Wiener’s problem for positive definite functions

We study the sharp constant \(W_{n}(D)\) in Wiener’s inequality for positive definite functions $$\begin{aligned} \int _{\mathbb {T}^{n}}|f|^{2}\,dx\le W_{n}(D)|D|^{-1}\int _{D}|f|^{2}\,dx,\quad D\subset \mathbb {T}^{n}. \end{aligned}$$ Wiener proved that \(W_{1}([-\delta ,\delta ])<\infty \), \(\delta \in (0,1/2)\). 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 \(\delta B^{n}\) and the cube \(\delta I^{n}\). In particular, we prove that \(W_{n}(\delta B^{n})\le 2^{(0.401\cdots +o(1))n}\). We also obtain a lower bound of \(W_{n}(D)\). Moreover, for a cube \(D=\frac{1}{q} I^{n}\) with \(q=3,4,\ldots ,\) we obtain that \(W_{n}(D)=2^{n}\). Our proofs are based on the interrelation between Wiener’s problem and the problems of Turan and Delsarte.