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''in the theory of functions of a complex variable''
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{{MSC|31A20}}
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\newcommand{\abs}[1]{\left|#1\right|}
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$
  
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
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The Fatou theorem is a theorem in the theory of functions of a complex variable: Suppose that the harmonic function $u(z)$, $z=r\mathrm{e}^{\mathrm{i}\phi}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral
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\[
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u(z) = \int
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\frac{1-r^2}{1-2r\cos(\theta-\phi)+r^2}
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\rd
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\mu(\zeta), \quad \zeta = \mathrm{e}^{\mathrm{i}\theta},
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\]
 +
where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}$, $\int\rd\mu(\xi)=1$. Then almost-everywhere with respect to the Lebesgue measure on $T$, $u(z)$ has [[Angular boundary value|angular boundary values]].
  
<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>
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This Fatou theorem can be generalized to harmonic functions $u(x)$, $x\in\R^n$, $n\geq2$, that can be represented by a Poisson–Stieltjes integral in Lyapunov domains $D\subset\R^n$ (see [REF], [REF]). For Fatou's theorem for [[Radial boundary value|radial boundary values]] of multiharmonic functions in the polydisc
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\[
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U^n = \left\{ z=(z_1,\ldots,z_n)\in\C^n : \abs{z_j}<1, j=1,\ldots,n \right\}
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\]
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see [REF], [REF].
  
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]]).
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If $f(z)$ is a bounded analytic function in $U$, then almost-everywhere with respect to the Lebesgue measure on $T$ it has angular boundary values.
  
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 , .
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This Fatou theorem can be generalized to [[Function of bounded characteristic|functions of bounded characteristic]] (see [REF]). Points $\zeta$ at which there is an angular boundary value $f(\zeta)$ are called Fatou points. Regarding generalizations of the Fatou theorem for analytic functions $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n\geq 2$, see [REF]; it turns out that for $n\geq 2$ there are also boundary values along complex tangent directions.
  
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.
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If the coefficients of a power series $\sum_{k=0}^\infty a_k z^k$ with unit disc of convergence $U$ tend to zero, $\lim_{k\rightarrow\infty a_k=0}$, then this series converges uniformly on every arc $\alpha\leq\theta\leq\beta$ of the circle $T$ consisting only of regular boundary points for the sum of the series.
  
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.
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If $\lim_{k\rightarrow\infty} a_k=0$ and the series converges uniformly on an arc $\alpha\leq\theta\leq\beta$, it does not follow that the points of this arc are regular for the sum of the series.
  
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.
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Theorems 1), 2) and 3) were proved by P. Fatou {{Cite|Fa}}.
  
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.
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====References====
 +
{|
 +
|-
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|valign="top"|{{Ref|Fa}}||valign="top"| P. Fatou, "Séries trigonométriques et séries de Taylor" ''Acta Math.'', '''30''' (1906) pp. 335–400
 +
|-
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|valign="top"|{{Ref|KhCh}}||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
 +
|-
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|valign="top"|{{Ref|Pr}}||valign="top"| I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
 +
|-
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|valign="top"|{{Ref|PrKu}}||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)
 +
|-
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|valign="top"|{{Ref|Ru}}||valign="top"| W. Rudin, "Function theory in polydiscs", Benjamin (1969)
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|-
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|valign="top"|{{Ref|So}}||valign="top"| E.D. Solomentsev, "On boundary values of subharmonic functions" ''Czechoslovak. Math. J.'', '''8''' (1958) pp. 520–536 (In Russian) (French summary)
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|-
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|valign="top"|{{Ref|Zy}}||valign="top"| A. Zygmund, "Trigonometric series", '''1–2''', Cambridge Univ. Press (1988)
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|-
 +
|}
  
Theorems 1), 2) and 3) were proved by P. Fatou [[#References|[1]]].
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====Comments====
 +
For Lyapunov domain see [[Lyapunov surfaces and curves]]. For Fatou theorems in $\C^n$ see {{Cite|Ru2}}, {{Cite|St}}, {{Cite|NaSt}}.
  
====References====
+
====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>
+
{|
 
+
|-
 
+
|valign="top"|{{Ref|Ho}}||valign="top"| K. Hoffman, "Banach spaces of analytic functions", Prentice-Hall (1962)
 
+
|-
====Comments====
+
|valign="top"|{{Ref|La}}||valign="top"| E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie", ''Das Kontinuum und andere Monographien'', Chelsea, reprint (1973)
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]]].
+
|-
 
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|valign="top"|{{Ref|NaSt}}||valign="top"| A. Nagel, E.M. Stein, "On certain maximal functions and approach regions" ''Adv. in Math.'', '''54''' (1984) pp. 83–106
====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>
+
|valign="top"|{{Ref|Ru2}}||valign="top"| W. Rudin, "Function theory in the unit ball in $\C^n$", Springer (1980)
 +
|-
 +
|valign="top"|{{Ref|St}}||valign="top"| E.M. Stein, "Boundary behavior of holomorphic functions of several complex variables", Princeton Univ. Press (1972)
 +
|-
 +
|}

Latest revision as of 21:19, 27 July 2012

2010 Mathematics Subject Classification: Primary: 31A20 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} $

The Fatou theorem is a theorem in the theory of functions of a complex variable: Suppose that the harmonic function $u(z)$, $z=r\mathrm{e}^{\mathrm{i}\phi}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral \[ u(z) = \int \frac{1-r^2}{1-2r\cos(\theta-\phi)+r^2} \rd \mu(\zeta), \quad \zeta = \mathrm{e}^{\mathrm{i}\theta}, \] where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}$, $\int\rd\mu(\xi)=1$. Then almost-everywhere with respect to the Lebesgue measure on $T$, $u(z)$ has angular boundary values.

This Fatou theorem can be generalized to harmonic functions $u(x)$, $x\in\R^n$, $n\geq2$, that can be represented by a Poisson–Stieltjes integral in Lyapunov domains $D\subset\R^n$ (see [REF], [REF]). For Fatou's theorem for radial boundary values of multiharmonic functions in the polydisc \[ U^n = \left\{ z=(z_1,\ldots,z_n)\in\C^n : \abs{z_j}<1, j=1,\ldots,n \right\} \] see [REF], [REF].

If $f(z)$ is a bounded analytic function in $U$, then almost-everywhere with respect to the Lebesgue measure on $T$ it has angular boundary values.

This Fatou theorem can be generalized to functions of bounded characteristic (see [REF]). Points $\zeta$ at which there is an angular boundary value $f(\zeta)$ are called Fatou points. Regarding generalizations of the Fatou theorem for analytic functions $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n\geq 2$, see [REF]; it turns out that for $n\geq 2$ there are also boundary values along complex tangent directions.

If the coefficients of a power series $\sum_{k=0}^\infty a_k z^k$ with unit disc of convergence $U$ tend to zero, $\lim_{k\rightarrow\infty a_k=0}$, then this series converges uniformly on every arc $\alpha\leq\theta\leq\beta$ of the circle $T$ consisting only of regular boundary points for the sum of the series.

If $\lim_{k\rightarrow\infty} a_k=0$ and the series converges uniformly on an arc $\alpha\leq\theta\leq\beta$, 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 [Fa].

References

[Fa] P. Fatou, "Séries trigonométriques et séries de Taylor" Acta Math., 30 (1906) pp. 335–400
[KhCh] 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
[Pr] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[PrKu] 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)
[Ru] W. Rudin, "Function theory in polydiscs", Benjamin (1969)
[So] E.D. Solomentsev, "On boundary values of subharmonic functions" Czechoslovak. Math. J., 8 (1958) pp. 520–536 (In Russian) (French summary)
[Zy] A. Zygmund, "Trigonometric series", 1–2, Cambridge Univ. Press (1988)

Comments

For Lyapunov domain see Lyapunov surfaces and curves. For Fatou theorems in $\C^n$ see [Ru2], [St], [NaSt].

References

[Ho] K. Hoffman, "Banach spaces of analytic functions", Prentice-Hall (1962)
[La] E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie", Das Kontinuum und andere Monographien, Chelsea, reprint (1973)
[NaSt] A. Nagel, E.M. Stein, "On certain maximal functions and approach regions" Adv. in Math., 54 (1984) pp. 83–106
[Ru2] W. Rudin, "Function theory in the unit ball in $\C^n$", Springer (1980)
[St] E.M. Stein, "Boundary behavior of holomorphic functions of several complex variables", Princeton Univ. Press (1972)
How to Cite This Entry:
Fatou theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fatou_theorem&oldid=27214
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article