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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202801.png" /> be the open unit disc in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202802.png" />. A [[Holomorphic function|holomorphic function]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202803.png" /></td> </tr></table>
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on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202804.png" /> is called a Blaschke factor if it occurs in a [[Blaschke product|Blaschke product]]
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Let $D$ be the open unit disc in the complex plane $\mathbf{C}$. A [[Holomorphic function|holomorphic function]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202805.png" /></td> </tr></table>
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\begin{equation*} f ( z ) = \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad | a | &lt; 1, \end{equation*}
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on $D$ is called a Blaschke factor if it occurs in a [[Blaschke product|Blaschke product]]
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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 } | ) &lt; \infty . \end{equation*}
  
 
The defining properties of a Blaschke factor are:
 
The defining properties of a Blaschke factor are:
  
a) a Blaschke factor has precisely one zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202806.png" />;
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a) a Blaschke factor has precisely one zero in $D$;
  
b) a Blaschke factor has norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202807.png" /> on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202808.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b1202809.png" /> in a [[Riemann surface|Riemann surface]] as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028010.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028011.png" /> is the [[Green function|Green function]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028014.png" /> is its (multiple-valued) harmonic conjugate. See [[#References|[a1]]] for the planar case.
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The properties a)–b) may be used to define Blaschke factors on a Dirichlet domain $\Omega$ in a [[Riemann surface|Riemann surface]] as $f ( z ) = e ^ { - ( G ( z , a ) + i \tilde{G} ( z , a ) ) }$. Here, $G$ is the [[Green function|Green function]] for $\Omega$ at $a \in \Omega$ and $\tilde { G }$ is its (multiple-valued) harmonic conjugate. See [[#References|[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.
 
Thus, in general a Blaschke factor will not be single valued, but it is single valued on simply-connected domains.
  
Next, for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028015.png" /> of the Nevanlinna class (cf. [[Boundary properties of analytic functions|Boundary properties of analytic functions]]), the term  "Blaschke factor"  is used to indicate the Blaschke product that has the same zeros as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028016.png" />. For example, on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028017.png" /> one has the decomposition formula
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Next, for functions $g$ of the Nevanlinna class (cf. [[Boundary properties of analytic functions|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028018.png" /></td> </tr></table>
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\begin{equation*} g = B . O . \frac { S _ { 1 } } { S _ { 2 } }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028019.png" /> is a Blaschke product or the Blaschke factor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028020.png" /> is the outer factor, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120280/b12028022.png" /> are singular inner functions; cf. [[#References|[a2]]], [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Hardy classes|Hardy classes]].
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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. [[#References|[a2]]], [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Hardy classes|Hardy classes]].
  
 
Similar decomposition theorems are known for domains in Riemann surfaces, cf. [[#References|[a3]]].
 
Similar decomposition theorems are known for domains in Riemann surfaces, cf. [[#References|[a3]]].
  
 
====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><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>

Revision as of 16:46, 1 July 2020

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=17194
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article