On uniqueness of the polynomial of best approximation of the function $\cos kx$ by trigonometric polynomials in the $L$ metric

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.