Namespaces
Variants
Actions

Difference between revisions of "Blaschke factor"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(details)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
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]]
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<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>
+
Out of 22 formulas, 22 were replaced by TEX code.-->
  
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]]
+
{{TEX|semi-auto}}{{TEX|done}}
 +
Let 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>
+
\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]]
 +
 
 +
\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:
  
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" />;
+
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" />.
+
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.
+
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
+
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>
+
\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]].
+
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>
+
<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>

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