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Constructive Approximation

, Volume 2, Issue 1, pp 303–329 | Cite as

Fractal functions and interpolation

  • Michael F. Barnsley
Article

Abstract

Let a data set {(xi,yi) ∈I×R;i=0,1,⋯,N} be given, whereI=[x0,xN]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsfIR, which interpolate the data according tof(xi)=yi fori ε {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.

AMS classification

26A 28A 41A 44A 60J 

Key words and phrases

Interpolation Fractals Moment theory 

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References

  1. [BD 1]
    M. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London, Ser. A,399:243–275.Google Scholar
  2. [BD 2]
    M. Barnsley, S. Demko (1984):Rational approximation of fractals. In: Rational Approximation and Interpolation (P. R. Graves-Morris, E. B. Saff, R. S. Varga, eds.). New York: Springer-Verlag.Google Scholar
  3. [BEHL]
    M. Barnsley, V. Ervin, D. Hardin, J. Lancaster (1986):Solution of an inverse problem for fractals and other sets. Proc. Nat. Acad. Sci. U.S.A.,83:1975–1977.Google Scholar
  4. [Be]
    J. Bellissard (1984): Stability and instability in quantum mechanics. C.N.R.S. (France). Preprint.Google Scholar
  5. [BlG]
    R. M. Blumenthal, R. G. Getoor (1960):Some theorems on stable processes. Trans. Amer. Math. Soc.,95:263–273.Google Scholar
  6. [BU]
    A. S. Besicovitch, H. D. Ursell (1937):Sets of fractional dimension. J. London Math. Soc.,12:18–25.Google Scholar
  7. [DHN]
    S. Demko, L. Hodges, B. Naylor (1985):Construction of fractal objects with iterated function systems. Computer Graphics,19:271–278.Google Scholar
  8. [Der]
    B. Derrida (1986):Real space renormalization and Julia sets in statistical physics. In: Chaotic Dynamics and Fractals (M. F. Barnsley, S. G. Demko, eds.). New York: Academic Press.Google Scholar
  9. [DS]
    P. Diaconis, M. Shashahani (1986):Products of random matrices and computer image generation. Contemp. Math.,50:173–182.Google Scholar
  10. [F]
    K. J. Falconer (1985): The Geometry of Fractal Sets. London: Cambridge University Press.Google Scholar
  11. [FFC]
    A. Fournier, D. Fussell, L. Carpenter (1982):Computer rendering of stochastic models. Comm. ACM,25.Google Scholar
  12. [G]
    A. M. Garcia (1962):Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc.,102:409–432.Google Scholar
  13. [H]
    J. Hutchinson (1981):Fractals and self-similarity. Indiana Univ. Math. J.,30:713–747.Google Scholar
  14. [M]
    B. Mandelbrot (1982): The Fractal Geometry of Nature. San Francisco: W. H. Freeman.Google Scholar
  15. [P]
    S. Pelikan (1984):Invariant densities for random maps of the interval. Trans. Amer. Math. Soc.,281:813–825.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Michael F. Barnsley
    • 1
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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