Wiener's problem for positive definite functions

Abstract

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 $W_1([-\delta ,\delta ])<\infty$, $\delta \in (0,1/2)$. 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 $W_n(B^n)\le 2^{(0.401\dots +o(1))n}$. We also obtain a lower bound of $W_n(D)$. Moreover, for a cube $D=\frac{1}{q}{I}^n$ with $q=3,4,\dots ,$ 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.

17 pages