(1,1)-Coherent Pairs on the Unit Circle

(1,1)-Coherent Pairs on the Unit Circle

A pair (𝒰,𝒱) of Hermitian regular linear functionals on the unit circle is said to be a (1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials {ϕn(x)}n≥0 and {ψn(x)}n≥0 satisfy ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z), an≠0, n≥1, where ϕn[1](z)=ϕn+1′(z)/(n+1). In this cont...

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Main Authors: Luis Garza, Francisco Marcellán, Natalia C. Pinzón-Cortés
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2013/307974
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spelling doaj-570da3aee52b4d9681ea7b8b063d428e2020-11-24T21:20:17ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/307974307974(1,1)-Coherent Pairs on the Unit CircleLuis Garza0Francisco Marcellán1Natalia C. Pinzón-Cortés2Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, 28045 Colima, COL, MexicoDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, SpainDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, SpainA pair (𝒰,𝒱) of Hermitian regular linear functionals on the unit circle is said to be a (1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials {ϕn(x)}n≥0 and {ψn(x)}n≥0 satisfy ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z), an≠0, n≥1, where ϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when 𝒰 is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where 𝒱 is associated with either a positive nontrivial measure or its rational spectral transformation.http://dx.doi.org/10.1155/2013/307974
collection DOAJ
language English
format Article
sources DOAJ
author Luis Garza
Francisco Marcellán
Natalia C. Pinzón-Cortés
spellingShingle Luis Garza
Francisco Marcellán
Natalia C. Pinzón-Cortés
(1,1)-Coherent Pairs on the Unit Circle
Abstract and Applied Analysis
author_facet Luis Garza
Francisco Marcellán
Natalia C. Pinzón-Cortés
author_sort Luis Garza
title (1,1)-Coherent Pairs on the Unit Circle
title_short (1,1)-Coherent Pairs on the Unit Circle
title_full (1,1)-Coherent Pairs on the Unit Circle
title_fullStr (1,1)-Coherent Pairs on the Unit Circle
title_full_unstemmed (1,1)-Coherent Pairs on the Unit Circle
title_sort (1,1)-coherent pairs on the unit circle
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2013-01-01
description A pair (𝒰,𝒱) of Hermitian regular linear functionals on the unit circle is said to be a (1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials {ϕn(x)}n≥0 and {ψn(x)}n≥0 satisfy ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z), an≠0, n≥1, where ϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when 𝒰 is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where 𝒱 is associated with either a positive nontrivial measure or its rational spectral transformation.
url http://dx.doi.org/10.1155/2013/307974
work_keys_str_mv AT luisgarza 11coherentpairsontheunitcircle
AT franciscomarcellan 11coherentpairsontheunitcircle
AT nataliacpinzoncortes 11coherentpairsontheunitcircle
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