Multivariate Davenport series

We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$ and $\{x\}$ is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.Comment: 43 page

Random Wavelet Series: Theory and Applications

Random Wavelet Series form a class of random processes with multifractal properties. We give three applications of this construction. First, we synthesize a random function having any given spectrum of singularities satisfying some conditions (but including non-concave spectra). Second, these processes provide examples where the multifractal spectrum coincides with the spectrum of large deviations, and we show how to recover it numerically. Finally, particular cases of these processes satisfy a generalized selfsimilarity relation proposed in the theory of fully developed turbulence.Comment: To appear in Annales Math\'ematiques Blaise Pasca

Analysis of the Lack of Compactness in the Critical Sobolev Embeddings

AbstractLet (un) be a bounded sequence inHs,p(Rd) (0<s<d/p). We show that (un) has a subsequence (u′n) such that eachu′ncan be expressed as a finite sum (plus a remainder) of translations/dilations of functionsφmand such that the remainder has arbitrary small norm inLq(1/q=(1/p)−(s/d)). This generalizes a result obtained by Patrick Gérard for the casep=2

Multifractal analysis of images: New connexions between analysis and geometry

Natural images can be modelled as patchworks of homogeneous textures with rough contours. The following stages play a key role in their analysis: - Separation of each component - Characterization of the corresponding textures - Determination of the geometric properties of their contours. Multifractal analysis proposes to classify functions by using as relevant parameters the dimensions of their sets of singularities. This framework can be used as a classification tool in the last two steps enumerated above. Several variants of multifractal analysis were introduced, depending on the notion of singularity which is used. We describe the variants based on Hölder and L^p regularity, and we apply these notions to the study of functions of bounded variation (indeed the BV setting is a standard functional assumption for modelling images, which is currently used in the first step for instance). We also develop a multifractal analysis adapted to contours, where the regularity exponent associated with points of the boundary is based on an accessibility condition. Its purpose is to supply classification tools for domains with fractal boundaries

Function spaces vs. Scaling functions: Some issues in image classification

Criteria based on the computation of fractal dimensions have been used in order to perform image analysis and classification; we show that such criteria often amount to deter- mine the regularity of the image in some classes of function spaces, and that looking for richer criteria naturally leads to the introduction of new classes of function spaces. We will investigate the properties of some of these classes, and show which type of additional information they yield for the initial image

Comprehensive multifractal analysis of turbulent velocity using wavelet leaders

International audienceThe multifractal (MF) framework relates the scaling properties of turbulence to its local regu- larity properties through a statistical description as a collection of local singularities. The MF properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the MF properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well- founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The MF formalism based on the wavelet leaders (WL) is a new MF formalism which is the first to meet all these requests. This paper aims at its description and at its application to experimental turbulent velocity data. After a detailed discussion of the practical use of the MF formalism based on the WL the following questions are carefully investigated: (1) What is the dependence of MF properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which MF model does correctly account for the observed MF properties? Results from several data set analyses are used to discuss the dependence of the computed MF properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulants exponents up to order 4 confirms that the log-normal model (with c2 = −0.025 ± 0.002) correctly accounts for the universal MF properties of turbulent velocity