Mean convergence of prolate spheroidal series and their extensions

TL;DR

This paper determines the p-range for which functions expanded in generalized prolate spheroidal wave functions converge in L p, extending classical results to new PSWF generalizations.

Contribution

It establishes the convergence range for generalized PSWF expansions in L p and introduces a general method to extend mean convergence results between bases.

Findings

Identifies p-ranges for convergence of PSWF expansions in L p.

Extends classical PSWF convergence results to generalized versions.

Provides a general framework for basis convergence extension.

Abstract

The aim of this paper is to establish the range of p's for which the expansion of a function f $\in$ 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 has been previously established by Barcel{\'o} 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).