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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

BMO rational approximation and one-dimensional Hausdorff content


Author: Joan Verdera
Journal: Trans. Amer. Math. Soc. 297 (1986), 283-304
MSC: Primary 30E10; Secondary 28A20, 30B40, 46E15
MathSciNet review: 849480
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Abstract: Let $ X \subset {\mathbf{C}}$ be compact and let $ f \in \operatorname{VMO} ({\mathbf{C}})$. We give necessary and sufficient conditions on $ f$ and $ X$ for $ {f_{\vert X}}$ to be the limit of a sequence of rational functions without poles on $ X$ in the norm of $ \operatorname{BMO} (X)$, the space of functions of bounded mean oscillation on $ X$. We also characterize those compact $ X \subset {\mathbf{C}}$ with the property that the restriction to $ X$ of each function in $ \operatorname{VMO} ({\mathbf{C}})$, which is holomorphic on $ \mathop X\limits^ \circ $, is the limit of a sequence of rational functions with poles off $ X$. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0849480-5
PII: S 0002-9947(1986)0849480-5
Keywords: Rational function, approximation, BMO, Hausdorff content
Article copyright: © Copyright 1986 American Mathematical Society