On positive definite distributions

Abstract

We provide necessary and sufficient conditions for a tempered distribution $F\in S'(R)$ to be positive definite. A generalized Cauchy transform $\widetilde{F}$ of $F$ is used as a numerical continuation of $F$ to the open upper and lower complex half-planes in $C$. In fact, our necessary and sufficient conditions for $F$ are determined completely by the properties of the restriction of $\widetilde{F}$ to the imaginary axis in $C$. The main result is given in terms of completely monotonic and absolutely monotonic functions.