A Unified Framework for the Study of the 2-microlocal and Large Deviation Multifractal Spectra

Abstract : The large deviation multifractal spectrum is a function of central importance in multifractal analysis. It allows a ne description of the distribution of the singularities of a function over a given domain. The 2-microlocal spectrum, on the other hand, provides an extremely precise picture of the regularity of a distribution at a point. These two spectra display a number of similarities: their de nitions use the same kind of ingredients; both functions are semi-continuous; the Legendre transform of the two spectra yields a function of independent interest: the 2-microlocal frontier in 2-microlocal analysis, and the "\tau " function in multifractal analysis. This paper investigates further these similarities by providing a common framework for the de nition and study of the spectra. As an application, we obtain slightly generalized versions of the 2-microlocal and weak multifractal formalisms (with simpler proofs), as well as results on the inverse problems for both spectra.
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Self similar processes and their applications, Jul 2009, Angers, France. 28, pp.13-44, 2012, Séminaires et Congrès. 〈http://smf4.emath.fr/Publications/SeminairesCongres/2012/28/html/smf_sem-cong_28_13-44.php〉
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Antoine Echelard, Jacques Lévy Véhel, Claude Tricot. A Unified Framework for the Study of the 2-microlocal and Large Deviation Multifractal Spectra. Self similar processes and their applications, Jul 2009, Angers, France. 28, pp.13-44, 2012, Séminaires et Congrès. 〈http://smf4.emath.fr/Publications/SeminairesCongres/2012/28/html/smf_sem-cong_28_13-44.php〉. 〈inria-00612342〉

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