Turán Extremal Problem for Periodic Functions with Small Support and Its Applications

We study a Turán extremal problem on the largest mean value of a 1-periodic even function with nonnegative Fourier coefficients, fixed value at zero, and support on a closed interval $[-h,h]$, $0<h\le1/2$. We show how the solution of this extremal problem for rational numbers $h=p/q$ is related to the solution of two finite-dimensional problems of linear programming. The solution of the Turán problem for rational numbers $h$ of the form $2/q$, $3/q$, $4/q$, is obtained. Applications of the Turán problem to analytic number theory are given.