Blaschke product

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In mathematics, the Blaschke product in complex analysis is an analytic function designed to have zeros at a (finite or infinite) sequence of prescribed complex numbers

a₀, a₁, ...

inside the unit disc. If the sequence is finite then the Blaschke product is also called a finite Blaschke product. Given such a sequence, subject to the condition that

Σ (1 − |a_n|)

is convergent, define

B(z) = Π B(a_n, z)

where the factor

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

provided a ≠ 0. Here a^* is the complex conjugate of a. When a = 0 take B(0,z) = z.

Then the Blaschke product B(z) is analytic in the open unit disc, and is zero at the a_n only (with multiplicity counted). It is named for Wilhelm Blaschke, who described it in a paper in 1915. This seemingly peculiar function takes on importance because of its relationship to the study of Hardy spaces.

The sequence of a_n satisfying the convergence criterion above is sometimes called a Blaschke sequence.

[edit] 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 with the following three properties:

f maps the open unit disc to itself, i.e. $|f(z)|<1 \,\, \mbox{if}\,\, |z|<1$ .
f has only finitely many zeros $a_1, \ldots, a_n$ inside the unit disc.
f can be extended to a continuous function on the closed unit disc $\overline{\Delta}= \{z \in \mathbb{C}\,|\, |z|\le 1\}$ which 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 $ζ$ 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 three conditions above and has no zeros inside the unit circle then f is constant (this fact is a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|f(z)|)).

Remark: It can be shown that the last of the three properties listed above entails the other two. The first is a consequence of the Maximum modulus principle for analytic functions, the second property can be deduced from the so-called identity principle which states that the set of zeros of a function holomorphic in a domain D in $\mathbb{C}$ is a discrete subset of D. If there were infinitely many zeros they would have an accumulation point (necessarily) on the boundary unit circle which contradicts the assumption that f is continuous on the closed unit disc.