المنشورات و المؤلفات
We define --anisotropic two-microlocal spaces by decay conditions on anisotropic wavelet coefficients on any --anisotropic wavelet basis of . We prove that these spaces allow the characterizing of pointwise anisotropic Hölder regularity. We also...
The notion of gentle spaces, introduced by Jaffard, describes what would be an “ideal” function space to work with wavelet coefficients. It is based mainly on the separability, the existence of bases, the homogeneity, and the γ-stability. We prove...
he study of d dimensional traces of functions of m several variables leads to directional behaviors. The purpose of this paper is two-fold. Firstly, we extend the notion of one direction pointwise Hölder regularity introduced by Jaffard to multi-...
Many natural mathematical objects, as well as many multi-dimensional signals and images from real physical problems, need to distinguish local directional behaviors (for tracking contours in image processing for example). Using some results of...
We study the de Rham function: the unique continuous (nowhere differentiable) function with satisfying the functional equation . We show that its pointwise Hölder regularity differs widely from point to point, and the values of fill an interval...
We characterize pointwise directional regularity by highly oriented multi-scaled wavelet coefficients.
In this paper, we prove that anisotropic homogeneous Besov spaces are gentle spaces, for all parameters s,p,q and all anisotropies . Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity. We...
Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is...
The oscillation spaces Os,s′p(Rd)Ops,s′(Rd) introduced by Jaffard are a variation on the definition of Besov spaces for either s ≥ 0 or s ≤ −d/p. On the contrary, the spaces Os,s′p(Rd)Ops,s′(Rd) for −d/p < s < 0 cannot be sharply imbedded...
The multifractal formalism for functions has been proved to be valid for a large class of selfsimilar functions. All the functions that have been studied are (or turned out to be) associated with a family of contractions which satisfies some...
A function f is self-similar if, modulo an error regular function g, it (i.e., f ) is invariant under specific transformations involving mainly dilations and translations (known as the piecewise linear dynamical systems). The multi-fractal formalism...
We review the solutions of some functional equations for which the differentiability was the aim of study in the beginning of the last century. By decomposing these functions on the Schauder basis (which are only Lipschitz of order 1), we determine...
The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the...
Selfsimilar functions can be written as the superposition of similar structures, at different scales, generated by a function g. Their expressions look like wavelet decompositions. In the case where g is regular, the multifractal formalism has been...
We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{ö}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness'' index, and finally we...
In this paper we prove that the conjectures of Frisch and Parisi and Arneodo et al. (called the multifractal formalism for functions) may fail for some non-homogeneous selfsimilar functions on ℝ2. In these cases, we compute the correct spectrum of...
Abstract
We study the Multifractal Formalism for certain classes of functions having selfsimilarity properties.
Summary The aim of this thesis is the multifractai analysis of selfsimilar functions and the study of the validity of the multifractai formalism. First, we determine the exact pointwise Holder regularity for functions such that locally the graph is...