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

Abstract

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∈[1,∞]. 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 Müntz–Szász 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.