Dimension analysis of continuous functions with unbounded variation. (English) Zbl 1371.28012
Summary: In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are 1. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is 1 also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be 1 when \(f(x)\) is self-similar.
References:
| [1] | Dai, M. F., Lipschitz equivalence of general sierpinski carpets, Fractals16 (2008) 379-388. · Zbl 1159.28004 |
| [2] | Tatom, F. B., The relationship between fractional calculus and fractals, Fractals03 (1995) 217-229. · Zbl 0877.28009 |
| [3] | Yao, K., Su, W. Y. and Zhou, S. P., On the fractional derivatives of a fractal function, Acta Math. Sin.22 (2006) 719-722. · Zbl 1106.28004 |
| [4] | Deliu, A. and Wingren, P., The Takagi operator, Bernoulli sequences, smoothness condition and fractal curves, Proc. Amer. Math. Soc.121 (1994) 893-895. · Zbl 0804.28004 |
| [5] | Oldham, K. B. and Spanier, J., The Fractional Calculus (Academic Press, New York, 1974). · Zbl 0292.26011 |
| [6] | Nasin akhtar, M. D., Guru prem prasad, M. and Navascués, M. A., Box dimension of \(\alpha \)-Fractal functions, Fractals24 (2016) 3. · Zbl 1354.28002 |
| [7] | Yao, K. and Su, W. Y., On the connection between the order of the fractional calculus and the dimension of a fractal function, Chaos Solitons Fractals23 (2005) 621-629. · Zbl 1062.28012 |
| [8] | Liang, Y. S., Box dimension of Riemann-Liouville fractional integrals of continuous functions of bounded variation, Nonlinear Anal.72 (2010) 4304-4306. · Zbl 1190.28001 |
| [9] | Liang, Y. S., Connection between the order of their fractional calculus and fractal dimensions of a type of fractal functions, Anal. Theory Appl.23 (2007) 354-363. · Zbl 1150.41006 |
| [10] | Liang, Y. S. and Su, W. Y., The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus, Chaos Solitons Fractals34 (2007) 682-692. · Zbl 1132.28305 |
| [11] | Falconer, F., Fractal Geometry: Mathematical Foundations and Applications (John Wiley Sons Inc., New York, 1990). · Zbl 0689.28003 |
| [12] | Zhang, Q., Some remarks on one-dimensional functions and their Riemann-Liouville fractional calculus, Acta Math. Sin.30 (2014) 517-524. · Zbl 1294.26010 |
| [13] | Zheng, W. X. and Wang, S. W., Real Function and Functional Analysis (Higher education publication, Beijing, 1980). |
| [14] | Almeida, R. and Torres, D. F. M., Fractional variational calculus for nondifferentiable functions, Comput. Math. Appl.61 (2011) 3097-3104. · Zbl 1222.49026 |
| [15] | Besicovitch, A. S. and Ursell, H. D., On dimensional numbers of some continuous curves, J. London Math. Soc.12 (1937) 18-25. · JFM 63.0184.04 |
| [16] | Pooseh, S., Almeida, R. and Torres, D. F. M., Approximation of fractional integrals by means of derivatives, Comp. Math. Appl.64 (2012) 3090-3100. · Zbl 1268.41024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
