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Comptes Rendus Mathématique
Volume 341, n° 6
pages 353-356 (septembre 2005)
Doi : 10.1016/j.crma.2005.06.029
Received : 1 April 2005 ;  accepted : 21 June 2005
Wavelet series built using multifractal measures
Séries dʼondelettes issues de mesures multifractales
 

Julien Barral , Stéphane Seuret
Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France 

Abstract

Let μ be a positive locally finite Borel measure on  . A natural way to construct multifractal wavelet series   is to set  , where  ,  . Under suitable conditions, the function   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   are studied. As an application, the multifractal spectrum of the celebrated  -cascades introduced by Arnéodo et al. is obtained. To cite this article: J. Barral, S. Seuret, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Étant donnée une mesure borélienne positive μ définie sur  , il est naturel de lui associer une série dʼondelettes   en prescrivant ses coefficients dʼondelettes de la façon suivante : on pose  , où  ,  . Nous montrons comment les propriétés multifractales de la mesure μ peuvent se transmettre à la série dʼondelettes  . Nous étudions la stabilité de la construction après perturbation des coefficients dʼondelettes. Ce travail permet de calculer le spectre multifractal des cascades aléatoires dʼondelettes dʼArnéodo et al. Pour citer cet article : J. Barral, S. Seuret, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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