Matrix-Valued Gegenbauer-Type polynomials

Autores
Koelink, Erik; de los Ríos, Ana M.; Román, Pablo Manuel
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν> 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν+ 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν= 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.
Fil: Koelink, Erik. Radboud Universiteit Nijmegen; Países Bajos
Fil: de los Ríos, Ana M.. Universidad de Sevilla; España
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
DARBOUX FACTORIZATION
GEGENBAUER POLYNOMIALS
MATRIX-VALUED DIFFERENTIAL OPERATORS
MATRIX-VALUED ORTHOGONAL POLYNOMIALS
SHIFT OPERATOR
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/60249
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/60249
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Matrix-Valued Gegenbauer-Type polynomialsKoelink, Erikde los Ríos, Ana M.Román, Pablo ManuelDARBOUX FACTORIZATIONGEGENBAUER POLYNOMIALSMATRIX-VALUED DIFFERENTIAL OPERATORSMATRIX-VALUED ORTHOGONAL POLYNOMIALSSHIFT OPERATORhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν> 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν+ 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν= 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.Fil: Koelink, Erik. Radboud Universiteit Nijmegen; Países BajosFil: de los Ríos, Ana M.. Universidad de Sevilla; EspañaFil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaSpringer2017-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/60249Koelink, Erik; de los Ríos, Ana M.; Román, Pablo Manuel; Matrix-Valued Gegenbauer-Type polynomials; Springer; Constructive Approximation; 46; 3; 12-2017; 459-4870176-42761432-0940CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00365-017-9384-4info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00365-017-9384-4info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:39:09Zoai:ri.conicet.gov.ar:11336/60249instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:39:09.812CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Matrix-Valued Gegenbauer-Type polynomials
title Matrix-Valued Gegenbauer-Type polynomials
spellingShingle Matrix-Valued Gegenbauer-Type polynomials
Koelink, Erik
DARBOUX FACTORIZATION
GEGENBAUER POLYNOMIALS
MATRIX-VALUED DIFFERENTIAL OPERATORS
MATRIX-VALUED ORTHOGONAL POLYNOMIALS
SHIFT OPERATOR
title_short Matrix-Valued Gegenbauer-Type polynomials
title_full Matrix-Valued Gegenbauer-Type polynomials
title_fullStr Matrix-Valued Gegenbauer-Type polynomials
title_full_unstemmed Matrix-Valued Gegenbauer-Type polynomials
title_sort Matrix-Valued Gegenbauer-Type polynomials
dc.creator.none.fl_str_mv Koelink, Erik
de los Ríos, Ana M.
Román, Pablo Manuel
author Koelink, Erik
author_facet Koelink, Erik
de los Ríos, Ana M.
Román, Pablo Manuel
author_role author
author2 de los Ríos, Ana M.
Román, Pablo Manuel
author2_role author
author
dc.subject.none.fl_str_mv DARBOUX FACTORIZATION
GEGENBAUER POLYNOMIALS
MATRIX-VALUED DIFFERENTIAL OPERATORS
MATRIX-VALUED ORTHOGONAL POLYNOMIALS
SHIFT OPERATOR
topic DARBOUX FACTORIZATION
GEGENBAUER POLYNOMIALS
MATRIX-VALUED DIFFERENTIAL OPERATORS
MATRIX-VALUED ORTHOGONAL POLYNOMIALS
SHIFT OPERATOR
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν> 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν+ 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν= 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.
Fil: Koelink, Erik. Radboud Universiteit Nijmegen; Países Bajos
Fil: de los Ríos, Ana M.. Universidad de Sevilla; España
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν> 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν+ 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν= 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.
publishDate 2017
dc.date.none.fl_str_mv 2017-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/60249
Koelink, Erik; de los Ríos, Ana M.; Román, Pablo Manuel; Matrix-Valued Gegenbauer-Type polynomials; Springer; Constructive Approximation; 46; 3; 12-2017; 459-487
0176-4276
1432-0940
CONICET Digital
CONICET
url http://hdl.handle.net/11336/60249
identifier_str_mv Koelink, Erik; de los Ríos, Ana M.; Román, Pablo Manuel; Matrix-Valued Gegenbauer-Type polynomials; Springer; Constructive Approximation; 46; 3; 12-2017; 459-487
0176-4276
1432-0940
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00365-017-9384-4
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00365-017-9384-4
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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