Numerical Analysis and Applications

, Volume 3, Issue 3, pp 199–207 | Cite as

Inverses of cyclic band matrices and the convergence of interpolation processes for derivatives of periodic interpolation splines



Entries of matrices inverse to cyclic band ones, as well as norms of the inverse matrices, are estimated. The estimates obtained are used to specify conditions under which interpolation processes with periodic odd degree splines converge for different derivatives. In particular, we give a positive solution to the C. de Boor problem on unconditional convergence of one of two middle derivatives without imposing any restrictions on a grid in the periodic case.

Key words

band(ed) matrix cyclic band(ed) matrix inverse matrix norm of matrix spline interpolation convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zav’yalov, Yu.S., Kvasov, B.I., and Miroshnichenko, V.L., Metody splain-funktsii (Spline-Function Methods), Moscow: Nauka, 1980.MATHGoogle Scholar
  2. 2.
    Volkov, Yu.S., Estimating Elements of the Inverse of a Cyclic Band Matrix, Sib. Zh. Vych. Mat., 2003, vol. 6, no. 3, pp. 263–267.MATHGoogle Scholar
  3. 3.
    Demko, S., Inverses of Band Matrices and Local Convergence of Spline Projections, SIAM J. Num. An., 1977, vol. 14, no. 4, pp. 616–619.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    de Boor, C., Odd-Degree Spline Interpolation at a Biinfinite Knot Sequence, Lect. Notes Math., vol. 556, pp. 30–53.Google Scholar
  5. 5.
    Volkov, Yu.S., Unconditional Convergence of One More Middle Derivative for Odd-Degree Spline Interpolation, Dokl. RAN, 2005, vol. 401, no. 5, pp. 592–594.Google Scholar
  6. 6.
    de Boor, C., On Bounding Spline Interpolation, J. Approx. Th., 1975, vol. 14, no. 3, pp. 191–203.MATHCrossRefGoogle Scholar
  7. 7.
    Higham, N.J., Estimating the Matrix p-Norm, Num. Math., 1992, vol. 62, no. 4, pp. 539–555.MATHMathSciNetGoogle Scholar
  8. 8.
    Ahlberg, J.H., Nilson, E.N., and Walsh, J.L., Teoriya splainov i ee prilozheniya (The Theory of Splines and Its Applications), Moscow: Mir, 1972.Google Scholar
  9. 9.
    Schumaker, L.L., Spline Functions: Basic Theory, New York: Wiley, 1981.MATHGoogle Scholar
  10. 10.
    Volkov, Yu.S., Constructing Interpolating Polynomial Splines, Vych. Sist., 1997, issue 159, pp. 3–18.Google Scholar
  11. 11.
    Volkov, Yu.S., Uniform Convergence of Derivatives of Odd-Degree Spline Interpolation, Preprint of Sobolev Inst. Math. SB RAS, Novosibirsk, 1984, no. 62.Google Scholar
  12. 12.
    Zmatrakov, N.L., Convergence of Interpolation Processes for Parabolic and Cubic Splines, Trudy Steklov Inst. Math., 1975, vol. 138, pp. 71–93.MATHMathSciNetGoogle Scholar
  13. 13.
    Zmatrakov, N.L., Uniform Convergence of Third Derivatives of Interpolating Cubic Splines, Vych. Sist., 1977, issue 72, pp. 10–29.Google Scholar
  14. 14.
    Shadrin, A.Yu., The L -Norm of the L 2-Spline Projector Is Bounded Independently of the Knot Sequence: A Proof of de Boor’s Conjecture, Acta Math., 2001, vol. 187, no. 1, pp. 59–137.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Volkov, Yu.S., Divergence of Odd-Degree Spline Interpolation, Vych. Sist., 1984, issue 106, pp. 41–56.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations