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

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 transform sampled on a rectangular lattice. We prove that this remains true if one replaces \(L^4([-B,B])\) by \(L^p([-B,B])\) with \(p \in [2,\infty]\). To do so, we adapt the original proof by Grohs and Liehr and use sampling results in Bernstein spaces with general integrability parameters. Furthermore, we present some modifications of a result of Müntz–Szász type first proven 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.