# Difference between revisions of "Fatou theorem"

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

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].

How to Cite This Entry:
Fatou theorem. Encyclopedia of Mathematics. URL: http://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