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, ..., zN and w1, ..., wN 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 [M], 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.

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.

Last modified on 22 July 2013, at 03:13