Box dimension of the graphs of the generalized Weierstrass-type functions

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2022-10-22 DOI:10.3934/dcds.2023068

Haojie Ren

引用次数: 0

Abstract

For a Lipschitz $\mathbb{Z}-$periodic function $\phi:\mathbb{R}\to \mathbb{R}^2$ satisfied that $\mathbb{R}^2\setminus\{\phi(x):x\in\mathbb{R}\}$ is not connected, an integer $b\ge 2$ and $\lambda\in (c/{b^{\frac12}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=\sum\limits_{n=0}^{\infty}{{\lambda}^n\phi(b^nx)}$: the box dimension of its graph is equal to $3+2\log_b\lambda$, where $c$ is a constant depending on $\phi$.

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广义weierstrass型函数图的盒维数

为了利普希茨 $\mathbb{Z}-$周期函数 $\phi:\mathbb{R}\to \mathbb{R}^2$ 满意了吗? $\mathbb{R}^2\setminus\{\phi(x):x\in\mathbb{R}\}$ 是未连接的，是整数吗 $b\ge 2$ 和 $\lambda\in (c/{b^{\frac12}},1)$，我们证明了广义weierstrass型函数的如下性质 $W(x)=\sum\limits_{n=0}^{\infty}{{\lambda}^n\phi(b^nx)}$:其图的盒维数为 $3+2\log_b\lambda$，其中 $c$ 常数是否取决于 $\phi$．

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.