On Fourier series of Jacobi-Sobolev orthogonal polynomials

Let a be the Jacobi measure on the interval [-1,1] and introduce the discrete Sobolev-type inner product <f, g> = integral(-1)(1)f(x)g(x)dmu(x) + Mf(c)g(c) + Nf'(c)g'(c) where c is an element of (1, infinity) 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 tern 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 p. No additional conditions are needed.