Some extremal problems for periodic functions with conditions on their values and Fourier coefficients

A solution of the discrete variant of the Fejér problem on the greatest value, at zero, of an even nonnegative trigonometric polynomial with fixed average is given. As a corollary, for all rational $h$, $0<h\le1/2$, the greatest averages are obtained for continuous 1-periodic even functions, with nonnegative Fourier coefficients and a fixed value at zero, equal to zero on the segment $[h,1-h]$ (the Turán problem) or nonpositive on this segment (the Delsarte problem). Similar problems are also solved in the discrete case. In addition, in one case, a solution of the extremal Montgomery problem for nonnegative trigonometric polynomials is given.