Advances in Computational Mathematics

, Volume 7, Issue 4, pp 455–545 | Cite as

Regularity of multiwavelets

  • C.A. Micchelli
  • Thomas Sauer
Article

Abstract

The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets.

The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary Lp spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.

subdivision refinable functions regularity multiwavelets 39B12 41A15 41A25 65D99 

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Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • C.A. Micchelli
    • 1
  • Thomas Sauer
    • 2
  1. 1.T.J. Watson Research CenterIBM Department of Mathematical SciencesYorktown HeightsUSA
  2. 2.Institute of MathematicsUniversity Erlangen-NurembergErlangenGermany

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