The approximation of indefinite Schur's functions

E. N. Andreishcheva

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2019), pp. 7-22

Résumé

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 $V$: $$ V=\begin{bmatrix} T\\ \\ [\cdot,v](0) \end{bmatrix}:\begin{pmatrix} \Pi_{\varkappa}\\ \\ \mathbb{C} \end{pmatrix}\rightarrow \begin{pmatrix} \Pi_{\varkappa}\\ \\ \mathbb{C} \end{pmatrix},$$ 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: $$ c_{\nu}= \left\{\begin{array}{l}\displaystyle{ \frac{1}{\overline{s_{k}(0)}}} \sum\limits_{i=1}^{\nu}C_{k-i}^{\nu-i}[(I-T)^{-(i +1)}T^{k+1}u,T^{k}u]-C_{k}^{\nu},~ 1\leqslant\nu k+1; \\ \displaystyle{\frac{1}{\overline{s_{k}(0)}}[(I-T)^{-(\nu+1)}T^{\nu}u,T^{k}u]},~~ k+1\leqslant\nu\leqslant n;\\ \displaystyle{\frac{1}{\overline{s_{k}(0)}}[(I-T)^{-(n+1)}T^{n}u,(I-T^{c})^{-(\nu-n)}T^{c(\nu-n)}T^{k}u]},\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n+1\leqslant\nu\leqslant 2n. \end{array}\right. $$

Détail

Keywords: Schur function, approximation, contraction, Pontryagin space, Cayley-Neumann transformation, indefinite metric, unitary realization, operator.
Mots-clés : kernel