Wiener's problem for positive definite functions

TL;DR

This paper investigates the sharp constants in Wiener's inequality for positive definite functions on the torus, providing new bounds for specific domains like balls and cubes, and connecting the problem to Turán and Delsarte problems.

Contribution

The authors sharpen existing bounds for Wiener's inequality constants for balls and cubes, and establish exact values for certain scaled cubes, linking the problem to Turán and Delsarte frameworks.

Findings

W_{n}(B^{n})  2^{(0.401...+o(1))n}

W_{n}(rac{1}{q}I^{n})=2^{n} for q=3,4,...

Established bounds and relations for Wiener's constants in various domains.

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 $W_{n}(B^{n})\le 2^{(0.401\ldots +o(1))n}$. 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 Tur\'an and Delsarte.