Wavelets on Fractals

We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on the line, there are also sharp contrasts. These are stated in our main result, a dichotomy theorem. The first section is the case of the middle-third Cantor set. This is followed by a review of the essentials on Hausdorff measure. The remaining sections of the paper cover multiresolutions in the general context of affine iterated function systems.
We have followed all the suggestions and corrections made by the referee. In the introduction, we added an overview of the three themes in the paper, and added some motivation. This was motivated by the comments and suggestions of the referee. We also rearranged the material as suggested by the referee. This makes the three themes in our paper more clear, we believe