A pure jump Markov process with a random singularity spectrum

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.Comment: 20 pages, 4 figure

Holder exponents of arbitrary functions

International audienceThe functional class of Holder exponents of continuous function has been completely characterized by P. Andersson, K. Daoudi, S. Jaffard, J. Levy Vehel and Y. Meyer [1, 2, 6, 9]; these authors have shown that this class exactly corresponds to that of the lower limits of the sequences of nonnegative continuous functions. The problem of determining whether or not the Holder exponents of discontinuous (and even unbounded) functions can belong to a larger class remained open during the last decade. The main goal of our article is to show that this is not the case: the latter Holder exponents can also be expressed as lower limits of sequences of continuous functions. Our proof mainly relies on a "wavelet-leader" reformulation of a nice characterization of pointwise Holder regularity due to P. Anderson

L'analyse par ondelettes

International audienc

Construction et proprietes des bases d'ondelettes. Remarques sur la controlabilite exacte

SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

New trends in applied harmonic analysis: sparse representations, compressed sensing, and multifractal analysis

This volume is a selection of written notes corresponding to courses taught at the CIMPA School: "New Trends in Applied Harmonic Analysis: Sparse Representations, Compressed Sensing and Multifractal Analysis". New interactions between harmonic analysis and signal and image processing have seen striking development in the last 10 years, and several technological deadlocks have been solved through the resolution of deep theoretical problems in harmonic analysis. New Trends in Applied Harmonic Analysis focuses on two particularly active areas that are representative of such advances: multifractal analysis, and sparse representation and compressed sensing. The contributions are written by leaders in these areas, and covers both theoretical aspects and applications. This work should prove useful not only to PhD students and postdocs in mathematics and signal and image processing, but also to researchers working in related topics