id	sid	tid	token	lemma	pos
ap-3450	1	1	acta	acta	PROPN
ap-3450	1	2	polytechnica	polytechnica	PROPN
ap-3450	1	3	doi:10.14311	doi:10.14311	PROPN
ap-3450	1	4	/	/	PROPN
ap-3450	1	5	ap.2016.56.0156	ap.2016.56.0156	PROPN
ap-3450	1	6	acta	acta	PROPN
ap-3450	1	7	polytechnica	polytechnica	PROPN
ap-3450	1	8	56(3):156–165	56(3):156–165	PROPN
ap-3450	1	9	,	,	PUNCT
ap-3450	1	10	2016	2016	NUM
ap-3450	1	11	©	©	PROPN
ap-3450	1	12	czech	czech	PROPN
ap-3450	1	13	technical	technical	PROPN
ap-3450	1	14	university	university	PROPN
ap-3450	1	15	in	in	ADP
ap-3450	1	16	prague	prague	PROPN
ap-3450	1	17	,	,	PUNCT
ap-3450	1	18	2016	2016	NUM
ap-3450	1	19	available	available	ADJ
ap-3450	1	20	online	online	ADV
ap-3450	1	21	at	at	ADP
ap-3450	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-3450	1	23	three	three	NUM
ap-3450	1	24	-	-	PUNCT
ap-3450	1	25	variable	variable	NOUN
ap-3450	1	26	alternating	alternate	VERB
ap-3450	1	27	trigonometric	trigonometric	ADJ
ap-3450	1	28	functions	function	NOUN
ap-3450	1	29	and	and	CCONJ
ap-3450	1	30	corresponding	corresponding	ADJ
ap-3450	1	31	fourier	fourier	NOUN
ap-3450	1	32	transforms	transform	VERB
ap-3450	1	33	agata	agata	NOUN
ap-3450	1	34	bezubika	bezubika	NOUN
ap-3450	1	35	,	,	PUNCT
ap-3450	1	36	severin	severin	PROPN
ap-3450	1	37	poštab,∗	poštab,∗	PROPN
ap-3450	1	38	a	a	DET
ap-3450	1	39	institute	institute	NOUN
ap-3450	1	40	of	of	ADP
ap-3450	1	41	mathematics	mathematics	PROPN
ap-3450	1	42	,	,	PUNCT
ap-3450	1	43	university	university	NOUN
ap-3450	1	44	of	of	ADP
ap-3450	1	45	białystok	białystok	PROPN
ap-3450	1	46	,	,	PUNCT
ap-3450	1	47	akademicka	akademicka	PROPN
ap-3450	1	48	2	2	NUM
ap-3450	1	49	,	,	PUNCT
ap-3450	1	50	15	15	NUM
ap-3450	1	51	-	-	SYM
ap-3450	1	52	267	267	NUM
ap-3450	1	53	białystok	białystok	NOUN
ap-3450	1	54	,	,	PUNCT
ap-3450	1	55	poland	poland	PROPN
ap-3450	1	56	b	b	PROPN
ap-3450	1	57	department	department	PROPN
ap-3450	1	58	of	of	ADP
ap-3450	1	59	mathematics	mathematic	NOUN
ap-3450	1	60	,	,	PUNCT
ap-3450	1	61	faculty	faculty	NOUN
ap-3450	1	62	of	of	ADP
ap-3450	1	63	nuclear	nuclear	ADJ
ap-3450	1	64	sciences	science	NOUN
ap-3450	1	65	and	and	CCONJ
ap-3450	1	66	physical	physical	ADJ
ap-3450	1	67	engineering	engineering	NOUN
ap-3450	1	68	,	,	PUNCT
ap-3450	1	69	czech	czech	PROPN
ap-3450	1	70	technical	technical	PROPN
ap-3450	1	71	university	university	PROPN
ap-3450	1	72	in	in	ADP
ap-3450	1	73	prague	prague	PROPN
ap-3450	1	74	,	,	PUNCT
ap-3450	1	75	trojanova	trojanova	X
ap-3450	1	76	13	13	NUM
ap-3450	1	77	,	,	PUNCT
ap-3450	1	78	120	120	NUM
ap-3450	1	79	00	00	NUM
ap-3450	1	80	prague	prague	PROPN
ap-3450	1	81	,	,	PUNCT
ap-3450	1	82	czech	czech	PROPN
ap-3450	1	83	republic	republic	NOUN
ap-3450	1	84	∗	∗	NOUN
ap-3450	1	85	corresponding	correspond	VERB
ap-3450	1	86	author	author	NOUN
ap-3450	1	87	:	:	PUNCT
ap-3450	1	88	severin.posta@fjfi.cvut.cz	severin.posta@fjfi.cvut.cz	NOUN
ap-3450	1	89	abstract	abstract	NOUN
ap-3450	1	90	.	.	PUNCT
ap-3450	2	1	the	the	DET
ap-3450	2	2	common	common	ADJ
ap-3450	2	3	trigonometric	trigonometric	ADJ
ap-3450	2	4	functions	function	NOUN
ap-3450	2	5	admit	admit	VERB
ap-3450	2	6	generalizations	generalization	NOUN
ap-3450	2	7	to	to	ADP
ap-3450	2	8	any	any	DET
ap-3450	2	9	higher	high	ADJ
ap-3450	2	10	dimension	dimension	NOUN
ap-3450	2	11	.	.	PUNCT
ap-3450	3	1	in	in	ADP
ap-3450	3	2	this	this	DET
ap-3450	3	3	paper	paper	NOUN
ap-3450	3	4	,	,	PUNCT
ap-3450	3	5	we	we	PRON
ap-3450	3	6	restrict	restrict	VERB
ap-3450	3	7	ourselves	ourselves	PRON
ap-3450	3	8	to	to	ADP
ap-3450	3	9	three	three	NUM
ap-3450	3	10	dimensional	dimensional	ADJ
ap-3450	3	11	generalization	generalization	NOUN
ap-3450	3	12	only	only	ADV
ap-3450	3	13	,	,	PUNCT
ap-3450	3	14	focusing	focus	VERB
ap-3450	3	15	on	on	ADP
ap-3450	3	16	alternating	alternate	VERB
ap-3450	3	17	case	case	NOUN
ap-3450	3	18	in	in	ADP
ap-3450	3	19	detail	detail	NOUN
ap-3450	3	20	.	.	PUNCT
ap-3450	4	1	many	many	ADJ
ap-3450	4	2	specific	specific	ADJ
ap-3450	4	3	properties	property	NOUN
ap-3450	4	4	of	of	ADP
ap-3450	4	5	this	this	DET
ap-3450	4	6	new	new	ADJ
ap-3450	4	7	class	class	NOUN
ap-3450	4	8	of	of	ADP
ap-3450	4	9	special	special	ADJ
ap-3450	4	10	functions	function	NOUN
ap-3450	4	11	are	be	AUX
ap-3450	4	12	studied	study	VERB
ap-3450	4	13	,	,	PUNCT
ap-3450	4	14	such	such	ADJ
ap-3450	4	15	as	as	ADP
ap-3450	4	16	the	the	DET
ap-3450	4	17	orthogonalities	orthogonality	NOUN
ap-3450	4	18	,	,	PUNCT
ap-3450	4	19	both	both	CCONJ
ap-3450	4	20	the	the	DET
ap-3450	4	21	continuous	continuous	ADJ
ap-3450	4	22	one	one	NOUN
ap-3450	4	23	and	and	CCONJ
ap-3450	4	24	the	the	DET
ap-3450	4	25	discrete	discrete	ADJ
ap-3450	4	26	one	one	NUM
ap-3450	4	27	on	on	ADP
ap-3450	4	28	the	the	DET
ap-3450	4	29	3d	3d	PROPN
ap-3450	4	30	lattice	lattice	NOUN
ap-3450	4	31	of	of	ADP
ap-3450	4	32	any	any	DET
ap-3450	4	33	density	density	NOUN
ap-3450	4	34	,	,	PUNCT
ap-3450	4	35	discrete	discrete	ADJ
ap-3450	4	36	and	and	CCONJ
ap-3450	4	37	continuous	continuous	ADJ
ap-3450	4	38	fourier	fourier	NOUN
ap-3450	4	39	transforms	transform	VERB
ap-3450	4	40	,	,	PUNCT
ap-3450	4	41	and	and	CCONJ
ap-3450	4	42	others	other	NOUN
ap-3450	4	43	.	.	PUNCT
ap-3450	5	1	rapidly	rapidly	ADV
ap-3450	5	2	increasing	increase	VERB
ap-3450	5	3	precision	precision	NOUN
ap-3450	5	4	of	of	ADP
ap-3450	5	5	the	the	DET
ap-3450	5	6	interpolation	interpolation	NOUN
ap-3450	5	7	with	with	ADP
ap-3450	5	8	increasing	increase	VERB
ap-3450	5	9	density	density	NOUN
ap-3450	5	10	of	of	ADP
ap-3450	5	11	the	the	DET
ap-3450	5	12	3d	3d	PROPN
ap-3450	5	13	lattice	lattice	NOUN
ap-3450	5	14	is	be	AUX
ap-3450	5	15	shown	show	VERB
ap-3450	5	16	in	in	ADP
ap-3450	5	17	an	an	DET
ap-3450	5	18	example	example	NOUN
ap-3450	5	19	.	.	PUNCT
ap-3450	6	1	keywords	keyword	NOUN
ap-3450	6	2	:	:	PUNCT
ap-3450	6	3	trigonometric	trigonometric	ADJ
ap-3450	6	4	functions	function	NOUN
ap-3450	6	5	;	;	PUNCT
ap-3450	6	6	orthogonal	orthogonal	ADJ
ap-3450	6	7	polynomials	polynomial	NOUN
ap-3450	6	8	.	.	PUNCT
ap-3450	7	1	1	1	X
ap-3450	7	2	.	.	X
ap-3450	7	3	introduction	introduction	NOUN
ap-3450	7	4	there	there	PRON
ap-3450	7	5	are	be	VERB
ap-3450	7	6	many	many	ADJ
ap-3450	7	7	applications	application	NOUN
ap-3450	7	8	of	of	ADP
ap-3450	7	9	functions	function	NOUN
ap-3450	7	10	which	which	PRON
ap-3450	7	11	are	be	AUX
ap-3450	7	12	symmetric	symmetric	ADJ
ap-3450	7	13	or	or	CCONJ
ap-3450	7	14	antisymmetric	antisymmetric	VERB
ap-3450	7	15	with	with	ADP
ap-3450	7	16	respect	respect	NOUN
ap-3450	7	17	to	to	ADP
ap-3450	7	18	the	the	DET
ap-3450	7	19	symmetry	symmetry	NOUN
ap-3450	7	20	group	group	PROPN
ap-3450	7	21	sn	sn	PROPN
ap-3450	7	22	in	in	ADP
ap-3450	7	23	mathematics	mathematic	NOUN
ap-3450	7	24	.	.	PUNCT
ap-3450	8	1	they	they	PRON
ap-3450	8	2	appear	appear	VERB
ap-3450	8	3	for	for	ADP
ap-3450	8	4	example	example	NOUN
ap-3450	8	5	in	in	ADP
ap-3450	8	6	quantum	quantum	ADJ
ap-3450	8	7	theory	theory	NOUN
ap-3450	8	8	or	or	CCONJ
ap-3450	8	9	in	in	ADP
ap-3450	8	10	theory	theory	NOUN
ap-3450	8	11	of	of	ADP
ap-3450	8	12	integrable	integrable	ADJ
ap-3450	8	13	systems	system	NOUN
ap-3450	8	14	.	.	PUNCT
ap-3450	9	1	in	in	ADP
ap-3450	9	2	[	[	X
ap-3450	9	3	1	1	NUM
ap-3450	9	4	]	]	PUNCT
ap-3450	9	5	(	(	PUNCT
ap-3450	9	6	anti)symmetric	anti)symmetric	ADJ
ap-3450	9	7	multivariate	multivariate	NOUN
ap-3450	9	8	exponential	exponential	NOUN
ap-3450	9	9	functions	function	NOUN
ap-3450	9	10	were	be	AUX
ap-3450	9	11	defined	define	VERB
ap-3450	9	12	and	and	CCONJ
ap-3450	9	13	corresponding	corresponding	ADJ
ap-3450	9	14	fourier	fourier	NOUN
ap-3450	9	15	transforms	transform	NOUN
ap-3450	9	16	were	be	AUX
ap-3450	9	17	given	give	VERB
ap-3450	9	18	.	.	PUNCT
ap-3450	10	1	these	these	DET
ap-3450	10	2	functions	function	NOUN
ap-3450	10	3	were	be	AUX
ap-3450	10	4	examined	examine	VERB
ap-3450	10	5	in	in	ADP
ap-3450	10	6	detail	detail	NOUN
ap-3450	10	7	in	in	ADP
ap-3450	10	8	dimension	dimension	NOUN
ap-3450	10	9	two	two	NUM
ap-3450	10	10	and	and	CCONJ
ap-3450	10	11	three	three	NUM
ap-3450	10	12	in	in	ADP
ap-3450	10	13	[	[	X
ap-3450	10	14	2	2	NUM
ap-3450	10	15	]	]	PUNCT
ap-3450	10	16	.	.	PUNCT
ap-3450	11	1	the	the	DET
ap-3450	11	2	natural	natural	ADJ
ap-3450	11	3	question	question	NOUN
ap-3450	11	4	which	which	PRON
ap-3450	11	5	arises	arise	VERB
ap-3450	11	6	is	be	AUX
ap-3450	11	7	the	the	DET
ap-3450	11	8	restriction	restriction	NOUN
ap-3450	11	9	of	of	ADP
ap-3450	11	10	the	the	DET
ap-3450	11	11	symmetry	symmetry	NOUN
ap-3450	11	12	to	to	ADP
ap-3450	11	13	the	the	DET
ap-3450	11	14	subgroup	subgroup	NOUN
ap-3450	11	15	an	an	PRON
ap-3450	11	16	of	of	ADP
ap-3450	11	17	sn	sn	NOUN
ap-3450	11	18	consisting	consist	VERB
ap-3450	11	19	of	of	ADP
ap-3450	11	20	transformations	transformation	NOUN
ap-3450	11	21	w	w	NOUN
ap-3450	11	22	with	with	ADP
ap-3450	11	23	detw	detw	NOUN
ap-3450	11	24	=	=	SYM
ap-3450	11	25	1	1	X
ap-3450	11	26	.	.	PUNCT
ap-3450	11	27	such	such	ADJ
ap-3450	11	28	functions	function	NOUN
ap-3450	11	29	were	be	AUX
ap-3450	11	30	examined	examine	VERB
ap-3450	11	31	in	in	ADP
ap-3450	11	32	[	[	X
ap-3450	11	33	3	3	NUM
ap-3450	11	34	]	]	PUNCT
ap-3450	11	35	for	for	ADP
ap-3450	11	36	an	an	DET
ap-3450	11	37	involving	involve	VERB
ap-3450	11	38	the	the	DET
ap-3450	11	39	general	general	ADJ
ap-3450	11	40	number	number	NOUN
ap-3450	11	41	n	n	CCONJ
ap-3450	11	42	,	,	PUNCT
ap-3450	11	43	and	and	CCONJ
ap-3450	11	44	in	in	ADP
ap-3450	11	45	[	[	X
ap-3450	11	46	4	4	X
ap-3450	11	47	]	]	PUNCT
ap-3450	11	48	in	in	ADP
ap-3450	11	49	greater	great	ADJ
ap-3450	11	50	detail	detail	NOUN
ap-3450	11	51	concerning	concern	VERB
ap-3450	11	52	the	the	DET
ap-3450	11	53	smallest	small	ADJ
ap-3450	11	54	possible	possible	ADJ
ap-3450	11	55	n	n	CCONJ
ap-3450	11	56	,	,	PUNCT
ap-3450	11	57	namely	namely	ADV
ap-3450	11	58	n	n	NOUN
ap-3450	11	59	=	=	SYM
ap-3450	11	60	3	3	X
ap-3450	11	61	.	.	PUNCT
ap-3450	12	1	the	the	DET
ap-3450	12	2	exponential	exponential	ADJ
ap-3450	12	3	functions	function	NOUN
ap-3450	12	4	are	be	AUX
ap-3450	12	5	not	not	PART
ap-3450	12	6	the	the	DET
ap-3450	12	7	only	only	ADJ
ap-3450	12	8	functions	function	NOUN
ap-3450	12	9	which	which	PRON
ap-3450	12	10	admit	admit	VERB
ap-3450	12	11	symmetric	symmetric	ADJ
ap-3450	12	12	,	,	PUNCT
ap-3450	12	13	antisymmetric	antisymmetric	ADJ
ap-3450	12	14	and	and	CCONJ
ap-3450	12	15	alternating	alternate	VERB
ap-3450	12	16	generalizations	generalization	NOUN
ap-3450	12	17	to	to	ADP
ap-3450	12	18	higher	high	ADJ
ap-3450	12	19	dimensions	dimension	NOUN
ap-3450	12	20	.	.	PUNCT
ap-3450	13	1	their	their	PRON
ap-3450	13	2	odd	odd	ADJ
ap-3450	13	3	and	and	CCONJ
ap-3450	13	4	even	even	ADV
ap-3450	13	5	counterparts	counterpart	NOUN
ap-3450	13	6	,	,	PUNCT
ap-3450	13	7	sines	sine	NOUN
ap-3450	13	8	and	and	CCONJ
ap-3450	13	9	cosines	cosine	NOUN
ap-3450	13	10	,	,	PUNCT
ap-3450	13	11	admit	admit	VERB
ap-3450	13	12	similar	similar	ADJ
ap-3450	13	13	generalizations	generalization	NOUN
ap-3450	13	14	and	and	CCONJ
ap-3450	13	15	fulfil	fulfil	VERB
ap-3450	13	16	many	many	ADJ
ap-3450	13	17	analogical	analogical	ADJ
ap-3450	13	18	properties	property	NOUN
ap-3450	13	19	as	as	ADP
ap-3450	13	20	pure	pure	ADJ
ap-3450	13	21	exponentials	exponential	NOUN
ap-3450	13	22	.	.	PUNCT
ap-3450	14	1	symmetric	symmetric	ADJ
ap-3450	14	2	and	and	CCONJ
ap-3450	14	3	antisymmetric	antisymmetric	ADJ
ap-3450	14	4	case	case	NOUN
ap-3450	14	5	in	in	ADP
ap-3450	14	6	general	general	ADJ
ap-3450	14	7	was	be	AUX
ap-3450	14	8	covered	cover	VERB
ap-3450	14	9	in	in	ADP
ap-3450	14	10	[	[	X
ap-3450	14	11	5	5	NUM
ap-3450	14	12	]	]	PUNCT
ap-3450	14	13	,	,	PUNCT
ap-3450	14	14	alternating	alternate	VERB
ap-3450	14	15	case	case	NOUN
ap-3450	14	16	was	be	AUX
ap-3450	14	17	covered	cover	VERB
ap-3450	14	18	in	in	ADP
ap-3450	14	19	[	[	X
ap-3450	14	20	6	6	NUM
ap-3450	14	21	]	]	PUNCT
ap-3450	14	22	.	.	PUNCT
ap-3450	15	1	two	two	NUM
ap-3450	15	2	-	-	PUNCT
ap-3450	15	3	dimensional	dimensional	ADJ
ap-3450	15	4	symmetric	symmetric	ADJ
ap-3450	15	5	and	and	CCONJ
ap-3450	15	6	antisymmetric	antisymmetric	ADJ
ap-3450	15	7	generalizations	generalization	NOUN
ap-3450	15	8	were	be	AUX
ap-3450	15	9	studied	study	VERB
ap-3450	15	10	in	in	ADP
ap-3450	15	11	detail	detail	NOUN
ap-3450	15	12	in	in	ADP
ap-3450	15	13	[	[	X
ap-3450	15	14	7	7	NUM
ap-3450	15	15	,	,	PUNCT
ap-3450	15	16	8	8	NUM
ap-3450	15	17	]	]	PUNCT
ap-3450	15	18	.	.	PUNCT
ap-3450	16	1	the	the	DET
ap-3450	16	2	aim	aim	NOUN
ap-3450	16	3	of	of	ADP
ap-3450	16	4	this	this	DET
ap-3450	16	5	paper	paper	NOUN
ap-3450	16	6	is	be	AUX
ap-3450	16	7	to	to	PART
ap-3450	16	8	study	study	VERB
ap-3450	16	9	in	in	ADP
ap-3450	16	10	detail	detail	NOUN
ap-3450	16	11	the	the	DET
ap-3450	16	12	alternating	alternate	VERB
ap-3450	16	13	generalization	generalization	NOUN
ap-3450	16	14	of	of	ADP
ap-3450	16	15	sine	sine	NOUN
ap-3450	16	16	and	and	CCONJ
ap-3450	16	17	cosine	cosine	NOUN
ap-3450	16	18	functions	function	NOUN
ap-3450	16	19	in	in	ADP
ap-3450	16	20	dimension	dimension	NOUN
ap-3450	16	21	three	three	NUM
ap-3450	16	22	.	.	PUNCT
ap-3450	17	1	(	(	PUNCT
ap-3450	17	2	anti)symmetric	anti)symmetric	ADJ
ap-3450	17	3	multivariate	multivariate	NOUN
ap-3450	17	4	sine	sine	NOUN
ap-3450	17	5	and	and	CCONJ
ap-3450	17	6	cosine	cosine	NOUN
ap-3450	17	7	functions	function	NOUN
ap-3450	17	8	,	,	PUNCT
ap-3450	17	9	considered	consider	VERB
ap-3450	17	10	in	in	ADP
ap-3450	17	11	[	[	X
ap-3450	17	12	5	5	NUM
ap-3450	17	13	,	,	PUNCT
ap-3450	17	14	7	7	NUM
ap-3450	17	15	,	,	PUNCT
ap-3450	17	16	8	8	NUM
ap-3450	17	17	]	]	PUNCT
ap-3450	17	18	,	,	PUNCT
ap-3450	17	19	as	as	ADV
ap-3450	17	20	well	well	ADV
ap-3450	17	21	as	as	ADP
ap-3450	17	22	alternating	alternate	VERB
ap-3450	17	23	multivariate	multivariate	NOUN
ap-3450	17	24	sine	sine	NOUN
ap-3450	17	25	and	and	CCONJ
ap-3450	17	26	cosine	cosine	NOUN
ap-3450	17	27	functions	function	NOUN
ap-3450	17	28	,	,	PUNCT
ap-3450	17	29	studied	study	VERB
ap-3450	17	30	in	in	ADP
ap-3450	17	31	[	[	X
ap-3450	17	32	6	6	NUM
ap-3450	17	33	]	]	PUNCT
ap-3450	17	34	and	and	CCONJ
ap-3450	17	35	in	in	ADP
ap-3450	17	36	this	this	DET
ap-3450	17	37	paper	paper	NOUN
ap-3450	17	38	,	,	PUNCT
ap-3450	17	39	are	be	AUX
ap-3450	17	40	closely	closely	ADV
ap-3450	17	41	related	relate	VERB
ap-3450	17	42	to	to	ADP
ap-3450	17	43	symmetric	symmetric	ADJ
ap-3450	17	44	and	and	CCONJ
ap-3450	17	45	antisymmetric	antisymmetric	ADJ
ap-3450	17	46	orbit	orbit	NOUN
ap-3450	17	47	functions	function	NOUN
ap-3450	17	48	studied	study	VERB
ap-3450	17	49	in	in	ADP
ap-3450	17	50	[	[	X
ap-3450	17	51	9–11	9–11	NOUN
ap-3450	17	52	]	]	PUNCT
ap-3450	17	53	.	.	PUNCT
ap-3450	18	1	because	because	SCONJ
ap-3450	18	2	the	the	DET
ap-3450	18	3	definition	definition	NOUN
ap-3450	18	4	of	of	ADP
ap-3450	18	5	the	the	DET
ap-3450	18	6	considered	consider	VERB
ap-3450	18	7	functions	function	NOUN
ap-3450	18	8	and	and	CCONJ
ap-3450	18	9	orbit	orbit	NOUN
ap-3450	18	10	functions	function	NOUN
ap-3450	18	11	are	be	AUX
ap-3450	18	12	similar	similar	ADJ
ap-3450	18	13	(	(	PUNCT
ap-3450	18	14	roughly	roughly	ADV
ap-3450	18	15	speaking	speak	VERB
ap-3450	18	16	,	,	PUNCT
ap-3450	18	17	the	the	DET
ap-3450	18	18	exponentials	exponential	NOUN
ap-3450	18	19	are	be	AUX
ap-3450	18	20	replaced	replace	VERB
ap-3450	18	21	by	by	ADP
ap-3450	18	22	sines	sine	NOUN
ap-3450	18	23	and	and	CCONJ
ap-3450	18	24	cosines	cosine	NOUN
ap-3450	18	25	)	)	PUNCT
ap-3450	18	26	,	,	PUNCT
ap-3450	18	27	they	they	PRON
ap-3450	18	28	satisfy	satisfy	VERB
ap-3450	18	29	the	the	DET
ap-3450	18	30	same	same	ADJ
ap-3450	18	31	or	or	CCONJ
ap-3450	18	32	similar	similar	ADJ
ap-3450	18	33	properties	property	NOUN
ap-3450	18	34	,	,	PUNCT
ap-3450	18	35	namely	namely	ADV
ap-3450	18	36	symmetry	symmetry	NOUN
ap-3450	18	37	relations	relation	NOUN
ap-3450	18	38	,	,	PUNCT
ap-3450	18	39	reductions	reduction	NOUN
ap-3450	18	40	to	to	ADP
ap-3450	18	41	the	the	DET
ap-3450	18	42	dominant	dominant	ADJ
ap-3450	18	43	or	or	CCONJ
ap-3450	18	44	semidominant	semidominant	ADJ
ap-3450	18	45	forms	form	NOUN
ap-3450	18	46	,	,	PUNCT
ap-3450	18	47	periodicity	periodicity	NOUN
ap-3450	18	48	etc	etc	X
ap-3450	18	49	.	.	PUNCT
ap-3450	19	1	the	the	DET
ap-3450	19	2	discrete	discrete	ADJ
ap-3450	19	3	fourier	fourier	NOUN
ap-3450	19	4	transforms	transform	VERB
ap-3450	19	5	of	of	ADP
ap-3450	19	6	sine	sine	NOUN
ap-3450	19	7	and	and	CCONJ
ap-3450	19	8	cosine	cosine	NOUN
ap-3450	19	9	functions	function	NOUN
ap-3450	19	10	can	can	AUX
ap-3450	19	11	be	be	AUX
ap-3450	19	12	derived	derive	VERB
ap-3450	19	13	with	with	ADP
ap-3450	19	14	the	the	DET
ap-3450	19	15	help	help	NOUN
ap-3450	19	16	of	of	ADP
ap-3450	19	17	relations	relation	NOUN
ap-3450	19	18	valid	valid	ADJ
ap-3450	19	19	for	for	ADP
ap-3450	19	20	exponential	exponential	ADJ
ap-3450	19	21	functions.[8	functions.[8	NOUN
ap-3450	19	22	]	]	PUNCT
ap-3450	19	23	the	the	DET
ap-3450	19	24	restriction	restriction	NOUN
ap-3450	19	25	of	of	ADP
ap-3450	19	26	the	the	DET
ap-3450	19	27	functions	function	NOUN
ap-3450	19	28	to	to	ADP
ap-3450	19	29	three	three	NUM
ap-3450	19	30	variables	variable	NOUN
ap-3450	19	31	allows	allow	VERB
ap-3450	19	32	us	we	PRON
ap-3450	19	33	to	to	PART
ap-3450	19	34	be	be	AUX
ap-3450	19	35	more	more	ADV
ap-3450	19	36	specific	specific	ADJ
ap-3450	19	37	about	about	ADP
ap-3450	19	38	the	the	DET
ap-3450	19	39	details	detail	NOUN
ap-3450	19	40	of	of	ADP
ap-3450	19	41	their	their	PRON
ap-3450	19	42	properties	property	NOUN
ap-3450	19	43	in	in	ADP
ap-3450	19	44	the	the	DET
ap-3450	19	45	following	follow	VERB
ap-3450	19	46	sections	section	NOUN
ap-3450	19	47	.	.	PUNCT
ap-3450	20	1	this	this	PRON
ap-3450	20	2	is	be	AUX
ap-3450	20	3	most	most	ADV
ap-3450	20	4	notable	notable	ADJ
ap-3450	20	5	when	when	SCONJ
ap-3450	20	6	describing	describe	VERB
ap-3450	20	7	their	their	PRON
ap-3450	20	8	discretization	discretization	NOUN
ap-3450	20	9	and	and	CCONJ
ap-3450	20	10	orthogonality	orthogonality	NOUN
ap-3450	20	11	relations	relation	NOUN
ap-3450	20	12	.	.	PUNCT
ap-3450	21	1	the	the	DET
ap-3450	21	2	paper	paper	NOUN
ap-3450	21	3	consists	consist	VERB
ap-3450	21	4	of	of	ADP
ap-3450	21	5	several	several	ADJ
ap-3450	21	6	parts	part	NOUN
ap-3450	21	7	.	.	PUNCT
ap-3450	22	1	after	after	ADP
ap-3450	22	2	the	the	DET
ap-3450	22	3	first	first	ADJ
ap-3450	22	4	part	part	NOUN
ap-3450	22	5	,	,	PUNCT
ap-3450	22	6	devoted	devote	VERB
ap-3450	22	7	to	to	ADP
ap-3450	22	8	the	the	DET
ap-3450	22	9	definition	definition	NOUN
ap-3450	22	10	of	of	ADP
ap-3450	22	11	three	three	NUM
ap-3450	22	12	dimensional	dimensional	ADJ
ap-3450	22	13	alternating	alternate	VERB
ap-3450	22	14	sine	sine	NOUN
ap-3450	22	15	and	and	CCONJ
ap-3450	22	16	cosine	cosine	NOUN
ap-3450	22	17	functions	function	NOUN
ap-3450	22	18	and	and	CCONJ
ap-3450	22	19	their	their	PRON
ap-3450	22	20	basic	basic	ADJ
ap-3450	22	21	properties	property	NOUN
ap-3450	22	22	,	,	PUNCT
ap-3450	22	23	come	come	VERB
ap-3450	22	24	the	the	DET
ap-3450	22	25	parts	part	NOUN
ap-3450	22	26	describing	describe	VERB
ap-3450	22	27	their	their	PRON
ap-3450	22	28	continuous	continuous	ADJ
ap-3450	22	29	orthogonality	orthogonality	NOUN
ap-3450	22	30	relations	relation	NOUN
ap-3450	22	31	,	,	PUNCT
ap-3450	22	32	their	their	PRON
ap-3450	22	33	decomposition	decomposition	NOUN
ap-3450	22	34	rules	rule	NOUN
ap-3450	22	35	and	and	CCONJ
ap-3450	22	36	the	the	DET
ap-3450	22	37	two	two	NUM
ap-3450	22	38	parts	part	NOUN
ap-3450	22	39	treating	treat	VERB
ap-3450	22	40	four	four	NUM
ap-3450	22	41	kinds	kind	NOUN
ap-3450	22	42	of	of	ADP
ap-3450	22	43	discrete	discrete	ADJ
ap-3450	22	44	cosine	cosine	NOUN
ap-3450	22	45	and	and	CCONJ
ap-3450	22	46	sine	sine	NOUN
ap-3450	22	47	transforms	transform	VERB
ap-3450	22	48	.	.	PUNCT
ap-3450	23	1	2	2	X
ap-3450	23	2	.	.	X
ap-3450	23	3	definition	definition	NOUN
ap-3450	23	4	three	three	NUM
ap-3450	23	5	dimensional	dimensional	ADJ
ap-3450	23	6	alternating	alternate	VERB
ap-3450	23	7	sine	sine	NOUN
ap-3450	23	8	functions	function	NOUN
ap-3450	23	9	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	23	10	)	)	PUNCT
ap-3450	23	11	:	:	PUNCT
ap-3450	23	12	r3	r3	PROPN
ap-3450	23	13	→	→	SYM
ap-3450	23	14	r	r	NOUN
ap-3450	23	15	have	have	VERB
ap-3450	23	16	the	the	DET
ap-3450	23	17	following	follow	VERB
ap-3450	23	18	explicit	explicit	ADJ
ap-3450	23	19	form	form	NOUN
ap-3450	23	20	[	[	X
ap-3450	23	21	6	6	NUM
ap-3450	23	22	]	]	X
ap-3450	23	23	:	:	PUNCT
ap-3450	23	24	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	PROPN
ap-3450	23	25	,	,	PUNCT
ap-3450	23	26	y	y	PROPN
ap-3450	23	27	,	,	PUNCT
ap-3450	23	28	z	z	NOUN
ap-3450	23	29	)	)	PUNCT
ap-3450	23	30	=	=	SYM
ap-3450	23	31	1	1	NUM
ap-3450	23	32	2	2	NUM
ap-3450	23	33	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-3450	23	34	sin	sin	NOUN
ap-3450	23	35	2πλx	2πλx	NUM
ap-3450	23	36	sin	sin	NOUN
ap-3450	23	37	2πλy	2πλy	PROPN
ap-3450	23	38	sin	sin	NOUN
ap-3450	23	39	2πλz	2πλz	NUM
ap-3450	23	40	sin	sin	NOUN
ap-3450	23	41	2πµx	2πµx	NUM
ap-3450	23	42	sin	sin	NOUN
ap-3450	23	43	2πµy	2πµy	PROPN
ap-3450	23	44	sin	sin	NOUN
ap-3450	23	45	2πµz	2πµz	NUM
ap-3450	23	46	sin	sin	NOUN
ap-3450	23	47	2πνx	2πνx	NUM
ap-3450	23	48	sin	sin	NOUN
ap-3450	24	1	2πνy	2πνy	NUM
ap-3450	24	2	sin	sin	NOUN
ap-3450	24	3	2πνz	2πνz	NUM
ap-3450	24	4	∣∣∣∣∣∣+	∣∣∣∣∣∣+	PROPN
ap-3450	24	5	1	1	NUM
ap-3450	24	6	2	2	NUM
ap-3450	24	7	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-3450	24	8	sin	sin	NOUN
ap-3450	24	9	2πλx	2πλx	NUM
ap-3450	24	10	sin	sin	NOUN
ap-3450	24	11	2πλy	2πλy	PROPN
ap-3450	24	12	sin	sin	NOUN
ap-3450	24	13	2πλz	2πλz	NUM
ap-3450	24	14	sin	sin	NOUN
ap-3450	24	15	2πµx	2πµx	NUM
ap-3450	24	16	sin	sin	NOUN
ap-3450	24	17	2πµy	2πµy	PROPN
ap-3450	24	18	sin	sin	NOUN
ap-3450	24	19	2πµz	2πµz	NUM
ap-3450	24	20	sin	sin	NOUN
ap-3450	24	21	2πνx	2πνx	NUM
ap-3450	24	22	sin	sin	NOUN
ap-3450	24	23	2πνy	2πνy	NUM
ap-3450	24	24	sin	sin	NOUN
ap-3450	24	25	2πνz	2πνz	NUM
ap-3450	24	26	∣∣∣∣∣∣	∣∣∣∣∣∣	ADJ
ap-3450	24	27	+	+	NOUN
ap-3450	24	28	=	=	PUNCT
ap-3450	24	29	sin	sin	NOUN
ap-3450	24	30	2πλx	2πλx	NUM
ap-3450	24	31	sin	sin	NOUN
ap-3450	24	32	2πµy	2πµy	PROPN
ap-3450	24	33	sin	sin	VERB
ap-3450	24	34	2πνz	2πνz	NUM
ap-3450	24	35	+	+	CCONJ
ap-3450	24	36	sin	sin	NOUN
ap-3450	24	37	2πλz	2πλz	NUM
ap-3450	24	38	sin	sin	NOUN
ap-3450	24	39	2πµx	2πµx	NUM
ap-3450	24	40	sin	sin	NOUN
ap-3450	24	41	2πνy	2πνy	NOUN
ap-3450	24	42	+	+	CCONJ
ap-3450	24	43	sin	sin	NOUN
ap-3450	24	44	2πλy	2πλy	PROPN
ap-3450	24	45	sin	sin	NOUN
ap-3450	24	46	2πµz	2πµz	NUM
ap-3450	24	47	sin	sin	NOUN
ap-3450	24	48	2πνx	2πνx	NUM
ap-3450	24	49	,	,	PUNCT
ap-3450	24	50	x	x	X
ap-3450	24	51	,	,	PUNCT
ap-3450	24	52	y	y	PROPN
ap-3450	24	53	,	,	PUNCT
ap-3450	24	54	z	z	PROPN
ap-3450	24	55	,	,	PUNCT
ap-3450	24	56	λ	λ	PROPN
ap-3450	24	57	,	,	PUNCT
ap-3450	24	58	µ	µ	NOUN
ap-3450	24	59	,	,	PUNCT
ap-3450	24	60	ν	ν	X
ap-3450	24	61	∈	∈	PROPN
ap-3450	24	62	r	r	NOUN
ap-3450	24	63	,	,	PUNCT
ap-3450	24	64	(	(	PUNCT
ap-3450	24	65	1	1	X
ap-3450	24	66	)	)	PUNCT
ap-3450	24	67	156	156	NUM
ap-3450	24	68	http://dx.doi.org/10.14311/ap.2016.56.0156	http://dx.doi.org/10.14311/ap.2016.56.0156	NOUN
ap-3450	24	69	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-3450	24	70	vol	vol	NOUN
ap-3450	24	71	.	.	PUNCT
ap-3450	25	1	56	56	NUM
ap-3450	25	2	no	no	NOUN
ap-3450	25	3	.	.	PUNCT
ap-3450	26	1	3/2016	3/2016	NUM
ap-3450	26	2	three	three	NUM
ap-3450	26	3	-	-	PUNCT
ap-3450	26	4	variable	variable	NOUN
ap-3450	26	5	alternating	alternate	VERB
ap-3450	26	6	trigonometric	trigonometric	ADJ
ap-3450	26	7	functions	function	NOUN
ap-3450	26	8	where	where	SCONJ
ap-3450	26	9	the	the	DET
ap-3450	26	10	second	second	ADJ
ap-3450	26	11	determinant	determinant	ADJ
ap-3450	26	12	with	with	ADP
ap-3450	26	13	superscipt	superscipt	NOUN
ap-3450	26	14	+	+	CCONJ
ap-3450	26	15	stands	stand	VERB
ap-3450	26	16	for	for	ADP
ap-3450	26	17	permanent	permanent	ADJ
ap-3450	26	18	[	[	X
ap-3450	26	19	12	12	NUM
ap-3450	26	20	]	]	PUNCT
ap-3450	26	21	,	,	PUNCT
ap-3450	26	22	which	which	PRON
ap-3450	26	23	is	be	AUX
ap-3450	26	24	symmetric	symmetric	ADJ
ap-3450	26	25	with	with	ADP
ap-3450	26	26	respect	respect	NOUN
ap-3450	26	27	to	to	ADP
ap-3450	26	28	permutations	permutation	NOUN
ap-3450	26	29	of	of	ADP
ap-3450	26	30	its	its	PRON
ap-3450	26	31	rows	row	NOUN
ap-3450	26	32	and	and	CCONJ
ap-3450	26	33	columns	column	NOUN
ap-3450	26	34	.	.	PUNCT
ap-3450	27	1	three	three	NUM
ap-3450	27	2	dimensional	dimensional	ADJ
ap-3450	27	3	alternating	alternate	VERB
ap-3450	27	4	cosine	cosine	NOUN
ap-3450	27	5	functions	function	NOUN
ap-3450	27	6	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	27	7	)	)	PUNCT
ap-3450	27	8	:	:	PUNCT
ap-3450	27	9	r3	r3	PROPN
ap-3450	27	10	→	→	SYM
ap-3450	27	11	r	r	NOUN
ap-3450	27	12	are	be	AUX
ap-3450	27	13	defined	define	VERB
ap-3450	27	14	similarly	similarly	ADV
ap-3450	27	15	:	:	PUNCT
ap-3450	27	16	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	27	17	,	,	PUNCT
ap-3450	27	18	y	y	PROPN
ap-3450	27	19	,	,	PUNCT
ap-3450	27	20	z	z	NOUN
ap-3450	27	21	)	)	PUNCT
ap-3450	27	22	=	=	SYM
ap-3450	27	23	1	1	NUM
ap-3450	27	24	2	2	NUM
ap-3450	27	25	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-3450	27	26	cos	cos	PROPN
ap-3450	27	27	2πλx	2πλx	PROPN
ap-3450	27	28	cos	cos	ADP
ap-3450	27	29	2πλy	2πλy	PROPN
ap-3450	27	30	cos	cos	ADP
ap-3450	27	31	2πλz	2πλz	NUM
ap-3450	27	32	cos	cos	PROPN
ap-3450	27	33	2πµx	2πµx	NUM
ap-3450	27	34	cos	cos	ADJ
ap-3450	27	35	2πµy	2πµy	PROPN
ap-3450	27	36	cos	cos	PROPN
ap-3450	27	37	2πµz	2πµz	NUM
ap-3450	27	38	cos	cos	PROPN
ap-3450	27	39	2πνx	2πνx	NOUN
ap-3450	28	1	cos	cos	ADP
ap-3450	28	2	2πνy	2πνy	NOUN
ap-3450	28	3	cos	cos	ADP
ap-3450	28	4	2πνz	2πνz	PROPN
ap-3450	28	5	∣∣∣∣∣∣+	∣∣∣∣∣∣+	PROPN
ap-3450	28	6	1	1	NUM
ap-3450	28	7	2	2	NUM
ap-3450	28	8	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ap-3450	28	9	cos	cos	PROPN
ap-3450	28	10	2πλx	2πλx	PROPN
ap-3450	28	11	cos	cos	ADP
ap-3450	28	12	2πλy	2πλy	PROPN
ap-3450	28	13	cos	cos	ADP
ap-3450	28	14	2πλz	2πλz	NUM
ap-3450	28	15	cos	cos	PROPN
ap-3450	28	16	2πµx	2πµx	NUM
ap-3450	28	17	cos	cos	ADJ
ap-3450	28	18	2πµy	2πµy	PROPN
ap-3450	28	19	cos	cos	PROPN
ap-3450	28	20	2πµz	2πµz	NUM
ap-3450	28	21	cos	cos	PROPN
ap-3450	28	22	2πνx	2πνx	NOUN
ap-3450	29	1	cos	cos	ADP
ap-3450	29	2	2πνy	2πνy	NOUN
ap-3450	29	3	cos	cos	ADP
ap-3450	29	4	2πνz	2πνz	PROPN
ap-3450	29	5	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ap-3450	29	6	+	+	X
ap-3450	29	7	=	=	SYM
ap-3450	29	8	cos	co	NOUN
ap-3450	29	9	2πλx	2πλx	PROPN
ap-3450	29	10	cos	cos	ADP
ap-3450	29	11	2πµy	2πµy	PROPN
ap-3450	29	12	cos	cos	ADP
ap-3450	29	13	2πνz	2πνz	PROPN
ap-3450	29	14	+	+	SYM
ap-3450	29	15	cos	cos	ADJ
ap-3450	29	16	2πλz	2πλz	PROPN
ap-3450	29	17	cos	cos	PROPN
ap-3450	29	18	2πµx	2πµx	NOUN
ap-3450	29	19	cos	co	NOUN
ap-3450	29	20	2πνy	2πνy	NOUN
ap-3450	29	21	+	+	CCONJ
ap-3450	29	22	cos	cos	PROPN
ap-3450	29	23	2πλy	2πλy	PROPN
ap-3450	29	24	cos	cos	ADP
ap-3450	29	25	2πµz	2πµz	PROPN
ap-3450	29	26	cos	cos	PROPN
ap-3450	29	27	2πνx	2πνx	PROPN
ap-3450	29	28	,	,	PUNCT
ap-3450	29	29	x	x	X
ap-3450	29	30	,	,	PUNCT
ap-3450	29	31	y	y	PROPN
ap-3450	29	32	,	,	PUNCT
ap-3450	29	33	z	z	PROPN
ap-3450	29	34	,	,	PUNCT
ap-3450	29	35	λ	λ	PROPN
ap-3450	29	36	,	,	PUNCT
ap-3450	29	37	µ	µ	NOUN
ap-3450	29	38	,	,	PUNCT
ap-3450	29	39	ν	ν	PROPN
ap-3450	29	40	∈	∈	PROPN
ap-3450	29	41	r.	r.	NOUN
ap-3450	29	42	(	(	PUNCT
ap-3450	29	43	2	2	NUM
ap-3450	29	44	)	)	PUNCT
ap-3450	29	45	from	from	ADP
ap-3450	29	46	(	(	PUNCT
ap-3450	29	47	1	1	X
ap-3450	29	48	)	)	PUNCT
ap-3450	29	49	we	we	PRON
ap-3450	29	50	immediately	immediately	ADV
ap-3450	29	51	see	see	VERB
ap-3450	29	52	the	the	DET
ap-3450	29	53	(	(	PUNCT
ap-3450	29	54	rotational	rotational	ADJ
ap-3450	29	55	)	)	PUNCT
ap-3450	29	56	symmetry	symmetry	NOUN
ap-3450	29	57	of	of	ADP
ap-3450	29	58	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	PROPN
ap-3450	29	59	,	,	PUNCT
ap-3450	29	60	y	y	PROPN
ap-3450	29	61	,	,	PUNCT
ap-3450	29	62	z	z	NOUN
ap-3450	29	63	)	)	PUNCT
ap-3450	29	64	with	with	ADP
ap-3450	29	65	respect	respect	NOUN
ap-3450	29	66	to	to	ADP
ap-3450	29	67	cyclic	cyclic	ADJ
ap-3450	29	68	permutations	permutation	NOUN
ap-3450	29	69	of	of	ADP
ap-3450	29	70	variables	variable	NOUN
ap-3450	29	71	(	(	PUNCT
ap-3450	29	72	x	x	X
ap-3450	29	73	,	,	PUNCT
ap-3450	29	74	y	y	PROPN
ap-3450	29	75	,	,	PUNCT
ap-3450	29	76	z	z	NOUN
ap-3450	29	77	)	)	PUNCT
ap-3450	29	78	and	and	CCONJ
ap-3450	29	79	(	(	PUNCT
ap-3450	29	80	λ	λ	PROPN
ap-3450	29	81	,	,	PUNCT
ap-3450	29	82	µ	µ	NOUN
ap-3450	29	83	,	,	PUNCT
ap-3450	29	84	ν	ν	NOUN
ap-3450	29	85	):	):	PUNCT
ap-3450	29	86	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	PROPN
ap-3450	29	87	,	,	PUNCT
ap-3450	29	88	y	y	PROPN
ap-3450	29	89	,	,	PUNCT
ap-3450	29	90	z	z	NOUN
ap-3450	29	91	)	)	PUNCT
ap-3450	29	92	=	=	SYM
ap-3450	30	1	sin(λ,µ,ν)(z	sin(λ,µ,ν)(z	PROPN
ap-3450	30	2	,	,	PUNCT
ap-3450	30	3	x	x	X
ap-3450	30	4	,	,	PUNCT
ap-3450	30	5	y	y	NOUN
ap-3450	30	6	)	)	PUNCT
ap-3450	30	7	=	=	SYM
ap-3450	31	1	sin(λ,µ,ν)(y	sin(λ,µ,ν)(y	NOUN
ap-3450	31	2	,	,	PUNCT
ap-3450	31	3	z	z	NOUN
ap-3450	31	4	,	,	PUNCT
ap-3450	31	5	x	x	X
ap-3450	31	6	)	)	PUNCT
ap-3450	31	7	=	=	SYM
ap-3450	31	8	sin(ν	sin(ν	PROPN
ap-3450	31	9	,	,	PUNCT
ap-3450	31	10	λ,µ)(x	λ,µ)(x	PROPN
ap-3450	31	11	,	,	PUNCT
ap-3450	31	12	y	y	PROPN
ap-3450	31	13	,	,	PUNCT
ap-3450	31	14	z	z	NOUN
ap-3450	31	15	)	)	PUNCT
ap-3450	31	16	=	=	SYM
ap-3450	31	17	sin(µ,ν	sin(µ,ν	NOUN
ap-3450	31	18	,	,	PUNCT
ap-3450	31	19	λ)(x	λ)(x	PROPN
ap-3450	31	20	,	,	PUNCT
ap-3450	31	21	y	y	PROPN
ap-3450	31	22	,	,	PUNCT
ap-3450	31	23	z	z	NOUN
ap-3450	31	24	)	)	PUNCT
ap-3450	31	25	.	.	PUNCT
ap-3450	32	1	from	from	ADP
ap-3450	32	2	(	(	PUNCT
ap-3450	32	3	2	2	X
ap-3450	32	4	)	)	PUNCT
ap-3450	32	5	we	we	PRON
ap-3450	32	6	have	have	AUX
ap-3450	32	7	similar	similar	ADJ
ap-3450	32	8	symmetries	symmetry	NOUN
ap-3450	32	9	for	for	ADP
ap-3450	32	10	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	32	11	,	,	PUNCT
ap-3450	32	12	y	y	PROPN
ap-3450	32	13	,	,	PUNCT
ap-3450	32	14	z	z	PROPN
ap-3450	32	15	):	):	PUNCT
ap-3450	32	16	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	32	17	,	,	PUNCT
ap-3450	32	18	y	y	PROPN
ap-3450	32	19	,	,	PUNCT
ap-3450	32	20	z	z	NOUN
ap-3450	32	21	)	)	PUNCT
ap-3450	32	22	=	=	PUNCT
ap-3450	32	23	cos(λ,µ,ν)(z	cos(λ,µ,ν)(z	PROPN
ap-3450	32	24	,	,	PUNCT
ap-3450	32	25	x	x	X
ap-3450	32	26	,	,	PUNCT
ap-3450	32	27	y	y	PROPN
ap-3450	32	28	)	)	PUNCT
ap-3450	32	29	=	=	SYM
ap-3450	32	30	cos(λ,µ,ν)(y	cos(λ,µ,ν)(y	PROPN
ap-3450	32	31	,	,	PUNCT
ap-3450	32	32	z	z	NOUN
ap-3450	32	33	,	,	PUNCT
ap-3450	32	34	x	x	NOUN
ap-3450	32	35	)	)	PUNCT
ap-3450	32	36	=	=	SYM
ap-3450	32	37	cos(ν	cos(ν	PROPN
ap-3450	32	38	,	,	PUNCT
ap-3450	32	39	λ,µ)(x	λ,µ)(x	PROPN
ap-3450	32	40	,	,	PUNCT
ap-3450	32	41	y	y	PROPN
ap-3450	32	42	,	,	PUNCT
ap-3450	32	43	z	z	NOUN
ap-3450	32	44	)	)	PUNCT
ap-3450	32	45	=	=	SYM
ap-3450	32	46	cos(µ,ν	cos(µ,ν	NOUN
ap-3450	32	47	,	,	PUNCT
ap-3450	32	48	λ)(x	λ)(x	PROPN
ap-3450	32	49	,	,	PUNCT
ap-3450	32	50	y	y	PROPN
ap-3450	32	51	,	,	PUNCT
ap-3450	32	52	z	z	NOUN
ap-3450	32	53	)	)	PUNCT
ap-3450	32	54	.	.	PUNCT
ap-3450	33	1	the	the	DET
ap-3450	33	2	rotational	rotational	ADJ
ap-3450	33	3	symmetries	symmetry	NOUN
ap-3450	33	4	above	above	ADV
ap-3450	33	5	allow	allow	VERB
ap-3450	33	6	us	we	PRON
ap-3450	33	7	to	to	PART
ap-3450	33	8	put	put	VERB
ap-3450	33	9	any	any	DET
ap-3450	33	10	real	real	ADJ
ap-3450	33	11	triple	triple	ADJ
ap-3450	33	12	(	(	PUNCT
ap-3450	33	13	λ	λ	PROPN
ap-3450	33	14	,	,	PUNCT
ap-3450	33	15	µ	µ	NOUN
ap-3450	33	16	,	,	PUNCT
ap-3450	33	17	ν	ν	NOUN
ap-3450	33	18	)	)	PUNCT
ap-3450	33	19	to	to	ADP
ap-3450	33	20	the	the	DET
ap-3450	33	21	canonical	canonical	ADJ
ap-3450	33	22	form	form	NOUN
ap-3450	33	23	.	.	PUNCT
ap-3450	34	1	for	for	ADP
ap-3450	34	2	example	example	NOUN
ap-3450	34	3	,	,	PUNCT
ap-3450	34	4	we	we	PRON
ap-3450	34	5	may	may	AUX
ap-3450	34	6	restrict	restrict	VERB
ap-3450	34	7	ourselves	ourselves	PRON
ap-3450	34	8	to	to	ADP
ap-3450	34	9	the	the	DET
ap-3450	34	10	functions	function	NOUN
ap-3450	34	11	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	34	12	)	)	PUNCT
ap-3450	34	13	and	and	CCONJ
ap-3450	34	14	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	34	15	)	)	PUNCT
ap-3450	34	16	with	with	ADP
ap-3450	34	17	so	so	ADV
ap-3450	34	18	called	call	VERB
ap-3450	34	19	semidominant	semidominant	PROPN
ap-3450	34	20	(	(	PUNCT
ap-3450	34	21	λ	λ	PROPN
ap-3450	34	22	,	,	PUNCT
ap-3450	34	23	µ	µ	NOUN
ap-3450	34	24	,	,	PUNCT
ap-3450	34	25	ν	ν	NOUN
ap-3450	34	26	)	)	PUNCT
ap-3450	34	27	,	,	PUNCT
ap-3450	34	28	that	that	ADV
ap-3450	34	29	is	is	ADV
ap-3450	34	30	,	,	PUNCT
ap-3450	34	31	triples	triple	NOUN
ap-3450	34	32	(	(	PUNCT
ap-3450	34	33	λ	λ	PROPN
ap-3450	34	34	,	,	PUNCT
ap-3450	34	35	µ	µ	NOUN
ap-3450	34	36	,	,	PUNCT
ap-3450	34	37	ν	ν	NOUN
ap-3450	34	38	)	)	PUNCT
ap-3450	34	39	where	where	SCONJ
ap-3450	34	40	λ	λ	PROPN
ap-3450	34	41	≥	≥	X
ap-3450	34	42	µ	µ	X
ap-3450	34	43	≥	≥	NOUN
ap-3450	34	44	ν	ν	NOUN
ap-3450	34	45	or	or	CCONJ
ap-3450	34	46	µ	µ	X
ap-3450	34	47	>	>	X
ap-3450	34	48	λ	λ	X
ap-3450	34	49	>	>	X
ap-3450	34	50	ν	ν	PROPN
ap-3450	34	51	holds	hold	NOUN
ap-3450	34	52	.	.	PUNCT
ap-3450	35	1	the	the	DET
ap-3450	35	2	functions	function	NOUN
ap-3450	35	3	sin(k	sin(k	PROPN
ap-3450	35	4	,	,	PUNCT
ap-3450	35	5	l	l	NOUN
ap-3450	35	6	,	,	PUNCT
ap-3450	35	7	m	m	NOUN
ap-3450	35	8	)	)	PUNCT
ap-3450	35	9	and	and	CCONJ
ap-3450	35	10	cos(k	cos(k	PROPN
ap-3450	35	11	,	,	PUNCT
ap-3450	35	12	l	l	NOUN
ap-3450	35	13	,	,	PUNCT
ap-3450	35	14	m	m	NOUN
ap-3450	35	15	)	)	PUNCT
ap-3450	35	16	with	with	ADP
ap-3450	35	17	k	k	PROPN
ap-3450	35	18	,	,	PUNCT
ap-3450	35	19	l	l	NOUN
ap-3450	35	20	,	,	PUNCT
ap-3450	35	21	m	m	VERB
ap-3450	35	22	∈	∈	PROPN
ap-3450	35	23	z	z	AUX
ap-3450	35	24	have	have	VERB
ap-3450	35	25	additional	additional	ADJ
ap-3450	35	26	symmetries	symmetry	NOUN
ap-3450	35	27	induced	induce	VERB
ap-3450	35	28	by	by	ADP
ap-3450	35	29	the	the	DET
ap-3450	35	30	periodicity	periodicity	NOUN
ap-3450	35	31	of	of	ADP
ap-3450	35	32	sine	sine	NOUN
ap-3450	35	33	and	and	CCONJ
ap-3450	35	34	cosine	cosine	NOUN
ap-3450	35	35	functions	function	NOUN
ap-3450	35	36	:	:	PUNCT
ap-3450	35	37	sin(k	sin(k	PROPN
ap-3450	35	38	,	,	PUNCT
ap-3450	35	39	l	l	NOUN
ap-3450	35	40	,	,	PUNCT
ap-3450	35	41	m)(x+	m)(x+	NOUN
ap-3450	35	42	r	r	NOUN
ap-3450	35	43	,	,	PUNCT
ap-3450	35	44	y	y	PROPN
ap-3450	35	45	+	+	PROPN
ap-3450	35	46	s	s	PROPN
ap-3450	35	47	,	,	PUNCT
ap-3450	35	48	z	z	PROPN
ap-3450	35	49	+	+	NUM
ap-3450	35	50	t	t	NOUN
ap-3450	35	51	)	)	PUNCT
ap-3450	35	52	=	=	SYM
ap-3450	35	53	sin(k	sin(k	PROPN
ap-3450	35	54	,	,	PUNCT
ap-3450	35	55	l	l	NOUN
ap-3450	35	56	,	,	PUNCT
ap-3450	35	57	m)(x	m)(x	PROPN
ap-3450	35	58	,	,	PUNCT
ap-3450	35	59	y	y	PROPN
ap-3450	35	60	,	,	PUNCT
ap-3450	35	61	z	z	NOUN
ap-3450	35	62	)	)	PUNCT
ap-3450	35	63	,	,	PUNCT
ap-3450	35	64	r	r	NOUN
ap-3450	35	65	,	,	PUNCT
ap-3450	35	66	s	s	PROPN
ap-3450	35	67	,	,	PUNCT
ap-3450	35	68	t	t	PROPN
ap-3450	35	69	∈	∈	PROPN
ap-3450	36	1	z	z	PROPN
ap-3450	36	2	,	,	PUNCT
ap-3450	36	3	cos(k	cos(k	PROPN
ap-3450	36	4	,	,	PUNCT
ap-3450	36	5	l	l	NOUN
ap-3450	36	6	,	,	PUNCT
ap-3450	36	7	m)(x+	m)(x+	NOUN
ap-3450	36	8	r	r	NOUN
ap-3450	36	9	,	,	PUNCT
ap-3450	36	10	y	y	PROPN
ap-3450	36	11	+	+	PROPN
ap-3450	36	12	s	s	PROPN
ap-3450	36	13	,	,	PUNCT
ap-3450	36	14	z	z	PROPN
ap-3450	36	15	+	+	NUM
ap-3450	36	16	t	t	NOUN
ap-3450	36	17	)	)	PUNCT
ap-3450	36	18	=	=	SYM
ap-3450	37	1	cos(k	cos(k	PROPN
ap-3450	37	2	,	,	PUNCT
ap-3450	37	3	l	l	NOUN
ap-3450	37	4	,	,	PUNCT
ap-3450	37	5	m)(x	m)(x	PROPN
ap-3450	37	6	,	,	PUNCT
ap-3450	37	7	y	y	PROPN
ap-3450	37	8	,	,	PUNCT
ap-3450	37	9	z	z	NOUN
ap-3450	37	10	)	)	PUNCT
ap-3450	37	11	,	,	PUNCT
ap-3450	37	12	r	r	NOUN
ap-3450	37	13	,	,	PUNCT
ap-3450	37	14	s	s	PROPN
ap-3450	37	15	,	,	PUNCT
ap-3450	37	16	t	t	PROPN
ap-3450	37	17	∈	∈	PROPN
ap-3450	37	18	z.	z.	PROPN
ap-3450	37	19	(	(	PUNCT
ap-3450	37	20	3	3	X
ap-3450	37	21	)	)	PUNCT
ap-3450	37	22	another	another	DET
ap-3450	37	23	symmetries	symmetry	NOUN
ap-3450	37	24	of	of	ADP
ap-3450	37	25	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	37	26	)	)	PUNCT
ap-3450	37	27	and	and	CCONJ
ap-3450	37	28	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	37	29	)	)	PUNCT
ap-3450	37	30	follow	follow	VERB
ap-3450	37	31	from	from	ADP
ap-3450	37	32	the	the	DET
ap-3450	37	33	well	well	ADV
ap-3450	37	34	known	know	VERB
ap-3450	37	35	fact	fact	NOUN
ap-3450	37	36	that	that	SCONJ
ap-3450	37	37	ordinary	ordinary	ADJ
ap-3450	37	38	sine	sine	NOUN
ap-3450	37	39	and	and	CCONJ
ap-3450	37	40	cosine	cosine	NOUN
ap-3450	37	41	functions	function	NOUN
ap-3450	37	42	of	of	ADP
ap-3450	37	43	one	one	NUM
ap-3450	37	44	variable	variable	NOUN
ap-3450	37	45	are	be	AUX
ap-3450	37	46	odd	odd	ADJ
ap-3450	37	47	and	and	CCONJ
ap-3450	37	48	even	even	ADV
ap-3450	37	49	functions	function	NOUN
ap-3450	37	50	,	,	PUNCT
ap-3450	37	51	respectively	respectively	ADV
ap-3450	37	52	.	.	PUNCT
ap-3450	38	1	therefore	therefore	ADV
ap-3450	38	2	for	for	ADP
ap-3450	38	3	all	all	DET
ap-3450	38	4	λ	λ	PROPN
ap-3450	38	5	,	,	PUNCT
ap-3450	38	6	µ	µ	NOUN
ap-3450	38	7	,	,	PUNCT
ap-3450	38	8	ν	ν	NOUN
ap-3450	38	9	and	and	CCONJ
ap-3450	38	10	x	x	NOUN
ap-3450	38	11	,	,	PUNCT
ap-3450	38	12	y	y	PROPN
ap-3450	38	13	,	,	PUNCT
ap-3450	38	14	z	z	VERB
ap-3450	38	15	we	we	PRON
ap-3450	38	16	have	have	VERB
ap-3450	38	17	sin(λ,µ,ν)(−x	sin(λ,µ,ν)(−x	PROPN
ap-3450	38	18	,	,	PUNCT
ap-3450	38	19	y	y	PROPN
ap-3450	38	20	,	,	PUNCT
ap-3450	38	21	z	z	NOUN
ap-3450	38	22	)	)	PUNCT
ap-3450	39	1	=	=	SYM
ap-3450	39	2	sin(λ,µ,ν)(x,−y	sin(λ,µ,ν)(x,−y	PROPN
ap-3450	39	3	,	,	PUNCT
ap-3450	39	4	z	z	NOUN
ap-3450	39	5	)	)	PUNCT
ap-3450	39	6	=	=	SYM
ap-3450	39	7	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	NOUN
ap-3450	39	8	,	,	PUNCT
ap-3450	39	9	y,−z	y,−z	NOUN
ap-3450	39	10	)	)	PUNCT
ap-3450	39	11	=	=	SYM
ap-3450	39	12	−	−	PROPN
ap-3450	39	13	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	PROPN
ap-3450	39	14	,	,	PUNCT
ap-3450	39	15	y	y	PROPN
ap-3450	39	16	,	,	PUNCT
ap-3450	39	17	z	z	NOUN
ap-3450	39	18	)	)	PUNCT
ap-3450	39	19	,	,	PUNCT
ap-3450	39	20	cos(λ,µ,ν)(−x	cos(λ,µ,ν)(−x	PROPN
ap-3450	39	21	,	,	PUNCT
ap-3450	39	22	y	y	PROPN
ap-3450	39	23	,	,	PUNCT
ap-3450	39	24	z	z	NOUN
ap-3450	39	25	)	)	PUNCT
ap-3450	39	26	=	=	SYM
ap-3450	39	27	cos(λ,µ,ν)(x,−y	cos(λ,µ,ν)(x,−y	NOUN
ap-3450	39	28	,	,	PUNCT
ap-3450	39	29	z	z	NOUN
ap-3450	39	30	)	)	PUNCT
ap-3450	39	31	=	=	SYM
ap-3450	39	32	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	39	33	,	,	PUNCT
ap-3450	39	34	y,−z	y,−z	NOUN
ap-3450	39	35	)	)	PUNCT
ap-3450	39	36	=	=	SYM
ap-3450	39	37	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	39	38	,	,	PUNCT
ap-3450	39	39	y	y	PROPN
ap-3450	39	40	,	,	PUNCT
ap-3450	39	41	z	z	NOUN
ap-3450	39	42	)	)	PUNCT
ap-3450	39	43	.	.	PUNCT
ap-3450	40	1	(	(	PUNCT
ap-3450	40	2	4	4	X
ap-3450	40	3	)	)	PUNCT
ap-3450	40	4	there	there	PRON
ap-3450	40	5	is	be	VERB
ap-3450	40	6	also	also	ADV
ap-3450	40	7	duality	duality	NOUN
ap-3450	40	8	of	of	ADP
ap-3450	40	9	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	40	10	)	)	PUNCT
ap-3450	40	11	and	and	CCONJ
ap-3450	40	12	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	40	13	)	)	PUNCT
ap-3450	40	14	functions	function	NOUN
ap-3450	40	15	,	,	PUNCT
ap-3450	40	16	sin(λ,µ,ν)(x	sin(λ,µ,ν)(x	NOUN
ap-3450	40	17	,	,	PUNCT
ap-3450	40	18	y	y	PROPN
ap-3450	40	19	,	,	PUNCT
ap-3450	40	20	z	z	NOUN
ap-3450	40	21	)	)	PUNCT
ap-3450	41	1	=	=	SYM
ap-3450	41	2	sin(x	sin(x	PROPN
ap-3450	41	3	,	,	PUNCT
ap-3450	41	4	y	y	PROPN
ap-3450	41	5	,	,	PUNCT
ap-3450	41	6	z)(λ	z)(λ	PROPN
ap-3450	41	7	,	,	PUNCT
ap-3450	41	8	µ	µ	NOUN
ap-3450	41	9	,	,	PUNCT
ap-3450	41	10	ν	ν	NOUN
ap-3450	41	11	)	)	PUNCT
ap-3450	41	12	,	,	PUNCT
ap-3450	41	13	cos(λ,µ,ν)(x	cos(λ,µ,ν)(x	PROPN
ap-3450	41	14	,	,	PUNCT
ap-3450	41	15	y	y	PROPN
ap-3450	41	16	,	,	PUNCT
ap-3450	41	17	z	z	NOUN
ap-3450	41	18	)	)	PUNCT
ap-3450	41	19	=	=	SYM
ap-3450	41	20	cos(x	cos(x	PROPN
ap-3450	41	21	,	,	PUNCT
ap-3450	41	22	y	y	PROPN
ap-3450	41	23	,	,	PUNCT
ap-3450	41	24	z)(λ	z)(λ	PROPN
ap-3450	41	25	,	,	PUNCT
ap-3450	41	26	µ	µ	NOUN
ap-3450	41	27	,	,	PUNCT
ap-3450	41	28	ν	ν	NOUN
ap-3450	41	29	)	)	PUNCT
ap-3450	41	30	.	.	PUNCT
ap-3450	42	1	(	(	PUNCT
ap-3450	42	2	5	5	X
ap-3450	42	3	)	)	PUNCT
ap-3450	42	4	moreover	moreover	ADV
ap-3450	42	5	,	,	PUNCT
ap-3450	42	6	sin(k	sin(k	PROPN
ap-3450	42	7	,	,	PUNCT
ap-3450	42	8	l	l	NOUN
ap-3450	42	9	,	,	PUNCT
ap-3450	42	10	m)(x	m)(x	PROPN
ap-3450	42	11	,	,	PUNCT
ap-3450	42	12	y	y	PROPN
ap-3450	42	13	,	,	PUNCT
ap-3450	42	14	z	z	NOUN
ap-3450	42	15	)	)	PUNCT
ap-3450	43	1	=	=	SYM
ap-3450	43	2	0	0	PUNCT
ap-3450	43	3	when	when	SCONJ
ap-3450	43	4	k	k	PROPN
ap-3450	43	5	=	=	PUNCT
ap-3450	43	6	0	0	NUM
ap-3450	43	7	or	or	CCONJ
ap-3450	43	8	l	l	NOUN
ap-3450	43	9	=	=	SYM
ap-3450	43	10	0	0	NUM
ap-3450	43	11	or	or	CCONJ
ap-3450	43	12	m	m	PROPN
ap-3450	43	13	=	=	ADJ
ap-3450	43	14	0	0	NUM
ap-3450	43	15	,	,	PUNCT
ap-3450	43	16	(	(	PUNCT
ap-3450	43	17	6	6	NUM
ap-3450	43	18	)	)	PUNCT
ap-3450	43	19	therefore	therefore	ADV
ap-3450	43	20	we	we	PRON
ap-3450	43	21	can	can	AUX
ap-3450	43	22	exclude	exclude	VERB
ap-3450	43	23	sin(k	sin(k	PROPN
ap-3450	43	24	,	,	PUNCT
ap-3450	43	25	l	l	NOUN
ap-3450	43	26	,	,	PUNCT
ap-3450	43	27	m	m	NOUN
ap-3450	43	28	)	)	PUNCT
ap-3450	43	29	with	with	ADP
ap-3450	43	30	at	at	ADV
ap-3450	43	31	least	least	ADV
ap-3450	43	32	one	one	NUM
ap-3450	43	33	index	index	NOUN
ap-3450	43	34	being	be	AUX
ap-3450	43	35	zero	zero	NUM
ap-3450	43	36	from	from	ADP
ap-3450	43	37	our	our	PRON
ap-3450	43	38	considerations	consideration	NOUN
ap-3450	43	39	.	.	PUNCT
ap-3450	44	1	the	the	DET
ap-3450	44	2	relations	relation	NOUN
ap-3450	44	3	(	(	PUNCT
ap-3450	44	4	3)–(6	3)–(6	X
ap-3450	44	5	)	)	PUNCT
ap-3450	44	6	imply	imply	VERB
ap-3450	44	7	that	that	SCONJ
ap-3450	44	8	it	it	PRON
ap-3450	44	9	is	be	AUX
ap-3450	44	10	sufficient	sufficient	ADJ
ap-3450	44	11	to	to	PART
ap-3450	44	12	restrict	restrict	VERB
ap-3450	44	13	to	to	ADP
ap-3450	44	14	the	the	DET
ap-3450	44	15	following	follow	VERB
ap-3450	44	16	families	family	NOUN
ap-3450	44	17	of	of	ADP
ap-3450	44	18	functions	function	NOUN
ap-3450	44	19	:	:	PUNCT
ap-3450	44	20	sin(k	sin(k	PROPN
ap-3450	44	21	,	,	PUNCT
ap-3450	44	22	l	l	NOUN
ap-3450	44	23	,	,	PUNCT
ap-3450	44	24	m	m	NOUN
ap-3450	44	25	)	)	PUNCT
ap-3450	44	26	,	,	PUNCT
ap-3450	44	27	k	k	PROPN
ap-3450	44	28	,	,	PUNCT
ap-3450	44	29	l	l	NOUN
ap-3450	44	30	,	,	PUNCT
ap-3450	44	31	m	m	PROPN
ap-3450	44	32	∈	∈	PROPN
ap-3450	44	33	n	n	CCONJ
ap-3450	44	34	,	,	PUNCT
ap-3450	44	35	k	k	PROPN
ap-3450	44	36	≥	≥	PROPN
ap-3450	44	37	l	l	PROPN
ap-3450	44	38	≥	≥	NOUN
ap-3450	44	39	m	m	PROPN
ap-3450	44	40	or	or	CCONJ
ap-3450	44	41	l	l	X
ap-3450	44	42	>	>	X
ap-3450	45	1	k	k	X
ap-3450	45	2	>	>	X
ap-3450	45	3	m	m	PROPN
ap-3450	45	4	,	,	PUNCT
ap-3450	45	5	and	and	CCONJ
ap-3450	45	6	cos(k	cos(k	PROPN
ap-3450	45	7	,	,	PUNCT
ap-3450	45	8	l	l	NOUN
ap-3450	45	9	,	,	PUNCT
ap-3450	45	10	m	m	NOUN
ap-3450	45	11	)	)	PUNCT
ap-3450	45	12	,	,	PUNCT
ap-3450	45	13	k	k	PROPN
ap-3450	45	14	,	,	PUNCT
ap-3450	45	15	l	l	NOUN
ap-3450	45	16	,	,	PUNCT
ap-3450	45	17	m	m	PROPN
ap-3450	45	18	∈	∈	PROPN
ap-3450	45	19	z	z	PROPN
ap-3450	45	20	,	,	PUNCT
ap-3450	45	21	k	k	PROPN
ap-3450	45	22	≥	≥	PROPN
ap-3450	45	23	l	l	PROPN
ap-3450	45	24	≥	≥	X
ap-3450	45	25	m	m	PROPN
ap-3450	45	26	≥	≥	NOUN
ap-3450	45	27	0	0	NUM
ap-3450	45	28	or	or	CCONJ
ap-3450	45	29	l	l	NOUN
ap-3450	45	30	>	>	X
ap-3450	46	1	k	k	X
ap-3450	46	2	>	>	X
ap-3450	46	3	m	m	PROPN
ap-3450	46	4	≥	≥	NOUN
ap-3450	46	5	0	0	NUM
ap-3450	46	6	,	,	PUNCT
ap-3450	46	7	and	and	CCONJ
ap-3450	46	8	to	to	PART
ap-3450	46	9	consider	consider	VERB
ap-3450	46	10	them	they	PRON
ap-3450	46	11	on	on	ADP
ap-3450	46	12	the	the	DET
ap-3450	46	13	closure	closure	NOUN
ap-3450	46	14	of	of	ADP
ap-3450	46	15	the	the	DET
ap-3450	46	16	fundamental	fundamental	ADJ
ap-3450	46	17	domain	domain	NOUN
ap-3450	46	18	f	f	X
ap-3450	46	19	(	(	PUNCT
ap-3450	46	20	ãaff	ãaff	X
ap-3450	46	21	3	3	NUM
ap-3450	46	22	)	)	PUNCT
ap-3450	47	1	[	[	X
ap-3450	47	2	6	6	NUM
ap-3450	47	3	]	]	PUNCT
ap-3450	47	4	only	only	ADV
ap-3450	47	5	.	.	PUNCT
ap-3450	48	1	this	this	DET
ap-3450	48	2	fundamental	fundamental	ADJ
ap-3450	48	3	domain	domain	NOUN
ap-3450	48	4	of	of	ADP
ap-3450	48	5	the	the	DET
ap-3450	48	6	dual	dual	ADJ
ap-3450	48	7	affine	affine	NOUN
ap-3450	48	8	group	group	NOUN
ap-3450	48	9	of	of	ADP
ap-3450	48	10	a3	a3	NOUN
ap-3450	48	11	can	can	AUX
ap-3450	48	12	be	be	AUX
ap-3450	48	13	chosen	choose	VERB
ap-3450	48	14	in	in	ADP
ap-3450	48	15	3d	3d	NUM
ap-3450	48	16	to	to	PART
ap-3450	48	17	be	be	AUX
ap-3450	48	18	equal	equal	ADJ
ap-3450	48	19	to	to	ADP
ap-3450	48	20	the	the	DET
ap-3450	48	21	part	part	NOUN
ap-3450	48	22	of	of	ADP
ap-3450	48	23	the	the	DET
ap-3450	48	24	cube	cube	NOUN
ap-3450	48	25	shown	show	VERB
ap-3450	48	26	in	in	ADP
ap-3450	48	27	fig	fig	NOUN
ap-3450	48	28	.	.	PUNCT
ap-3450	49	1	1	1	X
ap-3450	49	2	.	.	X
ap-3450	49	3	f	f	PROPN
ap-3450	49	4	(	(	PUNCT
ap-3450	49	5	ãaff	ãaff	X
ap-3450	49	6	3	3	X
ap-3450	49	7	)	)	PUNCT
ap-3450	49	8	=	=	PRON
ap-3450	49	9	{	{	PUNCT
ap-3450	49	10	(	(	PUNCT
ap-3450	49	11	x	x	NOUN
ap-3450	49	12	,	,	PUNCT
ap-3450	49	13	y	y	PROPN
ap-3450	49	14	,	,	PUNCT
ap-3450	49	15	z	z	NOUN
ap-3450	49	16	)	)	PUNCT
ap-3450	49	17	∈	∈	PROPN
ap-3450	49	18	(	(	PUNCT
ap-3450	49	19	0	0	NUM
ap-3450	49	20	,	,	PUNCT
ap-3450	49	21	1	1	NUM
ap-3450	49	22	2	2	NUM
ap-3450	49	23	)	)	PUNCT
ap-3450	49	24	×	×	NOUN
ap-3450	49	25	(	(	PUNCT
ap-3450	49	26	0	0	NUM
ap-3450	49	27	,	,	PUNCT
ap-3450	49	28	1	1	NUM
ap-3450	49	29	2	2	NUM
ap-3450	49	30	)	)	PUNCT
ap-3450	49	31	×	×	NOUN
ap-3450	49	32	(	(	PUNCT
ap-3450	49	33	0	0	NUM
ap-3450	49	34	,	,	PUNCT
ap-3450	49	35	1	1	NUM
ap-3450	49	36	2	2	NUM
ap-3450	49	37	)	)	PUNCT
ap-3450	49	38	|x	|x	NOUN
ap-3450	49	39	>	>	X
ap-3450	50	1	z	z	X
ap-3450	50	2	,	,	PUNCT
ap-3450	50	3	y	y	PROPN
ap-3450	50	4	>	>	X
ap-3450	50	5	z	z	PROPN
ap-3450	50	6	}	}	PUNCT
ap-3450	50	7	.	.	PUNCT
ap-3450	51	1	3	3	X
ap-3450	51	2	.	.	X
ap-3450	51	3	continuous	continuous	ADJ
ap-3450	51	4	orthogonality	orthogonality	NOUN
ap-3450	51	5	the	the	DET
ap-3450	51	6	alternating	alternate	VERB
ap-3450	51	7	trigonometric	trigonometric	ADJ
ap-3450	51	8	functions	function	NOUN
ap-3450	51	9	sin(k	sin(k	PROPN
ap-3450	51	10	,	,	PUNCT
ap-3450	51	11	l	l	NOUN
ap-3450	51	12	,	,	PUNCT
ap-3450	51	13	m	m	NOUN
ap-3450	51	14	)	)	PUNCT
ap-3450	51	15	resp	resp	NOUN
ap-3450	51	16	.	.	PUNCT
ap-3450	52	1	cos(k	cos(k	PROPN
ap-3450	52	2	,	,	PUNCT
ap-3450	52	3	l	l	NOUN
ap-3450	52	4	,	,	PUNCT
ap-3450	52	5	m	m	NOUN
ap-3450	52	6	)	)	PUNCT
ap-3450	52	7	are	be	AUX
ap-3450	52	8	pairwise	pairwise	NOUN
ap-3450	52	9	orthogonal	orthogonal	NOUN
ap-3450	52	10	on	on	ADP
ap-3450	52	11	f	f	PROPN
ap-3450	52	12	(	(	PUNCT
ap-3450	52	13	ãaff	ãaff	X
ap-3450	52	14	3	3	NUM
ap-3450	52	15	)	)	PUNCT
ap-3450	52	16	.	.	PUNCT
ap-3450	53	1	let	let	VERB
ap-3450	53	2	us	we	PRON
ap-3450	53	3	denote	denote	VERB
ap-3450	53	4	gklm	gklm	NOUN
ap-3450	54	1	=	=	PUNCT
ap-3450	54	2	{	{	PUNCT
ap-3450	54	3	3	3	NUM
ap-3450	54	4	if	if	SCONJ
ap-3450	54	5	k	k	NOUN
ap-3450	54	6	=	=	PUNCT
ap-3450	54	7	l	l	X
ap-3450	54	8	=	=	SYM
ap-3450	54	9	m	m	PROPN
ap-3450	54	10	,	,	PUNCT
ap-3450	54	11	1	1	NUM
ap-3450	54	12	otherwise	otherwise	ADV
ap-3450	54	13	.	.	PUNCT
ap-3450	55	1	157	157	NUM
ap-3450	55	2	agata	agata	PROPN
ap-3450	55	3	bezubik	bezubik	PROPN
ap-3450	55	4	,	,	PUNCT
ap-3450	55	5	severin	severin	PROPN
ap-3450	55	6	pošta	pošta	PROPN
ap-3450	55	7	acta	acta	PROPN
ap-3450	55	8	polytechnica	polytechnica	PROPN
ap-3450	55	9	xy	xy	PROPN
ap-3450	56	1	z	z	NOUN
ap-3450	56	2	0	0	NUM
ap-3450	56	3	1	1	NUM
ap-3450	56	4	2	2	NUM
ap-3450	56	5	1	1	NUM
ap-3450	56	6	2	2	NUM
ap-3450	56	7	1	1	NUM
ap-3450	56	8	2	2	NUM
ap-3450	56	9	figure	figure	NOUN
ap-3450	56	10	1	1	NUM
ap-3450	56	11	.	.	PUNCT
ap-3450	56	12	fundamental	fundamental	ADJ
ap-3450	56	13	domain	domain	NOUN
ap-3450	56	14	f	f	X
ap-3450	56	15	(	(	PUNCT
ap-3450	56	16	ãaff	ãaff	X
ap-3450	56	17	3	3	NUM
ap-3450	56	18	)	)	PUNCT
ap-3450	56	19	then	then	ADV
ap-3450	56	20	we	we	PRON
ap-3450	56	21	have	have	VERB
ap-3450	56	22	∫	∫	PROPN
ap-3450	56	23	f	f	PROPN
ap-3450	56	24	(	(	PUNCT
ap-3450	56	25	ãaff	ãaff	X
ap-3450	56	26	3	3	X
ap-3450	56	27	)	)	PUNCT
ap-3450	56	28	sin(k	sin(k	PROPN
ap-3450	56	29	,	,	PUNCT
ap-3450	56	30	l	l	NOUN
ap-3450	56	31	,	,	PUNCT
ap-3450	56	32	m)(x	m)(x	PROPN
ap-3450	56	33	,	,	PUNCT
ap-3450	56	34	y	y	PROPN
ap-3450	56	35	,	,	PUNCT
ap-3450	56	36	z	z	NOUN
ap-3450	56	37	)	)	PUNCT
ap-3450	56	38	sin(k′,l′,m′)(x	sin(k′,l′,m′)(x	NOUN
ap-3450	56	39	,	,	PUNCT
ap-3450	56	40	y	y	PROPN
ap-3450	56	41	,	,	PUNCT
ap-3450	56	42	z	z	NOUN
ap-3450	56	43	)	)	PUNCT
ap-3450	56	44	dxdy	dxdy	NOUN
ap-3450	56	45	dz	dz	PROPN
ap-3450	56	46	=	=	PUNCT
ap-3450	56	47	gklm	gklm	NOUN
ap-3450	56	48	64	64	NUM
ap-3450	56	49	δkk′δll′δmm′	δkk′δll′δmm′	ADJ
ap-3450	56	50	,	,	PUNCT
ap-3450	56	51	where	where	SCONJ
ap-3450	56	52	k	k	NOUN
ap-3450	56	53	,	,	PUNCT
ap-3450	56	54	l	l	NOUN
ap-3450	56	55	,	,	PUNCT
ap-3450	56	56	m	m	PROPN
ap-3450	56	57	,	,	PUNCT
ap-3450	56	58	k′	k′	PROPN
ap-3450	56	59	,	,	PUNCT
ap-3450	56	60	l′,m′	l′,m′	PROPN
ap-3450	56	61	∈	∈	PROPN
ap-3450	56	62	n	n	CCONJ
ap-3450	56	63	,	,	PUNCT
ap-3450	56	64	k	k	PROPN
ap-3450	56	65	≥	≥	PROPN
ap-3450	56	66	l	l	PROPN
ap-3450	56	67	≥	≥	NOUN
ap-3450	56	68	m	m	PROPN
ap-3450	56	69	or	or	CCONJ
ap-3450	56	70	l	l	X
ap-3450	56	71	>	>	X
ap-3450	57	1	k	k	X
ap-3450	57	2	>	>	X
ap-3450	57	3	m	m	PROPN
ap-3450	57	4	,	,	PUNCT
ap-3450	57	5	k′	k′	PROPN
ap-3450	57	6	≥	≥	NUM
ap-3450	57	7	l′	l′	PRON
ap-3450	57	8	≥	≥	PROPN
ap-3450	57	9	m′	m′	NUM
ap-3450	57	10	or	or	CCONJ
ap-3450	57	11	l′	l′	VERB
ap-3450	57	12	>	>	X
ap-3450	57	13	k′	k′	PROPN
ap-3450	57	14	>	>	X
ap-3450	57	15	m′	m′	PROPN
ap-3450	57	16	,	,	PUNCT
ap-3450	57	17	and∫	and∫	PROPN
ap-3450	57	18	f	f	X
ap-3450	57	19	(	(	PUNCT
ap-3450	57	20	ãaff	ãaff	X
ap-3450	57	21	3	3	X
ap-3450	57	22	)	)	PUNCT
ap-3450	57	23	cos(k	cos(k	PROPN
ap-3450	57	24	,	,	PUNCT
ap-3450	57	25	l	l	NOUN
ap-3450	57	26	,	,	PUNCT
ap-3450	57	27	m)(x	m)(x	PROPN
ap-3450	57	28	,	,	PUNCT
ap-3450	57	29	y	y	PROPN
ap-3450	57	30	,	,	PUNCT
ap-3450	57	31	z	z	NOUN
ap-3450	57	32	)	)	PUNCT
ap-3450	57	33	cos(k′,l′,m′)(x	cos(k′,l′,m′)(x	PROPN
ap-3450	57	34	,	,	PUNCT
ap-3450	57	35	y	y	PROPN
ap-3450	57	36	,	,	PUNCT
ap-3450	57	37	z	z	NOUN
ap-3450	57	38	)	)	PUNCT
ap-3450	57	39	dxdy	dxdy	NOUN
ap-3450	57	40	dz	dz	PROPN
ap-3450	57	41	=	=	PUNCT
ap-3450	57	42	gklm	gklm	NOUN
ap-3450	57	43	64	64	NUM
ap-3450	57	44	δkk′δll′δmm′	δkk′δll′δmm′	ADJ
ap-3450	57	45	,	,	PUNCT
ap-3450	57	46	where	where	SCONJ
ap-3450	57	47	k	k	NOUN
ap-3450	57	48	,	,	PUNCT
ap-3450	57	49	l	l	NOUN
ap-3450	57	50	,	,	PUNCT
ap-3450	57	51	m	m	PROPN
ap-3450	57	52	,	,	PUNCT
ap-3450	57	53	k′	k′	PROPN
ap-3450	57	54	,	,	PUNCT
ap-3450	57	55	l′,m′	l′,m′	PROPN
ap-3450	57	56	∈	∈	PROPN
ap-3450	57	57	{	{	PUNCT
ap-3450	57	58	0	0	NUM
ap-3450	57	59	,	,	PUNCT
ap-3450	57	60	1	1	NUM
ap-3450	57	61	,	,	PUNCT
ap-3450	57	62	2	2	NUM
ap-3450	57	63	,	,	PUNCT
ap-3450	57	64	.	.	PUNCT
ap-3450	57	65	.	.	PUNCT
ap-3450	57	66	.	.	PUNCT
ap-3450	58	1	}	}	PUNCT
ap-3450	58	2	,	,	PUNCT
ap-3450	58	3	k	k	X
ap-3450	58	4	≥	≥	PROPN
ap-3450	58	5	l	l	PROPN
ap-3450	58	6	≥	≥	NOUN
ap-3450	58	7	m	m	PROPN
ap-3450	58	8	or	or	CCONJ
ap-3450	58	9	l	l	X
ap-3450	58	10	>	>	X
ap-3450	58	11	k	k	X
ap-3450	58	12	>	>	X
ap-3450	58	13	m	m	PROPN
ap-3450	58	14	,	,	PUNCT
ap-3450	58	15	k′	k′	PROPN
ap-3450	58	16	≥	≥	NUM
ap-3450	58	17	l′	l′	PRON
ap-3450	58	18	≥	≥	PROPN
ap-3450	58	19	m′	m′	NUM
ap-3450	58	20	or	or	CCONJ
ap-3450	58	21	l′	l′	VERB
ap-3450	58	22	>	>	X
ap-3450	58	23	k′	k′	PROPN
ap-3450	58	24	>	>	X
ap-3450	58	25	m′.	m′.	PROPN
ap-3450	58	26	let	let	VERB
ap-3450	58	27	us	we	PRON
ap-3450	58	28	have	have	VERB
ap-3450	58	29	a	a	DET
ap-3450	58	30	function	function	NOUN
ap-3450	58	31	f	f	NOUN
ap-3450	58	32	:	:	PUNCT
ap-3450	58	33	r3	r3	PROPN
ap-3450	58	34	→	→	SYM
ap-3450	58	35	r	r	NOUN
ap-3450	58	36	with	with	ADP
ap-3450	58	37	the	the	DET
ap-3450	58	38	following	follow	VERB
ap-3450	58	39	properties	property	NOUN
ap-3450	58	40	:	:	PUNCT
ap-3450	58	41	rotational	rotational	ADJ
ap-3450	58	42	symmetry	symmetry	NOUN
ap-3450	58	43	,	,	PUNCT
ap-3450	58	44	that	that	ADV
ap-3450	58	45	is	is	ADV
ap-3450	58	46	,	,	PUNCT
ap-3450	58	47	f(x	f(x	PROPN
ap-3450	58	48	,	,	PUNCT
ap-3450	58	49	y	y	PROPN
ap-3450	58	50	,	,	PUNCT
ap-3450	58	51	z	z	NOUN
ap-3450	58	52	)	)	PUNCT
ap-3450	58	53	=	=	SYM
ap-3450	58	54	f(y	f(y	NOUN
ap-3450	58	55	,	,	PUNCT
ap-3450	58	56	z	z	NOUN
ap-3450	58	57	,	,	PUNCT
ap-3450	58	58	x	x	NOUN
ap-3450	58	59	)	)	PUNCT
ap-3450	58	60	=	=	SYM
ap-3450	58	61	f(z	f(z	NOUN
ap-3450	58	62	,	,	PUNCT
ap-3450	58	63	x	x	X
ap-3450	58	64	,	,	PUNCT
ap-3450	58	65	y	y	PROPN
ap-3450	58	66	)	)	PUNCT
ap-3450	58	67	;	;	PUNCT
ap-3450	58	68	periodic	periodic	NOUN
ap-3450	58	69	,	,	PUNCT
ap-3450	58	70	that	that	PRON
ap-3450	58	71	is	be	AUX
ap-3450	58	72	f(x+	f(x+	ADV
ap-3450	58	73	r	r	NOUN
ap-3450	58	74	,	,	PUNCT
ap-3450	58	75	y	y	PROPN
ap-3450	58	76	+	+	PROPN
ap-3450	58	77	s	s	PROPN
ap-3450	58	78	,	,	PUNCT
ap-3450	58	79	z	z	PROPN
ap-3450	58	80	+	+	NUM
ap-3450	58	81	t	t	NOUN
ap-3450	58	82	)	)	PUNCT
ap-3450	58	83	=	=	SYM
ap-3450	58	84	f(x	f(x	PROPN
ap-3450	58	85	,	,	PUNCT
ap-3450	58	86	y	y	PROPN
ap-3450	58	87	,	,	PUNCT
ap-3450	58	88	z	z	NOUN
ap-3450	58	89	)	)	PUNCT
ap-3450	58	90	for	for	ADP
ap-3450	58	91	all	all	DET
ap-3450	58	92	r	r	NOUN
ap-3450	58	93	,	,	PUNCT
ap-3450	58	94	s	s	PROPN
ap-3450	58	95	,	,	PUNCT
ap-3450	58	96	t	t	PROPN
ap-3450	58	97	∈	∈	PROPN
ap-3450	58	98	z	z	PROPN
ap-3450	58	99	,	,	PUNCT
ap-3450	58	100	and	and	CCONJ
ap-3450	58	101	odd	odd	ADJ
ap-3450	58	102	in	in	ADP
ap-3450	58	103	each	each	DET
ap-3450	58	104	variable	variable	NOUN
ap-3450	58	105	,	,	PUNCT
ap-3450	58	106	that	that	ADV
ap-3450	58	107	is	is	ADV
ap-3450	58	108	,	,	PUNCT
ap-3450	58	109	f(−x	f(−x	PUNCT
ap-3450	58	110	,	,	PUNCT
ap-3450	58	111	y	y	PROPN
ap-3450	58	112	,	,	PUNCT
ap-3450	58	113	z	z	NOUN
ap-3450	58	114	)	)	PUNCT
ap-3450	58	115	=	=	SYM
ap-3450	58	116	f(x,−y	f(x,−y	NOUN
ap-3450	58	117	,	,	PUNCT
ap-3450	58	118	z	z	NOUN
ap-3450	58	119	)	)	PUNCT
ap-3450	58	120	=	=	SYM
ap-3450	58	121	f(x	f(x	PROPN
ap-3450	58	122	,	,	PUNCT
ap-3450	58	123	y,−z	y,−z	NOUN
ap-3450	58	124	)	)	PUNCT
ap-3450	58	125	=	=	SYM
ap-3450	58	126	−f(x	−f(x	PROPN
ap-3450	58	127	,	,	PUNCT
ap-3450	58	128	y	y	PROPN
ap-3450	58	129	,	,	PUNCT
ap-3450	58	130	z	z	NOUN
ap-3450	58	131	)	)	PUNCT
ap-3450	58	132	.	.	PUNCT
ap-3450	59	1	let	let	VERB
ap-3450	59	2	f	f	PRON
ap-3450	59	3	be	be	AUX
ap-3450	59	4	sufficiently	sufficiently	ADV
ap-3450	59	5	smooth	smooth	ADJ
ap-3450	59	6	.	.	PUNCT
ap-3450	60	1	then	then	ADV
ap-3450	60	2	it	it	PRON
ap-3450	60	3	can	can	AUX
ap-3450	60	4	be	be	AUX
ap-3450	60	5	expanded	expand	VERB
ap-3450	60	6	in	in	ADP
ap-3450	60	7	terms	term	NOUN
ap-3450	60	8	of	of	ADP
ap-3450	60	9	the	the	DET
ap-3450	60	10	alternating	alternate	VERB
ap-3450	60	11	sine	sine	ADJ
ap-3450	60	12	functions	function	NOUN
ap-3450	60	13	,	,	PUNCT
ap-3450	60	14	respectively	respectively	ADV
ap-3450	60	15	,	,	PUNCT
ap-3450	60	16	using	use	VERB
ap-3450	60	17	formulas	formula	NOUN
ap-3450	60	18	f(x	f(x	PROPN
ap-3450	60	19	,	,	PUNCT
ap-3450	60	20	y	y	PROPN
ap-3450	60	21	,	,	PUNCT
ap-3450	60	22	z	z	NOUN
ap-3450	60	23	)	)	PUNCT
ap-3450	60	24	=	=	SYM
ap-3450	60	25	∑	∑	PUNCT
ap-3450	60	26	k	k	X
ap-3450	60	27	,	,	PUNCT
ap-3450	60	28	l	l	NOUN
ap-3450	60	29	,	,	PUNCT
ap-3450	60	30	m≥1	m≥1	PROPN
ap-3450	60	31	k≥l≥m	k≥l≥m	NOUN
ap-3450	60	32	or	or	CCONJ
ap-3450	60	33	l	l	NOUN
ap-3450	60	34	>	>	X
ap-3450	60	35	k	k	X
ap-3450	60	36	>	>	PROPN
ap-3450	60	37	m	m	PROPN
ap-3450	60	38	cklm	cklm	PROPN
ap-3450	60	39	sin(k	sin(k	PROPN
ap-3450	60	40	,	,	PUNCT
ap-3450	60	41	l	l	NOUN
ap-3450	60	42	,	,	PUNCT
ap-3450	60	43	m)(x	m)(x	PROPN
ap-3450	60	44	,	,	PUNCT
ap-3450	60	45	y	y	PROPN
ap-3450	60	46	,	,	PUNCT
ap-3450	60	47	z	z	NOUN
ap-3450	60	48	)	)	PUNCT
ap-3450	60	49	,	,	PUNCT
ap-3450	60	50	cklm	cklm	PROPN
ap-3450	61	1	=	=	PUNCT
ap-3450	62	1	64g−1	64g−1	NUM
ap-3450	62	2	klm	klm	PROPN
ap-3450	62	3	∫	∫	PROPN
ap-3450	62	4	f	f	PROPN
ap-3450	62	5	(	(	PUNCT
ap-3450	62	6	ãaff	ãaff	X
ap-3450	62	7	3	3	X
ap-3450	62	8	)	)	PUNCT
ap-3450	62	9	f(x	f(x	PROPN
ap-3450	62	10	,	,	PUNCT
ap-3450	62	11	y	y	PROPN
ap-3450	62	12	,	,	PUNCT
ap-3450	62	13	z	z	NOUN
ap-3450	62	14	)	)	PUNCT
ap-3450	62	15	sin(k	sin(k	PROPN
ap-3450	62	16	,	,	PUNCT
ap-3450	62	17	l	l	NOUN
ap-3450	62	18	,	,	PUNCT
ap-3450	62	19	m)(x	m)(x	PROPN
ap-3450	62	20	,	,	PUNCT
ap-3450	62	21	y	y	PROPN
ap-3450	62	22	,	,	PUNCT
ap-3450	62	23	z)dxdydz	z)dxdydz	PROPN
ap-3450	62	24	.	.	PUNCT
ap-3450	63	1	similarly	similarly	ADV
ap-3450	63	2	,	,	PUNCT
ap-3450	63	3	any	any	DET
ap-3450	63	4	function	function	NOUN
ap-3450	63	5	f̃	f̃	PROPN
ap-3450	63	6	:	:	PUNCT
ap-3450	63	7	r3	r3	PROPN
ap-3450	63	8	→	→	SYM
ap-3450	63	9	r	r	NOUN
ap-3450	63	10	with	with	ADP
ap-3450	63	11	the	the	DET
ap-3450	63	12	symmetries	symmetry	NOUN
ap-3450	63	13	f̃(x	f̃(x	PROPN
ap-3450	63	14	,	,	PUNCT
ap-3450	63	15	y	y	PROPN
ap-3450	63	16	,	,	PUNCT
ap-3450	63	17	z	z	NOUN
ap-3450	63	18	)	)	PUNCT
ap-3450	63	19	=	=	SYM
ap-3450	63	20	f(y	f(y	NOUN
ap-3450	63	21	,	,	PUNCT
ap-3450	63	22	z	z	NOUN
ap-3450	63	23	,	,	PUNCT
ap-3450	63	24	x	x	NOUN
ap-3450	63	25	)	)	PUNCT
ap-3450	63	26	=	=	SYM
ap-3450	63	27	f(z	f(z	NOUN
ap-3450	63	28	,	,	PUNCT
ap-3450	63	29	x	x	X
ap-3450	63	30	,	,	PUNCT
ap-3450	63	31	y	y	PROPN
ap-3450	63	32	)	)	PUNCT
ap-3450	63	33	,	,	PUNCT
ap-3450	63	34	f̃(x+	f̃(x+	NOUN
ap-3450	63	35	r	r	NOUN
ap-3450	63	36	,	,	PUNCT
ap-3450	63	37	y	y	PROPN
ap-3450	63	38	+	+	PROPN
ap-3450	63	39	s	s	PROPN
ap-3450	63	40	,	,	PUNCT
ap-3450	63	41	z	z	PROPN
ap-3450	63	42	+	+	NUM
ap-3450	63	43	t	t	NOUN
ap-3450	63	44	)	)	PUNCT
ap-3450	64	1	=	=	SYM
ap-3450	64	2	f(x	f(x	PROPN
ap-3450	64	3	,	,	PUNCT
ap-3450	64	4	y	y	PROPN
ap-3450	64	5	,	,	PUNCT
ap-3450	64	6	z	z	NOUN
ap-3450	64	7	)	)	PUNCT
ap-3450	64	8	for	for	ADP
ap-3450	64	9	all	all	DET
ap-3450	64	10	r	r	NOUN
ap-3450	64	11	,	,	PUNCT
ap-3450	64	12	s	s	PROPN
ap-3450	64	13	,	,	PUNCT
ap-3450	64	14	t	t	PROPN
ap-3450	64	15	∈	∈	PROPN
ap-3450	64	16	z	z	PROPN
ap-3450	64	17	,	,	PUNCT
ap-3450	64	18	f̃(−x	f̃(−x	PROPN
ap-3450	64	19	,	,	PUNCT
ap-3450	64	20	y	y	PROPN
ap-3450	64	21	,	,	PUNCT
ap-3450	64	22	z	z	NOUN
ap-3450	64	23	)	)	PUNCT
ap-3450	64	24	=	=	SYM
ap-3450	64	25	f(x,−y	f(x,−y	NOUN
ap-3450	64	26	,	,	PUNCT
ap-3450	64	27	z	z	NOUN
ap-3450	64	28	)	)	PUNCT
ap-3450	64	29	=	=	SYM
ap-3450	64	30	f(x	f(x	PROPN
ap-3450	64	31	,	,	PUNCT
ap-3450	64	32	y,−z	y,−z	NOUN
ap-3450	64	33	)	)	PUNCT
ap-3450	64	34	=	=	SYM
ap-3450	64	35	f(x	f(x	PROPN
ap-3450	64	36	,	,	PUNCT
ap-3450	64	37	y	y	PROPN
ap-3450	64	38	,	,	PUNCT
ap-3450	64	39	z	z	NOUN
ap-3450	64	40	)	)	PUNCT
ap-3450	64	41	,	,	PUNCT
ap-3450	64	42	and	and	CCONJ
ap-3450	64	43	sufficiently	sufficiently	ADV
ap-3450	64	44	smooth	smooth	ADJ
ap-3450	64	45	can	can	AUX
ap-3450	64	46	be	be	AUX
ap-3450	64	47	expanded	expand	VERB
ap-3450	64	48	in	in	ADP
ap-3450	64	49	terms	term	NOUN
ap-3450	64	50	of	of	ADP
ap-3450	64	51	the	the	DET
ap-3450	64	52	alternating	alternate	VERB
ap-3450	64	53	cosine	cosine	NOUN
ap-3450	64	54	functions	function	NOUN
ap-3450	64	55	.	.	PUNCT
ap-3450	65	1	the	the	DET
ap-3450	65	2	expansion	expansion	NOUN
ap-3450	65	3	is	be	AUX
ap-3450	65	4	done	do	VERB
ap-3450	65	5	using	use	VERB
ap-3450	65	6	the	the	DET
ap-3450	65	7	formulas	formula	NOUN
ap-3450	65	8	f̃(x	f̃(x	PROPN
ap-3450	65	9	,	,	PUNCT
ap-3450	65	10	y	y	PROPN
ap-3450	65	11	,	,	PUNCT
ap-3450	65	12	z	z	NOUN
ap-3450	65	13	)	)	PUNCT
ap-3450	65	14	=	=	SYM
ap-3450	65	15	∑	∑	PUNCT
ap-3450	65	16	k	k	X
ap-3450	65	17	,	,	PUNCT
ap-3450	65	18	l	l	NOUN
ap-3450	65	19	,	,	PUNCT
ap-3450	65	20	m≥0	m≥0	PROPN
ap-3450	65	21	k≥l≥m	k≥l≥m	NOUN
ap-3450	65	22	or	or	CCONJ
ap-3450	65	23	l	l	NOUN
ap-3450	65	24	>	>	X
ap-3450	65	25	k	k	X
ap-3450	65	26	>	>	X
ap-3450	65	27	m	m	PROPN
ap-3450	65	28	c̃klm	c̃klm	NOUN
ap-3450	65	29	cos(k	cos(k	PROPN
ap-3450	65	30	,	,	PUNCT
ap-3450	65	31	l	l	NOUN
ap-3450	65	32	,	,	PUNCT
ap-3450	65	33	m)(x	m)(x	PROPN
ap-3450	65	34	,	,	PUNCT
ap-3450	65	35	y	y	PROPN
ap-3450	65	36	,	,	PUNCT
ap-3450	65	37	z	z	NOUN
ap-3450	65	38	)	)	PUNCT
ap-3450	65	39	,	,	PUNCT
ap-3450	65	40	c̃klm	c̃klm	NOUN
ap-3450	66	1	=	=	SYM
ap-3450	66	2	64g−1	64g−1	NUM
ap-3450	66	3	klm	klm	PROPN
ap-3450	66	4	∫	∫	PROPN
ap-3450	66	5	f	f	PROPN
ap-3450	66	6	(	(	PUNCT
ap-3450	66	7	ãaff	ãaff	X
ap-3450	66	8	3	3	X
ap-3450	66	9	)	)	PUNCT
ap-3450	66	10	f̃(x	f̃(x	PROPN
ap-3450	66	11	,	,	PUNCT
ap-3450	66	12	y	y	PROPN
ap-3450	66	13	,	,	PUNCT
ap-3450	66	14	z	z	NOUN
ap-3450	66	15	)	)	PUNCT
ap-3450	66	16	cos(k	cos(k	PROPN
ap-3450	66	17	,	,	PUNCT
ap-3450	66	18	l	l	NOUN
ap-3450	66	19	,	,	PUNCT
ap-3450	66	20	m)(x	m)(x	PROPN
ap-3450	66	21	,	,	PUNCT
ap-3450	66	22	y	y	PROPN
ap-3450	66	23	,	,	PUNCT
ap-3450	66	24	z)dxdydz	z)dxdydz	PROPN
ap-3450	66	25	.	.	PUNCT
ap-3450	67	1	158	158	NUM
ap-3450	67	2	vol	vol	NOUN
ap-3450	67	3	.	.	PUNCT
ap-3450	68	1	56	56	NUM
ap-3450	68	2	no	no	NOUN
ap-3450	68	3	.	.	PUNCT
ap-3450	69	1	3/2016	3/2016	NUM
ap-3450	69	2	three	three	NUM
ap-3450	69	3	-	-	PUNCT
ap-3450	69	4	variable	variable	NOUN
ap-3450	69	5	alternating	alternate	VERB
ap-3450	69	6	trigonometric	trigonometric	ADJ
ap-3450	69	7	functions	function	NOUN
ap-3450	70	1	xy	xy	PROPN
ap-3450	70	2	z	z	NOUN
ap-3450	70	3	0	0	NUM
ap-3450	70	4	1	1	NUM
ap-3450	70	5	2	2	NUM
ap-3450	70	6	1	1	NUM
ap-3450	70	7	2	2	NUM
ap-3450	70	8	1	1	NUM
ap-3450	70	9	2	2	NUM
ap-3450	70	10	figure	figure	NOUN
ap-3450	70	11	2	2	NUM
ap-3450	70	12	.	.	PUNCT
ap-3450	70	13	finite	finite	PROPN
ap-3450	70	14	grid	grid	NOUN
ap-3450	70	15	of	of	ADP
ap-3450	70	16	points	point	NOUN
ap-3450	70	17	in	in	ADP
ap-3450	70	18	f	f	PROPN
ap-3450	70	19	(	(	PUNCT
ap-3450	70	20	ãaff	ãaff	X
ap-3450	70	21	3	3	NUM
ap-3450	70	22	)	)	PUNCT
ap-3450	70	23	4	4	NUM
ap-3450	70	24	.	.	PUNCT
ap-3450	70	25	product	product	NOUN
ap-3450	70	26	decomposition	decomposition	NOUN
ap-3450	70	27	the	the	DET
ap-3450	70	28	product	product	NOUN
ap-3450	70	29	of	of	ADP
ap-3450	70	30	two	two	NUM
ap-3450	70	31	alternating	alternate	VERB
ap-3450	70	32	cosines	cosine	NOUN
ap-3450	70	33	,	,	PUNCT
ap-3450	70	34	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	70	35	)	)	PUNCT
ap-3450	70	36	cos(λ′,µ′,ν′	cos(λ′,µ′,ν′	PROPN
ap-3450	70	37	)	)	PUNCT
ap-3450	70	38	(	(	PUNCT
ap-3450	70	39	we	we	PRON
ap-3450	70	40	omit	omit	VERB
ap-3450	70	41	general	general	ADJ
ap-3450	70	42	argument	argument	NOUN
ap-3450	70	43	(	(	PUNCT
ap-3450	70	44	x	x	X
ap-3450	70	45	,	,	PUNCT
ap-3450	70	46	y	y	PROPN
ap-3450	70	47	,	,	PUNCT
ap-3450	70	48	z	z	NOUN
ap-3450	70	49	)	)	PUNCT
ap-3450	70	50	here	here	ADV
ap-3450	70	51	)	)	PUNCT
ap-3450	70	52	can	can	AUX
ap-3450	70	53	be	be	AUX
ap-3450	70	54	decomposed	decompose	VERB
ap-3450	70	55	into	into	ADP
ap-3450	70	56	the	the	DET
ap-3450	70	57	sum	sum	NOUN
ap-3450	70	58	of	of	ADP
ap-3450	70	59	alternating	alternate	VERB
ap-3450	70	60	cosines	cosine	NOUN
ap-3450	70	61	using	use	VERB
ap-3450	70	62	the	the	DET
ap-3450	70	63	following	follow	VERB
ap-3450	70	64	formulas	formula	NOUN
ap-3450	70	65	.	.	PUNCT
ap-3450	71	1	the	the	DET
ap-3450	71	2	decomposition	decomposition	NOUN
ap-3450	71	3	can	can	AUX
ap-3450	71	4	serve	serve	VERB
ap-3450	71	5	e.	e.	PROPN
ap-3450	71	6	g.	g.	PROPN
ap-3450	71	7	for	for	ADP
ap-3450	71	8	the	the	DET
ap-3450	71	9	derivation	derivation	NOUN
ap-3450	71	10	of	of	ADP
ap-3450	71	11	recursion	recursion	NOUN
ap-3450	71	12	relations	relation	NOUN
ap-3450	71	13	for	for	ADP
ap-3450	71	14	the	the	DET
ap-3450	71	15	considered	consider	VERB
ap-3450	71	16	functions	function	NOUN
ap-3450	71	17	.	.	PUNCT
ap-3450	72	1	we	we	PRON
ap-3450	72	2	have	have	VERB
ap-3450	72	3	cos(λ,µ,ν	cos(λ,µ,ν	PROPN
ap-3450	72	4	)	)	PUNCT
ap-3450	72	5	cos(λ′,µ′,ν′	cos(λ′,µ′,ν′	NOUN
ap-3450	72	6	)	)	PUNCT
ap-3450	72	7	=	=	SYM
ap-3450	72	8	1	1	NUM
ap-3450	72	9	8	8	NUM
ap-3450	72	10	(	(	PUNCT
ap-3450	72	11	cos(λ−λ′,µ−µ′,ν−ν′	cos(λ−λ′,µ−µ′,ν−ν′	X
ap-3450	72	12	)	)	PUNCT
ap-3450	72	13	+	+	X
ap-3450	72	14	cos(λ−λ′,µ−µ′,ν+ν′	cos(λ−λ′,µ−µ′,ν+ν′	NOUN
ap-3450	72	15	)	)	PUNCT
ap-3450	72	16	+	+	CCONJ
ap-3450	72	17	cos(λ−λ′,µ+µ′,ν−ν′	cos(λ−λ′,µ+µ′,ν−ν′	VERB
ap-3450	72	18	)	)	PUNCT
ap-3450	73	1	+	+	X
ap-3450	73	2	cos(λ−λ′,µ+µ′,ν+ν′	cos(λ−λ′,µ+µ′,ν+ν′	X
ap-3450	73	3	)	)	PUNCT
ap-3450	73	4	+	+	NUM
ap-3450	73	5	cos(λ+λ′,µ−µ′,ν−ν′	cos(λ+λ′,µ−µ′,ν−ν′	X
ap-3450	73	6	)	)	PUNCT
ap-3450	73	7	+	+	CCONJ
ap-3450	73	8	cos(λ+λ′,µ−µ′,ν+ν′	cos(λ+λ′,µ−µ′,ν+ν′	X
ap-3450	73	9	)	)	PUNCT
ap-3450	74	1	+	+	CCONJ
ap-3450	74	2	cos(λ+λ′,µ+µ′,ν−ν′	cos(λ+λ′,µ+µ′,ν−ν′	X
ap-3450	74	3	)	)	PUNCT
ap-3450	74	4	+	+	SYM
ap-3450	74	5	cos(λ+λ′,µ+µ′,ν+ν′	cos(λ+λ′,µ+µ′,ν+ν′	NOUN
ap-3450	74	6	)	)	PUNCT
ap-3450	74	7	+	+	NUM
ap-3450	74	8	cos(λ−µ′,µ−ν′,ν−λ′	cos(λ−µ′,µ−ν′,ν−λ′	NOUN
ap-3450	74	9	)	)	PUNCT
ap-3450	75	1	+	+	CCONJ
ap-3450	75	2	cos(λ−µ′,µ−ν′,λ′+ν	cos(λ−µ′,µ−ν′,λ′+ν	ADJ
ap-3450	75	3	)	)	PUNCT
ap-3450	75	4	+	+	NOUN
ap-3450	75	5	cos(λ−µ′,µ+ν′,ν−λ′	cos(λ−µ′,µ+ν′,ν−λ′	NOUN
ap-3450	75	6	)	)	PUNCT
ap-3450	75	7	+	+	SYM
ap-3450	75	8	cos(λ−µ′,µ+ν′,λ′+ν	cos(λ−µ′,µ+ν′,λ′+ν	X
ap-3450	75	9	)	)	PUNCT
ap-3450	76	1	+	+	NUM
ap-3450	76	2	cos(λ+µ′,µ−ν′,ν−λ′	cos(λ+µ′,µ−ν′,ν−λ′	NOUN
ap-3450	76	3	)	)	PUNCT
ap-3450	77	1	+	+	NUM
ap-3450	77	2	cos(λ+µ′,µ−ν′,λ′+ν	cos(λ+µ′,µ−ν′,λ′+ν	NOUN
ap-3450	77	3	)	)	PUNCT
ap-3450	77	4	+	+	SYM
ap-3450	77	5	cos(λ+µ′,µ+ν′,ν−λ′	cos(λ+µ′,µ+ν′,ν−λ′	NOUN
ap-3450	77	6	)	)	PUNCT
ap-3450	77	7	+	+	NUM
ap-3450	77	8	cos(λ+µ′,µ+ν′,λ′+ν	cos(λ+µ′,µ+ν′,λ′+ν	NOUN
ap-3450	77	9	)	)	PUNCT
ap-3450	77	10	+	+	NUM
ap-3450	77	11	cos(λ−ν′,µ−λ′,ν−µ′	cos(λ−ν′,µ−λ′,ν−µ′	NOUN
ap-3450	77	12	)	)	PUNCT
ap-3450	78	1	+	+	NUM
ap-3450	78	2	cos(λ−ν′,µ−λ′,µ′+ν	cos(λ−ν′,µ−λ′,µ′+ν	NOUN
ap-3450	78	3	)	)	PUNCT
ap-3450	79	1	+	+	X
ap-3450	79	2	cos(λ−ν′,λ′+µ,ν−µ′	cos(λ−ν′,λ′+µ,ν−µ′	X
ap-3450	79	3	)	)	PUNCT
ap-3450	79	4	+	+	SYM
ap-3450	79	5	cos(λ−ν′,λ′+µ,µ′+ν	cos(λ−ν′,λ′+µ,µ′+ν	NOUN
ap-3450	79	6	)	)	PUNCT
ap-3450	79	7	+	+	NUM
ap-3450	79	8	cos(λ+ν′,µ−λ′,ν−µ′	cos(λ+ν′,µ−λ′,ν−µ′	NOUN
ap-3450	79	9	)	)	PUNCT
ap-3450	80	1	+	+	SYM
ap-3450	80	2	cos(λ+ν′,−λ′+µ,µ′+ν	cos(λ+ν′,−λ′+µ,µ′+ν	NOUN
ap-3450	80	3	)	)	PUNCT
ap-3450	80	4	+	+	ADJ
ap-3450	80	5	cos(λ+ν′,λ′+µ,ν−µ′	cos(λ+ν′,λ′+µ,ν−µ′	NOUN
ap-3450	80	6	)	)	PUNCT
ap-3450	80	7	+	+	SYM
ap-3450	80	8	cos(λ+ν′,λ′+µ,µ′+ν	cos(λ+ν′,λ′+µ,µ′+ν	NOUN
ap-3450	80	9	)	)	PUNCT
ap-3450	80	10	)	)	PUNCT
ap-3450	80	11	.	.	PUNCT
ap-3450	81	1	(	(	PUNCT
ap-3450	81	2	7	7	X
ap-3450	81	3	)	)	PUNCT
ap-3450	81	4	similarly	similarly	ADV
ap-3450	81	5	,	,	PUNCT
ap-3450	81	6	the	the	DET
ap-3450	81	7	product	product	NOUN
ap-3450	81	8	of	of	ADP
ap-3450	81	9	two	two	NUM
ap-3450	81	10	alternating	alternate	VERB
ap-3450	81	11	sines	sine	NOUN
ap-3450	81	12	,	,	PUNCT
ap-3450	81	13	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	81	14	)	)	PUNCT
ap-3450	81	15	sin(λ′,µ′,ν′	sin(λ′,µ′,ν′	NOUN
ap-3450	81	16	)	)	PUNCT
ap-3450	81	17	can	can	AUX
ap-3450	81	18	be	be	AUX
ap-3450	81	19	decomposed	decompose	VERB
ap-3450	81	20	into	into	ADP
ap-3450	81	21	the	the	DET
ap-3450	81	22	sum	sum	NOUN
ap-3450	81	23	of	of	ADP
ap-3450	81	24	alternating	alternate	VERB
ap-3450	81	25	cosines	cosine	NOUN
ap-3450	81	26	using	use	VERB
ap-3450	81	27	the	the	DET
ap-3450	81	28	following	follow	VERB
ap-3450	81	29	formula	formula	NOUN
ap-3450	81	30	:	:	PUNCT
ap-3450	81	31	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	81	32	)	)	PUNCT
ap-3450	81	33	sin(λ′,µ′,ν′	sin(λ′,µ′,ν′	NOUN
ap-3450	81	34	)	)	PUNCT
ap-3450	81	35	=	=	SYM
ap-3450	81	36	1	1	NUM
ap-3450	81	37	8	8	NUM
ap-3450	81	38	(	(	PUNCT
ap-3450	81	39	cos(λ−λ′,µ−µ′,ν−ν′)−	cos(λ−λ′,µ−µ′,ν−ν′)−	NUM
ap-3450	81	40	cos(λ−λ′,µ−µ′,ν+ν′)−	cos(λ−λ′,µ−µ′,ν+ν′)−	PROPN
ap-3450	81	41	cos(λ−λ′,µ+µ′,ν−ν′	cos(λ−λ′,µ+µ′,ν−ν′	VERB
ap-3450	81	42	)	)	PUNCT
ap-3450	82	1	+	+	CCONJ
ap-3450	82	2	cos(λ−λ′,µ+µ′,ν+ν′)−	cos(λ−λ′,µ+µ′,ν+ν′)−	X
ap-3450	82	3	cos(λ+λ′,µ−µ′,ν−ν′	cos(λ+λ′,µ−µ′,ν−ν′	X
ap-3450	82	4	)	)	PUNCT
ap-3450	83	1	+	+	CCONJ
ap-3450	83	2	cos(λ+λ′,µ−µ′,ν+ν′	cos(λ+λ′,µ−µ′,ν+ν′	X
ap-3450	83	3	)	)	PUNCT
ap-3450	84	1	+	+	NUM
ap-3450	84	2	cos(λ+λ′,µ+µ′,ν−ν′)−	cos(λ+λ′,µ+µ′,ν−ν′)−	PROPN
ap-3450	84	3	cos(λ+λ′,µ+µ′,ν+ν′	cos(λ+λ′,µ+µ′,ν+ν′	NOUN
ap-3450	84	4	)	)	PUNCT
ap-3450	85	1	+	+	NUM
ap-3450	85	2	cos(λ−µ′,µ−ν′,ν−λ′)−	cos(λ−µ′,µ−ν′,ν−λ′)−	NUM
ap-3450	85	3	cos(λ−µ′,µ−ν′,λ′+ν)−	cos(λ−µ′,µ−ν′,λ′+ν)−	PROPN
ap-3450	85	4	cos(λ−µ′,µ+ν′,ν−λ′	cos(λ−µ′,µ+ν′,ν−λ′	NOUN
ap-3450	85	5	)	)	PUNCT
ap-3450	86	1	+	+	CCONJ
ap-3450	86	2	cos(λ−µ′,µ+ν′,λ′+ν)−	cos(λ−µ′,µ+ν′,λ′+ν)−	NOUN
ap-3450	86	3	cos(λ+µ′,µ−ν′,ν−λ′	cos(λ+µ′,µ−ν′,ν−λ′	NOUN
ap-3450	86	4	)	)	PUNCT
ap-3450	87	1	+	+	NUM
ap-3450	87	2	cos(λ+µ′,µ−ν′,λ′+ν	cos(λ+µ′,µ−ν′,λ′+ν	NOUN
ap-3450	87	3	)	)	PUNCT
ap-3450	88	1	+	+	CCONJ
ap-3450	88	2	cos(λ+µ′,µ+ν′,ν−λ′)−	cos(λ+µ′,µ+ν′,ν−λ′)−	X
ap-3450	88	3	cos(λ+µ′,µ+ν′,λ′+ν	cos(λ+µ′,µ+ν′,λ′+ν	NOUN
ap-3450	88	4	)	)	PUNCT
ap-3450	89	1	+	+	CCONJ
ap-3450	89	2	cos(λ−ν′,µ−λ′,ν−µ′)−	cos(λ−ν′,µ−λ′,ν−µ′)−	NUM
ap-3450	89	3	cos(λ−ν′,µ−λ′,µ′+ν	cos(λ−ν′,µ−λ′,µ′+ν	NOUN
ap-3450	89	4	)	)	PUNCT
ap-3450	89	5	−	−	NOUN
ap-3450	89	6	cos(λ−ν′,λ′+µ,ν−µ′	cos(λ−ν′,λ′+µ,ν−µ′	ADJ
ap-3450	89	7	)	)	PUNCT
ap-3450	89	8	+	+	CCONJ
ap-3450	90	1	cos(λ−ν′,λ′+µ,µ′+ν)−	cos(λ−ν′,λ′+µ,µ′+ν)−	NUM
ap-3450	90	2	cos(λ+ν′,µ−λ′,ν−µ′	cos(λ+ν′,µ−λ′,ν−µ′	NOUN
ap-3450	90	3	)	)	PUNCT
ap-3450	90	4	+	+	CCONJ
ap-3450	90	5	cos(λ+ν′,µ−λ′,µ′+ν	cos(λ+ν′,µ−λ′,µ′+ν	NOUN
ap-3450	90	6	)	)	PUNCT
ap-3450	90	7	+	+	CCONJ
ap-3450	90	8	cos(λ+ν′,λ′+µ,ν−µ′	cos(λ+ν′,λ′+µ,ν−µ′	NOUN
ap-3450	90	9	)	)	PUNCT
ap-3450	90	10	−	−	PROPN
ap-3450	90	11	cos(λ+ν′,λ′+µ,µ′+ν	cos(λ+ν′,λ′+µ,µ′+ν	NOUN
ap-3450	90	12	)	)	PUNCT
ap-3450	90	13	)	)	PUNCT
ap-3450	90	14	.	.	PUNCT
ap-3450	91	1	(	(	PUNCT
ap-3450	91	2	8)	8)	NUM
ap-3450	91	3	for	for	ADP
ap-3450	91	4	the	the	DET
ap-3450	91	5	sake	sake	NOUN
ap-3450	91	6	of	of	ADP
ap-3450	91	7	completeness	completeness	NOUN
ap-3450	91	8	,	,	PUNCT
ap-3450	91	9	one	one	PRON
ap-3450	91	10	can	can	AUX
ap-3450	91	11	decompose	decompose	VERB
ap-3450	91	12	also	also	ADV
ap-3450	91	13	the	the	DET
ap-3450	91	14	product	product	NOUN
ap-3450	91	15	sin(λ,µ,ν	sin(λ,µ,ν	PROPN
ap-3450	91	16	)	)	PUNCT
ap-3450	91	17	cos(λ′,µ′,ν′	cos(λ′,µ′,ν′	NOUN
ap-3450	91	18	)	)	PUNCT
ap-3450	91	19	.	.	PUNCT
ap-3450	92	1	this	this	PRON
ap-3450	92	2	can	can	AUX
ap-3450	92	3	be	be	AUX
ap-3450	92	4	however	however	ADV
ap-3450	92	5	easily	easily	ADV
ap-3450	92	6	obtained	obtain	VERB
ap-3450	92	7	from	from	ADP
ap-3450	92	8	the	the	DET
ap-3450	92	9	decomposition	decomposition	NOUN
ap-3450	92	10	(	(	PUNCT
ap-3450	92	11	7	7	NUM
ap-3450	92	12	)	)	PUNCT
ap-3450	92	13	by	by	ADP
ap-3450	92	14	substitution	substitution	NOUN
ap-3450	92	15	cos→	cos→	ADJ
ap-3450	92	16	sin	sin	NOUN
ap-3450	92	17	on	on	ADP
ap-3450	92	18	the	the	DET
ap-3450	92	19	right	right	ADJ
ap-3450	92	20	hand	hand	NOUN
ap-3450	92	21	side	side	NOUN
ap-3450	92	22	.	.	PUNCT
ap-3450	93	1	5	5	X
ap-3450	93	2	.	.	X
ap-3450	93	3	discrete	discrete	ADJ
ap-3450	93	4	cosine	cosine	NOUN
ap-3450	93	5	transforms	transform	VERB
ap-3450	93	6	to	to	ADP
ap-3450	93	7	each	each	PRON
ap-3450	93	8	of	of	ADP
ap-3450	93	9	standard	standard	ADJ
ap-3450	93	10	discrete	discrete	ADJ
ap-3450	93	11	cosine	cosine	NOUN
ap-3450	93	12	transforms	transform	VERB
ap-3450	93	13	,	,	PUNCT
ap-3450	93	14	denoted	denote	VERB
ap-3450	93	15	in	in	ADP
ap-3450	93	16	the	the	DET
ap-3450	93	17	literature	literature	NOUN
ap-3450	93	18	often	often	ADV
ap-3450	93	19	as	as	ADP
ap-3450	93	20	dct-1	dct-1	NOUN
ap-3450	93	21	,	,	PUNCT
ap-3450	93	22	dct-2	dct-2	X
ap-3450	93	23	,	,	PUNCT
ap-3450	93	24	dct-3	dct-3	NUM
ap-3450	93	25	and	and	CCONJ
ap-3450	93	26	dct-4	dct-4	ADV
ap-3450	93	27	[	[	X
ap-3450	93	28	13	13	NUM
ap-3450	93	29	]	]	PUNCT
ap-3450	93	30	,	,	PUNCT
ap-3450	93	31	there	there	PRON
ap-3450	93	32	corresponds	correspond	VERB
ap-3450	93	33	an	an	DET
ap-3450	93	34	alternating	alternate	VERB
ap-3450	93	35	three	three	NUM
ap-3450	93	36	-	-	PUNCT
ap-3450	93	37	dimensional	dimensional	ADJ
ap-3450	93	38	discrete	discrete	ADJ
ap-3450	93	39	cosine	cosine	NOUN
ap-3450	93	40	transforms	transform	VERB
ap-3450	93	41	.	.	PUNCT
ap-3450	94	1	we	we	PRON
ap-3450	94	2	denote	denote	VERB
ap-3450	94	3	these	these	DET
ap-3450	94	4	corresponding	corresponding	ADJ
ap-3450	94	5	transforms	transform	NOUN
ap-3450	94	6	as	as	ADP
ap-3450	94	7	amdct-1	amdct-1	PROPN
ap-3450	94	8	,	,	PUNCT
ap-3450	94	9	amdct-2	amdct-2	NUM
ap-3450	94	10	,	,	PUNCT
ap-3450	94	11	amdct-3	amdct-3	NOUN
ap-3450	94	12	,	,	PUNCT
ap-3450	94	13	amdct-4	amdct-4	NUM
ap-3450	94	14	.	.	PROPN
ap-3450	95	1	5.1	5.1	NUM
ap-3450	95	2	.	.	PUNCT
ap-3450	96	1	amdct-1	amdct-1	AUX
ap-3450	96	2	for	for	ADP
ap-3450	96	3	given	give	VERB
ap-3450	96	4	n	n	PRON
ap-3450	96	5	∈	∈	NOUN
ap-3450	96	6	n	n	AUX
ap-3450	96	7	let	let	VERB
ap-3450	96	8	us	we	PRON
ap-3450	96	9	define	define	VERB
ap-3450	96	10	a	a	DET
ap-3450	96	11	lattice	lattice	NOUN
ap-3450	96	12	fn	fn	NOUN
ap-3450	96	13	=	=	PUNCT
ap-3450	96	14	{	{	PUNCT
ap-3450	96	15	(	(	PUNCT
ap-3450	96	16	r	r	NOUN
ap-3450	96	17	2n	2n	NUM
ap-3450	96	18	,	,	PUNCT
ap-3450	96	19	s	s	NOUN
ap-3450	96	20	2n	2n	NUM
ap-3450	96	21	,	,	PUNCT
ap-3450	96	22	t	t	PROPN
ap-3450	96	23	2n	2n	NUM
ap-3450	96	24	)	)	PUNCT
ap-3450	96	25	∣∣	∣∣	NUM
ap-3450	96	26	0	0	NUM
ap-3450	96	27	≤	≤	NOUN
ap-3450	96	28	r	r	NOUN
ap-3450	96	29	,	,	PUNCT
ap-3450	96	30	s	s	PROPN
ap-3450	96	31	,	,	PUNCT
ap-3450	96	32	t	t	PROPN
ap-3450	96	33	≤	≤	NUM
ap-3450	96	34	n	n	CCONJ
ap-3450	96	35	,	,	PUNCT
ap-3450	96	36	r	r	NOUN
ap-3450	96	37	≥	≥	NOUN
ap-3450	96	38	s	s	PART
ap-3450	96	39	≥	≥	NOUN
ap-3450	96	40	t	t	NOUN
ap-3450	96	41	or	or	CCONJ
ap-3450	96	42	s	s	VERB
ap-3450	96	43	>	>	X
ap-3450	96	44	r	r	X
ap-3450	96	45	>	>	X
ap-3450	96	46	t	t	PROPN
ap-3450	96	47	}	}	PUNCT
ap-3450	96	48	.	.	PUNCT
ap-3450	97	1	(	(	PUNCT
ap-3450	97	2	9	9	X
ap-3450	97	3	)	)	SYM
ap-3450	97	4	159	159	NUM
ap-3450	97	5	agata	agata	PROPN
ap-3450	97	6	bezubik	bezubik	PROPN
ap-3450	97	7	,	,	PUNCT
ap-3450	97	8	severin	severin	PROPN
ap-3450	97	9	pošta	pošta	PROPN
ap-3450	97	10	acta	acta	PROPN
ap-3450	97	11	polytechnica	polytechnica	PROPN
ap-3450	97	12	figure	figure	NOUN
ap-3450	97	13	3	3	NUM
ap-3450	97	14	.	.	PUNCT
ap-3450	98	1	the	the	DET
ap-3450	98	2	function	function	NOUN
ap-3450	98	3	(	(	PUNCT
ap-3450	98	4	10	10	NUM
ap-3450	98	5	)	)	PUNCT
ap-3450	98	6	used	use	VERB
ap-3450	98	7	in	in	ADP
ap-3450	98	8	the	the	DET
ap-3450	98	9	interpolation	interpolation	NOUN
ap-3450	98	10	example	example	NOUN
ap-3450	98	11	.	.	PUNCT
ap-3450	99	1	n	n	PRON
ap-3450	99	2	∫	∫	PROPN
ap-3450	99	3	f	f	PROPN
ap-3450	99	4	(	(	PUNCT
ap-3450	99	5	ãaff	ãaff	X
ap-3450	99	6	3	3	NUM
ap-3450	99	7	)	)	PUNCT
ap-3450	99	8	|f	|f	PUNCT
ap-3450	100	1	−	−	PROPN
ap-3450	100	2	fn	fn	NOUN
ap-3450	100	3	|	|	ADV
ap-3450	100	4	2	2	NUM
ap-3450	100	5	1	1	NUM
ap-3450	100	6	0.0486711	0.0486711	NUM
ap-3450	100	7	2	2	NUM
ap-3450	100	8	0.0391407	0.0391407	NUM
ap-3450	100	9	3	3	NUM
ap-3450	100	10	0.0096493	0.0096493	NUM
ap-3450	100	11	4	4	NUM
ap-3450	100	12	0.0029516	0.0029516	NUM
ap-3450	100	13	table	table	NOUN
ap-3450	100	14	1	1	NUM
ap-3450	100	15	.	.	PUNCT
ap-3450	100	16	integral	integral	ADJ
ap-3450	100	17	error	error	NOUN
ap-3450	100	18	estimates	estimate	NOUN
ap-3450	100	19	for	for	ADP
ap-3450	100	20	the	the	DET
ap-3450	100	21	approximations	approximation	NOUN
ap-3450	100	22	in	in	ADP
ap-3450	100	23	fig	fig	NOUN
ap-3450	100	24	.	.	PUNCT
ap-3450	101	1	4	4	X
ap-3450	101	2	.	.	X
ap-3450	101	3	figure	figure	NOUN
ap-3450	101	4	4	4	NUM
ap-3450	101	5	.	.	PUNCT
ap-3450	101	6	alternating	alternate	VERB
ap-3450	101	7	cosines	cosine	NOUN
ap-3450	101	8	approximations	approximation	NOUN
ap-3450	101	9	of	of	ADP
ap-3450	101	10	(	(	PUNCT
ap-3450	101	11	10	10	NUM
ap-3450	101	12	)	)	PUNCT
ap-3450	101	13	for	for	ADP
ap-3450	101	14	n	n	NOUN
ap-3450	101	15	=	=	SYM
ap-3450	101	16	1	1	NUM
ap-3450	101	17	,	,	PUNCT
ap-3450	101	18	2	2	NUM
ap-3450	101	19	,	,	PUNCT
ap-3450	101	20	3	3	NUM
ap-3450	101	21	,	,	PUNCT
ap-3450	101	22	4	4	NUM
ap-3450	101	23	.	.	PUNCT
ap-3450	102	1	this	this	DET
ap-3450	102	2	lattice	lattice	NOUN
ap-3450	102	3	is	be	AUX
ap-3450	102	4	chosen	choose	VERB
ap-3450	102	5	such	such	ADJ
ap-3450	102	6	way	way	NOUN
ap-3450	102	7	that	that	PRON
ap-3450	102	8	it	it	PRON
ap-3450	102	9	fulfills	fulfill	VERB
ap-3450	102	10	the	the	DET
ap-3450	102	11	whole	whole	ADJ
ap-3450	102	12	space	space	NOUN
ap-3450	102	13	when	when	SCONJ
ap-3450	102	14	symmetries	symmetry	NOUN
ap-3450	102	15	(	(	PUNCT
ap-3450	102	16	3	3	NUM
ap-3450	102	17	)	)	PUNCT
ap-3450	102	18	and	and	CCONJ
ap-3450	102	19	(	(	PUNCT
ap-3450	102	20	4	4	X
ap-3450	102	21	)	)	PUNCT
ap-3450	102	22	are	be	AUX
ap-3450	102	23	applied	apply	VERB
ap-3450	102	24	to	to	ADP
ap-3450	102	25	it	it	PRON
ap-3450	102	26	.	.	PUNCT
ap-3450	103	1	evidently	evidently	ADV
ap-3450	103	2	,	,	PUNCT
ap-3450	103	3	we	we	PRON
ap-3450	103	4	have	have	VERB
ap-3450	103	5	a	a	DET
ap-3450	103	6	partial	partial	ADJ
ap-3450	103	7	freedom	freedom	NOUN
ap-3450	103	8	which	which	DET
ap-3450	103	9	parts	part	NOUN
ap-3450	103	10	of	of	ADP
ap-3450	103	11	boundary	boundary	NOUN
ap-3450	103	12	to	to	PART
ap-3450	103	13	include	include	VERB
ap-3450	103	14	in	in	ADP
ap-3450	103	15	the	the	DET
ap-3450	103	16	lattice	lattice	NOUN
ap-3450	103	17	.	.	PUNCT
ap-3450	104	1	the	the	DET
ap-3450	104	2	lattice	lattice	NOUN
ap-3450	104	3	(	(	PUNCT
ap-3450	104	4	9	9	NUM
ap-3450	104	5	)	)	PUNCT
ap-3450	104	6	contains	contain	VERB
ap-3450	104	7	1	1	NUM
ap-3450	104	8	3	3	NUM
ap-3450	104	9	(	(	PUNCT
ap-3450	104	10	n3	n3	NOUN
ap-3450	104	11	+	+	CCONJ
ap-3450	104	12	3n2	3n2	NUM
ap-3450	104	13	+	+	NUM
ap-3450	104	14	5n	5n	NUM
ap-3450	104	15	+	+	CCONJ
ap-3450	104	16	3	3	NUM
ap-3450	104	17	)	)	PUNCT
ap-3450	104	18	points	point	NOUN
ap-3450	104	19	.	.	PUNCT
ap-3450	105	1	for	for	ADP
ap-3450	105	2	example	example	NOUN
ap-3450	105	3	,	,	PUNCT
ap-3450	105	4	f2	f2	PROPN
ap-3450	105	5	=	=	PUNCT
ap-3450	105	6	{	{	PUNCT
ap-3450	105	7	(	(	PUNCT
ap-3450	105	8	0	0	NUM
ap-3450	105	9	,	,	PUNCT
ap-3450	105	10	0	0	NUM
ap-3450	105	11	,	,	PUNCT
ap-3450	105	12	0	0	NUM
ap-3450	105	13	)	)	PUNCT
ap-3450	105	14	,	,	PUNCT
ap-3450	105	15	(	(	PUNCT
ap-3450	105	16	1	1	NUM
ap-3450	105	17	4	4	NUM
ap-3450	105	18	,	,	PUNCT
ap-3450	105	19	0	0	NUM
ap-3450	105	20	,	,	PUNCT
ap-3450	105	21	0	0	NUM
ap-3450	105	22	)	)	PUNCT
ap-3450	105	23	,	,	PUNCT
ap-3450	105	24	(	(	PUNCT
ap-3450	105	25	1	1	NUM
ap-3450	105	26	4	4	NUM
ap-3450	105	27	,	,	PUNCT
ap-3450	105	28	1	1	NUM
ap-3450	105	29	4	4	NUM
ap-3450	105	30	,	,	PUNCT
ap-3450	105	31	0	0	NUM
ap-3450	105	32	)	)	PUNCT
ap-3450	105	33	,	,	PUNCT
ap-3450	105	34	(	(	PUNCT
ap-3450	105	35	1	1	NUM
ap-3450	105	36	4	4	NUM
ap-3450	105	37	,	,	PUNCT
ap-3450	105	38	1	1	NUM
ap-3450	105	39	4	4	NUM
ap-3450	105	40	,	,	PUNCT
ap-3450	105	41	1	1	NUM
ap-3450	105	42	4	4	NUM
ap-3450	105	43	)	)	PUNCT
ap-3450	105	44	,	,	PUNCT
ap-3450	105	45	(	(	PUNCT
ap-3450	105	46	1	1	NUM
ap-3450	105	47	4	4	NUM
ap-3450	105	48	,	,	PUNCT
ap-3450	105	49	1	1	NUM
ap-3450	105	50	2	2	NUM
ap-3450	105	51	,	,	PUNCT
ap-3450	105	52	0	0	NUM
ap-3450	105	53	)	)	PUNCT
ap-3450	105	54	,	,	PUNCT
ap-3450	105	55	(	(	PUNCT
ap-3450	105	56	1	1	NUM
ap-3450	105	57	2	2	NUM
ap-3450	105	58	,	,	PUNCT
ap-3450	105	59	0	0	NUM
ap-3450	105	60	,	,	PUNCT
ap-3450	105	61	0	0	NUM
ap-3450	105	62	)	)	PUNCT
ap-3450	105	63	,	,	PUNCT
ap-3450	105	64	(	(	PUNCT
ap-3450	105	65	1	1	NUM
ap-3450	105	66	2	2	NUM
ap-3450	105	67	,	,	PUNCT
ap-3450	105	68	1	1	NUM
ap-3450	105	69	4	4	NUM
ap-3450	105	70	,	,	PUNCT
ap-3450	105	71	0	0	NUM
ap-3450	105	72	)	)	PUNCT
ap-3450	105	73	,	,	PUNCT
ap-3450	105	74	(	(	PUNCT
ap-3450	105	75	1	1	NUM
ap-3450	105	76	2	2	NUM
ap-3450	105	77	,	,	PUNCT
ap-3450	105	78	1	1	NUM
ap-3450	105	79	4	4	NUM
ap-3450	105	80	,	,	PUNCT
ap-3450	105	81	1	1	NUM
ap-3450	105	82	4	4	NUM
ap-3450	105	83	)	)	PUNCT
ap-3450	105	84	,	,	PUNCT
ap-3450	105	85	(	(	PUNCT
ap-3450	105	86	1	1	NUM
ap-3450	105	87	2	2	NUM
ap-3450	105	88	,	,	PUNCT
ap-3450	105	89	1	1	NUM
ap-3450	105	90	2	2	NUM
ap-3450	105	91	,	,	PUNCT
ap-3450	105	92	0	0	NUM
ap-3450	105	93	)	)	PUNCT
ap-3450	105	94	,	,	PUNCT
ap-3450	105	95	(	(	PUNCT
ap-3450	105	96	1	1	NUM
ap-3450	105	97	2	2	NUM
ap-3450	105	98	,	,	PUNCT
ap-3450	105	99	1	1	NUM
ap-3450	105	100	2	2	NUM
ap-3450	105	101	,	,	PUNCT
ap-3450	105	102	1	1	NUM
ap-3450	105	103	4	4	NUM
ap-3450	105	104	)	)	PUNCT
ap-3450	105	105	,	,	PUNCT
ap-3450	105	106	(	(	PUNCT
ap-3450	105	107	1	1	NUM
ap-3450	105	108	2	2	NUM
ap-3450	105	109	,	,	PUNCT
ap-3450	105	110	1	1	NUM
ap-3450	105	111	2	2	NUM
ap-3450	105	112	,	,	PUNCT
ap-3450	105	113	1	1	NUM
ap-3450	105	114	2	2	NUM
ap-3450	105	115	)	)	PUNCT
ap-3450	105	116	}	}	PUNCT
ap-3450	105	117	.	.	PUNCT
ap-3450	106	1	another	another	DET
ap-3450	106	2	example	example	NOUN
ap-3450	106	3	,	,	PUNCT
ap-3450	106	4	for	for	ADP
ap-3450	106	5	n	n	NOUN
ap-3450	106	6	=	=	SYM
ap-3450	106	7	5	5	NUM
ap-3450	106	8	,	,	PUNCT
ap-3450	106	9	is	be	AUX
ap-3450	106	10	shown	show	VERB
ap-3450	106	11	in	in	ADP
ap-3450	106	12	fig	fig	NOUN
ap-3450	106	13	.	.	PUNCT
ap-3450	107	1	2	2	X
ap-3450	107	2	.	.	X
ap-3450	107	3	the	the	DET
ap-3450	107	4	alternating	alternate	VERB
ap-3450	107	5	cosines	cosine	NOUN
ap-3450	107	6	cos(k	cos(k	PRON
ap-3450	107	7	,	,	PUNCT
ap-3450	107	8	l	l	NOUN
ap-3450	107	9	,	,	PUNCT
ap-3450	107	10	m	m	NOUN
ap-3450	107	11	)	)	PUNCT
ap-3450	107	12	are	be	AUX
ap-3450	107	13	pairwise	pairwise	NOUN
ap-3450	107	14	orthogonal	orthogonal	NOUN
ap-3450	107	15	on	on	ADP
ap-3450	107	16	this	this	DET
ap-3450	107	17	lattice	lattice	NOUN
ap-3450	107	18	,	,	PUNCT
ap-3450	107	19	namely	namely	ADV
ap-3450	107	20	we	we	PRON
ap-3450	107	21	have∑	have∑	VERB
ap-3450	107	22	0≤r	0≤r	PROPN
ap-3450	107	23	,	,	PUNCT
ap-3450	107	24	s	s	NOUN
ap-3450	107	25	,	,	PUNCT
ap-3450	107	26	t≤n	t≤n	ADJ
ap-3450	107	27	,	,	PUNCT
ap-3450	107	28	r≥s≥t	r≥s≥t	NOUN
ap-3450	107	29	or	or	CCONJ
ap-3450	107	30	s	s	NOUN
ap-3450	107	31	>	>	X
ap-3450	107	32	r	r	X
ap-3450	107	33	>	>	X
ap-3450	107	34	t	t	X
ap-3450	107	35	g−1	g−1	PROPN
ap-3450	107	36	rstcrcsct	rstcrcsct	PROPN
ap-3450	107	37	cos(k	cos(k	PROPN
ap-3450	107	38	,	,	PUNCT
ap-3450	107	39	l	l	NOUN
ap-3450	107	40	,	,	PUNCT
ap-3450	107	41	m	m	NOUN
ap-3450	107	42	)	)	PUNCT
ap-3450	107	43	(	(	PUNCT
ap-3450	107	44	r	r	NOUN
ap-3450	107	45	2n	2n	NUM
ap-3450	107	46	,	,	PUNCT
ap-3450	107	47	s	s	NOUN
ap-3450	107	48	2n	2n	NUM
ap-3450	107	49	,	,	PUNCT
ap-3450	108	1	t	t	PROPN
ap-3450	108	2	2n	2n	NUM
ap-3450	108	3	)	)	PUNCT
ap-3450	108	4	cos(k′,l′,m′	cos(k′,l′,m′	PROPN
ap-3450	108	5	)	)	PUNCT
ap-3450	108	6	(	(	PUNCT
ap-3450	109	1	r	r	NOUN
ap-3450	109	2	2n	2n	NUM
ap-3450	109	3	,	,	PUNCT
ap-3450	109	4	s	s	NOUN
ap-3450	109	5	2n	2n	NUM
ap-3450	109	6	,	,	PUNCT
ap-3450	109	7	t	t	PROPN
ap-3450	109	8	2n	2n	NUM
ap-3450	109	9	)	)	PUNCT
ap-3450	110	1	=	=	PUNCT
ap-3450	110	2	n3c̃k	n3c̃k	PROPN
ap-3450	110	3	c̃lc̃mgklmδ(k	c̃lc̃mgklmδ(k	PROPN
ap-3450	110	4	,	,	PUNCT
ap-3450	110	5	l	l	NOUN
ap-3450	110	6	,	,	PUNCT
ap-3450	110	7	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	110	8	)	)	PUNCT
ap-3450	110	9	,	,	PUNCT
ap-3450	110	10	where	where	SCONJ
ap-3450	110	11	cα	cα	NOUN
ap-3450	110	12	=	=	SYM
ap-3450	110	13	{	{	PUNCT
ap-3450	110	14	1	1	NUM
ap-3450	110	15	2	2	NUM
ap-3450	110	16	if	if	SCONJ
ap-3450	110	17	α	α	NOUN
ap-3450	110	18	=	=	NOUN
ap-3450	110	19	0	0	NUM
ap-3450	110	20	or	or	CCONJ
ap-3450	110	21	α	α	NOUN
ap-3450	110	22	=	=	SYM
ap-3450	110	23	n	n	CCONJ
ap-3450	110	24	,	,	PUNCT
ap-3450	110	25	1	1	NUM
ap-3450	110	26	otherwise	otherwise	ADV
ap-3450	110	27	,	,	PUNCT
ap-3450	110	28	c̃α	c̃α	NOUN
ap-3450	110	29	=	=	SYM
ap-3450	110	30	{	{	PUNCT
ap-3450	110	31	1	1	NUM
ap-3450	110	32	if	if	SCONJ
ap-3450	110	33	α	α	NOUN
ap-3450	110	34	=	=	NOUN
ap-3450	110	35	0	0	NUM
ap-3450	110	36	or	or	CCONJ
ap-3450	110	37	α	α	NOUN
ap-3450	110	38	=	=	SYM
ap-3450	110	39	n	n	CCONJ
ap-3450	110	40	,	,	PUNCT
ap-3450	110	41	1	1	NUM
ap-3450	110	42	2	2	NUM
ap-3450	110	43	otherwise	otherwise	ADV
ap-3450	110	44	,	,	PUNCT
ap-3450	110	45	and	and	CCONJ
ap-3450	110	46	0	0	NUM
ap-3450	110	47	≤	≤	NUM
ap-3450	111	1	k	k	NOUN
ap-3450	111	2	,	,	PUNCT
ap-3450	111	3	l	l	NOUN
ap-3450	111	4	,	,	PUNCT
ap-3450	111	5	m	m	VERB
ap-3450	111	6	≤	≤	NOUN
ap-3450	111	7	n	n	PRON
ap-3450	111	8	,	,	PUNCT
ap-3450	111	9	k	k	PROPN
ap-3450	111	10	≥	≥	PROPN
ap-3450	111	11	l	l	PROPN
ap-3450	111	12	≥	≥	NOUN
ap-3450	111	13	m	m	PROPN
ap-3450	111	14	or	or	CCONJ
ap-3450	111	15	l	l	X
ap-3450	111	16	>	>	X
ap-3450	112	1	k	k	X
ap-3450	112	2	>	>	X
ap-3450	112	3	m	m	PROPN
ap-3450	112	4	,	,	PUNCT
ap-3450	112	5	and	and	CCONJ
ap-3450	112	6	0	0	NUM
ap-3450	112	7	≤	≤	NOUN
ap-3450	112	8	k′	k′	PROPN
ap-3450	112	9	,	,	PUNCT
ap-3450	112	10	l′,m′	l′,m′	PROPN
ap-3450	112	11	≤	≤	PUNCT
ap-3450	112	12	n	n	CCONJ
ap-3450	112	13	,	,	PUNCT
ap-3450	112	14	k′	k′	PROPN
ap-3450	112	15	≥	≥	NUM
ap-3450	112	16	l′	l′	PRON
ap-3450	112	17	≥	≥	PROPN
ap-3450	112	18	m′	m′	NUM
ap-3450	112	19	or	or	CCONJ
ap-3450	112	20	l′	l′	VERB
ap-3450	112	21	>	>	X
ap-3450	112	22	k′	k′	PROPN
ap-3450	112	23	>	>	X
ap-3450	112	24	m′.	m′.	PROPN
ap-3450	112	25	let	let	VERB
ap-3450	112	26	f	f	PRON
ap-3450	112	27	be	be	AUX
ap-3450	112	28	a	a	DET
ap-3450	112	29	real	real	ADJ
ap-3450	112	30	function	function	NOUN
ap-3450	112	31	defined	define	VERB
ap-3450	112	32	on	on	ADP
ap-3450	112	33	fn	fn	PROPN
ap-3450	112	34	.	.	PUNCT
ap-3450	113	1	then	then	ADV
ap-3450	113	2	it	it	PRON
ap-3450	113	3	can	can	AUX
ap-3450	113	4	be	be	AUX
ap-3450	113	5	expanded	expand	VERB
ap-3450	113	6	into	into	ADP
ap-3450	113	7	a	a	DET
ap-3450	113	8	sum	sum	NOUN
ap-3450	113	9	of	of	ADP
ap-3450	113	10	alternating	alternate	VERB
ap-3450	113	11	cosines	cosine	NOUN
ap-3450	113	12	,	,	PUNCT
ap-3450	113	13	f(x	f(x	PROPN
ap-3450	113	14	,	,	PUNCT
ap-3450	113	15	y	y	PROPN
ap-3450	113	16	,	,	PUNCT
ap-3450	113	17	z	z	NOUN
ap-3450	113	18	)	)	PUNCT
ap-3450	113	19	=	=	PUNCT
ap-3450	114	1	∑	∑	PUNCT
ap-3450	114	2	0≤k	0≤k	PROPN
ap-3450	114	3	,	,	PUNCT
ap-3450	114	4	l	l	NOUN
ap-3450	114	5	,	,	PUNCT
ap-3450	114	6	m≤n	m≤n	PROPN
ap-3450	114	7	,	,	PUNCT
ap-3450	114	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	114	9	or	or	CCONJ
ap-3450	114	10	l	l	NOUN
ap-3450	114	11	>	>	X
ap-3450	114	12	k	k	PROPN
ap-3450	114	13	>	>	PROPN
ap-3450	114	14	m	m	PROPN
ap-3450	114	15	a(k	a(k	PROPN
ap-3450	114	16	,	,	PUNCT
ap-3450	114	17	l	l	NOUN
ap-3450	114	18	,	,	PUNCT
ap-3450	114	19	m	m	NOUN
ap-3450	114	20	)	)	PUNCT
ap-3450	114	21	cos(k	cos(k	PROPN
ap-3450	114	22	,	,	PUNCT
ap-3450	114	23	l	l	NOUN
ap-3450	114	24	,	,	PUNCT
ap-3450	114	25	m)(x	m)(x	PROPN
ap-3450	114	26	,	,	PUNCT
ap-3450	114	27	y	y	PROPN
ap-3450	114	28	,	,	PUNCT
ap-3450	114	29	z	z	NOUN
ap-3450	114	30	)	)	PUNCT
ap-3450	114	31	,	,	PUNCT
ap-3450	114	32	where	where	SCONJ
ap-3450	114	33	the	the	DET
ap-3450	114	34	coefficients	coefficient	NOUN
ap-3450	114	35	a(k	a(k	PROPN
ap-3450	114	36	,	,	PUNCT
ap-3450	114	37	l	l	NOUN
ap-3450	114	38	,	,	PUNCT
ap-3450	114	39	m	m	NOUN
ap-3450	114	40	)	)	PUNCT
ap-3450	114	41	are	be	AUX
ap-3450	114	42	given	give	VERB
ap-3450	114	43	by	by	ADP
ap-3450	114	44	the	the	DET
ap-3450	114	45	formula	formula	NOUN
ap-3450	114	46	a(k	a(k	PROPN
ap-3450	114	47	,	,	PUNCT
ap-3450	114	48	l	l	NOUN
ap-3450	114	49	,	,	PUNCT
ap-3450	114	50	m	m	NOUN
ap-3450	114	51	)	)	PUNCT
ap-3450	114	52	=	=	SYM
ap-3450	114	53	1	1	NUM
ap-3450	114	54	n3ckclcmgklm	n3ckclcmgklm	PROPN
ap-3450	114	55	∑	∑	PROPN
ap-3450	114	56	0≤r	0≤r	PROPN
ap-3450	114	57	,	,	PUNCT
ap-3450	114	58	s	s	PART
ap-3450	114	59	,	,	PUNCT
ap-3450	114	60	t≤n	t≤n	ADJ
ap-3450	114	61	,	,	PUNCT
ap-3450	114	62	r≥s≥t	r≥s≥t	NOUN
ap-3450	114	63	or	or	CCONJ
ap-3450	114	64	s	s	NOUN
ap-3450	114	65	>	>	X
ap-3450	114	66	r	r	X
ap-3450	114	67	>	>	X
ap-3450	114	68	t	t	X
ap-3450	114	69	c̃r	c̃r	X
ap-3450	114	70	c̃sc̃t	c̃sc̃t	NUM
ap-3450	114	71	grst	grst	NOUN
ap-3450	114	72	f	f	X
ap-3450	114	73	(	(	PUNCT
ap-3450	114	74	r	r	NOUN
ap-3450	114	75	2n	2n	NUM
ap-3450	114	76	,	,	PUNCT
ap-3450	114	77	s	s	NOUN
ap-3450	114	78	2n	2n	NUM
ap-3450	114	79	,	,	PUNCT
ap-3450	114	80	t	t	PROPN
ap-3450	114	81	2n	2n	NUM
ap-3450	114	82	)	)	PUNCT
ap-3450	114	83	cos(k	cos(k	PROPN
ap-3450	114	84	,	,	PUNCT
ap-3450	114	85	l	l	NOUN
ap-3450	114	86	,	,	PUNCT
ap-3450	114	87	m	m	NOUN
ap-3450	114	88	)	)	PUNCT
ap-3450	114	89	(	(	PUNCT
ap-3450	114	90	r	r	NOUN
ap-3450	114	91	2n	2n	NUM
ap-3450	114	92	,	,	PUNCT
ap-3450	114	93	s	s	NOUN
ap-3450	114	94	2n	2n	NUM
ap-3450	114	95	,	,	PUNCT
ap-3450	114	96	t	t	PROPN
ap-3450	114	97	2n	2n	NUM
ap-3450	114	98	)	)	PUNCT
ap-3450	114	99	.	.	PUNCT
ap-3450	115	1	as	as	ADP
ap-3450	115	2	an	an	DET
ap-3450	115	3	interpolation	interpolation	NOUN
ap-3450	115	4	example	example	NOUN
ap-3450	115	5	,	,	PUNCT
ap-3450	115	6	we	we	PRON
ap-3450	115	7	take	take	VERB
ap-3450	115	8	f(x	f(x	PROPN
ap-3450	115	9	,	,	PUNCT
ap-3450	115	10	y	y	PROPN
ap-3450	115	11	,	,	PUNCT
ap-3450	115	12	z	z	NOUN
ap-3450	115	13	)	)	PUNCT
ap-3450	115	14	=	=	SYM
ap-3450	116	1	cos	cos	PROPN
ap-3450	116	2	2πx	2πx	NOUN
ap-3450	116	3	cos	cos	PROPN
ap-3450	116	4	2πy	2πy	PROPN
ap-3450	116	5	cos	cos	PROPN
ap-3450	116	6	2πz	2πz	PROPN
ap-3450	116	7	cos	cos	PROPN
ap-3450	116	8	10(x+	10(x+	NUM
ap-3450	116	9	y	y	PROPN
ap-3450	116	10	+	+	PROPN
ap-3450	116	11	z	z	NOUN
ap-3450	116	12	)	)	PUNCT
ap-3450	116	13	,	,	PUNCT
ap-3450	116	14	(	(	PUNCT
ap-3450	116	15	10	10	NUM
ap-3450	116	16	)	)	PUNCT
ap-3450	116	17	and	and	CCONJ
ap-3450	116	18	compute	compute	VERB
ap-3450	116	19	four	four	NUM
ap-3450	116	20	interpolations	interpolation	NOUN
ap-3450	116	21	fn	fn	NOUN
ap-3450	116	22	for	for	ADP
ap-3450	116	23	n	n	NOUN
ap-3450	116	24	=	=	SYM
ap-3450	116	25	1	1	NUM
ap-3450	116	26	,	,	PUNCT
ap-3450	116	27	2	2	NUM
ap-3450	116	28	,	,	PUNCT
ap-3450	116	29	3	3	NUM
ap-3450	116	30	,	,	PUNCT
ap-3450	116	31	4	4	NUM
ap-3450	116	32	using	use	VERB
ap-3450	116	33	alternating	alternate	VERB
ap-3450	116	34	cosines	cosine	NOUN
ap-3450	116	35	.	.	PUNCT
ap-3450	117	1	the	the	DET
ap-3450	117	2	graph	graph	NOUN
ap-3450	117	3	of	of	ADP
ap-3450	117	4	the	the	DET
ap-3450	117	5	function	function	NOUN
ap-3450	117	6	(	(	PUNCT
ap-3450	117	7	10	10	NUM
ap-3450	117	8	)	)	PUNCT
ap-3450	117	9	is	be	AUX
ap-3450	117	10	shown	show	VERB
ap-3450	117	11	in	in	ADP
ap-3450	117	12	fig	fig	NOUN
ap-3450	117	13	.	.	PUNCT
ap-3450	118	1	3	3	NUM
ap-3450	118	2	and	and	CCONJ
ap-3450	118	3	four	four	NUM
ap-3450	118	4	approximations	approximation	NOUN
ap-3450	118	5	in	in	ADP
ap-3450	118	6	fig	fig	NOUN
ap-3450	118	7	.	.	PUNCT
ap-3450	119	1	4	4	X
ap-3450	119	2	.	.	X
ap-3450	119	3	by	by	ADP
ap-3450	119	4	visual	visual	ADJ
ap-3450	119	5	inspection	inspection	NOUN
ap-3450	119	6	,	,	PUNCT
ap-3450	119	7	approximations	approximation	VERB
ap-3450	119	8	fn	fn	AUX
ap-3450	119	9	get	get	VERB
ap-3450	119	10	closer	close	ADJ
ap-3450	119	11	to	to	ADP
ap-3450	119	12	f	f	PROPN
ap-3450	119	13	as	as	ADP
ap-3450	119	14	n	n	DET
ap-3450	119	15	increases	increase	NOUN
ap-3450	119	16	.	.	PUNCT
ap-3450	120	1	this	this	PRON
ap-3450	120	2	is	be	AUX
ap-3450	120	3	verified	verify	VERB
ap-3450	120	4	by	by	ADP
ap-3450	120	5	computing	compute	VERB
ap-3450	120	6	integral	integral	ADJ
ap-3450	120	7	error	error	NOUN
ap-3450	120	8	estimates	estimate	NOUN
ap-3450	120	9	shown	show	VERB
ap-3450	120	10	in	in	ADP
ap-3450	120	11	table	table	NOUN
ap-3450	120	12	1	1	NUM
ap-3450	120	13	.	.	NOUN
ap-3450	120	14	160	160	NUM
ap-3450	120	15	vol	vol	NOUN
ap-3450	120	16	.	.	PUNCT
ap-3450	121	1	56	56	NUM
ap-3450	121	2	no	no	NOUN
ap-3450	121	3	.	.	PUNCT
ap-3450	122	1	3/2016	3/2016	NUM
ap-3450	122	2	three	three	NUM
ap-3450	122	3	-	-	PUNCT
ap-3450	122	4	variable	variable	NOUN
ap-3450	122	5	alternating	alternate	VERB
ap-3450	122	6	trigonometric	trigonometric	ADJ
ap-3450	122	7	functions	function	NOUN
ap-3450	122	8	5.2	5.2	NUM
ap-3450	122	9	.	.	PUNCT
ap-3450	123	1	amdct-2	amdct-2	PROPN
ap-3450	123	2	alternating	alternate	VERB
ap-3450	123	3	discrete	discrete	ADJ
ap-3450	123	4	cosine	cosine	NOUN
ap-3450	123	5	transform	transform	NOUN
ap-3450	123	6	of	of	ADP
ap-3450	123	7	second	second	ADJ
ap-3450	123	8	kind	kind	NOUN
ap-3450	123	9	uses	use	VERB
ap-3450	123	10	a	a	DET
ap-3450	123	11	lattice	lattice	NOUN
ap-3450	123	12	f̃n	f̃n	ADV
ap-3450	123	13	=	=	PUNCT
ap-3450	123	14	{	{	PUNCT
ap-3450	123	15	(	(	PUNCT
ap-3450	123	16	r+1/2	r+1/2	PROPN
ap-3450	123	17	2n	2n	NUM
ap-3450	123	18	,	,	PUNCT
ap-3450	123	19	s+1/2	s+1/2	PROPN
ap-3450	123	20	2n	2n	NUM
ap-3450	123	21	,	,	PUNCT
ap-3450	123	22	t+1/2	t+1/2	PROPN
ap-3450	123	23	2n	2n	NUM
ap-3450	123	24	)	)	PUNCT
ap-3450	123	25	∣∣	∣∣	NUM
ap-3450	123	26	0	0	NUM
ap-3450	123	27	≤	≤	NOUN
ap-3450	123	28	r	r	NOUN
ap-3450	123	29	,	,	PUNCT
ap-3450	123	30	s	s	PROPN
ap-3450	123	31	,	,	PUNCT
ap-3450	123	32	t	t	NOUN
ap-3450	123	33	≤	≤	NUM
ap-3450	123	34	n	n	CCONJ
ap-3450	123	35	−	−	PROPN
ap-3450	123	36	1	1	NUM
ap-3450	123	37	,	,	PUNCT
ap-3450	123	38	r	r	NOUN
ap-3450	123	39	≥	≥	NOUN
ap-3450	123	40	s	s	PART
ap-3450	123	41	≥	≥	NOUN
ap-3450	123	42	t	t	NOUN
ap-3450	123	43	or	or	CCONJ
ap-3450	123	44	s	s	VERB
ap-3450	123	45	>	>	X
ap-3450	123	46	r	r	X
ap-3450	123	47	>	>	X
ap-3450	123	48	t	t	PROPN
ap-3450	123	49	}	}	PUNCT
ap-3450	123	50	.	.	PUNCT
ap-3450	124	1	(	(	PUNCT
ap-3450	124	2	11	11	NUM
ap-3450	124	3	)	)	PUNCT
ap-3450	124	4	this	this	DET
ap-3450	124	5	lattice	lattice	NOUN
ap-3450	124	6	contains	contain	VERB
ap-3450	124	7	1	1	NUM
ap-3450	124	8	3n(n2	3n(n2	NUM
ap-3450	124	9	+	+	CCONJ
ap-3450	124	10	2	2	NUM
ap-3450	124	11	)	)	PUNCT
ap-3450	124	12	points	point	NOUN
ap-3450	124	13	.	.	PUNCT
ap-3450	125	1	for	for	ADP
ap-3450	125	2	example	example	NOUN
ap-3450	125	3	,	,	PUNCT
ap-3450	125	4	f̃2	f̃2	PROPN
ap-3450	125	5	=	=	PRON
ap-3450	125	6	{	{	PUNCT
ap-3450	125	7	(	(	PUNCT
ap-3450	125	8	1	1	NUM
ap-3450	125	9	8	8	NUM
ap-3450	125	10	,	,	PUNCT
ap-3450	125	11	1	1	NUM
ap-3450	125	12	8	8	NUM
ap-3450	125	13	,	,	PUNCT
ap-3450	125	14	1	1	NUM
ap-3450	125	15	8	8	NUM
ap-3450	125	16	)	)	PUNCT
ap-3450	125	17	,	,	PUNCT
ap-3450	125	18	(	(	PUNCT
ap-3450	125	19	3	3	NUM
ap-3450	125	20	8	8	NUM
ap-3450	125	21	,	,	PUNCT
ap-3450	125	22	1	1	NUM
ap-3450	125	23	8	8	NUM
ap-3450	125	24	,	,	PUNCT
ap-3450	125	25	1	1	NUM
ap-3450	125	26	8	8	NUM
ap-3450	125	27	)	)	PUNCT
ap-3450	125	28	,	,	PUNCT
ap-3450	125	29	(	(	PUNCT
ap-3450	125	30	3	3	NUM
ap-3450	125	31	8	8	NUM
ap-3450	125	32	,	,	PUNCT
ap-3450	125	33	3	3	NUM
ap-3450	125	34	8	8	NUM
ap-3450	125	35	,	,	PUNCT
ap-3450	125	36	1	1	NUM
ap-3450	125	37	8	8	NUM
ap-3450	125	38	)	)	PUNCT
ap-3450	125	39	,	,	PUNCT
ap-3450	125	40	(	(	PUNCT
ap-3450	125	41	3	3	NUM
ap-3450	125	42	8	8	NUM
ap-3450	125	43	,	,	PUNCT
ap-3450	125	44	3	3	NUM
ap-3450	125	45	8	8	NUM
ap-3450	125	46	,	,	PUNCT
ap-3450	125	47	3	3	NUM
ap-3450	125	48	8	8	NUM
ap-3450	125	49	)	)	PUNCT
ap-3450	125	50	}	}	PUNCT
ap-3450	125	51	.	.	PUNCT
ap-3450	126	1	the	the	DET
ap-3450	126	2	alternating	alternate	VERB
ap-3450	126	3	cosines	cosine	NOUN
ap-3450	126	4	cos(k	cos(k	PRON
ap-3450	126	5	,	,	PUNCT
ap-3450	126	6	l	l	NOUN
ap-3450	126	7	,	,	PUNCT
ap-3450	126	8	m	m	NOUN
ap-3450	126	9	)	)	PUNCT
ap-3450	126	10	are	be	AUX
ap-3450	126	11	pairwise	pairwise	NOUN
ap-3450	126	12	orthogonal	orthogonal	NOUN
ap-3450	126	13	on	on	ADP
ap-3450	126	14	this	this	DET
ap-3450	126	15	lattice	lattice	NOUN
ap-3450	126	16	,	,	PUNCT
ap-3450	126	17	namely	namely	ADV
ap-3450	126	18	we	we	PRON
ap-3450	126	19	have∑	have∑	VERB
ap-3450	126	20	0≤r	0≤r	PROPN
ap-3450	126	21	,	,	PUNCT
ap-3450	126	22	s	s	X
ap-3450	126	23	,	,	PUNCT
ap-3450	126	24	t≤n−1	t≤n−1	ADJ
ap-3450	126	25	,	,	PUNCT
ap-3450	126	26	r≥s≥t	r≥s≥t	NOUN
ap-3450	126	27	or	or	CCONJ
ap-3450	126	28	s	s	NOUN
ap-3450	126	29	>	>	X
ap-3450	126	30	r	r	X
ap-3450	126	31	>	>	X
ap-3450	126	32	t	t	X
ap-3450	126	33	g−1	g−1	PROPN
ap-3450	126	34	rstcrcsct	rstcrcsct	PROPN
ap-3450	126	35	cos(k	cos(k	PROPN
ap-3450	126	36	,	,	PUNCT
ap-3450	126	37	l	l	NOUN
ap-3450	126	38	,	,	PUNCT
ap-3450	126	39	m	m	NOUN
ap-3450	126	40	)	)	PUNCT
ap-3450	127	1	(	(	PUNCT
ap-3450	127	2	r+1/2	r+1/2	PROPN
ap-3450	127	3	2n	2n	NUM
ap-3450	127	4	,	,	PUNCT
ap-3450	127	5	s+1/2	s+1/2	PROPN
ap-3450	127	6	2n	2n	NUM
ap-3450	127	7	,	,	PUNCT
ap-3450	127	8	t+1/2	t+1/2	PROPN
ap-3450	127	9	2n	2n	NUM
ap-3450	127	10	)	)	PUNCT
ap-3450	127	11	cos(k′,l′,m′	cos(k′,l′,m′	PROPN
ap-3450	127	12	)	)	PUNCT
ap-3450	127	13	(	(	PUNCT
ap-3450	127	14	r+1/2	r+1/2	PROPN
ap-3450	127	15	2n	2n	NUM
ap-3450	127	16	,	,	PUNCT
ap-3450	127	17	s+1/2	s+1/2	PROPN
ap-3450	127	18	2n	2n	NUM
ap-3450	127	19	,	,	PUNCT
ap-3450	127	20	t+1/2	t+1/2	PROPN
ap-3450	127	21	2n	2n	NUM
ap-3450	127	22	)	)	PUNCT
ap-3450	128	1	=	=	PUNCT
ap-3450	128	2	n3c̃k	n3c̃k	PROPN
ap-3450	128	3	c̃lc̃mgklmδ(k	c̃lc̃mgklmδ(k	PROPN
ap-3450	128	4	,	,	PUNCT
ap-3450	128	5	l	l	NOUN
ap-3450	128	6	,	,	PUNCT
ap-3450	128	7	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	128	8	)	)	PUNCT
ap-3450	128	9	,	,	PUNCT
ap-3450	128	10	where	where	SCONJ
ap-3450	128	11	0	0	NUM
ap-3450	128	12	≤	≤	NUM
ap-3450	128	13	k	k	NOUN
ap-3450	128	14	,	,	PUNCT
ap-3450	128	15	l	l	NOUN
ap-3450	128	16	,	,	PUNCT
ap-3450	128	17	m	m	VERB
ap-3450	128	18	≤	≤	NOUN
ap-3450	128	19	n	n	CCONJ
ap-3450	128	20	−	−	PROPN
ap-3450	128	21	1	1	NUM
ap-3450	128	22	,	,	PUNCT
ap-3450	128	23	k	k	X
ap-3450	128	24	≥	≥	PROPN
ap-3450	128	25	l	l	PROPN
ap-3450	128	26	≥	≥	NOUN
ap-3450	128	27	m	m	PROPN
ap-3450	128	28	or	or	CCONJ
ap-3450	128	29	l	l	X
ap-3450	128	30	>	>	X
ap-3450	129	1	k	k	X
ap-3450	129	2	>	>	X
ap-3450	129	3	m	m	PROPN
ap-3450	129	4	,	,	PUNCT
ap-3450	129	5	and	and	CCONJ
ap-3450	129	6	0	0	NUM
ap-3450	129	7	≤	≤	NOUN
ap-3450	129	8	k′	k′	PROPN
ap-3450	129	9	,	,	PUNCT
ap-3450	129	10	l′,m′	l′,m′	PROPN
ap-3450	129	11	≤	≤	NOUN
ap-3450	129	12	n	n	CCONJ
ap-3450	129	13	−	−	PROPN
ap-3450	129	14	1	1	NUM
ap-3450	129	15	,	,	PUNCT
ap-3450	129	16	k′	k′	PROPN
ap-3450	129	17	≥	≥	NUM
ap-3450	129	18	l′	l′	PRON
ap-3450	129	19	≥	≥	PROPN
ap-3450	129	20	m′	m′	NUM
ap-3450	129	21	or	or	CCONJ
ap-3450	129	22	l′	l′	VERB
ap-3450	129	23	>	>	X
ap-3450	129	24	k′	k′	PROPN
ap-3450	129	25	>	>	X
ap-3450	129	26	m′.	m′.	PROPN
ap-3450	129	27	let	let	VERB
ap-3450	129	28	f̃	f̃	PROPN
ap-3450	129	29	be	be	AUX
ap-3450	129	30	a	a	DET
ap-3450	129	31	real	real	ADJ
ap-3450	129	32	function	function	NOUN
ap-3450	129	33	defined	define	VERB
ap-3450	129	34	on	on	ADP
ap-3450	129	35	f̃n	f̃n	ADJ
ap-3450	129	36	.	.	PUNCT
ap-3450	130	1	then	then	ADV
ap-3450	130	2	it	it	PRON
ap-3450	130	3	can	can	AUX
ap-3450	130	4	be	be	AUX
ap-3450	130	5	expanded	expand	VERB
ap-3450	130	6	into	into	ADP
ap-3450	130	7	a	a	DET
ap-3450	130	8	sum	sum	NOUN
ap-3450	130	9	of	of	ADP
ap-3450	130	10	alternating	alternate	VERB
ap-3450	130	11	cosines	cosine	NOUN
ap-3450	130	12	,	,	PUNCT
ap-3450	130	13	f̃(x	f̃(x	PROPN
ap-3450	130	14	,	,	PUNCT
ap-3450	130	15	y	y	PROPN
ap-3450	130	16	,	,	PUNCT
ap-3450	130	17	z	z	NOUN
ap-3450	130	18	)	)	PUNCT
ap-3450	130	19	=	=	PUNCT
ap-3450	131	1	∑	∑	PUNCT
ap-3450	131	2	0≤k	0≤k	PROPN
ap-3450	131	3	,	,	PUNCT
ap-3450	131	4	l	l	NOUN
ap-3450	131	5	,	,	PUNCT
ap-3450	131	6	m≤n−1	m≤n−1	ADJ
ap-3450	131	7	,	,	PUNCT
ap-3450	131	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	131	9	or	or	CCONJ
ap-3450	131	10	l	l	NOUN
ap-3450	131	11	>	>	X
ap-3450	131	12	k	k	PROPN
ap-3450	131	13	>	>	PROPN
ap-3450	131	14	m	m	PROPN
ap-3450	131	15	b(k	b(k	PROPN
ap-3450	131	16	,	,	PUNCT
ap-3450	131	17	l	l	NOUN
ap-3450	131	18	,	,	PUNCT
ap-3450	131	19	m	m	NOUN
ap-3450	131	20	)	)	PUNCT
ap-3450	131	21	cos(k	cos(k	PROPN
ap-3450	131	22	,	,	PUNCT
ap-3450	131	23	l	l	NOUN
ap-3450	131	24	,	,	PUNCT
ap-3450	131	25	m)(x	m)(x	PROPN
ap-3450	131	26	,	,	PUNCT
ap-3450	131	27	y	y	PROPN
ap-3450	131	28	,	,	PUNCT
ap-3450	131	29	z	z	NOUN
ap-3450	131	30	)	)	PUNCT
ap-3450	131	31	,	,	PUNCT
ap-3450	131	32	where	where	SCONJ
ap-3450	131	33	the	the	DET
ap-3450	131	34	coefficients	coefficient	NOUN
ap-3450	131	35	b(k	b(k	PROPN
ap-3450	131	36	,	,	PUNCT
ap-3450	131	37	l	l	NOUN
ap-3450	131	38	,	,	PUNCT
ap-3450	131	39	m	m	NOUN
ap-3450	131	40	)	)	PUNCT
ap-3450	131	41	are	be	AUX
ap-3450	131	42	given	give	VERB
ap-3450	131	43	by	by	ADP
ap-3450	131	44	the	the	DET
ap-3450	131	45	formula	formula	NOUN
ap-3450	131	46	b(k	b(k	PROPN
ap-3450	131	47	,	,	PUNCT
ap-3450	131	48	l	l	NOUN
ap-3450	131	49	,	,	PUNCT
ap-3450	131	50	m	m	NOUN
ap-3450	131	51	)	)	PUNCT
ap-3450	131	52	=	=	SYM
ap-3450	131	53	1	1	NUM
ap-3450	131	54	n3ckclcmgklm	n3ckclcmgklm	PROPN
ap-3450	131	55	∑	∑	PROPN
ap-3450	131	56	0≤r	0≤r	PROPN
ap-3450	131	57	,	,	PUNCT
ap-3450	131	58	s	s	X
ap-3450	131	59	,	,	PUNCT
ap-3450	131	60	t≤n−1	t≤n−1	ADJ
ap-3450	131	61	,	,	PUNCT
ap-3450	131	62	r≥s≥t	r≥s≥t	NOUN
ap-3450	131	63	or	or	CCONJ
ap-3450	131	64	s	s	NOUN
ap-3450	131	65	>	>	X
ap-3450	131	66	r	r	X
ap-3450	131	67	>	>	X
ap-3450	131	68	t	t	X
ap-3450	131	69	c̃r	c̃r	X
ap-3450	131	70	c̃sc̃t	c̃sc̃t	NUM
ap-3450	131	71	grst	grst	NOUN
ap-3450	131	72	f̃	f̃	PROPN
ap-3450	131	73	(	(	PUNCT
ap-3450	131	74	r+1/2	r+1/2	PROPN
ap-3450	131	75	2n	2n	NUM
ap-3450	131	76	,	,	PUNCT
ap-3450	131	77	s+1/2	s+1/2	PROPN
ap-3450	131	78	2n	2n	NUM
ap-3450	131	79	,	,	PUNCT
ap-3450	131	80	t+1/2	t+1/2	PROPN
ap-3450	131	81	2n	2n	NUM
ap-3450	131	82	)	)	PUNCT
ap-3450	131	83	cos(k	cos(k	PROPN
ap-3450	131	84	,	,	PUNCT
ap-3450	131	85	l	l	NOUN
ap-3450	131	86	,	,	PUNCT
ap-3450	131	87	m	m	NOUN
ap-3450	131	88	)	)	PUNCT
ap-3450	131	89	(	(	PUNCT
ap-3450	131	90	r+1/2	r+1/2	PROPN
ap-3450	131	91	2n	2n	NUM
ap-3450	131	92	,	,	PUNCT
ap-3450	131	93	s+1/2	s+1/2	PROPN
ap-3450	131	94	2n	2n	NUM
ap-3450	131	95	,	,	PUNCT
ap-3450	131	96	t+1/2	t+1/2	PROPN
ap-3450	131	97	2n	2n	NUM
ap-3450	131	98	)	)	PUNCT
ap-3450	131	99	.	.	PUNCT
ap-3450	132	1	5.3	5.3	NUM
ap-3450	132	2	.	.	PUNCT
ap-3450	133	1	amdct-3	amdct-3	ADP
ap-3450	133	2	alternating	alternate	VERB
ap-3450	133	3	discrete	discrete	ADJ
ap-3450	133	4	cosine	cosine	NOUN
ap-3450	133	5	transform	transform	NOUN
ap-3450	133	6	of	of	ADP
ap-3450	133	7	the	the	DET
ap-3450	133	8	third	third	ADJ
ap-3450	133	9	kind	kind	NOUN
ap-3450	133	10	uses	use	VERB
ap-3450	133	11	the	the	DET
ap-3450	133	12	lattice	lattice	PROPN
ap-3450	133	13	f	f	PROPN
ap-3450	133	14	′n	′n	PROPN
ap-3450	133	15	=	=	X
ap-3450	133	16	{	{	PUNCT
ap-3450	133	17	(	(	PUNCT
ap-3450	133	18	r	r	NOUN
ap-3450	133	19	2n	2n	NUM
ap-3450	133	20	,	,	PUNCT
ap-3450	133	21	s	s	NOUN
ap-3450	133	22	2n	2n	NUM
ap-3450	133	23	,	,	PUNCT
ap-3450	133	24	t	t	PROPN
ap-3450	133	25	2n	2n	NUM
ap-3450	133	26	)	)	PUNCT
ap-3450	133	27	∣∣	∣∣	NUM
ap-3450	133	28	0	0	NUM
ap-3450	133	29	≤	≤	NOUN
ap-3450	133	30	r	r	NOUN
ap-3450	133	31	,	,	PUNCT
ap-3450	133	32	s	s	PROPN
ap-3450	133	33	,	,	PUNCT
ap-3450	133	34	t	t	NOUN
ap-3450	133	35	≤	≤	NUM
ap-3450	133	36	n	n	CCONJ
ap-3450	133	37	−	−	PROPN
ap-3450	133	38	1	1	NUM
ap-3450	133	39	,	,	PUNCT
ap-3450	133	40	r	r	NOUN
ap-3450	133	41	≥	≥	NOUN
ap-3450	133	42	s	s	PART
ap-3450	133	43	≥	≥	NOUN
ap-3450	133	44	t	t	NOUN
ap-3450	133	45	or	or	CCONJ
ap-3450	133	46	s	s	VERB
ap-3450	133	47	>	>	X
ap-3450	133	48	r	r	X
ap-3450	133	49	>	>	X
ap-3450	133	50	t	t	PROPN
ap-3450	133	51	}	}	PUNCT
ap-3450	133	52	.	.	PUNCT
ap-3450	134	1	instead	instead	ADV
ap-3450	134	2	of	of	ADP
ap-3450	134	3	alternating	alternate	VERB
ap-3450	134	4	cosines	cosine	NOUN
ap-3450	134	5	cos(k	cos(k	PROPN
ap-3450	134	6	,	,	PUNCT
ap-3450	134	7	l	l	NOUN
ap-3450	134	8	,	,	PUNCT
ap-3450	134	9	m	m	NOUN
ap-3450	134	10	)	)	PUNCT
ap-3450	134	11	,	,	PUNCT
ap-3450	134	12	shifted	shift	VERB
ap-3450	134	13	cosines	cosine	NOUN
ap-3450	134	14	cos(k+1/2,l+1/2,m+1/2	cos(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	134	15	)	)	PUNCT
ap-3450	134	16	are	be	AUX
ap-3450	134	17	used	use	VERB
ap-3450	134	18	.	.	PUNCT
ap-3450	135	1	they	they	PRON
ap-3450	135	2	have	have	VERB
ap-3450	135	3	similar	similar	ADJ
ap-3450	135	4	properties	property	NOUN
ap-3450	135	5	as	as	ADP
ap-3450	135	6	normal	normal	ADJ
ap-3450	135	7	alternating	alternate	VERB
ap-3450	135	8	cosines	cosine	NOUN
ap-3450	135	9	with	with	ADP
ap-3450	135	10	integer	integer	NOUN
ap-3450	135	11	arguments	argument	NOUN
ap-3450	135	12	.	.	PUNCT
ap-3450	136	1	for	for	ADP
ap-3450	136	2	scalar	scalar	ADJ
ap-3450	136	3	product	product	NOUN
ap-3450	136	4	we	we	PRON
ap-3450	136	5	obtain∑	obtain∑	VERB
ap-3450	136	6	0≤r	0≤r	NOUN
ap-3450	136	7	,	,	PUNCT
ap-3450	136	8	s	s	X
ap-3450	136	9	,	,	PUNCT
ap-3450	136	10	t≤n−1	t≤n−1	ADJ
ap-3450	136	11	,	,	PUNCT
ap-3450	136	12	r≥s≥t	r≥s≥t	NOUN
ap-3450	136	13	or	or	CCONJ
ap-3450	136	14	s	s	NOUN
ap-3450	136	15	>	>	X
ap-3450	136	16	r	r	X
ap-3450	136	17	>	>	X
ap-3450	136	18	t	t	PROPN
ap-3450	136	19	g−1	g−1	PROPN
ap-3450	136	20	rstcrcsct	rstcrcsct	PROPN
ap-3450	136	21	cos(k+1/2,l+1/2,m+1/2	cos(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	136	22	)	)	PUNCT
ap-3450	136	23	(	(	PUNCT
ap-3450	136	24	r	r	NOUN
ap-3450	136	25	2n	2n	NUM
ap-3450	136	26	,	,	PUNCT
ap-3450	136	27	s	s	NOUN
ap-3450	136	28	2n	2n	NUM
ap-3450	136	29	,	,	PUNCT
ap-3450	136	30	t	t	PROPN
ap-3450	136	31	2n	2n	NUM
ap-3450	136	32	)	)	PUNCT
ap-3450	136	33	cos(k′+1/2,l′+1/2,m′+1/2	cos(k′+1/2,l′+1/2,m′+1/2	PROPN
ap-3450	136	34	)	)	PUNCT
ap-3450	136	35	(	(	PUNCT
ap-3450	136	36	r	r	NOUN
ap-3450	136	37	2n	2n	NUM
ap-3450	136	38	,	,	PUNCT
ap-3450	136	39	s	s	NOUN
ap-3450	136	40	2n	2n	NUM
ap-3450	136	41	,	,	PUNCT
ap-3450	136	42	t	t	PROPN
ap-3450	136	43	2n	2n	NUM
ap-3450	136	44	)	)	PUNCT
ap-3450	137	1	=	=	PUNCT
ap-3450	137	2	n3c̃k	n3c̃k	PROPN
ap-3450	137	3	c̃lc̃mgklmδ(k	c̃lc̃mgklmδ(k	PROPN
ap-3450	137	4	,	,	PUNCT
ap-3450	137	5	l	l	NOUN
ap-3450	137	6	,	,	PUNCT
ap-3450	137	7	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	137	8	)	)	PUNCT
ap-3450	137	9	,	,	PUNCT
ap-3450	137	10	where	where	SCONJ
ap-3450	137	11	0	0	NUM
ap-3450	137	12	≤	≤	NUM
ap-3450	137	13	k	k	NOUN
ap-3450	137	14	,	,	PUNCT
ap-3450	137	15	l	l	NOUN
ap-3450	137	16	,	,	PUNCT
ap-3450	137	17	m	m	VERB
ap-3450	137	18	≤	≤	NOUN
ap-3450	137	19	n	n	CCONJ
ap-3450	137	20	−	−	PROPN
ap-3450	137	21	1	1	NUM
ap-3450	137	22	,	,	PUNCT
ap-3450	137	23	k	k	X
ap-3450	137	24	≥	≥	PROPN
ap-3450	137	25	l	l	PROPN
ap-3450	137	26	≥	≥	NOUN
ap-3450	137	27	m	m	PROPN
ap-3450	137	28	or	or	CCONJ
ap-3450	137	29	l	l	X
ap-3450	137	30	>	>	X
ap-3450	138	1	k	k	X
ap-3450	138	2	>	>	X
ap-3450	138	3	m	m	PROPN
ap-3450	138	4	,	,	PUNCT
ap-3450	138	5	and	and	CCONJ
ap-3450	138	6	0	0	NUM
ap-3450	138	7	≤	≤	NOUN
ap-3450	138	8	k′	k′	PROPN
ap-3450	138	9	,	,	PUNCT
ap-3450	138	10	l′,m′	l′,m′	PROPN
ap-3450	138	11	≤	≤	NOUN
ap-3450	138	12	n	n	CCONJ
ap-3450	138	13	−	−	PROPN
ap-3450	138	14	1	1	NUM
ap-3450	138	15	,	,	PUNCT
ap-3450	138	16	k′	k′	PROPN
ap-3450	138	17	≥	≥	NUM
ap-3450	138	18	l′	l′	PRON
ap-3450	138	19	≥	≥	PROPN
ap-3450	138	20	m′	m′	NUM
ap-3450	138	21	or	or	CCONJ
ap-3450	138	22	l′	l′	VERB
ap-3450	138	23	>	>	X
ap-3450	138	24	k′	k′	PROPN
ap-3450	138	25	>	>	X
ap-3450	138	26	m′.	m′.	PROPN
ap-3450	138	27	let	let	VERB
ap-3450	138	28	g	g	PRON
ap-3450	138	29	be	be	AUX
ap-3450	138	30	a	a	DET
ap-3450	138	31	real	real	ADJ
ap-3450	138	32	function	function	NOUN
ap-3450	138	33	defined	define	VERB
ap-3450	138	34	on	on	ADP
ap-3450	138	35	f	f	PROPN
ap-3450	138	36	′n	′n	PROPN
ap-3450	138	37	.	.	PUNCT
ap-3450	139	1	then	then	ADV
ap-3450	139	2	it	it	PRON
ap-3450	139	3	can	can	AUX
ap-3450	139	4	be	be	AUX
ap-3450	139	5	expanded	expand	VERB
ap-3450	139	6	into	into	ADP
ap-3450	139	7	a	a	DET
ap-3450	139	8	sum	sum	NOUN
ap-3450	139	9	of	of	ADP
ap-3450	139	10	shifted	shift	VERB
ap-3450	139	11	alternating	alternate	VERB
ap-3450	139	12	cosines	cosine	NOUN
ap-3450	139	13	,	,	PUNCT
ap-3450	139	14	g(x	g(x	PROPN
ap-3450	139	15	,	,	PUNCT
ap-3450	139	16	y	y	PROPN
ap-3450	139	17	,	,	PUNCT
ap-3450	139	18	z	z	NOUN
ap-3450	139	19	)	)	PUNCT
ap-3450	139	20	=	=	PUNCT
ap-3450	140	1	∑	∑	PUNCT
ap-3450	140	2	0≤k	0≤k	PROPN
ap-3450	140	3	,	,	PUNCT
ap-3450	140	4	l	l	NOUN
ap-3450	140	5	,	,	PUNCT
ap-3450	140	6	m≤n−1	m≤n−1	ADJ
ap-3450	140	7	,	,	PUNCT
ap-3450	140	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	140	9	or	or	CCONJ
ap-3450	140	10	l	l	NOUN
ap-3450	140	11	>	>	X
ap-3450	140	12	k	k	PROPN
ap-3450	140	13	>	>	NOUN
ap-3450	140	14	m	m	NOUN
ap-3450	140	15	c(k	c(k	NOUN
ap-3450	140	16	,	,	PUNCT
ap-3450	140	17	l	l	NOUN
ap-3450	140	18	,	,	PUNCT
ap-3450	140	19	m	m	NOUN
ap-3450	140	20	)	)	PUNCT
ap-3450	140	21	cos(k+1/2,l+1/2,m+1/2)(x	cos(k+1/2,l+1/2,m+1/2)(x	PROPN
ap-3450	140	22	,	,	PUNCT
ap-3450	140	23	y	y	PROPN
ap-3450	140	24	,	,	PUNCT
ap-3450	140	25	z	z	NOUN
ap-3450	140	26	)	)	PUNCT
ap-3450	140	27	,	,	PUNCT
ap-3450	140	28	where	where	SCONJ
ap-3450	140	29	the	the	DET
ap-3450	140	30	coefficients	coefficient	NOUN
ap-3450	140	31	c(k	c(k	PROPN
ap-3450	140	32	,	,	PUNCT
ap-3450	140	33	l	l	NOUN
ap-3450	140	34	,	,	PUNCT
ap-3450	140	35	m	m	NOUN
ap-3450	140	36	)	)	PUNCT
ap-3450	140	37	are	be	AUX
ap-3450	140	38	given	give	VERB
ap-3450	140	39	by	by	ADP
ap-3450	140	40	the	the	DET
ap-3450	140	41	formula	formula	NOUN
ap-3450	140	42	c(k	c(k	NOUN
ap-3450	140	43	,	,	PUNCT
ap-3450	140	44	l	l	NOUN
ap-3450	140	45	,	,	PUNCT
ap-3450	140	46	m	m	NOUN
ap-3450	140	47	)	)	PUNCT
ap-3450	140	48	=	=	SYM
ap-3450	140	49	1	1	NUM
ap-3450	140	50	n3ckclcmgklm	n3ckclcmgklm	PROPN
ap-3450	140	51	∑	∑	PROPN
ap-3450	140	52	0≤r	0≤r	PROPN
ap-3450	140	53	,	,	PUNCT
ap-3450	140	54	s	s	X
ap-3450	140	55	,	,	PUNCT
ap-3450	140	56	t≤n−1	t≤n−1	ADJ
ap-3450	140	57	,	,	PUNCT
ap-3450	140	58	r≥s≥t	r≥s≥t	NOUN
ap-3450	140	59	or	or	CCONJ
ap-3450	140	60	s	s	NOUN
ap-3450	140	61	>	>	X
ap-3450	140	62	r	r	X
ap-3450	140	63	>	>	X
ap-3450	140	64	t	t	X
ap-3450	140	65	c̃r	c̃r	X
ap-3450	140	66	c̃sc̃t	c̃sc̃t	NUM
ap-3450	140	67	grst	grst	NOUN
ap-3450	140	68	g	g	PROPN
ap-3450	140	69	(	(	PUNCT
ap-3450	140	70	r	r	NOUN
ap-3450	140	71	2n	2n	NUM
ap-3450	140	72	,	,	PUNCT
ap-3450	140	73	s	s	NOUN
ap-3450	140	74	2n	2n	NUM
ap-3450	140	75	,	,	PUNCT
ap-3450	140	76	t	t	PROPN
ap-3450	140	77	2n	2n	NUM
ap-3450	140	78	)	)	PUNCT
ap-3450	140	79	cos(k+1/2,l+1/2,m+1/2	cos(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	140	80	)	)	PUNCT
ap-3450	140	81	(	(	PUNCT
ap-3450	140	82	r	r	NOUN
ap-3450	140	83	2n	2n	NUM
ap-3450	140	84	,	,	PUNCT
ap-3450	140	85	s	s	NOUN
ap-3450	140	86	2n	2n	NUM
ap-3450	140	87	,	,	PUNCT
ap-3450	140	88	t	t	PROPN
ap-3450	140	89	2n	2n	NUM
ap-3450	140	90	)	)	PUNCT
ap-3450	140	91	.	.	PUNCT
ap-3450	141	1	5.4	5.4	NUM
ap-3450	141	2	.	.	PUNCT
ap-3450	142	1	amdct-4	amdct-4	NUM
ap-3450	142	2	alternating	alternate	VERB
ap-3450	142	3	discrete	discrete	ADJ
ap-3450	142	4	cosine	cosine	NOUN
ap-3450	142	5	transform	transform	NOUN
ap-3450	142	6	of	of	ADP
ap-3450	142	7	the	the	DET
ap-3450	142	8	fourth	fourth	ADJ
ap-3450	142	9	kind	kind	NOUN
ap-3450	142	10	uses	use	VERB
ap-3450	142	11	the	the	DET
ap-3450	142	12	lattice	lattice	NOUN
ap-3450	142	13	f̃n	f̃n	NOUN
ap-3450	142	14	and	and	CCONJ
ap-3450	142	15	shifted	shift	VERB
ap-3450	142	16	cosines	cosine	NOUN
ap-3450	142	17	as	as	ADP
ap-3450	142	18	in	in	ADP
ap-3450	142	19	the	the	DET
ap-3450	142	20	third	third	ADJ
ap-3450	142	21	case	case	NOUN
ap-3450	142	22	.	.	PUNCT
ap-3450	143	1	we	we	PRON
ap-3450	143	2	have∑	have∑	VERB
ap-3450	143	3	0≤r	0≤r	PROPN
ap-3450	143	4	,	,	PUNCT
ap-3450	143	5	s	s	X
ap-3450	143	6	,	,	PUNCT
ap-3450	143	7	t≤n−1	t≤n−1	ADJ
ap-3450	143	8	,	,	PUNCT
ap-3450	143	9	r≥s≥t	r≥s≥t	NOUN
ap-3450	143	10	or	or	CCONJ
ap-3450	143	11	s	s	NOUN
ap-3450	143	12	>	>	X
ap-3450	143	13	r	r	X
ap-3450	143	14	>	>	X
ap-3450	143	15	t	t	PROPN
ap-3450	143	16	g−1	g−1	PROPN
ap-3450	143	17	rstcrcsct	rstcrcsct	PROPN
ap-3450	143	18	cos(k+1/2,l+1/2,m+1/2	cos(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	143	19	)	)	PUNCT
ap-3450	143	20	(	(	PUNCT
ap-3450	143	21	r+1/2	r+1/2	PROPN
ap-3450	143	22	2n	2n	NUM
ap-3450	143	23	,	,	PUNCT
ap-3450	143	24	s+1/2	s+1/2	PROPN
ap-3450	143	25	2n	2n	NUM
ap-3450	143	26	,	,	PUNCT
ap-3450	143	27	t+1/2	t+1/2	PROPN
ap-3450	143	28	2n	2n	NUM
ap-3450	143	29	)	)	PUNCT
ap-3450	143	30	×	×	PROPN
ap-3450	143	31	cos(k′+1/2,l′+1/2,m′+1/2	cos(k′+1/2,l′+1/2,m′+1/2	NOUN
ap-3450	143	32	)	)	PUNCT
ap-3450	143	33	(	(	PUNCT
ap-3450	143	34	r+1/2	r+1/2	PROPN
ap-3450	143	35	2n	2n	NUM
ap-3450	143	36	,	,	PUNCT
ap-3450	143	37	s+1/2	s+1/2	PROPN
ap-3450	143	38	2n	2n	NUM
ap-3450	143	39	,	,	PUNCT
ap-3450	143	40	t+1/2	t+1/2	PROPN
ap-3450	143	41	2n	2n	NUM
ap-3450	143	42	)	)	PUNCT
ap-3450	144	1	=	=	PUNCT
ap-3450	144	2	n3c̃k	n3c̃k	PROPN
ap-3450	144	3	c̃lc̃mgklmδ(k	c̃lc̃mgklmδ(k	PROPN
ap-3450	144	4	,	,	PUNCT
ap-3450	144	5	l	l	NOUN
ap-3450	144	6	,	,	PUNCT
ap-3450	144	7	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	144	8	)	)	PUNCT
ap-3450	144	9	,	,	PUNCT
ap-3450	144	10	where	where	SCONJ
ap-3450	144	11	0	0	NUM
ap-3450	144	12	≤	≤	NUM
ap-3450	144	13	k	k	NOUN
ap-3450	144	14	,	,	PUNCT
ap-3450	144	15	l	l	NOUN
ap-3450	144	16	,	,	PUNCT
ap-3450	144	17	m	m	VERB
ap-3450	144	18	≤	≤	NOUN
ap-3450	144	19	n	n	CCONJ
ap-3450	144	20	−	−	PROPN
ap-3450	144	21	1	1	NUM
ap-3450	144	22	,	,	PUNCT
ap-3450	144	23	k	k	X
ap-3450	144	24	≥	≥	PROPN
ap-3450	144	25	l	l	PROPN
ap-3450	144	26	≥	≥	NOUN
ap-3450	144	27	m	m	PROPN
ap-3450	144	28	or	or	CCONJ
ap-3450	144	29	l	l	X
ap-3450	144	30	>	>	X
ap-3450	145	1	k	k	X
ap-3450	145	2	>	>	X
ap-3450	145	3	m	m	PROPN
ap-3450	145	4	,	,	PUNCT
ap-3450	145	5	and	and	CCONJ
ap-3450	145	6	0	0	NUM
ap-3450	145	7	≤	≤	NOUN
ap-3450	145	8	k′	k′	PROPN
ap-3450	145	9	,	,	PUNCT
ap-3450	145	10	l′,m′	l′,m′	PROPN
ap-3450	145	11	≤	≤	NOUN
ap-3450	145	12	n	n	CCONJ
ap-3450	145	13	−	−	PROPN
ap-3450	145	14	1	1	NUM
ap-3450	145	15	,	,	PUNCT
ap-3450	145	16	k′	k′	PROPN
ap-3450	145	17	≥	≥	NUM
ap-3450	145	18	l′	l′	PRON
ap-3450	145	19	≥	≥	PROPN
ap-3450	145	20	m′	m′	NUM
ap-3450	145	21	or	or	CCONJ
ap-3450	145	22	l′	l′	VERB
ap-3450	145	23	>	>	X
ap-3450	145	24	k′	k′	PROPN
ap-3450	145	25	>	>	X
ap-3450	145	26	m′.	m′.	PROPN
ap-3450	145	27	161	161	NUM
ap-3450	145	28	agata	agata	PROPN
ap-3450	145	29	bezubik	bezubik	PROPN
ap-3450	145	30	,	,	PUNCT
ap-3450	145	31	severin	severin	PROPN
ap-3450	145	32	pošta	pošta	PROPN
ap-3450	145	33	acta	acta	PROPN
ap-3450	145	34	polytechnica	polytechnica	PROPN
ap-3450	145	35	let	let	VERB
ap-3450	145	36	f̃	f̃	PROPN
ap-3450	145	37	be	be	AUX
ap-3450	145	38	a	a	DET
ap-3450	145	39	real	real	ADJ
ap-3450	145	40	function	function	NOUN
ap-3450	145	41	defined	define	VERB
ap-3450	145	42	on	on	ADP
ap-3450	145	43	f̃n	f̃n	ADJ
ap-3450	145	44	.	.	PUNCT
ap-3450	146	1	then	then	ADV
ap-3450	146	2	it	it	PRON
ap-3450	146	3	can	can	AUX
ap-3450	146	4	be	be	AUX
ap-3450	146	5	expanded	expand	VERB
ap-3450	146	6	into	into	ADP
ap-3450	146	7	a	a	DET
ap-3450	146	8	sum	sum	NOUN
ap-3450	146	9	f̃(x	f̃(x	PROPN
ap-3450	146	10	,	,	PUNCT
ap-3450	146	11	y	y	PROPN
ap-3450	146	12	,	,	PUNCT
ap-3450	146	13	z	z	NOUN
ap-3450	146	14	)	)	PUNCT
ap-3450	146	15	=	=	PUNCT
ap-3450	147	1	∑	∑	PUNCT
ap-3450	147	2	0≤k	0≤k	PROPN
ap-3450	147	3	,	,	PUNCT
ap-3450	147	4	l	l	NOUN
ap-3450	147	5	,	,	PUNCT
ap-3450	147	6	m≤n−1	m≤n−1	ADJ
ap-3450	147	7	,	,	PUNCT
ap-3450	147	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	147	9	or	or	CCONJ
ap-3450	147	10	l	l	NOUN
ap-3450	147	11	>	>	X
ap-3450	147	12	k	k	PROPN
ap-3450	147	13	>	>	PROPN
ap-3450	147	14	m	m	PROPN
ap-3450	147	15	d(k	d(k	PROPN
ap-3450	147	16	,	,	PUNCT
ap-3450	147	17	l	l	PROPN
ap-3450	147	18	,	,	PUNCT
ap-3450	147	19	m	m	NOUN
ap-3450	147	20	)	)	PUNCT
ap-3450	147	21	cos(k+1/2,l+1/2,m+1/2)(x	cos(k+1/2,l+1/2,m+1/2)(x	PROPN
ap-3450	147	22	,	,	PUNCT
ap-3450	147	23	y	y	PROPN
ap-3450	147	24	,	,	PUNCT
ap-3450	147	25	z	z	NOUN
ap-3450	147	26	)	)	PUNCT
ap-3450	147	27	,	,	PUNCT
ap-3450	147	28	where	where	SCONJ
ap-3450	147	29	the	the	DET
ap-3450	147	30	coefficients	coefficient	NOUN
ap-3450	147	31	d(k	d(k	PROPN
ap-3450	147	32	,	,	PUNCT
ap-3450	147	33	l	l	PROPN
ap-3450	147	34	,	,	PUNCT
ap-3450	147	35	m	m	NOUN
ap-3450	147	36	)	)	PUNCT
ap-3450	147	37	are	be	AUX
ap-3450	147	38	given	give	VERB
ap-3450	147	39	by	by	ADP
ap-3450	147	40	the	the	DET
ap-3450	147	41	formula	formula	NOUN
ap-3450	147	42	d(k	d(k	PROPN
ap-3450	147	43	,	,	PUNCT
ap-3450	147	44	l	l	NOUN
ap-3450	147	45	,	,	PUNCT
ap-3450	147	46	m	m	NOUN
ap-3450	147	47	)	)	PUNCT
ap-3450	147	48	=	=	SYM
ap-3450	147	49	1	1	NUM
ap-3450	147	50	n3ckclcmgklm	n3ckclcmgklm	PROPN
ap-3450	147	51	∑	∑	PROPN
ap-3450	147	52	0≤r	0≤r	PROPN
ap-3450	147	53	,	,	PUNCT
ap-3450	147	54	s	s	X
ap-3450	147	55	,	,	PUNCT
ap-3450	147	56	t≤n−1	t≤n−1	ADJ
ap-3450	147	57	,	,	PUNCT
ap-3450	147	58	r≥s≥t	r≥s≥t	NOUN
ap-3450	147	59	or	or	CCONJ
ap-3450	147	60	s	s	NOUN
ap-3450	147	61	>	>	X
ap-3450	147	62	r	r	X
ap-3450	147	63	>	>	X
ap-3450	147	64	t	t	X
ap-3450	147	65	c̃r	c̃r	X
ap-3450	147	66	c̃sc̃t	c̃sc̃t	NUM
ap-3450	147	67	grst	grst	NOUN
ap-3450	147	68	f̃	f̃	PROPN
ap-3450	147	69	(	(	PUNCT
ap-3450	147	70	r+1/2	r+1/2	PROPN
ap-3450	147	71	2n	2n	NUM
ap-3450	147	72	,	,	PUNCT
ap-3450	147	73	s+1/2	s+1/2	PROPN
ap-3450	147	74	2n	2n	NUM
ap-3450	147	75	,	,	PUNCT
ap-3450	147	76	t+1/2	t+1/2	PROPN
ap-3450	147	77	2n	2n	NUM
ap-3450	147	78	)	)	PUNCT
ap-3450	147	79	cos(k+1/2,l+1/2,m+1/2	cos(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	147	80	)	)	PUNCT
ap-3450	147	81	(	(	PUNCT
ap-3450	147	82	r+1/2	r+1/2	PROPN
ap-3450	147	83	2n	2n	NUM
ap-3450	147	84	,	,	PUNCT
ap-3450	147	85	s+1/2	s+1/2	PROPN
ap-3450	147	86	2n	2n	NUM
ap-3450	147	87	,	,	PUNCT
ap-3450	147	88	t+1/2	t+1/2	PROPN
ap-3450	147	89	2n	2n	NUM
ap-3450	147	90	)	)	PUNCT
ap-3450	147	91	.	.	PUNCT
ap-3450	148	1	6	6	X
ap-3450	148	2	.	.	X
ap-3450	148	3	discrete	discrete	ADJ
ap-3450	148	4	sine	sine	NOUN
ap-3450	148	5	transforms	transform	VERB
ap-3450	148	6	analogical	analogical	ADJ
ap-3450	148	7	formulas	formula	NOUN
ap-3450	148	8	for	for	ADP
ap-3450	148	9	discrete	discrete	ADJ
ap-3450	148	10	sine	sine	NOUN
ap-3450	148	11	transforms	transform	VERB
ap-3450	148	12	1–4	1–4	PROPN
ap-3450	148	13	can	can	AUX
ap-3450	148	14	be	be	AUX
ap-3450	148	15	derived	derive	VERB
ap-3450	148	16	.	.	PUNCT
ap-3450	149	1	to	to	ADP
ap-3450	149	2	each	each	PRON
ap-3450	149	3	of	of	ADP
ap-3450	149	4	standard	standard	ADJ
ap-3450	149	5	discrete	discrete	ADJ
ap-3450	149	6	sine	sine	NOUN
ap-3450	149	7	transforms	transform	VERB
ap-3450	149	8	,	,	PUNCT
ap-3450	149	9	denoted	denote	VERB
ap-3450	149	10	in	in	ADP
ap-3450	149	11	the	the	DET
ap-3450	149	12	literature	literature	NOUN
ap-3450	149	13	often	often	ADV
ap-3450	149	14	as	as	ADP
ap-3450	149	15	dst-1	dst-1	NUM
ap-3450	149	16	,	,	PUNCT
ap-3450	149	17	dst-2	dst-2	X
ap-3450	149	18	,	,	PUNCT
ap-3450	149	19	dst-3	dst-3	NUM
ap-3450	149	20	and	and	CCONJ
ap-3450	149	21	dst-4	dst-4	NOUN
ap-3450	149	22	,	,	PUNCT
ap-3450	149	23	there	there	PRON
ap-3450	149	24	corresponds	correspond	VERB
ap-3450	149	25	an	an	DET
ap-3450	149	26	alternating	alternate	VERB
ap-3450	149	27	threedimensional	threedimensional	ADJ
ap-3450	149	28	discrete	discrete	ADJ
ap-3450	149	29	sine	sine	NOUN
ap-3450	149	30	transforms	transform	VERB
ap-3450	149	31	.	.	PUNCT
ap-3450	150	1	we	we	PRON
ap-3450	150	2	denote	denote	VERB
ap-3450	150	3	these	these	DET
ap-3450	150	4	corresponding	corresponding	ADJ
ap-3450	150	5	transforms	transform	NOUN
ap-3450	150	6	as	as	ADP
ap-3450	150	7	amdst-1	amdst-1	PROPN
ap-3450	150	8	,	,	PUNCT
ap-3450	150	9	amdst-2	amdst-2	NUM
ap-3450	150	10	,	,	PUNCT
ap-3450	150	11	amdst-3	amdst-3	NUM
ap-3450	150	12	,	,	PUNCT
ap-3450	150	13	amdst-4	amdst-4	PROPN
ap-3450	150	14	.	.	PROPN
ap-3450	151	1	6.1	6.1	NUM
ap-3450	151	2	.	.	PUNCT
ap-3450	151	3	amdst-1	amdst-1	PRON
ap-3450	151	4	for	for	ADP
ap-3450	151	5	given	give	VERB
ap-3450	151	6	n	n	CCONJ
ap-3450	151	7	∈	∈	NOUN
ap-3450	152	1	n	n	AUX
ap-3450	152	2	let	let	VERB
ap-3450	152	3	us	we	PRON
ap-3450	152	4	define	define	VERB
ap-3450	152	5	a	a	DET
ap-3450	152	6	lattice	lattice	NOUN
ap-3450	152	7	f	f	X
ap-3450	152	8	(	(	PUNCT
ap-3450	152	9	s	s	NOUN
ap-3450	152	10	)	)	PUNCT
ap-3450	152	11	n	n	NOUN
ap-3450	152	12	=	=	PRON
ap-3450	152	13	{	{	PUNCT
ap-3450	152	14	(	(	PUNCT
ap-3450	152	15	r	r	NOUN
ap-3450	152	16	2n	2n	NUM
ap-3450	152	17	,	,	PUNCT
ap-3450	152	18	s	s	NOUN
ap-3450	152	19	2n	2n	NUM
ap-3450	152	20	,	,	PUNCT
ap-3450	152	21	t	t	PROPN
ap-3450	152	22	2n	2n	NUM
ap-3450	152	23	)	)	PUNCT
ap-3450	152	24	∣∣	∣∣	NUM
ap-3450	152	25	1	1	NUM
ap-3450	152	26	≤	≤	NOUN
ap-3450	152	27	r	r	NOUN
ap-3450	152	28	,	,	PUNCT
ap-3450	152	29	s	s	PROPN
ap-3450	152	30	,	,	PUNCT
ap-3450	152	31	t	t	NOUN
ap-3450	152	32	≤	≤	NUM
ap-3450	152	33	n	n	CCONJ
ap-3450	152	34	−	−	PROPN
ap-3450	152	35	1	1	NUM
ap-3450	152	36	,	,	PUNCT
ap-3450	152	37	r	r	NOUN
ap-3450	152	38	≥	≥	NOUN
ap-3450	152	39	s	s	PART
ap-3450	152	40	≥	≥	NOUN
ap-3450	152	41	t	t	NOUN
ap-3450	152	42	or	or	CCONJ
ap-3450	152	43	s	s	VERB
ap-3450	152	44	>	>	X
ap-3450	152	45	r	r	X
ap-3450	152	46	>	>	X
ap-3450	152	47	t	t	PROPN
ap-3450	152	48	}	}	PUNCT
ap-3450	152	49	.	.	PUNCT
ap-3450	153	1	this	this	DET
ap-3450	153	2	lattice	lattice	NOUN
ap-3450	153	3	contains	contain	VERB
ap-3450	153	4	1	1	NUM
ap-3450	153	5	3	3	NUM
ap-3450	153	6	(	(	PUNCT
ap-3450	153	7	n3	n3	NOUN
ap-3450	153	8	−	−	PROPN
ap-3450	153	9	3n2	3n2	NUM
ap-3450	153	10	+	+	NUM
ap-3450	153	11	5n	5n	NUM
ap-3450	153	12	−	−	NOUN
ap-3450	153	13	3	3	NUM
ap-3450	153	14	)	)	PUNCT
ap-3450	153	15	points	point	NOUN
ap-3450	153	16	.	.	PUNCT
ap-3450	154	1	for	for	ADP
ap-3450	154	2	example	example	NOUN
ap-3450	154	3	,	,	PUNCT
ap-3450	154	4	f	f	PROPN
ap-3450	154	5	(	(	PUNCT
ap-3450	154	6	s	s	NOUN
ap-3450	154	7	)	)	PUNCT
ap-3450	154	8	3	3	NUM
ap-3450	154	9	=	=	SYM
ap-3450	154	10	{	{	PUNCT
ap-3450	154	11	(	(	PUNCT
ap-3450	154	12	1	1	NUM
ap-3450	154	13	6	6	NUM
ap-3450	154	14	,	,	PUNCT
ap-3450	154	15	1	1	NUM
ap-3450	154	16	6	6	NUM
ap-3450	154	17	,	,	PUNCT
ap-3450	154	18	1	1	NUM
ap-3450	154	19	6	6	NUM
ap-3450	154	20	)	)	PUNCT
ap-3450	154	21	,	,	PUNCT
ap-3450	154	22	(	(	PUNCT
ap-3450	154	23	1	1	NUM
ap-3450	154	24	3	3	NUM
ap-3450	154	25	,	,	PUNCT
ap-3450	154	26	1	1	NUM
ap-3450	154	27	6	6	NUM
ap-3450	154	28	,	,	PUNCT
ap-3450	154	29	1	1	NUM
ap-3450	154	30	6	6	NUM
ap-3450	154	31	)	)	PUNCT
ap-3450	154	32	,	,	PUNCT
ap-3450	154	33	(	(	PUNCT
ap-3450	154	34	1	1	NUM
ap-3450	154	35	3	3	NUM
ap-3450	154	36	,	,	PUNCT
ap-3450	154	37	1	1	NUM
ap-3450	154	38	3	3	NUM
ap-3450	154	39	,	,	PUNCT
ap-3450	154	40	1	1	NUM
ap-3450	154	41	6	6	NUM
ap-3450	154	42	)	)	PUNCT
ap-3450	154	43	,	,	PUNCT
ap-3450	154	44	(	(	PUNCT
ap-3450	154	45	1	1	NUM
ap-3450	154	46	3	3	NUM
ap-3450	154	47	,	,	PUNCT
ap-3450	154	48	1	1	NUM
ap-3450	154	49	3	3	NUM
ap-3450	154	50	,	,	PUNCT
ap-3450	154	51	1	1	NUM
ap-3450	154	52	3	3	NUM
ap-3450	154	53	)	)	PUNCT
ap-3450	154	54	}	}	PUNCT
ap-3450	154	55	.	.	PUNCT
ap-3450	155	1	the	the	DET
ap-3450	155	2	alternating	alternate	VERB
ap-3450	155	3	sines	sine	NOUN
ap-3450	155	4	sin(k	sin(k	PROPN
ap-3450	155	5	,	,	PUNCT
ap-3450	155	6	l	l	NOUN
ap-3450	155	7	,	,	PUNCT
ap-3450	155	8	m	m	NOUN
ap-3450	155	9	)	)	PUNCT
ap-3450	155	10	are	be	AUX
ap-3450	155	11	pairwise	pairwise	NOUN
ap-3450	155	12	orthogonal	orthogonal	NOUN
ap-3450	155	13	on	on	ADP
ap-3450	155	14	this	this	DET
ap-3450	155	15	lattice	lattice	NOUN
ap-3450	155	16	,	,	PUNCT
ap-3450	155	17	namely	namely	ADV
ap-3450	155	18	we	we	PRON
ap-3450	155	19	have	have	VERB
ap-3450	155	20	∑	∑	ADV
ap-3450	155	21	1≤r	1≤r	NUM
ap-3450	155	22	,	,	PUNCT
ap-3450	155	23	s	s	NOUN
ap-3450	155	24	,	,	PUNCT
ap-3450	155	25	t≤n−1	t≤n−1	ADJ
ap-3450	155	26	,	,	PUNCT
ap-3450	155	27	r≥s≥t	r≥s≥t	NOUN
ap-3450	155	28	or	or	CCONJ
ap-3450	155	29	s	s	NOUN
ap-3450	155	30	>	>	X
ap-3450	155	31	r	r	X
ap-3450	155	32	>	>	X
ap-3450	155	33	t	t	X
ap-3450	155	34	g−1	g−1	PROPN
ap-3450	155	35	rst	rst	PROPN
ap-3450	155	36	sin(k	sin(k	PROPN
ap-3450	155	37	,	,	PUNCT
ap-3450	155	38	l	l	NOUN
ap-3450	155	39	,	,	PUNCT
ap-3450	155	40	m	m	NOUN
ap-3450	155	41	)	)	PUNCT
ap-3450	155	42	(	(	PUNCT
ap-3450	155	43	r	r	NOUN
ap-3450	155	44	2n	2n	NUM
ap-3450	155	45	,	,	PUNCT
ap-3450	155	46	s	s	NOUN
ap-3450	155	47	2n	2n	NUM
ap-3450	155	48	,	,	PUNCT
ap-3450	155	49	t	t	PROPN
ap-3450	155	50	2n	2n	NUM
ap-3450	155	51	)	)	PUNCT
ap-3450	155	52	sin(k′,l′,m′	sin(k′,l′,m′	PROPN
ap-3450	155	53	)	)	PUNCT
ap-3450	155	54	(	(	PUNCT
ap-3450	155	55	r	r	NOUN
ap-3450	155	56	2n	2n	NUM
ap-3450	155	57	,	,	PUNCT
ap-3450	155	58	s	s	NOUN
ap-3450	155	59	2n	2n	NUM
ap-3450	155	60	,	,	PUNCT
ap-3450	155	61	t	t	PROPN
ap-3450	155	62	2n	2n	NUM
ap-3450	155	63	)	)	PUNCT
ap-3450	156	1	=	=	PUNCT
ap-3450	156	2	(	(	PUNCT
ap-3450	156	3	n	n	ADV
ap-3450	156	4	2	2	NUM
ap-3450	156	5	)	)	PUNCT
ap-3450	156	6	3	3	NUM
ap-3450	157	1	gklmδ(k	gklmδ(k	PROPN
ap-3450	157	2	,	,	PUNCT
ap-3450	157	3	l	l	NOUN
ap-3450	157	4	,	,	PUNCT
ap-3450	157	5	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	157	6	)	)	PUNCT
ap-3450	157	7	,	,	PUNCT
ap-3450	157	8	where	where	SCONJ
ap-3450	157	9	1	1	NUM
ap-3450	157	10	≤	≤	NUM
ap-3450	157	11	k	k	X
ap-3450	157	12	,	,	PUNCT
ap-3450	157	13	l	l	NOUN
ap-3450	157	14	,	,	PUNCT
ap-3450	157	15	m	m	VERB
ap-3450	157	16	≤	≤	NOUN
ap-3450	157	17	n	n	CCONJ
ap-3450	157	18	−	−	PROPN
ap-3450	157	19	1	1	NUM
ap-3450	157	20	,	,	PUNCT
ap-3450	157	21	k	k	X
ap-3450	157	22	≥	≥	PROPN
ap-3450	157	23	l	l	PROPN
ap-3450	157	24	≥	≥	NOUN
ap-3450	157	25	m	m	PROPN
ap-3450	157	26	or	or	CCONJ
ap-3450	157	27	l	l	X
ap-3450	157	28	>	>	X
ap-3450	157	29	k	k	X
ap-3450	157	30	>	>	X
ap-3450	157	31	m	m	PROPN
ap-3450	157	32	,	,	PUNCT
ap-3450	157	33	and	and	CCONJ
ap-3450	157	34	1	1	NUM
ap-3450	157	35	≤	≤	NOUN
ap-3450	157	36	k′	k′	PROPN
ap-3450	157	37	,	,	PUNCT
ap-3450	157	38	l′,m′	l′,m′	PROPN
ap-3450	157	39	≤	≤	NOUN
ap-3450	157	40	n	n	CCONJ
ap-3450	157	41	−	−	PROPN
ap-3450	157	42	1	1	NUM
ap-3450	157	43	,	,	PUNCT
ap-3450	157	44	k′	k′	PROPN
ap-3450	157	45	≥	≥	NUM
ap-3450	157	46	l′	l′	PRON
ap-3450	157	47	≥	≥	PROPN
ap-3450	157	48	m′	m′	NUM
ap-3450	157	49	or	or	CCONJ
ap-3450	157	50	l′	l′	VERB
ap-3450	157	51	>	>	X
ap-3450	157	52	k′	k′	PROPN
ap-3450	157	53	>	>	X
ap-3450	157	54	m′.	m′.	PROPN
ap-3450	157	55	let	let	VERB
ap-3450	157	56	f	f	PROPN
ap-3450	157	57	(	(	PUNCT
ap-3450	157	58	s	s	X
ap-3450	157	59	)	)	PUNCT
ap-3450	157	60	be	be	AUX
ap-3450	157	61	a	a	DET
ap-3450	157	62	real	real	ADJ
ap-3450	157	63	function	function	NOUN
ap-3450	157	64	defined	define	VERB
ap-3450	157	65	on	on	ADP
ap-3450	157	66	f	f	PROPN
ap-3450	157	67	(	(	PUNCT
ap-3450	157	68	s	s	NOUN
ap-3450	157	69	)	)	PUNCT
ap-3450	157	70	n	n	NOUN
ap-3450	157	71	.	.	PUNCT
ap-3450	158	1	then	then	ADV
ap-3450	158	2	it	it	PRON
ap-3450	158	3	can	can	AUX
ap-3450	158	4	be	be	AUX
ap-3450	158	5	expanded	expand	VERB
ap-3450	158	6	into	into	ADP
ap-3450	158	7	a	a	DET
ap-3450	158	8	sum	sum	NOUN
ap-3450	158	9	of	of	ADP
ap-3450	158	10	alternating	alternate	VERB
ap-3450	158	11	sines	sine	NOUN
ap-3450	158	12	,	,	PUNCT
ap-3450	158	13	f	f	PROPN
ap-3450	158	14	(	(	PUNCT
ap-3450	158	15	s)(x	s)(x	PROPN
ap-3450	158	16	,	,	PUNCT
ap-3450	158	17	y	y	PROPN
ap-3450	158	18	,	,	PUNCT
ap-3450	158	19	z	z	NOUN
ap-3450	158	20	)	)	PUNCT
ap-3450	158	21	=	=	PUNCT
ap-3450	158	22	∑	∑	PUNCT
ap-3450	158	23	1≤k	1≤k	NUM
ap-3450	158	24	,	,	PUNCT
ap-3450	158	25	l	l	NOUN
ap-3450	158	26	,	,	PUNCT
ap-3450	158	27	m≤n−1	m≤n−1	ADJ
ap-3450	158	28	,	,	PUNCT
ap-3450	158	29	k≥l≥m	k≥l≥m	NOUN
ap-3450	158	30	or	or	CCONJ
ap-3450	158	31	l	l	NOUN
ap-3450	158	32	>	>	X
ap-3450	158	33	k	k	PROPN
ap-3450	158	34	>	>	PROPN
ap-3450	158	35	m	m	PROPN
ap-3450	158	36	a(k	a(k	PROPN
ap-3450	158	37	,	,	PUNCT
ap-3450	158	38	l	l	NOUN
ap-3450	158	39	,	,	PUNCT
ap-3450	158	40	m	m	NOUN
ap-3450	158	41	)	)	PUNCT
ap-3450	158	42	sin(k	sin(k	PROPN
ap-3450	158	43	,	,	PUNCT
ap-3450	158	44	l	l	NOUN
ap-3450	158	45	,	,	PUNCT
ap-3450	158	46	m)(x	m)(x	PROPN
ap-3450	158	47	,	,	PUNCT
ap-3450	158	48	y	y	PROPN
ap-3450	158	49	,	,	PUNCT
ap-3450	158	50	z	z	NOUN
ap-3450	158	51	)	)	PUNCT
ap-3450	158	52	,	,	PUNCT
ap-3450	158	53	where	where	SCONJ
ap-3450	158	54	the	the	DET
ap-3450	158	55	coefficients	coefficient	NOUN
ap-3450	158	56	a(s	a(	NOUN
ap-3450	158	57	)	)	PUNCT
ap-3450	158	58	(	(	PUNCT
ap-3450	158	59	k	k	X
ap-3450	158	60	,	,	PUNCT
ap-3450	158	61	l	l	NOUN
ap-3450	158	62	,	,	PUNCT
ap-3450	158	63	m	m	NOUN
ap-3450	158	64	)	)	PUNCT
ap-3450	158	65	are	be	AUX
ap-3450	158	66	given	give	VERB
ap-3450	158	67	by	by	ADP
ap-3450	158	68	the	the	DET
ap-3450	158	69	formula	formula	NOUN
ap-3450	158	70	a	a	DET
ap-3450	158	71	(	(	PUNCT
ap-3450	158	72	s	s	NOUN
ap-3450	158	73	)	)	PUNCT
ap-3450	158	74	(	(	PUNCT
ap-3450	158	75	k	k	X
ap-3450	158	76	,	,	PUNCT
ap-3450	158	77	l	l	NOUN
ap-3450	158	78	,	,	PUNCT
ap-3450	158	79	m	m	NOUN
ap-3450	158	80	)	)	PUNCT
ap-3450	158	81	=	=	SYM
ap-3450	158	82	8	8	NUM
ap-3450	158	83	n3gklm	n3gklm	PROPN
ap-3450	158	84	∑	∑	PROPN
ap-3450	158	85	1≤r	1≤r	NUM
ap-3450	158	86	,	,	PUNCT
ap-3450	158	87	s	s	NOUN
ap-3450	158	88	,	,	PUNCT
ap-3450	158	89	t≤n−1	t≤n−1	ADJ
ap-3450	158	90	,	,	PUNCT
ap-3450	158	91	r≥s≥t	r≥s≥t	NOUN
ap-3450	158	92	or	or	CCONJ
ap-3450	158	93	s	s	NOUN
ap-3450	158	94	>	>	X
ap-3450	158	95	r	r	X
ap-3450	158	96	>	>	X
ap-3450	158	97	t	t	PROPN
ap-3450	158	98	1	1	NUM
ap-3450	158	99	grst	grst	NOUN
ap-3450	158	100	f	f	X
ap-3450	158	101	(	(	PUNCT
ap-3450	158	102	s	s	NOUN
ap-3450	158	103	)	)	PUNCT
ap-3450	158	104	(	(	PUNCT
ap-3450	158	105	r	r	NOUN
ap-3450	158	106	2n	2n	NUM
ap-3450	158	107	,	,	PUNCT
ap-3450	158	108	s	s	NOUN
ap-3450	158	109	2n	2n	NUM
ap-3450	158	110	,	,	PUNCT
ap-3450	158	111	t	t	PROPN
ap-3450	158	112	2n	2n	NUM
ap-3450	158	113	)	)	PUNCT
ap-3450	158	114	sin(k	sin(k	PROPN
ap-3450	158	115	,	,	PUNCT
ap-3450	158	116	l	l	NOUN
ap-3450	158	117	,	,	PUNCT
ap-3450	158	118	m	m	NOUN
ap-3450	158	119	)	)	PUNCT
ap-3450	158	120	(	(	PUNCT
ap-3450	158	121	r	r	NOUN
ap-3450	158	122	2n	2n	NUM
ap-3450	158	123	,	,	PUNCT
ap-3450	158	124	s	s	NOUN
ap-3450	158	125	2n	2n	NUM
ap-3450	158	126	,	,	PUNCT
ap-3450	158	127	t	t	PROPN
ap-3450	158	128	2n	2n	NUM
ap-3450	158	129	)	)	PUNCT
ap-3450	158	130	.	.	PUNCT
ap-3450	159	1	as	as	ADP
ap-3450	159	2	an	an	DET
ap-3450	159	3	interpolation	interpolation	NOUN
ap-3450	159	4	example	example	NOUN
ap-3450	159	5	,	,	PUNCT
ap-3450	159	6	we	we	PRON
ap-3450	159	7	take	take	VERB
ap-3450	159	8	f	f	PROPN
ap-3450	159	9	(	(	PUNCT
ap-3450	159	10	s)(x	s)(x	PROPN
ap-3450	159	11	,	,	PUNCT
ap-3450	159	12	y	y	PROPN
ap-3450	159	13	,	,	PUNCT
ap-3450	159	14	z	z	NOUN
ap-3450	159	15	)	)	PUNCT
ap-3450	159	16	=	=	VERB
ap-3450	159	17	sin	sin	NOUN
ap-3450	159	18	2πx	2πx	ADJ
ap-3450	159	19	sin	sin	NOUN
ap-3450	159	20	2πy	2πy	NOUN
ap-3450	159	21	sin	sin	VERB
ap-3450	159	22	2πz	2πz	ADJ
ap-3450	159	23	sin	sin	NOUN
ap-3450	159	24	10(x+	10(x+	NUM
ap-3450	159	25	y	y	PROPN
ap-3450	160	1	+	+	PROPN
ap-3450	161	1	z	z	NOUN
ap-3450	161	2	)	)	PUNCT
ap-3450	162	1	,	,	PUNCT
ap-3450	162	2	(	(	PUNCT
ap-3450	162	3	12	12	NUM
ap-3450	162	4	)	)	PUNCT
ap-3450	162	5	and	and	CCONJ
ap-3450	162	6	compute	compute	VERB
ap-3450	162	7	four	four	NUM
ap-3450	162	8	interpolations	interpolation	NOUN
ap-3450	162	9	f	f	X
ap-3450	162	10	(	(	PUNCT
ap-3450	162	11	s	s	NOUN
ap-3450	162	12	)	)	PUNCT
ap-3450	162	13	n	n	X
ap-3450	162	14	for	for	ADP
ap-3450	162	15	n	n	NOUN
ap-3450	162	16	=	=	SYM
ap-3450	162	17	1	1	NUM
ap-3450	162	18	,	,	PUNCT
ap-3450	162	19	2	2	NUM
ap-3450	162	20	,	,	PUNCT
ap-3450	162	21	3	3	NUM
ap-3450	162	22	,	,	PUNCT
ap-3450	162	23	4	4	NUM
ap-3450	162	24	using	use	VERB
ap-3450	162	25	alternating	alternate	VERB
ap-3450	162	26	sines	sine	NOUN
ap-3450	162	27	.	.	PUNCT
ap-3450	163	1	picture	picture	NOUN
ap-3450	163	2	of	of	ADP
ap-3450	163	3	(	(	PUNCT
ap-3450	163	4	12	12	NUM
ap-3450	163	5	)	)	PUNCT
ap-3450	163	6	is	be	AUX
ap-3450	163	7	shown	show	VERB
ap-3450	163	8	in	in	ADP
ap-3450	163	9	fig	fig	NOUN
ap-3450	163	10	.	.	PUNCT
ap-3450	164	1	5	5	NUM
ap-3450	164	2	and	and	CCONJ
ap-3450	164	3	four	four	NUM
ap-3450	164	4	approximations	approximation	NOUN
ap-3450	164	5	in	in	ADP
ap-3450	164	6	fig	fig	NOUN
ap-3450	164	7	.	.	PUNCT
ap-3450	165	1	6	6	NUM
ap-3450	165	2	.	.	X
ap-3450	165	3	by	by	ADP
ap-3450	165	4	visual	visual	ADJ
ap-3450	165	5	inspection	inspection	NOUN
ap-3450	165	6	,	,	PUNCT
ap-3450	165	7	approximations	approximation	VERB
ap-3450	165	8	f	f	X
ap-3450	165	9	(	(	PUNCT
ap-3450	165	10	s	s	X
ap-3450	165	11	)	)	PUNCT
ap-3450	165	12	n	n	PRON
ap-3450	165	13	get	get	VERB
ap-3450	165	14	closer	close	ADJ
ap-3450	165	15	to	to	ADP
ap-3450	165	16	(	(	PUNCT
ap-3450	165	17	12	12	NUM
ap-3450	165	18	)	)	PUNCT
ap-3450	165	19	as	as	ADP
ap-3450	165	20	n	n	DET
ap-3450	165	21	increases	increase	NOUN
ap-3450	165	22	.	.	PUNCT
ap-3450	166	1	this	this	PRON
ap-3450	166	2	is	be	AUX
ap-3450	166	3	verified	verify	VERB
ap-3450	166	4	by	by	ADP
ap-3450	166	5	computing	compute	VERB
ap-3450	166	6	integral	integral	ADJ
ap-3450	166	7	error	error	NOUN
ap-3450	166	8	estimates	estimate	NOUN
ap-3450	166	9	shown	show	VERB
ap-3450	166	10	in	in	ADP
ap-3450	166	11	table	table	NOUN
ap-3450	166	12	2	2	NUM
ap-3450	166	13	.	.	NOUN
ap-3450	166	14	6.2	6.2	NUM
ap-3450	166	15	.	.	PUNCT
ap-3450	167	1	amdst-2	amdst-2	NUM
ap-3450	167	2	alternating	alternate	VERB
ap-3450	167	3	discrete	discrete	ADJ
ap-3450	167	4	sine	sine	NOUN
ap-3450	167	5	transform	transform	NOUN
ap-3450	167	6	of	of	ADP
ap-3450	167	7	second	second	ADJ
ap-3450	167	8	kind	kind	NOUN
ap-3450	167	9	uses	use	VERB
ap-3450	167	10	the	the	DET
ap-3450	167	11	same	same	ADJ
ap-3450	167	12	lattice	lattice	NOUN
ap-3450	167	13	as	as	ADP
ap-3450	167	14	amdct-2	amdct-2	PROPN
ap-3450	167	15	,	,	PUNCT
ap-3450	167	16	that	that	PRON
ap-3450	167	17	is	is	ADV
ap-3450	167	18	f̃	f̃	PROPN
ap-3450	167	19	(	(	PUNCT
ap-3450	167	20	s	s	NOUN
ap-3450	167	21	)	)	PUNCT
ap-3450	167	22	n	n	NOUN
ap-3450	167	23	=	=	SYM
ap-3450	167	24	f̃n	f̃n	NOUN
ap-3450	167	25	,	,	PUNCT
ap-3450	167	26	(	(	PUNCT
ap-3450	167	27	13	13	NUM
ap-3450	167	28	)	)	PUNCT
ap-3450	167	29	where	where	SCONJ
ap-3450	167	30	f̃n	f̃n	ADJ
ap-3450	167	31	is	be	AUX
ap-3450	167	32	given	give	VERB
ap-3450	167	33	by	by	ADP
ap-3450	167	34	(	(	PUNCT
ap-3450	167	35	11	11	NUM
ap-3450	167	36	)	)	PUNCT
ap-3450	167	37	.	.	PUNCT
ap-3450	168	1	162	162	NUM
ap-3450	168	2	vol	vol	NOUN
ap-3450	168	3	.	.	PUNCT
ap-3450	169	1	56	56	NUM
ap-3450	169	2	no	no	NOUN
ap-3450	169	3	.	.	PUNCT
ap-3450	170	1	3/2016	3/2016	NUM
ap-3450	170	2	three	three	NUM
ap-3450	170	3	-	-	PUNCT
ap-3450	170	4	variable	variable	NOUN
ap-3450	170	5	alternating	alternate	VERB
ap-3450	170	6	trigonometric	trigonometric	ADJ
ap-3450	170	7	functions	function	NOUN
ap-3450	170	8	figure	figure	VERB
ap-3450	170	9	5	5	NUM
ap-3450	170	10	.	.	PUNCT
ap-3450	171	1	the	the	DET
ap-3450	171	2	function	function	NOUN
ap-3450	171	3	(	(	PUNCT
ap-3450	171	4	12	12	NUM
ap-3450	171	5	)	)	PUNCT
ap-3450	171	6	used	use	VERB
ap-3450	171	7	in	in	ADP
ap-3450	171	8	interpolation	interpolation	NOUN
ap-3450	171	9	example	example	NOUN
ap-3450	171	10	.	.	PUNCT
ap-3450	172	1	n	n	PRON
ap-3450	172	2	∫	∫	PROPN
ap-3450	172	3	f	f	PROPN
ap-3450	172	4	(	(	PUNCT
ap-3450	172	5	ãaff	ãaff	X
ap-3450	172	6	3	3	NUM
ap-3450	172	7	)	)	PUNCT
ap-3450	172	8	|f	|f	PROPN
ap-3450	173	1	(	(	PUNCT
ap-3450	173	2	s	s	NOUN
ap-3450	173	3	)	)	PUNCT
ap-3450	174	1	−	−	PROPN
ap-3450	174	2	f	f	PROPN
ap-3450	174	3	(	(	PUNCT
ap-3450	174	4	s	s	NOUN
ap-3450	174	5	)	)	PUNCT
ap-3450	174	6	n	n	PRON
ap-3450	174	7	|2	|2	NUM
ap-3450	174	8	1	1	NUM
ap-3450	174	9	0.0482575	0.0482575	NUM
ap-3450	174	10	2	2	NUM
ap-3450	174	11	0.0282054	0.0282054	NUM
ap-3450	174	12	3	3	NUM
ap-3450	174	13	0.0078227	0.0078227	NUM
ap-3450	174	14	4	4	NUM
ap-3450	174	15	0.0006646	0.0006646	NUM
ap-3450	174	16	table	table	NOUN
ap-3450	174	17	2	2	NUM
ap-3450	174	18	.	.	PUNCT
ap-3450	174	19	integral	integral	ADJ
ap-3450	174	20	error	error	NOUN
ap-3450	174	21	estimates	estimate	NOUN
ap-3450	174	22	for	for	ADP
ap-3450	174	23	the	the	DET
ap-3450	174	24	approximations	approximation	NOUN
ap-3450	174	25	in	in	ADP
ap-3450	174	26	fig	fig	NOUN
ap-3450	174	27	.	.	PUNCT
ap-3450	175	1	6	6	X
ap-3450	175	2	.	.	X
ap-3450	175	3	figure	figure	NOUN
ap-3450	175	4	6	6	NUM
ap-3450	175	5	.	.	PUNCT
ap-3450	176	1	the	the	DET
ap-3450	176	2	alternating	alternate	VERB
ap-3450	176	3	sines	sine	NOUN
ap-3450	176	4	approximations	approximation	NOUN
ap-3450	176	5	of	of	ADP
ap-3450	176	6	(	(	PUNCT
ap-3450	176	7	12	12	NUM
ap-3450	176	8	)	)	PUNCT
ap-3450	176	9	for	for	ADP
ap-3450	176	10	n	n	NOUN
ap-3450	176	11	=	=	SYM
ap-3450	176	12	1	1	NUM
ap-3450	176	13	,	,	PUNCT
ap-3450	176	14	2	2	NUM
ap-3450	176	15	,	,	PUNCT
ap-3450	176	16	3	3	NUM
ap-3450	176	17	,	,	PUNCT
ap-3450	176	18	4	4	NUM
ap-3450	176	19	.	.	PUNCT
ap-3450	177	1	the	the	DET
ap-3450	177	2	alternating	alternate	VERB
ap-3450	177	3	sines	sine	NOUN
ap-3450	177	4	sin(k	sin(k	PROPN
ap-3450	177	5	,	,	PUNCT
ap-3450	177	6	l	l	NOUN
ap-3450	177	7	,	,	PUNCT
ap-3450	177	8	m	m	NOUN
ap-3450	177	9	)	)	PUNCT
ap-3450	177	10	are	be	AUX
ap-3450	177	11	pairwise	pairwise	NOUN
ap-3450	177	12	orthogonal	orthogonal	NOUN
ap-3450	177	13	on	on	ADP
ap-3450	177	14	this	this	DET
ap-3450	177	15	lattice	lattice	NOUN
ap-3450	177	16	,	,	PUNCT
ap-3450	177	17	namely	namely	ADV
ap-3450	177	18	we	we	PRON
ap-3450	177	19	have∑	have∑	VERB
ap-3450	177	20	0≤r	0≤r	PROPN
ap-3450	177	21	,	,	PUNCT
ap-3450	177	22	s	s	X
ap-3450	177	23	,	,	PUNCT
ap-3450	177	24	t≤n−1	t≤n−1	ADJ
ap-3450	177	25	,	,	PUNCT
ap-3450	177	26	r≥s≥t	r≥s≥t	NOUN
ap-3450	177	27	or	or	CCONJ
ap-3450	177	28	s	s	NOUN
ap-3450	177	29	>	>	X
ap-3450	177	30	r	r	X
ap-3450	177	31	>	>	X
ap-3450	177	32	t	t	X
ap-3450	177	33	g−1	g−1	PROPN
ap-3450	177	34	rst	rst	PROPN
ap-3450	177	35	sin(k	sin(k	PROPN
ap-3450	177	36	,	,	PUNCT
ap-3450	177	37	l	l	NOUN
ap-3450	177	38	,	,	PUNCT
ap-3450	177	39	m	m	NOUN
ap-3450	177	40	)	)	PUNCT
ap-3450	178	1	(	(	PUNCT
ap-3450	178	2	r+1/2	r+1/2	PROPN
ap-3450	178	3	2n	2n	NUM
ap-3450	178	4	,	,	PUNCT
ap-3450	178	5	s+1/2	s+1/2	PROPN
ap-3450	178	6	2n	2n	NUM
ap-3450	178	7	,	,	PUNCT
ap-3450	178	8	t+1/2	t+1/2	PROPN
ap-3450	178	9	2n	2n	NUM
ap-3450	178	10	)	)	PUNCT
ap-3450	178	11	sin(k′,l′,m′	sin(k′,l′,m′	PROPN
ap-3450	178	12	)	)	PUNCT
ap-3450	178	13	(	(	PUNCT
ap-3450	178	14	r+1/2	r+1/2	PROPN
ap-3450	178	15	2n	2n	NUM
ap-3450	178	16	,	,	PUNCT
ap-3450	178	17	s+1/2	s+1/2	PROPN
ap-3450	178	18	2n	2n	NUM
ap-3450	178	19	,	,	PUNCT
ap-3450	178	20	t+1/2	t+1/2	PROPN
ap-3450	178	21	2n	2n	NUM
ap-3450	178	22	)	)	PUNCT
ap-3450	179	1	=	=	PUNCT
ap-3450	179	2	(	(	PUNCT
ap-3450	179	3	n	n	ADV
ap-3450	179	4	2	2	NUM
ap-3450	179	5	)	)	PUNCT
ap-3450	179	6	3	3	NUM
ap-3450	179	7	ckclcmgklmδ(k	ckclcmgklmδ(k	PROPN
ap-3450	179	8	,	,	PUNCT
ap-3450	179	9	l	l	NOUN
ap-3450	179	10	,	,	PUNCT
ap-3450	179	11	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	179	12	)	)	PUNCT
ap-3450	179	13	,	,	PUNCT
ap-3450	179	14	where	where	SCONJ
ap-3450	179	15	cp	cp	PROPN
ap-3450	179	16	=	=	NOUN
ap-3450	179	17	1	1	NUM
ap-3450	179	18	2	2	NUM
ap-3450	179	19	for	for	ADP
ap-3450	179	20	p	p	NOUN
ap-3450	179	21	=	=	PUNCT
ap-3450	179	22	n	n	PROPN
ap-3450	179	23	and	and	CCONJ
ap-3450	179	24	cp	cp	INTJ
ap-3450	179	25	=	=	NOUN
ap-3450	179	26	1	1	NUM
ap-3450	179	27	otherwise	otherwise	ADV
ap-3450	179	28	and	and	CCONJ
ap-3450	179	29	1	1	NUM
ap-3450	179	30	≤	≤	NUM
ap-3450	180	1	k	k	NOUN
ap-3450	180	2	,	,	PUNCT
ap-3450	180	3	l	l	NOUN
ap-3450	180	4	,	,	PUNCT
ap-3450	180	5	m	m	VERB
ap-3450	180	6	≤	≤	NOUN
ap-3450	180	7	n	n	PRON
ap-3450	180	8	,	,	PUNCT
ap-3450	180	9	k	k	PROPN
ap-3450	180	10	≥	≥	PROPN
ap-3450	180	11	l	l	PROPN
ap-3450	180	12	≥	≥	NOUN
ap-3450	180	13	m	m	PROPN
ap-3450	180	14	or	or	CCONJ
ap-3450	180	15	l	l	X
ap-3450	180	16	>	>	X
ap-3450	181	1	k	k	X
ap-3450	181	2	>	>	X
ap-3450	181	3	m	m	PROPN
ap-3450	181	4	,	,	PUNCT
ap-3450	181	5	and	and	CCONJ
ap-3450	181	6	1	1	NUM
ap-3450	181	7	≤	≤	NOUN
ap-3450	181	8	k′	k′	PROPN
ap-3450	181	9	,	,	PUNCT
ap-3450	181	10	l′,m′	l′,m′	PROPN
ap-3450	181	11	≤	≤	PUNCT
ap-3450	181	12	n	n	CCONJ
ap-3450	181	13	,	,	PUNCT
ap-3450	181	14	k′	k′	PROPN
ap-3450	181	15	≥	≥	NUM
ap-3450	181	16	l′	l′	PRON
ap-3450	181	17	≥	≥	PROPN
ap-3450	181	18	m′	m′	NUM
ap-3450	181	19	or	or	CCONJ
ap-3450	181	20	l′	l′	VERB
ap-3450	181	21	>	>	X
ap-3450	181	22	k′	k′	PROPN
ap-3450	181	23	>	>	X
ap-3450	181	24	m′.	m′.	PROPN
ap-3450	181	25	let	let	VERB
ap-3450	181	26	f̃	f̃	PROPN
ap-3450	181	27	(	(	PUNCT
ap-3450	181	28	s	s	X
ap-3450	181	29	)	)	PUNCT
ap-3450	181	30	be	be	AUX
ap-3450	181	31	a	a	DET
ap-3450	181	32	real	real	ADJ
ap-3450	181	33	function	function	NOUN
ap-3450	181	34	defined	define	VERB
ap-3450	181	35	on	on	ADP
ap-3450	181	36	f̃	f̃	PROPN
ap-3450	181	37	(	(	PUNCT
ap-3450	181	38	s	s	NOUN
ap-3450	181	39	)	)	PUNCT
ap-3450	181	40	n	n	NOUN
ap-3450	181	41	.	.	PUNCT
ap-3450	182	1	then	then	ADV
ap-3450	182	2	it	it	PRON
ap-3450	182	3	can	can	AUX
ap-3450	182	4	be	be	AUX
ap-3450	182	5	expanded	expand	VERB
ap-3450	182	6	into	into	ADP
ap-3450	182	7	a	a	DET
ap-3450	182	8	sum	sum	NOUN
ap-3450	182	9	of	of	ADP
ap-3450	182	10	alternating	alternate	VERB
ap-3450	182	11	sines	sine	NOUN
ap-3450	182	12	,	,	PUNCT
ap-3450	182	13	f̃	f̃	PROPN
ap-3450	182	14	(	(	PUNCT
ap-3450	182	15	s)(x	s)(x	PROPN
ap-3450	182	16	,	,	PUNCT
ap-3450	182	17	y	y	PROPN
ap-3450	182	18	,	,	PUNCT
ap-3450	182	19	z	z	NOUN
ap-3450	182	20	)	)	PUNCT
ap-3450	182	21	=	=	PUNCT
ap-3450	182	22	∑	∑	PUNCT
ap-3450	182	23	1≤k	1≤k	NUM
ap-3450	182	24	,	,	PUNCT
ap-3450	182	25	l	l	NOUN
ap-3450	182	26	,	,	PUNCT
ap-3450	182	27	m≤n	m≤n	PROPN
ap-3450	182	28	,	,	PUNCT
ap-3450	182	29	k≥l≥m	k≥l≥m	NOUN
ap-3450	182	30	or	or	CCONJ
ap-3450	182	31	l	l	NOUN
ap-3450	182	32	>	>	X
ap-3450	182	33	k	k	X
ap-3450	182	34	>	>	PROPN
ap-3450	182	35	m	m	PROPN
ap-3450	182	36	b	b	PROPN
ap-3450	182	37	(	(	PUNCT
ap-3450	182	38	s	s	NOUN
ap-3450	182	39	)	)	PUNCT
ap-3450	182	40	(	(	PUNCT
ap-3450	182	41	k	k	X
ap-3450	182	42	,	,	PUNCT
ap-3450	182	43	l	l	NOUN
ap-3450	182	44	,	,	PUNCT
ap-3450	182	45	m	m	NOUN
ap-3450	182	46	)	)	PUNCT
ap-3450	182	47	sin(k	sin(k	PROPN
ap-3450	182	48	,	,	PUNCT
ap-3450	182	49	l	l	NOUN
ap-3450	182	50	,	,	PUNCT
ap-3450	182	51	m)(x	m)(x	PROPN
ap-3450	182	52	,	,	PUNCT
ap-3450	182	53	y	y	PROPN
ap-3450	182	54	,	,	PUNCT
ap-3450	182	55	z	z	NOUN
ap-3450	182	56	)	)	PUNCT
ap-3450	182	57	,	,	PUNCT
ap-3450	182	58	where	where	SCONJ
ap-3450	182	59	the	the	DET
ap-3450	182	60	coefficients	coefficient	NOUN
ap-3450	182	61	b(k	b(k	PROPN
ap-3450	182	62	,	,	PUNCT
ap-3450	182	63	l	l	NOUN
ap-3450	182	64	,	,	PUNCT
ap-3450	182	65	m	m	NOUN
ap-3450	182	66	)	)	PUNCT
ap-3450	182	67	are	be	AUX
ap-3450	182	68	given	give	VERB
ap-3450	182	69	by	by	ADP
ap-3450	182	70	the	the	DET
ap-3450	182	71	formula	formula	NOUN
ap-3450	182	72	b	b	X
ap-3450	182	73	(	(	PUNCT
ap-3450	182	74	s	s	NOUN
ap-3450	182	75	)	)	PUNCT
ap-3450	182	76	(	(	PUNCT
ap-3450	182	77	k	k	X
ap-3450	182	78	,	,	PUNCT
ap-3450	182	79	l	l	NOUN
ap-3450	182	80	,	,	PUNCT
ap-3450	182	81	m	m	NOUN
ap-3450	182	82	)	)	PUNCT
ap-3450	182	83	=	=	SYM
ap-3450	182	84	8	8	NUM
ap-3450	182	85	n3ckclcmgklm	n3ckclcmgklm	PROPN
ap-3450	182	86	∑	∑	PROPN
ap-3450	182	87	0≤r	0≤r	PROPN
ap-3450	182	88	,	,	PUNCT
ap-3450	182	89	s	s	X
ap-3450	182	90	,	,	PUNCT
ap-3450	182	91	t≤n−1	t≤n−1	ADJ
ap-3450	182	92	,	,	PUNCT
ap-3450	182	93	r≥s≥t	r≥s≥t	NOUN
ap-3450	182	94	or	or	CCONJ
ap-3450	182	95	s	s	NOUN
ap-3450	182	96	>	>	X
ap-3450	182	97	r	r	X
ap-3450	182	98	>	>	X
ap-3450	182	99	t	t	PROPN
ap-3450	182	100	1	1	NUM
ap-3450	182	101	grst	grst	NOUN
ap-3450	182	102	f̃	f̃	PROPN
ap-3450	182	103	(	(	PUNCT
ap-3450	182	104	s	s	NOUN
ap-3450	182	105	)	)	PUNCT
ap-3450	182	106	(	(	PUNCT
ap-3450	182	107	r+1/2	r+1/2	PROPN
ap-3450	182	108	2n	2n	NUM
ap-3450	182	109	,	,	PUNCT
ap-3450	182	110	s+1/2	s+1/2	PROPN
ap-3450	182	111	2n	2n	NUM
ap-3450	182	112	,	,	PUNCT
ap-3450	182	113	t+1/2	t+1/2	PROPN
ap-3450	182	114	2n	2n	NUM
ap-3450	182	115	)	)	PUNCT
ap-3450	182	116	sin(k	sin(k	PROPN
ap-3450	182	117	,	,	PUNCT
ap-3450	182	118	l	l	NOUN
ap-3450	182	119	,	,	PUNCT
ap-3450	182	120	m	m	NOUN
ap-3450	182	121	)	)	PUNCT
ap-3450	182	122	(	(	PUNCT
ap-3450	182	123	r+1/2	r+1/2	PROPN
ap-3450	182	124	2n	2n	NUM
ap-3450	182	125	,	,	PUNCT
ap-3450	182	126	s+1/2	s+1/2	PROPN
ap-3450	182	127	2n	2n	NUM
ap-3450	182	128	,	,	PUNCT
ap-3450	182	129	t+1/2	t+1/2	PROPN
ap-3450	182	130	2n	2n	NUM
ap-3450	182	131	)	)	PUNCT
ap-3450	182	132	.	.	PUNCT
ap-3450	183	1	6.3	6.3	NUM
ap-3450	183	2	.	.	PUNCT
ap-3450	184	1	amdst-3	amdst-3	NUM
ap-3450	184	2	alternating	alternate	VERB
ap-3450	184	3	discrete	discrete	ADJ
ap-3450	184	4	sine	sine	NOUN
ap-3450	184	5	transform	transform	NOUN
ap-3450	184	6	of	of	ADP
ap-3450	184	7	the	the	DET
ap-3450	184	8	third	third	ADJ
ap-3450	184	9	kind	kind	NOUN
ap-3450	184	10	uses	use	VERB
ap-3450	184	11	the	the	DET
ap-3450	184	12	lattice	lattice	PROPN
ap-3450	184	13	f	f	PROPN
ap-3450	184	14	′n	′n	PROPN
ap-3450	184	15	(	(	PUNCT
ap-3450	184	16	s	s	X
ap-3450	184	17	)	)	PUNCT
ap-3450	184	18	=	=	SYM
ap-3450	184	19	{	{	PUNCT
ap-3450	184	20	(	(	PUNCT
ap-3450	184	21	r	r	NOUN
ap-3450	184	22	2n	2n	NUM
ap-3450	184	23	,	,	PUNCT
ap-3450	184	24	s	s	NOUN
ap-3450	184	25	2n	2n	NUM
ap-3450	184	26	,	,	PUNCT
ap-3450	184	27	t	t	PROPN
ap-3450	184	28	2n	2n	NUM
ap-3450	184	29	)	)	PUNCT
ap-3450	184	30	∣∣	∣∣	NUM
ap-3450	184	31	1	1	NUM
ap-3450	184	32	≤	≤	NOUN
ap-3450	184	33	r	r	NOUN
ap-3450	184	34	,	,	PUNCT
ap-3450	184	35	s	s	PROPN
ap-3450	184	36	,	,	PUNCT
ap-3450	184	37	t	t	PROPN
ap-3450	184	38	≤	≤	NUM
ap-3450	184	39	n	n	CCONJ
ap-3450	184	40	,	,	PUNCT
ap-3450	184	41	r	r	NOUN
ap-3450	184	42	≥	≥	NOUN
ap-3450	184	43	s	s	PART
ap-3450	184	44	≥	≥	NOUN
ap-3450	184	45	t	t	NOUN
ap-3450	184	46	or	or	CCONJ
ap-3450	184	47	s	s	VERB
ap-3450	184	48	>	>	X
ap-3450	184	49	r	r	X
ap-3450	184	50	>	>	X
ap-3450	184	51	t	t	PROPN
ap-3450	184	52	}	}	PUNCT
ap-3450	184	53	.	.	PUNCT
ap-3450	185	1	instead	instead	ADV
ap-3450	185	2	of	of	ADP
ap-3450	185	3	alternating	alternate	VERB
ap-3450	185	4	sines	sine	NOUN
ap-3450	185	5	sin(k	sin(k	PROPN
ap-3450	185	6	,	,	PUNCT
ap-3450	185	7	l	l	NOUN
ap-3450	185	8	,	,	PUNCT
ap-3450	185	9	m	m	NOUN
ap-3450	185	10	)	)	PUNCT
ap-3450	185	11	,	,	PUNCT
ap-3450	185	12	shifted	shift	VERB
ap-3450	185	13	sines	sine	NOUN
ap-3450	185	14	sin(k+1/2,l+1/2,m+1/2	sin(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	185	15	)	)	PUNCT
ap-3450	185	16	are	be	AUX
ap-3450	185	17	used	use	VERB
ap-3450	185	18	.	.	PUNCT
ap-3450	186	1	they	they	PRON
ap-3450	186	2	have	have	VERB
ap-3450	186	3	similar	similar	ADJ
ap-3450	186	4	properties	property	NOUN
ap-3450	186	5	as	as	ADP
ap-3450	186	6	normal	normal	ADJ
ap-3450	186	7	alternating	alternate	VERB
ap-3450	186	8	sines	sine	NOUN
ap-3450	186	9	with	with	ADP
ap-3450	186	10	integer	integer	NOUN
ap-3450	186	11	arguments	argument	NOUN
ap-3450	186	12	.	.	PUNCT
ap-3450	187	1	for	for	ADP
ap-3450	187	2	scalar	scalar	ADJ
ap-3450	187	3	product	product	NOUN
ap-3450	187	4	we	we	PRON
ap-3450	187	5	obtain∑	obtain∑	VERB
ap-3450	187	6	1≤r	1≤r	NUM
ap-3450	187	7	,	,	PUNCT
ap-3450	187	8	s	s	NOUN
ap-3450	187	9	,	,	PUNCT
ap-3450	187	10	t≤n	t≤n	ADJ
ap-3450	187	11	,	,	PUNCT
ap-3450	187	12	r≥s≥t	r≥s≥t	NOUN
ap-3450	187	13	or	or	CCONJ
ap-3450	187	14	s	s	NOUN
ap-3450	187	15	>	>	X
ap-3450	187	16	r	r	X
ap-3450	187	17	>	>	X
ap-3450	187	18	t	t	X
ap-3450	187	19	g−1	g−1	PROPN
ap-3450	187	20	rstcrcsct	rstcrcsct	PROPN
ap-3450	187	21	sin(k+1/2,l+1/2,m+1/2	sin(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	187	22	)	)	PUNCT
ap-3450	187	23	(	(	PUNCT
ap-3450	187	24	r	r	NOUN
ap-3450	187	25	2n	2n	NUM
ap-3450	187	26	,	,	PUNCT
ap-3450	187	27	s	s	NOUN
ap-3450	187	28	2n	2n	NUM
ap-3450	187	29	,	,	PUNCT
ap-3450	187	30	t	t	PROPN
ap-3450	187	31	2n	2n	NUM
ap-3450	187	32	)	)	PUNCT
ap-3450	187	33	sin(k′+1/2,l′+1/2,m′+1/2	sin(k′+1/2,l′+1/2,m′+1/2	PROPN
ap-3450	187	34	)	)	PUNCT
ap-3450	187	35	(	(	PUNCT
ap-3450	187	36	r	r	NOUN
ap-3450	187	37	2n	2n	NUM
ap-3450	187	38	,	,	PUNCT
ap-3450	187	39	s	s	NOUN
ap-3450	187	40	2n	2n	NUM
ap-3450	187	41	,	,	PUNCT
ap-3450	187	42	t	t	PROPN
ap-3450	187	43	2n	2n	NUM
ap-3450	187	44	)	)	PUNCT
ap-3450	188	1	=	=	PUNCT
ap-3450	188	2	(	(	PUNCT
ap-3450	188	3	n	n	ADV
ap-3450	188	4	2	2	NUM
ap-3450	188	5	)	)	PUNCT
ap-3450	188	6	3	3	NUM
ap-3450	189	1	gklmδ(k	gklmδ(k	PROPN
ap-3450	189	2	,	,	PUNCT
ap-3450	189	3	l	l	NOUN
ap-3450	189	4	,	,	PUNCT
ap-3450	189	5	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	189	6	)	)	PUNCT
ap-3450	189	7	,	,	PUNCT
ap-3450	189	8	where	where	SCONJ
ap-3450	189	9	cp	cp	PROPN
ap-3450	189	10	=	=	NOUN
ap-3450	189	11	1	1	NUM
ap-3450	189	12	2	2	NUM
ap-3450	189	13	for	for	ADP
ap-3450	189	14	p	p	NOUN
ap-3450	189	15	=	=	PUNCT
ap-3450	189	16	n	n	PROPN
ap-3450	189	17	and	and	CCONJ
ap-3450	189	18	cp	cp	INTJ
ap-3450	189	19	=	=	NOUN
ap-3450	189	20	1	1	NUM
ap-3450	189	21	otherwise	otherwise	ADV
ap-3450	189	22	and	and	CCONJ
ap-3450	189	23	0	0	NUM
ap-3450	189	24	≤	≤	NUM
ap-3450	190	1	k	k	NOUN
ap-3450	190	2	,	,	PUNCT
ap-3450	190	3	l	l	NOUN
ap-3450	190	4	,	,	PUNCT
ap-3450	190	5	m	m	VERB
ap-3450	190	6	≤	≤	NOUN
ap-3450	190	7	n	n	CCONJ
ap-3450	190	8	−	−	PROPN
ap-3450	190	9	1	1	NUM
ap-3450	190	10	,	,	PUNCT
ap-3450	190	11	k	k	X
ap-3450	190	12	≥	≥	PROPN
ap-3450	190	13	l	l	PROPN
ap-3450	190	14	≥	≥	NOUN
ap-3450	190	15	m	m	PROPN
ap-3450	190	16	or	or	CCONJ
ap-3450	190	17	l	l	X
ap-3450	190	18	>	>	X
ap-3450	191	1	k	k	X
ap-3450	191	2	>	>	X
ap-3450	191	3	m	m	PROPN
ap-3450	191	4	,	,	PUNCT
ap-3450	191	5	and	and	CCONJ
ap-3450	191	6	0	0	NUM
ap-3450	191	7	≤	≤	NOUN
ap-3450	191	8	k′	k′	PROPN
ap-3450	191	9	,	,	PUNCT
ap-3450	191	10	l′,m′	l′,m′	PROPN
ap-3450	191	11	≤	≤	NOUN
ap-3450	191	12	n	n	CCONJ
ap-3450	191	13	−	−	PROPN
ap-3450	191	14	1	1	NUM
ap-3450	191	15	,	,	PUNCT
ap-3450	191	16	k′	k′	PROPN
ap-3450	191	17	≥	≥	NUM
ap-3450	191	18	l′	l′	PRON
ap-3450	191	19	≥	≥	PROPN
ap-3450	191	20	m′	m′	NUM
ap-3450	191	21	or	or	CCONJ
ap-3450	191	22	l′	l′	VERB
ap-3450	191	23	>	>	X
ap-3450	191	24	k′	k′	PROPN
ap-3450	191	25	>	>	X
ap-3450	191	26	m′.	m′.	PROPN
ap-3450	191	27	163	163	NUM
ap-3450	191	28	agata	agata	PROPN
ap-3450	191	29	bezubik	bezubik	PROPN
ap-3450	191	30	,	,	PUNCT
ap-3450	191	31	severin	severin	PROPN
ap-3450	191	32	pošta	pošta	PROPN
ap-3450	191	33	acta	acta	PROPN
ap-3450	191	34	polytechnica	polytechnica	PROPN
ap-3450	191	35	let	let	VERB
ap-3450	191	36	g(s	g(s	PROPN
ap-3450	191	37	)	)	PUNCT
ap-3450	191	38	be	be	AUX
ap-3450	191	39	a	a	DET
ap-3450	191	40	real	real	ADJ
ap-3450	191	41	function	function	NOUN
ap-3450	191	42	defined	define	VERB
ap-3450	191	43	on	on	ADP
ap-3450	191	44	f	f	PROPN
ap-3450	191	45	′n	′n	PROPN
ap-3450	191	46	(	(	PUNCT
ap-3450	191	47	s	s	NOUN
ap-3450	191	48	)	)	PUNCT
ap-3450	191	49	.	.	PUNCT
ap-3450	192	1	then	then	ADV
ap-3450	192	2	it	it	PRON
ap-3450	192	3	can	can	AUX
ap-3450	192	4	be	be	AUX
ap-3450	192	5	expanded	expand	VERB
ap-3450	192	6	into	into	ADP
ap-3450	192	7	a	a	DET
ap-3450	192	8	sum	sum	NOUN
ap-3450	192	9	of	of	ADP
ap-3450	192	10	shifted	shift	VERB
ap-3450	192	11	alternating	alternate	VERB
ap-3450	192	12	sines	sine	NOUN
ap-3450	192	13	,	,	PUNCT
ap-3450	192	14	g(s)(x	g(s)(x	PROPN
ap-3450	192	15	,	,	PUNCT
ap-3450	192	16	y	y	PROPN
ap-3450	192	17	,	,	PUNCT
ap-3450	192	18	z	z	NOUN
ap-3450	192	19	)	)	PUNCT
ap-3450	192	20	=	=	PUNCT
ap-3450	193	1	∑	∑	PUNCT
ap-3450	193	2	0≤k	0≤k	PROPN
ap-3450	193	3	,	,	PUNCT
ap-3450	193	4	l	l	NOUN
ap-3450	193	5	,	,	PUNCT
ap-3450	193	6	m≤n−1	m≤n−1	ADJ
ap-3450	193	7	,	,	PUNCT
ap-3450	193	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	193	9	or	or	CCONJ
ap-3450	193	10	l	l	NOUN
ap-3450	193	11	>	>	X
ap-3450	193	12	k	k	X
ap-3450	193	13	>	>	X
ap-3450	193	14	m	m	PROPN
ap-3450	193	15	c	c	NOUN
ap-3450	193	16	(	(	PUNCT
ap-3450	193	17	s	s	NOUN
ap-3450	193	18	)	)	PUNCT
ap-3450	193	19	(	(	PUNCT
ap-3450	193	20	k	k	X
ap-3450	193	21	,	,	PUNCT
ap-3450	193	22	l	l	NOUN
ap-3450	193	23	,	,	PUNCT
ap-3450	193	24	m	m	NOUN
ap-3450	193	25	)	)	PUNCT
ap-3450	193	26	sin(k+1/2,l+1/2,m+1/2)(x	sin(k+1/2,l+1/2,m+1/2)(x	PROPN
ap-3450	193	27	,	,	PUNCT
ap-3450	193	28	y	y	PROPN
ap-3450	193	29	,	,	PUNCT
ap-3450	193	30	z	z	NOUN
ap-3450	193	31	)	)	PUNCT
ap-3450	193	32	,	,	PUNCT
ap-3450	193	33	where	where	SCONJ
ap-3450	193	34	the	the	DET
ap-3450	193	35	coefficients	coefficient	NOUN
ap-3450	193	36	c(s	c(	VERB
ap-3450	193	37	)	)	PUNCT
ap-3450	193	38	(	(	PUNCT
ap-3450	193	39	k	k	X
ap-3450	193	40	,	,	PUNCT
ap-3450	193	41	l	l	NOUN
ap-3450	193	42	,	,	PUNCT
ap-3450	193	43	m	m	NOUN
ap-3450	193	44	)	)	PUNCT
ap-3450	193	45	are	be	AUX
ap-3450	193	46	given	give	VERB
ap-3450	193	47	by	by	ADP
ap-3450	193	48	the	the	DET
ap-3450	193	49	formula	formula	NOUN
ap-3450	193	50	c	c	NOUN
ap-3450	193	51	(	(	PUNCT
ap-3450	193	52	s	s	NOUN
ap-3450	193	53	)	)	PUNCT
ap-3450	193	54	(	(	PUNCT
ap-3450	193	55	k	k	X
ap-3450	193	56	,	,	PUNCT
ap-3450	193	57	l	l	NOUN
ap-3450	193	58	,	,	PUNCT
ap-3450	193	59	m	m	NOUN
ap-3450	193	60	)	)	PUNCT
ap-3450	193	61	=	=	SYM
ap-3450	193	62	8	8	NUM
ap-3450	193	63	n3gklm	n3gklm	PROPN
ap-3450	193	64	∑	∑	PROPN
ap-3450	193	65	1≤r	1≤r	NUM
ap-3450	193	66	,	,	PUNCT
ap-3450	193	67	s	s	NOUN
ap-3450	193	68	,	,	PUNCT
ap-3450	193	69	t≤n	t≤n	ADJ
ap-3450	193	70	,	,	PUNCT
ap-3450	193	71	r≥s≥t	r≥s≥t	NOUN
ap-3450	193	72	or	or	CCONJ
ap-3450	193	73	s	s	NOUN
ap-3450	193	74	>	>	X
ap-3450	193	75	r	r	PROPN
ap-3450	193	76	>	>	X
ap-3450	193	77	t	t	PROPN
ap-3450	193	78	crcsct	crcsct	PROPN
ap-3450	193	79	grst	grst	ADJ
ap-3450	193	80	g(s	g(s	PROPN
ap-3450	193	81	)	)	PUNCT
ap-3450	193	82	(	(	PUNCT
ap-3450	193	83	r	r	NOUN
ap-3450	193	84	2n	2n	NUM
ap-3450	193	85	,	,	PUNCT
ap-3450	193	86	s	s	NOUN
ap-3450	193	87	2n	2n	NUM
ap-3450	193	88	,	,	PUNCT
ap-3450	193	89	t	t	PROPN
ap-3450	193	90	2n	2n	NUM
ap-3450	193	91	)	)	PUNCT
ap-3450	193	92	sin(k+1/2,l+1/2,m+1/2	sin(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	193	93	)	)	PUNCT
ap-3450	193	94	(	(	PUNCT
ap-3450	193	95	r	r	NOUN
ap-3450	193	96	2n	2n	NUM
ap-3450	193	97	,	,	PUNCT
ap-3450	193	98	s	s	NOUN
ap-3450	193	99	2n	2n	NUM
ap-3450	193	100	,	,	PUNCT
ap-3450	193	101	t	t	PROPN
ap-3450	193	102	2n	2n	NUM
ap-3450	193	103	)	)	PUNCT
ap-3450	193	104	.	.	PUNCT
ap-3450	194	1	6.4	6.4	NUM
ap-3450	194	2	.	.	PUNCT
ap-3450	194	3	amdst-4	amdst-4	NUM
ap-3450	194	4	alternating	alternate	VERB
ap-3450	194	5	discrete	discrete	ADJ
ap-3450	194	6	sine	sine	NOUN
ap-3450	194	7	transform	transform	NOUN
ap-3450	194	8	of	of	ADP
ap-3450	194	9	the	the	DET
ap-3450	194	10	fourth	fourth	ADJ
ap-3450	194	11	kind	kind	NOUN
ap-3450	194	12	uses	use	VERB
ap-3450	194	13	the	the	DET
ap-3450	194	14	lattice	lattice	PROPN
ap-3450	194	15	f̃	f̃	PROPN
ap-3450	194	16	(	(	PUNCT
ap-3450	194	17	s	s	NOUN
ap-3450	194	18	)	)	PUNCT
ap-3450	194	19	n	n	CCONJ
ap-3450	194	20	given	give	VERB
ap-3450	194	21	by	by	ADP
ap-3450	194	22	(	(	PUNCT
ap-3450	194	23	13	13	NUM
ap-3450	194	24	)	)	PUNCT
ap-3450	194	25	and	and	CCONJ
ap-3450	194	26	shifted	shift	VERB
ap-3450	194	27	sines	sine	NOUN
ap-3450	194	28	as	as	ADP
ap-3450	194	29	in	in	ADP
ap-3450	194	30	the	the	DET
ap-3450	194	31	third	third	ADJ
ap-3450	194	32	case	case	NOUN
ap-3450	194	33	.	.	PUNCT
ap-3450	195	1	we	we	PRON
ap-3450	195	2	have∑	have∑	VERB
ap-3450	195	3	0≤r	0≤r	PROPN
ap-3450	195	4	,	,	PUNCT
ap-3450	195	5	s	s	X
ap-3450	195	6	,	,	PUNCT
ap-3450	195	7	t≤n−1	t≤n−1	ADJ
ap-3450	195	8	,	,	PUNCT
ap-3450	195	9	r≥s≥t	r≥s≥t	NOUN
ap-3450	195	10	or	or	CCONJ
ap-3450	195	11	s	s	NOUN
ap-3450	195	12	>	>	X
ap-3450	195	13	r	r	X
ap-3450	195	14	>	>	X
ap-3450	195	15	t	t	X
ap-3450	195	16	g−1	g−1	PROPN
ap-3450	195	17	rst	rst	PROPN
ap-3450	195	18	sin(k+1/2,l+1/2,m+1/2	sin(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	195	19	)	)	PUNCT
ap-3450	195	20	(	(	PUNCT
ap-3450	195	21	r+1/2	r+1/2	PROPN
ap-3450	195	22	2n	2n	NUM
ap-3450	195	23	,	,	PUNCT
ap-3450	195	24	s+1/2	s+1/2	PROPN
ap-3450	195	25	2n	2n	NUM
ap-3450	195	26	,	,	PUNCT
ap-3450	195	27	t+1/2	t+1/2	PROPN
ap-3450	195	28	2n	2n	NUM
ap-3450	195	29	)	)	PUNCT
ap-3450	195	30	sin(k′+1/2,l′+1/2,m′+1/2	sin(k′+1/2,l′+1/2,m′+1/2	PROPN
ap-3450	195	31	)	)	PUNCT
ap-3450	195	32	(	(	PUNCT
ap-3450	195	33	r+1/2	r+1/2	PROPN
ap-3450	195	34	2n	2n	NUM
ap-3450	195	35	,	,	PUNCT
ap-3450	195	36	s+1/2	s+1/2	PROPN
ap-3450	195	37	2n	2n	NUM
ap-3450	195	38	,	,	PUNCT
ap-3450	195	39	t+1/2	t+1/2	PROPN
ap-3450	195	40	2n	2n	NUM
ap-3450	195	41	)	)	PUNCT
ap-3450	196	1	=	=	PUNCT
ap-3450	196	2	(	(	PUNCT
ap-3450	196	3	n	n	ADV
ap-3450	196	4	2	2	NUM
ap-3450	196	5	)	)	PUNCT
ap-3450	196	6	3	3	NUM
ap-3450	197	1	gklmδ(k	gklmδ(k	PROPN
ap-3450	197	2	,	,	PUNCT
ap-3450	197	3	l	l	NOUN
ap-3450	197	4	,	,	PUNCT
ap-3450	197	5	m),(k′,l′,m′	m),(k′,l′,m′	ADJ
ap-3450	197	6	)	)	PUNCT
ap-3450	197	7	,	,	PUNCT
ap-3450	197	8	where	where	SCONJ
ap-3450	197	9	0	0	NUM
ap-3450	197	10	≤	≤	NUM
ap-3450	197	11	k	k	NOUN
ap-3450	197	12	,	,	PUNCT
ap-3450	197	13	l	l	NOUN
ap-3450	197	14	,	,	PUNCT
ap-3450	197	15	m	m	VERB
ap-3450	197	16	≤	≤	NOUN
ap-3450	197	17	n	n	CCONJ
ap-3450	197	18	−	−	PROPN
ap-3450	197	19	1	1	NUM
ap-3450	197	20	,	,	PUNCT
ap-3450	197	21	k	k	X
ap-3450	197	22	≥	≥	PROPN
ap-3450	197	23	l	l	PROPN
ap-3450	197	24	≥	≥	NOUN
ap-3450	197	25	m	m	PROPN
ap-3450	197	26	or	or	CCONJ
ap-3450	197	27	l	l	X
ap-3450	197	28	>	>	X
ap-3450	197	29	k	k	X
ap-3450	197	30	>	>	X
ap-3450	197	31	m	m	PROPN
ap-3450	197	32	,	,	PUNCT
ap-3450	197	33	and	and	CCONJ
ap-3450	197	34	0	0	NUM
ap-3450	197	35	≤	≤	NOUN
ap-3450	197	36	k′	k′	PROPN
ap-3450	197	37	,	,	PUNCT
ap-3450	197	38	l′,m′	l′,m′	PROPN
ap-3450	197	39	≤	≤	NOUN
ap-3450	197	40	n	n	CCONJ
ap-3450	197	41	−	−	PROPN
ap-3450	197	42	1	1	NUM
ap-3450	197	43	,	,	PUNCT
ap-3450	197	44	k′	k′	PROPN
ap-3450	197	45	≥	≥	NUM
ap-3450	197	46	l′	l′	PRON
ap-3450	197	47	≥	≥	PROPN
ap-3450	197	48	m′	m′	NUM
ap-3450	197	49	or	or	CCONJ
ap-3450	197	50	l′	l′	VERB
ap-3450	197	51	>	>	X
ap-3450	197	52	k′	k′	PROPN
ap-3450	197	53	>	>	X
ap-3450	197	54	m′.	m′.	PROPN
ap-3450	197	55	let	let	VERB
ap-3450	197	56	f̃	f̃	PROPN
ap-3450	197	57	(	(	PUNCT
ap-3450	197	58	s	s	X
ap-3450	197	59	)	)	PUNCT
ap-3450	197	60	be	be	AUX
ap-3450	197	61	a	a	DET
ap-3450	197	62	real	real	ADJ
ap-3450	197	63	function	function	NOUN
ap-3450	197	64	defined	define	VERB
ap-3450	197	65	on	on	ADP
ap-3450	197	66	f̃	f̃	PROPN
ap-3450	197	67	(	(	PUNCT
ap-3450	197	68	s	s	NOUN
ap-3450	197	69	)	)	PUNCT
ap-3450	197	70	n	n	NOUN
ap-3450	197	71	.	.	PUNCT
ap-3450	198	1	then	then	ADV
ap-3450	198	2	it	it	PRON
ap-3450	198	3	can	can	AUX
ap-3450	198	4	be	be	AUX
ap-3450	198	5	expanded	expand	VERB
ap-3450	198	6	into	into	ADP
ap-3450	198	7	a	a	DET
ap-3450	198	8	sum	sum	NOUN
ap-3450	198	9	f̃	f̃	PROPN
ap-3450	198	10	(	(	PUNCT
ap-3450	198	11	s)(x	s)(x	PROPN
ap-3450	198	12	,	,	PUNCT
ap-3450	198	13	y	y	PROPN
ap-3450	198	14	,	,	PUNCT
ap-3450	198	15	z	z	NOUN
ap-3450	198	16	)	)	PUNCT
ap-3450	198	17	=	=	PUNCT
ap-3450	199	1	∑	∑	PUNCT
ap-3450	199	2	0≤k	0≤k	PROPN
ap-3450	199	3	,	,	PUNCT
ap-3450	199	4	l	l	NOUN
ap-3450	199	5	,	,	PUNCT
ap-3450	199	6	m≤n−1	m≤n−1	ADJ
ap-3450	199	7	,	,	PUNCT
ap-3450	199	8	k≥l≥m	k≥l≥m	NOUN
ap-3450	199	9	or	or	CCONJ
ap-3450	199	10	l	l	NOUN
ap-3450	199	11	>	>	X
ap-3450	199	12	k	k	X
ap-3450	199	13	>	>	NOUN
ap-3450	199	14	m	m	PROPN
ap-3450	199	15	d	d	X
ap-3450	199	16	(	(	PUNCT
ap-3450	199	17	s	s	NOUN
ap-3450	199	18	)	)	PUNCT
ap-3450	199	19	(	(	PUNCT
ap-3450	199	20	k	k	X
ap-3450	199	21	,	,	PUNCT
ap-3450	199	22	l	l	NOUN
ap-3450	199	23	,	,	PUNCT
ap-3450	199	24	m	m	NOUN
ap-3450	199	25	)	)	PUNCT
ap-3450	199	26	sin(k+1/2,l+1/2,m+1/2)(x	sin(k+1/2,l+1/2,m+1/2)(x	PROPN
ap-3450	199	27	,	,	PUNCT
ap-3450	199	28	y	y	PROPN
ap-3450	199	29	,	,	PUNCT
ap-3450	199	30	z	z	NOUN
ap-3450	199	31	)	)	PUNCT
ap-3450	199	32	,	,	PUNCT
ap-3450	199	33	where	where	SCONJ
ap-3450	199	34	the	the	DET
ap-3450	199	35	coefficients	coefficient	NOUN
ap-3450	199	36	d(s	d(s	PROPN
ap-3450	199	37	)	)	PUNCT
ap-3450	199	38	(	(	PUNCT
ap-3450	199	39	k	k	X
ap-3450	199	40	,	,	PUNCT
ap-3450	199	41	l	l	NOUN
ap-3450	199	42	,	,	PUNCT
ap-3450	199	43	m	m	NOUN
ap-3450	199	44	)	)	PUNCT
ap-3450	199	45	are	be	AUX
ap-3450	199	46	given	give	VERB
ap-3450	199	47	by	by	ADP
ap-3450	199	48	the	the	DET
ap-3450	199	49	formula	formula	NOUN
ap-3450	199	50	d	d	X
ap-3450	199	51	(	(	PUNCT
ap-3450	199	52	s	s	NOUN
ap-3450	199	53	)	)	PUNCT
ap-3450	199	54	(	(	PUNCT
ap-3450	199	55	k	k	X
ap-3450	199	56	,	,	PUNCT
ap-3450	199	57	l	l	NOUN
ap-3450	199	58	,	,	PUNCT
ap-3450	199	59	m	m	NOUN
ap-3450	199	60	)	)	PUNCT
ap-3450	199	61	=	=	SYM
ap-3450	199	62	8	8	NUM
ap-3450	199	63	n3gklm	n3gklm	PROPN
ap-3450	199	64	∑	∑	PROPN
ap-3450	199	65	0≤r	0≤r	PROPN
ap-3450	199	66	,	,	PUNCT
ap-3450	199	67	s	s	X
ap-3450	199	68	,	,	PUNCT
ap-3450	199	69	t≤n−1	t≤n−1	ADJ
ap-3450	199	70	,	,	PUNCT
ap-3450	199	71	r≥s≥t	r≥s≥t	NOUN
ap-3450	199	72	or	or	CCONJ
ap-3450	199	73	s	s	NOUN
ap-3450	199	74	>	>	X
ap-3450	199	75	r	r	X
ap-3450	199	76	>	>	X
ap-3450	199	77	t	t	PROPN
ap-3450	199	78	1	1	NUM
ap-3450	199	79	grst	grst	NOUN
ap-3450	199	80	f̃	f̃	PROPN
ap-3450	199	81	(	(	PUNCT
ap-3450	199	82	r+1/2	r+1/2	PROPN
ap-3450	199	83	2n	2n	NUM
ap-3450	199	84	,	,	PUNCT
ap-3450	199	85	s+1/2	s+1/2	PROPN
ap-3450	199	86	2n	2n	NUM
ap-3450	199	87	,	,	PUNCT
ap-3450	199	88	t+1/2	t+1/2	PROPN
ap-3450	199	89	2n	2n	NUM
ap-3450	199	90	)	)	PUNCT
ap-3450	199	91	sin(k+1/2,l+1/2,m+1/2	sin(k+1/2,l+1/2,m+1/2	PROPN
ap-3450	199	92	)	)	PUNCT
ap-3450	199	93	(	(	PUNCT
ap-3450	200	1	r+1/2	r+1/2	PROPN
ap-3450	200	2	2n	2n	NUM
ap-3450	200	3	,	,	PUNCT
ap-3450	200	4	s+1/2	s+1/2	PROPN
ap-3450	200	5	2n	2n	NUM
ap-3450	200	6	,	,	PUNCT
ap-3450	200	7	t+1/2	t+1/2	PROPN
ap-3450	200	8	2n	2n	NUM
ap-3450	200	9	)	)	PUNCT
ap-3450	200	10	.	.	PUNCT
ap-3450	201	1	7	7	X
ap-3450	201	2	.	.	X
ap-3450	201	3	conclusion	conclusion	NOUN
ap-3450	201	4	we	we	PRON
ap-3450	201	5	have	have	AUX
ap-3450	201	6	presented	present	VERB
ap-3450	201	7	a	a	DET
ap-3450	201	8	detailed	detailed	ADJ
ap-3450	201	9	study	study	NOUN
ap-3450	201	10	of	of	ADP
ap-3450	201	11	alternating	alternate	VERB
ap-3450	201	12	three	three	NUM
ap-3450	201	13	dimensional	dimensional	ADJ
ap-3450	201	14	trigonometric	trigonometric	ADJ
ap-3450	201	15	functions	function	NOUN
ap-3450	201	16	and	and	CCONJ
ap-3450	201	17	their	their	PRON
ap-3450	201	18	properties	property	NOUN
ap-3450	201	19	including	include	VERB
ap-3450	201	20	product	product	NOUN
ap-3450	201	21	decomposition	decomposition	NOUN
ap-3450	201	22	,	,	PUNCT
ap-3450	201	23	continuous	continuous	ADJ
ap-3450	201	24	and	and	CCONJ
ap-3450	201	25	discrete	discrete	ADJ
ap-3450	201	26	orthogonality	orthogonality	NOUN
ap-3450	201	27	and	and	CCONJ
ap-3450	201	28	interpolation	interpolation	NOUN
ap-3450	201	29	problem	problem	NOUN
ap-3450	201	30	.	.	PUNCT
ap-3450	202	1	practical	practical	ADJ
ap-3450	202	2	computational	computational	ADJ
ap-3450	202	3	aspects	aspect	NOUN
ap-3450	202	4	of	of	ADP
ap-3450	202	5	the	the	DET
ap-3450	202	6	formalism	formalism	NOUN
ap-3450	202	7	presented	present	VERB
ap-3450	202	8	in	in	ADP
ap-3450	202	9	this	this	DET
ap-3450	202	10	paper	paper	NOUN
ap-3450	202	11	need	need	VERB
ap-3450	202	12	further	further	ADJ
ap-3450	202	13	investigation	investigation	NOUN
ap-3450	202	14	,	,	PUNCT
ap-3450	202	15	namely	namely	ADV
ap-3450	202	16	a	a	DET
ap-3450	202	17	fast	fast	ADJ
ap-3450	202	18	fourier	fourier	NOUN
ap-3450	202	19	transform	transform	NOUN
ap-3450	202	20	analog	analog	NOUN
ap-3450	202	21	and	and	CCONJ
ap-3450	202	22	comparing	compare	VERB
ap-3450	202	23	to	to	ADP
ap-3450	202	24	the	the	DET
ap-3450	202	25	usual	usual	ADJ
ap-3450	202	26	discrete	discrete	ADJ
ap-3450	202	27	fourier	fourier	NOUN
ap-3450	202	28	transform	transform	NOUN
ap-3450	202	29	in	in	ADP
ap-3450	202	30	three	three	NUM
ap-3450	202	31	dimensions	dimension	NOUN
ap-3450	202	32	.	.	PUNCT
ap-3450	203	1	acknowledgements	acknowledgement	NOUN
ap-3450	203	2	we	we	PRON
ap-3450	203	3	gratefully	gratefully	ADV
ap-3450	203	4	acknowledge	acknowledge	VERB
ap-3450	203	5	the	the	DET
ap-3450	203	6	support	support	NOUN
ap-3450	203	7	of	of	ADP
ap-3450	203	8	this	this	DET
ap-3450	203	9	work	work	NOUN
ap-3450	203	10	by	by	ADP
ap-3450	203	11	the	the	DET
ap-3450	203	12	doppler	doppler	NOUN
ap-3450	203	13	institute	institute	NOUN
ap-3450	203	14	of	of	ADP
ap-3450	203	15	the	the	DET
ap-3450	203	16	czech	czech	PROPN
ap-3450	203	17	technical	technical	PROPN
ap-3450	203	18	university	university	PROPN
ap-3450	203	19	in	in	ADP
ap-3450	203	20	prague	prague	PROPN
ap-3450	203	21	.	.	PUNCT
ap-3450	204	1	ab	ab	PROPN
ap-3450	204	2	is	be	AUX
ap-3450	204	3	grateful	grateful	ADJ
ap-3450	204	4	for	for	ADP
ap-3450	204	5	the	the	DET
ap-3450	204	6	hospitality	hospitality	NOUN
ap-3450	204	7	extended	extend	VERB
ap-3450	204	8	to	to	ADP
ap-3450	204	9	her	she	PRON
ap-3450	204	10	at	at	ADP
ap-3450	204	11	department	department	NOUN
ap-3450	204	12	of	of	ADP
ap-3450	204	13	mathematics	mathematics	PROPN
ap-3450	204	14	fnspe	fnspe	PROPN
ap-3450	204	15	ctu	ctu	PROPN
ap-3450	204	16	.	.	PROPN
ap-3450	204	17	sp	sp	PROPN
ap-3450	204	18	acknowledges	acknowledge	VERB
ap-3450	204	19	the	the	DET
ap-3450	204	20	support	support	NOUN
ap-3450	204	21	of	of	ADP
ap-3450	204	22	sgs15/215	sgs15/215	NOUN
ap-3450	204	23	/	/	SYM
ap-3450	204	24	ohk4/3t/14	ohk4/3t/14	PROPN
ap-3450	204	25	,	,	PUNCT
ap-3450	204	26	project	project	NOUN
ap-3450	204	27	of	of	ADP
ap-3450	204	28	the	the	DET
ap-3450	204	29	czech	czech	PROPN
ap-3450	204	30	technical	technical	PROPN
ap-3450	204	31	university	university	PROPN
ap-3450	204	32	in	in	ADP
ap-3450	204	33	prague	prague	PROPN
ap-3450	204	34	.	.	PUNCT
ap-3450	205	1	references	reference	NOUN
ap-3450	205	2	[	[	X
ap-3450	205	3	1	1	NUM
ap-3450	205	4	]	]	PUNCT
ap-3450	205	5	a.	a.	NOUN
ap-3450	205	6	klimyk	klimyk	PROPN
ap-3450	205	7	,	,	PUNCT
ap-3450	205	8	j.	j.	PROPN
ap-3450	205	9	patera	patera	PROPN
ap-3450	205	10	.	.	PUNCT
ap-3450	206	1	(	(	PUNCT
ap-3450	206	2	anti)symmetric	anti)symmetric	ADJ
ap-3450	206	3	multivariate	multivariate	NOUN
ap-3450	206	4	exponential	exponential	NOUN
ap-3450	206	5	functions	function	NOUN
ap-3450	206	6	and	and	CCONJ
ap-3450	206	7	corresponding	corresponding	ADJ
ap-3450	206	8	fourier	fourier	NOUN
ap-3450	206	9	transforms	transform	VERB
ap-3450	206	10	.	.	PUNCT
ap-3450	207	1	j	j	PROPN
ap-3450	207	2	phys	phy	NOUN
ap-3450	207	3	a	a	DET
ap-3450	207	4	40(34):10473–10489	40(34):10473–10489	NOUN
ap-3450	207	5	,	,	PUNCT
ap-3450	207	6	2007	2007	NUM
ap-3450	207	7	.	.	PUNCT
ap-3450	208	1	doi:10.1088/1751	doi:10.1088/1751	NOUN
ap-3450	208	2	-	-	PUNCT
ap-3450	208	3	8113/40/34/006	8113/40/34/006	PROPN
ap-3450	208	4	.	.	PUNCT
ap-3450	209	1	[	[	X
ap-3450	209	2	2	2	NUM
ap-3450	209	3	]	]	PUNCT
ap-3450	209	4	a.	a.	NOUN
ap-3450	209	5	bezubik	bezubik	PROPN
ap-3450	209	6	,	,	PUNCT
ap-3450	209	7	j.	j.	PROPN
ap-3450	209	8	hrivnák	hrivnák	PROPN
ap-3450	209	9	,	,	PUNCT
ap-3450	209	10	j.	j.	PROPN
ap-3450	209	11	patera	patera	PROPN
ap-3450	209	12	,	,	PUNCT
ap-3450	209	13	s.	s.	PROPN
ap-3450	209	14	pošta	pošta	PROPN
ap-3450	209	15	.	.	PUNCT
ap-3450	210	1	three	three	NUM
ap-3450	210	2	variable	variable	ADJ
ap-3450	210	3	symmetric	symmetric	NOUN
ap-3450	210	4	and	and	CCONJ
ap-3450	210	5	antisymmetric	antisymmetric	ADJ
ap-3450	210	6	exponential	exponential	ADJ
ap-3450	210	7	functions	function	NOUN
ap-3450	210	8	and	and	CCONJ
ap-3450	210	9	orthogonal	orthogonal	ADJ
ap-3450	210	10	polynomials	polynomial	NOUN
ap-3450	210	11	.	.	PUNCT
ap-3450	211	1	math	math	NOUN
ap-3450	211	2	slovaca	slovaca	PROPN
ap-3450	211	3	to	to	PART
ap-3450	211	4	appear	appear	VERB
ap-3450	211	5	.	.	PUNCT
ap-3450	212	1	[	[	X
ap-3450	212	2	3	3	NUM
ap-3450	212	3	]	]	PUNCT
ap-3450	212	4	a.	a.	NOUN
ap-3450	212	5	klimyk	klimyk	PROPN
ap-3450	212	6	,	,	PUNCT
ap-3450	212	7	j.	j.	PROPN
ap-3450	212	8	patera	patera	PROPN
ap-3450	212	9	.	.	PUNCT
ap-3450	213	1	alternating	alternate	VERB
ap-3450	213	2	group	group	NOUN
ap-3450	213	3	and	and	CCONJ
ap-3450	213	4	multivariate	multivariate	NOUN
ap-3450	213	5	exponential	exponential	ADJ
ap-3450	213	6	functions	function	NOUN
ap-3450	213	7	.	.	PUNCT
ap-3450	214	1	in	in	ADP
ap-3450	214	2	groups	group	NOUN
ap-3450	214	3	and	and	CCONJ
ap-3450	214	4	symmetries	symmetry	NOUN
ap-3450	214	5	,	,	PUNCT
ap-3450	214	6	vol	vol	NOUN
ap-3450	214	7	.	.	PUNCT
ap-3450	215	1	47	47	NUM
ap-3450	215	2	of	of	ADP
ap-3450	215	3	crm	crm	PROPN
ap-3450	215	4	proc	proc	NOUN
ap-3450	215	5	.	.	PUNCT
ap-3450	216	1	lecture	lecture	NOUN
ap-3450	216	2	notes	note	NOUN
ap-3450	216	3	,	,	PUNCT
ap-3450	216	4	pp	pp	ADV
ap-3450	216	5	.	.	PUNCT
ap-3450	217	1	233–246	233–246	NUM
ap-3450	217	2	.	.	PUNCT
ap-3450	218	1	amer	amer	PROPN
ap-3450	218	2	.	.	PUNCT
ap-3450	218	3	math	math	PROPN
ap-3450	218	4	.	.	PUNCT
ap-3450	219	1	soc	soc	PROPN
ap-3450	219	2	.	.	PUNCT
ap-3450	219	3	,	,	PUNCT
ap-3450	219	4	providence	providence	NOUN
ap-3450	219	5	,	,	PUNCT
ap-3450	219	6	ri	ri	NOUN
ap-3450	219	7	,	,	PUNCT
ap-3450	219	8	2009	2009	NUM
ap-3450	219	9	.	.	PUNCT
ap-3450	220	1	[	[	X
ap-3450	220	2	4	4	X
ap-3450	220	3	]	]	PUNCT
ap-3450	220	4	j.	j.	PROPN
ap-3450	220	5	hrivnák	hrivnák	PROPN
ap-3450	220	6	,	,	PUNCT
ap-3450	220	7	j.	j.	PROPN
ap-3450	220	8	patera	patera	PROPN
ap-3450	220	9	,	,	PUNCT
ap-3450	220	10	s.	s.	PROPN
ap-3450	220	11	pošta	pošta	PROPN
ap-3450	220	12	.	.	PUNCT
ap-3450	221	1	three	three	NUM
ap-3450	221	2	-	-	PUNCT
ap-3450	221	3	variable	variable	ADJ
ap-3450	221	4	exponential	exponential	ADJ
ap-3450	221	5	functions	function	NOUN
ap-3450	221	6	of	of	ADP
ap-3450	221	7	the	the	DET
ap-3450	221	8	alternating	alternate	VERB
ap-3450	221	9	group	group	NOUN
ap-3450	221	10	.	.	PUNCT
ap-3450	222	1	j	j	PROPN
ap-3450	222	2	phys	phy	NOUN
ap-3450	222	3	a	a	DET
ap-3450	222	4	45(4):045201	45(4):045201	NUM
ap-3450	222	5	,	,	PUNCT
ap-3450	222	6	10	10	NUM
ap-3450	222	7	,	,	PUNCT
ap-3450	222	8	2012	2012	NUM
ap-3450	222	9	.	.	PUNCT
ap-3450	223	1	doi:10.1088/1751	doi:10.1088/1751	ADV
ap-3450	223	2	-	-	PUNCT
ap-3450	223	3	8113/45/4/045201	8113/45/4/045201	NUM
ap-3450	223	4	.	.	PUNCT
ap-3450	224	1	[	[	X
ap-3450	224	2	5	5	NUM
ap-3450	224	3	]	]	PUNCT
ap-3450	224	4	a.	a.	NOUN
ap-3450	224	5	klimyk	klimyk	PROPN
ap-3450	224	6	,	,	PUNCT
ap-3450	224	7	j.	j.	PROPN
ap-3450	224	8	patera	patera	PROPN
ap-3450	224	9	.	.	PUNCT
ap-3450	225	1	(	(	PUNCT
ap-3450	225	2	anti)symmetric	anti)symmetric	ADJ
ap-3450	225	3	multivariate	multivariate	VERB
ap-3450	225	4	trigonometric	trigonometric	NOUN
ap-3450	225	5	functions	function	NOUN
ap-3450	225	6	and	and	CCONJ
ap-3450	225	7	corresponding	corresponding	ADJ
ap-3450	225	8	fourier	fourier	NOUN
ap-3450	225	9	transforms	transform	VERB
ap-3450	225	10	.	.	PUNCT
ap-3450	226	1	j	j	PROPN
ap-3450	226	2	math	math	PROPN
ap-3450	226	3	phys	phys	PROPN
ap-3450	226	4	48(9):093504	48(9):093504	NUM
ap-3450	226	5	,	,	PUNCT
ap-3450	226	6	24	24	NUM
ap-3450	226	7	,	,	PUNCT
ap-3450	226	8	2007	2007	NUM
ap-3450	226	9	.	.	PUNCT
ap-3450	227	1	doi:10.1063/1.2779768	doi:10.1063/1.2779768	NOUN
ap-3450	227	2	.	.	PUNCT
ap-3450	228	1	[	[	X
ap-3450	228	2	6	6	NUM
ap-3450	228	3	]	]	PUNCT
ap-3450	228	4	a.	a.	NOUN
ap-3450	228	5	klimyk	klimyk	PROPN
ap-3450	228	6	,	,	PUNCT
ap-3450	228	7	j.	j.	PROPN
ap-3450	228	8	patera	patera	PROPN
ap-3450	228	9	.	.	PUNCT
ap-3450	229	1	alternating	alternate	VERB
ap-3450	229	2	multivariate	multivariate	NOUN
ap-3450	229	3	trigonometric	trigonometric	NOUN
ap-3450	229	4	functions	function	NOUN
ap-3450	229	5	and	and	CCONJ
ap-3450	229	6	corresponding	corresponding	ADJ
ap-3450	229	7	fourier	fourier	NOUN
ap-3450	229	8	transforms	transform	VERB
ap-3450	229	9	.	.	PUNCT
ap-3450	230	1	j	j	PROPN
ap-3450	230	2	phys	phy	NOUN
ap-3450	230	3	a	a	DET
ap-3450	230	4	41(14):145205	41(14):145205	NUM
ap-3450	230	5	,	,	PUNCT
ap-3450	230	6	16	16	NUM
ap-3450	230	7	,	,	PUNCT
ap-3450	230	8	2008	2008	NUM
ap-3450	230	9	.	.	PUNCT
ap-3450	231	1	doi:10.1088/1751	doi:10.1088/1751	ADV
ap-3450	231	2	-	-	PUNCT
ap-3450	231	3	8113/41/14/145205	8113/41/14/145205	NUM
ap-3450	231	4	.	.	PUNCT
ap-3450	232	1	[	[	X
ap-3450	232	2	7	7	X
ap-3450	232	3	]	]	X
ap-3450	232	4	j.	j.	PROPN
ap-3450	232	5	hrivnák	hrivnák	PROPN
ap-3450	232	6	,	,	PUNCT
ap-3450	232	7	j.	j.	PROPN
ap-3450	232	8	patera	patera	PROPN
ap-3450	232	9	.	.	PUNCT
ap-3450	233	1	two	two	NUM
ap-3450	233	2	-	-	PUNCT
ap-3450	233	3	dimensional	dimensional	ADJ
ap-3450	233	4	symmetric	symmetric	ADJ
ap-3450	233	5	and	and	CCONJ
ap-3450	233	6	antisymmetric	antisymmetric	ADJ
ap-3450	233	7	generalizations	generalization	NOUN
ap-3450	233	8	of	of	ADP
ap-3450	233	9	exponential	exponential	ADJ
ap-3450	233	10	and	and	CCONJ
ap-3450	233	11	cosine	cosine	NOUN
ap-3450	233	12	functions	function	NOUN
ap-3450	233	13	.	.	PUNCT
ap-3450	234	1	j	j	PROPN
ap-3450	234	2	math	math	PROPN
ap-3450	234	3	phys	phy	NOUN
ap-3450	234	4	51(2):023515	51(2):023515	NUM
ap-3450	234	5	,	,	PUNCT
ap-3450	234	6	24	24	NUM
ap-3450	234	7	,	,	PUNCT
ap-3450	234	8	2010	2010	NUM
ap-3450	234	9	.	.	PUNCT
ap-3450	235	1	doi:10.1063/1.3282850	doi:10.1063/1.3282850	NOUN
ap-3450	235	2	.	.	PUNCT
ap-3450	236	1	164	164	NUM
ap-3450	236	2	http://dx.doi.org/10.1088/1751-8113/40/34/006	http://dx.doi.org/10.1088/1751-8113/40/34/006	NOUN
ap-3450	236	3	http://dx.doi.org/10.1088/1751-8113/45/4/045201	http://dx.doi.org/10.1088/1751-8113/45/4/045201	VERB
ap-3450	236	4	http://dx.doi.org/10.1063/1.2779768	http://dx.doi.org/10.1063/1.2779768	PRON
ap-3450	236	5	http://dx.doi.org/10.1088/1751-8113/41/14/145205	http://dx.doi.org/10.1088/1751-8113/41/14/145205	X
ap-3450	236	6	http://dx.doi.org/10.1063/1.3282850	http://dx.doi.org/10.1063/1.3282850	NOUN
ap-3450	236	7	vol	vol	NOUN
ap-3450	236	8	.	.	PUNCT
ap-3450	237	1	56	56	NUM
ap-3450	237	2	no	no	NOUN
ap-3450	237	3	.	.	PUNCT
ap-3450	238	1	3/2016	3/2016	NUM
ap-3450	238	2	three	three	NUM
ap-3450	238	3	-	-	PUNCT
ap-3450	238	4	variable	variable	NOUN
ap-3450	238	5	alternating	alternate	VERB
ap-3450	238	6	trigonometric	trigonometric	ADJ
ap-3450	238	7	functions	function	NOUN
ap-3450	238	8	[	[	X
ap-3450	238	9	8	8	NUM
ap-3450	238	10	]	]	PUNCT
ap-3450	238	11	j.	j.	PROPN
ap-3450	238	12	hrivnák	hrivnák	PROPN
ap-3450	238	13	,	,	PUNCT
ap-3450	238	14	l.	l.	PROPN
ap-3450	238	15	motlochová	motlochová	PROPN
ap-3450	238	16	,	,	PUNCT
ap-3450	238	17	j.	j.	PROPN
ap-3450	238	18	patera	patera	PROPN
ap-3450	238	19	.	.	PUNCT
ap-3450	239	1	two	two	NUM
ap-3450	239	2	-	-	PUNCT
ap-3450	239	3	dimensional	dimensional	ADJ
ap-3450	239	4	symmetric	symmetric	ADJ
ap-3450	239	5	and	and	CCONJ
ap-3450	239	6	antisymmetric	antisymmetric	ADJ
ap-3450	239	7	generalizations	generalization	NOUN
ap-3450	239	8	of	of	ADP
ap-3450	239	9	sine	sine	ADJ
ap-3450	239	10	functions	function	NOUN
ap-3450	239	11	.	.	PUNCT
ap-3450	240	1	j	j	PROPN
ap-3450	240	2	math	math	PROPN
ap-3450	240	3	phys	phy	NOUN
ap-3450	240	4	51(7):073509	51(7):073509	NUM
ap-3450	240	5	,	,	PUNCT
ap-3450	240	6	13	13	NUM
ap-3450	240	7	,	,	PUNCT
ap-3450	240	8	2010	2010	NUM
ap-3450	240	9	.	.	PUNCT
ap-3450	241	1	doi:10.1063/1.3430567	doi:10.1063/1.3430567	ADV
ap-3450	241	2	.	.	PUNCT
ap-3450	242	1	[	[	X
ap-3450	242	2	9	9	NUM
ap-3450	242	3	]	]	PUNCT
ap-3450	242	4	a.	a.	NOUN
ap-3450	242	5	klimyk	klimyk	PROPN
ap-3450	242	6	,	,	PUNCT
ap-3450	242	7	j.	j.	PROPN
ap-3450	242	8	patera	patera	PROPN
ap-3450	242	9	.	.	PUNCT
ap-3450	243	1	orbit	orbit	NOUN
ap-3450	243	2	functions	function	NOUN
ap-3450	243	3	.	.	PUNCT
ap-3450	244	1	sigma	sigma	PROPN
ap-3450	244	2	symmetry	symmetry	PROPN
ap-3450	244	3	integrability	integrability	PROPN
ap-3450	244	4	geom	geom	PROPN
ap-3450	244	5	methods	method	NOUN
ap-3450	244	6	appl	appl	PROPN
ap-3450	244	7	2	2	NUM
ap-3450	244	8	:	:	PUNCT
ap-3450	244	9	paper	paper	NOUN
ap-3450	244	10	006	006	NUM
ap-3450	244	11	,	,	PUNCT
ap-3450	244	12	60	60	NUM
ap-3450	244	13	,	,	PUNCT
ap-3450	244	14	2006	2006	NUM
ap-3450	244	15	.	.	PUNCT
ap-3450	245	1	doi:10.3842	doi:10.3842	NOUN
ap-3450	245	2	/	/	SYM
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ap-3450	245	4	.	.	PUNCT
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ap-3450	246	2	10	10	NUM
ap-3450	246	3	]	]	PUNCT
ap-3450	246	4	a.	a.	NOUN
ap-3450	246	5	klimyk	klimyk	PROPN
ap-3450	246	6	,	,	PUNCT
ap-3450	246	7	j.	j.	PROPN
ap-3450	246	8	patera	patera	PROPN
ap-3450	246	9	.	.	PUNCT
ap-3450	247	1	antisymmetric	antisymmetric	PROPN
ap-3450	247	2	orbit	orbit	NOUN
ap-3450	247	3	functions	function	NOUN
ap-3450	247	4	.	.	PUNCT
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ap-3450	248	2	symmetry	symmetry	PROPN
ap-3450	248	3	integrability	integrability	PROPN
ap-3450	248	4	geom	geom	PROPN
ap-3450	248	5	methods	method	NOUN
ap-3450	248	6	appl	appl	PROPN
ap-3450	248	7	3	3	NUM
ap-3450	248	8	:	:	PUNCT
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ap-3450	248	10	023	023	NUM
ap-3450	248	11	,	,	PUNCT
ap-3450	248	12	83	83	NUM
ap-3450	248	13	,	,	PUNCT
ap-3450	248	14	2007	2007	NUM
ap-3450	248	15	.	.	PUNCT
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ap-3450	249	2	/	/	SYM
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ap-3450	249	4	.	.	PUNCT
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ap-3450	250	2	11	11	NUM
ap-3450	250	3	]	]	PUNCT
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ap-3450	250	5	klimyk	klimyk	PROPN
ap-3450	250	6	,	,	PUNCT
ap-3450	250	7	j.	j.	PROPN
ap-3450	250	8	patera	patera	PROPN
ap-3450	250	9	.	.	PUNCT
ap-3450	251	1	e	e	X
ap-3450	251	2	-	-	NOUN
ap-3450	251	3	orbit	orbit	NOUN
ap-3450	251	4	functions	function	NOUN
ap-3450	251	5	.	.	PUNCT
ap-3450	252	1	sigma	sigma	PROPN
ap-3450	252	2	symmetry	symmetry	PROPN
ap-3450	252	3	integrability	integrability	PROPN
ap-3450	252	4	geom	geom	PROPN
ap-3450	252	5	methods	method	NOUN
ap-3450	252	6	appl	appl	PROPN
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ap-3450	252	8	:	:	PUNCT
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ap-3450	252	11	,	,	PUNCT
ap-3450	252	12	57	57	NUM
ap-3450	252	13	,	,	PUNCT
ap-3450	252	14	2008	2008	NUM
ap-3450	252	15	.	.	PUNCT
ap-3450	253	1	doi:10.3842	doi:10.3842	NOUN
ap-3450	253	2	/	/	SYM
ap-3450	253	3	sigma.2008.002	sigma.2008.002	NOUN
ap-3450	253	4	.	.	PUNCT
ap-3450	254	1	[	[	X
ap-3450	254	2	12	12	NUM
ap-3450	254	3	]	]	X
ap-3450	254	4	h.	h.	PROPN
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ap-3450	254	6	.	.	PUNCT
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ap-3450	254	8	,	,	PUNCT
ap-3450	254	9	vol	vol	NOUN
ap-3450	254	10	.	.	PROPN
ap-3450	254	11	9999	9999	NUM
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ap-3450	254	13	encyclopedia	encyclopedia	NOUN
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ap-3450	254	17	its	its	PRON
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ap-3450	254	19	.	.	PUNCT
ap-3450	255	1	addison	addison	PROPN
ap-3450	255	2	-	-	PUNCT
ap-3450	255	3	wesley	wesley	PROPN
ap-3450	255	4	publishing	publishing	PROPN
ap-3450	255	5	co.	co.	PROPN
ap-3450	255	6	,	,	PUNCT
ap-3450	255	7	reading	reading	NOUN
ap-3450	255	8	,	,	PUNCT
ap-3450	255	9	mass	mass	PROPN
ap-3450	255	10	.	.	PROPN
ap-3450	255	11	,	,	PUNCT
ap-3450	255	12	1978	1978	NUM
ap-3450	255	13	.	.	PUNCT
ap-3450	256	1	with	with	ADP
ap-3450	256	2	a	a	DET
ap-3450	256	3	foreword	foreword	NOUN
ap-3450	256	4	by	by	ADP
ap-3450	256	5	marvin	marvin	PROPN
ap-3450	256	6	marcus	marcus	PROPN
ap-3450	256	7	,	,	PUNCT
ap-3450	256	8	encyclopedia	encyclopedia	NOUN
ap-3450	256	9	of	of	ADP
ap-3450	256	10	mathematics	mathematic	NOUN
ap-3450	256	11	and	and	CCONJ
ap-3450	256	12	its	its	PRON
ap-3450	256	13	applications	application	NOUN
ap-3450	256	14	,	,	PUNCT
ap-3450	256	15	vol	vol	NOUN
ap-3450	256	16	.	.	PROPN
ap-3450	256	17	6	6	NUM
ap-3450	256	18	.	.	PUNCT
ap-3450	257	1	[	[	X
ap-3450	257	2	13	13	NUM
ap-3450	257	3	]	]	PUNCT
ap-3450	257	4	k.	k.	PROPN
ap-3450	257	5	r.	r.	PROPN
ap-3450	257	6	rao	rao	PROPN
ap-3450	257	7	,	,	PUNCT
ap-3450	257	8	p.	p.	PROPN
ap-3450	257	9	yip	yip	PROPN
ap-3450	257	10	.	.	PUNCT
ap-3450	258	1	discrete	discrete	ADJ
ap-3450	258	2	cosine	cosine	NOUN
ap-3450	258	3	transform	transform	NOUN
ap-3450	258	4	.	.	PUNCT
ap-3450	259	1	academic	academic	PROPN
ap-3450	259	2	press	press	PROPN
ap-3450	259	3	inc	inc	PROPN
ap-3450	259	4	.	.	PROPN
ap-3450	259	5	,	,	PUNCT
ap-3450	259	6	boston	boston	PROPN
ap-3450	259	7	,	,	PUNCT
ap-3450	259	8	ma	ma	PROPN
ap-3450	259	9	,	,	PUNCT
ap-3450	259	10	1990	1990	NUM
ap-3450	259	11	.	.	PUNCT
ap-3450	260	1	algorithms	algorithm	NOUN
ap-3450	260	2	,	,	PUNCT
ap-3450	260	3	advantages	advantage	NOUN
ap-3450	260	4	,	,	PUNCT
ap-3450	260	5	applications	application	NOUN
ap-3450	260	6	.	.	PUNCT
ap-3450	261	1	165	165	NUM
ap-3450	261	2	http://dx.doi.org/10.1063/1.3430567	http://dx.doi.org/10.1063/1.3430567	NOUN
ap-3450	261	3	http://dx.doi.org/10.3842/sigma.2006.006	http://dx.doi.org/10.3842/sigma.2006.006	NOUN
ap-3450	261	4	http://dx.doi.org/10.3842/sigma.2007.023	http://dx.doi.org/10.3842/sigma.2007.023	PRON
ap-3450	261	5	http://dx.doi.org/10.3842/sigma.2008.002	http://dx.doi.org/10.3842/sigma.2008.002	NOUN
ap-3450	261	6	acta	acta	PROPN
ap-3450	261	7	polytechnica	polytechnica	PROPN
ap-3450	261	8	56(3):156–165	56(3):156–165	PROPN
ap-3450	261	9	,	,	PUNCT
ap-3450	261	10	2016	2016	NUM
ap-3450	261	11	1	1	NUM
ap-3450	261	12	introduction	introduction	NOUN
ap-3450	261	13	2	2	NUM
ap-3450	261	14	definition	definition	NOUN
ap-3450	261	15	3	3	NUM
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ap-3450	261	17	orthogonality	orthogonality	NOUN
ap-3450	261	18	4	4	NUM
ap-3450	261	19	product	product	NOUN
ap-3450	261	20	decomposition	decomposition	NOUN
ap-3450	261	21	5	5	NUM
ap-3450	261	22	discrete	discrete	ADJ
ap-3450	261	23	cosine	cosine	NOUN
ap-3450	261	24	transforms	transform	VERB
ap-3450	261	25	5.1	5.1	NUM
ap-3450	261	26	amdct-1	amdct-1	NUM
ap-3450	261	27	5.2	5.2	NUM
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ap-3450	261	29	5.3	5.3	NUM
ap-3450	261	30	amdct-3	amdct-3	NUM
ap-3450	261	31	5.4	5.4	NUM
ap-3450	261	32	amdct-4	amdct-4	NUM
ap-3450	261	33	6	6	NUM
ap-3450	261	34	discrete	discrete	ADJ
ap-3450	261	35	sine	sine	NOUN
ap-3450	261	36	transforms	transform	VERB
ap-3450	261	37	6.1	6.1	NUM
ap-3450	261	38	amdst-1	amdst-1	NUM
ap-3450	261	39	6.2	6.2	NUM
ap-3450	261	40	amdst-2	amdst-2	NUM
ap-3450	261	41	6.3	6.3	NUM
ap-3450	261	42	amdst-3	amdst-3	NUM
ap-3450	261	43	6.4	6.4	NUM
ap-3450	261	44	amdst-4	amdst-4	NUM
ap-3450	261	45	7	7	NUM
ap-3450	261	46	conclusion	conclusion	NOUN
ap-3450	261	47	acknowledgements	acknowledgement	NOUN
ap-3450	261	48	references	reference	NOUN