Blaschke factor

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