Wiener's problem for positive definite functions

We study the sharp constant $W_n(D)$ in Wiener’s inequality for positive definite functions $${\int }_{{\mathbb{T}}^n}\mathrm{\mid }f{\mathrm{\mid }}^{2}dx\le W_n(D)\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\int }_D\mathrm{\mid }f{\mathrm{\mid }}^{2}dx,D\subset {\mathbb{T}}^n\mathrm{.}$$ N. Wiener proved that $W_1([-\delta ,\delta ])<\mathrm{\infty }$, $\delta \in (0,1\mathrm{/}2)$. E. Hlawka showed that $W_n(D)\le 2^n$, where $D$ is an origin-symmetric convex body. We sharpen Hlawka’s estimates for $D$ being the ball $B^n$ and the cube $I^n$. In particular, we prove that $W_{n}(B^{n})\le 2^{(0.401\ldots +o(1))n}$\mathrm{.}Wealsoobtainalowerboundof$W_{n}(D)$\mathrm{.}Moreover,foracube$ D=\frac1q I^{n}$with$q=3,4,\ldots,$weobtainthat$W_{n}(D)=2^{n}$. Our proofs are based on the interrelation between Wiener’s problem and the problems of Tur'an and Delsarte.