A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

TL;DR

This paper extends Gneiting's method to construct positive definite functions on product spaces using generalized functions, providing conditions for strict positive definiteness in metric spaces.

Contribution

It introduces a Gneiting-like approach utilizing generalized Stieltjes and Bernstein functions for positive definite functions on product metric spaces.

Findings

Provides necessary and sufficient conditions for strict positive definiteness.

Extends Gneiting's method to quasi-metric spaces.

Offers a framework for constructing space-time positive definite functions.

Abstract

This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.