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

Abstract

Abstract 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.