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System theory for lossless wave scattering. (English)
Operator theory and systems, Proc. Workshop, Amsterdam 1985, Oper. Theory, Adv. Appl. 19, 333-358 (1986).
[For the entire collection see Zbl 0594.00010.] The first part of this work deals with a specific class of discrete-time systems, called Lossless Wave Scattering Systems (LWSSs). These are systems whose inputs and outputs (which may be vector valued) are arranged as shown below, and the $L\sp 2$-norm of the input equals, by definition, the $L\sp 2$-norm of the output. Such systems arise in a number of engineering and mathematical applications such as, circuit/filter synthesis, modeling physical scattering media, modeling the second order properties of stationary stochastic sequences, Nevanlinna-Pick interpolation problems, and linear fractional transformations. It is a well known fact that such systems may be described by a transfer function, which mathematically is a J-inner matrix function, with J being a diagonal matrix containing $\pm 1’s$. Using the Nerode definition for the state and nullspaces of linear time-invariant systems, we deduce formulae for the state and nullspaces of a LWSS described by a J-inner function. These formulae are expressed in terms of the so called transmission zeros and the scattering function associated with the LWSS and clearly exhibit the N-P interpolation nature of LWSSs. The nullspaces are also characterized as shift-invariant subspaces in the indefinite inner product (Krein) spaces of vector valued analytic functions, while the state spaces are characterized as reproducing kernel de Branges spaces of analytic functions. The second part of the work deals with application of the above derived formulae to solve a problem known as Lossless Inverse Scattering Problem (LIS). In this problem, we are given scattering data, such as the values of a scattering function (a Schur function in mathematical language), at a number of complex points called interpolation points and we are required to find a LWSS whose intrinsic scattering function matches (interpolates) the given data. Such a problem arises in each of the contexts mentioned in the first paragraph. We solve this problem using state space concepts. We first construct a reproducing kernel Krein space using the data, and show this to be a state space of a LWSS. The LWSS itself is determined following a system realization approach, or a reproducing kernel approach or a recursive approach. What is not treated in this work is the case when the interpolation points are on the unit circle of the complex plane. The $L\sp 2$-theory as used here needs to be generalized.