Analytic balayage of measures, Carathéodory domains, and badly approximable functions in \(L^p\)
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 \(p\geq 1\).
scientific article; zbMATH DE number 6917464
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