Abstract
The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a “Weierstrass-like” fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.
Similar content being viewed by others
References
B. B. Mandelbrot,Fractals (Freeman, San Francisco, 1977).
J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617–656 (1985).
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman,Phys. Rev. A 33:1141 (1986).
L. S. Young,Ergod. Theor. Dyn. Syst. 2:109.
B. Schoeneberg, Elliptic modular functions, inDie Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 203.
D. Mumford,Tata Lectures on Theta 1, Birkhäuser, Boston.
T. M. Apostol,Modular Functions and Dirichlel Series in Number Theory (Springer-Verlag).
G. Doetsch,Handbuch der Laplace-Transformationen II (Birkhäuser, Basel, 1955).
P. Bessis, J. S. Geronimo, and P. Moussa,J. Stat. Phys. 34:75–110 (1984).
L. A. Smith, J. D. Fournier, and E. A. Spiegel,Phys. Lett. 114A:465–468 (1986).
A. Grossmann and J. Morlet,Decomposition of Functions into Wavelets of Constant Shape and Related Transforms (World Scientific, Singapore); A. Grossmann, J. Morlet, and T. Paul,J. Math. Phys. 26:2473–2479 (1985); I. Daubchies, A. Grossmann, and Y. Meyer,J. Math. Phys. 27:1271–1283 (1986).
R. Kronland-Martinet, J. Morlet, and A. Grossmann,J. Pattern Recognition Artificial Intelligence (Special Issue on Expert Systems and Pattern Analysis).
P. G. Lemarié and Y. Meyer,Rev. Iberoam. 1:1–18 (1986).
P. Collet, J. Lebowitz, and A. Porgio,J. Stat. Phys. 47:609–644 (1987).
T. Paul,Ann. Inst. H. Poincaré Phys. Theor.
E. M. Stein,Singular Integrals and Differentiability Properties of Functions (Princeton University Press).
M. V. Berry and Z. W. Lewis,Proc. R. Soc. Lond. A 370:459–484 (1980).
R. C. Gunning,Lectures on Modular Forms (Princeton University Press, 1962).
A. Zygmund,Trigonometric Series (Cambridge at the University Press, 1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holschneider, M. On the wavelet transformation of fractal objects. J Stat Phys 50, 963–993 (1988). https://doi.org/10.1007/BF01019149
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01019149