Abstract
The general formulas developed in the fourth paper in this series are applied to solve the inverse input scattering problem for canonical integral systems in the special cases that the input scattering matrix is ap×q matrix valued function in the Wiener class (and the associated pairs are homogeneous). These formulas are then further specialized to the rational case. Whenp=q, these formulas are connected to the earlier results of Alpay-Gohberg and Gohberg-Kaashoek-Sakhnovich, who studied inverse problems for a related system of differential equations.
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References
T. Aktosun, M. Klaus and C. van der Mee, Direct and inverse scattering for selfadjoint Hamiltonian systems on the line,Integral Equations Operator Theory, to appear.
D. Alpay and I. Gohberg, Inverse spectral problems for differential operators with rational scattering matrix functions,J. Differential Equations 118 (1995), 1–19.
D. Alpay and I. Gohberg, Inverse problems associated to a canonical differential system, preprint.
D. Z. Arov, Darlington realization of matrix valued functions,Math. USSR Izvestija 7 (1973), 1295–1326.
D. Z. Arov and H. Dym,J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations,Integral Equations Operator Theory 29 (1997), No. 4, 373–454.
D. Z. Arov and H. Dym,J-inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem,Integral Equations Operator Theory 36 (2000), No. 1, 11–70.
D. Z. Arov and H. Dym,J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem,Integral Equations Operator Theory 36 (2000), No. 2, 127–181.
D. Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problems, submitted.
H. Bart, I. Gohberg and M. A. Kaashoek, Convolution equations and linear systems,Integral Equations Operator Theory 5 (1982), 283–340.
R. Bellman, Functional equations in the theory of dynamic programming — VII. A partial differential equation for the Fredholm resolvent,Proc. Am. Math. Soc. 8 (1957), 435–440.
R. Bhatia,Matrix Analysis, Springer, New York, 1997.
P. Dewilde and H. Dym, Lossless chain scattering matrices and optimum linear prediction: the vector case,Internat. J. Circuit Theory Appl. 9 (1981), No. 2, 135–175.
H. Dym,J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series, number 71, Amer. Math. Soc., Providence, R.I., 1989.
H. Dym and I. Gohberg, On an extension problem, generalized Fourier analysis and an entropy formula,Integral Equations Operator Theory 3 (1980), 143–215.
H. Dym and A. Iacob, Positive definite extensions, canonical equations and inverse problems, in: Topics in Operator Theory, Systems and Networks (H. Dym and I. Gohberg, eds.),Oper. Theory Adv. Appl. 12, Birkhäuser, Basel, 1984, pp. 141–240.
F. Gesztesy, A. Kiselev and K. A. Makarov, Uniqueness results for matrix valued Schrödinger, Jacobi and Dirac-type operators,Math. Nachr., to appear.
I. Gohberg and M. G. Krein, Theory and applications of Volterra operators in Hilbert space,Trans. Math. Monographs 24 Amer. Math. Soc., Providence, R.I. (1970).
I. Gohberg and I. Koltracht, Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,Numer. Math. 47 (1985), 237–288.
I. Gohberg, M. A. Kaashoek and A. L. Sakhnovich, Canonical systems with rational spectral densities: Explicit formulas and applications,Math. Nachr. 149 (1998), 93–125.
I. Gohberg and S. Rubinstein, Proper contractions and their unitary minimal completions, in: Topics in Interpolation Theory of Rational Matrix Valued Functions, (I. Gohberg, ed.),Oper. Theory Adv. Appl. 33 Birkhäuser, Basel, (1988), pp. 223–247.
M. G. Krein, On integral equations leading to a second order differential equation,Dokl. Akad. Nauk SSSR 97 (1954), 21–24 (in Russian).
M. G. Krein, On a new method of solution of linear integral equations of the first and second kind,Dokl. Akad. Nauk SSSR 100 (1954), 413–416 (in Russian).
M. G. Krein, Continuous analogs of theorems on polynomials orthogonal on the unit circle,Dokl. Akad. Nauk SSSR 105 (1955), 433–436.
M. G. Krein and H. Langer, On some continuation problems which are closely related to the theory of operators in space IIk, IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions,J. Oper. Theory 13 (1985), 299–417.
F. E. Melik-Adamyan, On canonical differential operators in Hilbert space,Izv. Akad. Nauk Armyan SSR, Ser. Mat.12 (1977), 10–31.
L. A. Sakhnovich,Spectral Theory of Canonical Differential Systems. Method of Operator Identities, Birkhäuser, Basel, 1999.
A. J. F. Siegert, A systematic approach to a class of problems in the theory of noise. Part II.IRE Trans. Information Theory IT-3 (1957), 38–43.
S. L. Sobolev, Some remarks on numerical solution of integral equations, (Brief report),Usp. Mat. Nauk 9(3) (1954), 234–235.
S. L. Sobolev, Some remarks on numerical solution of integral equationsIzv. Akad. Nauk SSSR 20 (1956), 413–436.
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This research was partially supported by a Minerva Foundation grant that is acknowledged with thanks.
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Arov, D.Z., Dym, H. J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p×q input scattering matrices. Integr equ oper theory 43, 68–129 (2002). https://doi.org/10.1007/BF01196514
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DOI: https://doi.org/10.1007/BF01196514