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

$a_{0},\ a_{1},\ldots$

inside the unit disc, with the property that the magnitude of the function is constant along the boundary of the disc.

Blaschke product, $B(z)$, associated to 50 randomly chosen points in the unit disk. 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 $(a_{n})$ 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 $a\neq 0$. 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

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

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

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

where $\zeta$ lies on the unit circle and $m_{i}$ is the multiplicity of the zero $a_{i}$, $|a_{i}|<1$. 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 $\log(|f(z)|)$.

See also

• Hardy space
• Weierstrass product
• Complex Analysis

References

• Blaschke, W. (1915). "Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen". Berichte Math.-Phys. Kl. (in German). 67. Sächs. Gesell. der Wiss. Leipzig: 194–200.
• Colwell, Peter (1985). Blaschke Products. Ann Arbor, Michigan: University of Michigan Press. ISBN 0-472-10065-3. MR 0779463.
• Conway, John B. (1996). Functions of a Complex Variable II. Graduate Texts in Mathematics. Vol. 159. Springer-Verlag. pp. 273–274. ISBN 0-387-94460-5.
• Tamrazov, P.M. (2001) [1994]. "Blaschke product". Encyclopedia of Mathematics. EMS Press.