We establish novel uniqueness results for the Gabor phase retrieval problem: If $mathcal{G} : L^2(mathbb{R}) to L^2(mathbb{R}^2)$ denotes the Gabor transform then every $f in L^4[-c/2,c/2]$ is determined up to a global phase by the values $|mathcal{G}f(x,omega)|$ where $(x,omega)$ are points on the lattice $mathbb{Z} times (2c)^{-1}mathbb{Z}$. This for the first time shows that compactly supported functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Grochenig, Romero and Stockler, we prove analogous uniqueness results for functions in general shift-invariant spaces with Gaussian generator. Generalizations to nonuniform lattices are also presented. The results are based on a combination of certain Muntz-type theorems as well as sampling theorems in shift-invariant spaces.
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