Fractal dimensions and singularities of the Weierstrass type functions

Hu, Tian You; Lau, Ka-Sing

doi:10.1090/S0002-9947-1993-1076614-6

Fractal dimensions and singularities of the Weierstrass type functions
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by Tian You Hu and Ka-Sing Lau PDF

Trans. Amer. Math. Soc. 335 (1993), 649-665 Request permission

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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