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

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Last updated: 02 Mar 2021