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

DOI:
https://doi.org/10.1090/S0002-9947-1980-0583854-X

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 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 is a topological invariant (i.e., preserved by homeomorphisms of the pair ) when *M* is of finite genus but not in general, not even for quasi-conformal automorphisms of *M*. The principal result of this paper is that 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 , for ; *E* can be taken to be that portion of *M* which lies over a horizontal strip.

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0583854-X

Article copyright:
© Copyright 1980
American Mathematical Society