On the Stability of Fourier Phase Retrieval

Stefan Steinerberger

Published 2020-04-14Version 1

Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\widehat{f}|$. We study the stability properties of this problem in $L^p(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ for $1 \leq p < 2$. The simplest result is as follows: if $f \in L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ has a real-valued Fourier transform supported on a set of measure $L < \infty$, then for all all $g \in L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ $$ \| f-g\|_{L^2} \leq 2\cdot \| |\widehat{f}| - |\widehat{g}| \|_{L^2} + 30\sqrt{ L} \cdot \|f-g\|_{L^1}+ 2\| \Im \widehat{g} \|_{L^2}.$$ This is a form of stability of the phase retrieval problem for band-limited functions (up to the translation symmetry captured by the last term). The inequality follows from a general result for $f,g \in L^p(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$.