JournalsrmiVol. 22, No. 1pp. 131–180

Wavelets on Fractals

  • Dorin E. Dutkay

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

    University of Iowa, Iowa City, USA
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Abstract

We show that there are Hilbert spaces constructed from the Hausdorff measures Hs\mathcal{H}^{s} on the real line R\mathbb{R} with 0<s<10 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C[0,1]\mathbf{C}\subset \lbrack 0,1], the Hilbert space is a separable subspace of L2(R,(dx)s)L^{2}(\mathbb{R},(dx)^{s}) where s=log3(2)s=\log _{3}(2). While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale 33 which covers the traditional Cantor set C\mathbf{C}. Introducing

ψ1(x)=2χC(3x1)\mboxandψ2(x)=χC(3x)χC(3x2)\psi_{1}(x)=\sqrt{2}\chi _{\mathbf{C}}(3x-1) \qquad\mbox{and}\qquad \psi _{2}(x)=\chi _{\mathbf{C}}(3x)- \chi_{\mathbf{C}}(3x-2)

we first describe the subspace in L2(R,(dx)s)L^{2}(\mathbb{R},(dx)^{s}) which has the following family as an orthonormal basis (ONB):

ψi,j,k(x)=2j/2ψi(3jxk),\psi_{i,j,k}(x)=2^{j/2}\psi_{i}(3^{j}x-k)\text{,}

where i=1,2,ji=1,2,j, kZk\in \mathbb{Z}. Since the affine iteration systems of Cantor type arise from a certain algorithm in Rd\mathbb{R}^d 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