Blaschke product

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

:"a"₀, "a"₁, ...

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

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

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".

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 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 ƒ is equal to a finite Blaschke product

:$B(z)=zetaprod\_\{i=1\}^nleft($z-a_i}over {1-overline{a_i}z ight)^{m_i}

where $zeta$ lies on the unit circle and "m_i" is the multiplicity of the zero "a_i", . 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(|ƒ("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 ƒ is continuous on the closed unit disc.

ee also

* Hardy space
* Weierstrass product

References

* Peter Colwell, "Blaschke Products — Bounded Analytic Functions" (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3