Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions
Abstract
:1. Introduction
- (1)
- Both of two continuous functions have the Box dimension;
- (2)
- One continuous function has the Box dimension but the other one does not have the Box dimension;
- (3)
- Neither of two continuous functions has the Box dimension.
2. Preliminaries
2.1. Main Method
2.2. Basic Lemmas
2.3. Two Elementary Results
3. The Linear Combination of Two Functions with Different Fractal Dimensions
3.1. and
3.1.1. for
3.1.2.
- (1)
- If the Box dimension of exists, it may be any number between one and s.
- (2)
- If the Box dimension of does not exist,
3.2. and
3.2.1.
3.2.2.
3.2.3.
3.2.4.
3.2.5.
3.3. and
3.3.1.
3.3.2.
3.3.3.
3.3.4.
3.3.5.
3.3.6.
- (1)
- and are the only two elements in the set ;
- (2)
- is the only one element in the set ;
- (3)
- is the only one element in the set .
- (1)
- Three are only three elements , , in the set . Here, , and could be any number belonging to ;
- (2)
- is the only one element in the set ;
- (3)
- is the only one element in the set ;
- (4)
- is the only one element in the set .
- (1)
- and are the only two elements in the set ;
- (2)
- is the only one element in the set ;
- (3)
- is the only one element in the set .
- (1)
- If the Box dimension of exists,Here, u could be any number belonging to .
- (2)
- If the Box dimension of does not exist,Here, could be any numbers satisfying or .
3.3.7.
- (1)
- and are the only two elements in the set ;
- (2)
- is the only one element in the set ;
- (3)
- is the only one element in the set .
- (1)
- Three are only three elements , , in the set . Here, , and could be any number belonging to ;
- (2)
- is the only one element in the set ;
- (3)
- is the only one element in the set ;
- (4)
- is the only one element in the set .
- (1)
- If the Box dimension of exists,Here, u could be any number belonging to .
- (2)
- If the Box dimension of does not exist,Here, could be any numbers satisfying .
- (1)
- If the Box dimension of exists,Here, u could be any number belonging to .
- (2)
- If the Box dimension of does not exist,Here, could be any numbers satisfying .
4. Main Results
- (1)
- If ,
- (2)
- If ,
- (3)
- If ,
- (4)
- If ,
- (5)
- If ,
- (6)
- If ,
- (1)
- If ,
- (2)
- If ,
- (3)
- If ,
- (4)
- If ,
- (5)
- If ,
- (6)
- If ,
- (7)
- If ,
- (8)
- If ,
5. Conclusions
- (1)
- We put forward 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, which is the key work in the present paper.
- (2)
- We acquire several basic results for the lower and the upper Box dimension of the sum of two continuous functions in certain situations. If the upper Box dimensions of two continuous functions are not equal, the upper Box dimension of the sum of these two functions is equal to the maximum one of the upper Box dimensions of these two functions. If the upper Box dimension of one function is less than the lower Box dimension of another function, the lower Box dimension of the sum of these two functions is equal to the maximum one of the lower Box dimensions of these two functions.
- (3)
- We discuss the majority of possible cases of the sum of two continuous functions with different fractal dimensions. We divide the subjects into three broad categories to consider as follows:
- (i)
- Both of two continuous functions have the Box dimension;
- (ii)
- One continuous function has the Box dimension but the other one does not have the Box dimension;
- (iii)
- Neither of two continuous functions has the Box dimension.
Moreover, we obtain their corresponding fractal dimensions estimation by using the main method proposed in Section 2 and sum up the results for all the cases discussed above in the end. - (4)
- We find that the space which is consist of all continuous functions without Box dimension is not linear. We also find the fractal dimensions of the linear combination of continuous functions with certain fractal dimensions may be equal to arbitrary numbers belonging to a certain interval.
Question. What can the fractal dimensions of the product of two continuous functions be? Under what circumstances can the product of two continuous functions keep the fractal dimensions closed?
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Falconer, K.J. Fractal Geometry: Mathematical Foundations and Applications; John Wiley Sons Inc.: New York, NY, USA, 1990. [Google Scholar]
- Zhang, G.Q.; Lin, Y.Q. Lecture Notes on Functional Analysis; Peking University Publication House: Beijing, China, 2021. [Google Scholar]
- Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman and Company: San Francisco, CA, USA, 1983. [Google Scholar]
- Bedford, T.J. The box dimension of self-affine graphs and repellers. Nonlinearity 1989, 2, 53–71. [Google Scholar] [CrossRef]
- Ruan, H.J.; Su, W.Y.; Yao, K. Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory 2008, 161, 187–197. [Google Scholar] [CrossRef] [Green Version]
- Wen, Z.Y. Mathematical Foundations of Fractal Geometry; Science Technology Education Publication House: Shanghai, China, 2000. [Google Scholar]
- Wang, B.; Ji, W.L.; Zhang, L.G.; Li, X. The relationship between fractal dimensions of Besicovitch function and the order of Hadamard fractional integral. Fractals 2020, 28, 2050128. [Google Scholar] [CrossRef]
- Barnsley, M.F. Fractal functions and interpolation. Constr. Approx. 1986, 2, 303–329. [Google Scholar] [CrossRef]
- Xie, T.F.; Zhou, S.P. On a class of fractal functions with graph Box dimension 2. Chaos Solitons Fractals 2004, 22, 135–139. [Google Scholar] [CrossRef]
- Xie, T.F.; Zhou, S.P. On a class of singular continuous functions with graph Hausdorff dimension 2. Chaos Solitons Fractals 2007, 32, 1625–1630. [Google Scholar] [CrossRef]
- Zheng, W.X.; Wang, S.W. Real Function and Functional Analysis; High Education Publication: Beijing, China, 1980. [Google Scholar]
- Liang, Y.S. Definition and classification of one-dimensional continuous functions with unbounded variation. Fractals 2017, 25, 1750048. [Google Scholar] [CrossRef]
- Solomyak, B. On the random series ∑±λn. Ann. Math. 1995, 142, 611–625. [Google Scholar] [CrossRef]
- Besicovitch, A.S.; Ursell, H.D. Sets of fractional dimensions (V): On dimensional numbers of some continuous curves. J. Lond. Math. Soc. 1937, 1, 18–25. [Google Scholar] [CrossRef]
- Deliu, A.; Wingren, P. The Takagi operator, Bernoulli sequences, smoothness condition and fractal curves. Proc. Am. Math. Soc. 1994, 121, 871–881. [Google Scholar]
- Tricot, C. Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 1982, 91, 57–74. [Google Scholar] [CrossRef]
- Wang, J.; Yao, K. Dimension analysis of continuous functions with unbounded variation. Fractals 2017, 25, 1730001. [Google Scholar] [CrossRef]
- Wang, X.F.; Zhao, C.X. Fractal dimensions of linear combination of continuous functions with the same Box dimension. Fractals 2020, 28, 2050139. [Google Scholar] [CrossRef]
- Liu, N.; Yao, K. Fractal dimension of a special continuous function. Fractals 2018, 26, 1850048. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
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
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 StyleYu, 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 StyleYu, 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