Abstract
In this paper a simple algorithm based on harmonic wavelets is given for the generation of self similar functions. Due to their self similarity property and scale dependence, harmonic wavelets might offer a good approximation of fractals by a very few instances of the wavelet series and a more direct interpretation of the scale invariance for deterministic localized fractals.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abry, P., Goncalves, P., Lévy-Véhel, J.: Lois d’éschelle, Fractales et ondelettes, Hermes (2002)
Arnéodo, A., Grasseau, G., Holschneider, M.: Wavelet Transform of Multifractals. Phys. Review Letters 61(20), 2281–2284 (1988)
Borgnat, P., Flandrin, P.: On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation. Applied and Computational Harmonic Analysis 15, 134–146 (2003)
Cattani, C.: Harmonic Wavelets towards Solution of Nonlinear PDE. Computers and Mathematics with Applications 50(8-9), 1191–1210 (2005)
Cattani, C., Rushchitsky, J.J.: Wavelet and Wave Analysis as applied to Materials with Micro or Nanostructure. Series on Advances in Mathematics for Applied Sciences, p. 74. World Scientific, Singapore (2007)
Cattani, C.: Wavelet extraction of a pulse from a periodic signal. In: Gervasi, O., Murgante, B., Laganà, A., Taniar, D., Mun, Y., Gavrilova, M.L. (eds.) ICCSA 2008, Part I. LNCS, vol. 5072, pp. 1202–1211. Springer, Heidelberg (2008)
Cattani, C.: Shannon Wavelets Theory. Mathematical Problems in Engineering 2008, pgs. 24 (2008); Article ID 164808
Cattani, C.: Harmonic Wavelet Approximation of Random, Fractal and High Frequency Signals. Telecommunication Systems (to appear, 2009)
Chui, C.K.: An Introduction to Wavelets. Academic Press, New York (1992)
Daubechies, I.: Ten Lectures on wavelets. SIAM, Philadelphia (1992)
Dutkay, D.E., Jorgensen, P.E.T.: Wavelets on Fractals. Rev. Mat. Iberoamericana 22(1), 131–180 (2006)
Falconer, K.: Fractal Geometry. John Wiley, New York (1977)
Jorgensen, P.E.T.: Analysis and Probability, Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, Heidelberg (2006)
Mallat, S.: A Wavelet tour of signal processing. Academic Press, London (1998)
Muniandy, S.V., Moroz, I.M.: Galerkin modelling of the Burgers equation using harmonic wavelets. Phys. Lett. A 235, 352–356 (1997)
Newland, D.E.: Harmonic wavelet analysis. Proc. R. Soc. Lond. A 443, 203–222 (1993)
Wornell, G.: Signal Processing with Fractals: A Wavelet-Based Approach. Prentice-Hall, Englewood Cliffs (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cattani, C. (2009). Fractals Based on Harmonic Wavelets. In: Gervasi, O., Taniar, D., Murgante, B., Laganà, A., Mun, Y., Gavrilova, M.L. (eds) Computational Science and Its Applications – ICCSA 2009. ICCSA 2009. Lecture Notes in Computer Science, vol 5592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02454-2_56
Download citation
DOI: https://doi.org/10.1007/978-3-642-02454-2_56
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02453-5
Online ISBN: 978-3-642-02454-2
eBook Packages: Computer ScienceComputer Science (R0)