On 2D Inverse Problems/Pick-Nevanlinna interpolation

Certain questions about the layered inverse problems can be reduced to the Pick-Nevanlinna interpolation problem: given the values of a function at specific points of the domains D or C⁺, find its analytic continuation to an automorphism of the domain.

More formally, if z₁, ..., z_N and w₁, ..., w_N are collections of points in the unit disc or the complex right half-plane, one seeks an analytic function f defined in the whole domain, such that

$f:\mathbb {D} \to \mathbb {D} {\mbox{ or }}f:\mathbb {C} ^{+}\to \mathbb {C} ^{+}$,

and

$f(z_{k})=w_{k},k=1,2,\dots ,N.$

The function f can be chosen a rational Stieltjes continued fraction or the Blaschke product, depending on the domain in the problem. The interpolating function exists, see [M], if and only if the matrices

$\left({\frac {1-w_{k}{\overline {w_{l}}}}{1-z_{k}{\overline {z_{l}}}}}\right)_{k,l=1}^{N}{\mbox{ and }}\left({\frac {w_{k}+w_{l}}{z_{k}+z_{l}}}\right)_{k,l=1}^{N}$

are positive semi-definite, respectively. The interpolation function is unique if and only if the corresponding matrix is singular. If the matrix is not singular, then there're infinitely many interpolating continued fractions w/the number of levels larger than N.

Since the corresponding networks have equal Dirichle-to-Neumann operators, any pair of such networks can be transformed one to another by a finite sequence of Y-Δ moves. The intermidiate graphs do not have rotation symmetry, which provides an example of symmetry breaking.

Exercise (**). Using the solution for the Pick-Nevanlinna interpolation problem, find an algorithm for calculating the coefficients of the Stieltjes continued fraction from the interpolation data.