Stability estimates for reconstruction from the Fourier transform on the ball

TL;DR

This paper establishes stability estimates for reconstructing functions from their Fourier transform known on a ball, demonstrating the limits and optimality of such reconstructions.

Contribution

It provides the first Hölder-logarithmic stability estimates for the inverse Fourier problem on a ball, including instability examples showing the estimates are optimal.

Findings

Hölder-logarithmic stability estimates derived

Optimality of stability estimates demonstrated

Examples of instability provided

Abstract

We prove H\"{o}lder-logarithmic stability estimates for the problem of finding an integrable function $v$ on $\mathbb{R}^d$ with a super-exponential decay at infinity from its Fourier transform $\mathcal{F} v$ given on the ball $B_r$. These estimates arise from a H\"{o}lder-stable extrapolation of $\mathcal{F} v$ from $B_r$ to a larger ball. We also present instability examples showing an optimality of our results.