POSITIVE DEFINITE FUNCTIONS ON PRODUCTS OF METRIC SPACES VIA GENERALIZED STIELTJES FUNCTIONS

For quasi-metric spaces (X, rho) and (Y, sigma) and a positive real number lambda, we propose a model for generating positive definite functions G(r) : {rho(x, x') : x, x' is an element of X} x {sigma(y, y') : y, y' is an element of Y} bar right arrow R having the form G(r) (t, u) = 1/h(u)(r) f ((g(t)/h(u)). where r >= lambda, f belongs to a convex cone S-lambda(b) of bounded completely monotone functions, g is a nonnegative valued conditionally negative definite function on (X, rho), and h is a positive valued conditionally negative definite function on (Y, sigma). In the case where (X, rho) and (Y, sigma) are metric spaces, we determine necessary and sufficient conditions for the strict positive definiteness of the model. The cone S-lambda(b) possesses well-established stability properties that allow alternative formulations of the model leading to many classes of positive definite and strictly positive definite functions on X x Y. If X = R-d, Y = R, rho is the Euclidean distance on X, sigma(1/2) is the Euclidean distance on Y, g(t) = t(2), t >= 0, h is a positive valued function with a completely monotone derivative, and lambda = d/2, then {G(r) : r >= lambda} is a subset of the Gneiting's class of covariance space-time functions on X x Y frequently dealt with in the literature.