Random Wavelet Series

Abstract:

This paper concerns the study of functions which are known through the statistics of their wavelet coefficients. We first obtain sharp bounds on spectra of singularities and spectra of oscillating singularities, which are deduced from the sole knowledge of the wavelet histograms.

Then we study a mathematical model which has been considered both in the contexts of turbulence and signal processing: random wavelet series, obtained by picking independently wavelet coefficients at each scale, following a given sequence of probability laws. The sample paths of the processes thus constructed are almost surely multifractal functions, and their spectrum of singularities and their spectrum of oscillating singularities are determined. The bounds obtained in the first part are optimal, since they become equalities in the case of random wavelet series. This allows to derive a new multifractal formalism which has a wider range of validity than those that were previously proposed in the context of fully developed turbulence.