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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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