MULTIFRACTAL FORMALISM FOR FRACTAL SIGNALS - THE STRUCTURE-FUNCTION APPROACH VERSUS THE WAVELET-TRANSFORM MODULUS-MAXIMA METHOD

Several attempts have been made recently to generalize the multifracta l formalism, originally introduced for singular measures, to fractal s ignals. We report on a systematic comparison between the structure-fun ction approach, pioneered by Parisi and Frisch [in Proceedings of the International School on Turbulence and Predictability in Geophysical F luid Dynamics and Climate Dynamics, edited by M. Ghil, R. Benzi, and G . Parisi (North-Holland, Amsterdam, 1985), p. 84] to account for the m ultifractal nature of fully developed turbulent signals, and an altern ative method we have developed within the framework of the wavelet-tra nsform analysis. We comment on the intrinsic limitations of the struct ure-function approach; this technique has fundamental drawbacks and do es not provide a full characterization of the singularities of a signa l in many cases. We demonstrate that our method, based on the wavelet- transform modulus-maxima representation, works in most situations and is likely to be the ground of a unified multifractal description of se lf-affine distributions. Our theoretical considerations are both illus trated on pedagogical examples and supported by numerical simulations.