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
Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.
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.
info:eu-repo/semantics/publishedVersion
Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.
Matemática Pura
Materia
Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
Nivel de accesibilidad
acceso abierto
Condiciones de uso
Repositorio
Repositorio Digital Universitario (UNC)
Institución
Universidad Nacional de Córdoba
OAI Identificador
oai:rdu.unc.edu.ar:11086/555133
Acceder
id RDUUNC_7722575888cdea97a11ac35857675a58
oai_identifier_str oai:rdu.unc.edu.ar:11086/555133
network_acronym_str RDUUNC
repository_id_str 2572
network_name_str Repositorio Digital Universitario (UNC)
spelling Matrix-valued Gegenbauer-type polynomialsKoelink, ErikDe los Ríos, Ana M.Román, Pablo ManuelMatrix-valued orthogonal polynomialsMatrix-valued differential operatorsGegenbauer polynomialsShift operatorDarboux factorizationFil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.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.info:eu-repo/semantics/publishedVersionFil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.Matemática Purahttps://orcid.org/0000-0002-2791-385X2017info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfKoelink, E., De los Ríos, A. M. y Román, P. M. (2017). Matrix-valued Gegenbauer-type polynomials. Constructive Approximation, 46 (3), 459-487. https://doi.org/10.1007/s00365-017-9384-40176-4276http://hdl.handle.net/11086/5551331432-0940https://doi.org/10.1007/s00365-017-9384-4enginfo:eu-repo/semantics/openAccessreponame:Repositorio Digital Universitario (UNC)instname:Universidad Nacional de Córdobainstacron:UNC2025-11-06T09:37:14Zoai:rdu.unc.edu.ar:11086/555133Institucionalhttps://rdu.unc.edu.ar/Universidad públicaNo correspondehttp://rdu.unc.edu.ar/oai/snrdoca.unc@gmail.comArgentinaNo correspondeNo correspondeNo correspondeopendoar:25722025-11-06 09:37:14.319Repositorio Digital Universitario (UNC) - Universidad Nacional de Córdobafalse
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
Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
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.contributor.none.fl_str_mv https://orcid.org/0000-0002-2791-385X
dc.subject.none.fl_str_mv Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
topic Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
dc.description.none.fl_txt_mv Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.
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.
info:eu-repo/semantics/publishedVersion
Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina.
Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina.
Matemática Pura
description Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
publishDate 2017
dc.date.none.fl_str_mv 2017
dc.type.none.fl_str_mv info:eu-repo/semantics/publishedVersion
info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
status_str publishedVersion
format article
dc.identifier.none.fl_str_mv Koelink, E., De los Ríos, A. M. y Román, P. M. (2017). Matrix-valued Gegenbauer-type polynomials. Constructive Approximation, 46 (3), 459-487. https://doi.org/10.1007/s00365-017-9384-4
0176-4276
http://hdl.handle.net/11086/555133
1432-0940
https://doi.org/10.1007/s00365-017-9384-4
identifier_str_mv Koelink, E., De los Ríos, A. M. y Román, P. M. (2017). Matrix-valued Gegenbauer-type polynomials. Constructive Approximation, 46 (3), 459-487. https://doi.org/10.1007/s00365-017-9384-4
0176-4276
1432-0940
url http://hdl.handle.net/11086/555133
https://doi.org/10.1007/s00365-017-9384-4
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Repositorio Digital Universitario (UNC)
instname:Universidad Nacional de Córdoba
instacron:UNC
reponame_str Repositorio Digital Universitario (UNC)
collection Repositorio Digital Universitario (UNC)
instname_str Universidad Nacional de Córdoba
instacron_str UNC
institution UNC
repository.name.fl_str_mv Repositorio Digital Universitario (UNC) - Universidad Nacional de Córdoba
repository.mail.fl_str_mv oca.unc@gmail.com
_version_ 1848046202824687616
score 12.976206

Publicaciones similares