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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.

References [Enhancements On Off] (What's this?)

  • [Da] Guy David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479 (English, with English and French summaries). MR 1654535, 10.4171/RMI/242
  • [Ga] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • [To1] Xavier Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), no. 1, 105–149. MR 1982794, 10.1007/BF02393237
  • [To2] Xavier Tolsa, The semiadditivity of continuous analytic capacity and the inner boundary conjecture, Amer. J. Math. 126 (2004), no. 3, 523–567. MR 2058383
  • [Ve] Joan Verdera, Removability, capacity and approximation, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419–473. MR 1332967
  • [Vi] A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838

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Additional Information

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.