Blaschke product

Definition.

Suppose that $\{a_n\}$ is a sequence of complex numbers with and , then

$$B(z):=\prod _{n=1}^{\mathrm{\infty }}\frac{|a_n|}{a_n}\left(\frac{a_n-z}{1-{\overline{a}}_{n}z}\right)$$

is called the Blaschke product.

This product converges uniformly on compact subsets of the unit disc, and thus $B$ isa 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 unitdisc, $\left|B(z)\right|\le 1$.

Definition.

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

With this definition, the Blascke product becomes $B(z)={\prod }_{n=1}^{\mathrm{\infty }}\frac{|a_n|}{a_n}{B}_{a_n}(z)$.