Cardinality and Fractal Linear Subspace About Fractal Functions

Abstract

Since fractal functions are widely applied in dynamic systems and physics such as fractal growth and fractal antennas, this paper concerns fundamental problems of fractal continuous functions like cardinality of collection of fractal functions, box dimension of summation of fractal functions, and fractal linear space. After verifying that the cardinality of fractal continuous functions is the second category by Baire theory, we investigate the box dimension of sum of fractal continuous functions so as to discuss fractal linear space under fractal dimension. It is proved that the collection of 1-dimensional fractal continuous functions is a fractal linear space under usual addition and scale multiplication of functions. Particularly, it is revealed that the fractal function with the largest box dimension in the summation represents a fractal dimensional character whenever the other box dimension of functions exist or not. Simply speaking, the fractal function with the largest box dimension can absorb the other fractal features of functions in the summation.

Evaluation of maturity and moisture content in fresh jujube using fractal theory and tissue images

“…The calculation methods of fractal dimension include correlation dimension (Zhang et al, 2021), information dimension (Qiang et al, 2022), box dimension (Xiao, 2022) and so on. Among them, the box dimension is widely used because of its simple calculation process and easy operation.…”

Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform

“…In the paper [1], Xiao draws a conclusion that the cardinality of fractal function set is the same as that of real number set. So We strongly believe that fractal functions have the same status as real numbers, which is worth spending a lot of time to discuss.…”