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 product, B(z), associated to 50 randomly chosen points in the unit disk. . B(z) is represented as a Matplotlib plot, using a version of the Domain coloring method.

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

Definition[edit]

A sequence of points inside the unit disk is said to satisfy the Blaschke condition when

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

with factors

provided a ≠ 0. Here 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 an (with multiplicity counted): furthermore it is in the Hardy class .[1]

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

Szegő theorem[edit]

A theorem of Gábor Szegő states that if f is in , 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[edit]

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 mi is the multiplicity of the zero ai, |ai| < 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)|)).

See also[edit]

References[edit]

  1. ^ Conway (1996) 274
  • 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, 140 pages. ISBN 0-472-10065-3
  • Conway, John B. Functions of a Complex Variable II. Graduate Texts in Mathematics. 159. Springer-Verlag. pp. 273–274. ISBN 0-387-94460-5.
  • Tamrazov, P.M. (2001) [1994], "b/b016630", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4