Polynomial approximation on compact sets bounded by Dini-smooth arcs
type: Item
instance of: scholarly article
title: Polynomial approximation on compact sets bounded by Dini-smooth arcs
zbMATH Open document ID: 1059.30026
DOI: 10.1007/BF03321037
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41992
review text: In this paper, the authors use the Faber operator to derive several results on the error of uniform polynomial approximation on a compact subset \(K\) of the complex plane \textbf{C}. They consider the Banach space \(A(K)\) of functions that are continuous on \(K\) and holomorphic in the interior of \(K\) endowed with the maximum norm \(\| \cdot \| _K\), and study the quantity \(E_n(F,K) := \inf \{\| F - P\| _K: \, P\) is a polynomial of degree \(\leq n\}\) \((n \in\) \textbf{N} := \(\{1,2,...\})\). It is assumed that \(K\) is not a single point and that its complement in the extended plane \textbf{C}\(_\infty\) is a simply connected region, so that there exists a unique biholomorphic mapping \(\psi :\)\textbf{C}\(_\infty \setminus \overline{D} \to\) \textbf{C}\(_\infty \setminus K\) (\(D :=\) the open unit disk) with \(\psi (\infty ) = \infty\) and \(\psi '(\infty ) > 0\) and the Faber polynomials \(F_n\) with respect to \(K\) are well defined. Then \(K\) is called a Faber set if the mapping \(T:(P,\| \cdot \| _D) \to (P,\| \cdot \| _K)\) (\(P := \{\)the polynomials\(\}\)) given by \((Tp)(z) = \sum_{\nu = 0}^n a_\nu F_\nu (z)\) (if \(p(z) = \sum_{\nu = 0}^n a_\nu z^\nu\)) is continuous. In such a case \(T\) extends to a one-to-one continuous linear operator \(T:A(\overline{D}) \to A(\overline{K})\). By using the boundedness of \(T\), a large number of results concerning polynomial approximation had been made for the case that \(K\) is the closure of a Jordan domain. In the current paper, it is shown that the corresponding methods also work for compact sets \(K\) which are not the closure of Jordan domains. The following theorem is selected among the diverse statements proved in the paper: Let \(K\) be a Faber set such that \(\psi\) is \(\beta\)-lipschitzian on the boundary \(\partial D\) for some \(\beta > 0\), and let \(\alpha \leq 1\). Then there exists a constant \(c = c(K, \alpha \beta )\) such that \(E_n(F,K) \leq cM/n^{\alpha \beta}\) for all \(n \in\) \textb…
reviewed by: L. Bernal-González
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 2112938
zbMATH Keywords: Faber operators
zbMATH Keywords: Dini-smoothness
zbMATH Keywords: uniform approximation
zbMATH Keywords: touching domains
zbMATH Keywords: Jackson theorem
author: Leonhard Frerick
author: Jürgen Müller
MaRDI profile type: Publication
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OpenAlex ID: W2039618339
description: scientific article; zbMATH DE number 2112938