Numerical Algorithms

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

Using the refinement equation for the construction of pre-wavelets

  • Charles A. Micchelli


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 


Cube splines multiresolution analysis refinement equation wavelets 


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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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