Scale Invariance and Beyond: What Can We Learn from Wavelet Analysis ?

Abstract

In many situations in physics as well as in some applied sciences, one is faced to the problem of characterizing very irregular functions [1–8]. The examples range from plots of various kind of random walks, e.g. Brownian signals [9], to financial time-series [1], to geological shapes [1,6], to medical time-series [5], to interfaces developing in far from equilibrium growth processes [3,4,8], to turbulent velocity signals [7,10] and to “DNA walks” coding nucleotide sequences [11,12]. These functions can be qualified as fractal functions [1,2,9] whenever their graphs are fractal sets in ℝ ² (for our purpose here we will only consider functions from ℝ to ℝ). They are commonly called self-affine functions since their graphs are similar to themselves when transformed by anisotropic dilations: ∀ _x0 ∈ ℝ,∃H ∈ ℝ; such that for any λ > 0, one has

$$f\left( {{x_0} + \lambda x} \right) - f\left( {{x_0}} \right) \simeq {\lambda ^H}\left( {f\left( {{x_0} + x} \right) - f\left( {{x_0}} \right)} \right)$$

(1)