4.4 Approximation by holomorphic functions
4.4.2 Holomorphic approximation
Theorem 4.4.3. (see [41, §5.2]) Let E be a wig-set on a non-compact Riemann surface R. The following statements are equivalent:
(i) (R∪ {∗})\E is connected and locally connected at {∗}. (ii) A(E)⊂HR(E).
For the direction (ii)7→(i), we refer to Gauthier and Hengartner ([19]). So we only have to show the other direction.
Proof. Let K ⊂ R be compact and f ∈ A(E). If the complement of the set K ∩E
has a bounded component, then this component must be contained in E, sinceR\E
has no bounded components. Extend K to the set ˆK by taking the union of K with such components. Since f ∈A( ˆK ∩E), the approximation theorem of Bishop gives
f ∈HR( ˆK∩E) and hence, at fortiori,f ∈MR(K∩E). Thereforef satisfies condition (ii) of the Localization Theorem. Using Proposition 4.4.2, we can choose the function
gn in the proof of the Localization Theorem to be holomorphic on Dn−1∪En. This
implies that the limitg of the sequence {gn}is holomorphic on∪Dn−1 =R. Because
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Vita
Name: Nadya Askaripour
Post-secondary University of Western Ontario
Education and London, Canada
Degrees: 2006–2010, Ph.D.
Sharif University of Technology Tehran, Iran
2000–2002, M.Sc.
Shahid Chamran University Ahwaz, Iran
1996–2000, B.Sc.
Honors and Western Graduate Scholarship 2006–2010
Awards Ontario Graduate Scholarship 2007–2008
Related Work Teaching Assistant
Experience The University of Western Ontario, 2006–2010
Publications:
[1] (with T. Foth) On holomorphic k-differentials on open Riemann surfaces, sub- mitted, (October 2009)
[2] (with A. Boivin) Approximation on Riemann surfaces with infinite genus, in preparation