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
*King Saud University, Department of Mathematics, College of Science, P. O. Box 2455, Riyadh 11451, Saudi Arabia
†Faculty of Sciences of Monastir, Department of Mathematics, Computational Mathematics Laboratory, 5019, Monastir, Tunisia
‡Department of Mathematics, Preparatory Institute for Military Academies, Fondouk Jedid, 8012 Nabeul, Tunisia
Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector of functions, with , we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function has exactly a given value of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of . 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.


