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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
DOI: https://doi.org/10.1090/S0002-9947-1986-0849480-5
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 ${f_{|X}}$ 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 $\mathring {X}$, 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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Keywords: Rational function, approximation, BMO, Hausdorff content
Article copyright: © Copyright 1986 American Mathematical Society