Positive definite (p.d.) functions vs p.d. distributions

TL;DR

This paper develops explicit transforms and representation formulas for positive definite functions and distributions, extending classical theorems and applying to Gaussian processes, Dirac combs, and diffraction phenomena.

Contribution

It introduces explicit transforms for Hilbert spaces associated with positive definite functions and distributions, generalizes Bochner's theorem, and connects these to RKHSs and Gaussian processes.

Findings

Explicit transforms for p.d. functions and distributions

Representation formulas for p.d. tempered distributions

Applications to Gaussian processes and diffraction

Abstract

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, and the generalization to Bochner/Schwartz representations for positive definite tempered distributions on the other; in the latter case, via tempered positive measures. Via our transforms, we make precise the respective reproducing kernel Hilbert spaces (RKHSs), that of N. Aronszajn and that of L. Schwartz. Further applications are given to stationary-increment Gaussian processes.