A version of Turán's problem for positive definite functions of several variables

Abstract

Let G m(B) be the class of functions of m variables with support in the unit ball B centered at the origin of the space ℝ m, continuous in ℝ m, normed by the condition f(0) = 1, and having a nonnegative Fourier transform. In this paper, we study the problem of finding the maximum value Φ m(a) of normed integrals of functions from the class G m(B) over the sphere S a of radius a, 0 < a < 1, centered at the origin. It is proved that, in this problem, we may restrict our attention to spherically symmetric functions from G m(B). The existence of an extremal function is proved and a representation of this function as the self-convolution of a radial function is obtained. An integral equation is written for a solution of the problem for any m ≥ 3. The values Φ 3(a) are calculated for 1/3 ≤ a < 1. © 2012 Pleiades Publishing, Ltd.