  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.
