On 2D Inverse Problems/Stieltjes continued fractions

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

\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.