Approximation in Hankel Sobolev Space by Circular Prolate Spheroidal Series

Circular prolate spheroidal wave functions (CPSWFs) called also Hankel prolate denoted by \(\varphi _{n,c}^{(\alpha +p/2)}\) ${\phi }_{n,c}^{(\alpha +p/2)}$ , \(c > 0\) $c>0$ , \(p\in \mathbb {N}\) $p\in \mathbb{N}$ and \(\alpha \ge -1/2\) $\alpha \ge -1/2$ , are the radial part of the classical band limited and best essentially time functions defined on \(L^2(B_d)\) $L^2(B_d)$ , where \(B_d\) $B_d$ is the unit Ball of \(\mathbb {R}^d, d=p+2\ge 2\) ${\mathbb{R}}^d,d=p+2\ge 2$ , introduced by D. Slepian in the 60 s. The main issue of this work is to establish the \(L^2\) $L^2$ convergence quality of the truncated error given by the series expansion of a function \(f\in L^2[0,1]\) $f\in L^2[0,1]$ in CPSWFs basis, then to extend the previous challenge in the \(L^p\) $L^p$ norm, \(4/3<p<4\) $4/3<p<4$ . In the meantime, we will give two uniform approximations of the previous system by Bessel function of the first kind and a modified Jacobi polynomial which will be the important step to avoid all obstacles. Let recall that the Hankel prolate covers the classical PSWFs \((\alpha =\pm 1/2, p=0)\) $(\alpha =\pm 1/2,p=0)$ for which corresponding results have been established previously by many authors.