Abstract
Let G be a noncompact semisimple Lie group with finite centre. Let \(X=G/K\) be the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections associated to the Laplace-Beltrami operator are given by \(P_{\lambda }f =f*\Phi _{\lambda }\), where \(\Phi _{\lambda }\) are the elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for \(P_{\lambda }f\). Moreover, similar results are obtained in the case of generalized spectral projections associated to Dunkl Laplacian.
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Acknowledgements
The authors wish to thank the referee for the careful reading of the manuscript and for many useful suggestions. The first author is supported by Int. Ph.D. scholarship from Indian Institute of Science. The second author is supported by J. C. Bose Fellowship from D.S.T., Govt. of India.
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Ganguly, P., Thangavelu, S. An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type. Annali di Matematica 201, 289–311 (2022). https://doi.org/10.1007/s10231-021-01116-3
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DOI: https://doi.org/10.1007/s10231-021-01116-3
Keywords
- Chernoff’s theorem
- Riemannian symmetric spaces
- Helgason Fourier transform
- Dunkl transform
- Ingham’s theorem
- Spectral projections