Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

Abstract

We study the problem of recovering a function of the form $f(x)={\sum }_{k\in \mathbb{Z}}{c}_k{e}^{-(x-k{)}^2}$ from its phaseless samples $|f(\lambda )|$ on some arbitrary countable set $\Lambda \subseteq \mathbb{R}$. For real-valued functions this is possible up to a sign for every separated set with Beurling density $D^{-}(\Lambda )>2$. This result is sharp. For complex-valued functions we find all possible solutions with the same phaseless samples.