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Wavelet notes

May, 2003
81 pages
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Abstract:
Wavelets are a useful basis for constructing solutions of the integral and differential equations of scattering theory. Wavelet bases efficiently represent functions with smooth structures on different scales, and the matrix representation of operators in a wavelet basis are well-approximated by sparse matrices. The basis functions are related to solutions of a linear renormalization group equation, and the basis functions have structure on all scales. Numerical methods based on this renormalization group equation are discussed. These methods lead to accurate and efficient numerical approximations to the scattering equations. These notes provide a detailed introduction to the subject that focuses on numerical methods. We plan to provide periodic updates to these notes.
Note:
  • 80 pages, 4 figures
  • [1]
    Orthonormal bases of compactly supported wavelets
    • I. Daubechies
      • Commun.Pure Appl.Math. 41 (1988) 909
  • [2]

    Wavelets and Dilation Equations: A Brief Introduction

    • G. Strang
      • SIAM Rev. 31 4
  • [3]
    Ten Lectures on Wavelets Philadelphia, 1992
    • I. Daubechies
    • [4]
      A tutorial in Theory and Applications Press, 1992
      • C.K. Chui Wavelets
      • [5]

        Quadrature rules needed in Galerkin-wavelets methods

        • W.-C. Shann
        • [6]

          Quadratures involving polynomials and Daubechies’ wavelets

          • W.-C. Shann
            ,
          • J.-C. Yan
          • [7]
            to Wavelets, Birkhauser 1994
            • G. Kaiser
              ,
            • A. Friendly Guide
            • [8]

              Quadrature Formulae and Asymptotic Error Expansions for wavelet approximations of smooth functions

              • W. Sweldens
                ,
              • R. Piessens
                • SIAM J.Numer.Anal. 31 (1994) 1240
            • [9]
              Wavelet Analysis, The Scalable Structure of Information Verlag, NY
              • H.L. Resnikoff
                ,
              • R.O. Wells
              • [10]
                Wavelets through a Looking Glass, Birkhauser, 2002. In addition, some of the material in these notes is in our paper
                • O. Bratelli
                  ,
                • P. Jorgensen
                • These equations ensure that polynomials of degree < K - 1 can be locally represented by finite linear combinations of scaling functions on a fixed scale. This is a useful property for numerical approximations. The order scaling function has 2K scaling coefficients, with K = 1 corresponding to the Haar wavelets, and each additional value of K adds one more orthogonality condition. The scaling equation (95) and the moment conditions (104) for the mother wavelet function gives the additional equations necessary to find the Daubechies scaling coefficients, hl: 0 = (xn, ψ) = (Dxn, Dψ) = dxxn 2-n-1/2 m gmφ(x - m). This gives m dx(x + m)n gmφ(x) = 0. For n = 0 this gives (using the n = 0 equation) gm = 0, → m (-1)m hl-m = 0
                  • K-Daubechies