Skip to main content
SpringerLink
Log in

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

  • Published:
CALCOLO Aims and scope Submit manuscript

Abstract

In this article various existence results for theLDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by theLDU-factorizations of finite sections are discussed. The generalization of the results to theLDU-factorization of multi-index Toeplitz matrices is outlined.

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

Access this article

Log in via an institution

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. K. Amaratunga,A fast matlab routine for calculating Daubechies filters, Wavelet Digest4(4), (1995).

  2. F.L. Bauer,Ein direktes Iterations Verfahren zur Hurwitz-zerlegung eines Polynoms, Arch. Elektr. Uebertragung9, (1955) 285–290.

    Google Scholar 

  3. F.L. Bauer,Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen, ii. Direkte Faktorisierung eines Polynoms, Sitz. Ber. Bayer. Akad. Wiss., (1956) 163–203.

  4. B.P. Bogert, M.J.R. Healy, J.W. Tukey,The quefrency alanysis of time series for echoes: eepstrum pseudo-autocovariance, cross-cepstrum and saphe cracking, In M. Rosenblatt, editor, Proc. Symposium Time Series Analysis, pages 209–243, New York, 1963. John Wiley and Sons.

    Google Scholar 

  5. A. Calderón, F. Spitzer, H. Widom,Inversion of Toeplitz matrices, Illinois J. Math.3, (1959) 490–498.

    MATH  MathSciNet  Google Scholar 

  6. M. Cotronei, Multiwavelets: Analisi Teorica ed Algoritmi, PhD thesis, University of Messina, Italy, 1996.

    Google Scholar 

  7. I. Daubechies, Ten Lectures on Wavelets, volume 61 ofCBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1992.

  8. K.R. Davidson, Nest Algebras, volume 191 ofResearch Notes in Mathematics, Pitman London, 1988.

    Google Scholar 

  9. I. Gohberg, S. Goldberg, M. A. Kaashoek, Classes of Linear Operators, Vol. II, volume 63 ofOperator Theory: Advances and Applications, Birkhäuser, Basel-Boston, 1993.

    MATH  Google Scholar 

  10. I. Gohberg andM. A. Kaashoek,Projection methods for block Toeplitz operators with operator-valued symbols, In E.L. Basor and I. Gohberg, editors, Toeplitz Operators and Related Topics. The Harold Widom Anniversary Volume Workshop on Toeplitz and Wiener-Hopf Operators, Santa Cruz, California, September 20–22, 1992, volume 71 ofOperator Theory: Advances and Applications, pages 79–104, Basel-Boston, 1994. Birkhäuser.

    Google Scholar 

  11. I. Gohberg, P. Lancaster, L. Rodman,Spectral analysis of matrix polynomials. I. Canonical forms and divisors, Linear Algebra and Applications20, (1978) 1–44.

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

    MATH  Google Scholar 

  13. I. Gohberg andE.I. Sigal,An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Mat. Sbornik84(4), (1971) 607–629 (Russian, translated in Math. USSR Sbornik13(4), (1971) 603–625).

    MathSciNet  Google Scholar 

  14. I.C. Gohberg, I.A. Feldman, Convolution Equations and Projection Methods for their Solution, volume 41 ofTransl. Math. Monographs AMS, Providence, R.I., 1974.

    Google Scholar 

  15. I.C. Gohberg andM.G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, volume 24 ofTransl. Math. Monographs, AMS, Providence, R.I., 1970.

    MATH  Google Scholar 

  16. G.H. Golub andC.F. Van Loan, Matrix Computations, The John Hopkins University Press, Baltimore, third edition, 1996.

    MATH  Google Scholar 

  17. T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, S. Seatzu,On the Cholesky factorization of the Gram matrix of locally supported functions, BIT35(2), (1995) 233–257.

    Article  MATH  MathSciNet  Google Scholar 

  18. T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, S. Seatzu,On the limiting profile of exponentially decaying functions. Preprint, 1996.

  19. T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, S. Seatzu,Spectral factorization of Laurent polynomials. Adv. in Comp. Math. (1997). To appear.

  20. M.G. Krein,Integral equations on the half-line with kernel depending upon the difference of the arguments. Uspehi Mat. Nauk.13(5), (1958) 3–120 (Russian, translated in AMS Translations22, (1962) 163–288).

    MathSciNet  Google Scholar 

  21. S.L. Lee,B-splines for cardinal hermite interpolation, Linear Algebra and Applications12, (1975) 269–280.

    Article  MATH  Google Scholar 

  22. A.V. Oppenheim andR.W. Schafer, Discrete-Time Signal Processing, Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cliffs, NJ, 1989.

    Google Scholar 

  23. F. Riesz andB. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955.

    Google Scholar 

  24. S. Roch andB. Silbermann,C *-algebra techniques in numerical analysis, J. Operator Theory35, (1996), 241–280.

    MATH  MathSciNet  Google Scholar 

  25. L. Rodman, An Introduction to Operator Polynomials, volume 38 ofOperator Theory: Advances and Applications, Birkhäuser, Basel-Boston, 1989.

    Google Scholar 

  26. I.J. Schoenberg, Cardinal Spline Interpolation, volume 12 ofCBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1973.

  27. G. Strang andT. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996.

    Google Scholar 

  28. G. Wilson,Factorization of the covariance generating function of a pure moving average process, SIAM J. Numer. Anal.6(1), (1969) 1–7.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    S. Seatzu

Authors
  1. C. van der Mee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Rodriguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Seatzu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by the Italian Ministry of University and Scientific and Technological Research and by the Italian National Research Council

Rights and permissions

Reprints and permissions

About this article

Cite this article

van der Mee, C., Rodriguez, G. & Seatzu, S. LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices. Calcolo 33, 307–335 (1996). https://doi.org/10.1007/BF02576007

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02576007

Keywords

Search

Navigation