More on the analysis of local regularity through wavelets. (English) Zbl 0870.42009

Summary: In this paper, we extend the results of pointwise analysis through wavelet transforms to the class of functions where the local fluctuation is bounded by any submultiplicative function. This generalizes the results obtained before in the well-known case of Hölder regularity.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28A80 Fractals
Full Text: DOI


[1] E. M. Stein,Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, New Jersey, 1970). · Zbl 0207.13501
[2] S. Jaffard, Estimations Holderiennes ponctuelles des fonctions au moyen de leyrs coefficients d’ondelettes,C. R. Acad. Sci. Ser. I 308:7 (1989).
[3] M. Holschneider and P. Tchamitchihan, Pointwise regularity of Riemanns ”nowhere differentiable” function,Inventiones Mathematicae 105:157–175 (1991). · Zbl 0741.26004
[4] Y. Meyer,Ondelettes et Opérateurs (Hermann, Paris, 1990). · Zbl 0694.41037
[5] J. M. Combes, A. Grossmann, and P. Tchamitchian. eds.,Wavelets (Springer-Verlag, Berlin, 1989).
[6] I. Daubechies,Ten Lectures on Wavelets (CBMS-NSF Lecture Notes 61, SIAM, Philadelphia, 1992). · Zbl 0776.42018
[7] M. Holschneider, On the wavelet transformation of fractal objects,J. Stat. Phys. 50:953–993 (1988). · Zbl 1084.42518
[8] A. Arnéodo, G. Grasseau, and M. Holschneider, On the wavelet transform of multifractals,Phys. Rev. Lett. 61:2281–2284 (1988).
[9] A. Arnéodo, G. Grasseau, and M. Holschneider, Wavelet transform analysis of invariant measures of some dynamical systems, inWavelets, J. M. Combes, A. Grossmann, and P. Tchamitchian, eds. (Springer-Verlag, Berlin, 1989). · Zbl 0850.42005
[10] A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations I: General results,J. Math. Phys. 26:2473–2479 (1985). · Zbl 0571.22021
[11] M. Holschneider, Localization properties of wavelet transforms,J. Math. Phys. 34:3227–3244 (1993). · Zbl 0778.42027
[12] L. Hoermander,The Analysis of Linear Partial Differential Operators (Springer, 1982).
[13] A. Zygmund,Trigonometric Series (Cambridge University Press, Cambridge). · Zbl 0005.06303
[14] A. Grossmann, M. Holschneider, R. Kronland-Martinet, and J. Morlet, Detection of abrupt changes in sound signals with the help of wavelet transforms, inAdvances in Electronics and Electron Physics, Supplement 19,Inverse Problems (Academic Press, 1987).
[15] M. Holschneider,Wavelets: An Analysis Tool (Oxford University Press, Oxford, to appear). · Zbl 0952.42016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.