Wavelets pp 21-37 | Cite as

Orthonormal Wavelets

  • Y. Meyer
Part of the inverse problems and theoretical imaging book series (IPTI)

Abstract

The aim of this survey ori the theory of wavelets is to help the scientific community to use wavelets as an alternative to the standard Fourier analysis. This survey is a written and extended version of a lecture I have been asked to give at the international conference held at Marseille on ondelettes, méthodes temps-fréquences et espaces des phases (December 14–18, 1987). This conference was a remarkable success. People with distinct scientific educations could communicate and interact, using the new born concept of wavelets. As often the flexibility of the concept helped a lot and during the lunches and dinners, physicians, physicists and even mathematicians were surprised and delighted to understand each other. Now my task is less pleasant and I have to do my job, giving precise definitions and describing specific algorithms. The advantage will be to prepare the ground for programming these algorithms.

Keywords

Convolution Sine Stein Acoustics Metaphor 

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References

  1. [1]
    R.Balian. Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C.R.Acad.Sc.Paris, t.292, Série 11, 1357–1361, 1981.MathSciNetGoogle Scholar
  2. [2]
    G.Battle. A block spin construction of ondelettes. Part I: Lemarié functions. Commun. Math. Phys. 110, (1987), 601–615.CrossRefADSGoogle Scholar
  3. [3]
    G.Battle. A block spin construction of ondelettes. Part II: the QFT connection. Commun. Math. Phys. 114, (1988), 93–102.CrossRefADSGoogle Scholar
  4. [4]
    I.Daubechies. Orthonormal bases of compactly supported wavelets. (Preprint from AT&T Bell Laboratories, 600 Montain Avenue, Murray Hill, NJ 07974.Google Scholar
  5. [5]
    I.Daubechies. The wavelet transform, time-frequency localization and signal analysis (AT&T Bell).Google Scholar
  6. [6]
    K.Grochenig. Analyse multiéchelles et bases d’ondelettes. C.R.Acad.Sci.Paris, t.305, Série I, (1987), 13–17.MathSciNetGoogle Scholar
  7. [7]
    S.Jaffard et Y.Meyer. Bases d’ondelettes dans des ouverts de Rn. A paraître au Journal de Math. Pures et Appliquées.Google Scholar
  8. [8]
    P.G.Lemarié et Y. Meyer. Ondelettes et bases hilbertiennes. Revista Matematica Iberoamericana 2, (1986), 1–18.MathSciNetGoogle Scholar
  9. [9]
    S.Mallat. Multiresolution approximation and wavelets.(Preprint from GRASP lab, Dept. of Computer Science, University of Pennsylvania Philadelphia, PA 19104–6389.Google Scholar
  10. [10]
    Y.Meyer. Wavelets and operators. Cahiers mathématiques de la décision n° 8704.Google Scholar
  11. [11]
    Ondelettes, fonctions splines et analyses graduées. Cahiers n°8703.Google Scholar
  12. [12]
    H.C. Schweinler and E.P. Wigner. Orthogonalization methods. Journal of Mathematical Physics. Vol.II, (1970), 1693–1694.Google Scholar
  13. [13]
    J. O. Stromberg, A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces, Conference in Harmonic Analysis in honor of Antoni Zygmund, Vol.II, 475–493, edited by W.Beckner and al., Wadworth math, series.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Y. Meyer
    • 1
  1. 1.CeremadeUniversité Paris DauphineParis Cedex 16France

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