Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 117, 15 pages      arXiv:2006.12217      https://doi.org/10.3842/SIGMA.2020.117

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

Victor S. Barbosa a and Valdir A. Menegatto b
a) Centro Tecnológico de Joinville-UFSC, Rua Dona Francisca, 8300. Bloco U, 89219-600 Joinville SC, Brazil
b) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brazil

Received June 23, 2020, in final form November 07, 2020; Published online November 19, 2020

Abstract
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.

Key words: positive definite functions; generalized Stieltjes functions; Bernstein functions; Gneiting's model; products of metric spaces.

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