Using the Matrix Refinement Equation for the Construction of Wavelets II: Smooth Wavelets on [0,1]

  • Charles A. Micchelli
  • Yuesheng Xu
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


This paper continues the work in [4] on constructing orthogonal bases on the interval [0,1] by using the matrix refinement equation and the two basic operations of translation and scale. We call the elements of these bases wavelets. Here we amplify on the applicability of our method and construct smooth wavelets with and without boundary conditions. That is, we describe a procedure to recursively generate orthonormal bases with any prescribed number of continuous derivatives. As a caveat to the reader we reiterate our remark above that this paper is a continuation of our work in [4] and therefore some familiarity with [4] is assumed.


Spline Function Homogeneous Boundary Condition Spline Space Refinement Equation Refinement Operator 


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    Schumaker L. L. Spline Functions: Basic Theory. Wiley-Interscience, New York, 1981.MATHGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Charles A. Micchelli
    • 1
  • Yuesheng Xu
    • 2
  1. 1.IBM T. J. Watson Research CenterYorktown HeightUSA
  2. 2.Department of MathematicsNorth Dakota State UniversityFargoUSA

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