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
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by Koichi Saka

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

DOI: https://doi.org/10.1090/S0002-9939-04-07806-2

Published electronically: November 19, 2004

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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.

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