TITLE: Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$
QUESTION [9 upvotes]: For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$.
Question
Given $\epsilon> 0$, find a "low-degree" polynomial $P_\epsilon$ (the smaller the degree, the better!) which approximates $\sigma_R$ within $\epsilon$ w.r.t the sup-norm, ie.
$$\|\sigma_R-P_\epsilon\|_\infty := \max_{x \in [-R,R]}|\sigma_R(x)-P_\epsilon(x)| \le \epsilon.
$$
Observations
Here is a graphical illustration of such an approximation.

REPLY [12 votes]: Subtracting $x/2$, and rescaling, the problem is reduced to the best uniform approximation of $|x|$ on $[-1,1]$ by polynomials, a problem already considered by Bernstein.
According to Chebyshev's theory, the polynomial of degree $\le n$ that minimizes the uniform distance on $[-1,1]$ from the absolute value function is unique and is characterized by Chebyshev's equioscillation theorem. It is also even, by symmetrization argument. More details should be easily available in the literature; for instance, these notes devote the whole chapter 3 exactly to the problem.