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.

Abstract

Let {Qn(α,β)(x)}n≥0 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.