## BMO rational approximation and one-dimensional Hausdorff content

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- by Joan Verdera PDF
- Trans. Amer. Math. Soc.
**297**(1986), 283-304 Request permission

## 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.## References

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## Additional Information

- © Copyright 1986 American Mathematical Society
- 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