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Polynomials of Least Deviation from Zero in Sobolev p-Norm

Polynomials of Least Deviation from Zero in Sobolev p-Norm The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1<p<\infty $$\end{document}) for the case p=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=1$$\end{document}. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n-d∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n-\mathbf {d}^*$$\end{document} zeros on the convex hull of the support of the measure, where d∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbf {d}^*$$\end{document} denotes the number of terms in the discrete part. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2022
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-021-01233-5
Publisher site
See Article on Publisher Site

Abstract

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1<p<\infty $$\end{document}) for the case p=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=1$$\end{document}. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n-d∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n-\mathbf {d}^*$$\end{document} zeros on the convex hull of the support of the measure, where d∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbf {d}^*$$\end{document} denotes the number of terms in the discrete part.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Mar 1, 2022

Keywords: Polynomials of least deviation from zero; Extremal polynomials; Sobolev norm; Zero location; 30C10; 30C15; 33C47; 41A10; 41A50; 41A52; 46E35

References