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ON AN EXTREMAL PROBLEM FOR POLYNOMIALS WITH FIXED MEAN VALUE

Alexander G. Babenko     (Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences; Institute of Mathematics and Computer Science of the Ural Federal University, Ekaterinburg, Russian Federation)

Abstract


Let T+n be the set of nonnegative trigonometric polynomials τn of degree n that are strictly positive at zero. For 0≤α≤2π/(n+2), we find the minimum of the mean value of polynomial (cosα−cosx)τn(x)/τn(0) over τn∈T+n on the period [−π,π).


Keywords


Trigonometric polynomials, Extremal problem

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References


Krylov V.I. Approximate calculation of integrals, Fizmatgiz, Moscow, 1959. [Russian]

Pólya G., Szegő G. Problems and theorems in analysis, Vol. 2, Berlin: Springer, 1998.

Prudnikov A.P., Brychkov Yu. A., Marichev O.I. Integrals and series, Moscow: Nauka, 1981. [Russian]

Fejér L. Über trigonometrische Polynome, Gesammelte Arbeit. Budapest: Akad. Kiado, Bd. 1, 1970. P. 842–872.




DOI: http://dx.doi.org/10.15826/umj.2016.1.001

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