$a$ is a real number. Let $B_n(x)$ be the best uniform approximation degree-$n$ polynomial of $\sin(ax)$ on $[-1,1]$. Denote $R_n = ||B_n(x)-\sin(ax)||_{L^{\infty}}$. Does there exist some function $u(n), v(n)$ such that $$O(u(n))\le R_n\le O(v(n)),$$ and $u(n)$ and $v(n)$ are about the same (preferably $u(n)=v(n)$)?
Since $\sin(ax)$ is entire function, we know that its best uniform approximation error decays faster than any exponential functions of $n$. More precisely, we can deduce that $R_n<e^{ar/2}r^{-n}$ for any $r>1$, using the truncated Chebyshev series of $\sin(ax)$. Thus we can take the minimum on $r$ and get $$R_n\le (\frac{ea}{2n})^n.$$ (Taylor expansion can also yield similar upper bound.) However, I do not know how good is this upper bound, and I cannot get any meaningful lower bound of $R_n$. I guess this upper bound is not too bad and maybe there is a lower bound also in the form $(ka/n)^n$, where $k$ is not related to $n$ and $a$?
Any hint or references on this topic are welcome.
