Boundary approach filters for analytic functions

@article{Doob1973,
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 $D_1$.},
author = {Doob, J. L.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {187-213},
publisher = {Association des Annales de l'Institut Fourier},
title = {Boundary approach filters for analytic functions},
url = {http://eudml.org/doc/74137},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Doob, J. L.
TI - Boundary approach filters for analytic functions
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 3
SP - 187
EP - 213
AB - 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 $D_1$.
LA - eng
UR - http://eudml.org/doc/74137
ER -