Theorems of Ingham and Chernoff on Riemannian symmetric spaces of noncompact type

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

“…We remark that the above result has been discussed in [4,Theorem 4.2] for noncompact Riemannian symmetric spaces of arbitrary rank.…”

“…= ∞, combined with Lemma 3.3 in [4] it follows that ∑ ∞ m=1 M(2m) −1∕2m = ∞. So, by Theorem 2.2, and the remark following it, even polynomials are dense in L 1 e (ℝ, d f ) .…”

“…Hence in view of the inversion formula for the Jacobi transform we get Now considering the Jacobi operator L +p, +q with parameters + p, + q , from the Plancherel formula we obtain where d = + + p + q + 1. But from the hypothesis we have Hence we have Now under the assumption that ( ) ≥ 2 −1∕2 for ≥ 1 , it is a routine matter to check that ‖L m +p, +qf ,j ‖ 2 satisfies the Carleman condition, see e.g., [4]. Since f ,j vanishes in a neighbourhood of zero, from Theorem 2.7 we conclude that f ,j = 0.…”

“…Ingham's theorem has received considerable attention in recent years and analogues have been proved for Fourier series [3] and Fourier transforms on symmetric spaces [4] and nilpotent Lie groups [1,3]. In this note we would like to recast some of the results as uncertainty principles for generalized spectral projections associated to Laplace-Beltrami operators on Riemannian symmetric spaces.…”

“…We refer to Sect. 4 for more details on all the unexplained concepts and terms related to Dunkl analysis. We prove the following:…”