Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 128746, 22 pages
Research Article

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

1Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Prishtina, Mother Teresa 5, Prishtinë 10000, Kosovo
2Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

Received 5 May 2010; Accepted 24 August 2010

Academic Editor: J ózef Banaś

Copyright © 2010 Bujar Xh. Fejzullahu and Francisco Marcellán. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let {Qn(α,β)(x)}n0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product f,g=-11f(x)g(x)dμα,β(x)+λ-11f(x)g(x)dμα+1,β(x), where λ>0 and dμα,β(x)=(1-x)α(1+x)βdx with α>-1, β>-1. In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials {Qn(α,β)(x)}n. 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.