Hölder-logarithmic stability in Fourier synthesis

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