Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1016/j.apnum.2016.05.008 http://hdl.handle.net/11449/173102 |
Resumo: | When a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π-θ))=-dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn∗(z)} and {zΦn(z)-Φn∗(z)}, where Φn∗(z)=zn Φn(1/z/)/, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [-1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)-Φn∗(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn∗(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ-w|-2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn-1=-Φn(0)/, n≥1. Examples are also provided to justify all the results. |
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Repositório Institucional da UNESP |
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Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulasChain sequencesOrthogonal polynomials on the unit circlePara-orthogonal polynomialsWhen a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π-θ))=-dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn∗(z)} and {zΦn(z)-Φn∗(z)}, where Φn∗(z)=zn Φn(1/z/)/, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [-1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)-Φn∗(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn∗(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ-w|-2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn-1=-Φn(0)/, n≥1. Examples are also provided to justify all the results.Departamento de Matemática Aplicada IBILCE UNESP - Univ. Estadual PaulistaDepartment of Mathematics IIT RoorkeeDepartamento de Matemática Aplicada IBILCE UNESP - Univ. Estadual PaulistaUniversidade Estadual Paulista (Unesp)IIT RoorkeeBracciali, C. F. [UNESP]Sri Ranga, A. [UNESP]Swaminathan, A.2018-12-11T17:03:38Z2018-12-11T17:03:38Z2016-11-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article19-40application/pdfhttp://dx.doi.org/10.1016/j.apnum.2016.05.008Applied Numerical Mathematics, v. 109, p. 19-40.0168-9274http://hdl.handle.net/11449/17310210.1016/j.apnum.2016.05.0082-s2.0-849752516542-s2.0-84975251654.pdf83003224526224670000-0002-6823-4204Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengApplied Numerical Mathematics0,930info:eu-repo/semantics/openAccess2023-10-01T06:04:03Zoai:repositorio.unesp.br:11449/173102Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462023-10-01T06:04:03Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
title |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
spellingShingle |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas Bracciali, C. F. [UNESP] Chain sequences Orthogonal polynomials on the unit circle Para-orthogonal polynomials |
title_short |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
title_full |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
title_fullStr |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
title_full_unstemmed |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
title_sort |
Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas |
author |
Bracciali, C. F. [UNESP] |
author_facet |
Bracciali, C. F. [UNESP] Sri Ranga, A. [UNESP] Swaminathan, A. |
author_role |
author |
author2 |
Sri Ranga, A. [UNESP] Swaminathan, A. |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) IIT Roorkee |
dc.contributor.author.fl_str_mv |
Bracciali, C. F. [UNESP] Sri Ranga, A. [UNESP] Swaminathan, A. |
dc.subject.por.fl_str_mv |
Chain sequences Orthogonal polynomials on the unit circle Para-orthogonal polynomials |
topic |
Chain sequences Orthogonal polynomials on the unit circle Para-orthogonal polynomials |
description |
When a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π-θ))=-dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn∗(z)} and {zΦn(z)-Φn∗(z)}, where Φn∗(z)=zn Φn(1/z/)/, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [-1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)-Φn∗(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn∗(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ-w|-2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn-1=-Φn(0)/, n≥1. Examples are also provided to justify all the results. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016-11-01 2018-12-11T17:03:38Z 2018-12-11T17:03:38Z |
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/10.1016/j.apnum.2016.05.008 Applied Numerical Mathematics, v. 109, p. 19-40. 0168-9274 http://hdl.handle.net/11449/173102 10.1016/j.apnum.2016.05.008 2-s2.0-84975251654 2-s2.0-84975251654.pdf 8300322452622467 0000-0002-6823-4204 |
url |
http://dx.doi.org/10.1016/j.apnum.2016.05.008 http://hdl.handle.net/11449/173102 |
identifier_str_mv |
Applied Numerical Mathematics, v. 109, p. 19-40. 0168-9274 10.1016/j.apnum.2016.05.008 2-s2.0-84975251654 2-s2.0-84975251654.pdf 8300322452622467 0000-0002-6823-4204 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Applied Numerical Mathematics 0,930 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
19-40 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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1803649279298895872 |