Wavelets on Fractals

  • Dorin E. Dutkay

    Rutgers University, Piscataway, USA
  • Palle E.T. Jorgensen

    University of Iowa, Iowa City, USA


We show that there are Hilbert spaces constructed from the Hausdorff measures on the real line with which admit multiresolution wavelets. For the case of the middle-third Cantor set , the Hilbert space is a separable subspace of where . While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale which covers the traditional Cantor set . Introducing

we first describe the subspace in which has the following family as an orthonormal basis (ONB):

where , . Since the affine iteration systems of Cantor type arise from a certain algorithm in which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.

Cite this article

Dorin E. Dutkay, Palle E.T. Jorgensen, Wavelets on Fractals. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 131–180

DOI 10.4171/RMI/452