Positive definite functions on products of metric spaces by integral transforms

Abstract

An influential theorem proved by T. Gneiting in the beginning of the century provides a large class of continuous positive definite functions on the product Rd×R commonly used in theory and applications. It turns out that the positive definite functions given by this theorem fit into what is frequently called the scale mixture approach. In other words, they are generated by certain integral transforms defined by products of parameterized positive definite functions on Rd and R. In this paper, we consider positive definite functions on a product of metric spaces which are given by general integral transforms. We provide conditions under which the positive definite functions are either continuous or strictly positive definite. In the case in which one of the metric spaces is Rd, we offer constructions of continuous strictly positive definite functions defined by certain hypergeometric functions and conditionally negative definite functions. They complement and generalize the original Gneiting's contribution. Additionally, we present necessary and sufficient conditions for the strict positive definiteness of the aforementioned generalizations.