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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Scaling exponents of self-similar functions and wavelet analysis


Author: Koichi Saka
Journal: Proc. Amer. Math. Soc. 133 (2005), 1035-1045
MSC (2000): Primary 28A80; Secondary 42C40
Published electronically: November 19, 2004
MathSciNet review: 2117204
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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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Additional Information

Koichi Saka
Affiliation: Department of Mathematics, Akita University, Akita, 010-8502 Japan
Email: saka@math.akita-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07806-2
PII: S 0002-9939(04)07806-2
Keywords: Self-similar functions, scaling exponents, wavelet analysis
Received by editor(s): April 25, 2001
Received by editor(s) in revised form: July 8, 2003
Published electronically: November 19, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.