MULTIFRACTAL FORMALISM FOR FUNCTIONS .1. RESULTS VALID FOR ALL FUNCTIONS

The multifractal formalism for functions relates some functional norms of a signal to its ''Holder spectrum'' (which is the dimension of the set of points where the signal has a given Holder regularity). This f ormalism was initially introduced by Frisch and Parisi in order to num erically determine the spectrum of fully turbulent fluids: it was late r extended by Arneodo, Bacry, and Muzy using wavelet techniques and ha s since been used by many physicists. Until now, it has only been supp orted by heuristic arguments and verified for a few specific examples. Our purpose is to investigate the mathematical validity of these form ulas; in particular, we obtain the following results: The multifractal formalism yields for any function an upper bound of its spectrum. We introduce a ''case study,'' the self-similar functions; we prove that these functions have a concave spectrum (increasing and then decreasin g) and that the different formulas allow us to determine either the wh ole increasing part of their spectrum or a part of it. One of these me thods (the wavelet-maxima method) yields the complete spectrum of the self-similar functions. We also discuss the implications of these resu lts for fully developed turbulence.