I posted this question on [Math Stack Exchange](https://math.stackexchange.com/questions/3988744/find-the-maximum-trigonometric-polynomial-coefficient-a-k) 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.



Find the maximum trigonometric polynomial coefficient $A_{k}$
It is a known result:
$$\max A\_k = 2 \cos \frac{\pi}{\lfloor \frac{n}{k} \rfloor + 2}, \quad 1\le k \le n.$$


See:


1. Theorem 6 and the references therein, "Extremal Positive Trigonometric Polynomials", [https://www.dcce.ibilce.unesp.br/~dimitrov/papers/main.pdf](https://www.dcce.ibilce.unesp.br/%7Edimitrov/papers/main.pdf)
2. Theorem 16.2.4 in: Qazi Ibadur Rahman and Gerhard Schmeisser, “Analytic Theory of Polynomials”, 2002.


