The role of the endpoint in weighted polynomial approximation with varying weights
type: Item
instance of: scholarly article
title: The role of the endpoint in weighted polynomial approximation with varying weights
zbMATH Open document ID: 0864.41006
DOI: 10.1007/BF02433044
published in: Constructive Approximation
publication date: 1996-09-01T00:00:00Z
review text: Approximation by weighted polynomials of the form \(w^n\cdot p_n\) on an interval \([a,b]\) is considered, where \(w\) is a positive weight function and \(p_n\in\Pi_n\). The problem is to describe those functions \(f\), which can be approximated arbitrarily close by weighted polynomials of this type. Special emphasis is laid on the role of the interval endpoints and on that of internal zeros of the density function \(v\) of the extremal measure associated with \(w\). As a main result, it is shown that every function \(f\) which is approximable in the above sense must vanish in \(a\) if \(v(t)\) behaves like \((t-a)^\beta\) as \(t\to a\), with \(\beta>-1/2\). Furthermore, this result is applied to three types of illustrative examples, characterized by the Freud weights and two other types of weight functions, which lead to the incomplete polynomials resp. some fast decreasing polynomials.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 922397
zbMATH Keywords: weighted polynomials
zbMATH Keywords: Freud weights
zbMATH Keywords: incomplete polynomials
author: Arno B. J. Kuijlaars
reviewed by: Guido Walz
MaRDI profile type: Publication
cites work: Q5543516
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cites work: Christoffel functions and orthogonal polynomials for exponential weights on [-1,1]
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cites work: Weighted polynomial approximation with Freud weights
cites work: Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
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cites work: Weighted approximation with varying weight
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description: scientific article; zbMATH DE number 922397