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Multifractal Analysis Based on p-Exponents and Lacunarity Exponents

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Fractal Geometry and Stochastics V

Part of the book series: Progress in Probability ((PRPR,volume 70))

Abstract

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Hölder exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of “lacunarity” of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.

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Author information

Authors and Affiliations

  1. Signal, Systems and Physics, Physics Dept., CNRS UMR 5672, Ecole Normale Supérieure de Lyon, Lyon, France

    Patrice Abry & Roberto Leonarduzzi

  2. Université Paris Est, Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, UPEC, Créteil, France

    Stéphane Jaffard

  3. I2M, CNRS UMR 7373, Université Aix-Marseille, Marseille, France

    Clothilde Melot

  4. IRIT, CNRS UMR 5505, University of Toulouse, Toulouse, France

    Herwig Wendt

Authors
  1. Patrice Abry
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  2. Stéphane Jaffard
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  3. Roberto Leonarduzzi
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  4. Clothilde Melot
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  5. Herwig Wendt
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Corresponding author

Correspondence to Stéphane Jaffard .

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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany

    Christoph Bandt

  2. Mathematical Institute, University of St Andrews, St Andrews, United Kingdom

    Kenneth Falconer

  3. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Martina Zähle

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Abry, P., Jaffard, S., Leonarduzzi, R., Melot, C., Wendt, H. (2015). Multifractal Analysis Based on p-Exponents and Lacunarity Exponents. In: Bandt, C., Falconer, K., Zähle, M. (eds) Fractal Geometry and Stochastics V. Progress in Probability, vol 70. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18660-3_15

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