Skip to main content
SpringerLink
Log in

Dimensional Analysis of \(\alpha \)-Fractal Functions

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We provide a rigorous study on dimensions of fractal interpolation functions defined on a closed and bounded interval of \(\mathbb {R}\) which are associated to a continuous function with respect to a base function, scaling functions and a partition of the interval. In particular, we calculate an exact estimation of box dimension of \(\alpha \)-fractal functions under suitable hypotheses on the iterated function system.

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

Similar content being viewed by others

References

  1. Akhtar, Md.N., Prasad, M.G.P., Navascués, M.A.: Box dimensions of \(\alpha \)-fractal functions. Fractals 24(3), 1650037 (2016)

  2. Bandt, C., Graf, S.: Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114(4), 995–1001 (1992)

  3. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(4), 303–329 (1986)

    Article  MathSciNet  Google Scholar 

  4. Barnsley, M.F.: Fractals Everywhere. Academic Press, Boston, MA (1988)

    MATH  Google Scholar 

  5. Barnsley, M.F., Elton, J., Hardin, D., Massopust, P.: Hidden variable fractal interpolation functions. SIAM J. Math. Anal. 20(5), 1218–1242 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Carvalho, A.: Box dimension, oscillation and smoothness in function spaces. J. Funct. Spaces Appl. 3(3), 287–320 (2005)

    Article  MathSciNet  Google Scholar 

  8. Chand, A.K.B., Jha, S., Navascués, M.A.: Kantorovich–Bernstein \(\alpha \)-fractal functions in \(\cal{L}^p\) spaces. Quaest. Math. 43(2), 227–241 (2020)

    Article  MathSciNet  Google Scholar 

  9. Deliu, A., Jawerth, B.: Geometrical dimension versus smoothness. Constr. Approx. 8(2), 211–222 (1992)

    Article  MathSciNet  Google Scholar 

  10. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken (2003)

    Book  Google Scholar 

  11. Falconer, K.J.: Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106(2), 543–554 (1989)

    Article  MathSciNet  Google Scholar 

  12. Fan, A.H., Lau, K.S.: Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231(2), 319–344 (1999)

    Article  MathSciNet  Google Scholar 

  13. Gordon, R.A.: Real Analysis: A First Course, 2nd edn. Addison Wesley, London (2001)

    MATH  Google Scholar 

  14. Hardin, D.P., Massopust, P.R.: The capacity for a class of fractal functions. Comm. Math. Phys. 105(3), 455–460 (1986)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Lau, K.S., Rao, H., Ye, Y.L.: Corrigendum: “Iterated function system and Ruelle operator” [J. Math. Anal. Appl. 231 (1999), no. 2, 319–344; MR1669203 (2001a:37013)] by Lau and A. H. Fan. J. Math. Anal. Appl. 262(1), 446–451 (2001)

  17. 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  Google Scholar 

  18. Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, San Diego, CA (1994)

    MATH  Google Scholar 

  19. Massopust, P.R.: Interpolation and Approximation with Splines and Fractals. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Navascués, M.A.: Fractal approximation. Complex Anal. Oper. Theory 4(4), 953–974 (2010)

    Article  MathSciNet  Google Scholar 

  22. Peres, Y., Rams, M., Simon, K., Solomyak, B.: Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets. Proc. Amer. Math. Soc. 129(9), 2689–2699 (2001)

    Article  MathSciNet  Google Scholar 

  23. 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  Google Scholar 

  24. Schief, A.: Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122(1), 111–115 (1994)

    Article  MathSciNet  Google Scholar 

  25. Verma, S., Viswanathan, P.: A revisit to \(\alpha \)-fractal function and box dimension of its graph. Fractals 27(6), 1950090 (2019)

  26. Verma, S., Viswanathan, P.: A fractal operator associated with bivariate fractal interpolation functions on rectangular grids. Res. Math. 75(1), Paper No. 28, 26 (2020)

  27. Vijender, N.: Approximation by hidden variable fractal functions: a sequential approach. Res. Math. 74(4), Paper No. 192, 23 (2019)

  28. Vijender, N.: Bernstein fractal trigonometric approximation. Acta Appl. Math. 159, 11–27 (2019)

    Article  MathSciNet  Google Scholar 

  29. 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  Google Scholar 

  30. Ye, Y.-L.: Separation properties for self-conformal sets. Stud. Math. 152(1), 33–44 (2002)

Download references

Acknowledgements

The first author thanks Dr. P. Viswanathan for his suggestions, support and encouragement during preparation of the manuscript. We would like to thank the anonymous reviewers for the valuable and constructive suggestions that helped to improve the manuscript.

Author information

Author notes

  1. Both authors contributed equally in this manuscript.

Authors and Affiliations

  1. Department of Mathematics, NIT Rourkela, Rourkela, 769008, India

    S. Jha

  2. Department of Applied Sciences, IIIT Allahabad, Allahabad, 211015, India

    S. Verma

Authors
  1. S. Jha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Jha.

Ethics declarations

Conflict of interest

Both authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jha, S., Verma, S. Dimensional Analysis of \(\alpha \)-Fractal Functions. Results Math 76, 186 (2021). https://doi.org/10.1007/s00025-021-01495-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-021-01495-2

Keywords

Mathematics Subject Classification

Search

Navigation