On the Best Uniform Polynomial Approximation to the Checkmark Function

TL;DR

This paper analyzes the uniform polynomial approximation of the checkmark function, revealing the piecewise analytic nature of the minimax error and its V-shaped behavior as the parameter varies.

Contribution

It provides a detailed characterization of the minimax error function for polynomial approximation of the checkmark function, including its piecewise linear structure and the monotonicity of alternation points.

Findings

$E_n(eta)$ is piecewise analytic in $eta$.

$E_n(eta)$ has $n-1$ V-shaped segments.

For odd $n$, $E_n(eta)$ has a local maximum at $eta=0$.

Abstract

The best uniform polynomial approximation of the checkmark function $f(x)=|x-\alpha |$ is considered, as $\alpha$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (\alpha)$ is shown to be piecewise analytic in $\alpha$. In addition, $E_n(\alpha)$ is shown to feature $n-1$ piecewise linear decreasing/increasing sections, called V-shapes. The points of the alternation set are proven to be monotone increasing in $\alpha$ and their dynamics are completely characterized. We also prove a conjecture of Shekhtman that for odd $n$, $E_n(\alpha)$ has a local maximum at $\alpha=0$.