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

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Abstract

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 \([ - h,h],0 < h \leqslant 1/2\). 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, \(p/(2p + 1)\) is obtained. Applications of the Turán problem to analytic number theory are given.

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Authors and Affiliations

  1. Tula State University, Russia

    D. V. Gorbachev & A. S. Manoshina

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  1. D. V. Gorbachev
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  2. A. S. Manoshina
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Gorbachev, D.V., Manoshina, A.S. Turán Extremal Problem for Periodic Functions with Small Support and Its Applications. Mathematical Notes 76, 640–652 (2004). https://doi.org/10.1023/B:MATN.0000049663.45427.0f

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  • DOI: https://doi.org/10.1023/B:MATN.0000049663.45427.0f

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