User:Daviddaved/Blaschke products

From Wikibooks, open books for an open world
Jump to: navigation, search
Let a_i be a set of n points in the complex unit disc D. The corresponding Blaschke product is defined as
B(z)=\prod_k\frac{|a_k|}{a_k}(\frac{a_k-z}{1-\overline{a_k}z}).

If the set of points is finite, the function defines the n-to-1 map of the unit disc onto itself,

f:\mathbb{D}\xrightarrow[]{n\leftrightarrow 1}\mathbb{D}.

If the set of points is infinite, the product converges and defines an automorphism of the complex unit disc, given the Blaschke condition

\sum_k (1-|a_k|) <\infty.

The following fact will be useful in our calculations:

B(0)=\prod_n{|a_i|}.