In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on ℝ/[a,b], a better estimation of fractal dimension can be obtained. When a function is continuous on ℝ and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

中文翻译：

---

连续函数的上框维数与WEYL分数积分的阶之间的关系

本文主要研究了连续函数的分形维数与Weyl分数积分阶数的关系。如果定义在闭区间上的连续函数是有界变分的，那么它的外尔分数积分一定仍然是有界变分的连续函数。因此，它的 Weyl 分数积分和它本身都具有 Box 维数。如果一个连续函数满足Hölder条件，我们给出了它的Weyl分数积分的分形维数的估计。如果 Hölder 连续函数等于 0ℝ/[一种,b]，可以获得更好的分形维数估计。当一个函数是连续的ℝ并且它的Weyl分数积分是明确定义的，给出了该函数的Weyl分数积分的上Box维数的一般估计，它严格小于2。最后证明了连续函数的Weyl分数积分的上Box维数不超过原函数的上Box维数。