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Editorial

Special Issue on Rational Approximations for Systems

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Abstract

This special issue on rational approximations for systems presents a collection of papers which grew out of presentations held at a workshop at the Catholic University of Leuven in August 1981. The organizers of the workshop tried to bring a number of schools together, each school representing a different approach to rational approximations, with the hope that a convergence of the results would become apparent. With uncautious generalizations, the schools represented can be described as: (1) the ‘Padé-school’ where emphasis is placed on efficient recursive algorithms; (2) the ‘norm-approximation school’ where emphasis is put on approximation and convergence in a strong norm; and (3) the’ system-theory school’ with main emphasis on system estimation and representation. One may wonder how seemingly hyperclassical subjects can still muster interest. The reason is that in recent times many new results have appeared in the field, mainly due to cross-fertilization and to better understanding of the underlying mathematical techniques. The cross-fertilization referred to here is exemplified by papers like [1], [2], [3].

The workshop on rational approximations for systems and the present editorial work were partially supported by the NFWO (Brussels), the Belgian Ministry of Education and Culture, and the Third Cycle Research Fund of the Katholische Universiteit Leuven.

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References

  1. R. E. Kaiman, “On partial realization, transfer functions and canonical forms,” Acta Polytechnica Scandinavia, Vol. 31, 1979, pp. 9–32.

    Google Scholar 

  2. E. R. Berlekamp, “Algebraic coding theory,” McGraw Hill, N. Y., 1968.

    Google Scholar 

  3. J. L. Massey, “Shift register synthesis and BCH decoding,” IEEE Trans. on Information Theory,” Vol. IT-15, 1967, pp. 122–127.

    Google Scholar 

  4. V. M. Adamyan, D. Z. Arov, M. G. Krein, “Infinite Hankel matrices and generalized Carathéodory-Fejér and Riesz problems,” Function. Anal. Appl., Vol. 2, 1968, pp. 1–18.

    Article  Google Scholar 

  5. V. M. Adamyan, D. Z. Arov, M. G. Krein, “Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and I. Schur,” Funct. Anal. Appl., Vol. 2, 1968, pp. 269–281.

    Article  Google Scholar 

  6. V. M. Adamyan, D. Z. Arov, M. G. Krein, “Bounded operators that commute with a contraction of class Coo of unit rank of nonunitarity,” Funct. Anal. Appl., Vol. 3, 1969, pp. 242–243.

    Article  Google Scholar 

  7. V. M. Adamyan, D. Z. Arov, M. G. Krein, “Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem,” Math. Sbornik, Vol. 15, 1971, pp. 31–73.

    Article  Google Scholar 

  8. V. M. Adamyan, D. Z. Arov, M. G. Krein, “Infinite Hankel blockmatrices and related continuation problem,” Izv. Akad. Nauk Armjan SSR Ser. Mat. 6, 1971, pp. 87–112, and Amer. math. soc. Transi. (2), Vol. 111, 1978, pp. 133-156.

    Google Scholar 

  9. C. Brezinski, “The long history of continued fractions and Padé approximants,” in: Padé approximation and its applications, Lect. Notes Math. 888, Springer-Verlag, Berlin, 1981, pp. 1–27.

    Google Scholar 

  10. C. T. Mullis, R. A. Roberts, “The use of second order information in the approximation of discrete time linear systems,” IEEE Trans. on Acoustics, Speech and Signal Processing, Vol. ASSP-24, 1976, pp. 226–238.

    Article  Google Scholar 

  11. T. Kailath, S-Y. Kung, M. Morf, “Displacement ranks of matrices and linear equations,” Journ. of Math. Anal. Appl., Vol. 68, 1979, pp. 395–407.

    Article  Google Scholar 

  12. R. P. Brent, F. G. Gustavson, D. Y. Y. Yun, “Fast solution of Toeplitz systems of equations and Padé approximants,” J. Algorithms, Vol. 1, 1980, pp. 259–295.

    Article  Google Scholar 

  13. F. R. Gantmacher, “The theory of matrices,” Chelsea, N. Y., 1959.

    Google Scholar 

  14. W. F. Trench, “An algorithm for the inversion of finite Toeplitz matrices,” SIAM Journ. Appl. Math., Vol. 12, 1964, pp. 515–522.

    Article  Google Scholar 

  15. W. F. Trench, “An Algorithm for the inversion of finite Hankel matrices,” SIAM Journ. Appl. Math., Vol. 13, 1965, pp. 1102–1107.

    Article  Google Scholar 

  16. S. Zohar, “Toeplitz matrix inversion: The algorithm of W. F. Trench,” Journ. Assoc. Comp. Mach., Vol. 16, 1969, pp. 592–601.

    Article  Google Scholar 

  17. S. Zohar, “The solution of a Toeplitz set of linear equations,” Journ. Assoc. Comp. Mach., Vol. 21, 1975, pp. 272–276.

    Article  Google Scholar 

  18. C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral equations,” J. Res. Nat. Bur. Standards, Vol. 45, 1950, pp. 255–282.

    Article  Google Scholar 

  19. G. A. Baker, “Essentials of Padé approximants,” Academic Press, N. Y., 1975.

    Google Scholar 

  20. P. J. S. Watson, “An Algorithm for differentiation and integration,” in Padé approximations and their applications, P. R. Graves-Morris, (ed.) Academic Press, London, 1973, pp. 93–98.

    Google Scholar 

  21. G. A. Baker, P. R. Graves-Morris, “Padé approximants,” Addison-Wesley, London, 1981.

    Google Scholar 

  22. J. Rissanen, “Solution of linear equations with Hankel and Toeplitz matrices,” Num.Math., Vol. 22, 1974, pp. 361–366.

    Article  Google Scholar 

  23. A. N. Khovanskii, “The application of continued fractions and their generalization to problems in approximation theory,” P. Noordhoff N. V., Groningen, 1963.

    Google Scholar 

  24. W. B. Gragg, G. D. Johnson, “The Laurent-Padé table,” in Information Processing 74, North Holland, Amsterdam. pp. 632–637.

    Google Scholar 

  25. N. Levinson, “The Wiener RMS (Root Main Square) error criterion in filter design and prediction,” J. Math. Phys., Vol. 25, 1947, pp. 261–278.

    Google Scholar 

  26. I. Schur, “Über Potenzreihen die im Innern des Einheitskreises beschränkt sind,” Z. für die Reine und Angew. Math., Vol. 147, 1917, pp. 205–232, and Vol. 148, 1918, pp. 122-145.

    Google Scholar 

  27. T. Kailath, “A view of three decades of linear filtering theory,” IEEE Trans. on Information Theory, Vol. IT-20, 1974, pp. 146–181.

    Article  Google Scholar 

  28. G. Cybenko, “The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations,” SIAM Journ. Sci. Stat. Comput., Vol. 1, 1980, pp. 303–319.

    Article  Google Scholar 

  29. A. Bultheel, “Error analysis of incoming and outgoing schemes for the trigonometric moment problem,” in Padé approximation and its applications, Lect. Notes Math. 888, Springer-Verlag, Berlin, 1981, pp. 100–109.

    Google Scholar 

  30. W. J. Thron, “Two-point Padé approximants, T-fractions, and sequences of Schur,” in Padé and rational approximation, Academic Press, N. Y., 1977, pp. 215–226.

    Chapter  Google Scholar 

  31. P. R. Graves-Morris, “Practical, reliable, rational interpolation,” J. Inst. Maths Applics, Vol. 25, 1980, pp. 267–286.

    Article  Google Scholar 

  32. H. Werner, “Ein Algorithmus zur rationalen Interpolation,” Num. Meth. der Approx. Theorie, Vol. 5, Birkhäuser Verlag, Basel, 1980, pp. 319–337.

    Google Scholar 

  33. H. Arndt, “Ein verallgemeinerter Kettenbruch-Algorithmus zur rationalen Hermite-Interpolation,” Numer. Math., Vol. 36, 1980, pp. 99–107.

    Article  Google Scholar 

  34. D. Sarason, “Generalized interpolation in H ;,” Trans. Amer. Math. Soc, Vol. 127, 1967, pp. 179–203.

    Google Scholar 

  35. B. Sz.-Nagy, C. Foias, “Harmonie Analysis of operators on Hilbert space,” North Holland, Amsterdam, 1970.

    Google Scholar 

  36. V. P. Potapov, “The multiplicative structure of J-contractive matrix functions,” Amer. Math. Soc. Translations, ser. 2, Vol. 15, 1960, pp. 131–243.

    Google Scholar 

  37. M. G. Krein, “On the theory of entire matrix functions of exponential type (in Russian),” Ukrain Mat.Z., Vol. 3, 1951, pp. 164–173.

    Google Scholar 

  38. M. S. Livsic, “Operators, oscillations, waves, open systems,” Translat. Math. Monographs, Vol. 34.

    Google Scholar 

  39. W. B. Helton, “The distance of a function to H in the Poincaré metric,” J. Funct. Anal., Vol. 37, 1980, pp. 273–314.

    Article  Google Scholar 

  40. L. N. Trefethen, M. H. Gutknecht, “The Carathéodory-Fejér method for real rational approximation,” subm. SIAM J. Numer. Anal.

    Google Scholar 

  41. B. L. Ho, R. E. Kaiman, “Effective construction of linear state-variable models from input-output functions,” Regelungstechnik, Vol. 14, 1966, pp. 545–548.

    Google Scholar 

  42. W. B. Gragg, A. Lindquist, “On the partial realization problem,” Techn. Report, University of Kentucky, Lexington, 1982.

    Google Scholar 

  43. A. Bultheel, P. Dewilde, “On the Adamyan-Arov-Krein approximation, identification and balanced realization of a system,” Proc. ECCTD‘80 Warsaw, Vol. 2, 1980, pp. 186–191.

    Google Scholar 

  44. L. Silverman, M. Bettayeb, “Optimal approximation of linear systems,” Proc. Joint Autom. Contr. Conf., 1980, San Francisco, IEEE, New York.

    Google Scholar 

  45. S. Y. Kung, “Optimal Hankel-norm reduction: scalar systems,” Proc. Joint Autom. Contr. Conf., 1980, San Francisco, IEEE, New York.

    Google Scholar 

  46. Y. Genin, S. Y. Kung, “A two variable approach to the model reduction problem with Hankel-norm,” IEEE Trans. on Circuits and Systems, Vol. CAS-28, 1981, pp. 912–914.

    Article  Google Scholar 

  47. D. Lee, B. Friedlander, M. Morf, “Recursive ladder algorithms for ARMA modeling,” to appear, IEEE Tran. on Automatic Contr.

    Google Scholar 

  48. A. Bultheel, “Rekursieve rationale benaderingen,” Thesis, K. U. Leuven, 1979.

    Google Scholar 

  49. P. Dewilde, A. Vieira, T. Kailath, “On a generalized Szegö-Levinson realization algorithm for optimal linear predictors based on a network systhesis approach,” IEEE Trans. on Circuits and Systems, Vol. CAS-25, 1978, pp. 663–675.

    Article  Google Scholar 

  50. P. Dewilde, H. Dym, “Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences,” IEEE Trans. on Information Theory, Vol. IT-27, 1981, pp. 446–561.

    Article  Google Scholar 

  51. P. Dewilde, H. Dym, “Lossless chain scattering matrices and optimum linear prediction: the vector case,” Circuit Theory and Applications, Vol. 9, 135–175, 1981.

    Article  Google Scholar 

  52. P. Dewilde, J. T. Fokkema, I. Widya, “Inverse scattering and linear prediction, the time continuous case,” in Stochastic systems: The mathematics of filtering and identification and applications, pp. 351–382, 1981, D. Reidel, Dordrecht.

    Chapter  Google Scholar 

  53. U. Grenander, G. Szegö, “Toeplitz forms and their applications,” University of California Press, Berkeley, 1958.

    Google Scholar 

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Bultheel, A., Dewilde, P. (1983). Editorial. In: Bultheel, A., Dewilde, P. (eds) Rational Approximation in Systems Engineering. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6790-9_2

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  • DOI: https://doi.org/10.1007/978-1-4899-6790-9_2

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