Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

Formal orthogonal polynomials and Hankel/Toeplitz duality

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szegő recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. G.S. Ammar, W.B. Gragg and L. Reichel, Constructing a unitary Hessenberg matrix from spectral data, in:Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, eds. G. Golub and P. Van Dooren, vol. 70 of NATO-ASI Series, F: Computer and Systems Sciences (Springer, Berlin, 1991) pp. 385–395.

    Google Scholar 

  2. C. Brezinski,Padé-type Approximants and General Orthogonal Polynomials, vol. 50 of Int. Ser. of Numer. Math. (Birkhäuser, Basel, 1980).

    Google Scholar 

  3. C. Brezinski,Biorthogonality and Its Applications to Numerical Analysis, vol. 156 of Lecture Notes in Pure and Appl. Math. (Marcel Dekker, New York, 1991).

    Google Scholar 

  4. C. Brezinski, A unified approach to various orthogonalities, Ann. Fac. Sciences Toulouse, Ser. 3, 1(3) (1992) 227–292.

    Google Scholar 

  5. C. Brezinski, Formal orthogonal polynomials, NA Digest 95(08) (February 19, 1995).

  6. C. Brezinski and J. van Iseghem, Vector orthogonal polynomials of dimension-d, in:Approximation and Computation, ed. R. Zahar, vol. 119 of Int. Ser. of Numer. Math. (Birkhäuser, 1994) pp. 29–39.

  7. A. Bultheel, Division algorithms for continued fractions and the Padé table, J. Comput. Appl. Math. 6 (1980) 259–266.

    Article  Google Scholar 

  8. A. Bultheel,Laurent Series and Their Padé Approximations, vol. OT-27 of Oper. Theory: Adv. Appl. (Birkhäuser, Basel, Boston, 1987).

    Google Scholar 

  9. A. Bultheel and M. Van Barel, Euclid, Padé and Lanczos, another golden braid, Technical Report TW188, Department of Computer Science, K.U. Leuven (April 1993).

  10. A. Bultheel and M. Van Barel, Some applications of the Euclidean algorithm, in:Proc. 2nd Int. Colloquium on Numerical Analysis, eds. D. Bainov and V. Covachev, Zeist, The Netherlands (VSP Inl. Sci. Publ., 1994) pp. 45–54.

    Google Scholar 

  11. A. Bultheel and M. Van Barel, Vector orthogonal polynomials and least squares approximation, SIAM J. Matrix Anal. Appl. 16 (1995), to appear.

  12. J. Chun and T. Kailath, Generalized displacement structure for block-Toeplitz, Toeplitz-block, and Toeplitz-derived matrices, in:Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, eds. G. Golub and P. Van Dooren, vol. 70 of NATO-ASI Series, F: Computer and Systems Sciences (Springer, Berlin, 1991) pp. 215–236.

    Google Scholar 

  13. A. Draux,Polynomes Orthogonaux Formels—Applications, vol. 974 of Lecture Notes in Math, (Springer, Berlin, 1983).

    Google Scholar 

  14. R.W. Freund, G.H. Golub and N.M. Nachtigal, Iterative solution of linear systems, Acta Numerica 1 (1992) 57–100.

    Google Scholar 

  15. R.W. Freund and H. Zha, Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems, Lin. Alg. Appl. 188/189 (1993) 255–303.

    Article  Google Scholar 

  16. I. Gohberg (ed.),I. Schur Methods in Operator Theory and Signal Processing, vol. 18 of Oper. Theory: Adv. Appl. (Birkhäuser, Basel, 1986).

    Google Scholar 

  17. W.B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972) 1–62.

    Article  Google Scholar 

  18. W.B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mountain J. Math. 4 (1974) 213–225.

    Google Scholar 

  19. W.B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators and Gaussian quadrature on the unit circle, in:Numerical Methods in Linear Algebra, ed. E.S. Nikolaev (Moscow University Press, 1982) pp. 16–32 (in Russian).

  20. W.B. Gragg, The QR algorithm for unitary hessenberg Matrices, J. Comp. Appl. Math. 16 (1986) 1–8.

    Article  Google Scholar 

  21. W.B. Gragg and A. Lindquist, On the partial realization problem, Lin. Alg. Appl. 50 (1983) 277–319.

    Article  Google Scholar 

  22. M.H. Gutknecht, The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, the QD algorithm, biconjugate gradient squared algorithms and fast Hankel solvers, in:Proc. Copper Mountain Conf. on Iterative Methods (1990).

  23. M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Part I, SIAM J. Matrix Anal. Appl. 13 (1992) 594–639.

    Article  Google Scholar 

  24. M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Part II, SIAM J. Matrix Anal. Appl. 15 (1994) 15–58.

    Article  Google Scholar 

  25. M.H. Gutknecht and M. Hochbruck, Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems, Numer. Math. 70 (1995) 181–228.

    Article  Google Scholar 

  26. G. Heinig and K. Rost,Algebraic Methods for Toeplitz-like Matrices and Operators (Akademie Verlag, Berlin, 1984; also: Birkhäuser, Basel).

    Google Scholar 

  27. T. Kailath, Time-variant and time-invariant lattice filters for nonstationary processes, in:Outils et Modèles Mathématiques pour l'Automatique, l'Analyse de Systemès et le Traitement du Signal, vol. 2 (Editions du CNRS, Paris, 1982) pp. 417–464.

    Google Scholar 

  28. T. Kailath, S.Y. Kung and M. Morf, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl. 68 (1979) 395–407.

    Article  Google Scholar 

  29. H. Levi-Ari and T. Kailath, Triangular factorization of structured Hermitian matrices, in:I. Schur Methods in Operator Theory and Signal Processing, Oper. Theory: Adv. Appl., ed. I. Gohberg (Birkhäuser, Basel, 1986) pp. 301–324.

    Google Scholar 

  30. B.N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13 (1992) 567–593.

    Article  Google Scholar 

  31. A.H. Sayed, T. Constantinescu and T. Kailath, Time-variant displacement structure and interpolation problems, IEEE Trans. Auto. Contr. AC-39 (1994) 960–975.

    Article  Google Scholar 

  32. I. Schur, Uber Potenzreihen die im Innern des Einheitskreises Beschränkt sind, I, J. Reine Angew. Math. 147 (1917) 205–232, see also [16, pp. 31–59].

    Google Scholar 

  33. E.L. Stiefel, Kernel polynomials in linear algebra and their numerical applications, in:Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, vol. 49 of NBS Appl. Math. Series (NBS, 1958) pp 1–22.

  34. M. Van Barel and A. Bultheel, A parallel algorithm for discrete least squares rational approximation, Numer. Math. 63 (1992) 99–121.

    Article  Google Scholar 

  35. M. Van Barel and A. Bultheel, Discrete linearized least squares approximation on the unit circle, J. Comp. Appl. Math. 50 (1994) 545–563.

    Article  Google Scholar 

  36. M. Van Barel and A. Bultheel, Orthogonal polynomial vectors and least squares approximation for a discrete inner product, Electron. Trans. Numer. Anal. 3 (1995) 1–23.

    Google Scholar 

  37. J. van Iseghem, Approximants de Padé vectoriels, PhD thesis, Lille (1987).

  38. J. van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comp. Appl. Math. 19 (1987) 141–150.

    Google Scholar 

  39. H.S. Wall,Analytic Theory of Continued Fractions (Van Nostrand, Princeton, 1948).

    Google Scholar 

  40. P. Wynn, A general system of orthogonal polynomials, Quart. J. Math. 18 (1967) 81–96.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, K.U. Leuven, Leuven, Belgium

    Adhemar Bultheel & Marc Van Barel

Authors
  1. Adhemar Bultheel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Van Barel
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by C. Brezinski

This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bultheel, A., Van Barel, M. Formal orthogonal polynomials and Hankel/Toeplitz duality. Numer Algor 10, 289–335 (1995). https://doi.org/10.1007/BF02140773

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02140773

Keywords

Search

Navigation