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,

$B_b:𝔻\underset{}{\overset{n\leftrightarrow 1}{\to }}𝔻.$

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

$\sum _k(1-|b_k|)<\mathrm{\infty }.$

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. Template:BookCat