Abstract
This paper concerns systems of integro-differential equations of convolution type on the half-line for which the symbol is a rational matrix function. The equations are studied via certain singular input/output systems. For maximal operators associated with the equations, which act between certain Sobolev spaces on the half-line, explicit conditions for the operators to be invertible and to be Fredholm are derived, as well as explicit formulas for inverses, generalized inverses and the Fredholm characteristics. All conditions and formulas are expressed in terms of the matrices appearing in the singular system corresponding to the equations, and in matrices that are related to the system.
Research supported by the Netherlands organization for scientific research (NWO).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bart, H., Gohberg, I. and Kaashoek, M. A.: Wiener-Hopf integral equations, Toeplitz matrices and linear systems, in “Toeplitz Centennial” (Ed. I. Gohberg), Operator Theory: Advances and Applications, Vol. 4, Birkhäuser Verlag, Basel, 1982, pp. 85–135.
Bart, H., Gohberg, I. and Kaashoek, M. A.: Convolution equations and linear systems, Integral Equations and Operator Theory 5 (1982), 283–340.
Bart, H., Gohberg, I. and Kaashoek, M. A.: Explicit Wiener-Hopf factorization and realization, in “Constructive methods for Wiener-Hopf factorization” (Eds. I. Gohberg and M. A. Kaashoek), Operator Theory: Advances and Applications, Vol. 21, Birkhäuser Verlag, Basel, 1986, pp. 235–316.
Bart, H., Gohberg, I. and Kaashoek, M. A.: The state space method in problems of analysis, Proc. ICIAM 87, Paris-La Villette, 1987.
Gohberg, I., Goldberg, S. and Kaashoek, M. A.: Classes of Linear Operators Vol. 1, Operator Theory: Advances and Applications, Vol. 49, Birkhäuser Verlag, Basel 1990.
Gohberg, I. and Kaashoek, M. A.: Block Toeplitz operators with rational symbols, in “Contributions in Operator Theory and its Applications” (Eds. I. Gohberg, J.W. Helton, L. Rodman), Operator Theory: Advances and Applications, Vol. 35, Birkhäuser Verlag, Basel, 1988, pp. 385–440.
Gohberg, I. and Kaashoek, M. A.: The state space method for solving singular integral equations, in “Mathematical system theory, the influence of R. E. Kaiman” (Ed. A. C. Antoulas), Springer Verlag, Berlin, etc., 1991.
Gohberg, I., Kaashoek, M. A., Lerer, L. and Rodman, L.: Common multiples and common divisors of matrix polynomials I: spectral method, Ind. J. Math. 30, no. 3 (1981), 321–356.
Gohberg, I. and Kreĭn, M. G.: Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspehi Mat. Nauk 13 (1958), no. 2 (80), 3–72, (Russian) = Amer. Math. Soc. Transl (2) 14 (1960), 217–287.
Gohberg, I., Lancaster, P. and Rodman, L.: Matrix polynomials, Academic Press, New York, etc., 1982.
Hörmander, L.: Linear partial differential operators, Springer Verlag, Berlin, etc., 1963.
Kailath, T.: Linear systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
Kreĭn, M. G.: Integral equations on a half-line with kernel depending upon the difference of the arguments, Uspehi Mat. Nauk 13 (1958), no. 5 (83), 3–120, (Russian) = Amer. Math. Soc. Transl. (2) 22 (1962), 163–288.
Kuijper, A. B.: The state space method for integro-differential equations of convolution type with rational matrix symbols, Ph. D. thesis, Vrije Universiteit Amsterdam, Amsterdam, 1992.
Ramm, A. G.: Theory and Applications of some new classes of integral equations, Springer Verlag, New York, etc., 1980.
Ramm, A. G.: Random fields estimation theory, Mathl. Comput. Modelling 13, No. 4 (1990), 87–100.
Roozemond, L.: Systems of non-normal and Wiener-Hopf equations, Ph. D. thesis, Vrije Universiteit Amsterdam, Amsterdam, 1987.
Schwartz, L.: Théorie des distributions, Hermann, Paris, 1966.
Stummel, F.: Diskrete Konvergenz linearer Operatoren II, Math. Z. 120 (1971), 231–264.
Talenti, G.: Sulle equazioni integrali di Wiener-Hopf, Boll. Un. Mat. Ital. (4) 7, suppl. fasc. 1 (1973), 18–118.
Treves, F.: Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.
Volevich, L. R., and Gindikin, S. G.: The Cauchy problem and other related problems for convolution equations, Uspehi Mat. Nauk 27 (1972), no. 4 (166), 65–143 (Russian) = Russian Math. Surveys 27 (1972), 71–160.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation in generalized functions, Trudy Moskov. Mat. Obšč. 35 (1976), 165–214 (Russian) = Trans. Moscow Math. Soc. (1979), no. 1 (35), 167–216.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation in generalized functions that are smooth on the half line, Trudy Moskov. Mat. Obšč. 38 (1979), 29–74 (Russian) = Trans. Moscow Math. Soc. (1980), no. 2 (38), 25–70.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation for smooth and generalized functions in a half-space, Trudy Moskov. Mat. Obšč. 40 (1979), 241–295 (Russian) = Trans. Moscow Math. Soc. (1981), no. 2 (40), 231–284.
Volevich, L. R., and Gindikin, S. G.: Boundary value problems for the Wiener-Hopf equation in a half-space, Trudy Moskov. Mat. Obšč. 42 (1981) (Russian) — Trans. Moscow Math. Soc. (1982), no. 2 (42), 175–198.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Kuijper, A.B. (1992). The State Space Method for Integro-Differential Equations of Wiener-Hopf Type with Rational Matrix Symbols. In: Gohberg, I. (eds) Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations. Operator Theory: Advances and Applications, vol 58. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8596-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8596-6_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9695-5
Online ISBN: 978-3-0348-8596-6
eBook Packages: Springer Book Archive