Abstract
A classical result due to Paley and Wiener characterizes the existence of a non-zero function in \(L^2(\mathbb {R}),\) 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 unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equation on M(n).
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Acknowledgments
We would like to thank Swagato K. Ray for suggesting this problem and for the many useful discussions during the course of this work. We are grateful to the anonymous referees whose valuable suggestions helped to improve the exposition. The second author was supported by INSPIRE Faculty Award from Department of Science and Technology, India.
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Communicated by Hartmut Führ.
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Bhowmik, M., Sen, S. An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group. J Fourier Anal Appl 23, 1445–1464 (2017). https://doi.org/10.1007/s00041-016-9510-x
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DOI: https://doi.org/10.1007/s00041-016-9510-x