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 <math>a_0,\ a_1, \ldots </math> 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 Template:Harvs. They are related to Hardy spaces.
Definition
A sequence of points <math>(a_n)</math> inside the unit disk is said to satisfy the Blaschke condition when
- <math>\sum_n (1-|a_n|) <\infty.</math>
Given a sequence obeying the Blaschke condition, the Blaschke product is defined as
- <math>B(z)=\prod_n B(a_n,z)</math>
with factors
- <math>B(a,z)=\frac{|a|}{a}\;\frac{a-z}{1 - \overline{a}z}</math>
provided <math>a\neq 0</math>. Here <math>\overline{a}</math> is the complex conjugate of <math>a</math>. When <math>a=0</math> take <math>B(0,z)=z</math>.
The Blaschke product <math>B(z)</math> defines a function analytic in the open unit disc, and zero exactly at the <math>a_n</math> (with multiplicity counted): furthermore it is in the Hardy class <math>H^\infty</math>.<ref>Conway (1996) 274</ref>
The sequence of <math>a_n</math> satisfying the convergence criterion above is sometimes called a Blaschke sequence.
Szegő theorem
A theorem of Gábor Szegő states that if <math>f\in H^1</math>, the Hardy space with integrable norm, and if <math>f</math> is not identically zero, then the zeroes of <math>f</math> (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 <math>f</math> is an analytic function on the open unit disc such that <math>f</math> can be extended to a continuous function on the closed unit disc
- <math>\overline{\Delta}= \{z \in \mathbb{C} \mid |z|\le 1\} </math>
that maps the unit circle to itself. Then <math>f</math> is equal to a finite Blaschke product
- <math> B(z)=\zeta\prod_{i=1}^n\left({{z-a_i}\over {1-\overline{a_i}z}}\right)^{m_i} </math>
where <math>\zeta</math> lies on the unit circle and <math>m_i</math> is the multiplicity of the zero <math>a_i</math>, <math>|a_i|<1</math>. In particular, if <math>f</math> satisfies the condition above and has no zeros inside the unit circle, then <math>f</math> is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function <math>\log(|f(z)|)</math>.