From Multifractal Measures to Multifractal Wavelet Series

摘要

Given a positive locally finite Borel measure µ on R, a natural way to construct multifractal wavelet series $F_{mu}=sum_{jge0,kin Z}d_{j,k}psi_{j,k}(x)$ is to set $mid d_{j,k}mid = 2^{-j(s_0-1/p_0)}mu([k2^{-j},(k+1)2^{-j}))^{1/p_0}$ , where $s_0,p_0ge 0, s_0-1/p_0 >0$ . Indeed, under suitable conditions, it is shown that the function Fµ inherits the multifractal properties of µ. The transposition of multifractal properties works with many classes of statistically selfsimilar 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 Fµ are studied. As an application, multifractal Gaussian processes associated with Fµ are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arnéodo et al.