# Wavelets on Fractals

### Dorin E. Dutkay

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

University of Iowa, Iowa City, USA

## 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

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.

## 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