Blaschke product

Definition.

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

B(z):=\prod_{n=1}^{\infty}\frac{\lvert a_{n}\rvert}{a_{n}}\left(\frac{a_{n}-z}% {1-\bar{a}_{n}z}\right)

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.

Sometimes $B_{a}(z):=\frac{z-a}{1-\bar{a}z}$ is called the Blaschke factor.

With this definition, the Blascke product becomes $B(z)=\prod_{n=1}^{\infty}\frac{\lvert a_{n}\rvert}{a_{n}}B_{a_{n}}(z)$ .

References

1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title	Blaschke product
Canonical name	BlaschkeProduct
Date of creation	2013-03-22 14:19:35
Last modified on	2013-03-22 14:19:35
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Definition
Classification	msc 30C45
Defines	Blascke factor