View source for Fatou theorem

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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
\[
u(z) = \int 
\frac{1-r^2}{1-2r\cos(\theta-\phi)+r^2}
\rd 
\mu(\zeta), \quad \zeta = \mathrm{e}^{\mathrm{i}\theta},
\]
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 value|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 value|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 [[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 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 {{Cite|Fa}}.

====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)
|-
|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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|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)
|-
|valign="top"|{{Ref|Zy}}||valign="top"| A. Zygmund, "Trigonometric series", '''1–2''', Cambridge Univ. Press (1988)
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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==== 
{|
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|valign="top"|{{Ref|Ho}}||valign="top"| K. Hoffman, "Banach spaces of analytic functions", Prentice-Hall (1962)
|-
|valign="top"|{{Ref|La}}||valign="top"| E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie", ''Das Kontinuum und andere Monographien'', Chelsea, reprint (1973)
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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
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|valign="top"|{{Ref|Ru2}}||valign="top"| W. Rudin, "Function theory in the unit ball in $\C^n$", Springer (1980)
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|valign="top"|{{Ref|St}}||valign="top"| E.M.  Stein, "Boundary behavior of holomorphic functions of several complex variables", Princeton Univ. Press (1972)
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