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Classical results due to Ingham and Paley-Wiener characterize the existenceof nonzero functions supported on certain subsets of the real line in terms ofthe pointwise decay of the Fourier transforms. We view these results asuncertainty principles for Fourier transforms. We prove certain analogues ofthese uncertainty principles on the $n$-dimensional Euclidean space, the$n$-dimensional torus and connected, simply connected two step nilpotent Liegroups. We also use these results to show a unique continuation property ofsolutions to the initial value problem for time-dependent Schr\"odingerequations on the Euclidean space and a class of connected, simply connected twostep nilpotent Lie groups.

Around Uncertainty Principles of Ingham-type on $\R^n$, $\T^n$ and Two Step Nilpotent Lie Groups