# zbMATH — the first resource for mathematics

Brune sections in the non-stationary case. (English) Zbl 0998.93019
The classical inverse scattering problem consists in finding all representations of a given Schur-class function $$s$$ in the form $$s=T_{\Theta}(\sigma)=(\theta_{11}\sigma + \theta_{12})(\theta_{21}\sigma +\theta_{22})^{-1}$$ where $$\sigma$$ is also a Schur-class function, and $$\Theta=\left( \begin{smallmatrix} \theta_{11} & \theta_{12} \cr \theta_{21} & \theta_{22}\end{smallmatrix}\right)$$ is a meromorphic function in $${\mathbf D}$$ that is $$J$$-inner, i.e., $$\Theta(z)J\Theta(z)^*\leq J$$ at all points of holomorphy in $${\mathbf D}$$, and $$\Theta(z)J\Theta(z)^*= J$$ a.e. on $${\mathbf T}$$, with $$J=\left(\begin{smallmatrix} 1 & 0 \cr 0 & -1 \end{smallmatrix}\right)$$. The elementary such $$\Theta$$’s are of the form $$\Theta(z)=\left(\begin{smallmatrix} 1 & 0 \cr 0 & 1\end{smallmatrix}\right)-(1-z)/[\varepsilon(1-za^*)(1-a)]\left(\begin{smallmatrix} 1 & -k \cr k^* & -1\end{smallmatrix}\right)$$. For the Schur sections (called also Blaschke factors) one has $$a\in {\mathbf D}, k\in {\mathbf D}, \varepsilon=(1-|k|^2)/(1-|a|^2)$$, while in the case of Brune sections, $$a$$ and $$k$$ are on $${\mathbf T}$$ (with $$a\neq 1$$), and $$\varepsilon$$ is a strictly positive number. The inverse scattering problem is closely related to the theory of linear time-invariant dissipative systems, and it has numerous generalizations and ramifications. In the time-varying setting, Schur-class functions are replaced by upper triangular doubly infinite contractive matrices. The analogue of the Blaschke factor is known [see, e.g., D. Alpay, P. Dewilde and H. Dym, Oper. Theory, Adv. Appl. 47, 61-135 (1990; Zbl 0727.47005)]. Here, the authors introduce the time-varying counterpart of Brune sections. However, on the way some unbounded operators appear, and to remedy this dificulty, they use the so-called Zadeh extension. The theory results in a factorization theorem that generalizes the factorization of $$J$$-inner functions to the time-varying setting in the case of finitely specified systems.
##### MSC:
 93C05 Linear systems in control theory 30G30 Other generalizations of analytic functions (including abstract-valued functions) 15A06 Linear equations (linear algebraic aspects) 30E05 Moment problems and interpolation problems in the complex plane 47A57 Linear operator methods in interpolation, moment and extension problems 47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
Full Text:
##### References:
 [1] D. Alpay, Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes, Panoramas et Synthèses, vol. 6, Société Mathématique de France, Paris, 1998 [2] Alpay, D.; Bolotnikov, V.; Dewilde, P.; Dijksma, A., Sections de brune en théorie des systèmes non-stationnaires, C.R. acad. sci. Paris Sér. I. math., 330, 173-178, (2000) · Zbl 0952.93018 [3] D. Alpay, P. Dewilde, Time-varying signal approximation and estimation, in: M. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Signal Processing, Scattering and Operator Theory, and Numerical Methods, Amsterdam, 1989; Progress in Systems and Control Theory, vol. 5, Birkhäuser, Boston, MA, 1990, pp. 1-22 · Zbl 0722.93045 [4] D. Alpay, P. Dewilde, H. Dym, Lossless inverse scattering and reproducing kernels for upper triangular operators, in: Extension and Interpolation of Linear Operators and Matrix Functions, Birkhäuser, Basel, 1990, pp. 61-135 · Zbl 0727.47005 [5] D. Alpay, I. Gohberg, Unitary rational matrix functions, Operator Theory, Advances and Applications, vol. 33, Birkhäuser, Basel, 1988, pp. 175-222 [6] Belevitch, V., Classical network theory, (1968), Holden-Day San Francisco, CA · Zbl 0172.20404 [7] Dewilde, P.; Dym, H., Lossless inverse scattering, digital filters and estimation theory, IEEE trans. inform. theory, 30, 644-662, (1984) · Zbl 0568.94028 [8] P. Dewilde, H. Dym, Interpolation for upper triangular operators, in: I. Gohberg (Ed.), Time-variant Systems and Interpolation, Operator Theory, Advances and Applications, vol. 56, Birkhäuser, Basel, 1992, pp. 153-260 · Zbl 0788.47009 [9] Dewilde, P.; vanderVeen, Alle-Jan, Time-varying systems and computations, (1998), Kluwer Academic Publishers Boston, MA [10] J. Dieudonné, Eléments d’analyse, Tome 2: Chapitres XII à XV, Bordas, Paris, 1982 [11] H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989 · Zbl 0691.46013 [12] H. Dym, B. Freydin, Bitangential interpolation for triangular operators when the Pick operator is strictly positive, in: H. Dym, B. Fritzsche, V. Katsnelson, B. Kirstein (Eds.), Topics in Interpolation Theory, Operator Theory Advances and Applications, vol. 95, Birkhäuser, Basel, 1997, pp. 143-164 · Zbl 0951.47015 [13] H. Dym, B. Freydin, Bitangential interpolation for upper triangular operators, in: H. Dym, B. Fritzsche, V. Katsnelson, B. Kirstein (Eds.), Topics in Interpolation Theory, Operator Theory Advances and Applications, vol. 95, Birkhäuser, Basel, 1997, pp. 105-142 · Zbl 0951.47014 [14] Kimura, H., Directional interpolation approach to H∞ optimal control and robust stabilization, IEEE trans. automat. control, 32, 1085-1093, (1987) · Zbl 0639.93019 [15] Kovalishina, I.V., The Carathéodory Julia theorem for matrix-functions, Teor. funktsi$$ı̆$$ funktsional. anal. i prilozhen., 43, 70-82, (1985) · Zbl 0581.30042 [16] I.V. Kovalishina, Theory of a multiple j-elementary matrix-function with a pole on the boundary of the unit disk, Teor. Funktsi$$ı̆$$ Funktsional. Anal. i Prilozhen. 50(iii) (1988) pp. 62-74 · Zbl 0685.30014 [17] Kovalishina, I.V., A multiple boundary value interpolation problem for contracting matrix functions in the unit disk, Teor. funktsi$$ı̆$$ funktsional. anal. i prilozhen., 51, 38-55, (1989) · Zbl 0702.41006 [18] I. Schur, Über die Potenzreihen, die im Innern des Einheitkreises beschränkten sind, I, J. Reine Angew. Math. 147 (1917) 205-232 [English translation in: I. Schur, Methods in Operator Theory and Signal Processing, Operator Theory Advances and Applications OT, vol. 18, Birkhäuser, Basel, 1986]
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.