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

${\displaystyle \sum _{n}(1-|a_{n}|)<\infty .}$

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

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

with factors

${\displaystyle B(a,z)={\frac {|a|}{a}}\;{\frac {a-z}{1-{\overline {a}}z}}}$

provided a ≠ 0. Here ${\displaystyle {\overline {a}}}$ 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 ${\displaystyle H^{\infty }}$.[1]

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 ${\displaystyle H^{1}}$, 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

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

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

${\displaystyle B(z)=\zeta \prod _{i=1}^{n}\left({{z-a_{i}} \over {1-{\overline {a_{i}}}z}}\right)^{m_{i}}}$

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

1. Conway (1996) 274

• 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, 140 pages. ISBN 0-472-10065-3
• Conway, John B. Functions of a Complex Variable II. Graduate Texts in Mathematics. 159. Springer-Verlag. pp. 273–274. ISBN 0-387-94460-5.
• Tamrazov, P.M. (2001) [1994], "Blaschke product", Encyclopedia of Mathematics, EMS Press