  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 $d=2$.
