Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle

Detalhes bibliográficos
Autor(a) principal: Ranga, A. Sri [UNESP]
Data de Publicação: 2016
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.1090/proc12766
http://hdl.handle.net/11449/172434
Resumo: The principal objective here is to look at some algebraic properties of the orthogonal polynomials Ψn (b,s,t) n with respect to the Sobolev inner product on the unit circle <f,g>S (b,s,t) = (1 − t) <f,g>μ(b) + t f(1) g(1) + s <f', g'>μ(b+1), where <f, g> μ(b) = τ(b)/2π∫2π 0 f(eiθ) g(eiθ) (eπ−θ)Im(b)(sin2(θ/2))Re(b)dθ. Here, Re(b) > −1/2, 0 ≤ t < 1, s > 0 and τ(b) is taken to be such that <1, 1>μ(b) = 1. We show that, for example, the monic Sobolev orthogonal polynomials Ψ(b,s,t) n satisfy the recurrence Ψ(b,s,t) n (z)−β(b,s,t) n Ψ(b,s,t) n−1 (z) = Φ(b,t) n (z), n ≥ 1, where Φ(b,t) n are the monic orthogonal polynomials with respect to the inner product <f, g>μ(b,t) = (1 − t) <f, g> μ(b) + t f(1) g(1). Some related bounds and asymptotic properties are also given.
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spelling Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circleOrthogonal polynomials on the unit circlePara-orthogonal polynomialsPositive chain sequencesSobolev orthogonal polynomials on the unit circleThe principal objective here is to look at some algebraic properties of the orthogonal polynomials Ψn (b,s,t) n with respect to the Sobolev inner product on the unit circle <f,g>S (b,s,t) = (1 − t) <f,g>μ(b) + t f(1) g(1) + s <f', g'>μ(b+1), where <f, g> μ(b) = τ(b)/2π∫2π 0 f(eiθ) g(eiθ) (eπ−θ)Im(b)(sin2(θ/2))Re(b)dθ. Here, Re(b) > −1/2, 0 ≤ t < 1, s > 0 and τ(b) is taken to be such that <1, 1>μ(b) = 1. We show that, for example, the monic Sobolev orthogonal polynomials Ψ(b,s,t) n satisfy the recurrence Ψ(b,s,t) n (z)−β(b,s,t) n Ψ(b,s,t) n−1 (z) = Φ(b,t) n (z), n ≥ 1, where Φ(b,t) n are the monic orthogonal polynomials with respect to the inner product <f, g>μ(b,t) = (1 − t) <f, g> μ(b) + t f(1) g(1). Some related bounds and asymptotic properties are also given.Departamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaDepartamento de Matemática Aplicada IBILCE UNESP - Universidade Estadual PaulistaUniversidade Estadual Paulista (Unesp)Ranga, A. Sri [UNESP]2018-12-11T17:00:20Z2018-12-11T17:00:20Z2016-03-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article1129-1143application/pdfhttp://dx.doi.org/10.1090/proc12766Proceedings of the American Mathematical Society, v. 144, n. 3, p. 1129-1143, 2016.1088-68260002-9939http://hdl.handle.net/11449/17243410.1090/proc127662-s2.0-849545067962-s2.0-84954506796.pdfScopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengProceedings of the American Mathematical Society1,1831,183info:eu-repo/semantics/openAccess2023-12-27T06:21:52Zoai:repositorio.unesp.br:11449/172434Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T21:27:17.205403Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
title Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
spellingShingle Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
Ranga, A. Sri [UNESP]
Orthogonal polynomials on the unit circle
Para-orthogonal polynomials
Positive chain sequences
Sobolev orthogonal polynomials on the unit circle
title_short Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
title_full Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
title_fullStr Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
title_full_unstemmed Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
title_sort Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
author Ranga, A. Sri [UNESP]
author_facet Ranga, A. Sri [UNESP]
author_role author
dc.contributor.none.fl_str_mv Universidade Estadual Paulista (Unesp)
dc.contributor.author.fl_str_mv Ranga, A. Sri [UNESP]
dc.subject.por.fl_str_mv Orthogonal polynomials on the unit circle
Para-orthogonal polynomials
Positive chain sequences
Sobolev orthogonal polynomials on the unit circle
topic Orthogonal polynomials on the unit circle
Para-orthogonal polynomials
Positive chain sequences
Sobolev orthogonal polynomials on the unit circle
description The principal objective here is to look at some algebraic properties of the orthogonal polynomials Ψn (b,s,t) n with respect to the Sobolev inner product on the unit circle <f,g>S (b,s,t) = (1 − t) <f,g>μ(b) + t f(1) g(1) + s <f', g'>μ(b+1), where <f, g> μ(b) = τ(b)/2π∫2π 0 f(eiθ) g(eiθ) (eπ−θ)Im(b)(sin2(θ/2))Re(b)dθ. Here, Re(b) > −1/2, 0 ≤ t < 1, s > 0 and τ(b) is taken to be such that <1, 1>μ(b) = 1. We show that, for example, the monic Sobolev orthogonal polynomials Ψ(b,s,t) n satisfy the recurrence Ψ(b,s,t) n (z)−β(b,s,t) n Ψ(b,s,t) n−1 (z) = Φ(b,t) n (z), n ≥ 1, where Φ(b,t) n are the monic orthogonal polynomials with respect to the inner product <f, g>μ(b,t) = (1 − t) <f, g> μ(b) + t f(1) g(1). Some related bounds and asymptotic properties are also given.
publishDate 2016
dc.date.none.fl_str_mv 2016-03-01
2018-12-11T17:00:20Z
2018-12-11T17:00:20Z
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/proc12766
Proceedings of the American Mathematical Society, v. 144, n. 3, p. 1129-1143, 2016.
1088-6826
0002-9939
http://hdl.handle.net/11449/172434
10.1090/proc12766
2-s2.0-84954506796
2-s2.0-84954506796.pdf
url http://dx.doi.org/10.1090/proc12766
http://hdl.handle.net/11449/172434
identifier_str_mv Proceedings of the American Mathematical Society, v. 144, n. 3, p. 1129-1143, 2016.
1088-6826
0002-9939
10.1090/proc12766
2-s2.0-84954506796
2-s2.0-84954506796.pdf
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Proceedings of the American Mathematical Society
1,183
1,183
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 1129-1143
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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