Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation.
type: Item
instance of: scholarly article
title: Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation.
zbMATH Open document ID: 1043.30023
author: Daniyal M. Israfilov
published in: Journal of Approximation Theory
publication date: 2004-01-14T00:00:00Z
review text: Let \(\pi_n(z)\), \(n=1,2,...\) be the Bieberbach polynomials for \((G,z_0)\), where \(G\) is a finite simply connected domain in \(\mathbb{C}\) bounded by a rectifiable Jordan curve \(L\) and \(z_0\in G\). The paper deals with the uniform convergence on \(\overline{G}\) of the sequence of Bieberbach polynomials. The following definition, given here in a short version (the details can be seen in the paper), is crucial and it is added to the condition of \(L\) being smooth . ``The domain \(G\) is of bounded boundary rotation (BBR) if the tangent direction angle of \(L\) has bounded variation.'' The main result of the present paper is the following. Theorem. If \(G\) is a finite smooth domain of BBR, then for every \(\varepsilon >0\) there exists a constant \(c=c(\varepsilon)\) such that \[ \| \phi_0-\pi_n\| _{\overline{G}}:=\max_{z\in \overline{G}}| \phi_0(z)-\pi_n(z)| \leq \frac{c}{n^{1-\varepsilon}},\quad n\geq 1.\tag{1} \] Estimate (1) is an improvement of that obtained by Mergelyan in 1951 when \(L\) is a smooth Jordan curve.
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 2026874
zbMATH Keywords: Bieberbach polynomials
zbMATH Keywords: conformal mapping
zbMATH Keywords: smooth boundaries
zbMATH Keywords: bounded rotation
zbMATH Keywords: uniform convergence
zbMATH Keywords: rate of convergence
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: j.jat.2003.09.008
OpenAlex ID: W2010187965
cites work: About convergence of Bieberbach polynomials inside domains of the complex plane
cites work: Convergence of Bieberbach polynomials in domains with quasiconformal boundary
cites work: Q3697317
cites work: Convergence of Bieberbach polynomials in domains with interior cusps
cites work: Q4273944
cites work: Q5618011
cites work: Q3922909
cites work: On a polynomial lemma of Andrievskii
cites work: On the convergence of the Bieberbach polynomials in regions with corners
cites work: Q4020018
cites work: Polynomial approximation of conformal maps
cites work: Q5574105
cites work: Q4256531
cites work: Approximation by \(p\)-Faber polynomials in the weighted Smirnov class \(E^p(G,\omega)\) and the Bieberbach polynomials
cites work: On the boundary continuity of conformal maps
cites work: ON THE CONVERGENCE OF BIEBERBACH POLYNOMIALS IN THE CASE OF A LIPSCHITZ DOMAIN
cites work: Polynomials orthogonal over a region and Bieberbach polynomials
cites work: On conformal mapping of certain classes of Jordan domains
cites work: Q5533393
DOI: 10.1016/J.JAT.2003.09.008
description: scientific article; zbMATH DE number 2026874