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.

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

“…[17][18][19][20] for more details. More recent work about the fractal dimensions of the graph of continuous functions can be found in [21][22][23][24][25].…”

An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces

“…This problem can be traced back to the research made first by Falconer [33], who showed that the box dimension of the sum of two continuous functions equals the greater of the box dimensions of them. On this basis, a group of academic workers has pushed this study forward and obtained a series of preliminary conclusions, whose related progress can be found in [34][35][36][37][38][39][40]. So in this paper, we shall focus on the fractal dimension of the superposition of two fractal surfaces and investigate whether it has the same result as that of fractal curves.…”