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Volume 10
Issue 13
10.3390/math10132154
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Article

Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions

by
Binyan Yu
* and
Yongshun Liang
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(13), 2154; https://doi.org/10.3390/math10132154
Submission received: 12 May 2022 / Revised: 17 June 2022 / Accepted: 18 June 2022 / Published: 21 June 2022
(This article belongs to the Special Issue Advances in Fractals)
Versions Notes

Abstract

In the present paper, we try to estimate the fractal dimensions of the linear combination of continuous functions with different fractal dimensions. Initially, a general method to calculate the lower and the upper Box dimension of the sum of two continuous functions by classifying all the subsequences into different sets has been proposed. Further, we discuss the majority of possible cases of the sum of two continuous functions with different fractal dimensions and obtain their corresponding fractal dimensions estimation by using that general method. We prove that the linear combination of continuous functions having no Box dimension cannot keep the fractal dimensions closed. In this way, we have figured out how the fractal dimensions of the linear combination of continuous functions change with certain fractal dimensions.
Keywords: the fractal dimension; the linear combination; continuous functions the fractal dimension; the linear combination; continuous functions

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MDPI and ACS Style

Yu, B.; Liang, Y. Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions. Mathematics 2022, 10, 2154. https://doi.org/10.3390/math10132154

AMA Style

Yu B, Liang Y. Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions. Mathematics. 2022; 10(13):2154. https://doi.org/10.3390/math10132154

Chicago/Turabian Style

Yu, Binyan, and Yongshun Liang. 2022. "Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions" Mathematics 10, no. 13: 2154. https://doi.org/10.3390/math10132154

APA Style

Yu, B., & Liang, Y. (2022). Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions. Mathematics, 10(13), 2154. https://doi.org/10.3390/math10132154

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