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
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
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
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:
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.