On Fourier Series of a Discrete Jacobi–Sobolev Inner Product

AbstractLet μ be the Jacobi measure supported on the interval [−1, 1] and introduce the discrete Sobolev-type inner product 〈f,g〉=∫1−1f(x)g(x) dμ(x)+∑k=1K∑i=0NkMk,if(i)(ak)g(i)(ak), where ak, 1⩽k⩽K, are real numbers such that ∣ak∣>1 and Mk,i>0 for all k,i. This paper is a continuation of Marcellan et al. (On Fourier series of Jacobi–Sobolev orthogonal polynomials, J. Inequal. Appl., to appear) and our main purpose is to study the behaviour of the Fourier series associated with such a Sobolev inner product. For an appropriate function f, we prove here that the Fourier–Sobolev series converges to f on (−1,1) ∪Kk=1{ak}, and the derivatives of the series converge to f(i)(ak) for all i and k. Roughly speaking, the term appropriate means here the same as we need for a function f in order to have convergence for its Fourier series associated with the standard inner product given by the measure μ. No additional conditions are needed