Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
Autor(a) principal: | |
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Data de Publicação: | 2018 |
Outros Autores: | , |
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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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 |
|
_version_ |
1808128669306060800 |