# On 2D Inverse Problems/Blaschke products

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

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

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

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

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

${\displaystyle \sum _{k}(1-|b_{k}|)<\infty .}$
${\displaystyle \tau (z)={\frac {1-z}{1+z}}}$

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

${\displaystyle \tau \circ \tau =Id}$

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

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

and

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

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