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Journal of Fourier Analysis and Applications

, Volume 20, Issue 5, pp 961–984 | Cite as

A Family of Functions with Two Different Spectra of Singularities

  • Claire Coiffard
  • Clothilde Mélot
  • Thomas Willer
Article

Abstract

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\)-exponent. We prove that in the general case the Hölder and \(p\)-exponent are different 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 Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.

Keywords

Multifractal analysis Pointwise regularity Wavelet bases Fractionnal derivatives 

Mathematics Subject Classification

26A16 26A33 26B35 42C40 

Notes

Acknowledgments

The authors would like to thank Stéphane Jaffard and the two referees for their insightful and accurate comments, which helped to improve the paper significantly.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Claire Coiffard
    • 1
  • Clothilde Mélot
    • 1
  • Thomas Willer
    • 1
  1. 1.CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance

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