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ON FRACTAL DIMENSIONS OF FRACTAL FUNCTIONS USING FUNCTION SPACES

Part of: Approximations and expansions Classical measure theory

Published online by Cambridge University Press:  08 August 2022

SUBHASH CHANDRA  and
SYED ABBAS Open the ORCID record for SYED ABBAS [Opens in a new window]
SUBHASH CHANDRA
Affiliation:
School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh 175005, India e-mail: sahusubhash77@gmail.com
SYED ABBAS*
Affiliation:
School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh 175005, India
*
e-mail: sabbas.iitk@gmail.com
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Abstract

Based on the work of Mauldin and Williams [‘On the Hausdorff dimension of some graphs’, Trans. Amer. Math. Soc. 298(2) (1986), 793–803] on convex Lipschitz functions, we prove that fractal interpolation functions belong to the space of convex Lipschitz functions under certain conditions. Using this, we obtain some dimension results for fractal functions. We also give some bounds on the fractal dimension of fractal functions with the help of oscillation spaces.

Keywords

fractal interpolation functionsconvex-Lipschitz spaceoscillation spaceHausdorff dimensionbox dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 33C47: Other special orthogonal polynomials and functions 41A30: Approximation by other special function classes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 3 , December 2022 , pp. 470 - 480
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author received financial support from CSIR, India with grant no: 09/1058(0012)/2018-EMR-I.

References

Agrawal, V. and Som, T., ‘Fractal dimension of $\alpha$ -fractal function on the Sierpiński gasket’, Eur. Phys. J. Spec. Top. 230 (2021), 3781–3787.CrossRefGoogle Scholar
Agrawal, V. and Som, T., ‘ ${L}^p$ approximation using fractal functions on the Sierpiński gasket’, Results Math. 77 (2022), Article no. 74.CrossRefGoogle Scholar
Bárány, B., Rams, M. and Simon, K., ‘On the dimension of self-affine sets and measures with overlaps’, Proc. Amer. Math. Soc. 144 (2016), 4427–4440.CrossRefGoogle Scholar
Bárány, B., Rams, M. and Simon, K., ‘On the dimension of triangular self-affine sets’, Ergodic Theory Dynam. Systems 39(7) (2019), 1751–1783.CrossRefGoogle Scholar
Barnsley, M. F., ‘Fractal functions and interpolation’, Constr. Approx. 2 (1986), 303–329.CrossRefGoogle Scholar
Barnsley, M. F., Fractal Everywhere (Academic Press, Orlando, FL, 1988).Google Scholar
Carvalho, A., ‘Box dimension, oscillation and smoothness in function spaces’, J. Funct. Spaces Appl. 3(3) (2005), 287–320.CrossRefGoogle Scholar
Chandra, S. and Abbas, S., ‘Analysis of fractal dimension of mixed Riemann–Liouville integral’, Numer. Algorithms, to appear. Published online (29 March 2022). https://link.springer.com/article/10.1007/s11075-022-01290-2.CrossRefGoogle Scholar
Deliu, A. and Jawerth, B., ‘Geometrical dimension versus smoothness’, Constr. Approx. 8 (1992), 211–222.CrossRefGoogle Scholar
Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications (John Wiley Sons Inc., New York, 1999).Google Scholar
Hochman, M., ‘On self-similar sets with overlaps and inverse theorems for entropy’, Ann. of Math. (2) 180 (2014), 773–822.CrossRefGoogle Scholar
Hunt, B. R., ‘The Hausdorff dimension of graphs of Weierstrass functions’, Proc. Amer. Math. Soc. 126 (1998), 791–800.CrossRefGoogle Scholar
Jha, S. and Verma, S., ‘Dimensional analysis of $\alpha$ -fractal functions’, Results Math. 76(4) (2021), 1–24.CrossRefGoogle Scholar
Kong, Q. G., Ruan, H. J. and Zhang, S., ‘Box dimension of bilinear fractal interpolation surfaces’, Bull. Aust. Math. Soc. 98(1) (2018), 113–121.CrossRefGoogle Scholar
Liang, Y. S., ‘Box dimensions of Riemann–Liouville fractional integrals of continuous functions of bounded variation’, Nonlinear Anal. 72(11) (2010), 4304–4306.CrossRefGoogle Scholar
Liang, Y. S., ‘Fractal dimension of Riemann–Liouville fractional integral of $1$ -dimensional continuous functions’, Fract. Calc. Appl. Anal. 21(6) (2019), 1651–1658.CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C., ‘On the Hausdorff dimension of some graphs’, Trans. Amer. Math. Soc. 298(2) (1986), 793–803.CrossRefGoogle Scholar
Nussbaum, R. D., Priyadarshi, A. and Verduyn Lunel, S., ‘Positive operators and Hausdorff dimension of invariant sets’, Trans. Amer. Math. Soc. 364(2) (2012), 1029–1066.CrossRefGoogle Scholar
Pandey, M., Som, T. and Verma, S., ‘Set-valued $\alpha$ -fractal functions’, Preprint, 2022, arXiv:2207.02635.Google Scholar
Priyadarshi, A., ‘Continuity of the Hausdorff dimension for graph-directed systems’, Bull. Aust. Math. Soc. 94 (2016), 471–478.CrossRefGoogle Scholar
Ruan, H. J., Su, W. Y. and Yao, K., ‘Box dimension and fractional integral of linear fractal interpolation functions’, J. Approx. Theory 161 (2009), 187–197.CrossRefGoogle Scholar
Ruan, H. J., Xiao, J. and Yang, B., ‘Existence and box dimension of general recurrent fractal interpolation functions’, Bull. Aust. Math. Soc. 103(2) (2021), 278–290.CrossRefGoogle Scholar
Sahu, A. and Priyadarshi, A., ‘On the box-counting dimension of graphs of harmonic functions on the Sierpiński gasket’, J. Math. Anal. Appl. 487 (2020), 124036.CrossRefGoogle Scholar
Verma, S., ‘Hausdorff dimension and infinitesimal similitudes on complete metric spaces’, Preprint, 2021, arXiv:2101.07520.Google Scholar