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.

中文翻译：

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傅立叶合成中的 Hölder-对数稳定性

我们证明了一个 Holder-logarithmic 稳定性估计，用于根据 [−r, r]^d 上给出的傅立叶变换 Fv 在 R^d 上找到一个足够规则的紧支持函数 v。这个估计依赖于 Fv 从 [−r, r]^d 到更大域的 Holder 稳定延续。相关的重建程序基于切比雪夫多项式的截断级数。我们还给出了一个明确的例子，显示了我们稳定性估计的最优性。