On 2D Inverse Problems/Blaschke products

Let {b_k} be a set of n points in the complex unit disc D. The corresponding Blaschke product is defined as

$B_{b}(z)=\prod _{k}{\frac {|b_{k}|}{b_{k}}}({\frac {b_{k}-z}{1-{\overline {b_{k}}}z}}).$

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

$B_{b}:\mathbb {D} {\xrightarrow[{}]{n\leftrightarrow 1}}\mathbb {D} .$

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

$\sum _{k}(1-|b_{k}|)<\infty .$

The Cayley transform

$\tau (z)={\frac {1-z}{1+z}}$

provides a link between the Stieltjes continued fractions and Blaschke products and the Pick-Nevanlinna interpolation problem for the complex unit disc and the half-space.

Exercise(**). Prove that

$\tau \circ \tau =Id$

and every Stieltjes continued fraction is the conjugate of a Blaschke product w/real b_k's:

$\beta =\tau \circ (\pm B_{b})\circ \tau .$

and

$\prod _{\beta (\mu _{k})=1}{\frac {1-\mu _{k}}{1+\mu _{k}}}=\pm B_{b}(0)=\prod _{l}b_{l}=\pm {\frac {1-\beta (1)}{1+\beta (1)}}.$

(Hint.) Cayley transform is a 1-to-1 map between the complex unit disc and the half-space.