An extension of Aitken’s integral for Gaussians and positive definiteness

We recast an extension of Aitken’s integral frequently used in quantum field theory and use it to deduce general criteria for constructing positive definite and strictly positive definite kernels on a set $X$ making use of completely monotone functions and special multivariate conditionally negative definite kernels on $X$. The criteria extend to positive definite kernels on a cartesian product of sets. In particular, we obtain an extension of a classical model of $\mathrm{T}$. Gneiting for the construction of space-time covariance functions. New and old models are obtained as applications of the main results in the paper.

Keywords

positive definite, conditionally negative definite, Hadamard exponential, Schur product Theorem, Oppenheim’s inequality, Gneiting’s model

2010 Mathematics Subject Classification

26A48, 42A82, 62J10

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Received 15 January 2020

Accepted 25 June 2021

Published 1 March 2023