A theorem of Chernoff on quasi-analytic functions for Riemannian symmetric spaces

TL;DR

This paper generalizes Chernoff's theorem on quasi-analytic functions from Euclidean spaces to Riemannian symmetric spaces, providing a simplified proof and extending the scope to compact and noncompact types.

Contribution

It offers a simplified proof of Chernoff's $L^2$ quasi-analytic theorem and extends it to Riemannian symmetric spaces for $K$-biinvariant functions.

Findings

Generalized Chernoff's theorem to Riemannian symmetric spaces.

Provided a simpler proof applicable for all $p \, \in [1,2]$.

Extended results to both compact and noncompact symmetric spaces.

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 which 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.