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) = ∫ℝ_n_ eizξ dμ(ξ) for some positive Radon measure μ on ℝ_n_ such that ∫ exp M (L|ξ|) dμ(ξ) 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.