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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Approximating continuous functions by holomorphic and harmonic functions


Author: Christopher J. Bishop
Journal: Trans. Amer. Math. Soc. 311 (1989), 781-811
MSC: Primary 30E10; Secondary 31A05, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1989-0961619-2
MathSciNet review: 961619
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Abstract: If $\Omega$ is a Widom domain in the plane (e.g., finitely connected) and $f$ is any bounded harmonic function on $\Omega$ which is not holomorphic, then we prove the algebra ${H^\infty }(\Omega )[f]$ contains all the uniformly continuous functions on $\Omega$. The basic tools are the solution of the $\overline \partial$ equation with ${L^\infty }$ estimates and some estimates on the level sets of functions in BMOA.


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Keywords: Bounded holomorphic functions, function algebras, Widom domains, Wermer maximality, the Chang-Marshall theorem, the <!– MATH $\overline \partial$ –> <IMG WIDTH="18" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\overline \partial$"> equation, BMOA
Article copyright: © Copyright 1989 American Mathematical Society