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Luzin-type Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Paul M. Gauthier  and
Fatemeh Sharifi
Paul M. Gauthier
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal, H3C 3J7. e-mail: gauthier@dms.umontreal.ca
Fatemeh Sharifi
Affiliation:
Department of Mathematics, Middlesex College, The University of Western Ontario, London, Ontario, N6A 5B7. e-mail: fsharif8@uwo.ca
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Abstract

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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 $R$.

Keywords

30E1530F99Carleman approximationtangential approximationMyrberg surface

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 2 , 01 June 2017 , pp. 300 - 308
Copyright
Copyright © Canadian Mathematical Society 2017

References

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