Blaschke product

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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

a₀, a₁, ...

inside the unit disc.

Blaschke products were introduced by Wilhelm Blaschke (1915). They are related to Hardy spaces.

Contents 1 Definition 2 Szegő theorem 3 Finite Blaschke products 4 See also 5 References

[edit] Definition

A sequence is said to satisfy the Blaschke condition when

∑	(1 − | an | )
n

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

B(z) =	∏	B(an,z)
	n

with factors

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

The Blaschke product B(z) is analytic in the open unit disc, and is zero at the a_n only (with multiplicity counted).

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

[edit] Szegő theorem

A theorem of Gábor Szegő states that if (where H¹ is the Hardy space with integrable norm), and if f is not identically zero, then f has a countable number of zeros, and these zeros satisfy the Blaschke condition.

[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 such that f can be extended to a continuous function on the closed unit disc

which maps the unit circle to itself. Then ƒ is equal to a finite Blaschke product

where ζ lies on the unit circle and m_i is the multiplicity of the zero a_i, |a_i| < 1. In particular, if ƒ satisfies the condition above and has no zeros inside the unit circle then ƒ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|ƒ(z)|)).

[edit] See also

• Hardy space
• Weierstrass product

[edit] References

• W. Blaschke, Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig , 67 (1915) pp. 194–200
• Peter Colwell, Blaschke Products — Bounded Analytic Functions (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3
• Tamrazov, P.M. (2001), "Blaschke product", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104