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Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres
Ahmed Arafat, Pablo Gregori, Emilio Porcu · 2018-07-06 · via math.ST updates on arXiv.org

We consider the class $Ψ_d$ of continuous functions $ψ\colon [0,π] \to \mathbb{R}$, with $ψ(0)=1$ such that the associated isotropic kernel $C(ξ,η)= ψ(θ(ξ,η))$ ---with $ξ,η\in \mathbb{S}^d$ and $θ$ 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 $Ψ_d$ 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 $Ψ_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.

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