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}^+,

and


f(z_k) = w_k,k=1,2,\dots,N.

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


\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 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.

Y-Delta equivalent graphs

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

Expressing area of a triangle in terms of determinant of a matrix

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.