# Quasianalyticity of Positive Definite Continuous Functions

### Soon-Yeong Chung

Sogang University, Seoul, South Korea

## Abstract

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

- $f(x)$ is quasianalytic in some neighborhood of the origin.
- $f(x)$ can be expressed as an integral $f(x)=∫_{R_{n}}e_{izξ}dμ(ξ)$ for some positive Radon measure $μ$ on $R_{n}$ such that $∫expM(L∣ξ∣)dμ(ξ)$ is finite for some $L>0$ where the function $M(t)$ is a weight function corresponding to the quasaianalyticity.
- $f(x)$ is quasianalytic in $R_{n}$.

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

## Cite this article

Soon-Yeong Chung, Quasianalyticity of Positive Definite Continuous Functions. Publ. Res. Inst. Math. Sci. 38 (2002), no. 4, pp. 725–733

DOI 10.2977/PRIMS/1145476195