CONICET | Buscador de Institutos y Recursos Humanos

This book evolved from the written notes that were distributed to the students who participated in the CIMPA school: { em New Trends in Applied Harmonic Analysis:Sparse Representations, Compressed Sensing and Multifractal analysis } which took place in Mar del Plata (Argentina) in August 2013. This event was motivated by the recent interactions which developed between harmonic analysis, signal and image processing during the last 10 years. During that time, several technological deadlocks were solved through the resolution of deep theoretical problems in harmonic analysis. The purpose of this school was to focus on two particularly active areas which are representative of such advances. The courses were taught by leaders in these areas, and covered both theoretical aspects and applications. Most of the attendance was composed of PhD students and PostDocs from diverse backgrounds (mathematics, signal and image processing, ...), and the corresponding chapters of this book reflect the pedagogical care of the lecturers, in particular in the careful treatment of all needed prerequisites, and the illustration of the developments of each topic by several examples. Another original feature of this book is that some subjects overlap, with views taken from different perspectives, thus offering an in-depth picture of these scientific areas. Let us be more specific. Multifractal analysis offers new tools of classification for signals and images derived from their scaling invariance properties. The part of the book concerning this subject include the contribution of K. Hare, { em Multifractal analysis of Cantor-like measures}, which deals with basics of fractal analysis, and then focuses on the key example of Cantor-like measures. The contribution of Y. Heurteaux { em An introduction to Mandelbrot cascades} goes one step further in modeling complexity, and deals with the multifractal measures supplied by multiplicative cascades; a careful treatment of these examples is motivated both by the historical role played by these measures as models for the dissipation of energy in turbulent fluids, and by the importance that they have recently acquired in other areas of mathematics (fragmentation, coalescence, harmonic measure associated with fractal sets, Schramm-Loewner evolution,...). Finally, the contribution of Stéphane Seuret { em Multifractal analysis and Wavelets } deals with the extensions that these ideas have know in the setting of functions. The main tool here is wavelet analysis; a tool which is now prevalent in applied analysis, and reappears in several other chapters of this book. Here its role is to yield a characterization of both pointwise and global regularity of functions. This property explains the success of wavelets in applied multifractal analysis, since this subject can be seen as unfolding the relationships between pointwise and global regularity; and then deriving practical classification tools from these regularity characteristics. Recently, many powerful techniques have been developed emphasizing the role of sparsity in signal and image processing. These new methods have had a substantial impact in areas like sampling, data compression and representation, atomic decompositions, wavelets, frames and high dimensional data analysis. In particular compressed sensing represents a new paradigm in signal and image processing, allowing to reconstruct compressible data from the knowledge of an underdetermined system, through an $ell^1$ minimization. The mathematics behind these methods is rich and sophisticated and presents new challenges.Note that the contribution of Y. Heurteaux was not part of the courses taught at the CIMPA school of August 2013, but grew from the notes of another course taught at a fractal conference that took place in Porquerolles (France) in September 2013.

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