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

Published:2016-09-13
Printed: Jun 2017
• Paul M Gauthier,
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal, H3C3J7
• Fatemeh Sharifi,
Department of Mathematics, Middlesex College, The University of Western Ontario, London, Ontario, N6A 5B7
 Format: LaTeX MathJax PDF

## 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 $F\subset E,$ which approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$.
 Keywords: Carleman approximation, tangential approximation, Myrberg surface
 MSC Classifications: 30E15 - Asymptotic representations in the complex domain 30F99 - None of the above, but in this section

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