Wavelet methods for pointwise regularity and local oscillations of functions

Memoirs of the American Mathematical Society

Volumn 123, Issue 587, 1996, Pages

  Jaffard, Stéphane ^a   Meyer, Yves ^a  

^a NONE

Author keywords

Chirps; Hausdorff dimension; Littlewood Paley decomposition; Modulus of continuity; Riemann function; Selfsimilarity; Trace functions; Two microlocalization; Wavelets

Indexed keywords

EID: 0009402820     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0587     Document Type: Article

References (44)

1. Adams, Sobolev spaces. Academic press (1978)
2. A. Arneodo, E. Bacry and J.F. Muzy, Ondelettes, multifractales et turbulence, de l'ADN aux croissances cristallines . Collection Sciences en Actes, Diderot (1995).
3. J.M. Bony, Second microlocalization and propagation of singularities for semilinear hyperbolic equations. Tanaguchi Symp. HERT. Katata p. 11-49 (1984).
4. J.M. Bony, Interaction des singularités pour l'équation de Klein-Gordon non linéaire. Séminaire Goulaouic-Meyer-Schwartz, 1983-1984. Ecole Polytechnique.
5. G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures. Jour. Stat. Phys. T.66 PP.775-790 (1992).
6. A.P. Calderón and A. Zygmund, Singular integral operators and differential equations. American J. Math., Vol. 79, p. 901-921, (1957).
7. A.P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations. Studia Math. Vol. 20, p. 171-227 (1961).
8. L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. on Pure and Appl. Math., Vol. 35, p. 771 (1982).
9. I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. on Pure and Appl. Math., Vol. 41, (1988).
10. J.J. Duistermaat, Selfsimilarity of Riemann's non-differentiable function. Nieuw Archief voor Wiskunde 9 (1991) 303-337.
11. I. Falconer, Fractal geometry, Mathematical Foundations and applications. John Wiley and sons (1990).
12. P. Flandrin, Temps-Fréquence Hermes (1993).
13. J. Gerver, The differentiability of the Riemann function at certain rational multiples of π. Am. J. Math., 92, (1970).
14. P. Hajlasz and P. Strzelecki, A new proof of Reshetnyak's theorem concerning the pointwise differentiability of solutions of quasilinear equations. (Preprint).
15. G.H. Hardy, Weierstrass's non-differentiable function . Trans. AMS 17 (1916), 301-325 .
16. G.H. Hardy and J.E. Littlewood, Some problems of Diophantine approximation. Acta Mathematica 37 (1914), 193-239.
17. M. Holschneider, On the wavelet transformation of fractal objects Journal of Statistical Physics 5/6 (1988), 963-993.
18. M. Holschneider & P. Tchamitchian, Pointwise analysis of Riemann's "non differentiable" function. Inventiones Mathematicae 105, (1991), 157-176.
19. S. Itatsu, The differentiability of the Riemann function. Proc. Japan Acad., ser. A, Math. Sci., 57 (1981), n. 10, 492-495.
20. S. Jaffard, Exposants de Hölder en des points donnés et coefficients d'ondelettes. C.R.A.S. Vol.308 Série 1, p.79-81 (1989).
21. S. Jaffard, Sur la dimension de Hausdorff des points singuliers d'une fonction. C.R.A.S. t. 314 Série 1, p. 31-36 (1991).
22. S. Jaffard, Construction de fonctions multifractales ayant un spectre de singularités prescrit. C.R.A.S. t. 315 Série 1, p. 19-24 (1992).
23. S. Jaffard, The spectrum of singularities of Riemann's function, to appear in Revista Mathematica Iberoamericana.
24. S. Jaffard, Multifractal formalism for functions, Part 1: Results valid for all and Part 2: Selfsimilar functions. To appear in SIAM of mathematical Analysis.
25. S. Jaffard, Théorèmes de trace et "dimensions négatives". C.R.A.S. Vol.320 Série 1, p.409-413 (1995).
26. S. Jaffard and B.B. Mandelbrot, Local regularity of nonsmooth wavelet expansions and application to Polya's function To appear in Advances in Mathematics.
27. J.P. Kahane, Geza Freud and lacunary Fourier series. Journal of approximation Theory, Vol. 46, n.1, p. 51-57 (1986).
28. N. Lygeros, Exemple d'une courbe ni classique ni fractale. Gazette des mathématiciens, n. 49 (1991).
29. B. Mandelbrot, The fractal geometry of nature. W.H. Freeman ,(1982).
30. Y. Meyer, Ondelettes et opérateurs. Hermann (1990).
31. Y. Meyer, Les ondelettes, Algorithmes et applications. Armand Colin (1992).
32. H. Queffelec, Dérivabilité de certaines sommes de séries de Fourier lacunaires. Thèse (Orsay, 1971).
33. de Rham, Sur une courbe plane. Journal de mathématiques pures et appliquées. Vol. 35, p. 25-42 (1956).
34. E. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970).
35. R. Strichartz, Multipliers of fractional Sobolev spaces. Journal of Mathematics and Mechanics, Vol. 16 n. 9, p. 1031-1060 (1976).
36. P. Strzelecki, Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients. (Preprint).
37. [37 Ph. Tchamitchian & B. Torrésani, Ridge and Skeleton extraction from the wavelet transform. Wavelets and their applications, ed. by Mary Beth Ruskai, Jones and Bartlett Publ. Boston (1992).
38. C. Tricot, Rectifiable and fractal sets, in: Fractal Geometry and Analysis, Ed: J. Bélair and S.Dubuc, Nato ASI Series, Series C, Vol.346 (1991).
39. T. Vieseck, Fractal growth phenomena. World scientific (1989).
40. Visintin, Generalized coarea formula. (Preprint).
41. M. Weiss, On the law of the iterated logarithm for lacunary trigonometric series. Trans. A.M.S. Vol. 91, p. 444-469 (1959).
42. W.P. Ziemer, Weakly differentiate functions. Springer (1989).
43. Y. Zheng, Concentration in sequences of solutions to the nonlinear Klein-Gordon equation. Indiana University Math. Jour., Vol. 40 n. 1 (1991).
44. A. Zygmund, Trigonometric series. Cambridge Math. Lib. (1968).