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Luzin-type Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces
Published online by Cambridge University Press: 20 November 2018
Abstract
It is known that if $E$ is a closed subset of an open Riemann surface
$R$ and
$f$ is a holomorphic function on a neighbourhood of
$E$, then it is “usually” not possible to approximate
$f$ uniformly by functions holomorphic on all of
$R$. We show, however, that for every open Riemann surface
$R$ and every closed subset
$E\subset R$, there is closed subset
$F\subset E$ that approximates
$E$ extremely well, such that every function holomorphic on
$F$ can be approximated much better than uniformly by functions holomorphic on
$R$.
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- Research Article
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- Copyright © Canadian Mathematical Society 2017
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