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Volume 36, Issue 4
Hausdorff Dimension of a Class of Weierstrass Functions

Huojun Ruan & Na Zhang

Anal. Theory Appl., 36 (2020), pp. 482-496.

Published online: 2020-12

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  • Abstract

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.

  • Keywords

Hausdorff dimension, Weierstrass function, SRB measure.

  • AMS Subject Headings

28A80

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-36-482, author = {Huojun and Ruan and and 9752 and and Huojun Ruan and Na and Zhang and and 9753 and and Na Zhang}, title = {Hausdorff Dimension of a Class of Weierstrass Functions}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {4}, pages = {482--496}, abstract = {

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-SU8}, url = {http://global-sci.org/intro/article_detail/ata/18465.html} }
TY - JOUR T1 - Hausdorff Dimension of a Class of Weierstrass Functions AU - Ruan , Huojun AU - Zhang , Na JO - Analysis in Theory and Applications VL - 4 SP - 482 EP - 496 PY - 2020 DA - 2020/12 SN - 36 DO - http://doi.org/10.4208/ata.OA-SU8 UR - https://global-sci.org/intro/article_detail/ata/18465.html KW - Hausdorff dimension, Weierstrass function, SRB measure. AB -

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.

Huojun Ruan & Na Zhang. (2020). Hausdorff Dimension of a Class of Weierstrass Functions. Analysis in Theory and Applications. 36 (4). 482-496. doi:10.4208/ata.OA-SU8
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