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Luzin-type holomorphic approximation on closed subsets of open Riemann surfaces. (English) Zbl 1369.30039

Summary: 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 \(F\subset E,\) which approximates \(E\) extremely well, such that every function holomorphic on \(F\) can be approximated much better than uniformly by functions holomorphic on \(R\).

MSC:

30E15 Asymptotic representations in the complex plane
30F99 Riemann surfaces
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