Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions

We give new formulas for finding a compactly supported function $v$ on ${\mathbb{R}}^d$, $d\ge 1$, from its Fourier transform $\mathcal{F}v$ given within the ball $B_r$. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWFs). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.