频率快速增长的Weierstrass型函数图的维数
On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies
摘要：
我们确定了一类形式为$f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + θ_n)$的Weierstrass型函数的分形图的Hausdorff维数和盒维数，其中$g$是周期Lipschitz实函数，$a_{n+1}/a_n \to 0$，$b_{n+1}/b_n \to \infty$是$n \to \infty$ .此外，对于任何$H, B \in [1, 2]$、$H \leq B$，我们提供了具有$\dim_H(\graph f) = \underline{\dim}_B(\graph f) = H$、$\bar{\dim}_B(\graph f) = B$的此类函数的示例。
英文摘要：
We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + θ_n)$, where $g$ is a periodic Lipschitz real function and $a_{n+1}/a_n \to 0$, $b_{n+1}/b_n \to \infty$ as $n \to \infty$. Moreover, for any $H, B \in [1, 2]$, $H \leq B$ we provide examples of such functions with $\dim_H(\graph f) = \underline{\dim}_B(\graph f) = H$, $\bar{\dim}_B(\graph f) = B$.
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