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.
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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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Communicated by Albert Cohen.
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Coiffard, C., Mélot, C. & Willer, T. A Family of Functions with Two Different Spectra of Singularities. J Fourier Anal Appl 20, 961–984 (2014). https://doi.org/10.1007/s00041-014-9341-6
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DOI: https://doi.org/10.1007/s00041-014-9341-6