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Wavelets Techniques for Pointwise Anti-Hölderian Irregularity

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Abstract

In this paper, we introduce a notion of weak pointwise Hölder regularity, starting from the definition of the pointwise anti-Hölder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise Hölder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (referred to here as strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.

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Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050 du CNRS, Université Paris Est, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Marianne Clausel

  2. Institut de Mathématique, Grande Traverse, 12, Bâtiment B37, 4000, Liège (Sart-Tilman), Belgium

    Samuel Nicolay

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  1. Marianne Clausel
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  2. Samuel Nicolay
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Correspondence to Samuel Nicolay.

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Communicated by Stephane Jaffard.

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Clausel, M., Nicolay, S. Wavelets Techniques for Pointwise Anti-Hölderian Irregularity. Constr Approx 33, 41–75 (2011). https://doi.org/10.1007/s00365-010-9120-9

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  • DOI: https://doi.org/10.1007/s00365-010-9120-9

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