The Annals of Applied Probability

On lacunary wavelet series

Stéphane Jaffard

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We prove that the Hölder singularities of random lacunary wavelet series are chirps located on random fractal sets. We determine the Hausdorff dimensions of these singularities, and the a.e. modulus of continuity of the series. Lacunary wavelet series thus turn out to be a new example of multifractal functions.

Article information

Ann. Appl. Probab., Volume 10, Number 1 (2000), 313-329.

First available in Project Euclid: 25 April 2002

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Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 28A80: Fractals [See also 37Fxx]

Wavelet bases Hausdorff dimensions chirps Hölder regularity modulus of continuity


Jaffard, Stéphane. On lacunary wavelet series. Ann. Appl. Probab. 10 (2000), no. 1, 313--329. doi:10.1214/aoap/1019737675.

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  • [1] Arenedo, A., Bacry, E. and Muzy, J.-F. (1995). The thermodynamics of fractals revisited with wavelets. Phys. A 213 232-275.
  • [2] Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 19-90.
  • [3] DeVore, R., Jawerth, B. and Popov, V. (1992). Compression of wavelet decompositions Amer. J. Math. 114 737-795.
  • [4] DeVore, R. and Lucier, B. (1990). High order regularityfor conservation laws. Indiana Univ. Math. J. 39 413-430.
  • [5] DeVore, R. and Lucier, B. (1992). Fast wavelet techniques for near-optimal image processing. Proceedings IEEE Mil. Comm. Conf.
  • [6] Donoho, D. (1995). De-Noising via soft-thresholding. IEEE Trans. Inform. Theory 41 613- 627.
  • [7] Donoho, D., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc. Ser. B 57 301-369.
  • [8] Falconer, I. (1990). Fractal Geometry. Wiley, New York.
  • [9] Frisch, U. and Parisi, G. (1985). Fullydeveloped turbulence and intermittency. Proc. Internat. Summer School of Phys. Enrico Fermi 84-88. North-Holland, Amsterdom.
  • [10] Guiheneuf, B., Jaffard, S. and L´evy-V´ehel, J. (1998). Two results concerning chirps and 2-microlocal exponents prescription. Appl. Comput. Harmon. Anal. 5 487-492.
  • [11] Houdr´e, C. (1994). Wavelets, probabilityand statistics: some bridges. In Wavelets: Mathematics and Applications (J. J. Benedetto and M. Frazier, eds.) 365-398. CRC Press, Boca Raton, FL.
  • [12] Jaffard, S. (1991). Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ. Mat. 35 155-168.
  • [13] Jaffard, S. (1997). Multifractal formalism for functions. SIAM J. Math. Anal. 28 944-998.
  • [14] Jaffard, S. and Meyer, Y. (1996). Wavelet methods for pointwise regularityand local oscillations of functions. Mem. Amer. Math. Soc. 123.
  • [15] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press.
  • [16] Lemari´e, P.-G. and Meyer, Y. (1986). Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 1.
  • [17] Meyer, Y. (1990). Ondelettes et Op´erateurs. Hermann, Paris.
  • [18] Meyer, Y. and Xu, H. (1997). Wavelet analysis and chirps. Appl. Comput. Harmon. Anal. 4 366-379.