On an extremal problem for compactly supported positive definite functions

This article considers an extremal problem for positive definite functions onℝn with a fixed support and a fixed value at the origin (the class 𝔉 r (ℝn)). It is required to find the least upper bound of a special form functional over 𝔉 r (ℝn). This problem is a generalization of the Turán problem for functions with support in a ball. We have obtained a general solution to this problem forn≠2. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type.