和函数图像的盒维数的一个注记

Remark on the Box Dimension of Graphs of Sum Functions

主要研究两个函数和的图像的盒维数.定义连续函数图像,并给出连续函数图像的盒维数,利用连续函数图像的初等性质,对闭区间I上的两个连续函数,当这两个函数在该闭区间上图像的盒维数都存在且不相等时,那么这两个函数的和在该闭区间上图像的盒维数是存在的,且等于这两个函数图像盒维数的最大值.证明两个连续函数满足一致与反一致s阶赫尔德条件时,两个连续函数和图像的盒维数等于其中任何一个连续函数图像的盒维数.

In this paper,we study the Box dimension of graphs of two function sums.For any a continuous function,we give the definition of graphs of functions and the Box dimension of graphs of functions.When the Box dimensions of the graphs on the closed interval of these two functions exist and are not equal,the Box dimension of the sum of these two functions on the closed interval is existent,and is equal to the maximum of the Box dimension of the graphs of these two functions.It is proved that the box dimension of graphs of sum function is equal to the box dimension of graphs of any function when two functions satisfy order-s Holder condition and order-s reverse Holder condition.