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