Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
type: Item
instance of: scholarly article
title: Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
zbMATH Open document ID: 0582.41009
DOI: 10.1007/BF01890023
published in: Constructive Approximation
publication date: 1985-01-01T00:00:00Z
review text: A characterization is given of the sets supporting the uniform norms of weighted polynomials \([w(x)]^ nP_ n(x)\), where \(P_ n\) is any polynomial of degree at most n. The (closed) support \(\Sigma\) of w(x) may be bounded or unbounded; of special interest is the case when w(x) has a nonempty zero set Z. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of \(\Sigma\) \(\setminus Z.\) One main result of this paper states that there is a unique compact set (independent of n and \(P_ n)\) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights \([w(x)]^ n\) is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 3931644
zbMATH Keywords: characterization
zbMATH Keywords: weighted polynomials
zbMATH Keywords: distribution of the zeros of Chebyshev polynomials
zbMATH Keywords: weighted approximation
author: H. N. Mhaskar
author: E. B. Saff
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01890023
OpenAlex ID: W1978794151
description: scientific article; zbMATH DE number 3931644