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Difference between revisions of "Blaschke factor"

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Let $D$ be the open unit disc in the complex plane $\mathbf{C}$. A [[Holomorphic function|holomorphic function]]
 
Let $D$ be the open unit disc in the complex plane $\mathbf{C}$. A [[Holomorphic function|holomorphic function]]
  
\begin{equation*} f ( z ) = \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad | a | < 1, \end{equation*}
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\begin{equation*} f ( z ) = \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad | a | < 1, \end{equation*}
  
 
on $D$ is called a Blaschke factor if it occurs in a [[Blaschke product|Blaschke product]]
 
on $D$ is called a Blaschke factor if it occurs in a [[Blaschke product|Blaschke product]]
  
\begin{equation*} \prod _ { j = 1 } ^ { \infty } \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad \sum ( 1 - | a _ { j } | ) &lt; \infty . \end{equation*}
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\begin{equation*} \prod _ { j = 1 } ^ { \infty } \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad \sum ( 1 - | a _ { j } | ) < \infty . \end{equation*}
  
 
The defining properties of a Blaschke factor are:
 
The defining properties of a Blaschke factor are:
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====References====
 
====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  S.D. Fischer,  "Function thory on planar domains" , Wiley  (1983)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Voichick,  L. Zalcman,  "Inner and outer functions on Riemann Surfaces"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 1200–1204</td></tr></table>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top">  S.D. Fischer,  "Function thory on planar domains" , Wiley  (1983)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Voichick,  L. Zalcman,  "Inner and outer functions on Riemann Surfaces"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 1200–1204</td></tr>
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</table>

Latest revision as of 20:48, 22 January 2024

Let $D$ be the open unit disc in the complex plane $\mathbf{C}$. A holomorphic function

\begin{equation*} f ( z ) = \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad | a | < 1, \end{equation*}

on $D$ is called a Blaschke factor if it occurs in a Blaschke product

\begin{equation*} \prod _ { j = 1 } ^ { \infty } \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad \sum ( 1 - | a _ { j } | ) < \infty . \end{equation*}

The defining properties of a Blaschke factor are:

a) a Blaschke factor has precisely one zero in $D$;

b) a Blaschke factor has norm $1$ on the boundary of $D$.

The properties a)–b) may be used to define Blaschke factors on a Dirichlet domain $\Omega$ in a Riemann surface as $f ( z ) = e ^ { - ( G ( z , a ) + i \tilde{G} ( z , a ) ) }$. Here, $G$ is the Green function for $\Omega$ at $a \in \Omega$ and $\tilde { G }$ is its (multiple-valued) harmonic conjugate. See [a1] for the planar case.

Thus, in general a Blaschke factor will not be single valued, but it is single valued on simply-connected domains.

Next, for functions $g$ of the Nevanlinna class (cf. Boundary properties of analytic functions), the term "Blaschke factor" is used to indicate the Blaschke product that has the same zeros as $g$. For example, on the disc $D$ one has the decomposition formula

\begin{equation*} g = B . O . \frac { S _ { 1 } } { S _ { 2 } }, \end{equation*}

where $B$ is a Blaschke product or the Blaschke factor, $O$ is the outer factor, and $S _ { 1 }$, $S _ { 2 }$ are singular inner functions; cf. [a2], Boundary properties of analytic functions; Hardy classes.

Similar decomposition theorems are known for domains in Riemann surfaces, cf. [a3].

References

[a1] S.D. Fischer, "Function thory on planar domains" , Wiley (1983)
[a2] J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a3] M. Voichick, L. Zalcman, "Inner and outer functions on Riemann Surfaces" Proc. Amer. Math. Soc. , 16 (1965) pp. 1200–1204
How to Cite This Entry:
Blaschke factor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blaschke_factor&oldid=55294
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article