  We give new formulas for finding a compactly supported function $v$ on
$\mathbb{R}^d$, $d\geq 1$, 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.
