A Theorem of Chernoff on Quasi-analytic Functions for Riemannian Symmetric Spaces

Abstract

Abstract An $L^2$ version of the classical Denjoy–Carleman theorem regarding quasi-analytic functions was proved by P. Chernoff on $\mathbb {R}^n$ using iterates of the Laplacian. We give a simple proof of this theorem that generalizes the result on $\mathbb {R}^n$ for any $p\in [1, 2]$. We then extend this result to Riemannian symmetric spaces of compact and noncompact type for $K$-biinvariant functions.