  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.
