On an extremal problem for compactly supported positive definite functions

An extremal problem for positive definite functions on $\mathbb{R}^n$ with a fixed support and a fixed value at the origin (the class $\mathfrak{F}_r(\mathbb{R}^n))$ is considered. It is required to find the least upper bound for a special form functional over $\mathfrak{F}_r(\mathbb{R}^n))$. This problem is a generalization of the Turán problem for functions with support in a ball. A general solution to this problem for $n\ne2$ is obtained. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type.