## Fractal dimensions and singularities of the Weierstrass type functions

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- by Tian You Hu and Ka-Sing Lau
- Trans. Amer. Math. Soc.
**335**(1993), 649-665 - DOI: https://doi.org/10.1090/S0002-9947-1993-1076614-6
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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.## References

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## Bibliographic Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**335**(1993), 649-665 - MSC: Primary 28A75; Secondary 28A12
- DOI: https://doi.org/10.1090/S0002-9947-1993-1076614-6
- MathSciNet review: 1076614