Last modified on 26 January 2013, at 21:21

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

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 bk'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_lb_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.