Advertisement

Interpolation by Analytic Matrix Functions

Chapter
Part of the NATO ASI series book series (ASIC, volume 153)

Abstract

There are some classical interpolation problems for complex functions, solved many times over decades ago, which still generate dozens of new papers. One reason for this is strictly mathematical; the development of certain branches of operator theory has enabled us to view the problems in a different way and so understand some aspects of them better. There is another reason, which for me rings an even louder bell: contact with the theory of circuits and systems developed by engineers. An enormous range of worthwhile problems about complex functions comes from this source. Many of them are close to questions studied by the old masters of analysis: for example, variants of the Nevanlinna-Pick interpolation problem arise from a remarkable diversity of starting points. The earliest instance I know dates back to 1940 (see the account of F. Fenyves’ work in [9]) while, more recently, J.W. Helton’s far-reaching application of non-Euclidean functional analysis to electronics also centres around this problem [4]. However, the engineering slant generally calls for something slightly different from the old results.

Keywords

Minimal Norm Extremal Function Krein Space Orthogonal Projection Operator Finite Rank Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adamyan, V.M., Arov, D.Z. and Krein, M.G., Infinite Hankel block matrices and related extension problems, Amer. Math. Soc. Transl. (2) 111 (1978) pp. 133–156.zbMATHGoogle Scholar
  2. 2.
    Allison, A.C. and Young, N.J., Numerical algorithms for the Nevanlinna-Pick problem, Numer. Math. 42 (1983) pp. 125–145.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Foias, C., Contractive intertwining dilations and waves in layered media, Proceedings of the International Congress of Mathematicians, Helsinki (1978).Google Scholar
  4. 4.
    Helton, J.W., Non-euclidean functional analysis and electronics, Bull. Amer. Math. Soc. 7 (1982) pp. 1–64.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Kung, S.Y. and Lin, D.W., Optimal Eankel norm model reductions: multivariable systems, I.E.E.E. Transactions on Automatic Control 26 (1981) pp. 832–852.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Patel, R.V. and Munro, N., Multivariable system theory and design, Pergamon Press, Oxford 1981.Google Scholar
  7. 7.
    Sarason, D., Generalized interpolation in H , Trans. Amer. Math. Soc. 127 (1967) pp. 179–203.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Sarason, D., Operator-theoretic aspects of the Nevanlinna-Pick interpolation problem, this volume.Google Scholar
  9. 9.
    Solymosi, J., Interpolation with PR functions based on F. Fenyves’ method, Periodica Polytechnica (Budapest) 15 (1971) pp. 71–76.Google Scholar
  10. 10.
    Yeh, F.B., Numerical solution of matrix interpolation problems, Ph.D. Thesis, Glasgow University 1983.Google Scholar
  11. 11.
    Young, N.J., The Nevanlinna-Pick problem for matrix-valued functions, to be published.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  1. 1.University of GlasgowUK

Personalised recommendations