H\"older-logarithmic stability in Fourier synthesis

We prove a H"{o}lder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on ${\mathbb{R}}^d$ from its Fourier transform $\mathcal{F}v$ given on $[-r,r{]}^d$. This estimate relies on a H"{o}lder stable continuation of $\mathcal{F}v$ from $[-r,r{]}^d$ to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates.