On theorems of Jackson and Bernstein type in the complex plane
type: Item
instance of: scholarly article
title: On theorems of Jackson and Bernstein type in the complex plane
zbMATH Open document ID: 0639.30032
DOI: 10.1007/BF02075464
published in: Constructive Approximation
publication date: 1988-01-01T00:00:00Z
review text: The paper is concerned with the question: For which Jordan domains D are the generalizations of Jackson's and Bernstein's theorems valid? Let (J) denote the proposition: If \(f\in Lip \alpha\) in \(\bar D\) \((0<\alpha <1)\), then \(E_ n(f)=O(n^{-\alpha})\), and let (B) denote the converse proposition. Also, let \(\Phi\) be a conformal map from \(\{\) \(w: | w| >1\}\) onto \({\mathbb{C}}\setminus \bar D\) with \(\Phi (\infty)=\infty\), and let \(\Psi =\Phi^{-1}\). We then have the following four statements: (1) \(\Phi\in Lip 1\Rightarrow\) (J); (2) (J) \(\Rightarrow\Phi\in Lip 1\); (3) \(\Psi\in Lip 1\Rightarrow\) (B); (4) (B) \(\Rightarrow\Psi\in Lip 1.\) Of these, (3) was proved by Lebedev and Tamrazov in 1970, and (1) is true under slight additional assumptions on \(\Phi\) (Kövari, Belyi). Here it is shown that (2) and (4) are true under slight additional assumptions on \(\Phi\) or \(\Psi\). The authors conjecture that (1)-(4) are unconditionally true. - There are interesting side results: A local version of Bernstein's inequality, and results on the conformal mapping of strip domains.
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 4042255
zbMATH Keywords: Jackson's theorem
zbMATH Keywords: Bernstein's theorems
zbMATH Keywords: conformal mapping of strip domains
author: J. Milne Anderson
author: Aimo Hinkkanen
author: F. D. Lesley
reviewed by: Dieter Gaier
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02075464
OpenAlex ID: W2004226935
description: scientific article; zbMATH DE number 4042255