Boundary approach filters for analytic functions

Doob, J. L.

doi:10.5802/aif.476

Doob, J. L.

Annales de l'Institut Fourier, Tome 23 (1973) no. 3, pp. 187-213.

Résumé
Abstract

Soit $H^{\infty}$ l’espace des fonctions bornées holomorphes dans $D : | z | < 1$ , et soit $\bar{D}$ 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 $\bar{D} - D$ . Soit $D_{1}$ le sous-ensemble de $\bar{D}$ 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 $e^{i θ} A$ pour presque tout $θ$ . 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 ${Q}$ 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 $\bar{D}$ 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 $\bar{D} - D$ are studied. Let $D_{1}$ be the subset of $\bar{D}$ over the point 1. A subset $A$ of $D_{1}$ is a “Fatou set” if every $f$ in $H^{\infty}$ has a limit at $e^{i θ} A$ for almost every $θ$ . 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 $D_{1}$ .

MR Zbl

DOI : 10.5802/aif.476

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     author = {Doob, J. L.},
     title = {Boundary approach filters for analytic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {187--213},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {23},
     number = {3},
     year = {1973},
     doi = {10.5802/aif.476},
     mrnumber = {51 #3448},
     zbl = {0251.30034},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.476/}
}

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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://www.numdam.org/articles/10.5802/aif.476/

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