Multifractal analysis based on p-exponents and lacunarity exponents

Patrice Abry (1) , Stéphane Jaffard (2) , Roberto Leonarduzzi (1) , Clothilde Melot (3) , Herwig Wendt (4, 5)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 I2M - Institut de Mathématiques de Marseille
4 IRIT-SC - Signal et Communications
5 CNRS - Centre National de la Recherche Scientifique

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.

Mots clés: fractal; scale invariance; multifractal; Hausdorff dimension; Hölder regularity; wavelet; lacunarity exponent; p-exponent

Domaines: Analyse fonctionnelle [math.FA]

Patrice Abry, Stéphane Jaffard, Roberto Leonarduzzi, Clothilde Melot, Herwig Wendt. Multifractal analysis based on p-exponents and lacunarity exponents. Christoph Bandt et al., Eds. Fractal Geometry and Stochastics V, 70 (Part IV), Springer International Publishing, pp.279-313, 2016, Progress in Probability, 978-3-319-18659-7. ⟨10.1007/978-3-319-18660-3_15⟩. ⟨hal-01170037⟩