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An inductive proof of the Feinstein-Heath Swiss cheese ``Classicalisation'' theorem
Author(s):
J.
W. D.
Mason
Journal:
Proc. Amer. Math. Soc.
138
(2010),
4423-4432.
MSC (2010):
Primary 46J10;
Secondary 54H99
Posted:
June 16, 2010
MathSciNet review:
2680066
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Abstract:
A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.
References:
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Additional Information:
J.
W. D.
Mason
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
Email:
pmxjwdm@nottingham.ac.uk
DOI:
10.1090/S0002-9939-2010-10447-1
PII:
S 0002-9939(2010)10447-1
Keywords:
Swiss cheeses,
rational approximation,
uniform algebras
Received by editor(s):
October 30, 2009
Received by editor(s) in revised form:
February 12, 2010
Posted:
June 16, 2010
Additional Notes:
The author was supported by a Ph.D. grant from the EPSRC (UK)
Communicated by:
Nigel J. Kalton
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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