| Version 2 |
Version 1 |
| If $f \colon \mathbb{D} \to \mathbb{C}$ is an analytic function on the unit disc, we denote by |
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 |
$f^*(e^{i\theta})$ the radial limit of $f$ where it exists, that is |
| \begin{equation*} |
\begin{equation*} |
| f^*(e^{i\theta}) := \lim_{r\to 1, r<1} f(re^{i\theta}) . |
f^*(e^{i\theta}) := \lim_{r\to 1, r<1} f(re^{i\theta}) . |
| \end{equation*} |
\end{equation*} |
| A bounded analytic function on the disc will have radial limits almost everywhere (with respect to the Lebesgue measure on the $\partial \mathbb{D}$). |
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} |
\begin{defn} |
| A bounded analytic function $f$ is called an \emph{inner function} if $f^*(e^{i\theta}) = 1$ almost everywhere. If $f$ has no zeros on the unit disc, then $f$ is called a \emph{singular inner function}. |
A bounded analytic function $f$ is called an \emph{inner function} if $f^*(e^{i\theta}) = 1$ almost everywhere. If $f$ has no zeros on the unit disc, then $f$ is called a \emph{singular inner function}. |
| \end{defn} |
\end{defn} |
|
|
| \begin{thm} |
\begin{thm} |
| Every inner function can be written as |
Every inner function can be written as |
| \begin{equation*} |
\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) ,
|
f(z) := B(z) \exp \left( - \int \frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu(e^{i\theta}) \right) ,
|
| \end{equation*} |
\end{equation*} |
|
where $\mu$ is a positive singular measure on $\partial \mathbb{D}$, $B(z)$
|
where $\mu$ is a positive singular measure on $\partial \mathbb{D}$ and $B(z)$
|
|
is a Blaschke product and $\lvert \alpha \rvert = 1$ is a constant.
|
is a Blaschke productt.
|
| \end{thm} |
\end{thm} |
|
|
| Note that all the zeros of the function come from the Blaschke product. |
|
|
|
| \begin{defn} |
\begin{defn} |
| Let |
Let |
| \begin{equation*} |
\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) , |
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*} |
\end{equation*} |
| where $h$ is a real valued Lebesgue integrable function on the unit circle, then $f$ is called |
where $h$ is a real valued Lebesgue integrable function on the unit circle, then $f$ is called |
| an \emph{outer function}. |
an \emph{outer function}. |
| \end{defn} |
\end{defn} |
|
|
| The significance of these definitions is that every bounded function can be written as an inner function times an outer functions. See the \PMlinkname{factorization theorem for $H^\infty$ functions}{FactorizationTheoremForHinftyFunctions}. |
|
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Conway:complexII} |
\bibitem{Conway:complexII} |
| John~B. Conway. |
John~B. Conway. |
| {\em \PMlinkescapetext{Functions of One Complex Variable II}}. |
{\em \PMlinkescapetext{Functions of One Complex Variable II}}. |
| Springer-Verlag, New York, New York, 1995. |
Springer-Verlag, New York, New York, 1995. |
| \end{thebibliography} |
\end{thebibliography} |
