Wavelet leaders in multifractal analysis

The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Hölder singularities. The properties of this new multifractal formalism and of the alternative weak scaling exponent multifractal formalism are investigated. Examples based on known synthetic multifractal processes are illustrating its numerical implementation and abilities. ∗Address: Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France and Institut Universitaire de France. Email: [email protected] †Address: CNRS UMR 5672 Laboratoire de Physique, ENS de Lyon, 46, allée d’Italie, F-69364 Lyon cedex, France Email: [email protected] ‡Address: CNRS UMR 5672 Laboratoire de Physique, ENS de Lyon, 46, allée d’Italie, F-69364 Lyon cedex, France Email: [email protected]