# The approximation of indefinite Schur’s functions.

In the paper by M.G.Krein and H.Langer  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.

UDC:
517.58