Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences
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/10.1007/s40314-016-0392-y http://hdl.handle.net/11449/176364 |
Resumo: | It was shown recently that associated with a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {αn}n=0∞ associated with the orthogonal polynomials with respect to μ are given by the relation αn-1=τ¯n-1[1-2mn-icn1-icn],n≥1,where τ0= 1 , τn=∏k=1n(1-ick)/(1+ick), n≥ 1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {cn}n=1∞ and {mn}n=1∞. When the sequence {cn}n=1∞ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of z= - 1. Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences {cn}n=1∞ and {mn}n=1∞ with the additional restriction c2n=-c2n-1,n≥1. We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained. |
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Repositório Institucional da UNESP |
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Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequencesAlternating sign sequencesChain sequencesPara-orthogonal polynomialsPeriodic Verblunsky coefficientsProbability measuresIt was shown recently that associated with a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {αn}n=0∞ associated with the orthogonal polynomials with respect to μ are given by the relation αn-1=τ¯n-1[1-2mn-icn1-icn],n≥1,where τ0= 1 , τn=∏k=1n(1-ick)/(1+ick), n≥ 1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {cn}n=1∞ and {mn}n=1∞. When the sequence {cn}n=1∞ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of z= - 1. Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences {cn}n=1∞ and {mn}n=1∞ with the additional restriction c2n=-c2n-1,n≥1. We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Departamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaDepartamento de Matemática Universidade Federal do MaranhãoICTE Universidade Federal do Triângulo MineiroDepartamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaFAPESP: 2014/22571-2CNPq: 305073/2014-1CNPq: 305208/2015-2CNPq: 475502/2013-2Universidade Estadual Paulista (Unesp)Universidade Federal do MaranhãoUniversidade Federal do Triângulo MineiroBracciali, Cleonice F. [UNESP]Silva, Jairo S.Sri Ranga, A. [UNESP]Veronese, Daniel O.2018-12-11T17:20:29Z2018-12-11T17:20:29Z2018-05-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article1142-1161application/pdfhttp://dx.doi.org/10.1007/s40314-016-0392-yComputational and Applied Mathematics, v. 37, n. 2, p. 1142-1161, 2018.1807-03020101-8205http://hdl.handle.net/11449/17636410.1007/s40314-016-0392-y2-s2.0-850474341502-s2.0-85047434150.pdf83003224526224670000-0002-6823-4204Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengComputational and Applied Mathematics0,272info:eu-repo/semantics/openAccess2024-01-08T06:22:11Zoai:repositorio.unesp.br:11449/176364Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T22:23:53.348389Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
title |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
spellingShingle |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences Bracciali, Cleonice F. [UNESP] Alternating sign sequences Chain sequences Para-orthogonal polynomials Periodic Verblunsky coefficients Probability measures |
title_short |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
title_full |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
title_fullStr |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
title_full_unstemmed |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
title_sort |
Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences |
author |
Bracciali, Cleonice F. [UNESP] |
author_facet |
Bracciali, Cleonice F. [UNESP] Silva, Jairo S. Sri Ranga, A. [UNESP] Veronese, Daniel O. |
author_role |
author |
author2 |
Silva, Jairo S. Sri Ranga, A. [UNESP] Veronese, Daniel O. |
author2_role |
author author author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) Universidade Federal do Maranhão Universidade Federal do Triângulo Mineiro |
dc.contributor.author.fl_str_mv |
Bracciali, Cleonice F. [UNESP] Silva, Jairo S. Sri Ranga, A. [UNESP] Veronese, Daniel O. |
dc.subject.por.fl_str_mv |
Alternating sign sequences Chain sequences Para-orthogonal polynomials Periodic Verblunsky coefficients Probability measures |
topic |
Alternating sign sequences Chain sequences Para-orthogonal polynomials Periodic Verblunsky coefficients Probability measures |
description |
It was shown recently that associated with a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {αn}n=0∞ associated with the orthogonal polynomials with respect to μ are given by the relation αn-1=τ¯n-1[1-2mn-icn1-icn],n≥1,where τ0= 1 , τn=∏k=1n(1-ick)/(1+ick), n≥ 1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {cn}n=1∞ and {mn}n=1∞. When the sequence {cn}n=1∞ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of z= - 1. Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences {cn}n=1∞ and {mn}n=1∞ with the additional restriction c2n=-c2n-1,n≥1. We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained. |
publishDate |
2018 |
dc.date.none.fl_str_mv |
2018-12-11T17:20:29Z 2018-12-11T17:20:29Z 2018-05-01 |
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/10.1007/s40314-016-0392-y Computational and Applied Mathematics, v. 37, n. 2, p. 1142-1161, 2018. 1807-0302 0101-8205 http://hdl.handle.net/11449/176364 10.1007/s40314-016-0392-y 2-s2.0-85047434150 2-s2.0-85047434150.pdf 8300322452622467 0000-0002-6823-4204 |
url |
http://dx.doi.org/10.1007/s40314-016-0392-y http://hdl.handle.net/11449/176364 |
identifier_str_mv |
Computational and Applied Mathematics, v. 37, n. 2, p. 1142-1161, 2018. 1807-0302 0101-8205 10.1007/s40314-016-0392-y 2-s2.0-85047434150 2-s2.0-85047434150.pdf 8300322452622467 0000-0002-6823-4204 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Computational and Applied Mathematics 0,272 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
1142-1161 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_ |
1808129423934750720 |