Stability estimates for reconstruction from the Fourier transform on the ball

Abstract

We prove Hölder-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 Hölder-stable extrapolation of $\mathcal{F}v$ from $B_r$ to a larger ball. We also present instability examples showing an optimality of our results.