We give formulas for finding a compactly supported function v on R^d 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. In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. We also present a numerical implementation of these results. In particular, the results obtained give super-resolution reconstruction, that is, they allow recovering details beyond the diffraction limit, that is, details of size less than pi/r, where r is the radius of the ball mentioned above. This talk is based on the works Isaev, Novikov (2022 J. Math. Pures Appl. 163 318–333) and Isaev, Novikov, Sabinin (2022 Inverse Problems 38 105002)