Wavelets Techniques for Pointwise Anti-Hölderian Irregularity

Clausel, Marianne; Nicolay, Samuel

doi:10.1007/s00365-010-9120-9

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Wavelets Techniques for Pointwise Anti-Hölderian Irregularity

Clausel, Marianne; Nicolay, Samuel

2011 • In Constructive Approximation, 33, p. 41-75

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https://hdl.handle.net/2268/79758

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10.1007/s00365-010-9120-9

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Keywords :

Pointwise Hölder regularity; Wavelets; Spectrum of singularities; Multifractal formalism

Abstract :

[en] In this paper, we introduce a notion of weak pointwise Hölder regularity, starting from the definition of the pointwise anti-Hölder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise Hölder regularity.We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (referred to here as strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.

Disciplines :

Mathematics

Author, co-author :

Clausel, Marianne

Nicolay, Samuel ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Language :

English

Title :

Wavelets Techniques for Pointwise Anti-Hölderian Irregularity

Publication date :

2011

Journal title :

Constructive Approximation

ISSN :

0176-4276

eISSN :

1432-0940

Publisher :

Springer

Volume :

Pages :

41-75

Peer reviewed :

Peer Reviewed verified by ORBi

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since 18 December 2010

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Bibliography

Abry, P., Lashermes, B., Jaffard, S.: Wavelets leaders in multifractal analysis, wavelets analysis and applications. In: Tao, Q., Mang, I. V., Xu, Y. (eds.) Applied and Numerical Harmonic Analysis, pp. 219-264. Birkäuser, Basel (2006).
Abry, P., Jaffard, S., Vedel, B., Wendt, H.: The contribution of wavelets in multifractal analysis (submitted).
Adler, R. J.: The Geometry of Random Field. Wiley, New York (1981).
Aouidi, J., Slimane, M. B.: Multifractal formalism for quasi self-similar functions. J. Stat. Phys. 108, 541-589 (2002).
Arneodo, A., Bacry, E., Muzy, J.-F.: The thermodynamics of fractals revisited with wavelets. Physica A 213, 232-275 (1995).
Barral, J.: Moments, continuité et analyse multifractale des martingales de Mandelbrot. Probab. Theory Relat. Fields 113, 535-569 (1999).
Barral, J.: Continuity of the multifractal spectrum of a random statistically self-similar measures. J. Theor. Probab. 13, 1027-1060 (2000).
Barral, J., Seuret, S.: From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11(5), 589-614 (2005).
Berman, S. M.: Gaussian sample functions: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46, 63-86 (1972).
Berman, S. M.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23(1), 69-86 (1973).
Bousch, T., Heurteaux, Y.: On oscillations of Weierstrass-type functions. Manuscript (1999).
Bousch, T., Heurteaux, Y.: Caloric measure on domains bounded by Weierstrass type graphs. Ann. Acad. Sci. Fenn. Math. 25(2), 501-522 (2000).
Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775-790 (1992).
Clausel, M.: Etude de quelques notions d'irrégularité: le point de vue ondelettes. Thesis (2008).
Clausel, M., Nicolay, S.: Some prevalent results about strongly monoHölder functions. Nonlinearity 23, 2101-2116 (2010).
Collet, P., Koukiou, F.: Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329-342 (1992).
Daoudi, K., Lévy-Véhel, J., Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. 14, 349-385 (1998).
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909-996 (1988).
Demichel, Y., Tricot, C.: Analysis of fractal sum of pulses. Math. Proc. Camb. Philos. Soc. 141, 355-370 (2006).
DeVore, R. A., Sharpley, R. C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47 (1984).
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1990).
Geman, D., Horowitz, J.: Occupation densities. An. Probab. 8, 1-67 (1980).
Hardy, G. H.: Weierstrass's non differentiable function. Trans. Am. Math. Soc. 17, 301-325 (1916).
Heurteaux, Y.: Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. Henri Poincaré 34, 309-338 (1998).
Heurteaux, Y.: Weierstrass function with random phases. Trans. Am. Math. Soc. 355(8), 3065-3077 (2003).
Heurteaux, Y.: Weierstrass function in Zygmund's class. Proc. Am. Math. Soc. 133(9), 2711-2720 (2005).
Holley, R., Waymire, E. C.: Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab. 2, 819-845 (1992).
Jaffard, S.: The spectrum of singularities of Riemann's function. Rev. Mat. Iberoam. 12, 441-460 (1996).
Jaffard, S.: Multifractal formalism for functions. SIAM J. Math. Anal. 28, 944-998 (1997).
Jaffard, S.: Wavelet techniques in multifractal analysis, fractal geometry and applications. Proc. Symp. Pure Math. 72, 91-151 (2004).
Jaffard, S.: On Davenport expansions. Proc. Symp. Pure Math. 72, 91-151 (2004).
Jaffard, S., Nicolay, S.: Pointwise smoothness of space-filling functions. Appl. Comput. Harmon. Anal. 26(2), 181-199 (2009).
Kahane, J.-P.: Produits de poids aléatoires et indépendants et applications. In: Bélair, J., Dubuc, S. (eds.) Fractal Geometry and Analysis, pp. 277-324 (1991).
Kaplan, J. L., Mallet-Paret, J., Yorke, J. A.: The Lyapounov dimension of a nowhere differentiable torus. Ergod. Theory Dyn. Syst. 4, 261-281 (1984).
Krantz, S. G.: Lipschitz spaces, smoothness of functions, and approximation theory. Expo. Math. 1, 193-260 (1983).
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, New York (1998).
Mandelbrot, B. B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331-358 (1974).
Mauldin, R. D., Williams, S. C.: On the Hausdorff dimensions of some graphs. Trans. Am. Math. 298(2), 793-803 (1986).
Meyer, Y.: Ondelettes et opérateurs. Hermann, Paris (1990).
Michon, G.: Une construction des mesures de Gibbs sur certains ensembles de Cantor. C. R. Acad. Sci. Paris 308, 315-318 (1989).
Michon, G.: Mesures de Gibbs sur les Cantor réguliers. Ann. Inst. Henri Poincaré, Phys. Théor. 58(3), 267-285 (1993).
Molchan, G. M.: Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681-702 (1996).
Nicolay, S.: About the pointwise Hölder exponents of some functions. Preprint (2008).
Parisi, G., Frisch, U.: On the singularity spectrum of fully developped turbulence. In: Turbulence and Predictability in Geophysical Fluid Dynamics, Proceedings of the International Summer School in Physics Enrico Fermi, pp. 84-87. North-Holland, Amsterdam (1985).
Pitt, L. D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27, 309-330 (1978).
Przytycki, F., Urbanski, M.: On the Hausdorff dimension of some fractal sets. Stud. Math. 93, 155-186 (1989).
Tricot, C.: Courbes et Dimension Fractale. Springer, Berlin (1992).
Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields 109, 129-157 (1997).
Xiao, Y.: Properties of local nondeterminism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV, 157-193 (2005).
Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1962, pp. 145-212. Springer, New York (2009).