The approximation of indefinite Schur’s functions.

In the paper by M.G.Krein and H.Langer [18] researched the questions about aproximations of Nevanlinna functions. Our purpose is to get such result for Schur functions. A function $s(\lambda)$ is called a generalized Schur function if it is meromorphic in the open unit disk and the kernel $K_s(\lambda, \mu)=\frac{1-s(\lambda)\overline{s(\mu)}}{1-\lambda\overline{\mu}}$ has finite number of negative squares.

A set of all such functions forms the generalized Schur class.

As it is known, Schur function admits a unitary realization $s(\lambda)=s(0)+\lambda[(I-\lambda T)^{-1}u,v]$ or, in other words, it is a characteristic function for some unitary colligation $V$ :

$\begin{bmatrix}
T & u \\
[\cdot,v] & s(0)
\end{bmatrix}
:
\begin{pmatrix}
\Pi_\varkappa \\
\mathbb{C}
\end{pmatrix}
\rightarrow
\begin{pmatrix}
\Pi_\varkappa \\
\mathbb{C}
\end{pmatrix}$

Here $\Pi_\varkappa$ is a Pontryagin space with indefinite inner product $[\cdot,\cdot]$, $T$ is a contractive operator in $\Pi_\varkappa$ and $u, v\in \Pi_\varkappa$. Note that the unitary colligation must be chosen minimal what means that $\Pi_\varkappa=\overline{span}\{T^{n}u, (T^c)^mv: n,m=0,1,2,\cdots\}$, where $T^c$ is $\pi_\varkappa$-adjoint with $T$. Let $T$ be a contractive operator in $\Pi_\varkappa$. Then the element $u \in \Pi_\varkappa$ is called generating for operator $T$ if

$\Pi_\varkappa=\overline{span}\{(I-\lambda T)^{-1}u, \lambda\in\mathbb{D}, \frac{1}{\lambda} \notin \sigma_p(T) \}$.

By $W_\theta$ we denote a set of all $\beta \in \mathbb{C}\_$ such that $|\arg\beta+\frac{\pi}{2}|\leqslant\theta$, where $0\leqslant\theta < \frac{\pi}{2}$.
By $\Lambda_\theta$ denote a set of all $\lambda \in \mathbb{D}$, where $\lambda\in\mathbb{D}$, where $\mathbb{D}={\xi:|\xi|<1}$ such that
$\lambda=(\alpha -i)(\alpha -i)^{-1}, -\alpha \in W_\theta$.

The main result of this research is researched the question of the representation generalized Schur
function in the neighborhood of the unit.

Let $s(\lambda)=\lambda^{k}s_k(\lambda), s_k(0)\neq 0, k\leqslant n$. Then we have assertions

1. $s \in S_\varkappa$, where $S_\varkappa$ is a generalized Schur class;

2. for some integer $n>0$ there exist $2n$ numbers $c_1,c_2,\cdots,c_{2n}$ such that the following
equality is true: $s(\lambda)=1-\sum_{\nu=1}^{2n} c_\nu(\lambda-1)^\nu+O((\lambda-1)^{2n+1}), \lambda \to 1, \lambda \in \Lambda_\theta$

if and only if there exist a Pontryagin space $\Pi_\varkappa$, a contractive operator $T \in \Pi_\varkappa$, and a generative
element $u \in dom(I-T)^{-(n+1)}$ for operator $T$ such that:

$s(\lambda)=\lambda^k-\frac{1}{\overline{s_k(0)}}\lambda^k(\lambda-1)[(I-\lambda T)^{-1}(I-T)^{-1}T^{k+1}u, T^ku], \lambda\in\mathbb{D}, \frac{1}{\lambda} \notin \sigma_p(T)$

In this case we can express $c_\nu$ in such form:

$$c_\nu=\begin{cases}
\frac{1}{\overline{s_k(0)}}\sum_{i=1}^{\nu} C^{\nu-i}_{k-i}[(I-T)^{-(n+1)}T^{k+1}u, T^ku]-C^\nu_k, & 1\leqslant\nu\,{<}\,k+1; \\
\frac{1}{\overline{s_k(0)}} [(I-T)^{-(\nu+1)}T^{\nu}u, T^ku], & k+1\leqslant\nu\leqslant n; \\
\frac{1}{\overline{s_k(0)}} [(I-T)^{-(n+1)}T^{n}u, (I-T^c)^{-(\nu-n)}T^{c(\nu-n)}T^ku], & n+1\leqslant\nu\leqslant 2n; \\
\end{cases}$$
Keywords: Schur function, approximation, contraction, kernel, Pontryagin space, Cayley-Neumann transformation, indefinite metric, unitary realization, operator.