On 2D Inverse Problems/Pick-Nevanlinna interpolation

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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, ..., zN and w1, ..., wN 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}^+,


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 intermideate 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.