Boundary approach filters for analytic functions

J. L. Doob

Annales de l'institut Fourier (1973)

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

Abstract

top
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 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 .

How to cite

top

Doob, 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 $D:\vert z\vert &lt; 1$, 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 $D:\vert z\vert &lt; 1$, 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

top
  1. [1] M. BRELOT and J.L. DOOB, Limites angulaires et limites fines, Ann. Inst. Fourier, 13 (1963), 395-415. Zbl0132.33902MR33 #4299
  2. [2] J.L. DOOB, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431-458. Zbl0097.34004MR22 #844
  3. [3] Kenneth HOFFMAN, Banach spaces of analytic functions, Prentice Hall 1962. Zbl0117.34001
  4. [4] Kenneth HOFFMAN, Bounded analytic functions and Gleason parts, Ann. Math. 86 (1967), 74-111. Zbl0192.48302MR35 #5945
  5. [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. [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. [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. [8] M. TSUJI, Potential theory in modern function theory, Tokyo 1959. Zbl0087.28401MR22 #5712

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.