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Numerical algorithms for the Nevanlinna-Pick problem

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Summary

Based on the formulation in [14], efficient algorithms are devised for two variants of the Nevanlinna-Pick problem. Their implementation is validated by several examples. The method depends on establishing the equivalence of two different operator-theoretic approaches to the solution of classical interpolation problems [1, 14].

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References

  1. Adamjan, V.M., Arov, D.Z., Krein, M.G.: Analytic properties of Schmidt pairs and the generalised Schur-Takagi problem. Math. USSR Sbornik15, 31–73 (1971)

    Google Scholar 

  2. Akhiezer, N.I.: On a minimum problem in the theory of functions and on the number of roots of an algebraic equation which lie inside the unit circle. Izv. Akad. Nauk. SSSR Otd. Mat.9, 1169–1189 (1931)

    Google Scholar 

  3. Cohn, A.: Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z.14, 110–148 (1922)

    Google Scholar 

  4. Delsarte, Ph., Genin, Y., Kamp, Y.: On the role of the Nevanlinna-Pick problem in circuit and system theory. Circuit Theory and Applications9, 177–187 (1981)

    Google Scholar 

  5. Duren, P.: Theory ofH p spaces. New York: Academic Press 1970

    Google Scholar 

  6. Fenyves, F.: Beitrag zur Realisierung von Zweipolen mit vorgegebener Charakteristik. Budapest: Atheneum 1938

    Google Scholar 

  7. Gutknecht, M.H., Trefethen, L.M.: Recursive digital filter design by the Caratheodory-Fejér Method. Numer. Anal. Proj. Manuscript NA-80-01 1980

  8. Helton, J.W.: The distance of a function toH ∞ in the Poincaré metric: electrical power transfer. J. Functional Analysis38, 273–314 (1980)

    Google Scholar 

  9. Hoffman, K.: Banach Spaces of Analytic Functions. New Jersey: Prentice Hall 1962

    Google Scholar 

  10. Mal'cev, A.I.: Foundations of Linear Algebra. San Francisco: W.H. Freeman 1963

    Google Scholar 

  11. Nevanlinna, R.: Über beschränkte analytische. Funktionen. Ann. Acad. Sci. FennA 32, 1–75 (1929)

    Google Scholar 

  12. Pick, G.: Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden. Math. Ann.77, 7–23 (1916)

    Google Scholar 

  13. Pták, V., Young, N.J.: Functions of operators and the spectral radius. Linear Algebra and its Applications29, 357–392 (1980)

    Google Scholar 

  14. Sarason, D.: Generalised interpolation inH ∞. Trans. Amer. Math. Soc.127, 179–203 (1967)

    Google Scholar 

  15. Young, N.J.: The singular value decomposition of an infinite Hankel matrix. Linear Algebra and its Applications50, 639–656 (1983)

    Google Scholar 

  16. Young, N.J.: The solution of Lyapunov's equation by a geometric method. Proc. Roy. Soc. Edinburgh86 A, 347–354 (1980)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing Science, University of Glasgow, Scotland

    A. C. Allison

  2. Department of Mathematics, University of Glasgow, Scotland

    N. J. Young

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  1. A. C. Allison
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  2. N. J. Young
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Allison, A.C., Young, N.J. Numerical algorithms for the Nevanlinna-Pick problem. Numer. Math. 42, 125–145 (1983). https://doi.org/10.1007/BF01395306

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