Wavelet series built using multifractal measures

Julien Barral; Stéphane Seuret

doi:10.1016/j.crma.2005.06.029

Harmonic Analysis

Wavelet series built using multifractal measures

Presented by: Yves Meyer

Julien Barral ¹; Stéphane Seuret ¹

¹ Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 353-356.

Abstract
Résumé

Let μ be a positive locally finite Borel measure on $R$ . A natural way to construct multifractal wavelet series $F_{μ} (x) = \sum_{j ⩾ 0, k \in Z} d_{j, k} ψ_{j, k} (x)$ is to set $| d_{j, k} | = 2^{- j (s_{0} - 1 / p_{0})} μ ([k 2^{- j}, {(k + 1) 2^{- j}))}^{1 / p_{0}}$ , where $s_{0}, p_{0} ⩾ 0$ , $s_{0} - 1 / p_{0} > 0$ . Under suitable conditions, the function $F_{μ}$ 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 $F_{μ}$ 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 $F_{μ} (x) = \sum_{j ⩾ 0, k \in Z} d_{j, k} ψ_{j, k} (x)$ en prescrivant ses coefficients d'ondelettes de la façon suivante : on pose $| d_{j, k} | = 2^{- j (s_{0} - 1 / p_{0})} μ ([k 2^{- j}, {(k + 1) 2^{- j}))}^{1 / p_{0}}$ , où $s_{0}, p_{0} ⩾ 0$ , $s_{0} - 1 / p_{0} > 0$ . Nous montrons comment les propriétés multifractales de la mesure μ peuvent se transmettre à la série d'ondelettes $F_{μ}$ . 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.

Received: 2005-04-01
Accepted: 2005-06-21
Published online: 2005-09-01

DOI: 10.1016/j.crma.2005.06.029

Author's affiliations:

Julien Barral ¹; Stéphane Seuret ¹

¹ Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France

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     title = {Wavelet series built using multifractal measures},
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     pages = {353--356},
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     number = {6},
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Julien Barral; Stéphane Seuret. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.029/

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