Wavelet analysis of Self Similar Functions

The self-similarity property of some kind of fractals is sudied by using Harmonic Wavelets. The scale invariance of fractals is compared with the scale dependence of wavelets. Harmonic wavelets are complex values wavelets with sharp compact support in frequency domain, which show some kind of self-similarity with respect to scale changes. Due to their self similarity property and scale dependence, harmonic wavelets can offer an expedient tool for a good approximation and analysis of a suitable class of deterministic (finite energy) fractals. It is shown that (localized, deterministic) fractals are characterized by their wavelet coefficients. In particular, the Riemann-Weierstrass and Riemann-Cellerier function can be easily represented in terms of harmonic wavelets, in the sense that their wavelet coefficients can be easily (and analitically) computed. Some equations for generating self-similar functions are eventually given as well.