Skip to main content
Journal of Inequalities and Applications
Download PDF

Smooth fractal interpolation

Journal of Inequalities and Applications volume 2006, Article number: 78734 (2006) Cite this article

Abstract

Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in is proven.

[1234567891011121314]

References

  1. Ahlberg JH, Nilson EN, Walsh JL: The Theory of Splines and Their Applications. Academic Press, New York; 1967:xi+284.

    MATH  Google Scholar 

  2. Barnsley MF: Fractal functions and interpolation. Constructive Approximation 1986,2(4):303–329.

    Article  MATH  MathSciNet  Google Scholar 

  3. Barnsley MF: Fractals Everywhere. Academic Press, Massachusetts; 1988:xii+396.

    MATH  Google Scholar 

  4. Barnsley MF, Harrington AN: The calculus of fractal interpolation functions. Journal of Approximation Theory 1989,57(1):14–34. 10.1016/0021-9045(89)90080-4

    Article  MATH  MathSciNet  Google Scholar 

  5. Chand AKB, Kapoor GP: Generalized cubic spline fractal interpolation functions. SIAM Journal on Numerical Analysis 2006,44(2):655–676. 10.1137/040611070

    Article  MATH  MathSciNet  Google Scholar 

  6. Ciarlet PG, Schultz MH, Varga RS: Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem. Numerische Mathematik 1967,9(5):394–430. 10.1007/BF02162155

    Article  MATH  MathSciNet  Google Scholar 

  7. Hall CA, Meyer WW: Optimal error bounds for cubic spline interpolation. Journal of Approximation Theory 1976,16(2):105–122. 10.1016/0021-9045(76)90040-X

    Article  MATH  MathSciNet  Google Scholar 

  8. Hutchinson JE: Fractals and self-similarity. Indiana University Mathematics Journal 1981,30(5):713–747. 10.1512/iumj.1981.30.30055

    Article  MATH  MathSciNet  Google Scholar 

  9. Laurent P-J: Approximation et optimisation. Hermann, Paris; 1972:xiii+531.

    MATH  Google Scholar 

  10. Navascués MA, Sebastián MV: Some results of convergence of cubic spline fractal interpolation functions. Fractals 2003,11(1):1–7. 10.1142/S0218348X03001550

    Article  MATH  MathSciNet  Google Scholar 

  11. Navascués MA, Sebastián MV: Fitting curves by fractal interpolation: an application to the quantification of cognitive brain processes. In Thinking in Patterns: Fractals and Related Phenomena in Nature. Edited by: Novak MM. World Scientific, New Jersey; 2004:143–154.

    Chapter  Google Scholar 

  12. Navascués MA, Sebastián MV: Generalization of Hermite functions by fractal interpolation. Journal of Approximation Theory 2004,131(1):19–29. 10.1016/j.jat.2004.09.001

    Article  MATH  MathSciNet  Google Scholar 

  13. Navascués MA, Sebastián MV: A fractal version of Schultz's theorem. Mathematical Inequalities & Applications 2005,8(1):63–70.

    Article  MATH  MathSciNet  Google Scholar 

  14. Stoer J, Bulirsch R: Introduction to Numerical Analysis. Springer, New York; 1980:ix+609.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Zaragoza, C/María de Luna 3, Zaragoza, 50018, Spain

    M. A. Navascués

  2. Departamento de Matemáticas, Universidad de Zaragoza, Campus Plaza de San Francisco s/n, Zaragoza, 50009, Spain

    M. V. Sebastián

Authors
  1. M. A. Navascués
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Sebastián
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Navascués.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Navascués, M.A., Sebastián, M.V. Smooth fractal interpolation. J Inequal Appl 2006, 78734 (2006). https://doi.org/10.1155/JIA/2006/78734

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/JIA/2006/78734

Keywords

  • Classical Method
  • Interpolation Error
  • Fractal Function
  • General Frame
  • Fractal Technique