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Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Definitions and Formulas for the Spectral Matrix Functions
Abstract
We first review the theory of canonical differential expressions in the rational case. Then, we define and study the discrete
analogue of canonical differential expressions. We focus on the rational case. Two kinds of discrete systems are to be distinguished:
one-sided and two-sided. In both cases the analogue of the potential is a sequence of numbers in the open unit disk (Schur
coefficients). We define the characteristic spectral functions of the discrete systems and provide exact realization formulas
for them when the Schur coefficients are of a special form called strictly pseudo-exponential.
... Recall that discrete and continuous systems with the potentials, which belong to the subclass of the strictly pseudo-exponential potentials, have been actively studied in [1]- [6], [8]- [10]. In particular, direct and inverse problems for Szegö recurrence on the semiaxis with the scalar (p = 1) strictly pseudo-exponential potentials have been treated in [5,6]. Direct and inverse problems for the pseudo-exponential potentials (continuous case) have been studied in a series of Gohberg-Kaashoek-Sakhnovich papers [21,22] (see references therein and see also [17] for the case of the generalized pseudo-exponential potentials). ...
... The case of the discrete skewself-adjoint Dirac system have been studied in [24]. Notice that similar to [21,31] (see also [5,6,8,21,24]) we start our explicit constructions with the explicit formula for the fundamental solution. ...
... Notice that our matrices C k belong to the class of the so called pseudoexponential potentials. An important subclass of the strictly pseudoexponential potentials, that is, a subclass with an additional requirement σ(A) ⊂ C − (σ -spectrum), have been treated for p = 1 in [5,6]. In particular, for the strictly pseudoexponential subcase the inequality |ϕ(λ)| < 1 for λ ∈ C − is true. ...
Discrete Dirac type self-adjoint system is equivalent to the block Szeg\"o
recurrence. Representation of the fundamental solution is obtained, inverse
problems on the interval and semi-axis are solved. A Borg-Marchenko type result
is obtained too. Connections with the block Toeplitz matrices are treated.
... Remark 2.3 When m 1 = m 2 = 1, one easily removes the factor (1 − |ρ k | 2 ) −1/2 in (2.12) to obtain systems as in [4,5], where direct and inverse problems for the case of scalar strictly pseudo-exponential potentials have been treated. The square matrix version (i.e., the version where m 1 = m 2 ) of Szegö recurrence, its connections with Schur coefficients and applications are discussed in [8,9] (see also references therein). ...
A transfer matrix function representation of the fundamental solution of the
general-type discrete Dirac system, corresponding to rectangular Schur
coefficients and Weyl functions, is obtained. Connections with Szeg\"o
recurrence, Schur coefficients and structured matrices are treated.
Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the
interval and semiaxis are solved.
In this paper we consider classical Schur transformation and inverse Schur transformation for generalized Nevanlinna functions.
This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions.
One can associate to a rational function which is moreover strictly positive on the unit circle two sequences of numbers in
the open unit disk, called the Szegö sequence and the Nehari sequence. In the scalar case, they coincide up to multiplication
by −1. We study the corresponding result in the matrix-valued case.
We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation
is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation
problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces
and linear relations in these spaces.
We study the inverse problems associated to the characteristic spectral functions of first-order discrete systems. We focus
on the case where the coefficients defining the discrete system are strictly pseudo-exponential. The arguments use methods
from system theory. An important role is played by the description of the unitary solutions of a related Nehari interpolation
problem and by Hankel operators with unimodular symbols. An application to inverse problems for Jacobi matrices is also given.
We here extend the well known Positive Real Lemma (also known as the
Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive
rational function, when non-minimal realizations are considered. We then
exploit this result to provide an easy construction procedure of all (not
necessarily minimal) state space realizations of generalized positive
functions. As a by-product, we partition all state space realizations into
subsets: Each is identified with a set of matrices satisfying the same Lyapunov
inclusion and thus form a convex invertible cone, cic in short. Moreover, this
approach enables us to characterize systems which may be brought to be
generalized positive through static output feedback. The formulation through
Lyapunov inclusions suggests the introduction of an equivalence class of
rational functions of various dimensions associated with the same system
matrix.
This paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform, discrete transmission-line from its response to a given forcing function. This problem is the prototype for a wealth of one-dimensional inverse scattering problems arising in various fields. A unified, straightforward approach is presented, providing all known inversion procedures and some novel ones. The derivations readily follow from causality of signal propagation on the transmission-line. It is shown that a direct exploitation of the signal propagation model leads to recursive, computationally efficient and reliable inversion algorithms. In particular, the relationships between layer peeling (Schur type, difference equation) and layer-adjoining (Levinson type, integral equation) algorithms are clearly displayed.
In this paper we obtain explicit formulas for the coefficients of a second order difference block operator if its spectral or its scattering functions are rational matrix functions analytic and invertible on the unit circle. The solutions are given in terms of realizations of the spectral or scattering function.
In this volume three important papers of M.G. Krein appear for the first time in English translation. Each of them is a short self-contained monograph, each a masterpiece of exposition. Although two of them were written more than twenty years ago, the passage of time has not decreased their value. They are as fresh and vital as if they had been written only yesterday. These papers contain a wealth of ideas, and will serve as a source of stimulation and inspiration for experts and beginners alike. The first paper is dedicated to the theory of canonical linear differential equations, with periodic coefficients. It focuses on the study of linear Hamiltonian systems with bounded solutions which stay bounded under small perturbations of the system. The paper uses methods from operator theory in finite and infinite dimensional spaces and complex analysis. For an account of more recent literature which was generated by this paper see AMS Translations (2), Volume 93, 1970, pages 103-176 and Integral Equations and Operator Theory, Volume 5, Number 5, 1982, pages 718-757.
The paper contains characterizations of all rational matrix functions W(λ) which are contractions for every real λ and strict contractions at infinity. The results are obtained in terms of minimal realizations. A description of all minimal, unitary on the line completions of such functions is given. This description is used in the study of linear fractional decompositions of proper contractions W(λ).
This paper contains the theory of realization and minimal factorization of rational matrix valued functions unitary on the unit circle or on the imaginary line (in the framework of a, generally speaking, indefinite scalar product). The Blaschke-Potapov decomposition for unitary or J-inner rational matrix-valued functions is obtained as a corollary, and the inertia theorems are explained from the point of view of the theory of rational matrix-valued functions. The main results are also used to obtain decomposition theorems for selfadjoint matrix-valued functions.
1 Schur Parameters and Positive Block Matrices.- 1.1 Preliminaries.- 1.2 Renorming Hilbert Spaces and Elementary Rotations.- 1.3 Kolmogorov Decompositions. I.- 1.4 Row and Column Contractions.- 1.5 The Structure of Positive Definite Kernels.- 1.6 Kolmogorov Decompositions. II.- 1.7 Notes.- 2 Models for Triangular Contractions.- 2.1 Preliminaries.- 2.2 The Structure of Triangular Contractions.- 2.3 Realization of Triangular Contractions.- 2.4 Unitary Couplings and Operator Ranges.- 2.5 Modeling Families of Contractions.- 2.6 Notes.- 3 Moment Problems and Interpolation.- 3.1 A Survey on Completion Problems.- 3.2 Extensions of Partial Isometries.- 3.3 Krein's Formula.- 3.4 Moment Problems.- 3.5 The Commutant Lifting Method.- 3.6 Notes.- 4 Displacement Structures.- 4.1 Structured Matrices.- 4.2 Generalized Schur Algorithm.- 4.3 Discrete Transmission-Line Models.- 4.4 Displacement Structure and Completion Problems.- 4.5 Other Applications.- 4.6 Notes.- 5 Factorization of Positive Definite Kernels.- 5.1 Spectral Factors.- 5.2 Examples.- 5.3 Schur's Algorithm, Szego's Theory and Spectral Factors.- 5.4 Maximum Entropy.- 5.5 Notes.- 6 Nonstationary Processes.- 6.1 Modeling Nonstationary Processes.- 6.2 Kolmogorov-Wiener Prediction.- 6.3 Other Prediction Problems.- 6.4 Szego's Limit Theorems.- 6.5 Notes.- 7 Graphs and Completion Problems.- 7.1 Preliminaries.- 7.2 Completing Positive Partial Matrices. I.- 7.3 Completing Positive Partial Matrices. II.- 7.4 Completing Contractive Partial Matrices.- 7.5 Notes.- 8 Determinantal Formulae and Optimization.- 8.1 Determinantal Formulae.- 8.2 Maximum Determinant Formulae.- 8.3 Maximum Determinant for Nonchordal Graphs.- 8.4 Inheritance Principles.- 8.5 Notes.- References.
Semi-infinite block Toeplitz operators with rational matrix symbols are inverted explicitly by using the factorization method and by a method of reduction to singular systems with boundary conditions. The case of finite block Toeplitz matrices is also included. Fredholm characteristics are computed. All formulas are explicit and based on a realization of the symbol as a transfer function of a singular system.
The intimate connections between a form of the covariance extension problem and the spectral theory of canonical equations are developed and exploited in order to deduce a representation formula for the set of all solutions to the extension problem in an intuitively pleasing way. Enroute, an expository account of the forward and inverse spectral problem for canonical equations and the theory of Hilbert spaces of matrix valued entire functions is given and a number of related applications are discussed.
The string equation can be written in the form (0.1) where The equation (0.2) is said to be dual to equation (0.1). The notion of a dual string was investigated by I.S. Kac and M.G. Krein [1]. Kac and Krein obtained the dual string equation from the original by interchanging the variables x and M(x). Let us add conditions (0.3)
(0.4) to equations (0.1) and (0.2).
We obtain explicit formula for the reflexivity coefficient function (or potential) of an ordinary differential operator if its reflection coefficient is a rational matrix valued function. The solution is given in terms of a realization of the reflection coefficient function.
We solve the inverse problems associated with a differential expression of the form (1.1) for the Weyl coefficient function and the reflection coefficient function in the case of potentials k(t) of the special form (1.2).
A differential expression of the type (Dx)(r)=W * (r)J(d/dr)(W(r)x(r))(r∈[0,∞)); where J is (2n×2n)-matrix with the properties J * =-J, J 2 =-I 2n and W(r) is a continuous (2n×2n)-matrix function with J-unitary values (W * (r)JW(r)=J ∀r∈[0,∞)) is considered. An expression of this type generalizes the canonical differential expression (D v x)(r)=J(dx(r)/dr)-V(r)x(r) with Hermitian potential V(r)V * (r)=V(r). For the special class of equations (Dx)(r,λ)=λx(r,λ) which include equations (D v x)(r,λ)=λx(r,λ) with summable potential V(r) (V∈L 2n×2n 1 (0,∞)), the problems of scattering and spectral theory are solved.
A number of results in the theory of inverse problems for polynomials on the line and on the circle will be reviewed. Analogues of the methods of Gelfand-Levitan, Krein, and Marchenko will be presented with special emphasis on the role played by factorization and Toeplitz operators.
For rational and analytic matrix functions new formulas are obtained for the limits in the Szegö-Kac-Achiezer limit theorems. In the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions. These representations are known as realizations in mathematical systems theory.
The main theme of the first half of this paper rests upon the fact that there is a reproducing kernel Hilbert space of vector valued functions B (X) associated with each suitably restricted matrix valued analytic function X. The deep structural properties of certain classes of these spaces, and the theory of isometric and contractive inclusion of pairs of such spaces, which originates with de Branges, partially in collaboration with Rovnyak, is utilized to develop an algorithm for constructing a nested sequence B(X) ⊃ B(X
1) ⊃... of such spaces, each of which is included isometrically in its predecessor. This leads to a new and pleasing viewpoint of the Schur algorithm and various matrix generalizations thereof. The same methods are used to reinterpret the factorization of rational J inner matrices and a number of related issues, from the point of view of isometric inclusion of certain associated sequences of reproducing kernel Hilbert spaces.
We prove a trace formula for pairs of self-adjoint operators associated to canonical differential expressions. An important role is played by the associated Weyl function.
The Factorization of Rational Matrix Functions.- Decomposing Algebras of Matrix Functions.- Canonical Factorizations of Continuous Matrix Functions.- Factorization of Triangular Matrix Functions.- Factorization of Continuous Self-Adjoint Matrix Functions on the Unit Circle.- Miscellaneous Results on Factorization Relative to a Contour.- Generalized Factorization.- Further Results Concerning Generalized Factorization.- Local Principles in the Theory of Factorization.- Perturbations and Stability.
Explicit formulas are given for the solutions of the direct and inverse scattering problems for a canonical differential system with a strictly pseudo–exponential potential. The proofs are self–contained and employ state space techniques from mathematical system theory. The paper supplements an earlier paper of the first two authors where explicit formulas were given using Marchenko's approach, and an earlier paper of the last three authors where self–contained proofs were given for the corresponding direct and inverse spectral problems. Two types of factorizations of the scattering matrix function appear and connections between them are considered.
This paper solves explicitly the direct spectral problem of canonical differential systems for a special class of potentials. For a potential from this class the corresponding spectral function may have jumps and its absolutely continuous part has a rational derivative possibly with zeros on the real line. A direct and self-contained proof of the diagonalization of the associated differential operator is given, including explicit formulas for the diagonalizing operator and the spectral function. This proof also yields an explicit formula for the solution of the inverse problem. As an application new representations are derived for a large class of solutions of nonlinear integrable partial differential equations. The method employed is based on state space techniques and uses the idea of realization from mathematical system theory.
We study the connections between the Carathodory-Toeplitz extension problem and the Nehari extension problem in the discrete scalar case. This is the discrete counterpart of our previous paper [4].
The Schur algorithm and its time-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Chokesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.
This paper develops a method for obtaining linear fractional representations of a givennn matrix valued function which is analytic and contractive in either the unit disc or the open upper half plane. The method depends upon the theory of reproducing kernel Hilbert spaces of vector valued functions developed by de Branges. A self-contained account of the relevant aspects of these spaces to this study is included. In addition, the methods alluded to above are used in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric] operators with equal deficiency indices. A number of related issues and applications are also discussed.
This paper developes further the connections between linear systems and convolution equations. Here the emphasis is on equations on finite intervals. For these equations a new characteristic matrix (or operator) function is introduced, which contains all the important information about the equations and the corresponding operators. Explicit formulas for solutions and resolvent kernels are obtained. Convolution equations on the full line are also analyzed. Analogous results are derived for the inversion of finite and infinite Toeplitz matrices.
We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix
form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations
on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory
of one-dimensional Schrödinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson
theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated
by the Szegő equations with reflection coefficients having bounded variation.
The paper extends earlier results of the authors for canonical systems with spectral functions of which the absolutely continuous part has a rational derivative to a class of differential systems with skew selfadjoint potentials. The corresponding direct and inverse spectral problems are solved explicitly, using state space methods from mathematical system theory. Applications to nonlinear integrable partial differential equations are included.
We show that the multitude of applications of the Weyl–Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.
The main result is that forevery J-unitary 2×2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2×2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1–7] and studied in [Pisot and Salem Numbers, Birkhäuser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1–54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhäuser Verlag, Basel, 2000, p. 1; Monatshefte für Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions.
In this paper we study the inverse scattering problem for linear canonical differential equations of a special type in the case when the scattering function is a rational matrix—valued function. The main result here is the form of the potential in terms of a realisation of the scattering function. The difference between this publication and the previous one [3] consists in the fact that here is used the Marchenko method instead of Kreĭn’s method. For rational scattering functions we also prove the equivalence of the two methods.
"Lectures form the CBMS regional conference held at Case Western University, September 10-14, 1984." Incluye bibliografía
We present a novel technique to calculate the transmission and
reflection properties of inhomogeneous lossy transmission lines. It is
based on a perturbation expansion of the relevant telegrapher's equation
around the homogeneous solution. The method reveals that the
transmission properties are unaffected to first order in the impedance
and admittance, while reflection properties are modified by first-order
terms. Extending the analysis to second order, we find correction terms
for both transmission and reflection. These allow us to understand the
statistical properties of the information bearing capacity and the echo
of the inhomogeneous transmission line. The correction to the
transmission properties is fully reciprocal, despite the absence of any
symmetry properties. We compute scaling laws for echo and capacity
Schur parameters, factorization and dilation problems, volume 82 of Operator Theory: Advances and Applications
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Invariant subspaces of matrices with applications. Canadian Mathematical Society Series of Monographs and Advanced Texts
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On the spectral theory of a class of canonical systems of differential equations
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Toeplitz operators with rational symbols and realizations: an alternative version
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Über die Potenzreihen, die im Innern des Einheitkreises beschränkten sind, I. Journal für die Reine und Angewandte Mathematik English translation in: I. Schur methods in operator theory and signal processing. (Operator theory
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On a class of canonical differential operators. Izvestya Akademii Nauk English translation inMATHMathSciNet
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