A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product

A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product

<p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-128746-i1.gif"/></inline-formula> denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product <inline-formula> <graphic file="1029-242X-2010-12874...

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Main Authors: Marcell&#225;n Francisco, Fejzullahu BujarXh
Format: Article
Language:English
Published: SpringerOpen 2010-01-01
Series:Journal of Inequalities and Applications
Online Access:http://www.journalofinequalitiesandapplications.com/content/2010/128746
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spelling doaj-fe505844a94543a9a4e624ed293b254a2020-11-24T23:56:30ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2010-01-0120101128746A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner ProductMarcell&#225;n FranciscoFejzullahu BujarXh<p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-128746-i1.gif"/></inline-formula> denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product <inline-formula> <graphic file="1029-242X-2010-128746-i2.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2010-128746-i3.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-128746-i4.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2010-128746-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-128746-i6.gif"/></inline-formula>. In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials <inline-formula> <graphic file="1029-242X-2010-128746-i7.gif"/></inline-formula> Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved.</p>http://www.journalofinequalitiesandapplications.com/content/2010/128746
collection DOAJ
language English
format Article
sources DOAJ
author Marcell&#225;n Francisco
Fejzullahu BujarXh
spellingShingle Marcell&#225;n Francisco
Fejzullahu BujarXh
A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
Journal of Inequalities and Applications
author_facet Marcell&#225;n Francisco
Fejzullahu BujarXh
author_sort Marcell&#225;n Francisco
title A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
title_short A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
title_full A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
title_fullStr A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
title_full_unstemmed A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
title_sort cohen type inequality for fourier expansions of orthogonal polynomials with a nondiscrete jacobi-sobolev inner product
publisher SpringerOpen
series Journal of Inequalities and Applications
issn 1025-5834
1029-242X
publishDate 2010-01-01
description <p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-128746-i1.gif"/></inline-formula> denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product <inline-formula> <graphic file="1029-242X-2010-128746-i2.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2010-128746-i3.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-128746-i4.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2010-128746-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-128746-i6.gif"/></inline-formula>. In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials <inline-formula> <graphic file="1029-242X-2010-128746-i7.gif"/></inline-formula> Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved.</p>
url http://www.journalofinequalitiesandapplications.com/content/2010/128746
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