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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0961619-2
Keywords: Bounded holomorphic functions, function algebras, Widom domains, Wermer maximality, the Chang-Marshall theorem, the $ \overline \partial $ equation, BMOA
Article copyright: © Copyright 1989 American Mathematical Society

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