# 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 *z*_{1}, ..., *z*_{N} and *w*_{1}, ..., *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

- ,

and

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

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.