Wavelets on Fractals

Dutkay , Dorin E. ; Jorgensen , Palle E. T.

Rev. Mat. Iberoamericana, Tome 22 (2006) no. 2, p. 131-180 / Harvested from Project Euclid

Résumé

We show that there are Hilbert spaces constructed from the Hausdorff measures $\mathcal{H}^{s}$ on the real line $\mathbb{R}$ with $0 < s < 1$ which admit multiresolution wavelets. For the case of the middle-third Cantor set $\mathbf{C}\subset \lbrack 0,1]$, the Hilbert space is a separable subspace of $L^{2}(\mathbb{R},(dx)^{s})$ where $s=\log _{3}(2)$. While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale $3$ which covers the traditional Cantor set $\mathbf{C}$. Introducing \begin{equation*} \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) \end{equation*} we first describe the subspace in $L^{2}(\mathbb{R},(dx)^{s})$ which has the following family as an orthonormal basis (ONB): \begin{equation*} \psi_{i,j,k}(x)=2^{j/2}\psi_{i}(3^{j}x-k)\text{,} \end{equation*} where $i=1,2,j$, $k\in \mathbb{Z}$. Since the affine iteration systems of Cantor type arise from a certain algorithm in $\mathbb{R}^d$ which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.

Détail

Publié le : 2006-05-15
Classification:  Hausdorff measure,  Cantor sets,  iterated function systems (IFS),  fractal,  wavelets,  Hilbert space,  unitary operators,  orthonormal basis (ONB),  spectrum,  transfer operator,  cascade approximation,  scaling,  translation,  41A15,  42A16,  42A65,  42C40,  43A65,  46L60,  47D25,  46L45