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It was recently shown that functions in $L^4([-B,B])$ can be uniquelyrecovered up to a global phase factor from the absolute values of their Gabortransforms sampled on a rectangular lattice. We prove that this remains true ifone 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 samplingresult due to Beurling. Furthermore, we present a minor modification of aresult of M\"untz-Sz\'asz type by Zalik. Finally, we consider the implicationsof our results for more general function spaces obtained by applying thefractional Fourier transform to $L^p([-B,B])$ and for more general nonuniformsampling sets.

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