On 2D Inverse Problems/Blaschke products

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

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

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

The Cayley transform

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

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

and

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