Bochner-Riesz Means Convergence of Prolate Spheroidal Series and Their Extensions

TL;DR

This paper investigates the convergence of Bochner-Riesz means for prolate spheroidal series within weighted $L^p$ spaces, extending classical results to new spectral bases derived from Sturm-Liouville operators.

Contribution

It establishes convergence conditions for Bochner-Riesz expansions of functions in weighted $L^p$ spaces using generalized Slepian bases, extending prior spectral analysis results.

Findings

Convergence of Bochner-Riesz means under specific $L^p$ conditions

Extension of Slepian basis to new Sturm-Liouville spectra

Conditions for $L^p$-convergence of spectral expansions

Abstract

In this paper, we study the $L^p$-Bochner-Riesz mean summability problem related to the spectrum of some particular Sturm-Liouville operators in the weighted $L^p([a,b],\omega).$ Our purpose is to establish suitable conditions under which the Bochner-Riesz expansion of a function $f \in L^p([a,b],\omega)$,$1<p<\infty$, in two generalisations of Slepian's basis, converges to $f$ in $L^p([a,b],\omega)$.