MIXED MULTIFRACTAL ANALYSIS FOR FUNCTIONS: GENERAL UPPER BOUND AND OPTIMAL RESULTS FOR VECTORS OF SELF-SIMILAR OR QUASI-SELF-SIMILAR OF FUNCTIONS AND THEIR SUPERPOSITIONS

Mixed multifractal analysis for functions studies the Holder pointwise behavior of more than one single function. For a vector F = (f(1),..., f(L)) of L functions, with L >= 2, we are interested in the mixed Holder spectrum, which is the Hausdorff dimension of the set of points for which each function f(l) has exactly a given value al of pointwise Holder regularity. We will conjecture a formula which relates the mixed Holder spectrum to some mixed averaged wavelet quantities of F. We will prove an upper bound valid for any vector of uniform Holder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.