Abstract
In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the nth level. We discuss the oscillation space, \(\mathcal {L}_q\) space and some other function spaces on SG, and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1.
Similar content being viewed by others
References
Agrawal, V., Som, T.: \(\cal{L} ^p\)-approximation using fractal functions on the Sierpiński Gasket. Results Math. 77(2), 1–17 (2021)
Bandt, C., Barnsley, M., Hegland, M., Vince, A.: Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations. Chaos Solitons Fractals 91, 478–489 (2016)
Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)
Barnsley, M.F.: Fractal Everywhere. Academic Press, San Diego (1988)
Celik, D., Kocak, S., Özdemir, Y.: Fractal interpolation on the Sierpiński gasket. J. Math. Anal. Appl. 337(1), 343–347 (2008)
Chandra, S., Abbas, S.: The calculus of bivariate fractal interpolation surfaces. Fractals 29(3), 2150066 (2021)
Chandra, S., Abbas, S.: Analysis of fractal dimension of mixed Riemann-Liouville integral. Numer. Algorithms 91(3), 1021–1046 (2022)
Chandra, S., Abbas, S.: Box dimension of mixed Katugampola fractional integral of two-dimensional continuous functions. Fract. Calc. Appl. Anal. 25(3), 1022–1036 (2022). https://doi.org/10.1007/s13540-022-00050-2
Dalrymple, K., Strichartz, R.S., Vinson, J.P.: Fractal differential equations on the Sierpinski gasket. J. Fourier Anal. Appl. 5(2), 203–284 (1999)
Deliu, A., Jawerth, B.: Geometrical dimension versus smoothness. Constr. Approx. 8, 211–222 (1992)
Demir, B., Dzhafarov, V., Koçak, Ş, Üreyen, M.: Derivatives of the restrictions of harmonic functions on the Sierpinski gasket to segments. J. Math. Anal. Appl. 333(2), 817–822 (2007)
Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York (1999)
Falconer, K.J., Fraser, J.M.: The horizon problem for prevalent surfaces. Mathematical Proc. Cambridge Philos. Soc. 151(2), 355–372 (2011)
Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge, UK (2001)
Liang, Y.S.: Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal. 72(11), 4304–4306 (2010)
Liang, Y.S.: Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions. Fract. Calc. Appl. Anal. 21(6), 1651–1658 (2018). https://doi.org/10.1515/fca-2018-0087
Liang, Y.S., Su, W.Y.: Fractal dimensions of fractional integral of continuous functions. Acta Math. Sin. 32, 1494–1508 (2016)
Liang, Y. S.: Estimation of fractal dimension of fractional calculus of the Hölder continuous functions. Fractals, 28(07), 2050123 (6 pages) (2020)
Ri, S.G., Ruan, H.J.: Some properties of fractal interpolation functions on Sierpinski gasket. J. Math. Anal. Appl. 380(1), 313–322 (2011)
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)
Ruan, H.J.: Fractal interpolation functions on post critically finite self-similar sets. Fractals 18(1), 119–125 (2010)
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)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Switzerland (1993)
Strichartz, R.S.: Differential Equations on Fractals. Princeton University Press, Princeton, NJ (2006)
Tatom, F.B.: The relationship between fractional calculus and fractals. Fractals 03(01), 217–229 (1995)
Verma, S., Sahu, A.: Bounded variation on the Sierpiński gasket. Fractals 30(07), 1–12 (2022)
Acknowledgements
The first author thanks MHRD, India for financial support in the form of a Senior Research Fellowship at the Indian Institute of Technology Delhi.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The 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.
About this article
Cite this article
Verma, M., Priyadarshi, A. & Verma, S. Analytical and dimensional properties of fractal interpolation functions on the Sierpiński gasket. Fract Calc Appl Anal 26, 1294–1325 (2023). https://doi.org/10.1007/s13540-023-00148-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13540-023-00148-1
Keywords
- Fractal function
- Sierpiński gasket
- Hölder continuity
- Hausdorff dimension
- Box dimension
- Riemann-Liouville fractional integral