Quasianalyticity of Positive Definite Continuous Functions

Abstract

It is shown that for a positive definite continuous function $f(x)$ on $R^n$ the followings are equivalent:

1. $f(x)$ is quasianalytic in some neighborhood of the origin.
2. $f(x)$ can be expressed as an integral $f(x)={\int }_{R^n}{e}^{iz\xi }d\mu (\xi )$ for some positive Radon measure $\mu$ on $R^n$ such that $\int \exp M(L|\xi |)d\mu (\xi )$ is finite for some $L>0$ where the function $M(t)$ is a weight function corresponding to the quasaianalyticity.
3. $f(x)$ is quasianalytic in $R^n$.

Moreover, an analogue for the analyticity is also given as a corollary.