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Reports (Research Report) Year : 2003

2-microlocal Formalism


This paper is devoted to the study of a fine way to measure the local regularity of distributions. Starting from the 2-microlocal analysis introduce- d by J.M. Bony, we develop a 2-microlocal formalism, much in the spirit of the multifractal formalism. This allows to define a new regularity function, that we call the 2-microlocal spectrum. The 2-microlocal spectrum proves to be a powerful tool that we apply in three directions. First, it allows to recover all previously known results on local regularity exponents, as well as to discover new properties about them. Second, the 2-microlocal spectrum provides a deeper understanding of the 2-microlocal frontiers. It yields in particular a natural way of prescribing these frontiers on a countable dense set of points. Finally, we explore the close parallel between the multifractal and 2-microlocal formalisms. These applications are illustrated on examples such as the Weierstrass and the Riemann functions, as well as lacunary wavelet series.
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Dates and versions

inria-00071846 , version 1 (23-05-2006)


  • HAL Id : inria-00071846 , version 1


Jacques Lévy Véhel, Stéphane Seuret. 2-microlocal Formalism. [Research Report] RR-4741, INRIA. 2003. ⟨inria-00071846⟩
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