Harmonic Analysis
Wavelet series built using multifractal measures
Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 353-356.

Let μ be a positive locally finite Borel measure on R. A natural way to construct multifractal wavelet series Fμ(x)=j0,kZdj,kψj,k(x) is to set |dj,k|=2j(s01/p0)μ([k2j,(k+1)2j))1/p0, where s0,p00, s01/p0>0. Under suitable conditions, the function Fμ inherits the multifractal properties of μ. The transposition of multifractal properties works with most classes of statistically self-similar multifractal measures. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of Fμ are studied. As an application, the multifractal spectrum of the celebrated W-cascades introduced by Arnéodo et al. is obtained.

Étant donnée une mesure borélienne positive μ définie sur R, il est naturel de lui associer une série d'ondelettes Fμ(x)=j0,kZdj,kψj,k(x) en prescrivant ses coefficients d'ondelettes de la façon suivante : on pose |dj,k|=2j(s01/p0)μ([k2j,(k+1)2j))1/p0, où s0,p00, s01/p0>0. Nous montrons comment les propriétés multifractales de la mesure μ peuvent se transmettre à la série d'ondelettes Fμ. Nous étudions la stabilité de la construction après perturbation des coefficients d'ondelettes. Ce travail permet de calculer le spectre multifractal des cascades aléatoires d'ondelettes d'Arnéodo et al.

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DOI: 10.1016/j.crma.2005.06.029
Barral, Julien 1; Seuret, Stéphane 1

1 Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France
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Barral, Julien; Seuret, Stéphane. Wavelet series built using multifractal measures. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 353-356. doi : 10.1016/j.crma.2005.06.029. http://www.numdam.org/articles/10.1016/j.crma.2005.06.029/

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