Blaschke product

In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers

${\displaystyle a_{0},\ a_{1},\ldots }$

inside the unit disc, with the property that the magnitude of the function is constant along the boundary of the disc.

Blaschke product, ${\displaystyle B(z)}$, associated to 50 randomly chosen points in the unit disk. B(z) is represented as a Matplotlib plot, using a version of the Domain coloring method.

Blaschke products were introduced by Wilhelm Blaschke (1915). They are related to Hardy spaces.

Definition

A sequence of points ${\displaystyle (a_{n})}$ inside the unit disk is said to satisfy the Blaschke condition when

${\displaystyle \sum _{n}(1-|a_{n}|)<\infty .}$

Given a sequence obeying the Blaschke condition, the Blaschke product is defined as

${\displaystyle B(z)=\prod _{n}B(a_{n},z)}$

with factors

${\displaystyle B(a,z)={\frac {|a|}{a}}\;{\frac {a-z}{1-{\overline {a}}z}}}$

provided ${\displaystyle a\neq 0}$. Here ${\displaystyle {\overline {a}}}$ is the complex conjugate of ${\displaystyle a}$. When ${\displaystyle a=0}$ take ${\displaystyle B(0,z)=z}$.

The Blaschke product ${\displaystyle B(z)}$ defines a function analytic in the open unit disc, and zero exactly at the ${\displaystyle a_{n}}$ (with multiplicity counted): furthermore it is in the Hardy class ${\displaystyle H^{\infty }}$.^[1]

The sequence of ${\displaystyle a_{n}}$ satisfying the convergence criterion above is sometimes called a Blaschke sequence.

Szegő theorem

A theorem of Gábor Szegő states that if ${\displaystyle f\in H^{1}}$, the Hardy space with integrable norm, and if ${\displaystyle f}$ is not identically zero, then the zeroes of ${\displaystyle f}$ (certainly countable in number) satisfy the Blaschke condition.

Finite Blaschke products

Finite Blaschke products can be characterized (as analytic functions on the unit disc) in the following way: Assume that ${\displaystyle f}$ is an analytic function on the open unit disc such that ${\displaystyle f}$ can be extended to a continuous function on the closed unit disc

${\displaystyle {\overline {\Delta }}=\{z\in \mathbb {C} \mid |z|\leq 1\}}$

that maps the unit circle to itself. Then ${\displaystyle f}$ is equal to a finite Blaschke product

${\displaystyle B(z)=\zeta \prod _{i=1}^{n}\left({{z-a_{i}} \over {1-{\overline {a_{i}}}z}}\right)^{m_{i}}}$

where ${\displaystyle \zeta }$ lies on the unit circle and ${\displaystyle m_{i}}$ is the multiplicity of the zero ${\displaystyle a_{i}}$, ${\displaystyle |a_{i}|<1}$. In particular, if ${\displaystyle f}$ satisfies the condition above and has no zeros inside the unit circle, then ${\displaystyle f}$ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function ${\displaystyle \log(|f(z)|)}$.

See also

• Hardy space
• Weierstrass product
• Complex Analysis