6125	132	19	we	we	PRON
ap-6125	132	20	see	see	VERB
ap-6125	132	21	that	that	SCONJ
ap-6125	132	22	they	they	PRON
ap-6125	132	23	represent	represent	VERB
ap-6125	132	24	the	the	DET
ap-6125	132	25	inner	inner	ADJ
ap-6125	132	26	product	product	NOUN
ap-6125	132	27	and	and	CCONJ
ap-6125	132	28	norm	norm	NOUN
ap-6125	132	29	‖g‖l	‖g‖l	NOUN
ap-6125	132	30	in	in	ADP
ap-6125	132	31	the	the	DET
ap-6125	132	32	normed	normed	ADJ
ap-6125	132	33	space	space	NOUN
ap-6125	132	34	wl	wl	NOUN
ap-6125	132	35	=	=	SYM
ap-6125	132	36	w̃/pl−1	w̃/pl−1	NOUN
ap-6125	132	37	.	.	PUNCT
ap-6125	133	1	wl	wl	PROPN
ap-6125	133	2	is	be	AUX
ap-6125	133	3	the	the	DET
ap-6125	133	4	normed	normed	ADJ
ap-6125	133	5	space	space	NOUN
ap-6125	133	6	where	where	SCONJ
ap-6125	133	7	we	we	PRON
ap-6125	133	8	minimize	minimize	VERB
ap-6125	133	9	functionals	functional	NOUN
ap-6125	133	10	and	and	CCONJ
ap-6125	133	11	measure	measure	VERB
ap-6125	133	12	the	the	DET
ap-6125	133	13	smoothness	smoothness	NOUN
ap-6125	133	14	of	of	ADP
ap-6125	133	15	the	the	DET
ap-6125	133	16	interpolation	interpolation	NOUN
ap-6125	133	17	as	as	SCONJ
ap-6125	133	18	prescribed	prescribe	VERB
ap-6125	133	19	by	by	ADP
ap-6125	133	20	the	the	DET
ap-6125	133	21	choice	choice	NOUN
ap-6125	133	22	of	of	ADP
ap-6125	133	23	{	{	PUNCT
ap-6125	133	24	bα	bα	NOUN
ap-6125	133	25	}	}	PUNCT
ap-6125	133	26	.	.	PUNCT
ap-6125	134	1	we	we	PRON
ap-6125	134	2	complete	complete	VERB
ap-6125	134	3	the	the	DET
ap-6125	134	4	spacewl	spacewl	NOUN
ap-6125	134	5	in	in	ADP
ap-6125	134	6	the	the	DET
ap-6125	134	7	norm	norm	NOUN
ap-6125	134	8	‖	‖	PROPN
ap-6125	134	9	·	·	PUNCT
ap-6125	134	10	‖l	‖l	VERB
ap-6125	134	11	and	and	CCONJ
ap-6125	134	12	denote	denote	VERB
ap-6125	134	13	the	the	DET
ap-6125	134	14	completed	complete	VERB
ap-6125	134	15	space	space	NOUN
ap-6125	134	16	again	again	ADV
ap-6125	134	17	wl	wl	PROPN
ap-6125	134	18	.	.	PROPN
ap-6125	134	19	for	for	ADP
ap-6125	134	20	an	an	DET
ap-6125	134	21	arbitrary	arbitrary	ADJ
ap-6125	134	22	l	l	NOUN
ap-6125	134	23	≥	≥	NOUN
ap-6125	134	24	0	0	NUM
ap-6125	134	25	,	,	PUNCT
ap-6125	134	26	choose	choose	VERB
ap-6125	134	27	a	a	DET
ap-6125	134	28	basis	basis	NOUN
ap-6125	134	29	system	system	NOUN
ap-6125	134	30	of	of	ADP
ap-6125	134	31	functions	function	NOUN
ap-6125	134	32	{	{	PUNCT
ap-6125	134	33	gκ	gκ	NOUN
ap-6125	134	34	}	}	PUNCT
ap-6125	134	35	⊂wl	⊂wl	NOUN
ap-6125	134	36	that	that	PRON
ap-6125	134	37	is	be	AUX
ap-6125	134	38	complete	complete	ADJ
ap-6125	134	39	and	and	CCONJ
ap-6125	134	40	orthogonal	orthogonal	ADJ
ap-6125	134	41	(	(	PUNCT
ap-6125	134	42	in	in	ADP
ap-6125	134	43	the	the	DET
ap-6125	134	44	inner	inner	ADJ
ap-6125	134	45	product	product	NOUN
ap-6125	134	46	in	in	ADP
ap-6125	134	47	wl	wl	PROPN
ap-6125	134	48	)	)	PUNCT
ap-6125	134	49	,	,	PUNCT
ap-6125	134	50	i.e.	i.e.	X
ap-6125	134	51	,	,	PUNCT
ap-6125	134	52	if	if	SCONJ
ap-6125	134	53	κ	κ	X
ap-6125	134	54	=	=	PUNCT
ap-6125	134	55	(	(	PUNCT
ap-6125	134	56	κ1	κ1	NOUN
ap-6125	134	57	,	,	PUNCT
ap-6125	134	58	.	.	PUNCT
ap-6125	134	59	.	.	PUNCT
ap-6125	134	60	.	.	PUNCT
ap-6125	135	1	,	,	PUNCT
ap-6125	135	2	κn	κn	NOUN
ap-6125	135	3	)	)	PUNCT
ap-6125	135	4	and	and	CCONJ
ap-6125	135	5	µ	µ	X
ap-6125	135	6	=	=	SYM
ap-6125	135	7	(	(	PUNCT
ap-6125	135	8	µ1	µ1	PROPN
ap-6125	135	9	,	,	PUNCT
ap-6125	135	10	.	.	PUNCT
ap-6125	135	11	.	.	PUNCT
ap-6125	135	12	.	.	PUNCT
ap-6125	136	1	,	,	PUNCT
ap-6125	136	2	µn	µn	PROPN
ap-6125	136	3	)	)	PUNCT
ap-6125	136	4	are	be	AUX
ap-6125	136	5	nonnegative	nonnegative	ADJ
ap-6125	136	6	multiindices	multiindice	NOUN
ap-6125	136	7	then	then	ADV
ap-6125	136	8	(	(	PUNCT
ap-6125	136	9	gκ	gκ	INTJ
ap-6125	136	10	,	,	PUNCT
ap-6125	136	11	gµ)l	gµ)l	PROPN
ap-6125	136	12	=	=	NOUN
ap-6125	136	13	0	0	NUM
ap-6125	136	14	for	for	ADP
ap-6125	136	15	κ	κ	PROPN
ap-6125	136	16	6=	6=	PROPN
ap-6125	136	17	µ.	µ.	NOUN
ap-6125	136	18	(	(	PUNCT
ap-6125	136	19	12	12	NUM
ap-6125	136	20	)	)	PUNCT
ap-6125	136	21	if	if	SCONJ
ap-6125	136	22	l	l	PROPN
ap-6125	136	23	>	>	X
ap-6125	136	24	0	0	PUNCT
ap-6125	137	1	then	then	ADV
ap-6125	137	2	,	,	PUNCT
ap-6125	137	3	moreover	moreover	ADV
ap-6125	137	4	,	,	PUNCT
ap-6125	137	5	(	(	PUNCT
ap-6125	137	6	ϕα	ϕα	ADV
ap-6125	137	7	,	,	PUNCT
ap-6125	137	8	gκ)l	gκ)l	PROPN
ap-6125	137	9	=	=	SYM
ap-6125	137	10	0	0	NUM
ap-6125	137	11	for	for	ADP
ap-6125	137	12	a	a	DET
ap-6125	137	13	nonnegative	nonnegative	ADJ
ap-6125	137	14	multiindex	multiindex	NOUN
ap-6125	137	15	α	α	NOUN
ap-6125	137	16	,	,	PUNCT
ap-6125	137	17	|α|	|α|	PROPN
ap-6125	137	18	≤	≤	NUM
ap-6125	137	19	l−	l−	NOUN
ap-6125	137	20	1	1	NUM
ap-6125	137	21	.	.	PUNCT
ap-6125	138	1	(	(	PUNCT
ap-6125	138	2	13	13	NUM
ap-6125	138	3	)	)	PUNCT
ap-6125	138	4	the	the	DET
ap-6125	138	5	set	set	NOUN
ap-6125	138	6	{	{	PUNCT
ap-6125	138	7	ϕα	ϕα	NOUN
ap-6125	138	8	}	}	PUNCT
ap-6125	138	9	of	of	ADP
ap-6125	138	10	trend	trend	NOUN
ap-6125	138	11	functions	function	NOUN
ap-6125	138	12	is	be	AUX
ap-6125	138	13	empty	empty	ADJ
ap-6125	138	14	for	for	ADP
ap-6125	138	15	l	l	NOUN
ap-6125	138	16	=	=	SYM
ap-6125	139	1	0	0	X
ap-6125	139	2	.	.	PUNCT
ap-6125	139	3	definition	definition	NOUN
ap-6125	139	4	4	4	NUM
ap-6125	139	5	(	(	PUNCT
ap-6125	139	6	smooth	smooth	ADJ
ap-6125	139	7	interpolation	interpolation	NOUN
ap-6125	139	8	)	)	PUNCT
ap-6125	139	9	.	.	PUNCT
ap-6125	140	1	the	the	DET
ap-6125	140	2	problem	problem	NOUN
ap-6125	140	3	of	of	ADP
ap-6125	140	4	smooth	smooth	ADJ
ap-6125	140	5	interpolation	interpolation	NOUN
ap-6125	140	6	[	[	X
ap-6125	140	7	1	1	X
ap-6125	140	8	]	]	PUNCT
ap-6125	140	9	consists	consist	VERB
ap-6125	140	10	in	in	ADP
ap-6125	140	11	finding	find	VERB
ap-6125	140	12	the	the	DET
ap-6125	140	13	complex	complex	ADJ
ap-6125	140	14	coefficients	coefficient	NOUN
ap-6125	140	15	aκ	aκ	ADV
ap-6125	140	16	and	and	CCONJ
ap-6125	140	17	aα	aα	NOUN
ap-6125	140	18	of	of	ADP
ap-6125	140	19	the	the	DET
ap-6125	140	20	interpolant	interpolant	NOUN
ap-6125	140	21	z(x	z(x	NOUN
ap-6125	140	22	)	)	PUNCT
ap-6125	140	23	=	=	PUNCT
ap-6125	140	24	∑	∑	PUNCT
ap-6125	140	25	κ	κ	PROPN
ap-6125	140	26	aκgκ(x	aκgκ(x	PROPN
ap-6125	140	27	)	)	PUNCT
ap-6125	141	1	+	+	CCONJ
ap-6125	141	2	∑	∑	PROPN
ap-6125	141	3	|α|≤l−1	|α|≤l−1	PROPN
ap-6125	141	4	aαϕα(x	aαϕα(x	PROPN
ap-6125	141	5	)	)	PUNCT
ap-6125	141	6	(	(	PUNCT
ap-6125	141	7	14	14	NUM
ap-6125	141	8	)	)	PUNCT
ap-6125	141	9	with	with	ADP
ap-6125	141	10	nonnegative	nonnegative	ADJ
ap-6125	141	11	multiindices	multiindice	NOUN
ap-6125	141	12	κ	κ	NOUN
ap-6125	141	13	and	and	CCONJ
ap-6125	141	14	α	α	PRON
ap-6125	141	15	such	such	ADJ
ap-6125	141	16	that	that	SCONJ
ap-6125	141	17	z(xj	z(xj	NUM
ap-6125	141	18	)	)	PUNCT
ap-6125	142	1	=	=	SYM
ap-6125	142	2	fj	fj	PROPN
ap-6125	142	3	,	,	PUNCT
ap-6125	142	4	j	j	PROPN
ap-6125	142	5	=	=	SYM
ap-6125	142	6	1	1	NUM
ap-6125	142	7	,	,	PUNCT
ap-6125	142	8	.	.	PUNCT
ap-6125	142	9	.	.	PUNCT
ap-6125	142	10	.	.	PUNCT
ap-6125	143	1	,	,	PUNCT
ap-6125	143	2	n	n	CCONJ
ap-6125	143	3	,	,	PUNCT
ap-6125	143	4	(	(	PUNCT
ap-6125	143	5	15	15	NUM
ap-6125	143	6	)	)	PUNCT
ap-6125	143	7	and	and	CCONJ
ap-6125	143	8	the	the	DET
ap-6125	143	9	quantity	quantity	NOUN
ap-6125	143	10	‖z‖2	‖z‖2	NOUN
ap-6125	143	11	l	l	NOUN
ap-6125	143	12	attains	attain	VERB
ap-6125	143	13	its	its	PRON
ap-6125	143	14	minimum	minimum	NOUN
ap-6125	143	15	on	on	ADP
ap-6125	143	16	wl	wl	PROPN
ap-6125	143	17	.	.	PUNCT
ap-6125	144	1	(	(	PUNCT
ap-6125	144	2	16	16	NUM
ap-6125	144	3	)	)	PUNCT
ap-6125	144	4	the	the	DET
ap-6125	144	5	second	second	ADJ
ap-6125	144	6	sum	sum	NOUN
ap-6125	144	7	in	in	ADP
ap-6125	144	8	the	the	DET
ap-6125	144	9	interpolant	interpolant	NOUN
ap-6125	144	10	(	(	PUNCT
ap-6125	144	11	14	14	NUM
ap-6125	144	12	)	)	PUNCT
ap-6125	144	13	is	be	AUX
ap-6125	144	14	empty	empty	ADJ
ap-6125	144	15	for	for	ADP
ap-6125	144	16	l	l	NOUN
ap-6125	144	17	=	=	SYM
ap-6125	144	18	0	0	NUM
ap-6125	144	19	.	.	PUNCT
ap-6125	145	1	according	accord	VERB
ap-6125	145	2	to	to	ADP
ap-6125	145	3	(	(	PUNCT
ap-6125	145	4	10	10	NUM
ap-6125	145	5	)	)	PUNCT
ap-6125	145	6	,	,	PUNCT
ap-6125	145	7	the	the	DET
ap-6125	145	8	quantity	quantity	NOUN
ap-6125	145	9	‖z‖2	‖z‖2	NOUN
ap-6125	145	10	l	l	NOUN
ap-6125	145	11	is	be	AUX
ap-6125	145	12	the	the	DET
ap-6125	145	13	weighted	weighted	ADJ
ap-6125	145	14	sum	sum	NOUN
ap-6125	145	15	of	of	ADP
ap-6125	145	16	the	the	DET
ap-6125	145	17	squares	square	NOUN
ap-6125	145	18	of	of	ADP
ap-6125	145	19	l2	l2	NOUN
ap-6125	145	20	norms	norm	NOUN
ap-6125	145	21	of	of	ADP
ap-6125	145	22	the	the	DET
ap-6125	145	23	derivatives	derivative	NOUN
ap-6125	145	24	of	of	ADP
ap-6125	145	25	z	z	NOUN
ap-6125	145	26	of	of	ADP
ap-6125	145	27	all	all	DET
ap-6125	145	28	orders	order	NOUN
ap-6125	145	29	|α|	|α|	PROPN
ap-6125	145	30	with	with	ADP
ap-6125	145	31	weights	weight	NOUN
ap-6125	145	32	bα	bα	PROPN
ap-6125	145	33	.	.	PUNCT
ap-6125	146	1	putting	put	VERB
ap-6125	146	2	bα	bα	PROPN
ap-6125	146	3	>	>	X
ap-6125	146	4	0	0	PUNCT
ap-6125	146	5	for	for	ADP
ap-6125	146	6	some	some	DET
ap-6125	146	7	set	set	NOUN
ap-6125	146	8	of	of	ADP
ap-6125	146	9	multiindices	multiindice	NOUN
ap-6125	146	10	α	α	NUM
ap-6125	146	11	,	,	PUNCT
ap-6125	146	12	we	we	PRON
ap-6125	146	13	can	can	AUX
ap-6125	146	14	specify	specify	VERB
ap-6125	146	15	the	the	DET
ap-6125	146	16	partial	partial	ADJ
ap-6125	146	17	derivatives	derivative	NOUN
ap-6125	146	18	of	of	ADP
ap-6125	146	19	z	z	NOUN
ap-6125	146	20	whose	whose	DET
ap-6125	146	21	l2	l2	NOUN
ap-6125	146	22	norms	norm	NOUN
ap-6125	146	23	are	be	AUX
ap-6125	146	24	to	to	PART
ap-6125	146	25	be	be	AUX
ap-6125	146	26	minimized	minimize	VERB
ap-6125	146	27	,	,	PUNCT
ap-6125	146	28	i.e.	i.e.	X
ap-6125	146	29	,	,	PUNCT
ap-6125	146	30	the	the	DET
ap-6125	146	31	smoothness	smoothness	NOUN
ap-6125	146	32	of	of	ADP
ap-6125	146	33	the	the	DET
ap-6125	146	34	interpolant	interpolant	NOUN
ap-6125	146	35	z.	z.	PROPN
ap-6125	146	36	for	for	ADP
ap-6125	146	37	example	example	NOUN
ap-6125	146	38	,	,	PUNCT
ap-6125	146	39	if	if	SCONJ
ap-6125	146	40	n	n	NOUN
ap-6125	146	41	=	=	SYM
ap-6125	146	42	1	1	NUM
ap-6125	146	43	we	we	PRON
ap-6125	146	44	put	put	VERB
ap-6125	146	45	bk	bk	ADP
ap-6125	146	46	=	=	SYM
ap-6125	146	47	0	0	NUM
ap-6125	146	48	,	,	PUNCT
ap-6125	146	49	except	except	SCONJ
ap-6125	146	50	for	for	ADP
ap-6125	146	51	b2	b2	NOUN
ap-6125	146	52	=	=	SYM
ap-6125	146	53	1	1	NUM
ap-6125	146	54	(	(	PUNCT
ap-6125	146	55	i.e.	i.e.	X
ap-6125	146	56	l	l	X
ap-6125	146	57	=	=	SYM
ap-6125	146	58	2	2	NUM
ap-6125	146	59	)	)	PUNCT
ap-6125	146	60	,	,	PUNCT
ap-6125	146	61	and	and	CCONJ
ap-6125	146	62	minimize	minimize	VERB
ap-6125	146	63	the	the	DET
ap-6125	146	64	l2	l2	NOUN
ap-6125	146	65	150	150	NUM
ap-6125	146	66	vol	vol	NOUN
ap-6125	146	67	.	.	PUNCT
ap-6125	147	1	61	61	NUM
ap-6125	147	2	special	special	ADJ
ap-6125	147	3	issue/2021	issue/2021	NOUN
ap-6125	147	4	multivariate	multivariate	NOUN
ap-6125	147	5	interpolation	interpolation	NOUN
ap-6125	147	6	using	use	VERB
ap-6125	147	7	polyharmonic	polyharmonic	ADJ
ap-6125	147	8	splines	spline	NOUN
ap-6125	147	9	norm	norm	NOUN
ap-6125	147	10	of	of	ADP
ap-6125	147	11	the	the	DET
ap-6125	147	12	second	second	ADJ
ap-6125	147	13	derivative	derivative	NOUN
ap-6125	147	14	of	of	ADP
ap-6125	147	15	z	z	PROPN
ap-6125	147	16	,	,	PUNCT
ap-6125	147	17	which	which	PRON
ap-6125	147	18	corresponds	correspond	VERB
ap-6125	147	19	to	to	ADP
ap-6125	147	20	minimizing	minimize	VERB
ap-6125	147	21	its	its	PRON
ap-6125	147	22	curvature	curvature	NOUN
ap-6125	147	23	.	.	PUNCT
ap-6125	148	1	apparently	apparently	ADV
ap-6125	148	2	,	,	PUNCT
ap-6125	148	3	‖z‖2	‖z‖2	NOUN
ap-6125	148	4	l	l	NOUN
ap-6125	148	5	=	=	PUNCT
ap-6125	148	6	∑	∑	PUNCT
ap-6125	148	7	κ	κ	PROPN
ap-6125	148	8	aκa	aκa	PROPN
ap-6125	148	9	∗	∗	NOUN
ap-6125	148	10	κ‖gκ‖2	κ‖gκ‖2	NUM
ap-6125	148	11	l	l	NOUN
ap-6125	148	12	due	due	ADP
ap-6125	148	13	to	to	ADP
ap-6125	148	14	(	(	PUNCT
ap-6125	148	15	11	11	NUM
ap-6125	148	16	)	)	PUNCT
ap-6125	148	17	,	,	PUNCT
ap-6125	148	18	(	(	PUNCT
ap-6125	148	19	12	12	NUM
ap-6125	148	20	)	)	PUNCT
ap-6125	148	21	,	,	PUNCT
ap-6125	148	22	(	(	PUNCT
ap-6125	148	23	13	13	NUM
ap-6125	148	24	)	)	PUNCT
ap-6125	148	25	,	,	PUNCT
ap-6125	148	26	and	and	CCONJ
ap-6125	148	27	(	(	PUNCT
ap-6125	148	28	14	14	NUM
ap-6125	148	29	)	)	PUNCT
ap-6125	148	30	.	.	PUNCT
ap-6125	149	1	remark	remark	PROPN
ap-6125	149	2	1	1	NUM
ap-6125	149	3	.	.	PUNCT
ap-6125	149	4	with	with	ADP
ap-6125	149	5	a	a	DET
ap-6125	149	6	fixed	fixed	ADJ
ap-6125	149	7	n	n	CCONJ
ap-6125	149	8	,	,	PUNCT
ap-6125	149	9	it	it	PRON
ap-6125	149	10	is	be	AUX
ap-6125	149	11	easy	easy	ADJ
ap-6125	149	12	to	to	PART
ap-6125	149	13	employ	employ	VERB
ap-6125	149	14	the	the	DET
ap-6125	149	15	multinomial	multinomial	NOUN
ap-6125	149	16	theorem	theorem	VERB
ap-6125	149	17	to	to	PART
ap-6125	149	18	find	find	VERB
ap-6125	149	19	out	out	ADP
ap-6125	149	20	(	(	PUNCT
ap-6125	149	21	see	see	VERB
ap-6125	149	22	[	[	X
ap-6125	149	23	2	2	NUM
ap-6125	149	24	]	]	PUNCT
ap-6125	149	25	)	)	PUNCT
ap-6125	149	26	that	that	SCONJ
ap-6125	149	27	there	there	PRON
ap-6125	149	28	are	be	VERB
ap-6125	149	29	π(n	π(n	PROPN
ap-6125	149	30	,	,	PUNCT
ap-6125	149	31	|α|	|α|	PROPN
ap-6125	149	32	)	)	PUNCT
ap-6125	149	33	=	=	PUNCT
ap-6125	149	34	(	(	PUNCT
ap-6125	149	35	|α|+	|α|+	NOUN
ap-6125	149	36	n−	n−	PROPN
ap-6125	149	37	1	1	NUM
ap-6125	149	38	n−	n−	NOUN
ap-6125	149	39	1	1	NUM
ap-6125	149	40	)	)	PUNCT
ap-6125	149	41	mutually	mutually	ADV
ap-6125	149	42	different	different	ADJ
ap-6125	149	43	nonnegative	nonnegative	ADJ
ap-6125	149	44	multiindices	multiindice	NOUN
ap-6125	149	45	α	α	NOUN
ap-6125	149	46	of	of	ADP
ap-6125	149	47	n	n	PROPN
ap-6125	149	48	components	component	NOUN
ap-6125	149	49	with	with	ADP
ap-6125	149	50	|α|	|α|	PROPN
ap-6125	149	51	fixed	fix	VERB
ap-6125	149	52	.	.	PUNCT
ap-6125	150	1	the	the	DET
ap-6125	150	2	same	same	ADJ
ap-6125	150	3	is	be	AUX
ap-6125	150	4	the	the	DET
ap-6125	150	5	number	number	NOUN
ap-6125	150	6	of	of	ADP
ap-6125	150	7	the	the	DET
ap-6125	150	8	trend	trend	NOUN
ap-6125	150	9	functions	function	NOUN
ap-6125	150	10	ϕα	ϕα	ADV
ap-6125	150	11	with	with	ADP
ap-6125	150	12	|α|	|α|	PROPN
ap-6125	150	13	fixed	fix	VERB
ap-6125	150	14	and	and	CCONJ
ap-6125	150	15	t	t	PROPN
ap-6125	150	16	(	(	PUNCT
ap-6125	150	17	n	n	CCONJ
ap-6125	150	18	,	,	PUNCT
ap-6125	150	19	l	l	NOUN
ap-6125	150	20	)	)	PUNCT
ap-6125	150	21	=	=	SYM
ap-6125	151	1	∑	∑	PUNCT
ap-6125	151	2	|α|≤l−1	|α|≤l−1	NOUN
ap-6125	151	3	(	(	PUNCT
ap-6125	151	4	|α|+	|α|+	NOUN
ap-6125	151	5	n−	n−	PROPN
ap-6125	151	6	1	1	NUM
ap-6125	151	7	n−	n−	NOUN
ap-6125	151	8	1	1	NUM
ap-6125	151	9	)	)	PUNCT
ap-6125	151	10	=	=	PRON
ap-6125	151	11	(	(	PUNCT
ap-6125	151	12	l−	l−	NOUN
ap-6125	151	13	1	1	NUM
ap-6125	151	14	+	+	CCONJ
ap-6125	151	15	n	n	CCONJ
ap-6125	151	16	n	n	CCONJ
ap-6125	151	17	)	)	PUNCT
ap-6125	151	18	is	be	AUX
ap-6125	151	19	the	the	DET
ap-6125	151	20	total	total	ADJ
ap-6125	151	21	number	number	NOUN
ap-6125	151	22	of	of	ADP
ap-6125	151	23	the	the	DET
ap-6125	151	24	trend	trend	NOUN
ap-6125	151	25	functions	function	NOUN
ap-6125	151	26	ϕα	ϕα	ADV
ap-6125	151	27	,	,	PUNCT
ap-6125	151	28	|α|	|α|	PROPN
ap-6125	151	29	≤	≤	NUM
ap-6125	151	30	l−	l−	NOUN
ap-6125	151	31	1	1	NUM
ap-6125	151	32	.	.	PUNCT
ap-6125	151	33	to	to	PART
ap-6125	151	34	remove	remove	VERB
ap-6125	151	35	the	the	DET
ap-6125	151	36	inconvenient	inconvenient	ADJ
ap-6125	151	37	infinite	infinite	ADJ
ap-6125	151	38	sum	sum	NOUN
ap-6125	151	39	from	from	ADP
ap-6125	151	40	(	(	PUNCT
ap-6125	151	41	14	14	NUM
ap-6125	151	42	)	)	PUNCT
ap-6125	151	43	,	,	PUNCT
ap-6125	151	44	we	we	PRON
ap-6125	151	45	introduce	introduce	VERB
ap-6125	151	46	the	the	DET
ap-6125	151	47	generating	generate	VERB
ap-6125	151	48	function	function	NOUN
ap-6125	151	49	[	[	X
ap-6125	151	50	1	1	NUM
ap-6125	151	51	]	]	PUNCT
ap-6125	151	52	.	.	PUNCT
ap-6125	152	1	definition	definition	NOUN
ap-6125	152	2	5	5	NUM
ap-6125	152	3	(	(	PUNCT
ap-6125	152	4	generating	generate	VERB
ap-6125	152	5	function	function	NOUN
ap-6125	152	6	)	)	PUNCT
ap-6125	152	7	.	.	PUNCT
ap-6125	153	1	let	let	VERB
ap-6125	153	2	the	the	DET
ap-6125	153	3	basis	basis	NOUN
ap-6125	153	4	system	system	NOUN
ap-6125	153	5	of	of	ADP
ap-6125	153	6	functions	function	NOUN
ap-6125	153	7	{	{	PUNCT
ap-6125	153	8	gκ	gκ	NOUN
ap-6125	153	9	}	}	PUNCT
ap-6125	153	10	⊂wl	⊂wl	NOUN
ap-6125	153	11	,	,	PUNCT
ap-6125	153	12	where	where	SCONJ
ap-6125	153	13	κ	κ	NOUN
ap-6125	153	14	is	be	AUX
ap-6125	153	15	a	a	DET
ap-6125	153	16	nonnegative	nonnegative	ADJ
ap-6125	153	17	multiindex	multiindex	NOUN
ap-6125	153	18	,	,	PUNCT
ap-6125	153	19	be	be	AUX
ap-6125	153	20	complete	complete	ADJ
ap-6125	153	21	and	and	CCONJ
ap-6125	153	22	orthogonal	orthogonal	ADJ
ap-6125	153	23	in	in	ADP
ap-6125	153	24	wl	wl	PROPN
ap-6125	153	25	.	.	PUNCT
ap-6125	154	1	if	if	SCONJ
ap-6125	154	2	the	the	DET
ap-6125	154	3	series	series	PROPN
ap-6125	154	4	r(x	r(x	PROPN
ap-6125	154	5	,	,	PUNCT
ap-6125	154	6	y	y	NOUN
ap-6125	154	7	)	)	PUNCT
ap-6125	154	8	=	=	PUNCT
ap-6125	154	9	∑	∑	PUNCT
ap-6125	154	10	κ	κ	PROPN
ap-6125	154	11	gκ(x)g∗κ(y	gκ(x)g∗κ(y	PROPN
ap-6125	154	12	)	)	PUNCT
ap-6125	154	13	‖gκ‖2	‖gκ‖2	PROPN
ap-6125	154	14	l	l	NOUN
ap-6125	154	15	(	(	PUNCT
ap-6125	154	16	17	17	NUM
ap-6125	154	17	)	)	PUNCT
ap-6125	154	18	converges	converge	VERB
ap-6125	154	19	for	for	ADP
ap-6125	154	20	all	all	DET
ap-6125	154	21	x	x	NOUN
ap-6125	154	22	,	,	PUNCT
ap-6125	154	23	y	y	PROPN
ap-6125	154	24	∈	∈	PROPN
ap-6125	154	25	ω	ω	PROPN
ap-6125	154	26	and	and	CCONJ
ap-6125	154	27	is	be	AUX
ap-6125	154	28	continuous	continuous	ADJ
ap-6125	154	29	in	in	ADP
ap-6125	154	30	ω	ω	NUM
ap-6125	154	31	we	we	PRON
ap-6125	154	32	call	call	VERB
ap-6125	154	33	the	the	DET
ap-6125	154	34	fuction	fuction	NOUN
ap-6125	154	35	r(x	r(x	PROPN
ap-6125	154	36	,	,	PUNCT
ap-6125	154	37	y	y	PROPN
ap-6125	154	38	)	)	PUNCT
ap-6125	154	39	the	the	DET
ap-6125	154	40	generating	generate	VERB
ap-6125	154	41	function	function	NOUN
ap-6125	154	42	.	.	PUNCT
ap-6125	155	1	if	if	SCONJ
ap-6125	155	2	l	l	PROPN
ap-6125	155	3	>	>	X
ap-6125	155	4	0	0	NUM
ap-6125	155	5	,	,	PUNCT
ap-6125	155	6	introduce	introduce	VERB
ap-6125	155	7	an	an	DET
ap-6125	155	8	n	n	NUM
ap-6125	155	9	×	×	PROPN
ap-6125	155	10	t	t	PROPN
ap-6125	155	11	(	(	PUNCT
ap-6125	155	12	n	n	CCONJ
ap-6125	155	13	,	,	PUNCT
ap-6125	155	14	l	l	NOUN
ap-6125	155	15	)	)	PUNCT
ap-6125	155	16	matrix	matrix	NOUN
ap-6125	155	17	φ	φ	PROPN
ap-6125	155	18	with	with	ADP
ap-6125	155	19	entries	entry	NOUN
ap-6125	155	20	φjα	φjα	ADV
ap-6125	155	21	=	=	SYM
ap-6125	155	22	ϕα(xj	ϕα(xj	ADV
ap-6125	155	23	)	)	PUNCT
ap-6125	155	24	,	,	PUNCT
ap-6125	155	25	j	j	PROPN
ap-6125	156	1	=	=	SYM
ap-6125	156	2	1	1	NUM
ap-6125	156	3	,	,	PUNCT
ap-6125	156	4	.	.	PUNCT
ap-6125	156	5	.	.	PUNCT
ap-6125	157	1	.	.	PUNCT
ap-6125	158	1	,	,	PUNCT
ap-6125	158	2	n	n	CCONJ
ap-6125	158	3	,	,	PUNCT
ap-6125	158	4	|α|	|α|	PROPN
ap-6125	158	5	≤	≤	NUM
ap-6125	158	6	l−	l−	NOUN
ap-6125	158	7	1	1	NUM
ap-6125	158	8	.	.	PUNCT
ap-6125	159	1	the	the	DET
ap-6125	159	2	matrix	matrix	NOUN
ap-6125	159	3	φ	φ	PROPN
ap-6125	159	4	is	be	AUX
ap-6125	159	5	,	,	PUNCT
ap-6125	159	6	in	in	ADP
ap-6125	159	7	general	general	ADJ
ap-6125	159	8	,	,	PUNCT
ap-6125	159	9	rectangular	rectangular	ADJ
ap-6125	159	10	.	.	PUNCT
ap-6125	160	1	we	we	PRON
ap-6125	160	2	state	state	VERB
ap-6125	160	3	in	in	ADP
ap-6125	160	4	following	follow	VERB
ap-6125	160	5	theorem	theorem	VERB
ap-6125	160	6	2	2	NUM
ap-6125	160	7	that	that	SCONJ
ap-6125	160	8	a	a	DET
ap-6125	160	9	finite	finite	ADJ
ap-6125	160	10	linear	linear	ADJ
ap-6125	160	11	combination	combination	NOUN
ap-6125	160	12	of	of	ADP
ap-6125	160	13	the	the	DET
ap-6125	160	14	values	value	NOUN
ap-6125	160	15	of	of	ADP
ap-6125	160	16	the	the	DET
ap-6125	160	17	generating	generate	VERB
ap-6125	160	18	function	function	NOUN
ap-6125	160	19	r(x	r(x	PROPN
ap-6125	160	20	,	,	PUNCT
ap-6125	160	21	y	y	NOUN
ap-6125	160	22	)	)	PUNCT
ap-6125	160	23	at	at	ADP
ap-6125	160	24	nodes	node	NOUN
ap-6125	160	25	is	be	AUX
ap-6125	160	26	used	use	VERB
ap-6125	160	27	for	for	ADP
ap-6125	160	28	the	the	DET
ap-6125	160	29	practical	practical	ADJ
ap-6125	160	30	interpolation	interpolation	NOUN
ap-6125	160	31	instead	instead	ADV
ap-6125	160	32	of	of	ADP
ap-6125	160	33	the	the	DET
ap-6125	160	34	infinite	infinite	ADJ
ap-6125	160	35	linear	linear	ADJ
ap-6125	160	36	combination	combination	NOUN
ap-6125	160	37	of	of	ADP
ap-6125	160	38	the	the	DET
ap-6125	160	39	values	value	NOUN
ap-6125	160	40	of	of	ADP
ap-6125	160	41	the	the	DET
ap-6125	160	42	basis	basis	NOUN
ap-6125	160	43	functions	function	NOUN
ap-6125	160	44	in	in	ADP
ap-6125	160	45	(	(	PUNCT
ap-6125	160	46	14	14	NUM
ap-6125	160	47	)	)	PUNCT
ap-6125	160	48	.	.	PUNCT
ap-6125	161	1	theorem	theorem	NOUN
ap-6125	161	2	2	2	NUM
ap-6125	161	3	.	.	PUNCT
ap-6125	162	1	let	let	VERB
ap-6125	162	2	xi	xi	PROPN
ap-6125	162	3	6=	6=	ADP
ap-6125	162	4	xj	xj	PROPN
ap-6125	162	5	for	for	ADP
ap-6125	162	6	all	all	DET
ap-6125	162	7	i	i	PRON
ap-6125	162	8	6=	6=	PROPN
ap-6125	162	9	j.	j.	PROPN
ap-6125	162	10	assume	assume	VERB
ap-6125	162	11	that	that	SCONJ
ap-6125	162	12	the	the	DET
ap-6125	162	13	series	series	NOUN
ap-6125	162	14	(	(	PUNCT
ap-6125	162	15	17	17	NUM
ap-6125	162	16	)	)	PUNCT
ap-6125	162	17	converges	converge	VERB
ap-6125	162	18	for	for	ADP
ap-6125	162	19	all	all	DET
ap-6125	162	20	x	x	NOUN
ap-6125	162	21	,	,	PUNCT
ap-6125	162	22	y	y	PROPN
ap-6125	162	23	∈	∈	PROPN
ap-6125	162	24	ω	ω	PROPN
ap-6125	162	25	and	and	CCONJ
ap-6125	162	26	the	the	DET
ap-6125	162	27	generating	generate	VERB
ap-6125	162	28	function	function	NOUN
ap-6125	162	29	r(x	r(x	PROPN
ap-6125	162	30	,	,	PUNCT
ap-6125	162	31	y	y	NOUN
ap-6125	162	32	)	)	PUNCT
ap-6125	162	33	is	be	AUX
ap-6125	162	34	continuous	continuous	ADJ
ap-6125	162	35	in	in	ADP
ap-6125	162	36	ω	ω	NUM
ap-6125	162	37	.	.	PUNCT
ap-6125	163	1	moreover	moreover	ADV
ap-6125	163	2	,	,	PUNCT
ap-6125	163	3	let	let	VERB
ap-6125	163	4	rank	rank	PROPN
ap-6125	163	5	φ	φ	PROPN
ap-6125	163	6	=	=	SYM
ap-6125	163	7	t	t	PROPN
ap-6125	163	8	(	(	PUNCT
ap-6125	163	9	n	n	CCONJ
ap-6125	163	10	,	,	PUNCT
ap-6125	163	11	l	l	NOUN
ap-6125	163	12	)	)	PUNCT
ap-6125	163	13	.	.	PUNCT
ap-6125	164	1	then	then	ADV
ap-6125	164	2	the	the	DET
ap-6125	164	3	problem	problem	NOUN
ap-6125	164	4	(	(	PUNCT
ap-6125	164	5	14	14	NUM
ap-6125	164	6	)	)	PUNCT
ap-6125	164	7	,	,	PUNCT
ap-6125	164	8	(	(	PUNCT
ap-6125	164	9	15	15	NUM
ap-6125	164	10	)	)	PUNCT
ap-6125	164	11	,	,	PUNCT
ap-6125	164	12	and	and	CCONJ
ap-6125	164	13	(	(	PUNCT
ap-6125	164	14	16	16	NUM
ap-6125	164	15	)	)	PUNCT
ap-6125	164	16	of	of	ADP
ap-6125	164	17	smooth	smooth	ADJ
ap-6125	164	18	interpolation	interpolation	NOUN
ap-6125	164	19	has	have	VERB
ap-6125	164	20	the	the	DET
ap-6125	164	21	unique	unique	ADJ
ap-6125	164	22	solution	solution	NOUN
ap-6125	164	23	z(x	z(x	NUM
ap-6125	164	24	)	)	PUNCT
ap-6125	164	25	=	=	SYM
ap-6125	165	1	n∑	n∑	NOUN
ap-6125	165	2	j=1	j=1	PROPN
ap-6125	165	3	λjr(x	λjr(x	PROPN
ap-6125	165	4	,	,	PUNCT
ap-6125	165	5	xj	xj	PROPN
ap-6125	165	6	)	)	PUNCT
ap-6125	165	7	+	+	CCONJ
ap-6125	165	8	∑	∑	PROPN
ap-6125	165	9	|α|≤l−1	|α|≤l−1	PROPN
ap-6125	165	10	aαϕα(x	aαϕα(x	PROPN
ap-6125	165	11	)	)	PUNCT
ap-6125	165	12	,	,	PUNCT
ap-6125	165	13	(	(	PUNCT
ap-6125	165	14	18	18	NUM
ap-6125	165	15	)	)	PUNCT
ap-6125	165	16	where	where	SCONJ
ap-6125	165	17	the	the	DET
ap-6125	165	18	complex	complex	ADJ
ap-6125	165	19	,	,	PUNCT
ap-6125	165	20	in	in	ADP
ap-6125	165	21	general	general	ADJ
ap-6125	165	22	,	,	PUNCT
ap-6125	165	23	coefficients	coefficient	NOUN
ap-6125	165	24	λj	λj	PROPN
ap-6125	165	25	,	,	PUNCT
ap-6125	165	26	j	j	PROPN
ap-6125	165	27	=	=	SYM
ap-6125	165	28	1	1	NUM
ap-6125	165	29	,	,	PUNCT
ap-6125	165	30	.	.	PUNCT
ap-6125	165	31	.	.	PUNCT
ap-6125	165	32	.	.	PUNCT
ap-6125	165	33	,	,	PUNCT
ap-6125	165	34	n	n	CCONJ
ap-6125	165	35	,	,	PUNCT
ap-6125	165	36	and	and	CCONJ
ap-6125	165	37	aα	aα	NOUN
ap-6125	165	38	,	,	PUNCT
ap-6125	165	39	|α|	|α|	PROPN
ap-6125	165	40	≤	≤	NUM
ap-6125	165	41	l−	l−	NOUN
ap-6125	165	42	1	1	NUM
ap-6125	165	43	,	,	PUNCT
ap-6125	165	44	are	be	AUX
ap-6125	165	45	the	the	DET
ap-6125	165	46	unique	unique	ADJ
ap-6125	165	47	solution	solution	NOUN
ap-6125	165	48	of	of	ADP
ap-6125	165	49	the	the	DET
ap-6125	165	50	linear	linear	ADJ
ap-6125	165	51	algebraic	algebraic	ADJ
ap-6125	165	52	system	system	NOUN
ap-6125	165	53	n∑	n∑	PROPN
ap-6125	165	54	j=1	j=1	PROPN
ap-6125	165	55	λjr(xi	λjr(xi	X
ap-6125	165	56	,	,	PUNCT
ap-6125	165	57	xj	xj	NOUN
ap-6125	165	58	)	)	PUNCT
ap-6125	166	1	+	+	CCONJ
ap-6125	166	2	∑	∑	PROPN
ap-6125	166	3	|α|≤l−1	|α|≤l−1	NOUN
ap-6125	166	4	aαϕα(xi	aαϕα(xi	NOUN
ap-6125	166	5	)	)	PUNCT
ap-6125	166	6	=	=	PUNCT
ap-6125	167	1	fi	fi	NOUN
ap-6125	167	2	,	,	PUNCT
ap-6125	167	3	i	i	NOUN
ap-6125	167	4	=	=	NOUN
ap-6125	167	5	1	1	NUM
ap-6125	167	6	,	,	PUNCT
ap-6125	167	7	.	.	PUNCT
ap-6125	167	8	.	.	PUNCT
ap-6125	167	9	.	.	PUNCT
ap-6125	167	10	,	,	PUNCT
ap-6125	167	11	n	n	CCONJ
ap-6125	167	12	,	,	PUNCT
ap-6125	167	13	(	(	PUNCT
ap-6125	167	14	19	19	NUM
ap-6125	167	15	)	)	PUNCT
ap-6125	167	16	n∑	n∑	NOUN
ap-6125	167	17	j=1	j=1	PROPN
ap-6125	167	18	λjϕ	λjϕ	PROPN
ap-6125	167	19	∗	∗	X
ap-6125	167	20	α(xj	α(xj	PROPN
ap-6125	167	21	)	)	PUNCT
ap-6125	167	22	=	=	SYM
ap-6125	167	23	0	0	NUM
ap-6125	167	24	,	,	PUNCT
ap-6125	167	25	|α|	|α|	PROPN
ap-6125	167	26	≤	≤	NUM
ap-6125	167	27	l−	l−	NOUN
ap-6125	167	28	1	1	NUM
ap-6125	167	29	.	.	PUNCT
ap-6125	168	1	(	(	PUNCT
ap-6125	168	2	20	20	NUM
ap-6125	168	3	)	)	PUNCT
ap-6125	168	4	proof	proof	NOUN
ap-6125	168	5	.	.	PUNCT
ap-6125	169	1	the	the	DET
ap-6125	169	2	proof	proof	NOUN
ap-6125	169	3	is	be	AUX
ap-6125	169	4	given	give	VERB
ap-6125	169	5	in	in	ADP
ap-6125	169	6	[	[	X
ap-6125	169	7	2	2	NUM
ap-6125	169	8	]	]	PUNCT
ap-6125	169	9	.	.	PUNCT
ap-6125	170	1	note	note	VERB
ap-6125	170	2	that	that	SCONJ
ap-6125	170	3	we	we	PRON
ap-6125	170	4	have	have	VERB
ap-6125	170	5	to	to	PART
ap-6125	170	6	solve	solve	VERB
ap-6125	170	7	the	the	DET
ap-6125	170	8	linear	linear	ADJ
ap-6125	170	9	algebraic	algebraic	ADJ
ap-6125	170	10	system	system	NOUN
ap-6125	170	11	(	(	PUNCT
ap-6125	170	12	19	19	NUM
ap-6125	170	13	)	)	PUNCT
ap-6125	170	14	,	,	PUNCT
ap-6125	170	15	(	(	PUNCT
ap-6125	170	16	20	20	NUM
ap-6125	170	17	)	)	PUNCT
ap-6125	170	18	for	for	ADP
ap-6125	170	19	n	n	PROPN
ap-6125	170	20	+	+	X
ap-6125	170	21	t	t	PROPN
ap-6125	170	22	(	(	PUNCT
ap-6125	170	23	n	n	CCONJ
ap-6125	170	24	,	,	PUNCT
ap-6125	170	25	l	l	NOUN
ap-6125	170	26	)	)	PUNCT
ap-6125	170	27	unknowns	unknown	NOUN
ap-6125	170	28	.	.	PUNCT
ap-6125	171	1	the	the	DET
ap-6125	171	2	number	number	NOUN
ap-6125	171	3	of	of	ADP
ap-6125	171	4	unknowns	unknown	NOUN
ap-6125	171	5	(	(	PUNCT
ap-6125	171	6	and	and	CCONJ
ap-6125	171	7	equations	equation	NOUN
ap-6125	171	8	)	)	PUNCT
ap-6125	171	9	depends	depend	VERB
ap-6125	171	10	on	on	ADP
ap-6125	171	11	n	n	CCONJ
ap-6125	171	12	only	only	ADV
ap-6125	171	13	through	through	ADP
ap-6125	171	14	t	t	PROPN
ap-6125	171	15	(	(	PUNCT
ap-6125	171	16	n	n	CCONJ
ap-6125	171	17	,	,	PUNCT
ap-6125	171	18	l	l	NOUN
ap-6125	171	19	)	)	PUNCT
ap-6125	171	20	,	,	PUNCT
ap-6125	171	21	the	the	DET
ap-6125	171	22	number	number	NOUN
ap-6125	171	23	of	of	ADP
ap-6125	171	24	trend	trend	NOUN
ap-6125	171	25	functions	function	NOUN
ap-6125	171	26	.	.	PUNCT
ap-6125	172	1	the	the	DET
ap-6125	172	2	smooth	smooth	ADJ
ap-6125	172	3	interpolant	interpolant	NOUN
ap-6125	172	4	z	z	NOUN
ap-6125	172	5	given	give	VERB
ap-6125	172	6	by	by	ADP
ap-6125	172	7	(	(	PUNCT
ap-6125	172	8	14	14	NUM
ap-6125	172	9	)	)	PUNCT
ap-6125	172	10	can	can	AUX
ap-6125	172	11	now	now	ADV
ap-6125	172	12	be	be	AUX
ap-6125	172	13	rewritten	rewrite	VERB
ap-6125	172	14	for	for	ADP
ap-6125	172	15	the	the	DET
ap-6125	172	16	generating	generate	VERB
ap-6125	172	17	function	function	NOUN
ap-6125	172	18	r(x	r(x	PROPN
ap-6125	172	19	,	,	PUNCT
ap-6125	172	20	y	y	NOUN
ap-6125	172	21	)	)	PUNCT
ap-6125	172	22	in	in	ADP
ap-6125	172	23	the	the	DET
ap-6125	172	24	form	form	NOUN
ap-6125	172	25	(	(	PUNCT
ap-6125	172	26	18	18	NUM
ap-6125	172	27	)	)	PUNCT
ap-6125	172	28	.	.	PUNCT
ap-6125	173	1	6	6	X
ap-6125	173	2	.	.	X
ap-6125	173	3	a	a	DET
ap-6125	173	4	periodic	periodic	ADJ
ap-6125	173	5	basis	basis	NOUN
ap-6125	173	6	function	function	NOUN
ap-6125	173	7	system	system	NOUN
ap-6125	173	8	of	of	ADP
ap-6125	173	9	wl	wl	PROPN
ap-6125	173	10	let	let	VERB
ap-6125	173	11	the	the	DET
ap-6125	173	12	continuous	continuous	ADJ
ap-6125	173	13	function	function	NOUN
ap-6125	173	14	f(x	f(x	PROPN
ap-6125	173	15	)	)	PUNCT
ap-6125	173	16	=	=	SYM
ap-6125	174	1	f(x1	f(x1	NOUN
ap-6125	174	2	,	,	PUNCT
ap-6125	174	3	.	.	PUNCT
ap-6125	174	4	.	.	PUNCT
ap-6125	174	5	.	.	PUNCT
ap-6125	175	1	,	,	PUNCT
ap-6125	175	2	xn	xn	PROPN
ap-6125	175	3	)	)	PUNCT
ap-6125	175	4	,	,	PUNCT
ap-6125	175	5	to	to	PART
ap-6125	175	6	be	be	AUX
ap-6125	175	7	interpolated	interpolate	VERB
ap-6125	175	8	,	,	PUNCT
ap-6125	175	9	be	be	AUX
ap-6125	175	10	2π	2π	NOUN
ap-6125	175	11	-	-	NOUN
ap-6125	175	12	periodic	periodic	ADJ
ap-6125	175	13	in	in	ADP
ap-6125	175	14	each	each	DET
ap-6125	175	15	independent	independent	ADJ
ap-6125	175	16	variable	variable	PROPN
ap-6125	175	17	xs	xs	PROPN
ap-6125	175	18	,	,	PUNCT
ap-6125	175	19	s	s	PART
ap-6125	175	20	=	=	NOUN
ap-6125	175	21	1	1	NUM
ap-6125	175	22	,	,	PUNCT
ap-6125	175	23	.	.	PUNCT
ap-6125	175	24	.	.	PUNCT
ap-6125	176	1	.	.	PUNCT
ap-6125	177	1	,	,	PUNCT
ap-6125	177	2	n.	n.	NOUN
ap-6125	177	3	periodic	periodic	ADJ
ap-6125	177	4	functions	function	NOUN
ap-6125	177	5	with	with	ADP
ap-6125	177	6	other	other	ADJ
ap-6125	177	7	periods	period	NOUN
ap-6125	177	8	in	in	ADP
ap-6125	177	9	the	the	DET
ap-6125	177	10	individual	individual	ADJ
ap-6125	177	11	variables	variable	NOUN
ap-6125	177	12	can	can	AUX
ap-6125	177	13	be	be	AUX
ap-6125	177	14	formally	formally	ADV
ap-6125	177	15	transformed	transform	VERB
ap-6125	177	16	to	to	ADP
ap-6125	177	17	the	the	DET
ap-6125	177	18	period	period	NOUN
ap-6125	177	19	2π	2π	NOUN
ap-6125	177	20	.	.	PUNCT
ap-6125	178	1	let	let	VERB
ap-6125	178	2	us	we	PRON
ap-6125	178	3	consider	consider	VERB
ap-6125	178	4	f	f	PROPN
ap-6125	178	5	in	in	ADP
ap-6125	178	6	the	the	DET
ap-6125	178	7	cube	cube	NOUN
ap-6125	178	8	ω̃	ω̃	NUM
ap-6125	179	1	=	=	PUNCT
ap-6125	180	1	[	[	X
ap-6125	180	2	0	0	NUM
ap-6125	180	3	,	,	PUNCT
ap-6125	180	4	2π]n	2π]n	NUM
ap-6125	180	5	.	.	PUNCT
ap-6125	181	1	write	write	VERB
ap-6125	181	2	x	x	SYM
ap-6125	181	3	·	·	PUNCT
ap-6125	181	4	y	y	X
ap-6125	181	5	=	=	PUNCT
ap-6125	181	6	x1y1	x1y1	PUNCT
ap-6125	181	7	+	+	X
ap-6125	181	8	·	·	PUNCT
ap-6125	181	9	·	·	PUNCT
ap-6125	181	10	·	·	PUNCT
ap-6125	182	1	+	+	CCONJ
ap-6125	182	2	xnyn	xnyn	ADJ
ap-6125	182	3	for	for	ADP
ap-6125	182	4	the	the	DET
ap-6125	182	5	rn	rn	PROPN
ap-6125	182	6	inner	inner	ADJ
ap-6125	182	7	product	product	NOUN
ap-6125	182	8	of	of	ADP
ap-6125	182	9	vectors	vector	NOUN
ap-6125	182	10	x	x	PUNCT
ap-6125	182	11	and	and	CCONJ
ap-6125	182	12	y.	y.	NOUN
ap-6125	182	13	we	we	PRON
ap-6125	182	14	choose	choose	VERB
ap-6125	182	15	exponential	exponential	ADJ
ap-6125	182	16	functions	function	NOUN
ap-6125	182	17	of	of	ADP
ap-6125	182	18	a	a	DET
ap-6125	182	19	pure	pure	ADJ
ap-6125	182	20	imaginary	imaginary	ADJ
ap-6125	182	21	argument	argument	NOUN
ap-6125	182	22	for	for	ADP
ap-6125	182	23	the	the	DET
ap-6125	182	24	periodic	periodic	ADJ
ap-6125	182	25	basis	basis	NOUN
ap-6125	182	26	system	system	NOUN
ap-6125	182	27	{	{	PUNCT
ap-6125	182	28	gρ	gρ	NOUN
ap-6125	182	29	}	}	PUNCT
ap-6125	182	30	in	in	ADP
ap-6125	182	31	wl	wl	PROPN
ap-6125	182	32	,	,	PUNCT
ap-6125	182	33	where	where	SCONJ
ap-6125	182	34	gρ(x	gρ(x	VERB
ap-6125	182	35	)	)	PUNCT
ap-6125	182	36	=	=	PUNCT
ap-6125	183	1	exp(−iρ	exp(−iρ	PROPN
ap-6125	183	2	·	·	PUNCT
ap-6125	183	3	x	x	X
ap-6125	183	4	)	)	PUNCT
ap-6125	183	5	.	.	PUNCT
ap-6125	184	1	we	we	PRON
ap-6125	184	2	have	have	VERB
ap-6125	184	3	to	to	PART
ap-6125	184	4	change	change	VERB
ap-6125	184	5	the	the	DET
ap-6125	184	6	notation	notation	NOUN
ap-6125	184	7	properly	properly	ADV
ap-6125	184	8	with	with	ADP
ap-6125	184	9	respect	respect	NOUN
ap-6125	184	10	to	to	ADP
ap-6125	184	11	the	the	DET
ap-6125	184	12	fact	fact	NOUN
ap-6125	184	13	that	that	SCONJ
ap-6125	184	14	the	the	DET
ap-6125	184	15	integer	integer	NOUN
ap-6125	184	16	components	component	NOUN
ap-6125	184	17	of	of	ADP
ap-6125	184	18	the	the	DET
ap-6125	184	19	multiindex	multiindex	NOUN
ap-6125	184	20	ρ	ρ	PROPN
ap-6125	184	21	are	be	AUX
ap-6125	184	22	also	also	ADV
ap-6125	184	23	negative	negative	ADJ
ap-6125	184	24	.	.	PUNCT
ap-6125	185	1	the	the	DET
ap-6125	185	2	definition	definition	NOUN
ap-6125	185	3	(	(	PUNCT
ap-6125	185	4	3	3	NUM
ap-6125	185	5	)	)	PUNCT
ap-6125	185	6	of	of	ADP
ap-6125	185	7	the	the	DET
ap-6125	185	8	length	length	NOUN
ap-6125	185	9	|ρ|	|ρ|	NOUN
ap-6125	185	10	of	of	ADP
ap-6125	185	11	the	the	DET
ap-6125	185	12	multiindex	multiindex	NOUN
ap-6125	185	13	ρ	ρ	NOUN
ap-6125	185	14	remains	remain	VERB
ap-6125	185	15	without	without	ADP
ap-6125	185	16	change	change	NOUN
ap-6125	185	17	.	.	PUNCT
ap-6125	186	1	in	in	ADP
ap-6125	186	2	the	the	DET
ap-6125	186	3	definition	definition	NOUN
ap-6125	186	4	(	(	PUNCT
ap-6125	186	5	17	17	NUM
ap-6125	186	6	)	)	PUNCT
ap-6125	186	7	of	of	ADP
ap-6125	186	8	the	the	DET
ap-6125	186	9	generating	generate	VERB
ap-6125	186	10	function	function	NOUN
ap-6125	186	11	r(x	r(x	PROPN
ap-6125	186	12	,	,	PUNCT
ap-6125	186	13	y	y	PROPN
ap-6125	186	14	)	)	PUNCT
ap-6125	186	15	,	,	PUNCT
ap-6125	186	16	we	we	PRON
ap-6125	186	17	sum	sum	VERB
ap-6125	186	18	over	over	ADP
ap-6125	186	19	all	all	DET
ap-6125	186	20	multiindices	multiindice	NOUN
ap-6125	186	21	ρ	ρ	NOUN
ap-6125	186	22	(	(	PUNCT
ap-6125	186	23	not	not	PART
ap-6125	186	24	only	only	ADV
ap-6125	186	25	over	over	ADP
ap-6125	186	26	those	those	PRON
ap-6125	186	27	with	with	ADP
ap-6125	186	28	nonnegative	nonnegative	ADJ
ap-6125	186	29	components	component	NOUN
ap-6125	186	30	)	)	PUNCT
ap-6125	186	31	.	.	PUNCT
ap-6125	187	1	the	the	DET
ap-6125	187	2	following	follow	VERB
ap-6125	187	3	theorem	theorem	NOUN
ap-6125	187	4	shows	show	VERB
ap-6125	187	5	important	important	ADJ
ap-6125	187	6	properties	property	NOUN
ap-6125	187	7	of	of	ADP
ap-6125	187	8	the	the	DET
ap-6125	187	9	system	system	NOUN
ap-6125	187	10	{	{	PUNCT
ap-6125	187	11	gρ	gρ	NOUN
ap-6125	187	12	}	}	PUNCT
ap-6125	187	13	.	.	PUNCT
ap-6125	188	1	theorem	theorem	NOUN
ap-6125	188	2	3	3	X
ap-6125	188	3	.	.	PUNCT
ap-6125	189	1	let	let	VERB
ap-6125	189	2	there	there	PRON
ap-6125	189	3	be	be	AUX
ap-6125	189	4	an	an	DET
ap-6125	189	5	integer	integer	NOUN
ap-6125	189	6	u	u	NOUN
ap-6125	189	7	,	,	PUNCT
ap-6125	189	8	u	u	PROPN
ap-6125	189	9	≥	≥	NOUN
ap-6125	189	10	l	l	NOUN
ap-6125	189	11	,	,	PUNCT
ap-6125	189	12	such	such	ADJ
ap-6125	189	13	that	that	SCONJ
ap-6125	189	14	bα	bα	NOUN
ap-6125	190	1	=	=	NOUN
ap-6125	190	2	0	0	NUM
ap-6125	190	3	for	for	ADP
ap-6125	190	4	all	all	DET
ap-6125	190	5	|α|	|α|	PROPN
ap-6125	190	6	>	>	X
ap-6125	190	7	u	u	PROPN
ap-6125	190	8	in	in	ADP
ap-6125	190	9	wl	wl	PROPN
ap-6125	190	10	.	.	PUNCT
ap-6125	191	1	the	the	DET
ap-6125	191	2	system	system	NOUN
ap-6125	191	3	of	of	ADP
ap-6125	191	4	periodic	periodic	ADJ
ap-6125	191	5	exponential	exponential	ADJ
ap-6125	191	6	functions	function	NOUN
ap-6125	191	7	of	of	ADP
ap-6125	191	8	pure	pure	ADJ
ap-6125	191	9	imaginary	imaginary	ADJ
ap-6125	191	10	argument	argument	NOUN
ap-6125	191	11	gρ(x	gρ(x	PUNCT
ap-6125	191	12	)	)	PUNCT
ap-6125	191	13	=	=	PUNCT
ap-6125	192	1	exp(−iρ	exp(−iρ	PROPN
ap-6125	192	2	·	·	PUNCT
ap-6125	192	3	x	x	X
ap-6125	192	4	)	)	PUNCT
ap-6125	192	5	,	,	PUNCT
ap-6125	192	6	x	x	PUNCT
ap-6125	192	7	∈	∈	PROPN
ap-6125	192	8	ω̃	ω̃	PROPN
ap-6125	192	9	,	,	PUNCT
ap-6125	192	10	(	(	PUNCT
ap-6125	192	11	21	21	NUM
ap-6125	192	12	)	)	PUNCT
ap-6125	192	13	ρ	ρ	PROPN
ap-6125	192	14	being	be	AUX
ap-6125	192	15	a	a	DET
ap-6125	192	16	multiindex	multiindex	NOUN
ap-6125	192	17	with	with	ADP
ap-6125	192	18	integer	integer	NOUN
ap-6125	192	19	components	component	NOUN
ap-6125	192	20	ρs	ρs	ADV
ap-6125	192	21	=	=	SYM
ap-6125	192	22	0,±1,±2	0,±1,±2	PROPN
ap-6125	192	23	,	,	PUNCT
ap-6125	192	24	.	.	PUNCT
ap-6125	192	25	.	.	PUNCT
ap-6125	192	26	.	.	PUNCT
ap-6125	193	1	,	,	PUNCT
ap-6125	193	2	s	s	NOUN
ap-6125	193	3	=	=	NOUN
ap-6125	193	4	1	1	NUM
ap-6125	193	5	,	,	PUNCT
ap-6125	193	6	.	.	PUNCT
ap-6125	193	7	.	.	PUNCT
ap-6125	194	1	.	.	PUNCT
ap-6125	195	1	,	,	PUNCT
ap-6125	196	1	n	n	CCONJ
ap-6125	196	2	,	,	PUNCT
ap-6125	196	3	is	be	AUX
ap-6125	196	4	complete	complete	ADJ
ap-6125	196	5	and	and	CCONJ
ap-6125	196	6	orthogonal	orthogonal	ADJ
ap-6125	196	7	in	in	ADP
ap-6125	196	8	wl	wl	PROPN
ap-6125	196	9	.	.	PUNCT
ap-6125	197	1	proof	proof	NOUN
ap-6125	197	2	.	.	PUNCT
ap-6125	198	1	the	the	DET
ap-6125	198	2	proof	proof	NOUN
ap-6125	198	3	is	be	AUX
ap-6125	198	4	given	give	VERB
ap-6125	198	5	in	in	ADP
ap-6125	198	6	[	[	X
ap-6125	198	7	2	2	NUM
ap-6125	198	8	]	]	PUNCT
ap-6125	198	9	.	.	PUNCT
ap-6125	199	1	151	151	NUM
ap-6125	199	2	karel	karel	PROPN
ap-6125	199	3	segeth	segeth	PROPN
ap-6125	199	4	acta	acta	PROPN
ap-6125	199	5	polytechnica	polytechnica	PROPN
ap-6125	199	6	remark	remark	NOUN
ap-6125	199	7	2	2	NUM
ap-6125	199	8	.	.	PUNCT
ap-6125	199	9	note	note	VERB
ap-6125	199	10	that	that	SCONJ
ap-6125	199	11	on	on	ADP
ap-6125	199	12	the	the	DET
ap-6125	199	13	assumption	assumption	NOUN
ap-6125	199	14	of	of	ADP
ap-6125	199	15	theorem	theorem	NOUN
ap-6125	199	16	3	3	NUM
ap-6125	199	17	that	that	SCONJ
ap-6125	199	18	there	there	PRON
ap-6125	199	19	is	be	VERB
ap-6125	199	20	an	an	DET
ap-6125	199	21	integer	integer	NOUN
ap-6125	199	22	u	u	NOUN
ap-6125	199	23	of	of	ADP
ap-6125	199	24	required	require	VERB
ap-6125	199	25	properties	property	NOUN
ap-6125	199	26	,	,	PUNCT
ap-6125	199	27	bα	bα	PROPN
ap-6125	199	28	>	>	X
ap-6125	199	29	0	0	NUM
ap-6125	199	30	can	can	AUX
ap-6125	199	31	occur	occur	VERB
ap-6125	199	32	only	only	ADV
ap-6125	199	33	for	for	ADP
ap-6125	199	34	l	l	NOUN
ap-6125	199	35	≤	≤	NUM
ap-6125	199	36	|α|	|α|	PROPN
ap-6125	199	37	≤	≤	NUM
ap-6125	199	38	u	u	NOUN
ap-6125	199	39	.	.	PUNCT
ap-6125	200	1	we	we	PRON
ap-6125	200	2	will	will	AUX
ap-6125	200	3	keep	keep	VERB
ap-6125	200	4	this	this	DET
ap-6125	200	5	assumption	assumption	NOUN
ap-6125	200	6	in	in	ADP
ap-6125	200	7	the	the	DET
ap-6125	200	8	rest	rest	NOUN
ap-6125	200	9	of	of	ADP
ap-6125	200	10	the	the	DET
ap-6125	200	11	paper	paper	NOUN
ap-6125	200	12	.	.	PUNCT
ap-6125	201	1	we	we	PRON
ap-6125	201	2	further	far	ADV
ap-6125	201	3	follow	follow	VERB
ap-6125	201	4	[	[	X
ap-6125	201	5	2	2	NUM
ap-6125	201	6	]	]	PUNCT
ap-6125	201	7	.	.	PUNCT
ap-6125	202	1	for	for	ADP
ap-6125	202	2	the	the	DET
ap-6125	202	3	basis	basis	NOUN
ap-6125	202	4	system	system	NOUN
ap-6125	202	5	(	(	PUNCT
ap-6125	202	6	21	21	NUM
ap-6125	202	7	)	)	PUNCT
ap-6125	202	8	,	,	PUNCT
ap-6125	202	9	notice	notice	VERB
ap-6125	202	10	that	that	SCONJ
ap-6125	202	11	ρ	ρ	NOUN
ap-6125	202	12	is	be	AUX
ap-6125	202	13	not	not	PART
ap-6125	202	14	nonnegative	nonnegative	ADJ
ap-6125	202	15	and	and	CCONJ
ap-6125	202	16	the	the	DET
ap-6125	202	17	generating	generate	VERB
ap-6125	202	18	function	function	NOUN
ap-6125	202	19	r(x	r(x	PROPN
ap-6125	202	20	,	,	PUNCT
ap-6125	202	21	y	y	NOUN
ap-6125	202	22	)	)	PUNCT
ap-6125	202	23	=	=	SYM
ap-6125	203	1	∑	∑	PROPN
ap-6125	203	2	ρ	ρ	PROPN
ap-6125	203	3	gρ(x)g∗ρ(y	gρ(x)g∗ρ(y	NOUN
ap-6125	203	4	)	)	PUNCT
ap-6125	203	5	‖gρ‖2	‖gρ‖2	ADJ
ap-6125	203	6	l	l	NOUN
ap-6125	203	7	=	=	PUNCT
ap-6125	203	8	∑	∑	PUNCT
ap-6125	203	9	ρ	ρ	PROPN
ap-6125	203	10	exp(−iρ	exp(−iρ	PROPN
ap-6125	203	11	·	·	PUNCT
ap-6125	203	12	(	(	PUNCT
ap-6125	203	13	x−	x−	PROPN
ap-6125	203	14	y	y	PROPN
ap-6125	203	15	)	)	PUNCT
ap-6125	203	16	)	)	PUNCT
ap-6125	204	1	‖gρ‖2	‖gρ‖2	ADJ
ap-6125	204	2	l	l	NOUN
ap-6125	204	3	(	(	PUNCT
ap-6125	204	4	22	22	NUM
ap-6125	204	5	)	)	PUNCT
ap-6125	204	6	is	be	AUX
ap-6125	204	7	the	the	DET
ap-6125	204	8	n	n	ADV
ap-6125	204	9	-	-	PUNCT
ap-6125	204	10	dimensional	dimensional	ADJ
ap-6125	204	11	fourier	fourier	NOUN
ap-6125	204	12	series	series	NOUN
ap-6125	204	13	in	in	ADP
ap-6125	204	14	l2(ω̃	l2(ω̃	PROPN
ap-6125	204	15	)	)	PUNCT
ap-6125	204	16	with	with	ADP
ap-6125	204	17	the	the	DET
ap-6125	204	18	coefficients	coefficient	NOUN
ap-6125	204	19	‖gρ‖−2	‖gρ‖−2	PROPN
ap-6125	204	20	l	l	NOUN
ap-6125	204	21	,	,	PUNCT
ap-6125	204	22	where	where	SCONJ
ap-6125	204	23	‖gρ‖2	‖gρ‖2	ADJ
ap-6125	204	24	l	l	NOUN
ap-6125	204	25	=	=	SYM
ap-6125	204	26	(	(	PUNCT
ap-6125	204	27	2π)n	2π)n	NUM
ap-6125	204	28	u∑	u∑	ADJ
ap-6125	204	29	|α|=l	|α|=l	ADJ
ap-6125	204	30	bαρ	bαρ	NOUN
ap-6125	204	31	2α1	2α1	NOUN
ap-6125	204	32	1	1	NUM
ap-6125	204	33	.	.	PUNCT
ap-6125	204	34	.	.	PUNCT
ap-6125	204	35	.	.	PUNCT
ap-6125	205	1	ρ2αn	ρ2αn	PROPN
ap-6125	205	2	n	n	CCONJ
ap-6125	205	3	according	accord	VERB
ap-6125	205	4	to	to	ADP
ap-6125	205	5	(	(	PUNCT
ap-6125	205	6	10	10	NUM
ap-6125	205	7	)	)	PUNCT
ap-6125	205	8	.	.	PUNCT
ap-6125	206	1	let	let	VERB
ap-6125	206	2	now	now	ADV
ap-6125	206	3	the	the	DET
ap-6125	206	4	complex	complex	ADV
ap-6125	206	5	-	-	PUNCT
ap-6125	206	6	valued	value	VERB
ap-6125	206	7	function	function	NOUN
ap-6125	206	8	f	f	PROPN
ap-6125	206	9	,	,	PUNCT
ap-6125	206	10	to	to	PART
ap-6125	206	11	be	be	AUX
ap-6125	206	12	interpolated	interpolate	VERB
ap-6125	206	13	,	,	PUNCT
ap-6125	206	14	be	be	AUX
ap-6125	206	15	nonperiodic	nonperiodic	ADJ
ap-6125	206	16	in	in	ADP
ap-6125	206	17	rn	rn	PROPN
ap-6125	206	18	.	.	PROPN
ap-6125	206	19	redefine	redefine	VERB
ap-6125	206	20	the	the	DET
ap-6125	206	21	generating	generate	VERB
ap-6125	206	22	function	function	NOUN
ap-6125	206	23	r(x	r(x	PROPN
ap-6125	206	24	,	,	PUNCT
ap-6125	206	25	y	y	NOUN
ap-6125	206	26	)	)	PUNCT
ap-6125	207	1	=	=	SYM
ap-6125	207	2	∫	∫	PROPN
ap-6125	207	3	rn	rn	PROPN
ap-6125	207	4	exp(−iρ	exp(−iρ	PROPN
ap-6125	207	5	·	·	PUNCT
ap-6125	207	6	(	(	PUNCT
ap-6125	207	7	x−	x−	PROPN
ap-6125	207	8	y	y	PROPN
ap-6125	207	9	)	)	PUNCT
ap-6125	207	10	)	)	PUNCT
ap-6125	208	1	‖gρ‖2	‖gρ‖2	ADJ
ap-6125	208	2	l	l	NOUN
ap-6125	208	3	dρ	dρ	NOUN
ap-6125	208	4	=	=	SYM
ap-6125	208	5	f	f	PROPN
ap-6125	208	6	(	(	PUNCT
ap-6125	208	7	1	1	NUM
ap-6125	208	8	‖gρ‖2	‖gρ‖2	PROPN
ap-6125	208	9	l	l	NOUN
ap-6125	208	10	)	)	PUNCT
ap-6125	208	11	(	(	PUNCT
ap-6125	208	12	23	23	NUM
ap-6125	208	13	)	)	PUNCT
ap-6125	208	14	as	as	ADP
ap-6125	208	15	the	the	DET
ap-6125	208	16	n	n	ADV
ap-6125	208	17	-	-	PUNCT
ap-6125	208	18	dimensional	dimensional	ADJ
ap-6125	208	19	fourier	fourier	NOUN
ap-6125	208	20	transform	transform	NOUN
ap-6125	208	21	f	f	PROPN
ap-6125	208	22	of	of	ADP
ap-6125	208	23	the	the	DET
ap-6125	208	24	function	function	NOUN
ap-6125	208	25	‖gρ‖−2	‖gρ‖−2	PROPN
ap-6125	208	26	l	l	NOUN
ap-6125	208	27	of	of	ADP
ap-6125	208	28	n	n	CCONJ
ap-6125	208	29	continuous	continuous	ADJ
ap-6125	208	30	variables	variable	NOUN
ap-6125	208	31	ρ1	ρ1	NOUN
ap-6125	208	32	,	,	PUNCT
ap-6125	208	33	ρ2	ρ2	NOUN
ap-6125	208	34	,	,	PUNCT
ap-6125	208	35	.	.	PUNCT
ap-6125	208	36	.	.	PUNCT
ap-6125	208	37	.	.	PUNCT
ap-6125	209	1	,	,	PUNCT
ap-6125	209	2	ρn	ρn	INTJ
ap-6125	209	3	if	if	SCONJ
ap-6125	209	4	the	the	DET
ap-6125	209	5	integral	integral	ADJ
ap-6125	209	6	exists	exist	NOUN
ap-6125	209	7	[	[	X
ap-6125	209	8	4	4	NUM
ap-6125	209	9	]	]	PUNCT
ap-6125	209	10	.	.	PUNCT
ap-6125	210	1	employing	employ	VERB
ap-6125	210	2	the	the	DET
ap-6125	210	3	transition	transition	NOUN
ap-6125	210	4	from	from	ADP
ap-6125	210	5	the	the	DET
ap-6125	210	6	fourier	fourier	NOUN
ap-6125	210	7	series	series	NOUN
ap-6125	210	8	(	(	PUNCT
ap-6125	210	9	22	22	NUM
ap-6125	210	10	)	)	PUNCT
ap-6125	210	11	with	with	ADP
ap-6125	210	12	the	the	DET
ap-6125	210	13	coefficients	coefficient	NOUN
ap-6125	210	14	‖gρ‖−2	‖gρ‖−2	X
ap-6125	210	15	l	l	NOUN
ap-6125	210	16	to	to	ADP
ap-6125	210	17	the	the	DET
ap-6125	210	18	fourier	fourier	NOUN
ap-6125	210	19	transform	transform	NOUN
ap-6125	210	20	(	(	PUNCT
ap-6125	210	21	23	23	NUM
ap-6125	210	22	)	)	PUNCT
ap-6125	210	23	of	of	ADP
ap-6125	210	24	the	the	DET
ap-6125	210	25	function	function	NOUN
ap-6125	210	26	‖gρ‖−2	‖gρ‖−2	PROPN
ap-6125	210	27	l	l	NOUN
ap-6125	210	28	of	of	ADP
ap-6125	210	29	continuous	continuous	ADJ
ap-6125	210	30	variable	variable	ADJ
ap-6125	210	31	ρ	ρ	PROPN
ap-6125	210	32	∈	∈	PROPN
ap-6125	210	33	rn	rn	PROPN
ap-6125	210	34	(	(	PUNCT
ap-6125	210	35	cf	cf	NOUN
ap-6125	210	36	.	.	NOUN
ap-6125	210	37	,	,	PUNCT
ap-6125	210	38	e.g.	e.g.	ADV
ap-6125	210	39	,	,	PUNCT
ap-6125	210	40	[	[	X
ap-6125	210	41	5	5	NUM
ap-6125	210	42	]	]	NUM
ap-6125	210	43	)	)	PUNCT
ap-6125	210	44	,	,	PUNCT
ap-6125	210	45	we	we	PRON
ap-6125	210	46	have	have	AUX
ap-6125	210	47	transformed	transform	VERB
ap-6125	210	48	the	the	DET
ap-6125	210	49	basis	basis	NOUN
ap-6125	210	50	functions	function	NOUN
ap-6125	210	51	,	,	PUNCT
ap-6125	210	52	enriched	enrich	VERB
ap-6125	210	53	their	their	PRON
ap-6125	210	54	spectrum	spectrum	NOUN
ap-6125	210	55	,	,	PUNCT
ap-6125	210	56	and	and	CCONJ
ap-6125	210	57	released	release	VERB
ap-6125	210	58	the	the	DET
ap-6125	210	59	requirement	requirement	NOUN
ap-6125	210	60	of	of	ADP
ap-6125	210	61	periodicity	periodicity	NOUN
ap-6125	210	62	of	of	ADP
ap-6125	210	63	f	f	PROPN
ap-6125	210	64	.	.	PUNCT
ap-6125	211	1	moreover	moreover	ADV
ap-6125	211	2	,	,	PUNCT
ap-6125	211	3	if	if	SCONJ
ap-6125	211	4	the	the	DET
ap-6125	211	5	integral	integral	ADJ
ap-6125	211	6	(	(	PUNCT
ap-6125	211	7	23	23	NUM
ap-6125	211	8	)	)	PUNCT
ap-6125	211	9	does	do	AUX
ap-6125	211	10	not	not	PART
ap-6125	211	11	exist	exist	VERB
ap-6125	211	12	in	in	ADP
ap-6125	211	13	the	the	DET
ap-6125	211	14	usual	usual	ADJ
ap-6125	211	15	sense	sense	NOUN
ap-6125	211	16	,	,	PUNCT
ap-6125	211	17	in	in	ADP
ap-6125	211	18	many	many	ADJ
ap-6125	211	19	instances	instance	NOUN
ap-6125	211	20	,	,	PUNCT
ap-6125	211	21	we	we	PRON
ap-6125	211	22	can	can	AUX
ap-6125	211	23	calculate	calculate	VERB
ap-6125	211	24	r(x	r(x	PROPN
ap-6125	211	25	,	,	PUNCT
ap-6125	211	26	y	y	PROPN
ap-6125	211	27	)	)	PUNCT
ap-6125	211	28	as	as	SCONJ
ap-6125	211	29	the	the	DET
ap-6125	211	30	fourier	fourier	NOUN
ap-6125	211	31	transform	transform	NOUN
ap-6125	211	32	f	f	PROPN
ap-6125	211	33	of	of	ADP
ap-6125	211	34	the	the	DET
ap-6125	211	35	generalized	generalized	ADJ
ap-6125	211	36	function	function	NOUN
ap-6125	211	37	‖gρ‖−2	‖gρ‖−2	PROPN
ap-6125	211	38	l	l	NOUN
ap-6125	211	39	of	of	ADP
ap-6125	211	40	ρ	ρ	PROPN
ap-6125	211	41	.	.	PUNCT
ap-6125	212	1	the	the	DET
ap-6125	212	2	generating	generate	VERB
ap-6125	212	3	function	function	NOUN
ap-6125	212	4	r(x	r(x	PROPN
ap-6125	212	5	,	,	PUNCT
ap-6125	212	6	y	y	NOUN
ap-6125	212	7	)	)	PUNCT
ap-6125	212	8	given	give	VERB
ap-6125	212	9	by	by	ADP
ap-6125	212	10	(	(	PUNCT
ap-6125	212	11	23	23	NUM
ap-6125	212	12	)	)	PUNCT
ap-6125	212	13	depends	depend	VERB
ap-6125	212	14	on	on	ADP
ap-6125	212	15	x	x	PUNCT
ap-6125	212	16	and	and	CCONJ
ap-6125	212	17	y	y	PROPN
ap-6125	212	18	only	only	ADV
ap-6125	212	19	through	through	ADP
ap-6125	212	20	the	the	DET
ap-6125	212	21	distance	distance	NOUN
ap-6125	212	22	r(x	r(x	PROPN
ap-6125	212	23	,	,	PUNCT
ap-6125	212	24	y	y	PROPN
ap-6125	212	25	)	)	PUNCT
ap-6125	212	26	.	.	PUNCT
ap-6125	213	1	7	7	X
ap-6125	213	2	.	.	NOUN
ap-6125	213	3	polyharmonic	polyharmonic	ADJ
ap-6125	213	4	spline	spline	NOUN
ap-6125	213	5	interpolation	interpolation	NOUN
ap-6125	213	6	in	in	ADP
ap-6125	213	7	the	the	DET
ap-6125	213	8	notation	notation	NOUN
ap-6125	213	9	introduced	introduce	VERB
ap-6125	213	10	above	above	ADV
ap-6125	213	11	,	,	PUNCT
ap-6125	213	12	we	we	PRON
ap-6125	213	13	continue	continue	VERB
ap-6125	213	14	in	in	ADP
ap-6125	213	15	deriving	derive	VERB
ap-6125	213	16	the	the	DET
ap-6125	213	17	polyharmonic	polyharmonic	ADJ
ap-6125	213	18	spline	spline	NOUN
ap-6125	213	19	interpolation	interpolation	NOUN
ap-6125	213	20	according	accord	VERB
ap-6125	213	21	to	to	ADP
ap-6125	213	22	[	[	X
ap-6125	213	23	2	2	NUM
ap-6125	213	24	]	]	PUNCT
ap-6125	213	25	.	.	PUNCT
ap-6125	214	1	put	put	VERB
ap-6125	214	2	k(α	k(α	PROPN
ap-6125	214	3	)	)	PUNCT
ap-6125	214	4	=	=	SYM
ap-6125	214	5	|α|	|α|	PROPN
ap-6125	214	6	!	!	PUNCT
ap-6125	214	7	α1	α1	PROPN
ap-6125	214	8	!	!	PUNCT
ap-6125	214	9	.	.	PUNCT
ap-6125	214	10	.	.	PUNCT
ap-6125	214	11	.	.	PUNCT
ap-6125	215	1	αn	αn	INTJ
ap-6125	215	2	!	!	PUNCT
ap-6125	216	1	(	(	PUNCT
ap-6125	216	2	24	24	NUM
ap-6125	216	3	)	)	PUNCT
ap-6125	216	4	for	for	ADP
ap-6125	216	5	a	a	DET
ap-6125	216	6	nonnegative	nonnegative	ADJ
ap-6125	216	7	multiindex	multiindex	NOUN
ap-6125	216	8	α	α	PROPN
ap-6125	216	9	.	.	PUNCT
ap-6125	216	10	recall	recall	VERB
ap-6125	216	11	that	that	SCONJ
ap-6125	216	12	n	n	X
ap-6125	216	13	is	be	AUX
ap-6125	216	14	the	the	DET
ap-6125	216	15	dimension	dimension	NOUN
ap-6125	216	16	of	of	ADP
ap-6125	216	17	the	the	DET
ap-6125	216	18	problem	problem	NOUN
ap-6125	216	19	,	,	PUNCT
ap-6125	216	20	fix	fix	VERB
ap-6125	216	21	l	l	NOUN
ap-6125	216	22	>	>	X
ap-6125	216	23	0	0	NUM
ap-6125	216	24	,	,	PUNCT
ap-6125	216	25	and	and	CCONJ
ap-6125	216	26	put	put	VERB
ap-6125	216	27	bα	bα	NOUN
ap-6125	216	28	=	=	NOUN
ap-6125	216	29	0	0	NUM
ap-6125	216	30	for	for	ADP
ap-6125	216	31	all	all	DET
ap-6125	216	32	α	α	NOUN
ap-6125	216	33	,	,	PUNCT
ap-6125	216	34	|α|	|α|	PROPN
ap-6125	216	35	6=	6=	PROPN
ap-6125	216	36	l	l	NOUN
ap-6125	216	37	,	,	PUNCT
ap-6125	216	38	and	and	CCONJ
ap-6125	216	39	bα	bα	PROPN
ap-6125	216	40	=	=	SYM
ap-6125	216	41	k(α	k(α	PROPN
ap-6125	216	42	)	)	PUNCT
ap-6125	216	43	for	for	ADP
ap-6125	216	44	|α|	|α|	PROPN
ap-6125	216	45	=	=	SYM
ap-6125	216	46	l.	l.	PROPN
ap-6125	216	47	then	then	ADV
ap-6125	216	48	‖gρ‖2	‖gρ‖2	ADJ
ap-6125	216	49	l	l	NOUN
ap-6125	216	50	=	=	SYM
ap-6125	216	51	(	(	PUNCT
ap-6125	216	52	2π)n	2π)n	NUM
ap-6125	216	53	(	(	PUNCT
ap-6125	216	54	n∑	n∑	NOUN
ap-6125	216	55	s=1	s=1	X
ap-6125	216	56	ρ2	ρ2	NOUN
ap-6125	216	57	s	s	PART
ap-6125	216	58	)	)	PUNCT
ap-6125	216	59	l	l	NOUN
ap-6125	216	60	according	accord	VERB
ap-6125	216	61	to	to	ADP
ap-6125	216	62	the	the	DET
ap-6125	216	63	multinomial	multinomial	NOUN
ap-6125	216	64	theorem	theorem	NOUN
ap-6125	216	65	.	.	PROPN
ap-6125	217	1	in	in	ADP
ap-6125	217	2	tables	table	NOUN
ap-6125	217	3	(	(	PUNCT
ap-6125	217	4	e.g.	e.g.	ADV
ap-6125	217	5	[	[	X
ap-6125	217	6	6	6	NUM
ap-6125	217	7	]	]	NUM
ap-6125	217	8	)	)	PUNCT
ap-6125	217	9	,	,	PUNCT
ap-6125	217	10	we	we	PRON
ap-6125	217	11	easily	easily	ADV
ap-6125	217	12	find	find	VERB
ap-6125	217	13	r(x	r(x	PROPN
ap-6125	217	14	,	,	PUNCT
ap-6125	217	15	y	y	NOUN
ap-6125	217	16	)	)	PUNCT
ap-6125	217	17	=	=	SYM
ap-6125	218	1	f	f	X
ap-6125	218	2			PROPN
ap-6125	218	3	(	(	PUNCT
ap-6125	218	4	n∑	n∑	NOUN
ap-6125	218	5	s=1	s=1	X
ap-6125	218	6	ρ2	ρ2	NOUN
ap-6125	218	7	s	s	NOUN
ap-6125	218	8	)	)	PUNCT
ap-6125	218	9	−l	−l	NOUN
ap-6125	218	10	=	=	SYM
ap-6125	218	11	{	{	PUNCT
ap-6125	218	12	c1r	c1r	NOUN
ap-6125	218	13	2l−n	2l−n	NUM
ap-6125	218	14	for	for	ADP
ap-6125	218	15	n	n	PRON
ap-6125	218	16	odd	odd	ADJ
ap-6125	218	17	,	,	PUNCT
ap-6125	218	18	c21r	c21r	PROPN
ap-6125	218	19	2l−n	2l−n	NUM
ap-6125	218	20	ln	ln	NOUN
ap-6125	218	21	r	r	NOUN
ap-6125	218	22	+	+	PROPN
ap-6125	218	23	c22r	c22r	X
ap-6125	218	24	2l−n	2l−n	ADV
ap-6125	218	25	for	for	ADP
ap-6125	218	26	n	n	PRON
ap-6125	218	27	even	even	ADV
ap-6125	218	28	,	,	PUNCT
ap-6125	218	29	(	(	PUNCT
ap-6125	218	30	25	25	NUM
ap-6125	218	31	)	)	PUNCT
ap-6125	218	32	where	where	SCONJ
ap-6125	218	33	r	r	NOUN
ap-6125	218	34	=	=	SYM
ap-6125	218	35	r(x	r(x	PROPN
ap-6125	218	36	,	,	PUNCT
ap-6125	218	37	y	y	NOUN
ap-6125	218	38	)	)	PUNCT
ap-6125	218	39	is	be	AUX
ap-6125	218	40	given	give	VERB
ap-6125	218	41	by	by	ADP
ap-6125	218	42	(	(	PUNCT
ap-6125	218	43	2	2	NUM
ap-6125	218	44	)	)	PUNCT
ap-6125	218	45	and	and	CCONJ
ap-6125	218	46	c1	c1	PROPN
ap-6125	218	47	,	,	PUNCT
ap-6125	218	48	c21	c21	NOUN
ap-6125	218	49	,	,	PUNCT
ap-6125	218	50	and	and	CCONJ
ap-6125	218	51	c22	c22	NOUN
ap-6125	218	52	are	be	AUX
ap-6125	218	53	quantities	quantity	NOUN
ap-6125	218	54	depending	depend	VERB
ap-6125	218	55	only	only	ADV
ap-6125	218	56	on	on	ADP
ap-6125	218	57	n	n	PRON
ap-6125	218	58	and	and	CCONJ
ap-6125	218	59	l.	l.	PROPN
ap-6125	218	60	then	then	ADV
ap-6125	218	61	the	the	DET
ap-6125	218	62	generating	generate	VERB
ap-6125	218	63	function	function	NOUN
ap-6125	218	64	r(x	r(x	PROPN
ap-6125	218	65	,	,	PUNCT
ap-6125	218	66	y	y	NOUN
ap-6125	218	67	)	)	PUNCT
ap-6125	218	68	,	,	PUNCT
ap-6125	218	69	for	for	ADP
ap-6125	218	70	2l−	2l−	NUM
ap-6125	218	71	n	n	CCONJ
ap-6125	218	72	>	>	X
ap-6125	218	73	0	0	NUM
ap-6125	218	74	,	,	PUNCT
ap-6125	218	75	is	be	AUX
ap-6125	218	76	a	a	DET
ap-6125	218	77	radial	radial	ADJ
ap-6125	218	78	basis	basis	NOUN
ap-6125	218	79	function	function	NOUN
ap-6125	218	80	.	.	PUNCT
ap-6125	219	1	note	note	VERB
ap-6125	219	2	that	that	SCONJ
ap-6125	219	3	the	the	DET
ap-6125	219	4	function	function	NOUN
ap-6125	219	5	(	(	PUNCT
ap-6125	219	6	25	25	NUM
ap-6125	219	7	)	)	PUNCT
ap-6125	219	8	has	have	VERB
ap-6125	219	9	the	the	DET
ap-6125	219	10	form	form	NOUN
ap-6125	219	11	of	of	ADP
ap-6125	219	12	the	the	DET
ap-6125	219	13	polyharmonic	polyharmonic	ADJ
ap-6125	219	14	function	function	NOUN
ap-6125	219	15	(	(	PUNCT
ap-6125	219	16	6	6	NUM
ap-6125	219	17	)	)	PUNCT
ap-6125	219	18	only	only	ADV
ap-6125	219	19	if	if	SCONJ
ap-6125	219	20	the	the	DET
ap-6125	219	21	dimension	dimension	NOUN
ap-6125	219	22	n	n	ADV
ap-6125	219	23	is	be	AUX
ap-6125	219	24	odd	odd	ADJ
ap-6125	219	25	and	and	CCONJ
ap-6125	219	26	it	it	PRON
ap-6125	219	27	is	be	AUX
ap-6125	219	28	the	the	DET
ap-6125	219	29	sum	sum	NOUN
ap-6125	219	30	of	of	ADP
ap-6125	219	31	the	the	DET
ap-6125	219	32	polyharmonic	polyharmonic	ADJ
ap-6125	219	33	function	function	NOUN
ap-6125	219	34	(	(	PUNCT
ap-6125	219	35	7	7	NUM
ap-6125	219	36	)	)	PUNCT
ap-6125	219	37	and	and	CCONJ
ap-6125	219	38	c22r	c22r	X
ap-6125	219	39	2l−n	2l−n	NUM
ap-6125	219	40	if	if	SCONJ
ap-6125	219	41	n	n	PRON
ap-6125	219	42	is	be	AUX
ap-6125	219	43	even	even	ADV
ap-6125	219	44	.	.	PUNCT
ap-6125	220	1	using	use	VERB
ap-6125	220	2	lemmas	lemmas	PROPN
ap-6125	220	3	2	2	NUM
ap-6125	220	4	and	and	CCONJ
ap-6125	220	5	3	3	NUM
ap-6125	220	6	of	of	ADP
ap-6125	220	7	[	[	X
ap-6125	220	8	2	2	NUM
ap-6125	220	9	]	]	PUNCT
ap-6125	220	10	,	,	PUNCT
ap-6125	220	11	we	we	PRON
ap-6125	220	12	remove	remove	VERB
ap-6125	220	13	the	the	DET
ap-6125	220	14	term	term	NOUN
ap-6125	220	15	c22r	c22r	X
ap-6125	220	16	2l−n	2l−n	ADV
ap-6125	220	17	from	from	ADP
ap-6125	220	18	the	the	DET
ap-6125	220	19	formula	formula	NOUN
ap-6125	220	20	(	(	PUNCT
ap-6125	220	21	25	25	NUM
ap-6125	220	22	)	)	PUNCT
ap-6125	220	23	for	for	ADP
ap-6125	220	24	the	the	DET
ap-6125	220	25	generating	generate	VERB
ap-6125	220	26	function	function	NOUN
ap-6125	220	27	in	in	ADP
ap-6125	220	28	the	the	DET
ap-6125	220	29	case	case	NOUN
ap-6125	220	30	of	of	ADP
ap-6125	220	31	n	n	ADV
ap-6125	220	32	being	be	AUX
ap-6125	220	33	even	even	ADV
ap-6125	220	34	.	.	PUNCT
ap-6125	221	1	we	we	PRON
ap-6125	221	2	obtain	obtain	VERB
ap-6125	221	3	r(x	r(x	PROPN
ap-6125	221	4	,	,	PUNCT
ap-6125	221	5	y	y	NOUN
ap-6125	221	6	)	)	PUNCT
ap-6125	221	7	=	=	SYM
ap-6125	221	8	r2l−n	r2l−n	PROPN
ap-6125	221	9	for	for	ADP
ap-6125	221	10	n	n	PRON
ap-6125	221	11	odd	odd	ADJ
ap-6125	221	12	,	,	PUNCT
ap-6125	221	13	(	(	PUNCT
ap-6125	221	14	26	26	NUM
ap-6125	221	15	)	)	PUNCT
ap-6125	221	16	=	=	VERB
ap-6125	221	17	r2l−n	r2l−n	PROPN
ap-6125	221	18	ln	ln	ADJ
ap-6125	221	19	r	r	NOUN
ap-6125	221	20	for	for	ADP
ap-6125	221	21	n	n	PRON
ap-6125	221	22	even	even	ADV
ap-6125	221	23	.	.	PUNCT
ap-6125	222	1	(	(	PUNCT
ap-6125	222	2	27	27	NUM
ap-6125	222	3	)	)	PUNCT
ap-6125	222	4	for	for	ADP
ap-6125	222	5	2l	2l	NUM
ap-6125	222	6	−	−	NOUN
ap-6125	222	7	n	n	CCONJ
ap-6125	222	8	>	>	X
ap-6125	222	9	0	0	PROPN
ap-6125	222	10	,	,	PUNCT
ap-6125	222	11	the	the	DET
ap-6125	222	12	generating	generate	VERB
ap-6125	222	13	function	function	NOUN
ap-6125	222	14	r(x	r(x	PROPN
ap-6125	222	15	,	,	PUNCT
ap-6125	222	16	y	y	NOUN
ap-6125	222	17	)	)	PUNCT
ap-6125	222	18	is	be	AUX
ap-6125	222	19	the	the	DET
ap-6125	222	20	polyharmonic	polyharmonic	ADJ
ap-6125	222	21	spline	spline	NOUN
ap-6125	222	22	(	(	PUNCT
ap-6125	222	23	6	6	NUM
ap-6125	222	24	)	)	PUNCT
ap-6125	222	25	or	or	CCONJ
ap-6125	222	26	(	(	PUNCT
ap-6125	222	27	7	7	NUM
ap-6125	222	28	)	)	PUNCT
ap-6125	222	29	,	,	PUNCT
ap-6125	222	30	i.e.	i.e.	X
ap-6125	222	31	,	,	PUNCT
ap-6125	222	32	a	a	DET
ap-6125	222	33	radial	radial	ADJ
ap-6125	222	34	basis	basis	NOUN
ap-6125	222	35	function	function	NOUN
ap-6125	222	36	.	.	PUNCT
ap-6125	223	1	8	8	X
ap-6125	223	2	.	.	X
ap-6125	224	1	some	some	DET
ap-6125	224	2	properties	property	NOUN
ap-6125	224	3	of	of	ADP
ap-6125	224	4	the	the	DET
ap-6125	224	5	polyharmonic	polyharmonic	ADJ
ap-6125	224	6	interpolant	interpolant	NOUN
ap-6125	224	7	consider	consider	VERB
ap-6125	224	8	now	now	ADV
ap-6125	224	9	the	the	DET
ap-6125	224	10	interpolant	interpolant	NOUN
ap-6125	224	11	z	z	NOUN
ap-6125	224	12	given	give	VERB
ap-6125	224	13	by	by	ADP
ap-6125	224	14	(	(	PUNCT
ap-6125	224	15	18	18	NUM
ap-6125	224	16	)	)	PUNCT
ap-6125	224	17	where	where	SCONJ
ap-6125	224	18	the	the	DET
ap-6125	224	19	generating	generate	VERB
ap-6125	224	20	function	function	NOUN
ap-6125	224	21	r(x	r(x	PROPN
ap-6125	224	22	,	,	PUNCT
ap-6125	224	23	y	y	NOUN
ap-6125	224	24	)	)	PUNCT
ap-6125	224	25	is	be	AUX
ap-6125	224	26	the	the	DET
ap-6125	224	27	polyharmonic	polyharmonic	ADJ
ap-6125	224	28	spline	spline	NOUN
ap-6125	224	29	for	for	ADP
ap-6125	224	30	n	n	PRON
ap-6125	224	31	odd	odd	ADJ
ap-6125	224	32	(	(	PUNCT
ap-6125	224	33	26	26	NUM
ap-6125	224	34	)	)	PUNCT
ap-6125	224	35	or	or	CCONJ
ap-6125	224	36	for	for	ADP
ap-6125	224	37	n	n	PRON
ap-6125	224	38	even	even	ADV
ap-6125	224	39	(	(	PUNCT
ap-6125	224	40	27	27	NUM
ap-6125	224	41	)	)	PUNCT
ap-6125	224	42	,	,	PUNCT
ap-6125	224	43	n	n	PRON
ap-6125	224	44	fixed	fix	VERB
ap-6125	224	45	.	.	PUNCT
ap-6125	225	1	its	its	PRON
ap-6125	225	2	properties	property	NOUN
ap-6125	225	3	are	be	AUX
ap-6125	225	4	characterized	characterize	VERB
ap-6125	225	5	by	by	ADP
ap-6125	225	6	the	the	DET
ap-6125	225	7	following	follow	VERB
ap-6125	225	8	theorem	theorem	NOUN
ap-6125	225	9	and	and	CCONJ
ap-6125	225	10	lemma	lemma	PROPN
ap-6125	225	11	.	.	PUNCT
ap-6125	226	1	theorem	theorem	VERB
ap-6125	226	2	4	4	NUM
ap-6125	226	3	.	.	PUNCT
ap-6125	227	1	choose	choose	VERB
ap-6125	227	2	l	l	NOUN
ap-6125	227	3	such	such	ADJ
ap-6125	227	4	that	that	SCONJ
ap-6125	227	5	2l−n	2l−n	PROPN
ap-6125	227	6	>	>	X
ap-6125	227	7	0	0	X
ap-6125	227	8	.	.	PUNCT
ap-6125	228	1	let	let	VERB
ap-6125	228	2	the	the	DET
ap-6125	228	3	interpolant	interpolant	NOUN
ap-6125	228	4	z(x	z(x	NOUN
ap-6125	228	5	)	)	PUNCT
ap-6125	228	6	be	be	AUX
ap-6125	228	7	given	give	VERB
ap-6125	228	8	by	by	ADP
ap-6125	228	9	(	(	PUNCT
ap-6125	228	10	18	18	NUM
ap-6125	228	11	)	)	PUNCT
ap-6125	228	12	.	.	PUNCT
ap-6125	229	1	then	then	ADV
ap-6125	229	2	it	it	PRON
ap-6125	229	3	solves	solve	VERB
ap-6125	229	4	the	the	DET
ap-6125	229	5	polyharmonic	polyharmonic	ADJ
ap-6125	229	6	equation	equation	NOUN
ap-6125	229	7	(	(	PUNCT
ap-6125	229	8	8)	8)	NUM
ap-6125	229	9	of	of	ADP
ap-6125	229	10	order	order	NOUN
ap-6125	229	11	m	m	NOUN
ap-6125	229	12	=	=	SYM
ap-6125	229	13	l	l	NOUN
ap-6125	229	14	in	in	ADP
ap-6125	229	15	the	the	DET
ap-6125	229	16	set	set	ADJ
ap-6125	229	17	rnx	rnx	NOUN
ap-6125	229	18	=	=	SYM
ap-6125	229	19	rn	rn	PROPN
ap-6125	229	20	\	\	PROPN
ap-6125	229	21	⋃n	⋃n	PROPN
ap-6125	229	22	j=1{x	j=1{x	PROPN
ap-6125	229	23	=	=	SYM
ap-6125	229	24	xj	xj	PROPN
ap-6125	229	25	}	}	PUNCT
ap-6125	229	26	.	.	PUNCT
ap-6125	230	1	proof	proof	NOUN
ap-6125	230	2	.	.	PUNCT
ap-6125	231	1	according	accord	VERB
ap-6125	231	2	to	to	ADP
ap-6125	231	3	theorem	theorem	NOUN
ap-6125	231	4	1	1	NUM
ap-6125	231	5	,	,	PUNCT
ap-6125	231	6	the	the	DET
ap-6125	231	7	generating	generate	VERB
ap-6125	231	8	function	function	NOUN
ap-6125	231	9	r(x	r(x	PROPN
ap-6125	231	10	,	,	PUNCT
ap-6125	231	11	xj	xj	PROPN
ap-6125	231	12	)	)	PUNCT
ap-6125	231	13	=	=	SYM
ap-6125	231	14	r2l−n(x	r2l−n(x	PROPN
ap-6125	231	15	,	,	PUNCT
ap-6125	231	16	xj	xj	NOUN
ap-6125	231	17	)	)	PUNCT
ap-6125	231	18	for	for	ADP
ap-6125	231	19	n	n	PRON
ap-6125	231	20	odd	odd	ADJ
ap-6125	231	21	or	or	CCONJ
ap-6125	231	22	r(x	r(x	PROPN
ap-6125	231	23	,	,	PUNCT
ap-6125	231	24	xj	xj	PROPN
ap-6125	231	25	)	)	PUNCT
ap-6125	231	26	=	=	SYM
ap-6125	231	27	r2l−n(x	r2l−n(x	PROPN
ap-6125	231	28	,	,	PUNCT
ap-6125	231	29	xj	xj	ADJ
ap-6125	231	30	)	)	PUNCT
ap-6125	231	31	ln	ln	PROPN
ap-6125	231	32	r(x	r(x	PROPN
ap-6125	231	33	,	,	PUNCT
ap-6125	231	34	xj	xj	PROPN
ap-6125	231	35	)	)	PUNCT
ap-6125	231	36	for	for	ADP
ap-6125	231	37	n	n	PRON
ap-6125	231	38	even	even	ADV
ap-6125	231	39	,	,	PUNCT
ap-6125	231	40	it	it	PRON
ap-6125	231	41	is	be	AUX
ap-6125	231	42	the	the	DET
ap-6125	231	43	solution	solution	NOUN
ap-6125	231	44	of	of	ADP
ap-6125	231	45	the	the	DET
ap-6125	231	46	polyharmonic	polyharmonic	ADJ
ap-6125	231	47	equation	equation	NOUN
ap-6125	231	48	of	of	ADP
ap-6125	231	49	order	order	NOUN
ap-6125	231	50	m	m	VERB
ap-6125	231	51	=	=	NOUN
ap-6125	231	52	1	1	NUM
ap-6125	231	53	2	2	NUM
ap-6125	231	54	(	(	PUNCT
ap-6125	231	55	2l−	2l−	NUM
ap-6125	231	56	n+	n+	NUM
ap-6125	231	57	n	n	CCONJ
ap-6125	231	58	)	)	PUNCT
ap-6125	231	59	=	=	SYM
ap-6125	231	60	l	l	NOUN
ap-6125	231	61	in	in	ADP
ap-6125	231	62	rnxj	rnxj	NOUN
ap-6125	231	63	,	,	PUNCT
ap-6125	231	64	j	j	PROPN
ap-6125	231	65	=	=	SYM
ap-6125	231	66	1	1	NUM
ap-6125	231	67	,	,	PUNCT
ap-6125	231	68	.	.	PUNCT
ap-6125	231	69	.	.	PUNCT
ap-6125	232	1	.	.	PUNCT
ap-6125	233	1	,	,	PUNCT
ap-6125	233	2	n	n	X
ap-6125	233	3	.	.	PUNCT
ap-6125	234	1	moreover	moreover	ADV
ap-6125	234	2	,	,	PUNCT
ap-6125	234	3	the	the	DET
ap-6125	234	4	trend	trend	NOUN
ap-6125	234	5	functions	function	NOUN
ap-6125	234	6	ϕα(x	ϕα(x	NOUN
ap-6125	234	7	)	)	PUNCT
ap-6125	234	8	given	give	VERB
ap-6125	234	9	by	by	ADP
ap-6125	234	10	(	(	PUNCT
ap-6125	234	11	5	5	NUM
ap-6125	234	12	)	)	PUNCT
ap-6125	234	13	are	be	AUX
ap-6125	234	14	monomials	monomial	NOUN
ap-6125	234	15	of	of	ADP
ap-6125	234	16	a	a	DET
ap-6125	234	17	degree	degree	NOUN
ap-6125	234	18	at	at	ADP
ap-6125	234	19	most	most	ADJ
ap-6125	234	20	l−	l−	NOUN
ap-6125	234	21	1	1	NUM
ap-6125	234	22	.	.	PUNCT
ap-6125	235	1	they	they	PRON
ap-6125	235	2	satisfy	satisfy	VERB
ap-6125	235	3	the	the	DET
ap-6125	235	4	polyharmonic	polyharmonic	ADJ
ap-6125	235	5	equation	equation	NOUN
ap-6125	235	6	of	of	ADP
ap-6125	235	7	order	order	NOUN
ap-6125	235	8	m	m	VERB
ap-6125	235	9	=	=	ADJ
ap-6125	235	10	l	l	PROPN
ap-6125	235	11	in	in	ADP
ap-6125	235	12	rn	rn	PROPN
ap-6125	235	13	as	as	SCONJ
ap-6125	235	14	the	the	DET
ap-6125	235	15	operator	operator	NOUN
ap-6125	235	16	∆l	∆l	PROPN
ap-6125	235	17	is	be	AUX
ap-6125	235	18	a	a	DET
ap-6125	235	19	linear	linear	ADJ
ap-6125	235	20	combination	combination	NOUN
ap-6125	235	21	of	of	ADP
ap-6125	235	22	the	the	DET
ap-6125	235	23	derivatives	derivative	NOUN
ap-6125	235	24	of	of	ADP
ap-6125	235	25	order	order	NOUN
ap-6125	235	26	2l	2l	NOUN
ap-6125	235	27	and	and	CCONJ
ap-6125	235	28	,	,	PUNCT
ap-6125	235	29	according	accord	VERB
ap-6125	235	30	to	to	ADP
ap-6125	235	31	the	the	DET
ap-6125	235	32	multinomial	multinomial	NOUN
ap-6125	235	33	theorem	theorem	NOUN
ap-6125	235	34	,	,	PUNCT
ap-6125	235	35	each	each	PRON
ap-6125	235	36	of	of	ADP
ap-6125	235	37	these	these	DET
ap-6125	235	38	derivatives	derivative	NOUN
ap-6125	235	39	includes	include	VERB
ap-6125	235	40	a	a	DET
ap-6125	235	41	derivative	derivative	NOUN
ap-6125	235	42	of	of	ADP
ap-6125	235	43	order	order	NOUN
ap-6125	235	44	l	l	NOUN
ap-6125	235	45	with	with	ADP
ap-6125	235	46	respect	respect	NOUN
ap-6125	235	47	to	to	ADP
ap-6125	235	48	some	some	DET
ap-6125	235	49	particular	particular	ADJ
ap-6125	235	50	variable	variable	ADJ
ap-6125	235	51	xs	xs	PROPN
ap-6125	235	52	.	.	PUNCT
ap-6125	236	1	the	the	DET
ap-6125	236	2	coefficients	coefficient	NOUN
ap-6125	236	3	λj	λj	VERB
ap-6125	236	4	and	and	CCONJ
ap-6125	236	5	aα	aα	NOUN
ap-6125	236	6	are	be	AUX
ap-6125	236	7	complex	complex	ADJ
ap-6125	236	8	constants	constant	NOUN
ap-6125	236	9	.	.	PUNCT
ap-6125	237	1	therefore	therefore	ADV
ap-6125	237	2	,	,	PUNCT
ap-6125	237	3	the	the	DET
ap-6125	237	4	interpolant	interpolant	NOUN
ap-6125	237	5	(	(	PUNCT
ap-6125	237	6	18	18	NUM
ap-6125	237	7	)	)	PUNCT
ap-6125	237	8	satisfies	satisfy	VERB
ap-6125	237	9	the	the	DET
ap-6125	237	10	polyharmonic	polyharmonic	ADJ
ap-6125	237	11	equation	equation	NOUN
ap-6125	237	12	(	(	PUNCT
ap-6125	237	13	8)	8)	NUM
ap-6125	237	14	in	in	ADP
ap-6125	237	15	rnx	rnx	NOUN
ap-6125	237	16	.	.	PUNCT
ap-6125	238	1	152	152	NUM
ap-6125	238	2	vol	vol	NOUN
ap-6125	238	3	.	.	PUNCT
ap-6125	239	1	61	61	NUM
ap-6125	239	2	special	special	ADJ
ap-6125	239	3	issue/2021	issue/2021	NOUN
ap-6125	239	4	multivariate	multivariate	NOUN
ap-6125	239	5	interpolation	interpolation	NOUN
ap-6125	239	6	using	use	VERB
ap-6125	239	7	polyharmonic	polyharmonic	ADJ
ap-6125	239	8	splines	spline	NOUN
ap-6125	239	9	−1	−1	ADV
ap-6125	239	10	−0.8	−0.8	PROPN
ap-6125	239	11	−0.6	−0.6	PROPN
ap-6125	239	12	−0.4	−0.4	PUNCT
ap-6125	240	1	−0.2	−0.2	PROPN
ap-6125	240	2	0	0	NUM
ap-6125	240	3	0.2	0.2	NUM
ap-6125	240	4	0.4	0.4	NUM
ap-6125	240	5	0.6	0.6	NUM
ap-6125	240	6	0.8	0.8	NUM
ap-6125	240	7	1	1	NUM
ap-6125	240	8	−10	−10	NOUN
ap-6125	240	9	−5	−5	ADV
ap-6125	240	10	0	0	NUM
ap-6125	240	11	5	5	NUM
ap-6125	240	12	10	10	NUM
ap-6125	240	13	15	15	NUM
ap-6125	240	14	true	true	ADJ
ap-6125	240	15	degree=1	degree=1	PUNCT
ap-6125	240	16	degree=3	degree=3	PROPN
ap-6125	241	1	degree=5	degree=5	NOUN
ap-6125	241	2	figure	figure	NOUN
ap-6125	241	3	1	1	NUM
ap-6125	241	4	.	.	PUNCT
ap-6125	242	1	n	n	NOUN
ap-6125	242	2	=	=	SYM
ap-6125	242	3	5	5	X
ap-6125	242	4	.	.	PUNCT
ap-6125	243	1	the	the	DET
ap-6125	243	2	horizontal	horizontal	ADJ
ap-6125	243	3	axis	axis	NOUN
ap-6125	243	4	:	:	PUNCT
ap-6125	243	5	independent	independent	ADJ
ap-6125	243	6	variable	variable	NOUN
ap-6125	243	7	,	,	PUNCT
ap-6125	243	8	the	the	DET
ap-6125	243	9	vertical	vertical	ADJ
ap-6125	243	10	axis	axis	NOUN
ap-6125	243	11	:	:	PUNCT
ap-6125	243	12	the	the	DET
ap-6125	243	13	true	true	ADJ
ap-6125	243	14	function	function	NOUN
ap-6125	243	15	(	(	PUNCT
ap-6125	243	16	29	29	NUM
ap-6125	243	17	)	)	PUNCT
ap-6125	243	18	(	(	PUNCT
ap-6125	243	19	solid	solid	ADJ
ap-6125	243	20	line	line	NOUN
ap-6125	243	21	)	)	PUNCT
ap-6125	243	22	;	;	PUNCT
ap-6125	243	23	the	the	DET
ap-6125	243	24	interpolant	interpolant	NOUN
ap-6125	243	25	with	with	ADP
ap-6125	243	26	b1	b1	NOUN
ap-6125	243	27	=	=	SYM
ap-6125	243	28	1	1	NUM
ap-6125	243	29	(	(	PUNCT
ap-6125	243	30	dashed	dash	VERB
ap-6125	243	31	line	line	NOUN
ap-6125	243	32	,	,	PUNCT
ap-6125	243	33	piecewise	piecewise	NOUN
ap-6125	243	34	linear	linear	NOUN
ap-6125	243	35	)	)	PUNCT
ap-6125	243	36	,	,	PUNCT
ap-6125	243	37	b2	b2	NOUN
ap-6125	243	38	=	=	SYM
ap-6125	243	39	1	1	NUM
ap-6125	243	40	(	(	PUNCT
ap-6125	243	41	dotted	dotted	ADJ
ap-6125	243	42	line	line	NOUN
ap-6125	243	43	,	,	PUNCT
ap-6125	243	44	cubic	cubic	ADJ
ap-6125	243	45	spline	spline	NOUN
ap-6125	243	46	)	)	PUNCT
ap-6125	243	47	,	,	PUNCT
ap-6125	243	48	and	and	CCONJ
ap-6125	243	49	b3	b3	PROPN
ap-6125	243	50	=	=	SYM
ap-6125	243	51	1	1	NUM
ap-6125	243	52	(	(	PUNCT
ap-6125	243	53	dash	dash	NOUN
ap-6125	243	54	-	-	PUNCT
ap-6125	243	55	dot	dot	NOUN
ap-6125	243	56	line	line	NOUN
ap-6125	243	57	,	,	PUNCT
ap-6125	243	58	quintic	quintic	ADJ
ap-6125	243	59	spline	spline	NOUN
ap-6125	243	60	)	)	PUNCT
ap-6125	243	61	.	.	PUNCT
ap-6125	244	1	the	the	DET
ap-6125	244	2	scales	scale	NOUN
ap-6125	244	3	on	on	ADP
ap-6125	244	4	the	the	DET
ap-6125	244	5	xand	xand	PROPN
ap-6125	244	6	y	y	NOUN
ap-6125	244	7	-	-	PUNCT
ap-6125	244	8	axis	axis	NOUN
ap-6125	244	9	are	be	AUX
ap-6125	244	10	different	different	ADJ
ap-6125	244	11	.	.	PUNCT
ap-6125	245	1	we	we	PRON
ap-6125	245	2	have	have	AUX
ap-6125	245	3	just	just	ADV
ap-6125	245	4	proven	prove	VERB
ap-6125	245	5	that	that	SCONJ
ap-6125	245	6	the	the	DET
ap-6125	245	7	interpolant	interpolant	NOUN
ap-6125	245	8	(	(	PUNCT
ap-6125	245	9	18	18	NUM
ap-6125	245	10	)	)	PUNCT
ap-6125	245	11	with	with	ADP
ap-6125	245	12	the	the	DET
ap-6125	245	13	generating	generate	VERB
ap-6125	245	14	function	function	NOUN
ap-6125	245	15	(	(	PUNCT
ap-6125	245	16	26	26	NUM
ap-6125	245	17	)	)	PUNCT
ap-6125	245	18	or	or	CCONJ
ap-6125	245	19	(	(	PUNCT
ap-6125	245	20	27	27	NUM
ap-6125	245	21	)	)	PUNCT
ap-6125	245	22	is	be	AUX
ap-6125	245	23	polyharmonic	polyharmonic	ADJ
ap-6125	245	24	in	in	ADP
ap-6125	245	25	rnx	rnx	NOUN
ap-6125	245	26	.	.	PUNCT
ap-6125	246	1	moreover	moreover	ADV
ap-6125	246	2	,	,	PUNCT
ap-6125	246	3	in	in	ADP
ap-6125	246	4	the	the	DET
ap-6125	246	5	example	example	NOUN
ap-6125	246	6	we	we	PRON
ap-6125	246	7	will	will	AUX
ap-6125	246	8	use	use	VERB
ap-6125	246	9	its	its	PRON
ap-6125	246	10	another	another	DET
ap-6125	246	11	trivial	trivial	ADJ
ap-6125	246	12	property	property	NOUN
ap-6125	246	13	stated	state	VERB
ap-6125	246	14	in	in	ADP
ap-6125	246	15	the	the	DET
ap-6125	246	16	following	follow	VERB
ap-6125	246	17	lemma	lemma	PROPN
ap-6125	246	18	.	.	PUNCT
ap-6125	247	1	lemma	lemma	PROPN
ap-6125	247	2	1	1	X
ap-6125	247	3	.	.	PUNCT
ap-6125	248	1	let	let	VERB
ap-6125	248	2	the	the	DET
ap-6125	248	3	function	function	NOUN
ap-6125	248	4	u	u	NOUN
ap-6125	248	5	of	of	ADP
ap-6125	248	6	the	the	DET
ap-6125	248	7	variable	variable	NOUN
ap-6125	248	8	x	x	PUNCT
ap-6125	248	9	=	=	SYM
ap-6125	248	10	(	(	PUNCT
ap-6125	248	11	x1	x1	PROPN
ap-6125	248	12	,	,	PUNCT
ap-6125	248	13	x2	x2	PROPN
ap-6125	248	14	,	,	PUNCT
ap-6125	248	15	.	.	PUNCT
ap-6125	248	16	.	.	PUNCT
ap-6125	249	1	.	.	PUNCT
ap-6125	250	1	,	,	PUNCT
ap-6125	250	2	xn	xn	X
ap-6125	250	3	)	)	PUNCT
ap-6125	250	4	satisfy	satisfy	VERB
ap-6125	250	5	the	the	DET
ap-6125	250	6	polyharmonic	polyharmonic	ADJ
ap-6125	250	7	equation	equation	NOUN
ap-6125	250	8	of	of	ADP
ap-6125	250	9	order	order	NOUN
ap-6125	250	10	m	m	VERB
ap-6125	250	11	in	in	ADP
ap-6125	250	12	the	the	DET
ap-6125	250	13	set	set	NOUN
ap-6125	250	14	ψ	ψ	PROPN
ap-6125	250	15	⊂	⊂	PROPN
ap-6125	250	16	rn	rn	PROPN
ap-6125	250	17	.	.	PUNCT
ap-6125	251	1	then	then	ADV
ap-6125	251	2	it	it	PRON
ap-6125	251	3	satisfies	satisfy	VERB
ap-6125	251	4	the	the	DET
ap-6125	251	5	polyharmonic	polyharmonic	ADJ
ap-6125	251	6	equation	equation	NOUN
ap-6125	251	7	of	of	ADP
ap-6125	251	8	order	order	NOUN
ap-6125	251	9	m+	m+	NUM
ap-6125	251	10	l	l	NOUN
ap-6125	251	11	for	for	ADP
ap-6125	251	12	any	any	DET
ap-6125	251	13	positive	positive	ADJ
ap-6125	251	14	l	l	NOUN
ap-6125	251	15	in	in	ADP
ap-6125	251	16	the	the	DET
ap-6125	251	17	same	same	ADJ
ap-6125	251	18	set	set	NOUN
ap-6125	251	19	.	.	PUNCT
ap-6125	252	1	9	9	X
ap-6125	252	2	.	.	NOUN
ap-6125	252	3	example	example	NOUN
ap-6125	252	4	in	in	ADP
ap-6125	252	5	fig	fig	NOUN
ap-6125	252	6	.	.	PUNCT
ap-6125	253	1	1	1	NUM
ap-6125	253	2	,	,	PUNCT
ap-6125	253	3	we	we	PRON
ap-6125	253	4	show	show	VERB
ap-6125	253	5	results	result	NOUN
ap-6125	253	6	of	of	ADP
ap-6125	253	7	a	a	DET
ap-6125	253	8	simple	simple	ADJ
ap-6125	253	9	computation	computation	NOUN
ap-6125	253	10	:	:	PUNCT
ap-6125	253	11	the	the	DET
ap-6125	253	12	polyharmonic	polyharmonic	ADJ
ap-6125	253	13	spline	spline	NOUN
ap-6125	253	14	interpolation	interpolation	NOUN
ap-6125	253	15	for	for	ADP
ap-6125	253	16	n	n	NOUN
ap-6125	253	17	=	=	SYM
ap-6125	253	18	1	1	NUM
ap-6125	253	19	,	,	PUNCT
ap-6125	253	20	i.e.	i.e.	X
ap-6125	253	21	the	the	DET
ap-6125	253	22	modification	modification	NOUN
ap-6125	253	23	z(x	z(x	NOUN
ap-6125	253	24	)	)	PUNCT
ap-6125	253	25	=	=	SYM
ap-6125	254	1	n∑	n∑	NOUN
ap-6125	254	2	j=1	j=1	PROPN
ap-6125	254	3	λjr(x	λjr(x	PROPN
ap-6125	254	4	,	,	PUNCT
ap-6125	254	5	xj	xj	PROPN
ap-6125	254	6	)	)	PUNCT
ap-6125	254	7	+	+	CCONJ
ap-6125	254	8	l−1∑	l−1∑	ADJ
ap-6125	254	9	k=0	k=0	PROPN
ap-6125	254	10	akϕk(x	akϕk(x	PROPN
ap-6125	254	11	)	)	PUNCT
ap-6125	254	12	(	(	PUNCT
ap-6125	254	13	28	28	NUM
ap-6125	254	14	)	)	PUNCT
ap-6125	254	15	of	of	ADP
ap-6125	254	16	the	the	DET
ap-6125	254	17	formula	formula	NOUN
ap-6125	254	18	(	(	PUNCT
ap-6125	254	19	18	18	NUM
ap-6125	254	20	)	)	PUNCT
ap-6125	254	21	with	with	ADP
ap-6125	254	22	r(x	r(x	PROPN
ap-6125	254	23	,	,	PUNCT
ap-6125	254	24	y	y	NOUN
ap-6125	254	25	)	)	PUNCT
ap-6125	254	26	given	give	VERB
ap-6125	254	27	by	by	ADP
ap-6125	254	28	(	(	PUNCT
ap-6125	254	29	26	26	NUM
ap-6125	254	30	)	)	PUNCT
ap-6125	254	31	.	.	PUNCT
ap-6125	255	1	note	note	VERB
ap-6125	255	2	that	that	SCONJ
ap-6125	255	3	α	α	PRON
ap-6125	255	4	=	=	X
ap-6125	255	5	k	k	PROPN
ap-6125	255	6	is	be	AUX
ap-6125	255	7	now	now	ADV
ap-6125	255	8	a	a	DET
ap-6125	255	9	simple	simple	ADJ
ap-6125	255	10	index	index	NOUN
ap-6125	255	11	.	.	PUNCT
ap-6125	256	1	we	we	PRON
ap-6125	256	2	consider	consider	VERB
ap-6125	256	3	three	three	NUM
ap-6125	256	4	cases	case	NOUN
ap-6125	256	5	:	:	PUNCT
ap-6125	256	6	the	the	DET
ap-6125	256	7	minimization	minimization	NOUN
ap-6125	256	8	of	of	ADP
ap-6125	256	9	the	the	DET
ap-6125	256	10	l2	l2	NOUN
ap-6125	256	11	norm	norm	NOUN
ap-6125	256	12	of	of	ADP
ap-6125	256	13	the	the	DET
ap-6125	256	14	1st	1st	NOUN
ap-6125	256	15	,	,	PUNCT
ap-6125	256	16	or	or	CCONJ
ap-6125	256	17	2nd	2nd	NOUN
ap-6125	256	18	,	,	PUNCT
ap-6125	256	19	or	or	CCONJ
ap-6125	256	20	3rd	3rd	ADJ
ap-6125	256	21	derivative	derivative	NOUN
ap-6125	256	22	of	of	ADP
ap-6125	256	23	the	the	DET
ap-6125	256	24	interpolant	interpolant	NOUN
ap-6125	256	25	.	.	PUNCT
ap-6125	257	1	we	we	PRON
ap-6125	257	2	interpolate	interpolate	VERB
ap-6125	257	3	the	the	DET
ap-6125	257	4	third	third	ADJ
ap-6125	257	5	degree	degree	NOUN
ap-6125	257	6	polynomial	polynomial	ADJ
ap-6125	257	7	f(x	f(x	PROPN
ap-6125	257	8	)	)	PUNCT
ap-6125	258	1	=	=	PUNCT
ap-6125	258	2	8x3	8x3	NOUN
ap-6125	258	3	+	+	CCONJ
ap-6125	258	4	6x2	6x2	NUM
ap-6125	258	5	+	+	CCONJ
ap-6125	258	6	2x−	2x−	NUM
ap-6125	258	7	1	1	NUM
ap-6125	258	8	(	(	PUNCT
ap-6125	258	9	29	29	NUM
ap-6125	258	10	)	)	PUNCT
ap-6125	258	11	on	on	ADP
ap-6125	258	12	ω	ω	NUM
ap-6125	258	13	=	=	PUNCT
ap-6125	259	1	[	[	X
ap-6125	259	2	−1	−1	NOUN
ap-6125	259	3	,	,	PUNCT
ap-6125	259	4	1	1	NUM
ap-6125	259	5	]	]	PUNCT
ap-6125	259	6	(	(	PUNCT
ap-6125	259	7	solid	solid	ADJ
ap-6125	259	8	line	line	NOUN
ap-6125	259	9	in	in	ADP
ap-6125	259	10	fig	fig	NOUN
ap-6125	259	11	.	.	PUNCT
ap-6125	260	1	1	1	NUM
ap-6125	260	2	)	)	PUNCT
ap-6125	260	3	with	with	ADP
ap-6125	260	4	n	n	NOUN
ap-6125	260	5	=	=	SYM
ap-6125	260	6	3	3	NUM
ap-6125	260	7	,	,	PUNCT
ap-6125	260	8	i.e.	i.e.	X
ap-6125	260	9	using	use	VERB
ap-6125	260	10	the	the	DET
ap-6125	260	11	nodes	node	NOUN
ap-6125	260	12	x1	x1	PROPN
ap-6125	260	13	=	=	SYM
ap-6125	260	14	−1	−1	NOUN
ap-6125	260	15	,	,	PUNCT
ap-6125	261	1	x2	x2	PROPN
ap-6125	261	2	=	=	SYM
ap-6125	261	3	0	0	NUM
ap-6125	261	4	,	,	PUNCT
ap-6125	261	5	and	and	CCONJ
ap-6125	261	6	x3	x3	ADJ
ap-6125	261	7	=	=	SYM
ap-6125	262	1	1	1	X
ap-6125	262	2	.	.	PUNCT
ap-6125	263	1	we	we	PRON
ap-6125	263	2	employ	employ	VERB
ap-6125	263	3	the	the	DET
ap-6125	263	4	formula	formula	NOUN
ap-6125	263	5	(	(	PUNCT
ap-6125	263	6	24	24	NUM
ap-6125	263	7	)	)	PUNCT
ap-6125	263	8	for	for	ADP
ap-6125	263	9	k(α	k(α	NOUN
ap-6125	263	10	)	)	PUNCT
ap-6125	263	11	in	in	ADP
ap-6125	263	12	case	case	NOUN
ap-6125	263	13	n	n	NOUN
ap-6125	263	14	=	=	SYM
ap-6125	263	15	1	1	NUM
ap-6125	263	16	,	,	PUNCT
ap-6125	263	17	i.e.	i.e.	X
ap-6125	263	18	,	,	PUNCT
ap-6125	263	19	k(l	k(l	PROPN
ap-6125	263	20	)	)	PUNCT
ap-6125	263	21	=	=	SYM
ap-6125	263	22	1	1	NUM
ap-6125	263	23	,	,	PUNCT
ap-6125	263	24	and	and	CCONJ
ap-6125	263	25	put	put	VERB
ap-6125	263	26	bk	bk	ADP
ap-6125	263	27	=	=	SYM
ap-6125	263	28	0	0	NUM
ap-6125	263	29	for	for	ADP
ap-6125	263	30	all	all	DET
ap-6125	263	31	k	k	PROPN
ap-6125	263	32	,	,	PUNCT
ap-6125	263	33	k	k	PROPN
ap-6125	263	34	6=	6=	PROPN
ap-6125	263	35	l	l	PROPN
ap-6125	263	36	,	,	PUNCT
ap-6125	263	37	and	and	CCONJ
ap-6125	263	38	bl	bl	X
ap-6125	263	39	=	=	SYM
ap-6125	263	40	1	1	X
ap-6125	263	41	.	.	PUNCT
ap-6125	264	1	if	if	SCONJ
ap-6125	264	2	we	we	PRON
ap-6125	264	3	put	put	VERB
ap-6125	264	4	l	l	NOUN
ap-6125	264	5	=	=	SYM
ap-6125	264	6	1	1	NUM
ap-6125	264	7	,	,	PUNCT
ap-6125	264	8	b1	b1	NOUN
ap-6125	264	9	=	=	SYM
ap-6125	264	10	1	1	NUM
ap-6125	264	11	,	,	PUNCT
ap-6125	264	12	and	and	CCONJ
ap-6125	264	13	bk	bk	VERB
ap-6125	264	14	=	=	SYM
ap-6125	264	15	0	0	NUM
ap-6125	264	16	otherwise	otherwise	ADV
ap-6125	264	17	to	to	PART
ap-6125	264	18	minimize	minimize	VERB
ap-6125	264	19	the	the	DET
ap-6125	264	20	l2	l2	NOUN
ap-6125	264	21	norm	norm	NOUN
ap-6125	264	22	of	of	ADP
ap-6125	264	23	the	the	DET
ap-6125	264	24	1st	1st	ADJ
ap-6125	264	25	derivative	derivative	NOUN
ap-6125	264	26	of	of	ADP
ap-6125	264	27	the	the	DET
ap-6125	264	28	interpolant	interpolant	NOUN
ap-6125	264	29	according	accord	VERB
ap-6125	264	30	to	to	ADP
ap-6125	264	31	(	(	PUNCT
ap-6125	264	32	16	16	NUM
ap-6125	264	33	)	)	PUNCT
ap-6125	264	34	then	then	ADV
ap-6125	264	35	the	the	DET
ap-6125	264	36	generating	generate	VERB
ap-6125	264	37	function	function	NOUN
ap-6125	264	38	r(x	r(x	PROPN
ap-6125	264	39	,	,	PUNCT
ap-6125	264	40	y	y	NOUN
ap-6125	264	41	)	)	PUNCT
ap-6125	265	1	=	=	SYM
ap-6125	265	2	r2l−n	r2l−n	NOUN
ap-6125	265	3	=	=	SYM
ap-6125	265	4	r	r	NOUN
ap-6125	265	5	(	(	PUNCT
ap-6125	265	6	a	a	DET
ap-6125	265	7	piecewise	piecewise	NOUN
ap-6125	265	8	linear	linear	NOUN
ap-6125	265	9	function	function	NOUN
ap-6125	265	10	given	give	VERB
ap-6125	265	11	by	by	ADP
ap-6125	265	12	(	(	PUNCT
ap-6125	265	13	2	2	NUM
ap-6125	265	14	)	)	PUNCT
ap-6125	265	15	)	)	PUNCT
ap-6125	265	16	and	and	CCONJ
ap-6125	265	17	the	the	DET
ap-6125	265	18	trend	trend	NOUN
ap-6125	265	19	function	function	NOUN
ap-6125	265	20	is	be	AUX
ap-6125	265	21	a	a	DET
ap-6125	265	22	polynomial	polynomial	ADJ
ap-6125	265	23	of	of	ADP
ap-6125	265	24	degree	degree	NOUN
ap-6125	265	25	l−	l−	NOUN
ap-6125	265	26	1	1	NUM
ap-6125	265	27	=	=	SYM
ap-6125	265	28	0	0	NUM
ap-6125	265	29	,	,	PUNCT
ap-6125	265	30	i.e.	i.e.	X
ap-6125	265	31	a	a	DET
ap-6125	265	32	constant	constant	ADJ
ap-6125	265	33	.	.	PUNCT
ap-6125	266	1	the	the	DET
ap-6125	266	2	interpolant	interpolant	NOUN
ap-6125	266	3	(	(	PUNCT
ap-6125	266	4	28	28	NUM
ap-6125	266	5	)	)	PUNCT
ap-6125	266	6	solves	solve	VERB
ap-6125	266	7	the	the	DET
ap-6125	266	8	equation	equation	NOUN
ap-6125	266	9	(	(	PUNCT
ap-6125	266	10	8)	8)	NUM
ap-6125	266	11	,	,	PUNCT
ap-6125	266	12	which	which	PRON
ap-6125	266	13	is	be	AUX
ap-6125	266	14	now	now	ADV
ap-6125	266	15	harmonic	harmonic	ADJ
ap-6125	266	16	(	(	PUNCT
ap-6125	266	17	m	m	NOUN
ap-6125	266	18	=	=	PUNCT
ap-6125	266	19	l	l	NOUN
ap-6125	266	20	=	=	SYM
ap-6125	266	21	1	1	NUM
ap-6125	266	22	)	)	PUNCT
ap-6125	266	23	,	,	PUNCT
ap-6125	266	24	according	accord	VERB
ap-6125	266	25	to	to	ADP
ap-6125	266	26	theorem	theorem	ADJ
ap-6125	266	27	4	4	NUM
ap-6125	266	28	,	,	PUNCT
ap-6125	266	29	everywhere	everywhere	ADV
ap-6125	266	30	in	in	ADP
ap-6125	266	31	r1	r1	PROPN
ap-6125	266	32	except	except	SCONJ
ap-6125	266	33	for	for	ADP
ap-6125	266	34	the	the	DET
ap-6125	266	35	points	point	NOUN
ap-6125	266	36	x	x	X
ap-6125	266	37	=	=	SYM
ap-6125	266	38	xj	xj	PROPN
ap-6125	266	39	,	,	PUNCT
ap-6125	266	40	j	j	PROPN
ap-6125	266	41	=	=	SYM
ap-6125	266	42	1	1	NUM
ap-6125	266	43	,	,	PUNCT
ap-6125	266	44	2	2	NUM
ap-6125	266	45	,	,	PUNCT
ap-6125	266	46	3	3	NUM
ap-6125	266	47	(	(	PUNCT
ap-6125	266	48	cf	cf	NOUN
ap-6125	266	49	.	.	PUNCT
ap-6125	267	1	sec	sec	PROPN
ap-6125	267	2	.	.	PROPN
ap-6125	268	1	8)	8)	NUM
ap-6125	268	2	.	.	PUNCT
ap-6125	269	1	the	the	DET
ap-6125	269	2	constant	constant	ADJ
ap-6125	269	3	trend	trend	NOUN
ap-6125	269	4	function	function	NOUN
ap-6125	269	5	satisfies	satisfy	VERB
ap-6125	269	6	the	the	DET
ap-6125	269	7	equation	equation	NOUN
ap-6125	269	8	(	(	PUNCT
ap-6125	269	9	8)	8)	NUM
ap-6125	269	10	everywhere	everywhere	ADV
ap-6125	269	11	in	in	ADP
ap-6125	269	12	r1	r1	PROPN
ap-6125	269	13	.	.	PUNCT
ap-6125	270	1	from	from	ADP
ap-6125	270	2	the	the	DET
ap-6125	270	3	form	form	NOUN
ap-6125	270	4	of	of	ADP
ap-6125	270	5	the	the	DET
ap-6125	270	6	interpolation	interpolation	NOUN
ap-6125	270	7	formula	formula	NOUN
ap-6125	270	8	(	(	PUNCT
ap-6125	270	9	28	28	NUM
ap-6125	270	10	)	)	PUNCT
ap-6125	270	11	we	we	PRON
ap-6125	270	12	see	see	VERB
ap-6125	270	13	that	that	SCONJ
ap-6125	270	14	the	the	DET
ap-6125	270	15	interpolant	interpolant	NOUN
ap-6125	270	16	z(x	z(x	NOUN
ap-6125	270	17	)	)	PUNCT
ap-6125	270	18	(	(	PUNCT
ap-6125	270	19	dashed	dash	VERB
ap-6125	270	20	line	line	NOUN
ap-6125	270	21	in	in	ADP
ap-6125	270	22	fig	fig	NOUN
ap-6125	270	23	.	.	PUNCT
ap-6125	271	1	1	1	NUM
ap-6125	271	2	)	)	PUNCT
ap-6125	271	3	does	do	AUX
ap-6125	271	4	not	not	PART
ap-6125	271	5	satisfy	satisfy	VERB
ap-6125	271	6	the	the	DET
ap-6125	271	7	equation	equation	NOUN
ap-6125	271	8	(	(	PUNCT
ap-6125	271	9	8)	8)	NUM
ap-6125	271	10	at	at	ADP
ap-6125	271	11	the	the	DET
ap-6125	271	12	three	three	NUM
ap-6125	271	13	nodes	node	NOUN
ap-6125	271	14	x1	x1	PROPN
ap-6125	271	15	,	,	PUNCT
ap-6125	271	16	x2	x2	PROPN
ap-6125	271	17	,	,	PUNCT
ap-6125	271	18	x3	x3	ADJ
ap-6125	271	19	.	.	PUNCT
ap-6125	272	1	this	this	PRON
ap-6125	272	2	is	be	AUX
ap-6125	272	3	not	not	PART
ap-6125	272	4	important	important	ADJ
ap-6125	272	5	at	at	ADP
ap-6125	272	6	the	the	DET
ap-6125	272	7	first	first	ADJ
ap-6125	272	8	and	and	CCONJ
ap-6125	272	9	last	last	ADJ
ap-6125	272	10	node	node	NOUN
ap-6125	272	11	where	where	SCONJ
ap-6125	272	12	the	the	DET
ap-6125	272	13	value	value	NOUN
ap-6125	272	14	prescribed	prescribe	VERB
ap-6125	272	15	can	can	AUX
ap-6125	272	16	be	be	AUX
ap-6125	272	17	understood	understand	VERB
ap-6125	272	18	as	as	ADP
ap-6125	272	19	a	a	DET
ap-6125	272	20	boundary	boundary	ADJ
ap-6125	272	21	condition	condition	NOUN
ap-6125	272	22	.	.	PUNCT
ap-6125	273	1	we	we	PRON
ap-6125	273	2	can	can	AUX
ap-6125	273	3	thus	thus	ADV
ap-6125	273	4	claim	claim	VERB
ap-6125	273	5	that	that	SCONJ
ap-6125	273	6	the	the	DET
ap-6125	273	7	interpolant	interpolant	NOUN
ap-6125	273	8	(	(	PUNCT
ap-6125	273	9	28	28	NUM
ap-6125	273	10	)	)	PUNCT
ap-6125	273	11	satisfies	satisfy	VERB
ap-6125	273	12	the	the	DET
ap-6125	273	13	harmonic	harmonic	ADJ
ap-6125	273	14	equation	equation	NOUN
ap-6125	273	15	(	(	PUNCT
ap-6125	273	16	8)	8)	NUM
ap-6125	273	17	on	on	ADP
ap-6125	273	18	(	(	PUNCT
ap-6125	273	19	−1	−1	NOUN
ap-6125	273	20	,	,	PUNCT
ap-6125	273	21	1	1	NUM
ap-6125	273	22	)	)	PUNCT
ap-6125	273	23	\	\	NOUN
ap-6125	273	24	{	{	PUNCT
ap-6125	273	25	0	0	NUM
ap-6125	273	26	}	}	PUNCT
ap-6125	273	27	.	.	PUNCT
ap-6125	274	1	moreover	moreover	ADV
ap-6125	274	2	,	,	PUNCT
ap-6125	274	3	according	accord	VERB
ap-6125	274	4	to	to	ADP
ap-6125	274	5	lemma	lemma	PROPN
ap-6125	274	6	1	1	NUM
ap-6125	274	7	,	,	PUNCT
ap-6125	274	8	the	the	DET
ap-6125	274	9	interpolant	interpolant	NOUN
ap-6125	274	10	z(x	z(x	NUM
ap-6125	274	11	)	)	PUNCT
ap-6125	274	12	given	give	VERB
ap-6125	274	13	by	by	ADP
ap-6125	274	14	(	(	PUNCT
ap-6125	274	15	28	28	NUM
ap-6125	274	16	)	)	PUNCT
ap-6125	274	17	also	also	ADV
ap-6125	274	18	satisfies	satisfy	VERB
ap-6125	274	19	the	the	DET
ap-6125	274	20	polyharmonic	polyharmonic	ADJ
ap-6125	274	21	equation	equation	NOUN
ap-6125	274	22	(	(	PUNCT
ap-6125	274	23	8)	8)	NUM
ap-6125	274	24	of	of	ADP
ap-6125	274	25	any	any	DET
ap-6125	274	26	order	order	NOUN
ap-6125	274	27	m	m	VERB
ap-6125	274	28	>	>	X
ap-6125	274	29	1	1	NUM
ap-6125	274	30	in	in	ADP
ap-6125	274	31	the	the	DET
ap-6125	274	32	same	same	ADJ
ap-6125	274	33	set	set	NOUN
ap-6125	274	34	as	as	ADP
ap-6125	274	35	the	the	DET
ap-6125	274	36	equation	equation	NOUN
ap-6125	274	37	of	of	ADP
ap-6125	274	38	order	order	NOUN
ap-6125	274	39	1	1	NUM
ap-6125	274	40	,	,	PUNCT
ap-6125	274	41	i.e.	i.e.	X
ap-6125	274	42	on	on	ADP
ap-6125	274	43	(	(	PUNCT
ap-6125	274	44	−1	−1	NOUN
ap-6125	274	45	,	,	PUNCT
ap-6125	274	46	1)\{0	1)\{0	NUM
ap-6125	274	47	}	}	PUNCT
ap-6125	274	48	.	.	PUNCT
ap-6125	275	1	if	if	SCONJ
ap-6125	275	2	we	we	PRON
ap-6125	275	3	further	far	ADV
ap-6125	275	4	put	put	VERB
ap-6125	275	5	l	l	NOUN
ap-6125	275	6	=	=	SYM
ap-6125	275	7	2	2	NUM
ap-6125	275	8	,	,	PUNCT
ap-6125	275	9	b2	b2	NOUN
ap-6125	275	10	=	=	SYM
ap-6125	275	11	1	1	NUM
ap-6125	275	12	,	,	PUNCT
ap-6125	275	13	and	and	CCONJ
ap-6125	275	14	bk	bk	VERB
ap-6125	275	15	=	=	SYM
ap-6125	275	16	0	0	NUM
ap-6125	275	17	otherwise	otherwise	ADV
ap-6125	275	18	to	to	PART
ap-6125	275	19	minimize	minimize	VERB
ap-6125	275	20	the	the	DET
ap-6125	275	21	l2	l2	NOUN
ap-6125	275	22	norm	norm	NOUN
ap-6125	275	23	of	of	ADP
ap-6125	275	24	the	the	DET
ap-6125	275	25	2nd	2nd	ADJ
ap-6125	275	26	derivative	derivative	NOUN
ap-6125	275	27	of	of	ADP
ap-6125	275	28	the	the	DET
ap-6125	275	29	interpolant	interpolant	NOUN
ap-6125	275	30	,	,	PUNCT
ap-6125	275	31	then	then	ADV
ap-6125	275	32	r(x	r(x	PROPN
ap-6125	275	33	,	,	PUNCT
ap-6125	275	34	y	y	NOUN
ap-6125	275	35	)	)	PUNCT
ap-6125	275	36	=	=	SYM
ap-6125	275	37	r2l−n	r2l−n	NOUN
ap-6125	275	38	=	=	SYM
ap-6125	275	39	r3	r3	PROPN
ap-6125	275	40	(	(	PUNCT
ap-6125	275	41	the	the	DET
ap-6125	275	42	well	well	ADV
ap-6125	275	43	-	-	PUNCT
ap-6125	275	44	known	know	VERB
ap-6125	275	45	cubic	cubic	ADJ
ap-6125	275	46	spline	spline	NOUN
ap-6125	275	47	)	)	PUNCT
ap-6125	275	48	,	,	PUNCT
ap-6125	275	49	the	the	DET
ap-6125	275	50	trend	trend	NOUN
ap-6125	275	51	functions	function	NOUN
ap-6125	275	52	are	be	AUX
ap-6125	275	53	a	a	DET
ap-6125	275	54	constant	constant	ADJ
ap-6125	275	55	and	and	CCONJ
ap-6125	275	56	linear	linear	ADJ
ap-6125	275	57	function	function	NOUN
ap-6125	275	58	.	.	PUNCT
ap-6125	276	1	the	the	DET
ap-6125	276	2	interpolant	interpolant	NOUN
ap-6125	276	3	(	(	PUNCT
ap-6125	276	4	28	28	NUM
ap-6125	276	5	)	)	PUNCT
ap-6125	276	6	153	153	NUM
ap-6125	276	7	karel	karel	PROPN
ap-6125	276	8	segeth	segeth	PROPN
ap-6125	276	9	acta	acta	PROPN
ap-6125	276	10	polytechnica	polytechnica	PROPN
ap-6125	276	11	(	(	PUNCT
ap-6125	276	12	dotted	dotted	ADJ
ap-6125	276	13	line	line	NOUN
ap-6125	276	14	in	in	ADP
ap-6125	276	15	fig	fig	NOUN
ap-6125	276	16	.	.	PUNCT
ap-6125	277	1	1	1	X
ap-6125	277	2	)	)	PUNCT
ap-6125	277	3	solves	solve	VERB
ap-6125	277	4	the	the	DET
ap-6125	277	5	biharmonic	biharmonic	NOUN
ap-6125	277	6	equation	equation	NOUN
ap-6125	277	7	(	(	PUNCT
ap-6125	277	8	8)	8)	NUM
ap-6125	277	9	as	as	ADP
ap-6125	277	10	m	m	NOUN
ap-6125	277	11	=	=	NOUN
ap-6125	277	12	l	l	NOUN
ap-6125	277	13	=	=	SYM
ap-6125	277	14	2	2	NUM
ap-6125	277	15	according	accord	VERB
ap-6125	277	16	to	to	ADP
ap-6125	277	17	theorem	theorem	ADJ
ap-6125	277	18	4	4	NUM
ap-6125	277	19	,	,	PUNCT
ap-6125	277	20	everywhere	everywhere	ADV
ap-6125	277	21	in	in	ADP
ap-6125	277	22	r1	r1	PROPN
ap-6125	277	23	except	except	SCONJ
ap-6125	277	24	for	for	ADP
ap-6125	277	25	the	the	DET
ap-6125	277	26	points	point	NOUN
ap-6125	277	27	x	x	X
ap-6125	277	28	=	=	SYM
ap-6125	277	29	xj	xj	PROPN
ap-6125	277	30	,	,	PUNCT
ap-6125	277	31	j	j	PROPN
ap-6125	277	32	=	=	SYM
ap-6125	277	33	1	1	NUM
ap-6125	277	34	,	,	PUNCT
ap-6125	277	35	2	2	NUM
ap-6125	277	36	,	,	PUNCT
ap-6125	277	37	3	3	NUM
ap-6125	277	38	.	.	PUNCT
ap-6125	278	1	the	the	DET
ap-6125	278	2	constant	constant	ADJ
ap-6125	278	3	and	and	CCONJ
ap-6125	278	4	linear	linear	ADJ
ap-6125	278	5	trend	trend	NOUN
ap-6125	278	6	functions	function	NOUN
ap-6125	278	7	satisfy	satisfy	VERB
ap-6125	278	8	the	the	DET
ap-6125	278	9	equation	equation	NOUN
ap-6125	278	10	(	(	PUNCT
ap-6125	278	11	8)	8)	NUM
ap-6125	278	12	everywhere	everywhere	ADV
ap-6125	278	13	in	in	ADP
ap-6125	278	14	r1	r1	PROPN
ap-6125	278	15	.	.	PUNCT
ap-6125	279	1	as	as	ADP
ap-6125	279	2	in	in	ADP
ap-6125	279	3	the	the	DET
ap-6125	279	4	previous	previous	ADJ
ap-6125	279	5	case	case	NOUN
ap-6125	279	6	,	,	PUNCT
ap-6125	279	7	we	we	PRON
ap-6125	279	8	see	see	VERB
ap-6125	279	9	that	that	SCONJ
ap-6125	279	10	the	the	DET
ap-6125	279	11	interpolant	interpolant	NOUN
ap-6125	279	12	(	(	PUNCT
ap-6125	279	13	28	28	NUM
ap-6125	279	14	)	)	PUNCT
ap-6125	279	15	satisfies	satisfy	VERB
ap-6125	279	16	the	the	DET
ap-6125	279	17	biharmonic	biharmonic	NOUN
ap-6125	279	18	equation	equation	NOUN
ap-6125	279	19	(	(	PUNCT
ap-6125	279	20	8)	8)	NUM
ap-6125	279	21	on	on	ADP
ap-6125	279	22	(	(	PUNCT
ap-6125	279	23	−1	−1	NOUN
ap-6125	279	24	,	,	PUNCT
ap-6125	279	25	1	1	NUM
ap-6125	279	26	)	)	PUNCT
ap-6125	279	27	\	\	NOUN
ap-6125	279	28	{	{	PUNCT
ap-6125	279	29	0	0	NUM
ap-6125	279	30	}	}	PUNCT
ap-6125	279	31	.	.	PUNCT
ap-6125	280	1	again	again	ADV
ap-6125	280	2	,	,	PUNCT
ap-6125	280	3	according	accord	VERB
ap-6125	280	4	to	to	ADP
ap-6125	280	5	lemma	lemma	PROPN
ap-6125	280	6	1	1	NUM
ap-6125	280	7	,	,	PUNCT
ap-6125	280	8	the	the	DET
ap-6125	280	9	interpolant	interpolant	NOUN
ap-6125	280	10	z(x	z(x	NOUN
ap-6125	280	11	)	)	PUNCT
ap-6125	280	12	also	also	ADV
ap-6125	280	13	satisfies	satisfy	VERB
ap-6125	280	14	the	the	DET
ap-6125	280	15	polyharmonic	polyharmonic	ADJ
ap-6125	280	16	equation	equation	NOUN
ap-6125	280	17	(	(	PUNCT
ap-6125	280	18	8)	8)	NUM
ap-6125	280	19	of	of	ADP
ap-6125	280	20	any	any	DET
ap-6125	280	21	order	order	NOUN
ap-6125	280	22	m	m	VERB
ap-6125	280	23	>	>	X
ap-6125	280	24	2	2	NUM
ap-6125	280	25	in	in	ADP
ap-6125	280	26	the	the	DET
ap-6125	280	27	same	same	ADJ
ap-6125	280	28	set	set	NOUN
ap-6125	280	29	.	.	PUNCT
ap-6125	281	1	if	if	SCONJ
ap-6125	281	2	we	we	PRON
ap-6125	281	3	finally	finally	ADV
ap-6125	281	4	put	put	VERB
ap-6125	281	5	l	l	NOUN
ap-6125	281	6	=	=	SYM
ap-6125	281	7	3	3	NUM
ap-6125	281	8	,	,	PUNCT
ap-6125	281	9	b3	b3	NOUN
ap-6125	281	10	=	=	SYM
ap-6125	281	11	1	1	NUM
ap-6125	281	12	,	,	PUNCT
ap-6125	281	13	and	and	CCONJ
ap-6125	281	14	bk	bk	VERB
ap-6125	281	15	=	=	SYM
ap-6125	281	16	0	0	NUM
ap-6125	281	17	otherwise	otherwise	ADV
ap-6125	281	18	to	to	PART
ap-6125	281	19	minimize	minimize	VERB
ap-6125	281	20	the	the	DET
ap-6125	281	21	l2	l2	NOUN
ap-6125	281	22	norm	norm	NOUN
ap-6125	281	23	of	of	ADP
ap-6125	281	24	the	the	DET
ap-6125	281	25	3rd	3rd	ADJ
ap-6125	281	26	derivative	derivative	NOUN
ap-6125	281	27	of	of	ADP
ap-6125	281	28	the	the	DET
ap-6125	281	29	interpolant	interpolant	NOUN
ap-6125	281	30	,	,	PUNCT
ap-6125	281	31	then	then	ADV
ap-6125	281	32	r(x	r(x	PROPN
ap-6125	281	33	,	,	PUNCT
ap-6125	281	34	y	y	NOUN
ap-6125	281	35	)	)	PUNCT
ap-6125	281	36	=	=	SYM
ap-6125	281	37	r2l−n	r2l−n	NOUN
ap-6125	281	38	=	=	SYM
ap-6125	281	39	r5	r5	PROPN
ap-6125	281	40	(	(	PUNCT
ap-6125	281	41	quintic	quintic	ADJ
ap-6125	281	42	spline	spline	NOUN
ap-6125	281	43	)	)	PUNCT
ap-6125	281	44	,	,	PUNCT
ap-6125	281	45	the	the	DET
ap-6125	281	46	trend	trend	NOUN
ap-6125	281	47	functions	function	NOUN
ap-6125	281	48	are	be	AUX
ap-6125	281	49	a	a	DET
ap-6125	281	50	constant	constant	ADJ
ap-6125	281	51	,	,	PUNCT
ap-6125	281	52	linear	linear	ADJ
ap-6125	281	53	function	function	NOUN
ap-6125	281	54	,	,	PUNCT
ap-6125	281	55	and	and	CCONJ
ap-6125	281	56	a	a	DET
ap-6125	281	57	quadratic	quadratic	ADJ
ap-6125	281	58	function	function	NOUN
ap-6125	281	59	.	.	PUNCT
ap-6125	282	1	the	the	DET
ap-6125	282	2	interpolant	interpolant	NOUN
ap-6125	282	3	(	(	PUNCT
ap-6125	282	4	28	28	NUM
ap-6125	282	5	)	)	PUNCT
ap-6125	282	6	(	(	PUNCT
ap-6125	282	7	dash	dash	NOUN
ap-6125	282	8	-	-	PUNCT
ap-6125	282	9	dot	dot	NOUN
ap-6125	282	10	line	line	NOUN
ap-6125	282	11	in	in	ADP
ap-6125	282	12	fig	fig	NOUN
ap-6125	282	13	.	.	PUNCT
ap-6125	283	1	1	1	X
ap-6125	283	2	)	)	PUNCT
ap-6125	283	3	solves	solve	VERB
ap-6125	283	4	the	the	DET
ap-6125	283	5	triharmonic	triharmonic	ADJ
ap-6125	283	6	equation	equation	NOUN
ap-6125	283	7	(	(	PUNCT
ap-6125	283	8	8)	8)	NUM
ap-6125	283	9	as	as	ADP
ap-6125	283	10	m	m	NOUN
ap-6125	283	11	=	=	NOUN
ap-6125	283	12	l	l	NOUN
ap-6125	283	13	=	=	SYM
ap-6125	283	14	3	3	NUM
ap-6125	283	15	by	by	ADP
ap-6125	283	16	theorem	theorem	NOUN
ap-6125	283	17	4	4	NUM
ap-6125	283	18	everywhere	everywhere	ADV
ap-6125	283	19	in	in	ADP
ap-6125	283	20	r1	r1	PROPN
ap-6125	283	21	except	except	SCONJ
ap-6125	283	22	for	for	ADP
ap-6125	283	23	the	the	DET
ap-6125	283	24	points	point	NOUN
ap-6125	283	25	x	x	X
ap-6125	283	26	=	=	SYM
ap-6125	283	27	xj	xj	PROPN
ap-6125	283	28	,	,	PUNCT
ap-6125	283	29	j	j	PROPN
ap-6125	283	30	=	=	SYM
ap-6125	283	31	1	1	NUM
ap-6125	283	32	,	,	PUNCT
ap-6125	283	33	2	2	NUM
ap-6125	283	34	,	,	PUNCT
ap-6125	283	35	3	3	NUM
ap-6125	283	36	.	.	X
ap-6125	283	37	all	all	DET
ap-6125	283	38	the	the	DET
ap-6125	283	39	trend	trend	NOUN
ap-6125	283	40	functions	function	NOUN
ap-6125	283	41	satisfy	satisfy	VERB
ap-6125	283	42	the	the	DET
ap-6125	283	43	equation	equation	NOUN
ap-6125	283	44	(	(	PUNCT
ap-6125	283	45	8)	8)	NUM
ap-6125	283	46	everywhere	everywhere	ADV
ap-6125	283	47	in	in	ADP
ap-6125	283	48	r1	r1	PROPN
ap-6125	283	49	.	.	PUNCT
ap-6125	284	1	as	as	ADP
ap-6125	284	2	in	in	ADP
ap-6125	284	3	the	the	DET
ap-6125	284	4	previous	previous	ADJ
ap-6125	284	5	case	case	NOUN
ap-6125	284	6	,	,	PUNCT
ap-6125	284	7	we	we	PRON
ap-6125	284	8	see	see	VERB
ap-6125	284	9	that	that	SCONJ
ap-6125	284	10	the	the	DET
ap-6125	284	11	interpolant	interpolant	NOUN
ap-6125	284	12	(	(	PUNCT
ap-6125	284	13	28	28	NUM
ap-6125	284	14	)	)	PUNCT
ap-6125	284	15	satisfies	satisfy	VERB
ap-6125	284	16	the	the	DET
ap-6125	284	17	triharmonic	triharmonic	ADJ
ap-6125	284	18	equation	equation	NOUN
ap-6125	284	19	(	(	PUNCT
ap-6125	284	20	8)	8)	NUM
ap-6125	284	21	on	on	ADP
ap-6125	284	22	(	(	PUNCT
ap-6125	284	23	−1	−1	NOUN
ap-6125	284	24	,	,	PUNCT
ap-6125	284	25	1	1	NUM
ap-6125	284	26	)	)	PUNCT
ap-6125	284	27	\	\	NOUN
ap-6125	284	28	{	{	PUNCT
ap-6125	284	29	0	0	NUM
ap-6125	284	30	}	}	PUNCT
ap-6125	284	31	.	.	PUNCT
ap-6125	285	1	according	accord	VERB
ap-6125	285	2	to	to	ADP
ap-6125	285	3	lemma	lemma	PROPN
ap-6125	285	4	1	1	NUM
ap-6125	285	5	,	,	PUNCT
ap-6125	285	6	the	the	DET
ap-6125	285	7	interpolant	interpolant	NOUN
ap-6125	285	8	z(x	z(x	NOUN
ap-6125	285	9	)	)	PUNCT
ap-6125	285	10	satisfies	satisfy	VERB
ap-6125	285	11	the	the	DET
ap-6125	285	12	polyharmonic	polyharmonic	ADJ
ap-6125	285	13	equation	equation	NOUN
ap-6125	285	14	(	(	PUNCT
ap-6125	285	15	8)	8)	NUM
ap-6125	285	16	of	of	ADP
ap-6125	285	17	any	any	DET
ap-6125	285	18	order	order	NOUN
ap-6125	285	19	m	m	VERB
ap-6125	285	20	>	>	X
ap-6125	285	21	3	3	NUM
ap-6125	285	22	in	in	ADP
ap-6125	285	23	the	the	DET
ap-6125	285	24	same	same	ADJ
ap-6125	285	25	set	set	NOUN
ap-6125	285	26	.	.	PUNCT
ap-6125	286	1	the	the	DET
ap-6125	286	2	results	result	NOUN
ap-6125	286	3	agree	agree	VERB
ap-6125	286	4	with	with	ADP
ap-6125	286	5	general	general	ADJ
ap-6125	286	6	expectations	expectation	NOUN
ap-6125	286	7	.	.	PUNCT
ap-6125	287	1	the	the	DET
ap-6125	287	2	interpolation	interpolation	NOUN
ap-6125	287	3	conditions	condition	NOUN
ap-6125	287	4	(	(	PUNCT
ap-6125	287	5	1	1	X
ap-6125	287	6	)	)	PUNCT
ap-6125	287	7	are	be	AUX
ap-6125	287	8	satisfied	satisfied	ADJ
ap-6125	287	9	.	.	PUNCT
ap-6125	288	1	for	for	ADP
ap-6125	288	2	n	n	NOUN
ap-6125	288	3	=	=	SYM
ap-6125	288	4	1	1	NUM
ap-6125	288	5	,	,	PUNCT
ap-6125	288	6	the	the	DET
ap-6125	288	7	formula	formula	NOUN
ap-6125	288	8	(	(	PUNCT
ap-6125	288	9	2	2	X
ap-6125	288	10	)	)	PUNCT
ap-6125	288	11	gives	give	VERB
ap-6125	288	12	r(x	r(x	PROPN
ap-6125	288	13	,	,	PUNCT
ap-6125	288	14	y	y	NOUN
ap-6125	288	15	)	)	PUNCT
ap-6125	288	16	=	=	SYM
ap-6125	289	1	|x−y|	|x−y|	ADJ
ap-6125	289	2	,	,	PUNCT
ap-6125	289	3	i.e.	i.e.	X
ap-6125	289	4	the	the	DET
ap-6125	289	5	absolute	absolute	ADJ
ap-6125	289	6	value	value	NOUN
ap-6125	289	7	of	of	ADP
ap-6125	289	8	the	the	DET
ap-6125	289	9	difference	difference	NOUN
ap-6125	289	10	x	x	INTJ
ap-6125	289	11	−	−	PROPN
ap-6125	289	12	y.	y.	NOUN
ap-6125	289	13	naturally	naturally	ADV
ap-6125	289	14	,	,	PUNCT
ap-6125	289	15	minimizing	minimize	VERB
ap-6125	289	16	the	the	DET
ap-6125	289	17	l2	l2	NOUN
ap-6125	289	18	norm	norm	NOUN
ap-6125	289	19	of	of	ADP
ap-6125	289	20	the	the	DET
ap-6125	289	21	first	first	ADJ
ap-6125	289	22	derivative	derivative	NOUN
ap-6125	289	23	of	of	ADP
ap-6125	289	24	the	the	DET
ap-6125	289	25	interpolant	interpolant	NOUN
ap-6125	289	26	(	(	PUNCT
ap-6125	289	27	b1	b1	NOUN
ap-6125	289	28	=	=	SYM
ap-6125	289	29	1	1	X
ap-6125	289	30	)	)	PUNCT
ap-6125	289	31	gives	give	VERB
ap-6125	289	32	a	a	DET
ap-6125	289	33	broken	broken	ADJ
ap-6125	289	34	line	line	NOUN
ap-6125	289	35	.	.	PUNCT
ap-6125	290	1	minimizing	minimize	VERB
ap-6125	290	2	the	the	DET
ap-6125	290	3	same	same	ADJ
ap-6125	290	4	norm	norm	NOUN
ap-6125	290	5	of	of	ADP
ap-6125	290	6	the	the	DET
ap-6125	290	7	second	second	ADJ
ap-6125	290	8	derivative	derivative	NOUN
ap-6125	290	9	of	of	ADP
ap-6125	290	10	the	the	DET
ap-6125	290	11	interpolant	interpolant	NOUN
ap-6125	290	12	(	(	PUNCT
ap-6125	290	13	its	its	PRON
ap-6125	290	14	curvature	curvature	NOUN
ap-6125	290	15	)	)	PUNCT
ap-6125	290	16	using	use	VERB
ap-6125	290	17	b2	b2	NOUN
ap-6125	290	18	=	=	SYM
ap-6125	290	19	1	1	NUM
ap-6125	290	20	leads	lead	VERB
ap-6125	290	21	to	to	ADP
ap-6125	290	22	a	a	DET
ap-6125	290	23	cubic	cubic	ADJ
ap-6125	290	24	spline	spline	NOUN
ap-6125	290	25	.	.	PUNCT
ap-6125	291	1	if	if	SCONJ
ap-6125	291	2	we	we	PRON
ap-6125	291	3	put	put	VERB
ap-6125	291	4	b3	b3	NOUN
ap-6125	291	5	=	=	SYM
ap-6125	291	6	1	1	NUM
ap-6125	291	7	,	,	PUNCT
ap-6125	291	8	we	we	PRON
ap-6125	291	9	minimize	minimize	VERB
ap-6125	291	10	the	the	DET
ap-6125	291	11	l2	l2	NOUN
ap-6125	291	12	norm	norm	NOUN
ap-6125	291	13	of	of	ADP
ap-6125	291	14	the	the	DET
ap-6125	291	15	third	third	ADJ
ap-6125	291	16	derivative	derivative	NOUN
ap-6125	291	17	of	of	ADP
ap-6125	291	18	the	the	DET
ap-6125	291	19	interpolant	interpolant	NOUN
ap-6125	291	20	by	by	ADP
ap-6125	291	21	a	a	DET
ap-6125	291	22	quintic	quintic	ADJ
ap-6125	291	23	spline	spline	NOUN
ap-6125	291	24	.	.	PUNCT
ap-6125	292	1	apparently	apparently	ADV
ap-6125	292	2	,	,	PUNCT
ap-6125	292	3	it	it	PRON
ap-6125	292	4	is	be	AUX
ap-6125	292	5	not	not	PART
ap-6125	292	6	possible	possible	ADJ
ap-6125	292	7	to	to	PART
ap-6125	292	8	draw	draw	VERB
ap-6125	292	9	principal	principal	ADJ
ap-6125	292	10	conclusions	conclusion	NOUN
ap-6125	292	11	from	from	ADP
ap-6125	292	12	a	a	DET
ap-6125	292	13	single	single	ADJ
ap-6125	292	14	1d	1d	NUM
ap-6125	292	15	example	example	NOUN
ap-6125	292	16	.	.	PUNCT
ap-6125	293	1	2d	2d	NOUN
ap-6125	293	2	and	and	CCONJ
ap-6125	293	3	3d	3d	NUM
ap-6125	293	4	cases	case	NOUN
ap-6125	293	5	are	be	AUX
ap-6125	293	6	more	more	ADV
ap-6125	293	7	interesting	interesting	ADJ
ap-6125	293	8	and	and	CCONJ
ap-6125	293	9	can	can	AUX
ap-6125	293	10	be	be	AUX
ap-6125	293	11	applied	apply	VERB
ap-6125	293	12	to	to	ADP
ap-6125	293	13	many	many	ADJ
ap-6125	293	14	problems	problem	NOUN
ap-6125	293	15	of	of	ADP
ap-6125	293	16	practice	practice	NOUN
ap-6125	293	17	.	.	PUNCT
ap-6125	294	1	10	10	X
ap-6125	294	2	.	.	PUNCT
ap-6125	295	1	conclusion	conclusion	NOUN
ap-6125	295	2	using	use	VERB
ap-6125	295	3	the	the	DET
ap-6125	295	4	general	general	ADJ
ap-6125	295	5	theory	theory	NOUN
ap-6125	295	6	of	of	ADP
ap-6125	295	7	smooth	smooth	ADJ
ap-6125	295	8	interpolation	interpolation	NOUN
ap-6125	295	9	we	we	PRON
ap-6125	295	10	constructed	construct	VERB
ap-6125	295	11	a	a	DET
ap-6125	295	12	radial	radial	ADJ
ap-6125	295	13	basis	basis	NOUN
ap-6125	295	14	interpolant	interpolant	NOUN
ap-6125	295	15	as	as	ADP
ap-6125	295	16	a	a	DET
ap-6125	295	17	linear	linear	ADJ
ap-6125	295	18	combination	combination	NOUN
ap-6125	295	19	of	of	ADP
ap-6125	295	20	the	the	DET
ap-6125	295	21	values	value	NOUN
ap-6125	295	22	of	of	ADP
ap-6125	295	23	a	a	DET
ap-6125	295	24	polyharmonic	polyharmonic	ADJ
ap-6125	295	25	spline	spline	NOUN
ap-6125	295	26	of	of	ADP
ap-6125	295	27	fixed	fix	VERB
ap-6125	295	28	order	order	NOUN
ap-6125	295	29	and	and	CCONJ
ap-6125	295	30	a	a	DET
ap-6125	295	31	linear	linear	ADJ
ap-6125	295	32	combination	combination	NOUN
ap-6125	295	33	of	of	ADP
ap-6125	295	34	the	the	DET
ap-6125	295	35	trend	trend	NOUN
ap-6125	295	36	functions	function	NOUN
ap-6125	295	37	.	.	PUNCT
ap-6125	296	1	the	the	DET
ap-6125	296	2	construction	construction	NOUN
ap-6125	296	3	shows	show	VERB
ap-6125	296	4	how	how	SCONJ
ap-6125	296	5	to	to	PART
ap-6125	296	6	choose	choose	VERB
ap-6125	296	7	the	the	DET
ap-6125	296	8	functional	functional	ADJ
ap-6125	296	9	applied	apply	VERB
ap-6125	296	10	to	to	ADP
ap-6125	296	11	the	the	DET
ap-6125	296	12	formula	formula	NOUN
ap-6125	296	13	in	in	ADP
ap-6125	296	14	order	order	NOUN
ap-6125	296	15	to	to	PART
ap-6125	296	16	minimize	minimize	VERB
ap-6125	296	17	particular	particular	ADJ
ap-6125	296	18	derivatives	derivative	NOUN
ap-6125	296	19	of	of	ADP
ap-6125	296	20	the	the	DET
ap-6125	296	21	interpolant	interpolant	NOUN
ap-6125	296	22	,	,	PUNCT
ap-6125	296	23	i.e.	i.e.	X
ap-6125	296	24	,	,	PUNCT
ap-6125	296	25	to	to	PART
ap-6125	296	26	get	get	VERB
ap-6125	296	27	the	the	DET
ap-6125	296	28	smoothness	smoothness	NOUN
ap-6125	296	29	of	of	ADP
ap-6125	296	30	these	these	DET
ap-6125	296	31	derivatives	derivative	NOUN
ap-6125	296	32	.	.	PUNCT
ap-6125	297	1	moreover	moreover	ADV
ap-6125	297	2	,	,	PUNCT
ap-6125	297	3	the	the	DET
ap-6125	297	4	interpolant	interpolant	NOUN
ap-6125	297	5	is	be	AUX
ap-6125	297	6	proven	prove	VERB
ap-6125	297	7	to	to	PART
ap-6125	297	8	be	be	AUX
ap-6125	297	9	piecewise	piecewise	NOUN
ap-6125	297	10	polyharmonic	polyharmonic	NOUN
ap-6125	297	11	,	,	PUNCT
ap-6125	297	12	which	which	PRON
ap-6125	297	13	can	can	AUX
ap-6125	297	14	be	be	AUX
ap-6125	297	15	considered	consider	VERB
ap-6125	297	16	advantageous	advantageous	ADJ
ap-6125	297	17	in	in	ADP
ap-6125	297	18	some	some	DET
ap-6125	297	19	cases	case	NOUN
ap-6125	297	20	.	.	PUNCT
ap-6125	298	1	note	note	VERB
ap-6125	298	2	that	that	SCONJ
ap-6125	298	3	the	the	DET
ap-6125	298	4	problem	problem	NOUN
ap-6125	298	5	considered	consider	VERB
ap-6125	298	6	is	be	AUX
ap-6125	298	7	n	n	ADV
ap-6125	298	8	-	-	PUNCT
ap-6125	298	9	dimensional	dimensional	ADJ
ap-6125	298	10	and	and	CCONJ
ap-6125	298	11	that	that	SCONJ
ap-6125	298	12	the	the	DET
ap-6125	298	13	number	number	NOUN
ap-6125	298	14	of	of	ADP
ap-6125	298	15	equations	equation	NOUN
ap-6125	298	16	of	of	ADP
ap-6125	298	17	the	the	DET
ap-6125	298	18	linear	linear	ADJ
ap-6125	298	19	algebraic	algebraic	ADJ
ap-6125	298	20	system	system	NOUN
ap-6125	298	21	to	to	PART
ap-6125	298	22	be	be	AUX
ap-6125	298	23	solved	solve	VERB
ap-6125	298	24	is	be	AUX
ap-6125	298	25	the	the	DET
ap-6125	298	26	number	number	NOUN
ap-6125	298	27	n	n	ADP
ap-6125	298	28	of	of	ADP
ap-6125	298	29	nodes	node	NOUN
ap-6125	298	30	of	of	ADP
ap-6125	298	31	the	the	DET
ap-6125	298	32	measurement	measurement	NOUN
ap-6125	298	33	plus	plus	CCONJ
ap-6125	298	34	the	the	DET
ap-6125	298	35	number	number	NOUN
ap-6125	298	36	of	of	ADP
ap-6125	298	37	trends	trend	NOUN
ap-6125	298	38	(	(	PUNCT
ap-6125	298	39	that	that	PRON
ap-6125	298	40	depends	depend	VERB
ap-6125	298	41	on	on	ADP
ap-6125	298	42	the	the	DET
ap-6125	298	43	dimension	dimension	NOUN
ap-6125	298	44	)	)	PUNCT
ap-6125	298	45	.	.	PUNCT
ap-6125	299	1	acknowledgements	acknowledgement	VERB
ap-6125	299	2	the	the	DET
ap-6125	299	3	author	author	NOUN
ap-6125	299	4	was	be	AUX
ap-6125	299	5	supported	support	VERB
ap-6125	299	6	by	by	ADP
ap-6125	299	7	rvo	rvo	PROPN
ap-6125	299	8	67985840	67985840	NUM
ap-6125	299	9	and	and	CCONJ
ap-6125	299	10	by	by	ADP
ap-6125	299	11	czech	czech	PROPN
ap-6125	299	12	science	science	NOUN
ap-6125	299	13	foundation	foundation	NOUN
ap-6125	299	14	grant	grant	VERB
ap-6125	299	15	18	18	NUM
ap-6125	299	16	-	-	NOUN
ap-6125	299	17	09628s	09628s	NUM
ap-6125	299	18	.	.	PUNCT
ap-6125	300	1	references	reference	NOUN
ap-6125	300	2	[	[	X
ap-6125	300	3	1	1	NUM
ap-6125	300	4	]	]	PUNCT
ap-6125	300	5	a.	a.	NOUN
ap-6125	300	6	talmi	talmi	PROPN
ap-6125	300	7	,	,	PUNCT
ap-6125	300	8	g.	g.	PROPN
ap-6125	300	9	gilat	gilat	PROPN
ap-6125	300	10	.	.	PUNCT
ap-6125	301	1	method	method	PROPN
ap-6125	301	2	for	for	ADP
ap-6125	301	3	smooth	smooth	ADJ
ap-6125	301	4	approximation	approximation	NOUN
ap-6125	301	5	of	of	ADP
ap-6125	301	6	data	data	PROPN
ap-6125	301	7	.	.	PUNCT
ap-6125	302	1	j	j	PROPN
ap-6125	302	2	comput	comput	PROPN
ap-6125	302	3	phys	phy	NOUN
ap-6125	302	4	23:93–123	23:93–123	NUM
ap-6125	302	5	,	,	PUNCT
ap-6125	302	6	1977	1977	NUM
ap-6125	302	7	.	.	PUNCT
ap-6125	303	1	doi:10.1016/0021	doi:10.1016/0021	VERB
ap-6125	303	2	-	-	PUNCT
ap-6125	303	3	9991(77)90115	9991(77)90115	NUM
ap-6125	303	4	-	-	SYM
ap-6125	303	5	2	2	NUM
ap-6125	303	6	.	.	PUNCT
ap-6125	304	1	[	[	X
ap-6125	304	2	2	2	NUM
ap-6125	304	3	]	]	PUNCT
ap-6125	304	4	k.	k.	PROPN
ap-6125	304	5	segeth	segeth	PROPN
ap-6125	304	6	.	.	PUNCT
ap-6125	305	1	polyharmonic	polyharmonic	ADJ
ap-6125	305	2	splines	spline	NOUN
ap-6125	305	3	generated	generate	VERB
ap-6125	305	4	by	by	ADP
ap-6125	305	5	multivariate	multivariate	NOUN
ap-6125	305	6	smooth	smooth	ADJ
ap-6125	305	7	interpolation	interpolation	NOUN
ap-6125	305	8	.	.	PUNCT
ap-6125	306	1	comput	comput	PROPN
ap-6125	306	2	math	math	PROPN
ap-6125	306	3	appl	appl	PROPN
ap-6125	306	4	78:3067–3076	78:3067–3076	PROPN
ap-6125	306	5	,	,	PUNCT
ap-6125	306	6	2019	2019	NUM
ap-6125	306	7	.	.	PUNCT
ap-6125	307	1	doi:10.1016	doi:10.1016	PROPN
ap-6125	307	2	/	/	SYM
ap-6125	307	3	j.camwa.2019.04.018	j.camwa.2019.04.018	PROPN
ap-6125	307	4	.	.	PUNCT
ap-6125	308	1	[	[	X
ap-6125	308	2	3	3	X
ap-6125	308	3	]	]	PUNCT
ap-6125	308	4	k.	k.	PROPN
ap-6125	308	5	segeth	segeth	PROPN
ap-6125	308	6	.	.	PUNCT
ap-6125	309	1	some	some	DET
ap-6125	309	2	splines	spline	NOUN
ap-6125	309	3	produced	produce	VERB
ap-6125	309	4	by	by	ADP
ap-6125	309	5	smooth	smooth	ADJ
ap-6125	309	6	interpolation	interpolation	NOUN
ap-6125	309	7	.	.	PUNCT
ap-6125	310	1	appl	appl	PROPN
ap-6125	310	2	math	math	PROPN
ap-6125	310	3	comput	comput	PROPN
ap-6125	310	4	319:387–394	319:387–394	NUM
ap-6125	310	5	,	,	PUNCT
ap-6125	310	6	2018	2018	NUM
ap-6125	310	7	.	.	PUNCT
ap-6125	311	1	doi:10.1016	doi:10.1016	PROPN
ap-6125	311	2	/	/	SYM
ap-6125	311	3	j.amc.2017.04.022	j.amc.2017.04.022	PROPN
ap-6125	311	4	.	.	PUNCT
ap-6125	312	1	[	[	X
ap-6125	312	2	4	4	NUM
ap-6125	312	3	]	]	X
ap-6125	312	4	l.	l.	PROPN
ap-6125	312	5	mitáš	mitáš	PROPN
ap-6125	312	6	,	,	PUNCT
ap-6125	312	7	h.	h.	PROPN
ap-6125	312	8	mitášová	mitášová	PROPN
ap-6125	312	9	.	.	PUNCT
ap-6125	313	1	general	general	ADJ
ap-6125	313	2	variational	variational	ADJ
ap-6125	313	3	approach	approach	NOUN
ap-6125	313	4	to	to	ADP
ap-6125	313	5	the	the	DET
ap-6125	313	6	interpolation	interpolation	NOUN
ap-6125	313	7	problem	problem	NOUN
ap-6125	313	8	.	.	PUNCT
ap-6125	314	1	comput	comput	PROPN
ap-6125	314	2	math	math	PROPN
ap-6125	314	3	appl	appl	PROPN
ap-6125	314	4	16:983–992	16:983–992	PROPN
ap-6125	314	5	,	,	PUNCT
ap-6125	314	6	1988	1988	NUM
ap-6125	314	7	.	.	PUNCT
ap-6125	315	1	doi:10.1016/0898	doi:10.1016/0898	PROPN
ap-6125	315	2	-	-	PUNCT
ap-6125	315	3	1221(88)90255	1221(88)90255	NUM
ap-6125	315	4	-	-	SYM
ap-6125	315	5	6	6	NUM
ap-6125	315	6	.	.	PUNCT
ap-6125	316	1	[	[	X
ap-6125	316	2	5	5	X
ap-6125	316	3	]	]	PUNCT
ap-6125	316	4	k.	k.	PROPN
ap-6125	316	5	segeth	segeth	PROPN
ap-6125	316	6	.	.	PUNCT
ap-6125	317	1	a	a	DET
ap-6125	317	2	periodic	periodic	ADJ
ap-6125	317	3	basis	basis	NOUN
ap-6125	317	4	system	system	NOUN
ap-6125	317	5	of	of	ADP
ap-6125	317	6	the	the	DET
ap-6125	317	7	smooth	smooth	ADJ
ap-6125	317	8	approximation	approximation	NOUN
ap-6125	317	9	space	space	NOUN
ap-6125	317	10	.	.	PUNCT
ap-6125	318	1	appl	appl	PROPN
ap-6125	318	2	math	math	PROPN
ap-6125	318	3	comput	comput	PROPN
ap-6125	318	4	267:436–444	267:436–444	NUM
ap-6125	318	5	,	,	PUNCT
ap-6125	318	6	2015	2015	NUM
ap-6125	318	7	.	.	PUNCT
ap-6125	319	1	doi:10.1016	doi:10.1016	PROPN
ap-6125	319	2	/	/	SYM
ap-6125	319	3	j.amc.2015.01.120	j.amc.2015.01.120	PROPN
ap-6125	319	4	.	.	PUNCT
ap-6125	320	1	[	[	X
ap-6125	320	2	6	6	NUM
ap-6125	320	3	]	]	PUNCT
ap-6125	320	4	s.	s.	PROPN
ap-6125	320	5	g.	g.	PROPN
ap-6125	320	6	krĕın	krĕın	PROPN
ap-6125	320	7	(	(	PUNCT
ap-6125	320	8	ed	ed	NOUN
ap-6125	320	9	.	.	PUNCT
ap-6125	320	10	)	)	PUNCT
ap-6125	320	11	.	.	PUNCT
ap-6125	321	1	functional	functional	ADJ
ap-6125	321	2	analysis	analysis	NOUN
ap-6125	321	3	(	(	PUNCT
ap-6125	321	4	russian	russian	NOUN
ap-6125	321	5	)	)	PUNCT
ap-6125	321	6	.	.	PUNCT
ap-6125	322	1	1st	1st	PROPN
ap-6125	322	2	edition	edition	PROPN
ap-6125	322	3	.	.	PUNCT
ap-6125	323	1	nauka	nauka	PROPN
ap-6125	323	2	,	,	PUNCT
ap-6125	323	3	moskva	moskva	PROPN
ap-6125	323	4	,	,	PUNCT
ap-6125	323	5	1964	1964	NUM
ap-6125	323	6	.	.	PUNCT
ap-6125	324	1	154	154	NUM
ap-6125	324	2	http://dx.doi.org/10.1016/0021-9991(77)90115-2	http://dx.doi.org/10.1016/0021-9991(77)90115-2	PROPN
ap-6125	324	3	http://dx.doi.org/10.1016/j.camwa.2019.04.018	http://dx.doi.org/10.1016/j.camwa.2019.04.018	PROPN
ap-6125	324	4	http://dx.doi.org/10.1016/j.amc.2017.04.022	http://dx.doi.org/10.1016/j.amc.2017.04.022	NOUN
ap-6125	324	5	http://dx.doi.org/10.1016/0898-1221(88)90255-6	http://dx.doi.org/10.1016/0898-1221(88)90255-6	NOUN
ap-6125	324	6	http://dx.doi.org/10.1016/j.amc.2015.01.120	http://dx.doi.org/10.1016/j.amc.2015.01.120	PROPN
ap-6125	324	7	acta	acta	PROPN
ap-6125	324	8	polytechnica	polytechnica	PROPN
ap-6125	324	9	61(si):148–154	61(si):148–154	NOUN
ap-6125	324	10	,	,	PUNCT
ap-6125	324	11	2021	2021	NUM
ap-6125	324	12	1	1	NUM
ap-6125	324	13	introduction	introduction	NOUN
ap-6125	324	14	2	2	NUM
ap-6125	324	15	problem	problem	NOUN
ap-6125	324	16	of	of	ADP
ap-6125	324	17	data	datum	NOUN
ap-6125	324	18	interpolation	interpolation	NOUN
ap-6125	324	19	3	3	NUM
ap-6125	324	20	interpolation	interpolation	NOUN
ap-6125	324	21	with	with	ADP
ap-6125	324	22	radial	radial	ADJ
ap-6125	324	23	basis	basis	NOUN
ap-6125	324	24	functions	function	NOUN
ap-6125	324	25	4	4	NUM
ap-6125	324	26	polyharmonic	polyharmonic	ADJ
ap-6125	324	27	splines	spline	NOUN
ap-6125	324	28	5	5	NUM
ap-6125	324	29	smooth	smooth	ADJ
ap-6125	324	30	interpolation	interpolation	NOUN
ap-6125	324	31	6	6	NUM
ap-6125	324	32	a	a	DET
ap-6125	324	33	periodic	periodic	ADJ
ap-6125	324	34	basis	basis	NOUN
ap-6125	324	35	function	function	NOUN
ap-6125	324	36	system	system	NOUN
ap-6125	324	37	of	of	ADP
ap-6125	324	38	wl	wl	PROPN
ap-6125	324	39	7	7	NUM
ap-6125	324	40	polyharmonic	polyharmonic	ADJ
ap-6125	324	41	spline	spline	NOUN
ap-6125	324	42	interpolation	interpolation	NOUN
ap-6125	324	43	8	8	NUM
ap-6125	324	44	some	some	DET
ap-6125	324	45	properties	property	NOUN
ap-6125	324	46	of	of	ADP
ap-6125	324	47	the	the	DET
ap-6125	324	48	polyharmonic	polyharmonic	ADJ
ap-6125	324	49	interpolant	interpolant	NOUN
ap-6125	324	50	9	9	NUM
ap-6125	324	51	example	example	NOUN
ap-6125	324	52	10	10	NUM
ap-6125	324	53	conclusion	conclusion	NOUN
ap-6125	324	54	acknowledgements	acknowledgement	NOUN
ap-6125	324	55	references	reference	NOUN