Schur Transformation for the Generalized Caratheodory Function on Circle.

In the present paper we consider essentially Caratheodory class of scalar functions. This class consists of the meromorphic functions $f(z)$ on the open unit disc $\mathbb{D}$ for which the kernel
$$K_f (z, \omega) = \frac{f(z) + f(\omega)^∗}{1 − z\omega^∗} , z, \omega ∈ \textbf{hol(f)}$$
has a finite number $\varkappa$ of negative squares(here $\textbf{hol(f)}$ is the domain of homomorphy of $f(z)$). This is equivalent to the fact that the function $f(z)$ has $\varkappa$ poles in $\mathbb{D}$ but the metric constraint of being not expansive on the unit circle $\mathbb{T}$ . We call these functions $f(z)$ generalized Caratheodory functions with $\varkappa$ negative squares.
The approach to the Schur transformation in the indefinite case in the given paper is based on the theory of reproducing kernel Pontryagin spaces for the scalar and matrix functions, associated with a Caratheodory function $f(z)$ and a $2 \times 2$ matrix function $Θ(z)$ through the reproducing kernels
$$K_f (z, ω) = \frac{f(z) + f(\omega)^∗}{1 − z\omega^∗}, K_Θ(z, \omega) = \frac{J_f − Θ(z)J_f Θ(\omega)^∗}{ 1 − zω^∗}, J_f = \begin{pmatrix} 0 & −1 \\ −1 & 0 \end{pmatrix}.$$
In positive case, they have been first introduced by L. de Branges and J. Rovnyak in [9] and [10]. They play an important role in operator models and interpolation theory, see, for instance,[3] and [4]. In the indefinite case equivalent spaces were introduced in the papers by M.G. Krein
and H. Langer [16],[17] and [18] mentioned earlier.
The transformation $s(z) \mapsto \hat{s}(z)$ was defined and studied by I.Schur in 1917 – 1918 in the paper [21] and is called the Schur transformation. The starting point is a function $s(z)$ which is analytic and contractive in the open unit disk $\mathbb{D}$ ; we call such functions Schur functions. The Schur transformation maps the set of Schur functions which are not identically equal to a unimodular constant into the set of Schur functions. In this way, I. Schur associated with a Schur function $s(z)$ a finite or infinite sequence of numbers, called Schur coefficients, via the formulas
$$s_0(z) = s(z), s_{j+1}(z) = \hat{s}_j(z) = \frac{1}{b(z)} \frac{s_j(z) − s_j(z_1)}{1 − s_j(z)s_j(z_1)^∗} , b(z) = \frac{z − z_1}{1 − zz^*_1} , z_1 ∈ \mathbb{D} , j = 0, 1, ... $$
This recursion is called the Schur algorithm.
In given paper we consider this transformation to an indefinite setting for generalized Caratheodory functions centered at $z_1 ∈ \mathbb{T}$ . We called this transformation the generalized Schur transformation for Caratheodory functions.
The generalized Schur transformation can be written as linear fractional transform $\hat{f}(z) = χ_{\Theta^{−1}} (f(z))$, for any matrix polynomial $\Theta(z)$ and generalized Caratheodory function $f(z)$. If $z_1 ∈ \mathbb{T}$ we consider functions $f(z)$ which have an asymptotic expansion of the form
$$f(z) = \tau_{0} + \sum_{i=1}^{2p−1} \tau_i(z − z_1)^i + O((z − z_1)^{2p} ), z → z_1.$$
We define the vector function $R(z)$, fix some normalization point $z_0 ∈ \mathbb{T} , z_0 \not= z_1$, and introduce the polynomial $p(z)$. Then the Schur transformation $\hat{f}(z)$ for generalized Caratheodory function $f(z)$ is defined by the formula
$$\hat{f}(z) = \chi_{\Theta(z)^{−1}}(f(z)) = \frac{\{(1 − zz^∗_1 )^k − \tau_0(1 − zz^∗_0 )p(z)\}f(z) − {\tau}_0\tau^∗_0 (1 − zz^∗_0 )p(z)} {−(1 − zz^∗_0 )p(z)f(z) + \{(1 − zz^∗_1)^k − \tau^∗_0 (1 − zz^∗_0 )p(z)\}}.$$
In this paper we also consider the basic boundary interpolation problem for generalized
Caratheodory functions and factorization of the rational matrix functions which are $J$-unitary
on $\mathbb{T} \backslash\{z_1\}$ and have a unique pole in $z_1$.
Keywords: indefinite metrics, Pontryagin space, Schur transformation, generalised Carateodori
function, interpolation problem, factorization of rational matrix function