Skip to main content
Log in

Using the refinement equation for the construction of pre-wavelets

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

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

Access this article

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. Battle, A block spin construction of ondelettes. Part III: A note on pre-ondellettes, preprint, 1990.

  2. G. Battle, A block spin construction of ondelettes. Part I: Lemarié functions, Commun. Math. Phys. 110 (1987) 601–615.

    Google Scholar 

  3. A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc., to appear.

  4. C.K. Chui and J.Z. Wang, On compactly supported spline wavelets and duality principle, CAT Report 213, 1990.

  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.

  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. 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. 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. I. Daubechies, Orthonormal basis of compactly supported wavelets, Communications on Pure and Applied Mathematics 41 (1988) 909–996.

    Google Scholar 

  10. F.D. Gantmacher,The Theory of Matrices, Vol. II (Chelsea Publishing Co., New York, 1959).

    Google Scholar 

  11. T.N.T. Goodman and C.A. Micchelli, On refinement equations determined by Pólya frequency sequences, IBM Research Report No. 16275, 1990.

  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.

  13. S. Karlin,Total Positivity (Stanford University Press, Stanford, California, 1968).

    Google Scholar 

  14. T.Y. Lam, Serre's conjecture, Lecture Notes in Mathematics 635 (Springer Verlag, 1978).

  15. P.G. Lemarié, Ondelettes a localisation exponentielle, J. Math. Pures et Appli. 67 (1988) 227–236.

    Google Scholar 

  16. S.G. Mallat, Multiresolution approximations and wavelet orthonormal bais ofL 2 (R), Trans. Amer. Math. Soc. 315 (1989) 69–87.

    Google Scholar 

  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. L.H. Rowen,Ring Theory, Vol. II, Pure and Applied Mathematics, Vol. 128 (Academic Press Inc., Boston, 1988).

    Google Scholar 

  19. W. Rudin,Functional Analysis (McGraw-Hill Book Company, New York, 1973).

    Google Scholar 

  20. I.J. Schoenberg, Cardinal spline interpolation, CBMS Vol. 12, SIAM, Philadelphia, 1973.

    Google Scholar 

  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. B. Sturmfels, An algorithmic proof of the Quillen-Suslin theorem, IMA Preprint Series No. 409, 1988.

  23. R.J. Walker,Algebraic Curves (Princeton University Press, Princeton, 1950).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research partially supported by DARPA and NSF Grant INT-87-12424

Rights and permissions

Reprints and permissions

About this article

Cite this article

Micchelli, C.A. Using the refinement equation for the construction of pre-wavelets. Numer Algor 1, 75–116 (1991). https://doi.org/10.1007/BF02145583

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02145583

AMS(NOS) Classification

Keywords

Navigation