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

Let H be the class of bounded analytic functions on D:|z|<1, and let D ¯ be the set of maximal ideals of the algebra H , a compactification of D. The relations between functions in H and their cluster values on D ¯-D are studied. Let D 1 be the subset of D ¯ over the point 1. A subset A of D 1 is a “Fatou set” if every f in H has a limit at e iθ 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 .

Soit H l’espace des fonctions bornées holomorphes dans D:|z|<1, et soit D ¯ l’espace des idéaux maximaux de l’algèbre H , une compactification de D. On étudie les relations entre les fonctions de H et leurs valeurs limites sur D ¯-D. Soit D 1 le sous-ensemble de D ¯ sur le point 1. Un sous-ensemble A de D 1 est un “ensemble de Fatou” si tout f dans H a une limite en e iθ 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 .

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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},
     volume = {23},
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     doi = {10.5802/aif.476},
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     language = {en},
     url = {http://archive.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, Volume 23 (1973) no. 3, pp. 187-213. doi : 10.5802/aif.476. http://archive.numdam.org/articles/10.5802/aif.476/

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