Integer square root

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 Template:Harvs. 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

$\sum _n(1-|a_n|)<\mathrm{\infty }.$

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

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

with factors

$B(a,z)=\frac{\left|a\right|}{a}\frac{a-z}{1-\bar{a}z}$

provided a ≠ 0. Here $\bar{a}$ 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 $H^{\mathrm{\infty }}$.

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

$\bar{\mathrm{\Delta }}=\{z\in \mathbb{ℂ}|\left|z\right|\le 1\}$

which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product

$B(z)=\zeta \prod _{i=1}^n{\left(\frac{z-a_i}{1-\overline{a_i}z}\right)}^{m_i}$

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

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• 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
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