Geodesic
On uniqueness of the polynomial of best approximation of the function $\cos kx$ by trigonometric polynomials in the $L$ metric
Matematičeskie zametki, Tome 15 (1974) no. 5, pp. 729-737.

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In this paper we clarify a problem concerning uniqueness of the polynomial which best approximates $\cos kx$ in the $L$ metric with respect to a trigonometric system of order $n$ in which $\cos kx$ is absent. We prove uniqueness in the case $n=(2l +1)k$. In the remaining cases there is no uniqueness. An analogous problem in the $C$ metric is solved and the relationship between $n$ and $k$ in the case of uniqueness ia distinguished from the conditions in the $L$ metric. 
@article{MZM_1974_15_5_a8,
     author = {V. N. Temlyakov},
     title = {On uniqueness of the polynomial of best approximation of the function $\cos kx$ by trigonometric polynomials in the $L$ metric},
     journal = {Matemati\v{c}eskie zametki},
     pages = {729--737},
     publisher = {mathdoc},
     volume = {15},
     number = {5},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_5_a8/}
}
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V. N. Temlyakov. On uniqueness of the polynomial of best approximation of the function $\cos kx$ by trigonometric polynomials in the $L$ metric. Matematičeskie zametki, Tome 15 (1974) no. 5, pp. 729-737. http://geodesic.mathdoc.fr/item/MZM_1974_15_5_a8/