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In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ toprove an $L^2$ version of the Denjoy-Carleman theorem which provides asufficient condition for a smooth function on $\mathbb{R}^n$ to bequasi-analytic. In this paper, we prove an exact analogue of Chernoff's theoremfor all rank one Riemannian symmetric spaces (of noncompact and compact types)using iterates of the associated Laplace-Beltrami operators.

On a theorem of Chernoff on rank one Riemannian symmetric spaces