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

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 $L^p([a,b],\omega)$.