Approximating continuous functions by holomorphic and harmonic functions

Bishop, Christopher J.

doi:10.1090/S0002-9947-1989-0961619-2

Approximating continuous functions by holomorphic and harmonic functions
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by Christopher J. Bishop

Trans. Amer. Math. Soc. 311 (1989), 781-811

DOI: https://doi.org/10.1090/S0002-9947-1989-0961619-2

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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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