Holder-logarithmic stability in Fourier synthesis

We prove a Holder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function v on R-d from its Fourier transform Fv given on [-r, r](d). This estimate relies on a Holder stable continuation of Fv 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.