This book aims to present the theory of interpolation for rational matrix functions as a recently matured independent mathematical subject with its own problems, methods and applications. The authors decided to start working on this book during the regional CBMS conference in Lincoln, Nebraska organized by F. Gilfeather and D. Larson. The principal lecturer, J. William Helton, presented ten lectures on operator and systems theory and the interplay between them. The conference was very stimulating and helped us to decide that the time was ripe for a book on interpolation for matrix valued functions (both rational and non-rational). When the work started and the first partial draft of the book was ready it became clear that the topic is vast and that the rational case by itself with its applications is already enough material for an interesting book. In the process of writing the book, methods for the rational case were developed and refined. As a result we are now able to present the rational case as an independent theory. After two years a major part of the first draft was prepared.
Then a long period of revising the original draft and introducing recently acquired results and methods followed. There followed a period of polishing and of 25 chapters and the appendix commuting at various times somewhere between Williamsburg, Blacksburg, Tel Aviv, College Park and Amsterdam (sometimes with one or two of the authors).
I Homogeneous Interpolation Problems with Standard Data.- 1. Null Structure for Analytic Matrix Functions.- 2. Null Structure and Interpolation Problems for Matrix Polynomials.- 3. Local Data for Meromorphic Matrix Functions.- 4. Rational Matrix Functions.- 5. Rational Matrix Functions with Null and Pole Structure at Infinity.- 6. Rational Matrix Functions with J-Unitary Values on the Imaginary Line.- 7. Rational Matrix Functions with J-Unitary Values on the Unit Circle.- II Homogeneous Interpolation Problems with Other Forms of Local Data.- 8. Interpolation Problems with Null and Pole Pairs.- 9. Interpolation Problems for Rational Matrix Functions Based on Divisibility.- 10. Polynomial Interpolation Problems Based on Divisibility.- 11. Coprime Representations and an Interpolation Problem.- III Subspace Interpolation Problems.- 12. Null-Pole Subspaces: Elementary Properties.- 13. Null-Pole Subspaces for Matrix Functions with J-Unitary Values on the Imaginary Axis or Unit Circle.- 14. Subspace Interpolation Problems.- 15. Subspace Interpolation with Data at Infinity.- IV Nonhomogeneous Interpolation Problems.- 16. Interpolation Problems for Matrix Polynomials and Rational Matrix Functions.- 17. Partial Realization as an Interpolation Problem.- V Nonhomogeneous Interpolation Problems with Metric Constraints.- 18. Matrix Nevanlinna-Pick Interpolation and Generalizations.- 19. Matrix Nevanlinna-Pick-Takagi Interpolation.- 20. Nehari Interpolation Problem.- 21. Boundary Nevanlinna-Pick Interpolation.- 22. Caratheodory-Toeplitz Interpolation.- VI Some Applications to Control and Systems Theory.- 23. Sensitivity Minimization.- 24. Model Reduction.- 25. Robust Stabilizations.- Appendix. Sylvester, Lyapunov and Stein Equations.- A.1 Sylvester equations.- A.2 Stein equations.- A.3 Lyapunov and symmetric Stein equations.- Notes.- References.- Notations and Conventions.
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