Advertisement

Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 11–27 | Cite as

Bernstein Fractal Trigonometric Approximation

  • N. VijenderEmail author
Article
First Online:
  • 88 Downloads

Abstract

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

Keywords

\(\alpha \)-Fractal functions Bernstein polynomials Fractal approximation Convergence Fractal trigonometric approximation Fractal Fourier series 

Mathematics Subject Classification

30E10 28A80 41A30 41A17 65D017 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The author would like to thank the anonymous reviewers for helpful and constructive comments that greatly contributed to improve the quality and presentation of the manuscript.

References

  1. 1.
    Uber die analytische Darsstellbarkeit Sogenannter Willkurlicher Dunctionen einer reellen Veranderlichen. Sitz-Ber. Akad. d. Wiss. Berlin, 633–639 u. 789–805 Google Scholar
  2. 2.
    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chand, A.K.B., Kapoor, G.P.: Generalized cubic spline fractal interpolation functions. SIAM J. Numer. Anal. 44(2), 655–676 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chand, A.K.B., Kapoor, G.P.: Hidden variable bivariate fractal interpolation surfaces. Fractals 11(3), 277–288 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chand, A.K.B., Viswanathan, P.: A constructive approach to cubic Hermite fractal interpolation function and its constrained aspects. BIT Numer. Math. 53(4), 841–865 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chand, A.K.B., Vijender, N., Navascués, M.A.: Shape preservation of scientific data through rational fractal splines. Calcolo 51, 329–362 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Navascués, M.A.: Non-smoothpolynomials. Int. J. Math. Anal. 1, 159–174 (2007) MathSciNetGoogle Scholar
  8. 8.
    Navascués, M.A.: Fractal approximation. Complex Anal. Oper. Theory 4, 953–974 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Navascués, M.A.: Fractal bases of \(L_{p}\) spaces. Fractals 20, 141–148 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Navascués, M.A., Chand, A.K.B.: Fundamental sets of fractal functions. Acta Appl. Math. 100, 247–261 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Navascués, M.A., Sebastián, M.V.: Smooth fractal interpolation. J. Inequal. Appl. 2006, 78734 (2006).  https://doi.org/10.1155/JIA/2006/78734 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Navascués, M.A.: Fractal trigonometric approximation. Electron. Trans. Numer. Anal. 20, 64–74 (2005) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chand, A.K.B., Navascués, M.A., Viswanathan, P., Katiyar, S.K.: Fractal trigonometric polynomials for restricted range approximation. Fractals 24, 1650022 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chand, A.K.B., Tyda, K.R.: Partially blended constrained rational cubic trigonometric fractal interpolation surfaces. Fractals 24, 1650027 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gal, S.G.: Shape Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston (2008) CrossRefzbMATHGoogle Scholar
  16. 16.
    Nasim Akhtar, M.D., Guru Prem Prasad, M., Navascués, M.A.: Box dimensions of \(\alpha \)-fractal functions. Fractals 24(3), 1650037 (2016).  https://doi.org/10.1142/S0218348X16500377 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1996) zbMATHGoogle Scholar
  18. 18.
    Hall, C.A., Meyer, W.W.: Optimal error bounds for cubic spline interpolation. J. Approx. Theory 16(2), 105–122 (1976) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematics Division, School of Advanced SciencesVIT University ChennaiChennaiIndia

Personalised recommendations