An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

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.