On 2D Inverse Problems/Pick-Nevanlinna interpolation

Certain questions about the layered inverse problems can be reduced to the Pick-Nevanlinna interpolation problem. That is, given the values of a function at specific points of the domains D or C⁺, one looks for its analytic continuation to a 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, the Nevanlinna–Pick problem is the problem of finding an analytic function f such that

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

and

$f(z_i) = w_i.$

The rational function f can be chosen to be a Stieltjes continued fraction or the Blaschke product, depending on the domain in the problem. The interpolating function exists, see [24], if and only if the matrices

$\left( \frac{1-w_i \overline{w_j}}{1-z_i \overline{z_j}} \right)_{i,j=1}^N \mbox { and } \left( \frac{w_i + w_j}{z_i + z_j} \right)_{i,j=1}^N$

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 floors 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

$c = \frac{\det\begin{pmatrix}a_1 & b_1 \\ a_2 & b_2\end{pmatrix}}{(a_1-a_2)+(b_2-b_1)}.$

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

$\det(A-x1)=0,$

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:

$A-x1 = \begin{pmatrix} A & 1 \\ 1 & 1/x \end{pmatrix} / (1/x)$

and, therefore,

$\det(A-x1)=x\det \begin{pmatrix} A & 1 \\ 1 & 1/x \end{pmatrix}.$

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.