Asymptotics of matrix valued orthogonal polynomials on [−1,1]

Autores
Deaño, Alfredo; Kuijlaars, Arno B. J.; Román, Pablo Manuel
Año de publicación
2023
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
Fil: Deaño, Alfredo. Universidad Carlos III de Madrid. Instituto de Salud; España
Fil: Kuijlaars, Arno B. J.. Katholikie Universiteit Leuven; Bélgica
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
ASYMPTOTIC ANALYSIS
MATRIX ORTHOGONAL POLYNOMIALS
RIEMANN-HILBERT PROBLEMS
STEEPEST DESCENT METHOD
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/226119
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/226119
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Asymptotics of matrix valued orthogonal polynomials on [−1,1]Deaño, AlfredoKuijlaars, Arno B. J.Román, Pablo ManuelASYMPTOTIC ANALYSISMATRIX ORTHOGONAL POLYNOMIALSRIEMANN-HILBERT PROBLEMSSTEEPEST DESCENT METHODhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.Fil: Deaño, Alfredo. Universidad Carlos III de Madrid. Instituto de Salud; EspañaFil: Kuijlaars, Arno B. J.. Katholikie Universiteit Leuven; BélgicaFil: 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; ArgentinaAcademic Press Inc Elsevier Science2023-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/226119Deaño, Alfredo; Kuijlaars, Arno B. J.; Román, Pablo Manuel; Asymptotics of matrix valued orthogonal polynomials on [−1,1]; Academic Press Inc Elsevier Science; Advances in Mathematics; 423; 6-2023; 1-610001-8708CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S000187082300186Xinfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2023.109043info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T15:04:33Zoai:ri.conicet.gov.ar:11336/226119instacron: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:34982025-10-15 15:04:33.333CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Asymptotics of matrix valued orthogonal polynomials on [−1,1]
title Asymptotics of matrix valued orthogonal polynomials on [−1,1]
spellingShingle Asymptotics of matrix valued orthogonal polynomials on [−1,1]
Deaño, Alfredo
ASYMPTOTIC ANALYSIS
MATRIX ORTHOGONAL POLYNOMIALS
RIEMANN-HILBERT PROBLEMS
STEEPEST DESCENT METHOD
title_short Asymptotics of matrix valued orthogonal polynomials on [−1,1]
title_full Asymptotics of matrix valued orthogonal polynomials on [−1,1]
title_fullStr Asymptotics of matrix valued orthogonal polynomials on [−1,1]
title_full_unstemmed Asymptotics of matrix valued orthogonal polynomials on [−1,1]
title_sort Asymptotics of matrix valued orthogonal polynomials on [−1,1]
dc.creator.none.fl_str_mv Deaño, Alfredo
Kuijlaars, Arno B. J.
Román, Pablo Manuel
author Deaño, Alfredo
author_facet Deaño, Alfredo
Kuijlaars, Arno B. J.
Román, Pablo Manuel
author_role author
author2 Kuijlaars, Arno B. J.
Román, Pablo Manuel
author2_role author
author
dc.subject.none.fl_str_mv ASYMPTOTIC ANALYSIS
MATRIX ORTHOGONAL POLYNOMIALS
RIEMANN-HILBERT PROBLEMS
STEEPEST DESCENT METHOD
topic ASYMPTOTIC ANALYSIS
MATRIX ORTHOGONAL POLYNOMIALS
RIEMANN-HILBERT PROBLEMS
STEEPEST DESCENT METHOD
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 analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
Fil: Deaño, Alfredo. Universidad Carlos III de Madrid. Instituto de Salud; España
Fil: Kuijlaars, Arno B. J.. Katholikie Universiteit Leuven; Bélgica
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 analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
publishDate 2023
dc.date.none.fl_str_mv 2023-06
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/11336/226119
Deaño, Alfredo; Kuijlaars, Arno B. J.; Román, Pablo Manuel; Asymptotics of matrix valued orthogonal polynomials on [−1,1]; Academic Press Inc Elsevier Science; Advances in Mathematics; 423; 6-2023; 1-61
0001-8708
CONICET Digital
CONICET
url http://hdl.handle.net/11336/226119
identifier_str_mv Deaño, Alfredo; Kuijlaars, Arno B. J.; Román, Pablo Manuel; Asymptotics of matrix valued orthogonal polynomials on [−1,1]; Academic Press Inc Elsevier Science; Advances in Mathematics; 423; 6-2023; 1-61
0001-8708
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S000187082300186X
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2023.109043
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
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 13.22299

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