Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres

TL;DR

This paper studies positive definite functions on spheres, providing new formulas for Schoenberg coefficients and improved bounds for curvature at the origin, with implications for spatial statistics and kernel design.

Contribution

It extends the understanding of Schoenberg coefficients for isotropic kernels on spheres and offers new recurrence formulas and bounds for curvature, addressing open problems.

Findings

Expressed $d$-Schoenberg coefficients as combinations of 1-Schoenberg coefficients.

Derived recurrence formulas for Schoenberg coefficients of exponential and Askey families.

Improved bounds for the curvature at the origin of kernels with local support.

Abstract

We consider the class $\Psi_d$ of continuous functions $\psi \colon [0,\pi] \to \mathbb{R}$, with $\psi(0)=1$ such that the associated isotropic kernel $C(\xi,\eta)= \psi(\theta(\xi,\eta))$ ---with $\xi,\eta \in \mathbb{S}^d$ and $\theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $\mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $\Psi_d$ as combinations of $1$-Schoenberg coefficients. We also give expressions for the computation of Schoenberg coefficients of the exponential and Askey families for all even dimensions through recurrence formula. Problem 3 regards the curvature at the origin of members of $\Psi_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.