Quasianalyticity of positive definite continuous functions (Q1869588)

Summary: It is shown that for a positive definite continuous function \(f(x)\) on \(\mathbb{R}^n\) the following statements are equivalent: (i) \(f(x)\) is quasianalytic in some neighborhood of the origin. (ii) \(f(x)\) can be expressed as an integral \(f(x)= \int_{\mathbb{R}^n} e^{ix\xi}d\mu(\xi)\) for some positive Radon measure \(\mu\) on \(\mathbb{R}^n\) 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 quasianalyticity. (iii) \(f(x)\) is quasianalytic everywhere in \(\mathbb{R}^n\). Moreover, an analogue for the analyticity is also given as a corollary.