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 formalism was initially introduced by Frisch and Parisi in order to numerically determine the spectrum of fully turbulent fluids: it was later extended by Arneodo, Bacry, and Muzy using wavelet techniques and has since been used by many physicists. Until now, it has only been supported by heuristic arguments and verified for a few specific examples. Our purpose is to investigate the mathematical validity of these formulas; 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 decreasing) and that the different formulas allow us to determine either the whole increasing part of their spectrum or a part of it. One of these methods (the wavelet-maxima method) yields the complete spectrum of the self-similar functions. We also discuss the implications of these results for fully developed turbulence.