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

Let $K\subset \mathbb{N}$ and $\mathbf{T}(K)$ is a set of trigonometric polynomials $$T(x)=T_0+\sum _{k\in K,k\le H}{T}_k\cos (2\pi kx),H>1,$$ $T(x)\ge 0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le 1\mathrm{/}2$ and $K(h)$ is the class of functions $$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n\cos (2\pi nx)$$ satisfying the following conditions: $a_n\ge 0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le \mathrm{\mid }x\mathrm{\mid }\le 1\mathrm{/}2$. We consider an relation between extremum problem $$\delta (K)=\underset{T\in \mathbf{T}(K)}{\inf }{T}_0$$ and Tur'an extremum problem $$A(h)=\underset{f\in K(h)}{\sup }{a}_0=\underset{f\in K(h)}{\sup }{\int }_{-h}^{h}f(x)dx$$ for rational numbers $h=p\mathrm{/}q$ and set $K={\bigcup }_{\nu =0}^{\mathrm{\infty }}\{q\nu +p,\mathrm{.}\mathrm{.}\mathrm{.},q\nu +q-p\}$. The problem $\delta (K)$ is connection with van der Korput sets. Van der Korput sets study in analytic number theory.