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

It was recently shown that functions in L4([-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 L4([-B, B]) by Lp([-B, B]) with p & ISIN; [1, & INFIN;]. 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 Muntz-Szasz type by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to Lp([-B, B]) and for more general nonuniform sampling sets.& COPY; 2023 Elsevier Inc. All rights reserved.