Let $p\_{n,K}(x)$ be the polynomial of degree $n$ that minimizes $\epsilon\_{n,K} = ||p\_{n,K}(x) - sin(x)||\_\infty$ on the interval $[-K, K]$. Question: what is the asymptotic behavior of $\epsilon\_{n,K}$ where $n \to \infty$? 



Minimax Approximation to Sine Function on interval [-K, K]
The rate of polynomial approximation 
$$E\_{n}(f,\mathcal{K}):=\inf\{\max\_{z\in \mathcal{K}}|f(z)-P\_{n}(z)|,~\text{deg}~P\_{n}\leq n\}$$
to an entire function $f$ on a compact set $\mathcal{K}$ of the complex plane of positive capacity was derived by A. Batyrev in



> 
> Batyrev, A. V., On the best approximation of analytic functions by
>  polynomials. Doklady Akad. Nauk SSSR (N.S.) 76, (1951) 173--175.
> 
> 
> 


His theorem states that
$$
\operatorname { limsup } \_ { n \rightarrow \infty } n \left[ E \_ { n } ( f , \mathcal{K} ) \right] ^ { \rho / n } = e \rho \tau ^ { \rho } \sigma.
$$
Here $\tau$ is the capacity of $\mathcal{K}$, $\rho$ and $\sigma$ are the order and type of $f$ respectively.


Since the segment $\mathcal{K}=[-K,K]$ has capacity $K/2$, and the sine function is of order 1 and type 1, one gets that $E\_{n}(\sin,\mathcal{K})$ decreases like 
$$\left(\frac{eK}{2n}\right)^{n},$$
as $n$ tends to infinity.


Another reference, possibly easier to get, is



> 
> Giroux A., Approximation of entire functions over bounded domains. J.
>  Approx. Theory 28 (1980), no. 1, 45-53.
> 
> 
> 


which extends Batyrev's result to $L^p$ norms, $2\leq p\leq\infty$.


