Almost Everywhere Convergence of Prolate Spheroidal Series

Abstract

In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $P{W}_c^p\subset L^p(\mathcal{R})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley-Wiener like spaces $B_{\alpha ,c}^p\subset L^p(0,\infty )$ of functions whose Hankel transform ${\mathcal{H}}^{\alpha }$ is supported in $[0,c]$.As a side product, we show the continuity of the projection operator $P_c^{\alpha }f:={\mathcal{H}}^{\alpha }({\chi }_{[0,c]}\cdot {\mathcal{H}}^{\alpha }f)$ from $L^p(0,\infty )$ to $L^q(0,\infty )$, $1<p\le q<\infty$.