Single wavelets in n-dimensions

  • Paolo M. Soardi
  • David Weiland
Article

Abstract

Under very minimal regularity assumptions, it can be shown that 2n−1 functions are needed to generate an orthonormal wavelet basis for L2(ℝn). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝn such that the single function ψ, defined by\(\hat \psi = \chi K\), is an orthonormal wavelet for L2(ℝn). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.

Keywords

Tight Frame Orthonormal Wavelet Wavelet Family Dyadic Wavelet Orthonormal Wavelet Basis 

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References

  1. [1]
    Auscher, P. (1995). Solution of two problems on wavelets,J. Geom. Anal.,5(2), 181–236.MATHMathSciNetGoogle Scholar
  2. [2]
    David, G. (1991). Wavelets and singular integrals on curves and surfaces,Lecture Notes in Mathematics 1465, Springer-Verlag.Google Scholar
  3. [3]
    Dai, X., Larson, D., and Speegle, D.Wavelet sets inn, preprint.Google Scholar
  4. [4]
    Frazier, M., Garrigós, G., Wang, K., and Weiss, G.A characterization of functions that generate wavelet and related expansions, preprint.Google Scholar
  5. [5]
    Fang, X. and Wang, X. (1996). Construction of minimally supported frequency wavelets,J. Fourier Anal. Appl.,2(4), 315–327.MATHMathSciNetGoogle Scholar
  6. [6]
    Garrigós, G. (1996). Personal communication.Google Scholar
  7. [7]
    Gripenberg, G. (1995). A necessary and sufficient condition for the existence of a father wavelet,Studia Math.,114, 207–226.MATHMathSciNetGoogle Scholar
  8. [8]
    Hernández, E. and Weiss, G. (1996).A First Course on Wavelets, CRC Press, Boca Raton, FL.MATHGoogle Scholar
  9. [9]
    Hernández, E., Wang, X., and Weiss, G. (1996). Smoothing minimally supported frequency wavelets. Part I,J. Fourier Anal. Appl.,2(4), 329–340.MATHMathSciNetGoogle Scholar
  10. [10]
    Meyer, Y. (1990).Ondelettes et Opérateurs, Hermann, Paris.MATHGoogle Scholar
  11. [11]
    Ron, A. and Shen, Z.Affine systems in L 2(ℝd):the analysis of the analysis operator, preprint.Google Scholar
  12. [12]
    Wang, X.The study of wavelets from the properties of their Fourier transforms, Ph.D. thesis, Washington University, St. Louis, MO.Google Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Paolo M. Soardi
    • 1
  • David Weiland
    • 2
  1. 1.Dipartimento di MatematicaMilanoItaly
  2. 2.Department of MathematicsUniversity of Texas at AustinAustin

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