Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure

Detalhes bibliográficos
Autor(a) principal: Martínez-Finkelshtein, A.
Data de Publicação: 2018
Outros Autores: Sri Ranga, A. [UNESP], Veronese, D. O.
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.1090/mcom/3210
http://hdl.handle.net/11449/175682
Resumo: Given a nontrivial Borel measure μ on the unit circle T{double-struck}, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, {cn} and {dn}, where {dn} is additionally a positive chain sequence. Coefficients (cn, dn) provide a parametrization of a family of measures related to μ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (cn, dn)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of μ at z = 1. These results are easily reformulated in order to find gaps in the support of μ at any other z ∈ T{double-struck}. We provide also some examples showing that the bounds are tight and illustrate their computational applications.
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spelling Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measureOrthogonal polynomials on the unit circlePara-orthogonal polynomials on the unit circlePositive chain sequencesThree term recurrenceGiven a nontrivial Borel measure μ on the unit circle T{double-struck}, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, {cn} and {dn}, where {dn} is additionally a positive chain sequence. Coefficients (cn, dn) provide a parametrization of a family of measures related to μ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (cn, dn)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of μ at z = 1. These results are easily reformulated in order to find gaps in the support of μ at any other z ∈ T{double-struck}. We provide also some examples showing that the bounds are tight and illustrate their computational applications.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Departamento de Matemáticas Universidad de AlmeríaInstituto Carlos I de Física Teórica and Computacional Granada UniversityDepartamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaICTE UFTM - Universidade Federal do Triangulo MineiroDepartamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaFAPESP: 2009/13832-9CNPq: 305073/2014-1CNPq: 475502/2013-2Universidad de AlmeríaGranada UniversityUniversidade Estadual Paulista (Unesp)UFTM - Universidade Federal do Triangulo MineiroMartínez-Finkelshtein, A.Sri Ranga, A. [UNESP]Veronese, D. O.2018-12-11T17:17:04Z2018-12-11T17:17:04Z2018-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article261-288application/pdfhttp://dx.doi.org/10.1090/mcom/3210Mathematics of Computation, v. 87, n. 309, p. 261-288, 2018.0025-5718http://hdl.handle.net/11449/17568210.1090/mcom/32102-s2.0-850389515082-s2.0-85038951508.pdfScopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengMathematics of Computation1,939info:eu-repo/semantics/openAccess2023-10-31T06:11:40Zoai:repositorio.unesp.br:11449/175682Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T16:32:51.128829Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
title Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
spellingShingle Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
Martínez-Finkelshtein, A.
Orthogonal polynomials on the unit circle
Para-orthogonal polynomials on the unit circle
Positive chain sequences
Three term recurrence
title_short Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
title_full Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
title_fullStr Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
title_full_unstemmed Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
title_sort Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
author Martínez-Finkelshtein, A.
author_facet Martínez-Finkelshtein, A.
Sri Ranga, A. [UNESP]
Veronese, D. O.
author_role author
author2 Sri Ranga, A. [UNESP]
Veronese, D. O.
author2_role author
author
dc.contributor.none.fl_str_mv Universidad de Almería
Granada University
Universidade Estadual Paulista (Unesp)
UFTM - Universidade Federal do Triangulo Mineiro
dc.contributor.author.fl_str_mv Martínez-Finkelshtein, A.
Sri Ranga, A. [UNESP]
Veronese, D. O.
dc.subject.por.fl_str_mv Orthogonal polynomials on the unit circle
Para-orthogonal polynomials on the unit circle
Positive chain sequences
Three term recurrence
topic Orthogonal polynomials on the unit circle
Para-orthogonal polynomials on the unit circle
Positive chain sequences
Three term recurrence
description Given a nontrivial Borel measure μ on the unit circle T{double-struck}, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, {cn} and {dn}, where {dn} is additionally a positive chain sequence. Coefficients (cn, dn) provide a parametrization of a family of measures related to μ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (cn, dn)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of μ at z = 1. These results are easily reformulated in order to find gaps in the support of μ at any other z ∈ T{double-struck}. We provide also some examples showing that the bounds are tight and illustrate their computational applications.
publishDate 2018
dc.date.none.fl_str_mv 2018-12-11T17:17:04Z
2018-12-11T17:17:04Z
2018-01-01
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://dx.doi.org/10.1090/mcom/3210
Mathematics of Computation, v. 87, n. 309, p. 261-288, 2018.
0025-5718
http://hdl.handle.net/11449/175682
10.1090/mcom/3210
2-s2.0-85038951508
2-s2.0-85038951508.pdf
url http://dx.doi.org/10.1090/mcom/3210
http://hdl.handle.net/11449/175682
identifier_str_mv Mathematics of Computation, v. 87, n. 309, p. 261-288, 2018.
0025-5718
10.1090/mcom/3210
2-s2.0-85038951508
2-s2.0-85038951508.pdf
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Mathematics of Computation
1,939
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 261-288
application/pdf
dc.source.none.fl_str_mv Scopus
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv
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