The high-dimensional Weierstrass functions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916199364820992 |
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| author | Ren, Haojie Shen, Weixiao |
| author_facet | Ren, Haojie Shen, Weixiao |
| contents | For a real analytic periodic function $ϕ:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $λ\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function $W(x)=\sum_{n=0}^{\infty}{λ^nϕ(b^nx)}$ are both equal to $$\min\left\{\log_{λ^{-1}}b,\,1+\left(\,d-q\,\right)\left(1+\log_bλ\right)\right\},$$ where $q = q(ϕ, b, λ)$ denotes the maximum dimension of all linear spaces $V < \mathbb{R}^d$ such that the projection $π_V W$ is Lipschitz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06778 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The high-dimensional Weierstrass functions Ren, Haojie Shen, Weixiao Classical Analysis and ODEs Dynamical Systems For a real analytic periodic function $ϕ:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $λ\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function $W(x)=\sum_{n=0}^{\infty}{λ^nϕ(b^nx)}$ are both equal to $$\min\left\{\log_{λ^{-1}}b,\,1+\left(\,d-q\,\right)\left(1+\log_bλ\right)\right\},$$ where $q = q(ϕ, b, λ)$ denotes the maximum dimension of all linear spaces $V < \mathbb{R}^d$ such that the projection $π_V W$ is Lipschitz. |
| title | The high-dimensional Weierstrass functions |
| topic | Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2404.06778 |