On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

Abstract

Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of f+g by classifying all the subsequences tending to zero into different sets. Using this method, we explore the lower and upper Box dimensions of the graph of f+g when the Box dimension of the graph of g is between the lower and upper Box dimensions of the graph of f. In this case, we prove that the upper Box dimension of the graph of f+g is just equal to the upper Box dimension of the graph of f. We also prove that the lower Box dimension of the graph of f+g could be an arbitrary number belonging to a certain interval. In addition, some other cases when the Box dimension of the graph of g is equal to the lower or upper Box dimensions of the graph of f have also been studied.

Keywords

• fractal dimension
• lower Box dimension
• upper Box dimension
• sum of two continuous functions