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International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 2, Pages 363-370

Functions in the space R2(E) at boundary points of the interior

Department of Mathematics, Marshall University, Huntington 25701, West Virginia, USA

Received 15 September 1982

Copyright © 1983 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let E be a compact subset of the complex plane . We denote by R(E) the algebra consisting of (the restrictions to E of) rational functions with poles off E. Let m denote 2-dimensional Lebesgue measure. For p1, let Rp(E) be the closure of R(E) in Lp(E,dm).

In this paper we consider the case p=2. Let x ϵ E be a bounded point evaluation for R2(E). Suppose there is a C>0 such that x is a limit point of the set s={y|y ϵ Int E,Dist(y,E)C|yx|}. For those y ϵ S sufficiently near x we prove statements about |f(y)f(x)| for all f ϵ R(E).