An extremal problem for positive definite functions with support in a ball

The extremal problem under consideration is related to the set of continuous positive definite functions on $\mathbb{R}^n$ with support in a closed ball of radius $r>0$ and fixed value at the origin (the class $\mathfrak{F}_r(\mathbb{R}^n)$).
Given $r>0$, the problem consists in finding the supremum on $\mathfrak{F}_r(\mathbb{R}^n)$ of a functional of a special form.
A general solution to this problem is obtained for $n\neq2$. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type $\leqslant r$.
Bibliography: 24 titles.