Geodesic
Some extremal problems for periodic functions with conditions on their values and Fourier coefficients
Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 92-111

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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 $h$, $0$, 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. 
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     author = {V. I. Ivanov and D. V. Gorbachev and Yu. D. Rudomazina},
     title = {Some extremal problems for periodic functions with conditions on their values and {Fourier} coefficients},
     journal = {Trudy Instituta matematiki i mehaniki},
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     url = {http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a7/}
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V. I. Ivanov; D. V. Gorbachev; Yu. D. Rudomazina. Some extremal problems for periodic functions with conditions on their values and Fourier coefficients. Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 92-111. http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a7/