Geodesic
On an extremal problem for polynomials with fixed mean value
Ural mathematical journal, Tome 2 (2016) no. 1, pp. 3-8

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Let $T_n^+$ be the set of nonnegative trigonometric polynomials $\tau_n$ of degree $n$ that are strictly positive at zero. For $0\le\alpha\le2\pi/(n+2),$ we find the minimum of the mean value of polynomial $(\cos\alpha-\cos{x})\tau_n(x)/\tau_n(0)$ over $\tau_n\in{T_n^+}$ on the period $[-\pi,\pi).$ The paper was originally published in a hard accessible collection of articles Approximation of Functions by Polynomials and Splines (The Ural Scientific Center of the Academy of Sciences of the USSR, Sverdlovsk, 1985), p. 15–22 (in Russian).
Keywords: Trigonometric polynomials, Extremal problem. 
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Alexander G. Babenko. On an extremal problem for polynomials with fixed mean value. Ural mathematical journal, Tome 2 (2016) no. 1, pp. 3-8. http://geodesic.mathdoc.fr/item/UMJ_2016_2_1_a0/