# Analogs of certain quasi-analiticity results on Riemannian symmetric spaces of noncompact type

## TL;DR

This paper extends classical results relating Fourier decay and quasi-analyticity from Euclidean spaces to Riemannian symmetric spaces of noncompact type, establishing new analogs and connections between these theorems.

## Contribution

It introduces analogs of Chernoff and Ingham theorems on noncompact symmetric spaces, linking decay of Fourier transforms to quasi-analyticity in this setting.

## Key findings

- Extended Chernoff's $L^2$ quasi-analyticity theorem to symmetric spaces.
- Derived Ingham's theorem as a consequence of Chernoff's result in this context.
- Established the relationship between Fourier decay and quasi-analyticity on noncompact symmetric spaces.

## Abstract

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.