Almost Everywhere Convergence of Prolate Spheroidal Series

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 $PW_c^p\subset L^p(\mathcal{R})$ 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 $L^q(0,\infty)$, $1<p\leq q<\infty$.