On the Stability of Fourier Phase Retrieval

Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $\mathrm{\mid }\widehat{f}\mathrm{\mid }$. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$\mathrm{\parallel }f-g{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}\le 2\cdot \mathrm{\parallel }\mathrm{\mid }\widehat{f}\mathrm{\mid }-\mathrm{\mid }\widehat{g}\mathrm{\mid }{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}+h_f(\mathrm{\parallel }f-g{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}^{})+J(\widehat{f},\widehat{g}),$$ where $1\le p<2$, $h_f$ 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 $1\le p<2$ while, usually, $L^p$ cannot be used to control $L^2$, the stability estimate has the flavor of an inverse H"older inequality. It seems conceivable that the estimate is optimal up to constants.