Wavelet techniques for pointwise regularity

Stéphane Jaffard

doi:10.5802/afst.1111

Stéphane Jaffard ¹

¹ Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex (France).

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 3-33.

Résumé
Abstract

Soit $E$ un espace de Banach (ou un quasi-Banach) invariant par translation et dilatation (typiquement un espace de Besov ou de Sobolev homogène). Nous introduisons une définition générale de régularité ponctuelle associée à $E$ , et notée $C_{E}^{α} (x_{0})$ . Nous montrons comment les propriétés de $E$ se traduisent en propriétés de $C_{E}^{α} (x_{0})$ . Nous donnons également des application en analyse multifractale.

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$ , and denoted by $C_{E}^{α} (x_{0})$ . We show how properties of $E$ are transferred into properties of $C_{E}^{α} (x_{0})$ . Applications are given in multifractal analysis.

DOI : 10.5802/afst.1111

Affiliations des auteurs :

Stéphane Jaffard ¹

¹ Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex (France).

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Stéphane Jaffard. Wavelet techniques for pointwise regularity. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 1, pp. 3-33. doi : 10.5802/afst.1111. https://afst.centre-mersenne.org/articles/10.5802/afst.1111/

Bibliographie
Cité par

[1] A. Arneodo; B. Audit; N. Decoster; J.-F. Muzy; C. Vaillant; A. Bunde; J. Kropp; H. J. Schellnhuber. Wavelet-based multifractal formalism: Applications to DNA sequences, satellite images of the cloud structure and stock market data, The Science of Disasters, Springer, 2002, pp. 27-102

[2] F. Autin Point de vue maxiset en estimation non paramétrique, Université Paris 7 (2004) (Ph. D. Thesis)

[3] B. Beauzamy Introduction to Banach spaces and their geometry, 68, North-Holland Publishing Co., Amsterdam, 1985 | MR | Zbl

[4] A. Benassi; S. Cohen; J. Istas Identifying the multifractional function of a Gaussian process, Stat. Proba. letters., Volume 39 (1998), pp. 337-345 | MR | Zbl

[5] A. Benassi; S. Jaffard; D. Roux Elliptic Gaussian random processes, Rev. Mat. Iberoam., Volume 13 (1997), pp. 19-90 | MR | Zbl

[6] J.-M. Bony Second microlocalization and propagation of singularities for semilinear hyperbolic equations, Hyperbolic equations and related topics, Academic Press, 1986, pp. 11-49 (Katata/Kyoto, 1984) | MR | Zbl

[7] G. Bourdaud Réalisations des espaces de Besov homogènes, Arkiv för Mat., Volume 26 (1988), pp. 41-54 | MR | Zbl

[8] H. Brezis Analyse fonctionnelle, Masson, 1983 | MR | Zbl

[9] A. P. Caldéron; A. Zygmund Local properties of solutions of elliptic partial differential equations, Studia Math., Volume 20 (1961), pp. 171-227 | MR | Zbl

[10] A. Cohen; W. Dahmen; I. Daubechies; R. DeVore Harmonic analysis of the space BV, Rev. Mat. Iberoam., Volume 19 (2003), pp. 235-263 | MR | Zbl

[11] A. Cohen; W. Dahmen; R. DeVore Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comput., Volume 70 (2001) no. 233, pp. 27-75 | MR | Zbl

[12] S. Cohen Liens entre densité spectrale et autosimilarité asymptotique dans certains modèles gaussiens (might appear in Ann. Univ. Blaise Pascal)

[13] D. Donoho; I. M. Johnstone; G. Kerkyacharian; D. Picard Wavelet shrinkage: Asymptopia?, J. R. Stat. Soc., Ser. B, Volume 57 (1995), pp. 301-369 | MR | Zbl

[14] H. Feichtinger; G. Zimmermann An exotic minimal Banach space of functions, Math. Nachr., Volume 239-240 (2002), pp. 42-61 | MR | Zbl

[15] M. Frazier; B. Jawerth; G. Weiss Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, 79, AMS, 1991 | Zbl

[16] S. Jaffard; C. Melot Wavelet analysis of fractal Boundaries, Part 1: Local regularity and Part 2: Multifractal formalism, Comm. Math. Phys., Volume 258 (2005) no. 3, pp. 513-565 | MR | Zbl

[17] S. Jaffard; Y. Meyer; R. Ryan Wavelets: Tools for Science and Technology, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001 | MR | Zbl

[18] S. Jaffard; Y. Meyer Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc., Volume 123 (1996) no. 587, pp. x+110 | MR | Zbl

[19] S. Jaffard Pointwise smoothness, two-microlocalization and wavelet coefficients, Publications Matematiques, Volume 35 (1991), pp. 155-168 | MR | Zbl

[20] S. Jaffard Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., Volume 29 (1992), pp. 965-986 | MR | Zbl

[21] S. Jaffard Local behavior of Riemann’s function, Harmonic analysis and operator theory (Caracas, 1994) (Contemporary Mathematics), Volume 189, Amer. Math. Soc., Providence, RI, 1995, pp. 287-307 | MR | Zbl

[22] S. Jaffard Oscillation spaces: Properties and applications to fractal and multifractal functions, J. Math. Phys., Volume 39 (1998), pp. 4129-4141 | MR | Zbl

[23] S. Jaffard Sur la dimension de boîte des graphes, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998) no. 5, pp. 555-560 | MR | Zbl

[24] S. Jaffard Pointwise regularity criteria, C.R.A.S., Série 1, Volume 339 (2004) no. 11, pp. 757-762 | MR | Zbl

[25] S. Jaffard; M. Lapidus; M. van Frankenhuijsen Wavelet techniques in multifractal analysis, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2 (Proc. Sympos. Pure Math.), Volume 72, Amer. Math. Soc., Providence, RI, 2004, pp. 91-151 | MR | Zbl

[26] S. Jaffard Beyond Besov spaces Part 2: Oscillation spaces, Constr. Approx., Volume 21 (2005) no. 1, pp. 29-61 | MR | Zbl

[27] S. Mallat A Wavelet Tour of Signal Processing, Academic Press, 1998 | MR | Zbl

[28] Y. Meyer; H. Xu Wavelet analysis and chirps, Appl. Comput. Harmon. Anal., Volume 4 (1997) no. 4, pp. 366-379 | Zbl

[29] Y. Meyer La minimalité de l’espace de Besov $B_{1}^{0, 1}$ et la continuité des opérateurs définis par des intégrales singulières, Monografias de Matematicas, Univ. Autonoma de Madrid, 1986 no. 4 | MR | Zbl

[30] Y. Meyer Ondelettes et opérateurs, Hermann, 1990 | MR | Zbl

[31] Y. Meyer Wavelet analysis, local Fourier analysis and 2-microlocalization, Harmonic Analysis and Operator Theory (Caracas, 1994) (Contemp. Math.), Volume 189, Amer. Math. Soc., Providence, RI, 1995, pp. 393-401 | MR | Zbl

[32] Y. Meyer Wavelets, Vibrations and Scalings, CRM Monograph Series, 9, American Mathematical Society, 1998 | MR | Zbl

[33] S. Moritoh; T. Yamada Two-microlocal Besov spaces and wavelets, Rev. Mat. Iberoamericana, Volume 20 (2004), pp. 277-283 | MR | Zbl

[34] G. Parisi; U. Frisch On the singularity spectrum of fully developped turbulence, Turbulence and predictability in geophysical fluid dynamics (1985), pp. 84-87

[35] I. Singer Bases in Banach spaces 1, Springer-Verlag, 1970 | MR | Zbl

[36] H. Triebel Wavelet frames for distributions; local and pointwise regularity, Studia Math., Volume 154 (2003), pp. 59-88 | MR | Zbl

[37] B. Vedel Règlement de la divergence infra-rouge dans des bases d’ondelettes adaptées, Université d’Amiens (2004) (Ph. D. Thesis)

[38] P. Wojtaszczyk Banach spaces for analysts, Cambridge Univ. Press, 1991 | MR | Zbl

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