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.