On positive definite distributions

TL;DR

This paper characterizes positive definite tempered distributions using their generalized Cauchy transform, linking their properties to monotonic functions and providing necessary and sufficient conditions based on complex analysis.

Contribution

It offers a complete characterization of positive definite distributions via their Cauchy transform and monotonicity properties, advancing the understanding of distribution positivity.

Findings

Necessary and sufficient conditions for positive definiteness of distributions.

Use of generalized Cauchy transform to analyze distributions.

Characterization in terms of monotonic functions.

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.