We give new formulas for finding a compactly supported function v on R^d, d≥1, from its Fourier transform Fv given within the ball B_r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). 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.