Difference between revisions of "Blaschke factor"
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+ | Let 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*} | |
+ | |||
+ | 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 | + | a) a Blaschke factor has precisely one zero in D; |
− | b) a Blaschke factor has norm | + | 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 | + | 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 | + | 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 |
− | + | \begin{equation*} g = B . O . \frac { S _ { 1 } } { S _ { 2 } }, \end{equation*} | |
− | where | + | 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>< | + | <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 |
Blaschke factor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blaschke_factor&oldid=17194