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

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

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 BlaschkeProduct 2013-03-22 14:19:35 2013-03-22 14:19:35 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 30C45 Blascke factor