# Boundary approach filters for analytic functions

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 3, page 187-213
- ISSN: 0373-0956

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topDoob, J. L.. "Boundary approach filters for analytic functions." Annales de l'institut Fourier 23.3 (1973): 187-213. <http://eudml.org/doc/74137>.

@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 -

## References

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- [3] Kenneth HOFFMAN, Banach spaces of analytic functions, Prentice Hall 1962. Zbl0117.34001
- [4] Kenneth HOFFMAN, Bounded analytic functions and Gleason parts, Ann. Math. 86 (1967), 74-111. Zbl0192.48302MR35 #5945
- [5] L. LUMER-NAÏM, Sur le rôle de la frontière de R.S. Martin dans la théorie du potentiel, Ann. Inst. Fourier 7 (1957), 183-281. Zbl0086.30603MR20 #6608
- [6] Gabriel MOKODOBZKI, Ultrafiltres rapides sur N. Construction d'une densité relative de deux potentiels comparables, Séminaire Théorie Potentiel Brelot-Choquet-Deny 1967/1968 Exp. 12. Zbl0177.37701
- [7] M. ROSENFELD and MAX L. Weiss, A function algebra approach to a theorem of Lindelöf, J. London Math. Soc. (2) 2 (1970), 209-215. Zbl0193.10301
- [8] M. TSUJI, Potential theory in modern function theory, Tokyo 1959. Zbl0087.28401MR22 #5712

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