On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

TL;DR

This paper explores the fractal dimensions of linear combinations of fractal continuous functions, revealing that certain classes form fractal linear spaces while others do not, depending on their fractal dimensions.

Contribution

It proves that specific classes of fractal continuous functions form fractal linear spaces, while others are non-fractal linear spaces, providing new insights into their fractal properties.

Findings

All fractal continuous functions with bounded variation form a fractal linear space.

Functions with Box dimension one also form a fractal linear space.

Functions with Box dimension s (1<s≤2) form a non-fractal linear space, with complex dimension behavior.

Abstract

This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) $BV_{I}$ all fractal continuous functions with bounded variation is fractal linear space; (2) ${}^{1}D_{I}$ all fractal continuous functions with Box dimension one is a fractal linear space; (3) ${}^{s}D_{I}$ all fractal continuous functions with identical Box dimension $s(1<s\leq 2)$ is surprisingly a non-fractal linear space, even non-fractal linear manifold, beyond our initial expectation, because the Box dimension of linear combination of fractal continuous functions can take any real number in $[1,s)$ if it exists, and some different upper and lower Box dimension if it does not exit. This attracts our interests to probe into fractal characteristics of ${}^{s}D_{I}$, and get some suggesting results.