On the Decomposition of Continuous Functions and Dimensions

Abstract

In this paper, we consider decomposition of continuous functions in [Formula: see text] in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers [Formula: see text], any real-valued continuous function in [Formula: see text] can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension [Formula: see text] and lower box dimension [Formula: see text]. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

Graphs of continuous functions and fractal dimension

“…In this paper, we proved the decomposition results for the upper box dimension in terms of product, which is a analogue of Theorem 1.3, but the following analogous decomposition results regarding Hausdorff dimension [9, Theorem 1.2], packing dimension [10, Theorem 1.7] and lower box dimension [11,Theorem 1.4] are still open.…”

“…Proposition 2.10. [11] Let X be a compact subset of [0, 1]. Then for each continuous function f on X, we have…”

“…Recently in 2020, J. Liu and D. Liu [11] establised the decomposition Result (1.1) for lower box dimension and β ∈ [1,2]. There are many literature available related to decomposition of a continuous function into sum of two continuous functions, where each graph has pre-decided fractal dimensions.…”