A family of functions with two different spectra of singularities

Claire Coiffard (1) , Clothilde Melot (2) , Thomas Willer (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 I2M - Institut de Mathématiques de Marseille

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coeffi cients. We compute at each point the pointwise Holder exponent of these functions and also their local Lp regularity, computing the so-called p-exponent. We prove that in the general case the Holder and p exponent are di fferent at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Holder and Lp local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for the functions in that family.

Mots clés: analyse multifractale; ondelettes; régularité ponctuelle; oscillations; espaces de Besov; formalisme multifractal

Domaines: Mathématiques générales [math.GM]

Claire Coiffard, Clothilde Melot, Thomas Willer. A family of functions with two different spectra of singularities. Journal of Fourier Analysis and Applications, 2014, 20 (5), pp.961-984. ⟨10.1007/s00041-014-9341-6⟩. ⟨hal-00831404v2⟩