From multifractal measures to multifractal wavelet series

Given a positive locally finite Borel measure A on R, a natural way to construct multifractal wavelet series F mu(x) = Sigma(-)j >= 0,k is an element of z(d)j,k Psi j,k(x) is to set vertical bar(d)jk vertical bar = 2(-j(s0-1/P0)) It ([k2(-j), (k+1)2(-j)))(1/P0), where s(0), p(0) >= 0, s(0)-1/p(0) > 0. Indeed, under suitable conditions, it is shown that the function F-mu inherits the multifractal properties of mu. The transposition of multifractal properties works with many classes of statistically self-similar multifractal measures, enlarging the class of processes which have self-similarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of F-mu are studied. As an application, multifractal Gaussian processes associated with F-mu are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arneodo et al.