On the Stability of Fourier Phase Retrieval

2020/04/14 by Stefan Steinerberger, Steinerberger, Stefan
Medicine · Physics and Astronomy · #Advanced X-ray Imaging Techniques #Aortic aneurysm repair treatments #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2004.06671

openalex publication_date 2020/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Phase retrieval is concerned with recovering a function f from the absolute value of its Fourier transform |\widehatf|. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that ‖ f-g‖_L²(ℝⁿ) ≤ 2⋅ ‖ |\widehatf| - |\widehatg| ‖_L²(ℝⁿ) + h_f( ‖f-g‖_L^p(ℝⁿ)) + J(\widehatf, \widehatg), where 1 ≤ p < 2, h_f is an explicit nonlinear function depending on the smoothness of f and J is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of L^p for 1 ≤ p < 2 while, usually, L^p cannot be used to control L², the stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants.

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