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

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 $d=2$.