Geodesic
A version of the Turan problem for positive definite functions of several variables
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 3, pp. 136-154

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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 $1/3\le a1$.
Keywords: Turan problem, positive definite functions, multidimensional functions. 
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     author = {A. V. Efimov},
     title = {A version of the {Turan} problem for positive definite functions of several variables},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {136--154},
     publisher = {mathdoc},
     volume = {17},
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     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2011_17_3_a14/}
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A. V. Efimov. A version of the Turan problem for positive definite functions of several variables. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 3, pp. 136-154. http://geodesic.mathdoc.fr/item/TIMM_2011_17_3_a14/