Numerical calculations of Hölder exponents for the Weierstrass functions with (min,+)-wavelets

Abdelouahab KENOUFI, Michel GONDRAN


For all function f : Rn to R one introduces (min; +)-wavelets which are lower and upper hulls build from (min; +) analysis.One shows at theoretical level and on numerical applications for the Weierstrass functions, that (min, +)-wavelets decomposition opens a non-linear branch to the multi-resolution analysis of a signal, in particular for the Hölder exponents calculation and Empirical Mode Decomposition (EMD).

Full Text:



[AAB+95] A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J.-F. Muzy. Ondelettes,

multifractales et turbulence. Diderot, Paris, 1995.

[Alt05] M. V. Altaisky. Wavelets, Theory, Applications and Implementation.

Universities Press, 2005.

[Ast94] N.M. Astaf'eva. Wavelet analysis : basic theory and some applications.

Uspekhi zicheskih nauk, 166(11) :11461170, 1994. (in Russian).

[Chu92] C.K. Chui. An Introduction to Wavelets. Academic Press Inc., 1992.

[Dau88] I. Daubechies. Orthonormal bases of compactly supported wavelets.

Comm. Pure. Apl. Math., 41 :909996, 1988.

[Dau92] I. Daubechies. Ten lectures on wavelets. S.I.A.M., Philadelphie, 1992.

[Far92] M. Farge. Wavelets and their application to turbulence. Annual review

of uid mechanics, 24 :395457, 1992.

[Gab46] D. Gabor. Theory of communication. Proc. IEE, 93 :429457, 1946.

[GGM85] P. Goupillaud, A. Grossmann, and J. Morlet. Cycle-octave and related

transforms in seismic signal analysis. Geoexploration, 23 :85102,



[GM84] A. Grossmann and J. Morlet. Decomposition of Hardy functions into

square-integrable wavelets of constant shape. SIAM J. Math. Anal.,

(4) :723736, 1984.

[Gon96] M. Gondran. Analyse minplus. C. R. Acad. Sci. Paris 323, (323) :371

, 1996.

[Gon99] M. Gondran. Convergences de fonctions à valeurs dans Rk et analyse

minplus complexe. C. R. Acad. Sci. Paris, (329) :783788, 1999.

[GRG03] P. Flandrin G. Rilling and Paulo Goncalvès. On empirical mode decomposition

and its algorithms. IEEE-EURASIP Workshop on Nonlinear

Signal and Image Processing NSIP-03, Grado (I), 2003.

[HL98] S. Long M. Wu H. Shih Q. Zheng N. Yen C. Tung . Huang, Z. Shen and

H. Liu. The empirical mode decomposition and the hilbert spectrum

for nonlinear and non-stationary time series analysis. Proceedings of the

Royal Society : Mathematical, Physical and Engineering Sciences, 454,

:903995, 1998.

[Hun98] B. R. Hunt. The hausdor dimension of graphs of weierstrass functions.

Proceedings of the American Mathematical Society, 126 :791800, 1998.

[Jaf00] S. Jaard. Décompositions en ondelettes,"developments of mathematics

-2000" (j-p. pier ed.). pages 609634, 2000.

[JLKY84] J. Mallet-Paret J. L. Kaplan and J. A. Yorke. The lyapounov dimension

of nowhere dierentianle attracting torus. Ergod. Th. Dynam. Sys.,

:261281, 1984.

[MAF82] J. Morlet, G. Arens, and I. Fourgeau. Wave propagation and sampling

theory. Geophysics, 47 :203226, 1982.

[MG08] M. Minoux M. Gondran. Graphs, Dioids and Semirings. Springer, 2008.

[Mor81] J. Morlet. Sampling theory and wave propagation. In Proc. 51st Annu.

Meet. Soc. Explor. Geophys., Los-Angeles, 1981.

[SF98] K. Shneider and M. Farge. Wavelet approach for modelling and computing

turbulence, volume 1998-05 of Advances in turbulence modelling.

Von Karman Institute for Fluid Dynamics, Bruxelles, 1998.

[Tri93] C. Tricot. Courbes et dimension fractale. Springer-Verlag, 1993.

[Wei67] K. Weierstrass. über continuirliche funktionen eines reellen arguments,

die für keinen werth des letzteren einen bestimmten dierentialquotienten

besitzen. Karl Weiertrass Mathematische Werke, Abhandlungen II,

Gelesen in der Königl. Akademie der Wissenchaften am 18 Juli 1872,

[Zim81] V.D. Zimin. Hierarchic turbulence model. Izv. Atmos. Ocean. Phys.,

:941946, 1981.


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:



Desenvolvido por:

Logomarca da Lepidus Tecnologia