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Explicit solutions using realizations

  • Harm Bart
  • Marinus A. Kaashoek
  • André C. M. Ran
Part of the Operator Theory: Advances and Applications book series (OT, volume 200)

Abstract

As we have seen in Chapter 1, canonical factorization serves as a tool to solve Wiener-Hopf integral equations, their discrete analogues, and the more general singular integral equations. In this chapter the state space factorization method developed in Chapter 2 is used to solve the problem of canonical factorization (necessary and sufficient conditions for its existence) and to derive explicit formulas for its factors. This is done in Section 3.1 for rational matrix functions and later in Section 7.1 for operator-valued transfer functions that are analytic on an open neighborhood of a curve. The results are applied to invert Wiener-Hopf integral equations with a rational matrix symbol (Section 3.2), block Toeplitz operators (Section 3.3) and singular integral equations (Section 3.4). The methods developed in this chapter also allow us to deal with the Riemann-Hilbert boundary value problem. This is done in the final section which also contains material on the homogeneous Wiener-Hopf equation.

Keywords

Unit Circle Open Neighborhood Matrix Function Half Plane Explicit Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser/Springer Basel AG 2010

Authors and Affiliations

  • Harm Bart
    • 1
  • Marinus A. Kaashoek
    • 2
  • André C. M. Ran
    • 2
  1. 1.Econometrisch InstituutErasumus Universiteit RotterdamRotterdamThe Netherlands
  2. 2.Department of Mathematics, FEWVrije UniversiteitAmsterdamThe Netherlands

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