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

a₀, a₁, ...

inside the unit disc.

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 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 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

$\overline{\Delta}= \{z \in \mathbb{C}\,|\, |z|\le 1\}$

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 m_i is the multiplicity of the zero a_i, |a_i| < 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)|)).

See also

• Hardy space
• Weierstrass product

References

• W. Blaschke, Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig, 67 (1915) pp. 194–200
• Peter Colwell, Blaschke Products — Bounded Analytic Functions (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3