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Wavelet transforms of self-similar processes. (English) Zbl 0735.76039

Summary: This paper deals with the wavelet transforms of self-similar random processes, of the kind assumed in the Kolmogorov (1941) theory of turbulence. It is shown that, after suitable rescaling, the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform. The rescaling depends on the scaling exponent. The statistical properties of the resulting random fluctuations, such as its correlation function, depend on the choice of the analyzing wavelet.
Some implications are: (i) the presence of fluctuations in \(\log-\log\) plots of the absolute value of wavelet transforms versus the scale; (ii) an estimate of the small scale fluctuations, generalizing the iterated logarithm law of Brownian motion; (iii) an ultraviolet ergodic formula giving the scaling exponents in terms of zoom-averages. Some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data.

MSC:

76F20 Dynamical systems approach to turbulence
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[1] Arnéodo, A.; Grasseau, G.; Holschneider, M., Phys. Rev. Lett., 61, 2287 (1988)
[2] Farge, M.; Rabreau, S., Compt. Rend. Acad. Sci. Paris II, 307, 1479 (1988)
[3] Argoul, F.; Arnéodo, A.; Grasseau, G.; Gagne, Y.; Hopfinger, E. J.; Frisch, U., Nature, 338, 51 (1989)
[4] Everson, R.; Sirovich, L., Report 89-182 (1989), Center for Fluid Mechanics, Brown University
[5] Bacry, E.; Arnéodo, A.; Frisch, U.; Gagne, Y.; Hopfinger, E. J., (Métais, O.; Lesieur, M., Turbulence and Coherent Structures (1990), Kluwer: Kluwer Dordrecht), 203
[6] Yamada, M.; Ohkitani, K., Progr. Theor. Phys., 83, 819 (1990)
[7] Everson, R.; Sirovich, L.; Sreenivasan, K. R., Phys. Lett. A, 145, 314 (1990)
[8] Farge, M.; Guezennec, Y.; Ho, C. M.; Menevau, C., Continuous Wavelet Analysis of Coherent Structures, Center for Turbulence Research, (Proceedings of the Summer Program (1990)), 1
[9] Menevau, C. M., Analysis of turbulence in the orthonormal wavelet representation (1990), preprint
[10] Benzi, R.; Vergassola, M., Optimal wavelet transform and its application to two dimensional turbulence, (Proc. Int. Workshop on Novel Experiments. Proc. Int. Workshop on Novel Experiments, Fluid Dyn. Res., 8 (1991)), 117
[11] Kolmogorov, A. N., Dokl. Akad. Nauk SSSR, 30, 301 (1941)
[12] Mandelbrot, B. B., Compt. Rend. Acad. Sci. Paris, 260, 3274 (1965)
[13] Flandrin, P., Wavelet analysis and synthesis of fractional brownian motion (1991), preprint
[14] Lévy, P., Processus Stochastiques et Mouvement Brownien (1965), Gauthiers-Villars: Gauthiers-Villars Paris
[15] Monin, A. S.; Yaglom, A. M., (Lumley, J., Statistical Fluid Mechanics, Vol. 2 (1975), MIT Press: MIT Press Cambridge, MA)
[16] Meyer, Y., Ondelettes (1990), Hermann: Hermann Paris · Zbl 0646.42015
[17] Holschneider, M.; Tchamitchian, P., (Lemarié, P. G., Les Ondelettes en 1989 (1990), Springer: Springer Berlin), 102
[18] Kahane, J. P., Some Random Series of Functions (1985), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[19] Smith, L. A.; Fournier, J. D.; Spiegel, E. A., Phys. Lett. A, 114, 465 (1986)
[20] Anselmet, F.; Gagne, Y.; Hopfinger, E. J.; Antonia, R. A., J. Fluid Mech., 140, 63 (1984)
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