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

Abstract

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector [Formula: see text] of [Formula: see text] functions, with [Formula: see text], we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function [Formula: see text] has exactly a given value [Formula: see text] of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of [Formula: see text]. We will prove an upper bound valid for any vector of uniform Hölder 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.