Geodesic
Tur\'an Extremal Problem for Periodic Functions with Small Support and Its Applications
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 688-700

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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 $[-h,h]$, $0$. 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. 
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     author = {D. V. Gorbachev and A. S. Manoshina},
     title = {Tur\'an {Extremal} {Problem} for {Periodic} {Functions} with {Small} {Support} and {Its} {Applications}},
     journal = {Matemati\v{c}eskie zametki},
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     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a4/}
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D. V. Gorbachev; A. S. Manoshina. Tur\'an Extremal Problem for Periodic Functions with Small Support and Its Applications. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 688-700. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a4/