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Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.2010 Mathematics Subject Classification. Primary 22E30; Secondary 22E46, 43A80.

A classical result due to Paley and Wiener characterizes the existence of a nonzero function in L 2 (R), supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group M (n). We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schrödinger equation on M (n).MSC 2010 : Primary 22E30; Secondary 43A80.

A classical result of N. Levinson characterizes the existence of a nonzero integrable function vanishing on a nonempty open subset of the real line in terms of the pointwise decay of its Fourier transform. We prove an analogue of this result for Riemannian symmetric spaces of noncompact type.