# 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 $H_{s}$ on the real line $R$ with $0<s<1$ which admit multiresolution wavelets. For the case of the middle-third Cantor set $C⊂[0,1]$, the Hilbert space is a separable subspace of $L_{2}(R,(dx)_{s})$ where $s=g_{3}(2)$. 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 $C$. Introducing

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

where $i=1,2,j$, $k∈Z$. Since the affine iteration systems of Cantor type arise from a certain algorithm in $R_{d}$ 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