The approximation of indefinite Schur's functions
E. N. Andreishcheva Black Sea Higher Naval School
Abstract:
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 $\displaystyle{K_{s}(\lambda,\mu)=\frac{1-s(\lambda)\overline{s(\mu)}}{1-\lambda\overline{\mu}}}$ 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
$V$:
$$ V=\begin{bmatrix}
T&u\\
\\
[\cdot,v]&s(0)
\end{bmatrix}:\begin{pmatrix}
\Pi_{\varkappa}\\
\\
\mathbb{C}
\end{pmatrix}\rightarrow
\begin{pmatrix}
\Pi_{\varkappa}\\
\\
\mathbb{C}
\end{pmatrix},$$
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
$\Pi_{\varkappa}=\overline{span}
\{T^{n}u,(T^{c})^{m}v:n,m=0,1,2,\ldots\},$ 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
$$ \Pi_{\varkappa}=\overline{span}
\displaystyle{\{(I-\lambda T)^{-1}u,~~\lambda\in
\mathbb{D},~~\frac{1}{\lambda}\notin\sigma_{p}(T)\}}.$$
By
$W_{\theta}$ we denote a set of all
$\beta\in
\mathbb{C_{-}}$ such that
$\displaystyle{|\arg\beta+\frac{\pi}{2}|\leqslant\theta},$ where $\displaystyle{0\leqslant\theta<\frac{\pi}{2}}.$
By
$\Lambda_{\theta}$ denote a set of all
$\lambda\in\mathbb{D}$,
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
- $s\in S_{\varkappa}$, where $S_{\varkappa}$ is a generalized Schur class;
- for some integer $n > 0$ there exist $2n$ numbers $c_{1},c_{2},\ldots,c_{2n}$ such that
the following equality is true:
$ \displaystyle{s(\lambda)=1-\sum_{\nu=1}^{2n}{c_{\nu}(\lambda-1)^{\nu}}+
O((\lambda-1)^{2n+1}),~~\lambda\rightarrow1,~\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:
$$ s(\lambda)=\lambda^{k}-
\frac{1}{\overline{s_{k}(0)}}\lambda^{k}(\lambda-1)[(I-\lambda
T)^{-1}(I-T)^{-1}T^{k+1}u,T^{k}u],~\\ \lambda\in \mathbb{D},~
\frac{1}{\lambda}\notin\sigma_{p}(T)$$
In this case we can express
$c_{\nu}$ in such form:
$$
c_{\nu}= \left\{
\begin{array}{l}\displaystyle{
\frac{1}{\overline{s_{k}(0)}}}
\sum\limits_{i=1}^{\nu}C_{k-i}^{\nu-i}[(I-T)^{-(i
+1)}T^{k+1}u,T^{k}u]-C_{k}^{\nu},~ 1\leqslant\nu< k+1; \\
\displaystyle{\frac{1}{\overline{s_{k}(0)}}[(I-T)^{-(\nu+1)}T^{\nu}u,T^{k}u]},~~
k+1\leqslant\nu\leqslant
n;\\
\displaystyle{\frac{1}{\overline{s_{k}(0)}}[(I-T)^{-(n+1)}T^{n}u,(I-T^{c})^{-(\nu-n)}T^{c(\nu-n)}T^{k}u]},\\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n+1\leqslant\nu\leqslant
2n.
\end{array}
\right.
$$
Keywords:
Schur function, approximation, contraction, kernel, Pontryagin space, Cayley-Neumann transformation, indefinite metric, unitary
realization, operator.
UDC:
517.58
MSC: 47A58