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DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR ${\boldsymbol H}^{\boldsymbol{\infty}}$ MAPS
Published online by Cambridge University Press: 12 May 2021
Abstract
Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .
Keywords
MSC classification
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 113 , Issue 3 , December 2022 , pp. 289 - 317
- Copyright
- © 2021 Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Finnur Larusson
This research is supported in part by NSERC Grant No. 10010444.
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