Advertisement

Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

  • Arne Barinka
  • Titus Barsch
  • Stephan Dahlke
  • Mario Mommer
  • Michael Konik
Article
  • 52 Downloads

Abstract

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.

Gauss quadrature scaling functions wavelets splines error estimates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    L. Anderson, N. Hall, B. Jawerth, and G. Peters, Wavelets on closed subsets of the real line, in Topics in the Theory and Applications of Wavelets (L. L. Schumaker and G. Webb, eds.), Academic Press, Boston, 1994, pp. 1–61.Google Scholar
  2. 2.
    A. Barinka, T. Barsch, S. Dahlke, and M. Konik, Some remarks on quadrature formulas for refinable functions and wavelets, iZAMM 81(2001) 12, 839–855.Google Scholar
  3. 3.
    A. Barinka, T. Barsch, S. Dahlke, M. Konik, and M. Mommer, The IGPM quadrature machine, RWTH Aachen, 2000, http://www.igpm.rwth-aachen.de/barinka/mattoys/ soft.htmlGoogle Scholar
  4. 4.
    C. K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.Google Scholar
  5. 5.
    A. Cohen, I. Daubechies, and J. Feauveau, -orthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45, 485–560 (1992).Google Scholar
  6. 6.
    A. Cohen and I. Daubechies, Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9, 51–137 (1993).Google Scholar
  7. 7.
    A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast algorithms on an interval, C.R. Acad. Sci. Paris 316, 417–421 (1993).Google Scholar
  8. 8.
    W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica 6, Cambridge University Press, Cambridge, 1997, pp. 55–228.Google Scholar
  9. 9.
    W. Dahmen, A. Kunoth, and K. Urban, Biorthogonal spline-wavelets on the interval — Stability and moment conditions, Appl. Comp. Harm. Anal. 6, 1–52 (1992).Google Scholar
  10. 10.
    W. Dahmen and C. A. Micchelli, Recent progresses in multivariate splines, in Approximation Theory IV (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, New York, 1983, pp. 27–121.Google Scholar
  11. 11.
    W. Dahmen and C. A. Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal. 30(2), 507–537 (1993).Google Scholar
  12. 12.
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Math. 61, SIAM, Philadelphia, 1992.Google Scholar
  13. 13.
    P. Davis and P. Rabinowitz, Methods of Numerical Integration (W. Rheinboldt, ed.), Academic Press, New York, 1975.Google Scholar
  14. 14.
    R. Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of prewavelets II: Powers of two, in Curves and Surfaces (P. J. Laurent, A. LeMèhautè, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 209–246.Google Scholar
  15. 15.
    J.-P. Kahane and P.-G. Lemariè-Rieusset, Fourier Series and Wavelets, Gordon and Breach Science Publishers, Luxembourg, 1995.Google Scholar
  16. 16.
    V. I. Krylov, Approximate Calculation of Integrals, translated by A. H. Stroud, Macmillan, New York, 1962.Google Scholar
  17. 17.
    Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics 37, Cambridge, 1992.Google Scholar
  18. 18.
    J Stoer, Numerische Mathematik 1 (2nd ed.), Springer, Berlin, 1993.Google Scholar
  19. 19.
    A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, New Jersey, 1971.Google Scholar
  20. 20.
    A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, New Jersey, 1966.Google Scholar
  21. 21.
    G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications 13, AMS, Rhode Island, 1939.Google Scholar
  22. 22.
    P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, 1997.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Arne Barinka
    • 1
  • Titus Barsch
    • 1
  • Stephan Dahlke
    • 1
  • Mario Mommer
    • 1
  • Michael Konik
    • 2
  1. 1.RWTH AachenInstitut für Geometrie und Praktische MathematikAachenGermany
  2. 2.Fakultauml;t für MathematikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations