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