Wiener's problem for positive definite functions

We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions $\int_{\mathbb{T}^{n}}|f|^{2}\,dx\le W_{n}(D)|D|^{-1}\int_{D}|f|^{2}\,dx,\quad D\subset \mathbb{T}^{n}.$ N. Wiener proved that $W_{1}([-δ,δ])<\infty$, $δ\in (0,1/2)$. 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 $W_{n}(B^{n})\le 2^{(0.401\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. Moreover, for a cube $D=\frac1q I^{n}$ with $q=3,4,\ldots,$ we obtain that $W_{n}(D)=2^{n}$. Our proofs are based on the interrelation between Wiener's problem and the problems of Turán and Delsarte.

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