# 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 to be 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 transforms. The intermideate graphs do not have rotation symmetry, which provides an example of **symmetry breaking**.

**Exercise (**).** Find a sequence of ten Y-Δ transforms b/w the following two planar graphs w/natural boundary.

The following exercise plays an important role in the algorithm of interpolation.

**Exercise (*).** Prove that on the following picture the areas of the triangles are equal and

Note, that the picture is not symmetric w/respect to the *x = y* line.

**Exercise (**).** Let *A* be a square *n* by *n* non-singular matrix. Prove that there is a unique number *x* such that

where *1* is an *n* by *n* matrix consisting of all ones.

(Hint.)The matrix *A-x1* is the Schur complement of the following block matrix:

and, therefore,

**Exercise (**)**. Using the previous exercise and the existence and uniqueness criteria for the Pick-Nevanlinna interpolation find an algorithm for calculating the coefficients of the Stieltjes continued fraction from the interpolation data.