Part 1: Fractal behavior in time series The functions f(t) typically studied in mathematical analysis are continuous and have continuous derivatives. Hence, they can be approximated in the vicinity of some time t_{i} by a socalled Taylor series or power series 

(eqn. 1)  
For small regions around t_{i}, just a few terms of the expansion (eqn. 1) are necessary to approximate the function f(t). In contrast, most time series f(t) found in "reallife" applications appear quite noisy (Fig. 1). Therefore, at almost every point in time, they cannot be approximated either by Taylor series (or by Fourier series) of just a few terms. Moreover, many experimental or empirical time series have fractal featuresi.e., for some times t_{i}, the series f(t) displays singular behavior. By this, we mean that at those times t_{i}, the signal has components with noninteger powers of time which appear as steplike or cusplike features, the socalled singularities, in the signal (see Figs. 1b,c).  
Formally, one can write (2, 4): 

(eqn. 2)  
where t is inside a small vicinity of t_{i}, and h_{i} is a noninteger number quantifying the local singularity of f(t) at t = t_{i}.  
