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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 inthe essay Gneiting (2013b). We have considered an extension that encompassesthe solution of Problem 1 solved in Fiedler (2013), regarding the expression ofthe $d$-Schoenberg coefficients of members of $\Psi_d$ as combinations of$1$-Schoenberg coefficients. We also give expressions for the computation ofSchoenberg coefficients of the exponential and Askey families for all evendimensions through recurrence formula. Problem 3 regards the curvature at theorigin of members of $\Psi_d$ of local support. We have improved the currentbounds for determining this curvature, which is of applied interest at leastfor $d=2$.

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