On 2D Inverse Problems/Stieltjes continued fractions

Let $\{a_k\}$ be a set of n positive numbers. The Stieltjes continued fraction is an expression of the form, see also [19],

$\beta_a(z) = a_nz + \cfrac{1}{a_{n-1}z + \cfrac{1}{ \ddots + \cfrac{1}{a_1 z} }}$

or its reciprocal

$\beta^*_a(z) = \frac{1}{\beta_a}.$

The function defines a rational n-to-1 map of the right half of the complex plane onto itself,

$\beta_a,1/\beta_a:\mathbb{C^+}\xrightarrow[]{n\leftrightarrow 1}\mathbb{C^+},$

since

$\begin{cases} Re(z_1), Re(z_2) > 0 \implies Re(z_1+z_2) > 0, \\ Re(z) \implies Re(1/z) > 0, \\ Re(z) > 0, a > 0 \implies Re(az) > 0. \end{cases}$

Exercise(***). Use the mapping properties of a Stieltjes continued fractions to prove that it's a rational functions w/a zero or a pole at the origin w/simple, symmetric, interlacing zeros and poles lying on the imaginary axes and that the above properties characterize the continued fractions.

Exercise(**). Prove that the continued fractions have the following representation:

$\beta_a(z) = z(\xi_{\infty} + \sum_k\frac{\xi_k}{z^2+\theta_k^2}),$

where

$\xi_{\infty}, \xi_k \mbox{ and } \theta_k, k\in\mathbb{N}$

are non-negative real numbers and the converse of the statement is true.

The function $\beta_a$ is determined by the pre-image set $\Mu=\{\mu_k\}$ (of size n, counting multiplicities) of the point {z = 1}, since

$\beta_a(z) = \frac{p(z^2)}{zq(z^2)}=1 \iff p(z^2)-zq(z^2) = 0,$

and a complex polynomial is determined by its roots up to a multiplicative constant by the fundamental theorem of algebra.

Let $\sigma_l$ be the elementary symmetric functions of the set $\Mu$. That is,

$\prod_k (z-\mu_k) = \sum_k \sigma_{n-k} z^k.$

Then, the coefficients $a_k$ of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following n by n square Hurwitz matrix:

$H_\Mu = \begin{pmatrix} \sigma_1 & \sigma_3 & \sigma_5 & \sigma_7 & \ldots & 0\\ 1 & \sigma_2 & \sigma_4 & \sigma_6& \ldots & 0\\ 0 & \sigma_1 & \sigma_3 & \sigma_5& \ldots & 0\\ 0 & 1 & \sigma_2 & \sigma_4& \ldots & 0\\ 0 & 0 & \sigma_1 & \sigma_3& \ldots & 0\\ \vdots & \vdots & \vdots & \vdots& \ddots& \vdots\\ 0 & 0 & 0 & 0& \ldots& \sigma_n\\ \end{pmatrix}$

and, therefore, can be expressed as the ratios of monomials of the determinants of the blocks of $\Mu$.

Exercise (**). Prove that

$a_1 = 1/ \sigma_1, a_2 = \frac{\sigma_1^2}{\det\begin{pmatrix}\sigma_1 & \sigma_3 \\ 1 & \sigma_2 \end{pmatrix}}, a_3 = \frac{\det \begin{pmatrix}\sigma_1 & \sigma_3 \\ 1 & \sigma_2 \end{pmatrix}^2}{\sigma_1\det \begin{pmatrix}\sigma_1 & \sigma_3 & 0 \\ 1 & \sigma_2 & \sigma_4 \\ 0 & \sigma_1 & \sigma_3 \end{pmatrix}}, \ldots$

Exercise (*). Use the previous exercise to prove that

$\prod_k a_k = \frac{1}{\prod_k \mu_k} = 1/\sigma_n.$

Exercise (**). Let A be a diagonal matrix with the alternating in sign diagonal entries:

$A = \begin{pmatrix} \pm a_1 & 0 & 0 & \ldots & 0 \\ 0 & \mp a_2 & 0 & \ldots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & -a_{n-1} & 0 \\ 0 & 0 & \ldots & 0 & a_n \\ \end{pmatrix}$

and D the (0,1)-matrix

$D = \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \\ 0 & 1 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 0 & 1 \\ 0 & 0 & \ldots & 1 & 1 \\ \end{pmatrix} .$

Prove that

$\beta_a(\mu) = 1 \iff \mu \in \sigma(A^{-1}D) = \sigma(DA^{-1}).$

That is

$const(p(z^2)-zq(z^2)) = \rho_{A^{-1}D}(z)=\rho_{DA^{-1}}(z).$

Exercise (*). Find the constant in the equation above.