Uniform algebras generated by holomorphic and close-to-harmonic functions

Details Uniform algebras generated by holomorphic and close-to-harmonic functions Uniform algebras generated by holomorphic and close-to-harmonic functions

2010-07-26

arxiv.org/abs/1007.4430v2

Mathematics

Complex Variables

6 pages; added an important remark on page 2 and a reference; to appear in Proc. Amer. Math. Soc

Scientific paper

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 closure(D) generated by z and h --- where h is a nowhere-holomorphic harmonic function on D that is continuous up to the boundary --- equals the algebra of continuous functions on closure(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.

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