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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fractal dimensions and singularities of the Weierstrass type functions

Authors: Tian You Hu and Ka-Sing Lau
Journal: Trans. Amer. Math. Soc. 335 (1993), 649-665
MSC: Primary 28A75; Secondary 28A12
MathSciNet review: 1076614
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Abstract: A new type of fractal measures ${\mathcal {K}^s}$, $1 \leq s \leq 2$, defined on the subsets of the graph of a continuous function is introduced. The $\mathcal {K}$-dimension defined by this measure is ’closer’ to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by $W(x) = \sum \nolimits _0^\infty {{\lambda ^{ - \alpha i}}g({\lambda ^i}x)}$, where $\lambda > 1$, $0 < \alpha < 1$, and $g$ is an almost periodic Lipschitz function of order greater than $\alpha$, it is shown that the $\mathcal {K}$-dimension of the graph of $W$ equals to $2 - \alpha$, this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the ’knot’ points and the nondifferentiability of $W$ are also discussed.

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Keywords: Almost periodic functions, box dimension, Hausdorff dimension, knot points, nondifferentiability, Weierstrass type functions
Article copyright: © Copyright 1993 American Mathematical Society