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

A. V. Efimov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 3, pp. 136-154

Résumé

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 $a$, $0$, 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 $1/3\le a1$.

Détail

Keywords: Turan problem, positive definite functions, multidimensional functions.

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