Numerical Algorithms

, Volume 1, Issue 1, pp 75–116 | Cite as

Using the refinement equation for the construction of pre-wavelets

  • Charles A. Micchelli
Article

Abstract

A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarié, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method “can produce only the Lemarié Ondelettes”. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.

AMS(NOS) Classification

41A15 41A65 42B99 65D07 

Keywords

Cube splines multiresolution analysis refinement equation wavelets 

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References

  1. [1]
    G. Battle, A block spin construction of ondelettes. Part III: A note on pre-ondellettes, preprint, 1990.Google Scholar
  2. [2]
    G. Battle, A block spin construction of ondelettes. Part I: Lemarié functions, Commun. Math. Phys. 110 (1987) 601–615.Google Scholar
  3. [3]
    A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc., to appear.Google Scholar
  4. [4]
    C.K. Chui and J.Z. Wang, On compactly supported spline wavelets and duality principle, CAT Report 213, 1990.Google Scholar
  5. [5]
    C.K. Chui and J.Z. Wang, A cardinal spline approach to wavelets, talk presented at NSF/CBMS Conference on Wavelets, Lowell, Mass., 1990.Google Scholar
  6. [6]
    W. Dahmen and C.A. Micchelli, On stationary subdivision and the construction of compactly supported orthonormal wavelets, multivariate approximation and interpolation, eds. N. Haussmann and K. Jetter, ISNM Series in Mathematics (Birkhäuser Verlag, Basel 1990) 69–90.Google Scholar
  7. [7]
    W. Dahmen and C.A. Micchelli, On the local linear independence of translates of a box spline, Studied Mathematica 82 (1985) 243–263.Google Scholar
  8. [8]
    W. Dahmen and C.A. Micchelli, Subdivision algorithms for the generation of box spline surfaces, Computer Aided Geometric Design 1 (1984) 115–129.Google Scholar
  9. [9]
    I. Daubechies, Orthonormal basis of compactly supported wavelets, Communications on Pure and Applied Mathematics 41 (1988) 909–996.Google Scholar
  10. [10]
    F.D. Gantmacher,The Theory of Matrices, Vol. II (Chelsea Publishing Co., New York, 1959).Google Scholar
  11. [11]
    T.N.T. Goodman and C.A. Micchelli, On refinement equations determined by Pólya frequency sequences, IBM Research Report No. 16275, 1990.Google Scholar
  12. [12]
    R.Q. Jia and C.A. Micchelli, On linear independence for integer translates of a finite number of functions, University of Waterloo Research Report, CS-90-10, 1990.Google Scholar
  13. [13]
    S. Karlin,Total Positivity (Stanford University Press, Stanford, California, 1968).Google Scholar
  14. [14]
    T.Y. Lam, Serre's conjecture, Lecture Notes in Mathematics 635 (Springer Verlag, 1978).Google Scholar
  15. [15]
    P.G. Lemarié, Ondelettes a localisation exponentielle, J. Math. Pures et Appli. 67 (1988) 227–236.Google Scholar
  16. [16]
    S.G. Mallat, Multiresolution approximations and wavelet orthonormal bais ofL 2 (R), Trans. Amer. Math. Soc. 315 (1989) 69–87.Google Scholar
  17. [17]
    C.A. Micchelli, CardinalL-splines, in:Studies in Splines and Approximation Theory, eds. S. Karlin, C.A. Micchelli, A. Pinkus and I.J. Schoenberg Academic Press, New York, 1976) p. 203–250.Google Scholar
  18. [18]
    L.H. Rowen,Ring Theory, Vol. II, Pure and Applied Mathematics, Vol. 128 (Academic Press Inc., Boston, 1988).Google Scholar
  19. [19]
    W. Rudin,Functional Analysis (McGraw-Hill Book Company, New York, 1973).Google Scholar
  20. [20]
    I.J. Schoenberg, Cardinal spline interpolation, CBMS Vol. 12, SIAM, Philadelphia, 1973.Google Scholar
  21. [21]
    J. Stromberg, A modified Franklin system and higher-order systems ofR n as unconditional basis for Hardy spaces,Conf. in Harmonic Analysis in Honor of A. Zygmund, Vol. 2, (Wadsworth Inc., Belmont, California, 1983) p. 475–493.Google Scholar
  22. [22]
    B. Sturmfels, An algorithmic proof of the Quillen-Suslin theorem, IMA Preprint Series No. 409, 1988.Google Scholar
  23. [23]
    R.J. Walker,Algebraic Curves (Princeton University Press, Princeton, 1950).Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1991

Authors and Affiliations

  • Charles A. Micchelli
    • 1
  1. 1.Department of Mathematical SciencesIBM T.J. Watson Research CenterYorktown HeightsUSA

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