Relation between Tur\'an extremum problem and van der Corput sets

Abstract

Let $K\subset\mathbb N$ and $\mathbf T(K)$ is a set of trigonometric polynomials \[ T(x)=T_0+\sum_{k\in K, k\le H}T_k\cos(2\pi kx), \qquad H>1, \] $T(x)\ge0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le1/2$ and $K(h)$ is the class of functions \[ f(x)=\sum_{n=0}^{\infty}a_n\cos(2\pi nx) \] satisfying the following conditions: $a_n\ge0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le|x|\le1/2$. We consider an relation between extremum problem \[ \delta(K)=\inf_{T\in\mathbf T(K)}T_0 \] and Tur\'an extremum problem \[ A(h)=\sup_{f\in K(h)}a_0=\sup_{f\in K(h)}\int_{-h}^hf(x) dx \] for rational numbers $h=p/q$ and set $K=\bigcup\limits_{\nu=0}^\infty\{q\nu+p,...,q\nu+q-p\}$. The problem $\delta(K)$ is connection with van der Korput sets. Van der Korput sets study in analytic number theory.