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

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 f = f * Φ , where Φ are the elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for P f . Moreover, similar results are obtained in the case of generalized spectral projections associated to Dunkl Laplacian.

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

“…As we have already mentioned, ρ λ k (f )e n−1 k,λ (z, t) are eigenfunctions of L and hence the above theorem is a version of Ingham's theorem for the spectral projections. Earlier we have proved such theorems for spectral projections associated to certain elliptic differential operators; see [11] for on noncompact Riemannian symmetric spaces and [10] for the Hermite and special Hermite operators and on compact symmetric spaces.…”

Theorems of Chernoff and Ingham for certain eigenfunction expansions

“…These spherical functions are known explicitly. They are expressible in terms of Jacobi polynomials (See Helgason [9]). In fact, ϕ λ (a r ) = P (α,β) λ (r) where P (α,β) λ (r) are Jacobi polynomials with parameters (α, β) associated to the symmetric space G/K.…”

“…We remark that analogues of Theorem 1.4 can be established for the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces and the Dunkl Laplacian on R n , see [8].…”

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

“…Chernoff's theorem for ∆ α . As in our earlier works [9,10] we make use of the following result of de Jeu [13] which is a generalisation of a theorem of Carleman in the one dimensional case. Theorem 3.1.…”

“…As we have already mentioned, ρ λ k (f )e n−1 k,λ (z, t) are eigenfunctions of L and hence the above theorem is a version of Ingham's theorem for the spectral projections. Earlier we have proved such theorems for spectral projections associated to certain elliptic differential operators, see [10] for ∆ on non compact Riemannian symmetric spaces and [9] for the Hermite and special Hermite operators and ∆ on compact symmetric spaces.…”