For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$.


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. $$\|\sigma_R-P_\epsilon\|_\infty := \max_{x \in [-R,R]}|\sigma_R(x)-P_\epsilon(x)| \le \epsilon. $$


Here is a graphical illustration of such an approximation.

enter image description here


2 Answers 2


This is done by using Remes's algorithm. It is implemented in Mathematica's command MiniMaxApproximation[]. Below is an image of the corresponding Mathematica notebook, giving a polynomial approximation of degree $10$. (Click on the image to see it better.) The quality of the approximation seems substantially better than in your picture.

enter image description here

  • $\begingroup$ Thanks for the response, and for Ramez algorithm. My interest for the question is almost purely analytical. In particular, for a fixed sup-norm accuracy $\epsilon$, I'm interested in degree of the best polynomial approximation (+ existence), in terms of $R$ and $epsilon$. Intuitively it should be something like $\text{poly}(R,1/\epsilon)$ or even $\text{poly}(R,\log(1/\epsilon))$. $\endgroup$
    – dohmatob
    Jul 8, 2019 at 21:45
  • 4
    $\begingroup$ According to Bernstein, for polynomials of degree $2n$, and $R=1$, one has $\epsilon\sim C/n$ with $0.278\le C\le0.286$ (check the link in my answer) $\endgroup$ Jul 8, 2019 at 22:48
  • $\begingroup$ Great. Thanks for the comment. $\endgroup$
    – dohmatob
    Jul 8, 2019 at 22:59
  • 1
    $\begingroup$ (so on $[-R,R]$ the degree is $\sim 2CR/\epsilon$) $\endgroup$ Jul 9, 2019 at 10:05

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.


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