Further properties of ball prolates and approximation of related almost band-limited functions

In this paper we aim to give various explicit and local estimates of ball prolate spheroidal wave functions defined in [25] as eigenfunctions of both finite Fourier transform and some differential operator. In particular, we give further refined bounds of these functions and their related eigenvalues. As consequence, we show that ball PSWFs are well adapted for the approximation of almost band-limited functions and we compare this result with the one related to the ball polynomials.