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_ 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.
- 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. 725–733DOI 10.2977/PRIMS/1145476195