一类单无界变差点的连续函数的分形维数估计
Estimation of Fractal Dimension of a Classof Continuous Functionwith SingleUnbounded Variable Difference
DOI: 10.12677/PM.2023.135137, PDF, HTML,    国家自然科学基金支持
作者: 任倩倩*, 梁永顺：南京理工大学, 数学与统计学院, 江苏 南京
关键词: 分形维数；无界变差点；Weierstrass 函数；Fractal Dimension； Unbounded Variation； Weierstrass Function
摘要: 在本文中，我们主要在闭区间上构造了仅有一个无界变差点的连续函数。接着讨论了它的分形维数，该函数图像的分形维数严格大于其拓扑维数。尽管该连续函数只在零点处不可微，但仍具有明显的分形特征。
Abstract: In this paper, we mainly construct a continuous function with only one unbounded variable difference on closed intervals. Then we discuss its fractal dimension. The fractal dimension of the function image is strictly larger than its topological dimension. Although the given function is only nondifferentiable at the zero point, it still has obvious fractal characteristics.
文章引用：任倩倩, 梁永顺. 一类单无界变差点的连续函数的分形维数估计[J]. 理论数学, 2023, 13(5): 1341-1354. https://doi.org/10.12677/PM.2023.135137

参考文献

[1] Falconer, K.J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York.
https://doi.org/10.2307/2532125
[2] Besicovitch, A.S. and Ursell, H.D. (1937) Sets of Fractional Dimensions V: On Dimensional Numbers of Some Continuous Curves. Journal of the London Mathematical Society, 12, 18-25.
https://doi.org/10.1112/jlms/s1-12.45.18
[3] Barnsley, M.F. (1986) Fractal Functions and Interpolation. Constructive Approximation, 2, 303-329.
https://doi.org/10.1007/BF01893434
[4] Bedford, T.J. (1989) The Box Dimension of Self-Affine Graphs and Repellers. Nonlinearity, 2, 53-71.
https://doi.org/10.1088/0951-7715/2/1/005
[5] Ruan, H.J., Su, W.Y. and Yao, K. (2009) Box Dimension and Fractional Integral of Linear Fractal Interpolation Functions. Journal of Approximation Theory, 161, 187-197.
https://doi.org/10.1016/j.jat.2008.08.012
[6] Liang, Y.S. (2017) Definition and Classification of One-Dimensional Continuous Functions with Unbounded Variation. Fractals, 25, Article 17500487.
https://doi.org/10.1142/S0218348X17500487
[7] Xie, T.F. and Zhou, S.P. (2004) On a Class of Fractal Functions with Graph Box Dimension 2. Chaos, Solitons and Fractals, 22, 135-139.
https://doi.org/10.1016/j.chaos.2003.12.100
[8] Xie, T.F. and Zhou, S.P. (2007) On a Class of Singular Continuous Functions with Graph Hausdorff Dimension 2. Chaos, Solitons and Fractals, 32, 1625-1630.
https://doi.org/10.1016/j.chaos.2005.12.038
[9] Zheng, W.X. and Wang, S.W. (1980) Real Function and Functional Analysis. Higher Education Press, Beijing.
[10] 王宏勇, 陈刚. Bush 型函数的分形维数及其奇异性 [J]. 数学研究, 1996, 29(1): 87-92.
[11] Shen, W.X. (2018) Hausdorff Dimension of the Graphs of the Classical Weierstrass Functions. Mathematische Zeitschrift, 289, 223-266.
https://doi.org/10.1007/s00209-017-1949-1
[12] Zhang, Q. (2014) Some Remarks on One-Dimensional Functions and Their Riemann-Liouville Fractional Calculus. Acta Mathematica Sinica, English Series, 30, 517-524.
https://doi.org/10.1007/s10114-013-2044-0
[13] Wen, Z.Y. (2000) Mathematical Foundations of Fractal Geometry: Science Technology Education Publication House, Shanghai.

为你推荐