ArticleNo Access

A REVIEW OF FRACTAL FUNCTIONS AND APPLICATIONS

XUEFEI WANG

https://orcid.org/0000-0002-0869-3644

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

E-mail Address: wxf@njust.edu.cn

Corresponding author.

Search for more papers by this author

CHUNXIA ZHAO

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

Search for more papers by this author

, and

XIA YUAN

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

Search for more papers by this author

https://doi.org/10.1142/S0218348X22501134Cited by:5 (Source: Crossref)

Abstract

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

Keywords:

References

1. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, Sanfrancisco, 1982). Google Scholar
2. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (John Wiley & Sons, New York, 1990). Google Scholar
3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons, New York, 1976). Google Scholar
4. G. Q. Zhang and Y. Q. Lin, Functional Analysis (Peking University Publication House, Beijing, 2001) (in Chinese). Google Scholar
5. A. S. Besicovitch and H. D. Besicovitch, Sets of fractional dimensions V: On dimensional numbers of some continuous curves, J. London Math. Soc. 12(2) (1937) 18–25. Crossref, Google Scholar
6. M. F. Barnsley, Fractal functions and interpolation, Construct. Approx. 2(8) (1986) 303–329. Crossref, Web of Science, Google Scholar
7. K. A. Bush, Continuous functions without derivatives, Amer. Math. Mon. 59(7) (1952) 222–225. Crossref, Google Scholar
8. P. R. Massopust, On local fractal functions in Besov and Triebel–Lizorkin spaces, J. Math. Anal. Appl. 4(36) (2016) 393–407. Crossref, Web of Science, Google Scholar
9. W. X. Zheng and S. W. Wang, Real Function and Functional Analysis (High Education Publication House, Beijing, 1980) (in Chinese). Google Scholar
10. W. X. Shen, Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Z. 28(9) (2018) 223–266. Crossref, Web of Science, Google Scholar
11. T. J. Bedford, The box dimension of self-affine graphs and repellers, Nonlinearity 2(7) (1989) 53–71. Crossref, Web of Science, Google Scholar
12. M. F. Barnsley, Fractals Everywhere (Academic Press, San Diego, 1988). Google Scholar
13. T. Takagi, A simple example of the continuous function without derivative, Proc. Physico-Math. Soc. Japan 1(9) (1903) 176–177. Google Scholar
14. J. Hyde, V. Laschos and L. Olsen, On the box dimensions of graphs of typical continuous functions, J. Math. Anal. Appl. 39(1) (2012) 567–581. Crossref, Web of Science, Google Scholar
15. G. Peano, Sur une courbe, qui remplit toute une aire plane, Math. Ann. 36(1) (1890) 157–160. Crossref, Google Scholar
16. Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sinica Engl. Ser. 30(4) (2014) 517–524. Crossref, Web of Science, Google Scholar
17. Y. S. Liang and Q. Zhang, 1-Dimensional continuous functions with uncountable unbounded variation points, Chin. J. Comtemp. Math. 3(9) (2018) 129–136. Google Scholar
18. T. F. Xie and S. P. Zhou, On a class of fractal functions with graph Box dimension 2, Chaos Solitons Fractals 22(3) (2004) 135–139. Crossref, Web of Science, Google Scholar
19. T. F. Xie and S. P. Zhou, On a class of singular continuous functions with graph Hausdorff dimension 2, Chaos Solitons Fractals 32(7) (2007) 1625–1630. Crossref, Web of Science, Google Scholar
20. P. Humke and G. Petruska, The packing dimension of a typical continuous function is 2, Real Anal. Exchange 14(6) (1989) 345–358. Google Scholar
21. D. C. Sun and Z. Y. Wen, The Bouligand dimension of the graphs of functions defined by lacunary series, Bull. Sci. Sinica 36(8) (1993) 487–490. Google Scholar
22. G. Z. Ma, G. X. Yuan and Z. W. Cui, Upper bound and lower bound estimate of monotone increasing fractal function, Chin. Quart. J. Math. 23(7) (2008) 188–194. Google Scholar
23. Z. Y. Wen, Mathematical Foundations of Fractal Geometry (Science Technology Education Publication House, Shanghai, 2000) (in Chinese). Google Scholar
24. S. V. Astashkin, Rademacher functions in symmetric spaces, J. Math. Sci. 16(9) (2010) 725–881. Crossref, Google Scholar
25. B. Solomyak, On the random series $\sum \pm λ^{n}$ $\sum \pm λ^{n}$ , Ann. Math. 14(2) (1995) 611–625. Crossref, Web of Science, Google Scholar
26. M. A. Navascués, Fractal polynomial interpolation, J. Anal. Appl. 24(7) (2005) 401–418. Google Scholar
27. M. A. Navascués, Fractal trigonometric approximation, Electron. Trans. Numer. Anal. 20(9) (2005) 64–74. Google Scholar
28. A. K. Chand, M. A. Navascues and P. Viswanathan, Fractal trigonometric polynomials for restric-ted range approximation, Fractals 24(2) (2016) 1650022, https://doi.org/10.1142/S0218348X16500225. Link, Web of Science, Google Scholar
29. M. A. Navascués, Fractal approximation, Compl. Anal. Operator Theory 4(3) (2010) 953–974. Crossref, Web of Science, Google Scholar
30. P. Viswanathan, M. A. Navascués and A. K. Chand, Associate fractal functions in $L^{p}$ $L^{p}$ -spaces and in one-sided uniform approximation, J. Math. Anal. Appl. 43(3) (2016) 862–876. Crossref, Web of Science, Google Scholar
31. F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 3(8) (1995) 217–229. Link, Web of Science, Google Scholar
32. M. Zähle, Fractional differentiation in the self-affine case V-the local degree of differentiability, Math. Nachr. 18(5) (1997) 297–306. Google Scholar
33. M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math. 7(6) (1996) 265–275. Crossref, Web of Science, Google Scholar
34. R. R. Nigmatullin, W. Zhang and I. Gubaidullin, Accurate relationship between fractals and fractional integrals: New approaches and evaluations, Fract. Calcul. Appl. Anal. 20(6) (2017) 1263–1280. Crossref, Web of Science, Google Scholar
35. S. Butera and M. D. Paola, A physically based connection between fractional calculus and fractal geometry, Ann. Phys. 35(10) (2014) 146–158. Crossref, Web of Science, Google Scholar
36. S. Butera and M. D. Paola, A physical approach to the connection between fractal geometry and fractional calculus, in Int. Conf. Fractional Differentiation and Its Applications (2014), https://doi.org/10.1109/ICFDA.2014.6967378. Crossref, Google Scholar
37. J. T. Machado, F. Mainardi and V. Kiryakova, Fractional calculus: Quo Vadimus? (Where are we going?), Fract. Calcul. Appl. Anal. 18(6) (2015) 495–526. Crossref, Web of Science, Google Scholar
38. A. Gowrisankar and M. Prasad, Riemann–Liouville calculus on quadratic fractal interpolation function with variable scaling factors, J. Anal. 27(2) (2019) 347–363. Crossref, Web of Science, Google Scholar
39. Y. Li and Y. S. Liang, Upper bound estimation of fractal dimension of fractional calculus of continuous functions, Adv. Anal. 2(2) (2017) 121–128. Google Scholar
40. K. Yao, W. Y. Su and S. P. Zhou, On the fractional calculus of a type of Weierstrass function, Chin. Ann. Math. 25(7) (2004) 711–716. Google Scholar
41. Y. S. Liang, The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus, Appl. Math. Comput. 20(8) (2008) 197–207. Google Scholar
42. H. J. Ruan, W. Y. Su and K. Yao, Box dimension and fractional integral of linear fractal interpolation functions, J. Approx. Theory 16(1) (2009) 187–197. Crossref, Web of Science, Google Scholar
43. Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals 27(5) (2019) 1950084, https://doi.org/10.1142/S0218348X19500841. Link, Web of Science, Google Scholar
44. X. F. Wang and C. X. Zhao, Fractal dimensions of linear combination of continuous functions with the same Box dimension, Fractals 28(7) (2020) 2050139, https://doi.org/10.1142/S0218348X2050139X. Link, Web of Science, Google Scholar
45. P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets (Academic Press, München, 2017). Google Scholar

Vol. 30, No. 06

Metrics

Downloaded 213 times

History

Received 23 November 2021

Accepted 13 April 2022

Published: September 12, 2022

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

A REVIEW OF FRACTAL FUNCTIONS AND APPLICATIONS

Abstract

References

FRACTAL DIMENSION OF PRODUCT OF CONTINUOUS FUNCTIONS WITH BOX DIMENSION

PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

FRACTAL DIMENSIONS OF THE LOGARITHM OF CONTINUOUS FUNCTIONS

REMARKS ON THE LOCAL STRUCTURE OF REGULAR FRACTAL FUNCTIONS WITH FRACTAL DIMENSIONS

THE CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR FRACTIONAL INTEGRAL

ESTIMATION OF FRACTAL DIMENSION OF WEYL FRACTIONAL INTEGRAL OF CERTAIN CONTINUOUS FUNCTIONS

CARDINALITY AND FRACTAL LINEAR SUBSPACE ABOUT FRACTAL FUNCTIONS

APPROXIMATION WITH FRACTAL FUNCTIONS BY FRACTAL DIMENSION

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

A REVIEW OF FRACTAL FUNCTIONS AND APPLICATIONS

Abstract

Recommended

FRACTAL DIMENSION OF PRODUCT OF CONTINUOUS FUNCTIONS WITH BOX DIMENSION

PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

FRACTAL DIMENSIONS OF THE LOGARITHM OF CONTINUOUS FUNCTIONS

REMARKS ON THE LOCAL STRUCTURE OF REGULAR FRACTAL FUNCTIONS WITH FRACTAL DIMENSIONS

THE CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR FRACTIONAL INTEGRAL

ESTIMATION OF FRACTAL DIMENSION OF WEYL FRACTIONAL INTEGRAL OF CERTAIN CONTINUOUS FUNCTIONS

CARDINALITY AND FRACTAL LINEAR SUBSPACE ABOUT FRACTAL FUNCTIONS

APPROXIMATION WITH FRACTAL FUNCTIONS BY FRACTAL DIMENSION