A version of the Turan problem for positive definite functions of several variables

Let $G_m(\mathbb B)$ be the class of functions of $m$ variables with support in the unit ball $\mathbb B$ centered at the origin of the space $\mathbb R^m$, continuous on the space $\mathbb R^m$, 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 $\Phi_m(a)$ of normed integrals of functions from the class $G_m(\mathbb B)$ over the sphere $\mathbb S_a$ of radius $a$, $0<a<1$, centered at the origin. It is proved that we may consider spherically symmetric functions only. The existence of an extremal function is proved and a presentation of such a function as the self-convolution of a radial function is obtained. An integral equation is written for a solution of the problem for any $m\ge3$. The values $\Phi_3(a)$ are obtained for $1/3\le a<1$.