From Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class

TL;DR

This paper explores the mathematical properties of variograms, introducing new classes and demonstrating stability under products, thereby advancing the theoretical understanding of rotationally invariant variograms.

Contribution

It introduces new classes of Schoenberg-Lévy type kernels and shows that a large subclass of variograms is closed under products, enhancing the theoretical framework.

Findings

Large subclass of variograms closed under products

Introduction of new Schoenberg-Lévy type kernels

Demonstration of stability properties in variogram classes

Abstract

We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L\'{e}vy type and demonstrate some important properties of rotationally invariant variograms.