Multifractal analysis based on p-exponents and lacunarity exponents Chapitre d’ouvrage - Janvier 2016

Patrice Abry, Stéphane Jaffard, Roberto Leonarduzzi, Clothilde Melot, Herwig Wendt

Patrice Abry, Stéphane Jaffard, Roberto Leonarduzzi, Clothilde Melot, Herwig Wendt, « Multifractal analysis based on p-exponents and lacunarity exponents  », in Christoph Bandt et al., Eds. (ed.), Fractal Geometry and Stochastics V, 2016, pp. 279-313. ISBN 978-3-319-18659-7

Abstract

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the H\"older 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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