Skip to main content
SpringerLink
Log in

Dimensions of new fractal functions and associated measures

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we obtain a vector-valued fractal interpolation function in a more general setting by using the Rakotch fixed point theory and the iterated function system. We also show the existence of the Borel probability measure, widely known as a fractal measure, supported on the graph of this vector-valued fractal interpolation function. We obtain the Hausdorff dimension and the box-counting dimension of the graph of this more general fractal function using some function spaces. We obtain bounds for the fractal dimension of the graph of the Riemann-Liouville fractional integral of this fractal function. Using our techniques, we also calculate the fractal dimensions of the graphs of some fractal interpolation functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Agrawal, V., Som, T.: Lp-approximation using fractal functions on the Sierpiński Gasket. Results Math. 77(2), 1–17 (2022)

    Article  MATH  Google Scholar 

  2. Akhtar, M.N., Prasad, M.G.P., Navascués, M.A.: Box dimensions of \(\alpha \)-fractal functions. Fractals 24(03), 1650037 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(1), 303–329 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barnsley, M.F.: Fractal Everywhere. Academic Press, Orlando (1988)

    MATH  Google Scholar 

  5. Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57(1), 14–34 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barnsley, M.F., Massopust, P.R.: Bilinear fractal interpolation and box dimension. J. Approx. Theory 192, 362–378 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bogachev, V.: Measure Theory. Springer, Berlin, Heidelberg, New York (2007)

    Book  MATH  Google Scholar 

  8. S. Chandra, S. Abbas, Analysis of fractal dimension of mixed Riemann-Liouville integral, Numer. Algorithms (2022) 1–26

  9. S. Chandra, S. Abbas, Box dimension of mixed Katugampola fractional integral of two-dimensional continuous functions, Fract. Calc. Appl. Anal. (2022) 1–15

  10. S. Chandra, S. Abbas, On fractal dimensions of fractal functions using functions spaces, Bull. Aust. Math. Soc. (2022) 1–11

  11. Dalla, L., Drakopoulos, V., Prodromou, M.: On the box dimension for a class of non-affine fractal interpolation functions. Anal. Theory Appl. 19(3), 220–233 (2003)

    Article  MathSciNet  Google Scholar 

  12. Drakopoulos, V., Bouboulis, P., Theodoridis, S.: Image compression using affine fractal interpolation on rectangular lattices. Fractals 14(04), 259–269 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Falconer, K.J.: Techniques in Fractal Geometry. Wiley, New York (1997)

    MATH  Google Scholar 

  14. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley & Sons Inc., Hoboken, NJ (2003)

    Book  MATH  Google Scholar 

  15. Hardin, D.P., Massopust, P.R.: Fractal interpolation functions from \(\mathbb{R} ^n\) to \(\mathbb{R} ^m\) and their projections. Z. Anal. Anwend. 12(3), 535–548 (1993)

    Article  MATH  Google Scholar 

  16. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jachymski, J.: Equivalence of some contractivity properties over metrical structures. Proc. Amer. Math. Soc. 125(8), 2327–2335 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jachymski, J., Jóźwik, I.: Nonlinear contractive conditions: a comparison and related problems. Banach Center Publ. 77, 123–146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jha, S., Verma, S.: Dimensional analysis of \(\alpha \)-fractal functions. Results Math. 76(4), 1–24 (2021)

    Article  MathSciNet  Google Scholar 

  20. Leśniak, K., Snigireva, N., Strobin, F.: Weakly contractive iterated function systems and beyond: a manual. J. Difference Equ. Appl. 26(8), 1114–1173 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liang, Y.S.: Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal. 72(11), 4304–4306 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liang, Y.S.: Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions. Fract. Calc. Appl. Anal. 21(6), 1651–1658 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. S. Liang, Estimation of fractal dimension of fractional calculus of the Hölder continuous functions, Fractals, 28(07) (2020) 2050123 (6 pages)

  24. Massopust, P.R.: Vector-valued fractal interpolation functions and their box dimension. Aequationes Math. 42(1), 1–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Massopust, P.R.: Non-stationary fractal interpolation. Mathematics 7(8), 666 (2019)

    Article  Google Scholar 

  26. Mauldin, R.D., Williams, S.C.: On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298(2), 793–803 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Navascués, M.A.: Fractal polynomial interpolation. Z. Anal. Anwend. 24(2), 401–418 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Navascués, M.A., Verma, S.: Non-stationary alpha-fractal surfaces. Mediterr. J. Math. 20(1), 48 (2023)

    Article  MATH  Google Scholar 

  29. R. D. Nussbaum, A. Priyadarshi and S. Verduyn Lunel, Positive operators and Hausdorff dimension of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029–1066

  30. Priyadarshi, A.: Lower bound on the Hausdorff dimension of a set of complex continued fractions. J. Math. Anal. Appl. 449(1), 91–95 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rakotch, E.: A note on contractive mappings. Proc. Amer. Math. Soc. 13(3), 459–465 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ri, S.: A new idea to construct the fractal interpolation function. Indag. Math. 29(3), 962–971 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ruan, H.J., Su, W.Y., Yao, K.: Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory 161(1), 187–197 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sahu, A., Priyadarshi, A.: On the box-counting dimension of graphs of harmonic functions on the Sierpiński gasket. J. Math. Anal. Appl. 487(2), 124036 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tatom, F.B.: The relationship between fractional calculus and fractals. Fractals 3(01), 217–229 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Verma, A. Priyadarshi, Graphs of continuous functions and fractal dimension, https://arxiv.org/abs/2202.11502

  37. S. Verma, P. R. Massopust, Dimension preserving approximation, Aequationes Math. (2022) 1–15

  38. S. Verma, Hausdorff dimension and infinitesimal-similitudes on complete metric spaces, https://arxiv.org/abs/2101.07520

  39. Wang, H.Y., Yu, J.S.: Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their valuable comments and suggestions that substantially improved the presentation of this paper.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Manuj Verma & Amit Priyadarshi

Authors
  1. Manuj Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amit Priyadarshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to the manuscript.

Corresponding author

Correspondence to Amit Priyadarshi.

Ethics declarations

Ethical Approval

Not applicable.

Competing Interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Verma, M., Priyadarshi, A. Dimensions of new fractal functions and associated measures. Numer Algor 94, 817–846 (2023). https://doi.org/10.1007/s11075-023-01521-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01521-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation