Asymptotically optimal estimates of the \(n\)-widths of bounded analytic functions
type: Item
instance of: scholarly article
title: Asymptotically optimal estimates of the \(n\)-widths of bounded analytic functions
zbMATH Open document ID: 0816.41025
author: Klaus Wilderotter
published in: Journal of Complexity
publication date: 1995-02-16T00:00:00Z
review text: In this paper, optimal lower and upper bounds fort the \(n\)-width of bounded analytic functions are given. The \(n\)-width of a subset \(A\) of a Banach space \(X\) is defined by \[ d_ n(A,X) : = \inf_{X_ n} \sup_{x \in A} \inf_{y \in X_ n} | x - y |, \] where \(X_ n\) runs over all \(n\)-dimensional subspaces of \(X\). Let \(\Delta\) be the unit disk in the complex plane \(\mathbb{C}\), and let \(H^ \infty\) be the Hardy space consisting of all bounded analytic functions on \(\Delta\). With \(A\) we denote the unit ball in \(H^ \infty\). Assume that \(E\) is an arbitrary, simply connected subset of \(\Delta\), and that the boundary \(\delta E\) belongs to \(C^{1, \alpha}\) \((0 < \alpha \leq 1)\). Then, we have \[ \exp \left( - {n \over \text{cap} (E, \Delta)} \right) \leq d_ n \bigl( A,C (E) \bigr) \leq \exp \left( - {n \over \text{cap} (E, \Delta)} \right) \quad \biggl( 1 + O \bigl( \ln (n)/n^{\alpha} \bigr) \biggr) \] for all \(n \in \mathbb{N}\), where \(\text{cap} (E, \Delta)\) is the Green capacity of \(E\) with respect to \(\Delta\). The lower bound follows straightforwardly from \textit{H. Widom} [J. Approximation Theory 5, 343-361 (1972; Zbl 0234.30023)] and \textit{S. D. Fisher} and \textit{Ch. A. Micchelli} [Duke Math. J. 47, 789-801 (1980; Zbl 0451.30023)]. The upper bound is proved using some techniques (introduced in Fisher, Micchelli (1980)) from the theory of Faber polynomials to the setting of Blaschke products in the unit disk.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 710611
zbMATH Keywords: \(n\)-width of bounded analytic functions
zbMATH Keywords: Faber polynomials
zbMATH Keywords: Blaschke products
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: jcom.1994.1026
OpenAlex ID: W4210274615
DOI: 10.1006/JCOM.1994.1026
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description: scientific article; zbMATH DE number 710611