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

TL;DR

This paper presents a numerical method for reconstructing compactly supported functions from their Fourier transform within a ball, utilizing prolate spheroidal wave functions and Radon transform inversion, demonstrating super-resolution capabilities.

Contribution

The paper introduces a numerical implementation of formulas for function reconstruction from Fourier data using prolate spheroidal wave functions and Radon transform inversion, including super-resolution examples.

Findings

Successful numerical reconstruction of functions from Fourier data.

Demonstration of super-resolution beyond the diffraction limit.

Effective use of prolate spheroidal wave functions in multidimensional inversion.

Abstract

We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882] for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} [v]$ 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.