Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions

TL;DR

This paper introduces new formulas for reconstructing compactly supported functions from their Fourier transform within a ball, utilizing prolate spheroidal wave functions in one dimension and Radon transform theory in higher dimensions.

Contribution

It provides novel reconstruction formulas based on PSWFs for 1D and extends the approach to higher dimensions using Radon transform theory.

Findings

New formulas for 1D reconstruction using PSWFs

Extension to multidimensional cases via Radon transform

Results on stability and convergence rates

Abstract

We give new formulas 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 (PSWFs). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.