Weighted polynomials on discrete sets
type: Item
instance of: scholarly article
title: Weighted polynomials on discrete sets
zbMATH Open document ID: 1036.42001
published in: Monatshefte für Mathematik
publication date: 2003-05-04T00:00:00Z
review text: Let \(I\) be a real interval of positive length. A weighted polynomial of degree at most \(n\geq 1\) on \(I\) is an expression of the form \(P_nw^n\) where \(P_n\) is an algebraic polynomial of degree at most \(n\geq 1\) and \(w:I\rightarrow [0,\infty)\) is a positive, not identically zero continuous weight on \(I\). In this paper, the author proves a necessary and sufficient condition which ensures that the continuous \(L_p\) (\(0<p\leq \infty\)) norm of a weighted polynomial, \(P_nw^n\), \(\deg P_n\leq n\), \(n\geq 1,\) is in an \(n\)th root sense, controlled by its corresponding discrete Hölder norm on a very general class of discrete subsets of \(I\). As a consequence, he establishes Nikol'skij inequalities and theorems dealing with zero distribution, zero location and sup and \(L_p\) infinite-finite range inequalities.
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 1903791
zbMATH Keywords: asymptotics
zbMATH Keywords: discrete norm
zbMATH Keywords: extremal polynomial
zbMATH Keywords: infinite-finite range inequality
zbMATH Keywords: \(L_p\) norm
zbMATH Keywords: \(L_\infty\) norm
zbMATH Keywords: Nikol'skij inequality
zbMATH Keywords: potential theory
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: weighted approximation
zbMATH Keywords: zero distribution
author: S. B. Damelin
reviewed by: Youssèf Ben Cheikh
MaRDI profile type: Publication
full work available at URL: s00605-002-0517-9
OpenAlex ID: W2084071444
DOI: 10.1007/S00605-002-0517-9
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description: scientific article; zbMATH DE number 1903791