Blaschke product


In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a sequence of prescribed complex numbers
inside the unit disc.
plot, using a version of the Domain coloring method.
Blaschke products were introduced by. They are related to Hardy spaces.

Definition

A sequence of points inside the unit disk is said to satisfy the Blaschke condition when
Given a sequence obeying the Blaschke condition, the Blaschke product is defined as
with factors
provided a ≠ 0. Here is the complex conjugate of a. When a = 0 take B = z.
The Blaschke product B defines a function analytic in the open unit disc, and zero exactly at the an : furthermore it is in the Hardy class.
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, the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f satisfy the Blaschke condition.

Finite Blaschke products

Finite Blaschke products can be characterized 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
which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product
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.