Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres
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We consider the class Ψ_d of continuous functions ψ [0,π] →R, with ψ(0)=1 such that the associated isotropic kernel C(ξ,η)= ψ(θ(ξ,η)) ---with ξ,η∈S^d and θ the geodesic distance--- is positive definite on the product of two d-dimensional spheres 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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