Wavelets on Fractals

Abstract

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.