Skip to main content
Log in

On the Hankel-norm approximation of upper-triangular operators and matrices

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. G. Golub and C.F. Van Loan,Matrix Computations. The Johns Hopkins University Press, 1984.

  2. V.M. Adamjan, D.Z. Arov, and M.G. Krein, “Analytic Properties of Schmidt Pairs for a Hankel Operator and the Generalized Schur-Takagi Problem,”Math. USSR Sbornik, vol. 15, no. 1, pp. 31–73, 1971. (transl. ofIz. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971)).

    Google Scholar 

  3. L. Kronecker, “Algebraische Reduction der Schaaren Bilinearer Formen,”S.B. Akad. Berlin, pp. 663–776, 1890.

  4. Y.V. Genin and S.Y. Kung, “A Two-Variable Approach to the Model Reduction Problem with Hankel Norm Criterion,”IEEE Trans. Circuits Syst., vol. 28, no. 9, pp. 912–924, 1981.

    Google Scholar 

  5. Z. Nehari, “On Bounded Bilinear Forms,”Ann. of Math., vol. 65, no. 2, pp. 153–162, 1957.

    Google Scholar 

  6. A. Bultheel and P.M. Dewilde, “On the Adamjan-Arov-Krein Approximation, Identification, and Balanced Realization,” inProc. 1980 Eur. Conf. on Circ. Th. and Design, vol. 2, pp. 186–191, 1980.

  7. K. Glover, “All Optimal Hankel Norm Approximations of Linear Multi-Variable Systems and theirL -error Bounds,”Int. J. Control., vol. 39, no. 6, pp. 1115–1193, 1984.

    Google Scholar 

  8. S.Y. Kung and D.W. Lin, “Optimal Hankel Norm Model Reductions: Multi-variable Systems,”IEEE Trans. Automat. Control, vol. 26, pp. 832–852, Aug. 1981.

    Google Scholar 

  9. D.J.N. Limebeer and M. Green, “Parametric Interpolation,H -Control and Model Reduction,”Int. J. Control, vol. 52, no. 2, pp. 293–318, 1990.

    Google Scholar 

  10. P. Dewilde, A.C. Vieira, and T. Kailath, “On a Generalized Szegö-Levinson Realization Algorithm for Optimal Linear Predictors Based on a Network Synthesis Approach,”IEEE Trans. Circuits Syst., vol. 25, pp. 663–675, Sept. 1978.

    Google Scholar 

  11. P. Dewilde and H. Dym, “Schur Recursions, Error Formulas, and Convergence of Rational Estimators for Stationary Stochastic Sequences,”IEEE Trans. Informat. Th., vol. 27, pp. 446–461, July 1981.

    Google Scholar 

  12. P. Dewilde and H. Dym, “Lossless Chain Scattering Matrices and Optimum Linear Prediction: The Vector Case,”Circuit Theory and Appl., vol. 9, pp. 135–175, 1981.

    Google Scholar 

  13. P. Dewilde and H. Dym, “Lossless Inverse Scattering, Digital Filters, and Estimation Theory,”IEEE Trans. Informat. Th., vol. 30, pp. 644–662, July 1984.

    Google Scholar 

  14. H. Dym,J-Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. No. 71 in CBMS regional conference series, Providence: American Math. Soc., 1989.

    Google Scholar 

  15. D. Alpay and P. Dewilde, “Time-varying Signal Approximation and Estimation,” inSignal Processing, Scattering and Operator Theory, and Numerical Methods (M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, eds.), vol. III ofProc. Int. Symp. MTNS-89, pp. 1–22, Birkhäuser Verlag, 1990.

  16. D. Alpay, P. Dewilde, and H. Dym, “Lossless Inverse Scattering and Reproducing Kernels for Upper Triangular Operators,” inExtension and Interpolation of Linear Operators and Matrix Functions (I. Gohberg, ed.), vol. 47 ofOperator Theory, Advances and Applications, pp. 61–135, Birkhäuser Verlag, 1990.

  17. J.A. Ball, I. Gohberg, and L. Rodman,Interpolation of Rational Matrix Functions, vol. 45 ofOperator Theory: Advances and Applications. Birkhäuser Verlag, 1990.

  18. G. Zames, “Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Semi-Norms, and Approximate Inverses,”IEEE Trans. Automat. Control, vol. 23, pp. 301–320, 1982.

    Google Scholar 

  19. J.W. Helton, “Non-Euclidean Functional Analysis and Electronics,”Bull. of the AMS, vol. 7, No. 1, pp. 1–64, 1982.

    Google Scholar 

  20. J.W. Helton, “Orbit Structure of the Möbius transformation Semigroup Acting onH (Broadband Matching),” inTopics in Functional Analysis, vol. 3 ofAdv. in Math. Suppl. Studies, pp. 129–133, Academic Press, 1978.

  21. P. Dewilde and E. Deprettere, “Approximative Inversion of Positive Matrices with Applications to Modeling,” inNATO ASI Series, Vol. F34 on Modeling, Robustness and Sensitivity Reduction in Control Systems, Berlin: Springer Verlag, 1987.

    Google Scholar 

  22. P. Dewilde and E. Deprettere, “The Generalized Schur Algorithm: Approximation and Hierarchy” inOperator Theory: Advances and Applications, vol. 29, pp. 97–116, Birkhäuser Verlag, 1988.

  23. P. Dewilde and H. Dym. “Interpolation for Upper Triangular Operators,” inTime-Variant Systems and Interpolation (I. Gohberg, ed.), vol. 56 ofOperator Theory: Advances and Applications, pp. 153–260, Birkhäuser Verlag, 1992.

  24. A.J. van der Veen and P.M. Dewilde, “Time-Varying System Theory for Computational Networks,” inAlgorithms and Parallel VLSI Architectures, II (P. Quinton and Y. Robert, eds.), pp. 103–127, Elsevier, 1991.

  25. B.L. Ho and R.E. Kalman, “Effective Construction of Linear, State-Variable Models from Input/Output Functions,”Regelungstechnik, vol. 14, pp. 545–548, 1966.

    Google Scholar 

  26. L.A. Zadeh, “Time-Varying Networks, I,”Proc. IRE, vol. 49, pp. 1488–1503, Oct. 1961.

    Google Scholar 

  27. A. Feintuch and R. Saeks,System Theory: A Hilbert Space Approach. Academic Press, 1982.

  28. E.W. Kamen, P.P. Khargonekar, and K.R. Poolla, “A Transfer-Function Approach to Linear Time-Varying Discrete-Time Systems,”SIAM J. Control and Optimization, vol. 23, pp. 550–565, July 1985.

    Google Scholar 

  29. P.P. Khargonekar and K. Poolla, “On Polynomial Matrix Fraction Representations for Linear Time-Varying Discrete-Time Systems,”Lin. Alg. Appl., vol. 80, pp. 1–37, 1986.

    Google Scholar 

  30. K. Poolla and P. Khargonekar, “Stabilizability and Stable-Proper Factorizations for Linear Time-Varying Systems,”SIAM J. Control and Optimization, vol. 25, pp. 723–736, May 1987.

    Google Scholar 

  31. I. Gohberg, M.J. Kaashoek, and H.J. Woerdeman, “The Band Method for Positive and Strictly Contractive Extension Problems: an Alternative Version and New Applications,”Integral Eq. Operator Th., vol. 12, pp. 343–382, 1989.

    Google Scholar 

  32. I. Gohberg, M.A.Kaashoek, and H.J. Woerdeman, “Time Variant Extension Problems of Nehari Type and the Band Method,” inH -Control Theory (lectures given at the 2nd session of C.I.M.E., Como, June 18–26, 1990) (C. Foias, B. Francis, and J.W. Helton, eds.), Lecture Notes Math. 1496, pp. 309–323, Springer Verlag, 1991.

  33. I. Schur, “Uber Potenzreihen, die im Innern des Einheitskreises Beschränkt Sind, I,”J. Reine Angew. Math., vol. 147, pp. 205–232, 1917. Eng. Transl.Operator Theory: Adv. Appl., vol 18, pp. 31–59, Birkhäuser Verlag, 1986.

    Google Scholar 

  34. N.I. Akhiezer,The Classical Moment Problem. Edinburgh: Oliver and Boyd, 1965.

    Google Scholar 

  35. H. Woerdeman,Matrix and Operator Extensions. PhD thesis, Dept. Math. Comp Sci., Free University, Amsterdam, The Netherlands, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dewilde, P., van der Veen, A.J. On the Hankel-norm approximation of upper-triangular operators and matrices. Integr equ oper theory 17, 1–45 (1993). https://doi.org/10.1007/BF01322544

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01322544

MCS 1991

Navigation