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Blaschke product
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(Definition)
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Definition 1 Suppose that $\{ a_n \}$ is a sequence of complex numbers with $0 < \lvert a_n \rvert < 1$ and $\sum_{n=1}^\infty (1 - \lvert a_n \rvert) < \infty$ , then \begin{equation*} B(z) := \prod_{n=1}^\infty \frac{\lvert a_n \rvert}{a_n} \left( \frac{a_n - z}{1-\bar{a}_nz} \right) \end{equation*}is called the Blaschke product.
This product converges uniformly on compact subsets of the unit disc, and thus $B$ is a holomorhic function on the unit disc. Further it is the function on the disc that has zeros exactly at $\{ a_n \}$ . And finally for $z$ in the unit disc, $\left\lvert B(z) \right\rvert \leq 1$ .
Definition 2 Sometimes $B_a(z) := \frac{z-a}{1-\bar{a}z}$ is called the Blaschke factor.
With this definition, the Blascke product becomes $B(z) = \prod_{n=1}^\infty \frac{\lvert a_n \rvert}{a_n} B_{a_n}(z)$ .
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Blaschke product" is owned by jirka.
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Blascke factor |
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Cross-references: factor, disc, function, unit disc, compact subsets, converges uniformly, product, complex numbers, sequence
There is 1 reference to this entry.
This is version 5 of Blaschke product, born on 2004-04-22, modified 2005-12-07.
Object id is 5795, canonical name is BlaschkeProduct.
Accessed 4809 times total.
Classification:
| AMS MSC: | 30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions ) |
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Pending Errata and Addenda
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