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

 

Distance estimates and pointwise bounded density


Authors: A. M. Davie, T. W. Gamelin and J. Garnett
Journal: Trans. Amer. Math. Soc. 175 (1973), 37-68
MSC: Primary 30A78; Secondary 30A98, 46J15
MathSciNet review: 0313514
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Abstract: Let U be a bounded open subset of the complex plane, and let H be a closed subalgebra of $ {H^\infty }(U)$, the bounded analytic functions on U. If E is a subset of $ \partial U$, let $ {L_E}$ be the algebra of all bounded continuous functions on U which extend continuously to E, and set $ {H_E} = H \cap {L_E}$. This paper relates distance estimates of the form $ d(h,H) = d(h,{H_E})$, for all $ h \in {L_E}$, to pointwise bounded density of $ {H_E}$ in H. There is also a discussion of the linear space $ H + {L_E}$, which turns out often to be a closed algebra.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0313514-8
PII: S 0002-9947(1973)0313514-8
Keywords: Bounded analytic functions, pointwise bounded approximation, uniform approximation, distance estimates
Article copyright: © Copyright 1973 American Mathematical Society