The Faber operator and its boundedness
type: Item
instance of: scholarly article
title: The Faber operator and its boundedness
zbMATH Open document ID: 0943.30025
published in: Journal of Approximation Theory
publication date: 2000-08-30T00:00:00Z
review text: Let \(G\) be a domain bounded by a rectifiable Jordan curve \(\Gamma\), and let \(A(G) := \mathcal O(G) \cap\mathcal C(\overline G)\) be the Banach space of functions holomorphic in \(G\) and continuous on \(\overline G\). Let \[ \psi(w) = bw +b_o+ {{b_1}\over w} + \dots ,\quad |w|>1, \] be the normalized exterior mapping which maps \(\{w:|w|>1\}\) onto the exterior of \(\Gamma \). Let \(\mathbb D\) be the unit disk. The Faber operator \(T:A(\mathbb D) \mapsto \mathcal O(G)\) is defined by the formula \[ \left (Tf\right)(z) := {{1}\over {2\pi i}}\int_{\partial \mathbb D} f(w) {{\psi'(w)}\over {\psi(w)-z}} dw, \quad z \in G, \quad f\in A(\mathbb D). \] Put \(\|T\|:= \sup \{\|Tf\|_{G}/\|f\|_{\mathbb D}; f \in A(\mathbb D)\}\). If \(T\) is bounded, i.e. \(\|T\|<\infty \), then \(T(A(D)) \subset A(G)\). The author shows that \(\|T\|<\infty\), if \(\Gamma\) is piecewise Dini-smooth, and gives an example of a quasicircle \(\Gamma\) for which \(T\) is unbounded.
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 1398585
zbMATH Keywords: complex approximation
zbMATH Keywords: Faber polynomials
zbMATH Keywords: Faber operator
author: Dieter Gaier
reviewed by: Józef Siciak
MaRDI profile type: Publication
full work available at URL: jath.1999.3400
OpenAlex ID: W2068174450
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DOI: 10.1006/JATH.1999.3400
description: scientific article; zbMATH DE number 1398585