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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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/

[1] Konyagin S. V., Predstavlenie funktsii trigonometricheskimi ryadami, Dis. $\dots$ dokt. fiz.-mat. nauk, MGU, M., 1989

[2] Odlyzko A. M., “Minima of cosine sums and maxima of polynomials on the unit circle”, J. London Math. Soc., 26:3 (1982), 412–420 | DOI | MR | Zbl

[3] Askey R., Steinig J., “Some positive trigonometric sums”, Trans. Amer. Math. Soc., 187:1 (1974), 295–307 | DOI | MR | Zbl

[4] Szegő G., “Inequalities for the zeros of Legendre polynomials and related functions”, Trans. Amer. Math. Soc., 39:1 (1936), 1–17 | DOI | MR | Zbl

[5] Zigmund A., Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR

[6] Konyagin S. V., “O probleme Littlvuda”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 243–265 | MR | Zbl

[7] Mc Gehee O. C., Pigno L., Smith B., “Hardy's inequality and the $L^1$ norm of exponential sums”, Ann. Math. Ser. 2, 113:3 (1981), 613–618 | DOI | MR

[8] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl