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

02/24/2022

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by   Mikhail Isaev, et al.

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We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882] 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.