Theorems of Chernoff and Ingham for certain eigenfunction expansions

Abstract

We prove an uncertainty principle for certain eigenfunction expansions on L 2 (R + , w(r)dr) and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on C n and Hermite operator on R n .

On a theorem of Chernoff on rank one Riemannian symmetric spaces

“…This is the analogue of Theorem 3.1 for Jacobi polynomial expansions which plays an important role in proving Theorem 1.9 for compact Riemannian symmetric spaces. This result has been essentially proved in [11,Theorem 2.4] where the vanishing condition assumed is much stronger. However, one can imitate the proof given in [11,Theorem 2.4] to get a stronger result with a weaker vanishing condition as in Theorem 4.1.…”

“…This result has been essentially proved in [11,Theorem 2.4] where the vanishing condition assumed is much stronger. However, one can imitate the proof given in [11,Theorem 2.4] to get a stronger result with a weaker vanishing condition as in Theorem 4.1. Nevertheless, for the sake of convenience of the reader, we provide a proof here.…”

“…In Section 4, after recalling necessary results from the theory of spherical harmonics and Jacobi polynomials and setting up the notations, we prove Theorem 1.9. We refer the reader to the papers [11] and [12] for related ideas.…”

“…Here M (m) denotes the mth order moment of the measure ν f . Finally, proceeding as in [11,Theorem 2.4] one easily checks that the moments of the measure ν f satisfy the Carleman condition i.e., ∞ m=1 M (2m) −1/2m = ∞. Therefore, using [15,Theorem 2.3] we infer that polynomials are dense in L 1 (R, dν f ).…”

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 [10] for ∆ on non compact Riemannian symmetric spaces and [9] for the Hermite and special Hermite operators and ∆ on compact symmetric spaces.…”