With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article J. L. DoobBoundary approach filters for analytic functionsAnnales de l'institut Fourier, 23 no. 3 (1973), p. 187-213, doi: 10.5802/aif.476 Article PDF | Reviews MR 51 #3448 | Zbl 0251.30034 Résumé - AbstractLet $H^\infty$ 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 $D_1$. Bibliography[1] M. BRELOT and J.L. DOOB, Limites angulaires et limites fines, Ann. Inst. Fourier, 13 (1963), 395-415. Cedram |  MR 33 #4299 |  Zbl 0132.33902[2] J.L. DOOB, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431-458. Numdam |  MR 22 #844 |  Zbl 0097.34004[3] Kenneth HOFFMAN, Banach spaces of analytic functions, Prentice Hall 1962.  Zbl 0117.34001[4] Kenneth HOFFMAN, Bounded analytic functions and Gleason parts, Ann. Math. 86 (1967), 74-111.  MR 35 #5945 |  Zbl 0192.48302[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. Cedram |  MR 20 #6608 |  Zbl 0086.30603[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. Numdam |  Zbl 0177.37701[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.  Zbl 0193.10301[8] M. TSUJI, Potential theory in modern function theory, Tokyo 1959.  MR 22 #5712 |  Zbl 0087.28401 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310