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 $1<p<\mathrm{\infty }$, 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 $B_{\alpha ,c}^p\subset L^p(0,\mathrm{\infty })$ 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)$,$1<p\leq q<\infty$.