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

Abstract

An L2 version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff on Rd using iterates of the Laplacian. In 1934 Ingham used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of an integrable function on R. In this paper, we prove analogues of the theorems of Chernoff and Ingham for Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.