Mean convergence of prolate spheroidal series and their extensions

The aim of this paper is to establish the range of p's for which the expansion of a function f is an element of L-P in a generalized prolate spheroidal wave function (PSWFs) basis converges to f in L-P. Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases cover the classical PSWFs for which the corresponding results have been previously established by Barcelo and Cordoba. To establish those results, we prove a general result that allows to extend mean convergence in a given basis (e.g. Jacobi polynomials or Bessel basis) to mean convergence in a second basis (here the generalized PSWFs). (C) 2019 Elsevier Inc. All rights reserved.