On the Best Uniform Polynomial Approximation to the Checkmark Function

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 $\alpha =0$.

arXiv comment23 pages, 6 figures, now accepted for publication in Journal of Approximation Theory