Scaling exponents of self-similar functions and wavelet analysis

Saka, Koichi

doi:10.1090/S0002-9939-04-07806-2

Scaling exponents of self-similar functions and wavelet analysis
HTML articles powered by AMS MathViewer

by Koichi Saka PDF

Proc. Amer. Math. Soc. 133 (2005), 1035-1045 Request permission

Abstract:

In this paper we give estimations of the pointwise scaling exponents of self-similar functions on the $n$-dimensional Euclidean space ${\mathbb R}^{n}$. These estimations are derived by using a technique based on wavelet analysis. Examples of such self-similar functions include indefinite integrals of self-similar measures on ${\mathbb R}$, and they also include widely oscillatory functions (e.g. the Takagi function, the Weierstrass function and Lévy’s function). Pointwise scaling exponents provide an objective description of an irregularity of a function at a point. Our results are applied to compute the scaling exponents of several oscillatory functions.

References

Mourad Ben Slimane, Multifractal formalism and anisotropic selfsimilar functions, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 2, 329–363. MR 1631127, DOI 10.1017/S0305004198002710
M. Ben Slimane, Multifractal formalism for self-similar functions under the action of nonlinear dynamical systems, Constr. Approx. 15 (1999), no. 2, 209–240. MR 1668925, DOI 10.1007/s003659900105
Stéphane Jaffard, Old friends revisited: the multifractal nature of some classical functions, J. Fourier Anal. Appl. 3 (1997), no. 1, 1–22. MR 1428813, DOI 10.1007/s00041-001-4047-y
S. Jaffard, Multifractal formalism for functions. I. Results valid for all functions, SIAM J. Math. Anal. 28 (1997), no. 4, 944–970. MR 1453315, DOI 10.1137/S0036141095282991
S. Jaffard, Multifractal formalism for functions. I. Results valid for all functions, SIAM J. Math. Anal. 28 (1997), no. 4, 944–970. MR 1453315, DOI 10.1137/S0036141095282991
Stéphane Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, 1998. MR 1614527
Yves Meyer, Wavelets, vibrations and scalings, CRM Monograph Series, vol. 9, American Mathematical Society, Providence, RI, 1998. With a preface in French by the author. MR 1483896, DOI 10.1090/crmm/009
Robert S. Strichartz, Self-similar measures and their Fourier transforms. I, Indiana Univ. Math. J. 39 (1990), no. 3, 797–817. MR 1078738, DOI 10.1512/iumj.1990.39.39038
Robert S. Strichartz, Self-similar measures and their Fourier transforms. II, Trans. Amer. Math. Soc. 336 (1993), no. 1, 335–361. MR 1081941, DOI 10.1090/S0002-9947-1993-1081941-2
Robert S. Strichartz, Self-similar measures and their Fourier transforms. III, Indiana Univ. Math. J. 42 (1993), no. 2, 367–411. MR 1237052, DOI 10.1512/iumj.1993.42.42018
H. Triebel, Theory of function spaces, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 38, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983. MR 730762, DOI 10.1007/978-3-0346-0416-1