关于某些连续函数的分数阶微积分的分形维数估计

A Remark on Fractal Dimension Estimation of Fractional Calculus of Certain Continuous Functions

在本文中,我们对分形函数的定义进行了初步的研究,接着讨论了分形函数分数阶微积分的分形维数估计。我们使用新方法进行的估计表明分形函数的分形维数和分数阶微积分的阶之间存在一定关系。如果分形函数满足 Hölder 条件,则这种分形函数的 Riemann-Liouville 分数阶积分的上 Box 维数小于这些分形函数的上 Box 维数。这就意味着一个重要的结论:分形函数的 Riemann-Liouville 分数阶微积分的上 Box 维数不会增加。

In the present paper, we make research on the definition of fractal functions elementary. Then we discuss the fractal dimensions of fractional calculus of fractal functions. The estimation using a new method shows certain relationship between the fractal dimensions of fractal functions and orders of fractional calculus. If the fractal function satisfies the Hölder condition, the upper Box dimension of the Riemann-Liouville fractional integral of such fractal functions has been proved to be less than the upper Box dimension of those fractal functions. This means an important conclusion that the upper Box dimension of the Riemann-Liouville fractional integral of such fractal functions will not increase.