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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.

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