Weighted holomorphic polynomial approximation
type: Item
instance of: scholarly article
title: Weighted holomorphic polynomial approximation
zbMATH Open document ID: 1543.41003
author: Stéphane Charpentier
author: Norm Levenberg
author: F. Wielonsky
published in: Analysis and Mathematical Physics
publication date: 2024-08-06T00:00:00Z
review text: The paper is concerned with the weighted holomorphic polynomial approximation. In the late 1990's \textit{I. E. Pritsker} and \textit{R. S. Varga} [Constr. Approx. 14, No. 4, 475--492 (1998; Zbl 0920.30028)] showed that any function holomorphic in an open set \(G\) in \(\mathbb{C}\) can be locally uniformly approximated in \(G\) by weighted holomorphic polynomials \(\{W(z)^{n}p_{n}(z)\}, \deg (p_{n})\le n\), where \(W\) is a non-vanishing holomorphic function in \(G\). In this paper this theory has been developed by proving a quantitative Bernstein-Walsh type theorem for certain pairs \((G,W)\). A special case has been considered where \(W(z)=\frac{1}{(z+1)}\) and \(G\) is a loop of the lemniscate \(\{z\in\mathbb{C}:|z(z+1)|=\frac{1}{4}\}\). It has been shown that the normalized measures associated to the zeros of the \(n^{th}\) order Taylor polynomial about \(0\) of the function \((1+z)^{-n}\) converge to the weighted equilibrium measure of \({G}\) with weight \(|W|\) as \(n\rightarrow\infty\). In the last weighted holomorphic polynomial approximation has been studied in \(\mathbb{C}^{n}\), \(n>1\).
reviewed by: Devendra Kumar
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 7892769
zbMATH Keywords: weighted holomorphic polynomial approximation
zbMATH Keywords: Taylor polynomials
zbMATH Keywords: weighted potential theory
zbMATH Keywords: Bernstein-Walsh theorem
MaRDI profile type: Publication
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DOI: 10.1007/S13324-024-00943-W
description: scientific article; zbMATH DE number 7892769