Analytic balayage of measures, Carathéodory domains, and badly approximable functions in \(L^p\)
type: Item
instance of: scholarly article
title: Analytic balayage of measures, Carathéodory domains, and badly approximable functions in \(L^p\)
zbMATH Open document ID: 1402.30034
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-08-14T00:00:00Z
review text: The analytic balayage \(\nu\) of a measure \(\mu\) with support inside a compact \(X \subset\mathbb C\) is the maximal measure on the boundary \(\partial X\) such that \(\int f\mathrm d(\mu-\nu)=0\) for all \(f\in R(X)\) where \(R(X)\) is the closure of the span of all rational functions with poles outside \(X\). In [\textit{J. J. Carmona} and \textit{K. Yu. Fedorovskiy}, in: Selected topics in complex analysis. The S. Ya. Khavinson memorial volume. Basel: Birkhäuser. 109--130 (2005; Zbl 1089.30035)] a characterization of \(\nu\) was given in terms of an extremal problem which is the \(H^p\) approximation of an \(L^p\) function for \(p=1\). This paper is basically about the case where the solution of the general \(L^p\) problem is zero. In that case the \(L^p\) function is called \(p\)-badly approximable. The main theorem of the paper gives an explicit characterization of \(p\)-badly approximable functions for \(p\geq 1\).
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 6917464
zbMATH Keywords: analytic balayage
zbMATH Keywords: approximation in \(L^p\)
zbMATH Keywords: \(p\)-badly approximable function
zbMATH Keywords: outer function
zbMATH Keywords: inner function
author: Evgeny Abakumov
author: Konstantin Yu. Fedorovskiy
MaRDI profile type: Publication
full work available at URL: j.crma.2018.05.018
OpenAlex ID: W2810408192
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DOI: 10.1016/J.CRMA.2018.05.018
description: scientific article; zbMATH DE number 6917464