  An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding
quasi-analytic functions was proved by Chernoff \cite{CR} on $\mathbb R^d$
using iterates of the Laplacian. In $1934$ Ingham \cite{I} used the classical
Denjoy-Carleman theorem to relate the decay of Fourier transform and
quasi-analyticity of integrable functions on $\mathbb R$. In this paper we
extend both these theorems to Riemannian symmetric spaces of noncompact type
and show that the theorem of Ingham follows from that of Chernoff.
