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

We consider the class ${\mathrm{\Psi }}_d$ of continuous functions $\psi \colon [0,\pi] \to $\mathbb{R}$$,with$\psi(0)=1$suchthattheassociatedisotropickernel$C(\xi,\eta)= \psi(\theta(\xi,\eta))$---with$\xi,\eta \in $\mathbb{S}$^d$and$\theta$thegeodesicdistance---ispositivedefiniteontheproductoftwo$d$-dimensionalspheres$$\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 ${\mathrm{\Psi }}_d$ 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 ${\mathrm{\Psi }}_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.