On Fourier series of a discrete Jacobi-Sobolev inner product

Abstract

Let $\mu$ be the Jacobi measure supported on the interval $[-1,1]$ and introduce the discrete Sobolev-type inner product $$\langle f,g\rangle= \int _{-1} f(x) g(x) d\mu(x)+ \sum _{k=1} \sum N_k}_{i=0} M_{k,i} f (i)}(a_k) g (i)}(a_k),$$ where $a_k$, $1\le k\le K$, are real numbers such that $ _k 1$ and $M_{k,i}> 0$ for all $k$, $i$. This paper is a continuation of [{\it F. Marcellán}, {\it B. P. Osilenker} and {\it I. A. Rocha}, "On Fourier series of Jacobi-Sobolev orthogonal polynomials", J. Inequal. Appl. 7, 673-699 (2002; Zbl 1016.42014)] 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 $(-1,1)\bigcup _{k=1}\{a_k\}$, and the derivatives of the series converge to $f (i)}(a_k)$ 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 $\mu$. No additional conditions are needed.