Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions

TL;DR

This paper extends the injectivity results of Gabor phase retrieval from $L^4$ spaces to all $L^p$ spaces with $p$ in [1, e], using classical sampling theorems and adaptations of existing proofs.

Contribution

It generalizes the phase retrieval injectivity results to a broader class of function spaces $L^p$, including nonuniform sampling sets and fractional Fourier transforms.

Findings

Injectivity holds for all $L^p$ spaces with $p \,\in\, [1,\infty]$.

Uses Beurling's sampling theorem and adaptations of Mfntz-Sze1sz type results.

Implications for fractional Fourier transforms and nonuniform sampling sets.

Abstract

It was recently shown that functions in $L^4([-B,B])$ can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transforms sampled on a rectangular lattice. We prove that this remains true if one replaces $L^4([-B,B])$ by $L^p([-B,B])$ with $p \in [1,\infty]$. To do so, we adapt the original proof by Grohs and Liehr and use a classical sampling result due to Beurling. Furthermore, we present a minor modification of a result of M\"untz-Sz\'asz type by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to $L^p([-B,B])$ and for more general nonuniform sampling sets.