Regular ArticleFree Access

RESEARCH ON THE $K$ -DIMENSION OF THE SUM OF TWO CONTINUOUS FUNCTIONS AND ITS APPLICATION

Y. X. CAO

https://orcid.org/0000-0002-0281-1259

Fundamental Education Department, Army Engineering University of PLA, Nanjing 211101, P. R. China

Search for more papers by this author

N. LIU

https://orcid.org/0000-0003-0980-4040

Fundamental Education Department, Army Engineering University of PLA, Nanjing 211101, P. R. China

E-mail Address: ningliu1@tom.com

Corresponding author.

Search for more papers by this author

, and

Y. S. LIANG

https://orcid.org/0000-0002-9394-6502

School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0218348X24500300Cited by:0 (Source: Crossref)

Abstract

In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different $K$ -dimensions and approximation of $s$ -dimensional fractal functions. We first investigate the $K$ -dimension of the linear combination of fractal function whose $K$ -dimension is $s$ and the function satisfying Lipschitz condition is still $s$ -dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the $s$ -dimensional continuous function is given, which is composed of finite triangular series and partial Weierstrass function. In addition, some preliminary results on the approximation of one-dimensional and two-dimensional fractal continuous functions have been given.

Keywords:

References

1. K. J. Falconer , Fractal Geometry: Mathematical Foundations and Applications (John Wiley Sons, New York, 1990). Google Scholar
2. T. Y. Hu and K. S. Lau , Fractal dimensions and singularities of the Weierstrass type functions, Trans. Amer. Math. Soc. 335(2) (1993) 649–665. Crossref, Web of Science, Google Scholar
3. S. J. Wang and J. H. Yu , A simple proof on the $K$ -dimension of Weierstrass type functions graph, J. Harbin Univ. Sci. Technol. 4(2) (1999) 91–94. Google Scholar
4. K. Yao, W. Y. Su and S. P. Zhou , On the connection between the order of fractional calculus and the dimensions of a fractal function, Chaos Solitons Fractals 23(2) (2005) 621–629. Crossref, Web of Science, Google Scholar
5. K. Yao, W. Y. Su and S. P. Zhou , The fractional derivatives of a fractal function, Acta Math. Sin. (Engl. Ser.) 22(3) (2006) 719–722. Crossref, Web of Science, Google Scholar
6. Y. P. Wu, K. Yao and X. Zhang , The Hadamard fractional calculus of a fractal function, Fractals 26(3) (2018) 1850025. Link, Web of Science, Google Scholar
7. T. F. Xie and S. P. Zhou , On a class of fractal functions with graph Hausdorff dimension 2, Chaos Solitons Fractals 32(5) (2007) 1750048. Crossref, Web of Science, Google Scholar
8. N. Liu, K. Yao and Y. S. Liang , Dimension of Riemann-Liouville fractional integral of Takagi function, 2018 Chinese Control And Decision Conference (CCDC), Shenyang, China, 2018, pp. 2376–2380. Crossref, Google Scholar
9. Y. S. Liang , Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals 25(5) (2017) 1750048. Link, Web of Science, Google Scholar
10. Y. S. Liang , Approximation of the same Box dimension in continuous functions space, Fractals 30(3) (2022) 2250039. Link, Web of Science, Google Scholar
11. S. Verma and P. R. Massopust , Dimension preserving approximation, Aequationes Math. 96(6) (2022) 1233–1247. Crossref, Web of Science, Google Scholar
12. T. Takagi , A Simple Example of a Continuous Function Without Derivative (Mathematical Society of Japan, Tokyo, 1903). Google Scholar
13. J. O. Smith , Mathematics of the Discrete Fourier Transform (DFT) (W3K Publishing, Stanford, California, 2007). Google Scholar