  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.
