# ANNALES DE L'INSTITUT FOURIER

Boundary approach filters for analytic functions
Annales de l'Institut Fourier, Tome 23 (1973) no. 3, pp. 187-213.

Soit ${H}^{\infty }$ l’espace des fonctions bornées holomorphes dans $D:|z|<1$, et soit $\overline{D}$ l’espace des idéaux maximaux de l’algèbre ${H}^{\infty }$, une compactification de $D$. On étudie les relations entre les fonctions de ${H}^{\infty }$ et leurs valeurs limites sur $\overline{D}-D$. Soit ${D}_{1}$ le sous-ensemble de $\overline{D}$ sur le point 1. Un sous-ensemble $A$ de ${D}_{1}$ est un “ensemble de Fatou” si tout $f$ dans ${H}^{\infty }$ a une limite en ${e}^{i\theta }A$ pour presque tout $\theta$. Le sous-ensemble nontangentiel est un ensemble de Fatou d’après le théorème de Fatou. Il y a beaucoup d’ensembles de Fatou plus grands, par exemple le sous-ensemble de ${D}_{1}$ des points fixes, mais il n’y a pas un ensemble de Fatou maximal. L’ensemble des points $Q$ de ${D}_{1}$ dont $\left\{Q\right\}$ est un ensemble de Fatou est dense dans ${D}_{1}$.

Let ${H}^{\infty }$ be the class of bounded analytic functions on $D:|z|<1$, 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 ${D}_{1}$.

@article{AIF_1973__23_3_187_0,
author = {Doob, J. L.},
title = {Boundary approach filters for analytic functions},
journal = {Annales de l'Institut Fourier},
pages = {187--213},
publisher = {Imprimerie Louis-Jean},
volume = {23},
number = {3},
year = {1973},
doi = {10.5802/aif.476},
zbl = {0251.30034},
mrnumber = {51 \#3448},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1973__23_3_187_0/}
}
Doob, J. L. Boundary approach filters for analytic functions. Annales de l'Institut Fourier, Tome 23 (1973) no. 3, pp. 187-213. doi : 10.5802/aif.476. https://aif.centre-mersenne.org/item/AIF_1973__23_3_187_0/

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