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 $\mathbf{C}\subset [0,1]$, 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

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

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.