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

Abstract

We consider the class ${\Psi }_d$ of continuous functions $\psi :[0,\pi ]\to \mathbb{R}$, 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 $d=2$.

18 pages, 3 figures