The Turán extremal problem for periodic functions with small support and its applications. (Q556585)

The author considers the extremal problem of estimating \[ A(h)=\sup_{f\in K(h)}\int_{-h}^h f(x)\,dx, \] where \(K(h)\) is the (non-empty) class of continuous periodic even functions \(f\) with non-negative Fourier coefficients, supported on \((-h,h),\) and satisfying \(f(0)=1.\) Such a problem was suggested by P. Turán in 1970 in connection with certain applications to number theory. S. B. Stechkin proved in 1972 that \(A(1/q)=1/q,\) \(q=2,3,\dots\) . In this paper, the problem is studied for arbitrary rational \(h=p/q.\) It is shown that the solution is reduced to that of two problems of linear programming. For \(h=2/q,\) \(3/q,\) \(4/q,\) and \(p/(2p+1)\) a complete solution is obtained. Applications to analytic number theory are given.