# Blaschke product

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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

a0, a1, ...

inside the unit disc. Blaschke product, B(z), associated to 50 randomly chosen points in the unit disk. $\zeta =e^{2\pi i/3}$ . 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 $(a_{n})$ inside the unit disk is said to satisfy the Blaschke condition when

$\sum _{n}(1-|a_{n}|)<\infty .$ Given a sequence obeying the Blaschke condition, the Blaschke product is defined as

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

$B(a,z)={\frac {|a|}{a}}\;{\frac {a-z}{1-{\overline {a}}z}}$ provided a ≠ 0. Here ${\overline {a}}$ is the complex conjugate of a. When a = 0 take B(0,z) = z.

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

The sequence of an satisfying the convergence criterion above is sometimes called a Blaschke sequence.

## Szegő theorem

A theorem of Gábor Szegő states that if f is in $H^{1}$ , the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of 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 f is an analytic function on the open unit disc such that f can be extended to a continuous function on the closed unit disc

${\overline {\Delta }}=\{z\in \mathbb {C} \,|\,|z|\leq 1\}$ which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product

$B(z)=\zeta \prod _{i=1}^{n}\left({{z-a_{i}} \over {1-{\overline {a_{i}}}z}}\right)^{m_{i}}$ where ζ lies on the unit circle and mi is the multiplicity of the zero ai, |ai| < 1. In particular, if ƒ satisfies the condition above and has no zeros inside the unit circle then ƒ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|ƒ(z)|)).