Abstract

It was recently shown in [10] 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 \( L^p([-B,B]) \), for \( p \in [2,\infty] \). To do so, we adapt the original proof and use sampling results in Bernstein spaces with general integrability parameters. Furthermore, we present some modifications of a result of Müntz--Szász type first presented in [17]. 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 non-uniform sampling sets.