Skip to main content
Log in

Hankel operators and best Hankel approximation on the half-plane

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The objective of this paper is to give an interrelation between Hankel oeprators on the unit disc and Hankel operators on the half-plane. As an application, the AAK result on the half-plane is established and the rate of best Hankel approximation on the halfplane is derived.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. V.M. Adamyan, D.Z. Arov and M.G. Krein, Analytic properties of Schmidt pairs of a Hankel operator and the generalized Schur—Takagi problem, Math. USSR-Sb. 15 (1971) 31–73.

    Article  Google Scholar 

  2. R.F. Curtain,Modeling, Robustness and Sensitive Reduction in Control Systems (Springer, New York, 1987).

    Google Scholar 

  3. C.K. Chui and G. Chen,Signal Processing and Systems Theory: Selected Topics (Springer, New York, 1992).

    MATH  Google Scholar 

  4. R.F. Curtain and K. Glover, Robust stabilization of infinite dimensional systems by finite dimensional controllers, Syst. Contr. Lett. 7 (1986) 41–47.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Carathéodory and L. Fejér, Über den Zusammenhang der Extremen, von harmonischen Funktionen mit ihren Koeffizienten and über der Picard—Landauschen Satz, Rent. Circ. Mat. Palermo 32 (1991) 218–239.

    Article  Google Scholar 

  6. C.K. Chui and X. Li, Continuity of best Hankel approximation and convergence of near-best approximants, SIAM J. Contr. Optim., to appear.

  7. C.K. Chui, X. Li and J.D. Ward, System reduction via Hankel matrices, Math. Contr. Sign. Sys. 4 (1991) 161–175.

    MATH  MathSciNet  Google Scholar 

  8. C.K. Chui, X. Li and J.D. Ward, Rate of uniform convergence of rational functions corresponding to best approximants of truncated Hankel operators. Math. Contr. Sign. Sys. 5 (1992) 67–79.

    Article  MATH  MathSciNet  Google Scholar 

  9. B.A. Francis,A Course in H Control (Springer, New York, 1986).

    Google Scholar 

  10. F.R. Gentmacher,Theory of Matrices, 2nd ed. (“Nauka”, Moscow, 1966; English transl. of 1st ed.: Chelsea, New York 1959).

    Google Scholar 

  11. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and theirL -error bounds, Int. J. Contr. 39 (1984) 1115–1193.

    MATH  MathSciNet  Google Scholar 

  12. I.C. Gohberg and M.G. Krein,Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Spaces (Amer. Math. Soc., Providence, 1969).

    Google Scholar 

  13. E. Hayashi, L.N. Trefethen and M.H. Gutknecht, The CF table, Constr. Approx. Theory 6 (1990) 195–223.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958) 862–866.

    Article  MathSciNet  Google Scholar 

  15. S.Y. Kung, Optimal Hankel-norm model reduction: scalar systems, in:Proc. 1980 Joint Automatic Control Conf. (1980).

  16. D.W. Lin and S.Y. Kung, Optimal Hankel-norm model reduction: multicariable systems, IEEE Trans. Auto. Contr. AC-26 (1981) 832–852.

    Article  MATH  MathSciNet  Google Scholar 

  17. Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153–162.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Nevanlinna, Über beschränkte analytische Funcktionen, Ann. Acad. Sci. Fennica A 32 (1929) 1–75.

    Google Scholar 

  19. J.R. Partington,An Introduction to Hankel Operators (Cambridge University Press, England, 1998).

    MATH  Google Scholar 

  20. V.V. Peller, Hankel operators and continuity properties of operators of best approximation, Leningrad Math. J. 2 (1991) 139–160.

    MathSciNet  Google Scholar 

  21. S.C. Power,Hankel operators on Hilbert Space (Pitman, Boston, 1982).

    MATH  Google Scholar 

  22. I. Schur, Über Potenzreihen, die im innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. 148 (1918) 122–145.

    MATH  Google Scholar 

  23. T. Takagi, On an algebraic problem related to an analytic theorem of Carathéodory and Fejér, Japan J. Math. 1 (1924) and 2 (1925) 13–17.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research partially supported by the University Research Grants and Fellowship Committee at UNLV.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X. Hankel operators and best Hankel approximation on the half-plane. Adv Comput Math 2, 343–355 (1994). https://doi.org/10.1007/BF02521115

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02521115

Key words

AMS(MOS) subject classification

Navigation