Blaschke product

Short description: Concept in complex analysis

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

$a_0,a_1,\dots$

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

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

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

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

with factors

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

provided $a\ne 0$. 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 $f\in H^1$, 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

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

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

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

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" (in de). Berichte Math.-Phys. Kl. (Sächs. Gesell. der Wiss. Leipzig) 67: 194–200. https://babel.hathitrust.org/cgi/pt?id=mdp.39015036849837;view=1up;seq=226. 
• Colwell, Peter (1985). Blaschke Products. Ann Arbor, Michigan: University of Michigan Press. ISBN 0-472-10065-3. 
• Conway, John B. (1996). Functions of a Complex Variable II. Graduate Texts in Mathematics. 159. Springer-Verlag. pp. 273–274. ISBN 0-387-94460-5. https://books.google.com/books/about/Functions_of_One_Complex_Variable_II.html?id=JN0hz3qO1eMC&pg=PA273. 
• Hazewinkel, Michiel, ed. (2001), "Blaschke product", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page