Positive definite functions on products of metric spaces via generalized Stieltjes functions

Menegatto, V.

doi:10.1090/proc/15137

Positive definite functions on products of metric spaces via generalized Stieltjes functions
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Proc. Amer. Math. Soc. 148 (2020), 4781-4795 Request permission

Abstract:

For quasi-metric spaces $(X,\rho )$ and $(Y,\sigma )$ and a positive real number $\lambda$, we propose a model for generating positive definite functions $G_r: \{\rho (x,x’):x,x’\in X\} \times \{\sigma (y,y’):y,y’ \in Y\} \mapsto \mathbb {R}$ having the form \begin{equation*} G_r(t,u)=\frac {1}{h(u)^r} f\left (\frac {g(t)}{h(u)}\right ), \end{equation*} where $r\geq \lambda$, $f$ belongs to a convex cone $\mathcal {S}_\lambda ^b$ 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 $\mathcal {S}_\lambda ^b$ 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 $X\times Y$. If $X=\mathbb {R}^d$, $Y=\mathbb {R}$, $\rho$ is the Euclidean distance on $X$, $\sigma ^{1/2}$ is the Euclidean distance on $Y$, $g(t)=t^2$, $t\geq 0$, $h$ is a positive valued function with a completely monotone derivative, and $\lambda =d/2$, then $\{G_r:r\geq \lambda \}$ is a subset of the Gneiting’s class of covariance space-time functions on $X\times Y$ frequently dealt with in the literature.

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