Turan Extremum Problem for Periodic Function with Small Support

We consider an extremum problem posed by Turan. The aim of this problem is to find a maximum mean value of 1-periodic continuous even function such that sum of Fourier coefficient modules for this function is equal to 1 and support of this function lies in $[-h,h]$, $0<h\le 1/2$. We show that this extremum problem for rational $h=p/q$ is equivalent two finite-dimensional linear programming problems. Here there are exact results for rational $h=2/q$, $h=p/(2p+1)$, $h=3/q$, and asymptotic equalities.