Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions

We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318-33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform 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, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.