On fourier series of Jacobi-Sobolev orthogonal polynomials

Abstract

Let be the Jacobi measure on the interval and introduce the discrete Sobolev-type inner product where and , are non negative constants such that . 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 , we prove here that the Fourier-Sobolev series converges to on the interval as well as to and the derivative of the series converges to . The term appropriate means here, in general, the same as we need for a function in order to have convergence for the series of associated to the standard inner product given by the measure . No additional conditions are needed.

Keywords

• Orthogonal polynomials
• Sobolev inner product
• Fourier series