Journal of Inequalities and Applications 
Volume 7 (2002), Issue 5, Pages 673-699
doi:10.1155/S1025583402000358

On fourier series of Jacobi-Sobolev orthogonal polynomials

F. Marcellán,1 B. P. Osilenker,2 and I. A. Rocha3

 1Dpto. de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 20, 28911 Leganés, Madrid, Spain
2Dep. of Math., Moscow State Civil Engineering University, Moscow, Russia
3Dpto. de Matemática Aplicada, E.U.l.T. Telecomunicación, Universidad Politécnica de Madrid, Ctra. de Valencia Km. 7, Madrid 28031, Spain
Received 16 October 2000; Revised 26 February 2001

Abstract

Let μ be the Jacobi measure on the interval [−1,1] and introduce the discrete Sobolev-type inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+Mf(c)g(c)+Nf′(c)g′(c) where c∈(1,∞) and M, N are non negative constants such that M+N>0. The main purpose of this paper is to study the behaviour of the Fourier series in terms of the polynomials associated to the Sobolev inner product. For an appropriate function f, we prove here that the Fourier-Sobolev series converges to f on the interval (−1,1) as well as to f(c) and the derivative of the series converges to f′(c). The term appropriate means here, in general, the same as we need for a function f(x) in order to have convergence for the series of f(x) associated to the standard inner product given by the measure μ. No additional conditions are needed.