Strictly and non-strictly positive definite functions on spheres

TL;DR

This paper reviews positive definite functions on spheres, their characterizations, and applications in spatial statistics and approximation theory, highlighting new insights into dimension walks and parametric classes of functions.

Contribution

It provides a comprehensive review of characterizations of positive definite functions on spheres and introduces new dimension walk techniques and parametric classes for applications.

Findings

Monotonicity of Gegenbauer coefficients guarantees positive definiteness in higher dimensions.

Euclidean space functions can be directly substituted with great circle distances on spheres.

Completely monotone functions are positive definite on spheres of any dimension.

Abstract

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.