A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I)

Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the
set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many real-life signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the so-called thermodynamic multifractal formalism, based on the Frish-Parisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal.
In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens.