Comptes Rendus
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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.06.029
Julien Barral 1; Stéphane Seuret 1

1 Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France
@article{CRMATH_2005__341_6_353_0,
     author = {Julien Barral and St\'ephane Seuret},
     title = {Wavelet series built using multifractal measures},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {353--356},
     publisher = {Elsevier},
     volume = {341},
     number = {6},
     year = {2005},
     doi = {10.1016/j.crma.2005.06.029},
     language = {en},
}
TY  - JOUR
AU  - Julien Barral
AU  - Stéphane Seuret
TI  - Wavelet series built using multifractal measures
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 353
EP  - 356
VL  - 341
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2005.06.029
LA  - en
ID  - CRMATH_2005__341_6_353_0
ER  - 
%0 Journal Article
%A Julien Barral
%A Stéphane Seuret
%T Wavelet series built using multifractal measures
%J Comptes Rendus. Mathématique
%D 2005
%P 353-356
%V 341
%N 6
%I Elsevier
%R 10.1016/j.crma.2005.06.029
%G en
%F CRMATH_2005__341_6_353_0
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/

[1] A. Arnéodo; E. Bacry; J.F. Muzy Random cascades on wavelet dyadic trees, J. Math. Phys., Volume 39 (1998) no. 8, pp. 4142-4264

[2] J. Barral, S. Seuret, From multifractal measures to multifractal wavelet series, J. Fourier Anal. Appl. (2005) in press

[3] J. Barral, S. Seuret, Inside singularity sets of random Gibbs measures, J. Stat. Phys. (2005), in press; J. Barral, S. Seuret, Renewal of singularity sets of statistically self-similar measures, Preprint, 2004

[4] G. Brown; G. Michon; J. Peyrière On the multifractal analysis of measures, J. Statist. Phys., Volume 66 (1992) no. 3–4, pp. 775-790

[5] A.H. Fan Limsup deviations on trees, Anal. Theory Appl., Volume 20 (2004), pp. 113-148

[6] S. Jaffard On lacunary wavelet series, Ann. Appl. Probab., Volume 10 (2000) no. 1, pp. 313-329

[7] S. Jaffard Wavelet techniques in multifractal analysis, Proc. Symp. Pure Math., Volume 72 (2004) no. 2

[8] J.-P. Kahane; J. Peyrière Sur certaines martingales de Benoît Mandelbrot, Adv. Math., Volume 22 (1976), pp. 131-145

[9] B. Mandelbrot Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid. Mech., Volume 62 (1974), pp. 331-358

[10] Y. Meyer Ondelettes et Opérateurs, Hermann, 1990

[11] S. Seuret, Analyse de régularité locale. quelques applications à l'analyse multifractale, Thèse, Ecole Polytechnique, 2003

Cited by Sources:

Comments - Policy


Articles of potential interest

Sums of Dirac masses and conditioned ubiquity

Julien Barral; Stéphane Seuret

C. R. Math (2004)


A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades

Julien Barral; Stéphane Seuret

C. R. Math (2005)


On multifractal time subordination

Stéphane Seuret

C. R. Math (2008)