On the best uniform polynomial approximation to the checkmark function

In this talk we shall consider the best uniform polynomial approximation of the checkmark function $f(x)=|x-\alpha |$ as $\alpha$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (\alpha)$ is shown to be piecewise analytic in $\alpha$ and the best approximants stem from classical polynomials of Chebyshev, Zolotarev, Krein, etc. 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 $\alpha=0$.
This is a joint work with Alan Legg (PUFW) and Ramon Orive (ULL), arXiv:2102.09502.