I posted this question on [Math Stack Exchange] but did not get any answer. I am trying my luck here.

>Let $n,k$ be  given positive integers and $n>k$. If for all real numbers $x$ we have $$A_{1}\cos{x}+A_{2}\cos{(2x)}+\cdots+A_{n}\cos{(nx)}\le 1$$
Find the maximum value of $A_{k}$.

I don't know if this question has been studied

If  $n=2$ it is easy to solve it.


[Math Stack Exchange]: https://math.stackexchange.com/questions/3988744/find-the-maximum-trigonometric-polynomial-coefficient-a-k