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Failure of rational approximation on some Cantor type sets

Author: Albert Mas-Blesa
Journal: Proc. Amer. Math. Soc. 137 (2009), 635-640
MSC (2000): Primary 30C85; Secondary 31A15
Published electronically: June 20, 2008
MathSciNet review: 2448585
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Abstract: Let $ A(K)$ be the algebra of continuous functions on a compact set $ K\subset\mathbb{C}$ which are analytic on the interior of $ K$, and let $ R(K)$ be the closure (with respect to uniform convergence on $ K$) of the functions that are analytic on a neighborhood of $ K$. A counterexample of a question posed by A. O'Farrell about the equality of the algebras $ R(K)$ and $ A(K)$ when $ K=(K_{1}\times[0,1])\cup([0,1]\times K_{2})\subseteq\mathbb{C}$, with $ K_{1}$ and $ K_{2}$ compact subsets of $ [0,1]$, is given. Also, the equality is proved with the assumption that $ K_{1}$ has no interior.

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Albert Mas-Blesa
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Keywords: Rational approximation, analytic capacity, Cantor sets
Received by editor(s): February 6, 2008
Published electronically: June 20, 2008
Additional Notes: This work was supported by grant AP2006-02416 (Programa FPU del MEC, España), and also partially supported by grants 2005SGR-007749 (Generalitat de Catalunya) and MTM2007-62817 (MEC, España)
Communicated by: Mario Bonk
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.