Blaschke product explained

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 products were introduced by . 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|)} < inf ty.

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) = (|a|)/(a) (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} .^[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\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 C \mid |z|\le 1 \}

that maps the unit circle to itself. Then

f is equal to a finite Blaschke productB(z) = \zeta\prod ^{n\left({{z - a} _{i}\over} { 1- \overline{a ^{m _{i}} _{i}z}}\right)}

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

• W.. Blaschke. Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen. de. Berichte Math.-Phys. Kl.. Sächs. Gesell. der Wiss. Leipzig. 67 . 1915. 194–200.
• Book: Colwell, Peter . 0-472-10065-3 . 779463 . University of Michigan Press . Ann Arbor, Michigan . Blaschke Products . 1985.
• Book: Conway, John B. . John B. Conway

. Functions of a Complex Variable II . 1996 . 159 . . . 0-387-94460-5 . John B. Conway . 273–274 .

Notes and References

1. Conway (1996) 274