Uniform algebras generated by holomorphic and close-to-harmonic functions
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- by Gautam Bharali and Sushil Gorai PDF
- Proc. Amer. Math. Soc. 139 (2011), 2183-2189 Request permission
Abstract:
The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc $\overline {\mathbb {D}}$ generated by $z$ and $h$, where $h$ is a nowhere-holomorphic harmonic function on $\mathbb {D}$ that is continuous up to $\partial {\mathbb {D}}$, equals $\mathcal {C}(\overline {\mathbb {D}})$. The abstract tools used by Axler and Shields make harmonicity of $h$ an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if $h$ is replaced by $h+R$, where $R$ is a non-harmonic perturbation whose Laplacian is “small” in a certain sense.References
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Additional Information
- Gautam Bharali
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
- Email: bharali@math.iisc.ernet.in
- Sushil Gorai
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
- Email: sushil@math.iisc.ernet.in
- Received by editor(s): January 20, 2010
- Received by editor(s) in revised form: May 13, 2010, and June 15, 2010
- Published electronically: November 30, 2010
- Additional Notes: The first author is supported by the DST via the Fast Track grant SR/FTP/MS-12/2007
The second author is supported by CSIR-UGC fellowship 09/079(2063). Support is also provided by the UGC under DSA-SAP, Phase IV - Communicated by: Franc Forstneric
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2183-2189
- MSC (2010): Primary 30E10, 32E20, 32U05, 46J15
- DOI: https://doi.org/10.1090/S0002-9939-2010-10708-6
- MathSciNet review: 2775396