Blaschke product

Definition

A sequence of points (a_n) inside the unit disk is said to satisfy the Blaschke condition when

:\sum_n (1-|a_n|)

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\neq 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 a_n (with multiplicity counted): furthermore it is in the Hardy class H^\infty.

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

Szegő theorem

A theorem of Gábor Szegő states that if f\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} \mid |z|\le 1}

that maps the unit circle to itself. Then f is equal to a finite Blaschke product

: B(z)=\zeta\prod_{i=1}^n\left(\right)^{m_i}

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

References

• {{cite book

References

1. Conway (1996) 274