Distance estimates and pointwise bounded density

Davie, A. M.; Gamelin, T. W.; Garnett, J.

doi:10.1090/S0002-9947-1973-0313514-8

Distance estimates and pointwise bounded density
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by A. M. Davie, T. W. Gamelin and J. Garnett

Trans. Amer. Math. Soc. 175 (1973), 37-68

DOI: https://doi.org/10.1090/S0002-9947-1973-0313514-8

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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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