On the Best Uniform Polynomial Approximation to the Checkmark Function

The best uniform polynomial approximation of the checkmark function $f(x)=|x-α|$ is considered, as $α$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (α)$ is shown to be piecewise analytic in $α$. In addition, $E_n(α)$ 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 $α$ and their dynamics are completely characterized. We also prove a conjecture of Shekhtman that for odd $n$, $E_n(α)$ has a local maximum at $α=0$.

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