Geodesic
On~an~extremal problem on the minimum of a trigonometric polynomial
Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 593-606

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The exact value of the quantity $$ M(n)=\min\biggl\{-\min_x\sum_{k=1}^na_k\cos(kx)\colon a_1\geqslant 1,\dots ,a_n\geqslant 1\biggr\} $$ is found for any positive integer $n$. It is proved that an extremal trigonometric polynomial on which this minimum is attained is unique. Some properties of these extremal polynomials are studied. 
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     author = {A. S. Belov},
     title = {On~an~extremal problem on the minimum of a trigonometric polynomial},
     journal = {Izvestiya. Mathematics },
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     volume = {43},
     number = {3},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a10/}
}
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A. S. Belov. On~an~extremal problem on the minimum of a trigonometric polynomial. Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 593-606. http://geodesic.mathdoc.fr/item/IM2_1994_43_3_a10/