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 \(n \ne 2\) is obtained. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type.