inner functionInnerFunction2005-12-07 12:23:132007-08-22 08:49:16Definitionsingular inner functionouter function% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{rmk}{Remark}If $f \colon \mathbb{D} \to \mathbb{C}$ is an analytic function on the unit disc, we denote by
$f^*(e^{i\theta})$ the radial limit of $f$ where it exists, that is
\begin{equation*}
f^*(e^{i\theta}) := \lim_{r\to 1, r<1} f(re^{i\theta}) .
\end{equation*}
A bounded analytic function on the disc will have radial limits almost everywhere (with respect to the Lebesgue measure on the $\partial \mathbb{D}$).
\begin{defn}
A bounded analytic function $f$ is called an \emph{inner function} if $\lvert f^*(e^{i\theta}) \rvert = 1$ almost everywhere. If $f$ has no zeros on the unit disc, then $f$ is called a \emph{singular inner function}.
\end{defn}
\begin{thm}
Every inner function can be written as
\begin{equation*}
f(z) := \alpha B(z) \exp \left( - \int \frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu(e^{i\theta}) \right) ,
\end{equation*}
where $\mu$ is a positive singular measure on $\partial \mathbb{D}$, $B(z)$
is a Blaschke product and $\lvert \alpha \rvert = 1$ is a constant.
\end{thm}
Note that all the zeros of the function come from the Blaschke product.
\begin{defn}
Let
\begin{equation*}
f(z) := \exp \left(\int \frac{e^{i\theta}+z}{e^{i\theta}-z}h(e^{i\theta})dm(e^{i\theta}) \right) ,
\end{equation*}
where $h$ is a real valued Lebesgue integrable function on the unit circle and $m$ is the Lebesgue measure. Then $f$ is called
an \emph{outer function}.
\end{defn}
The significance of these definitions is that every bounded holomorphic function can be written as an inner function times an outer function. See the \PMlinkname{factorization theorem for $H^\infty$ functions}{FactorizationTheoremForHinftyFunctions}.
\begin{thebibliography}{9}
\bibitem{Conway:complexII}
John~B. Conway.
{\em \PMlinkescapetext{Functions of One Complex Variable II}}.
Springer-Verlag, New York, New York, 1995.
\end{thebibliography}