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.
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© 2010 Birkhäuser/Springer Basel AG
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Bart, H., Kaashoek, M.A., Ran, A.C.M. (2010). Explicit solutions using realizations. In: A State Space Approach to Canonical Factorization with Applications. Operator Theory: Advances and Applications, vol 200. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8753-2_4
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DOI: https://doi.org/10.1007/978-3-7643-8753-2_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8752-5
Online ISBN: 978-3-7643-8753-2
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