View source for Fatou theorem

''in the theory of functions of a complex variable''

Suppose that the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382502.png" />, can be represented in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382503.png" /> by a Poisson–Stieltjes integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382504.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382505.png" /> is a Borel measure concentrated on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382507.png" />. Then almost-everywhere with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f0382509.png" /> has angular boundary values (cf. [[Angular boundary value|Angular boundary value]]).

This Fatou theorem can be generalized to harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825012.png" />, that can be represented by a Poisson–Stieltjes integral in Lyapunov domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825013.png" /> (see , ). For Fatou's theorem for radial boundary values (cf. [[Radial boundary value|Radial boundary value]]) of multiharmonic functions in the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825014.png" /> see , .

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825015.png" /> is a bounded analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825016.png" />, then almost-everywhere with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825017.png" /> it has angular boundary values.

This Fatou theorem can be generalized to functions of bounded characteristic (cf. [[Function of bounded characteristic|Function of bounded characteristic]]) (see ). Points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825018.png" /> at which there is an angular boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825019.png" /> are called Fatou points. Regarding generalizations of the Fatou theorem for analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825020.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825022.png" />, see ; it turns out that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825023.png" /> there are also boundary values along complex tangent directions.

If the coefficients of a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825024.png" /> with unit disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825025.png" /> tend to zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825026.png" />, then this series converges uniformly on every arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825027.png" /> of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825028.png" /> consisting only of regular boundary points for the sum of the series.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825029.png" /> and the series converges uniformly on an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825030.png" />, it does not follow that the points of this arc are regular for the sum of the series.

Theorems 1), 2) and 3) were proved by P. Fatou [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Fatou,   "Séries trigonométriques et séries de Taylor"  ''Acta Math.'' , '''30'''  (1906)  pp. 335–400</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,   P.I. Kuznetsov,   "On boundary problems and various classes of harmonic and subharmonic functions on an arbitrary domain"  ''Mat. Sb.'' , '''6''' :  3  (1939)  pp. 345–376  (In Russian)  (French summary)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.D. Solomentsev,   "On boundary values of subharmonic functions"  ''Czechoslovak. Math. J.'' , '''8'''  (1958)  pp. 520–536  (In Russian)  (French summary)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,   "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.M. Khenkin,   E.M. Chirka,   "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5''' :  5  (1976)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''4'''  (1975)  pp. 13–142</TD></TR></table>



====Comments====
For Lyapunov domain see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]. For Fatou theorems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825031.png" /> see [[#References|[a3]]]–[[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Landau,   "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Hoffman,   "Banach spaces of analytic functions" , Prentice-Hall  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038250/f03825032.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.M. Stein,   "Boundary behavior of holomorphic functions of several complex variables" , Princeton Univ. Press  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Nagel,   E.M. Stein,   "On certain maximal functions and approach regions"  ''Adv. in Math.'' , '''54'''  (1984)  pp. 83–106</TD></TR></table>