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arXiv · 2102.09502

On the Best Uniform Polynomial Approximation to the Checkmark Function

Peter D. Dragnev·Alan R. Legg·Ramon Orive
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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 $\alpha=0$.

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Peter D. Dragnev, Alan R. Legg, Ramon Orive. 2021-02-18. On the Best Uniform Polynomial Approximation to the Checkmark Function. https://doi.org/10.1016/j.jat.2022.105698

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