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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Noninvariance of an approximation property for closed subsets of Riemann surfaces

Author: Stephen Scheinberg
Journal: Trans. Amer. Math. Soc. 262 (1980), 245-258
MSC: Primary 30E10; Secondary 30F99
MathSciNet review: 583854
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Abstract: A closed subset E of an open Riemann surface M is said to have the approximation property $ \mathcal{a}$ if each continuous function on E which is analytic at all interior points of E can be approximaed uniformly on E by functions which are everywhere analytic on M. It is known that $ \mathcal{a}$ is a topological invariant (i.e., preserved by homeomorphisms of the pair $ (M,E)$) when M is of finite genus but not in general, not even for $ {C^\infty }$ quasi-conformal automorphisms of M. The principal result of this paper is that $ \mathcal{a}$ is not invariant even under a real-analytic isotopy of quasi-conformal automorphisms (of a certain M). M is constructed as the two-sheeted unbranched cover of the plane minus a certain discrete subset of the real axis, and the isotopy is induced by $ (x + \,iy,\,t) \mapsto x + \,ity$, for $ t > 0$; E can be taken to be that portion of M which lies over a horizontal strip.

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Article copyright: © Copyright 1980 American Mathematical Society