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 En(α) is shown to be piecewise analytic in α. In addition, En(α) is shown to feature n−1 piecewise linear decreasing/increasing sections, ca...

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Bibliographic Details
Published inJournal of approximation theory Vol. 275; p. 105698
Main Authors Dragnev, P.D., Legg, A.R., Orive, R.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.03.2022
Subjects
41A10
41A50
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Summary: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 En(α) is shown to be piecewise analytic in α. In addition, En(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 piecewise analytic and monotone increasing in α and their dynamics are completely characterized. We also prove a conjecture of Shekhtman that for odd n, En(α) has a local maximum at α=0.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2022.105698