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J. L. Doob
Boundary approach filters for analytic functions
Annales de l'institut Fourier, 23 no. 3 (1973), p. 187-213, doi: 10.5802/aif.476
Article PDF | Reviews MR 51 #3448 | Zbl 0251.30034

Résumé - Abstract

Let $H^\infty $ be the class of bounded analytic functions on $D:\vert z\vert < 1$, and let $\overline{D}$ be the set of maximal ideals of the algebra $H^\infty $, a compactification of $D$. The relations between functions in $H^\infty $ and their cluster values on $\overline{D} -D$ are studied. Let $D_1$ be the subset of $\overline{D}$ over the point 1. A subset $A$ of $D_1$ is a ``Fatou set" if every $f$ in $H^\infty $ has a limit at $e^{i\theta }A$ for almost every $\theta $. The nontangential subset of $D_1$ is a Fatou set according to the Fatou theorem. There are many larger Fatou sets, for example the fine topology subset of $D_1$ but there is no largest Fatou set. The set of those points of $D_1$ which are Fatou singletons is dense in $D_1$.

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