Constructive Approximation

, Volume 38, Issue 2, pp 311–337 | Cite as

Fractal Continuation

  • Michael F. BarnsleyEmail author
  • Andrew Vince


A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.


Fractal function Analytic continuation Iterated function system Interpolation 

Mathematics Subject Classification

28A80 26E05 26A30 41A05 



We thank Louisa Barnsley for help with the illustrations and Mike Eastwood for helpful discussions.


  1. 1.
    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barnsley, M.F.: Fractals Everywhere. Academic Press, San Diego (1988). 2nd edn., Morgan Kaufmann (1993); 3rd edn., Dover (2012) zbMATHGoogle Scholar
  3. 3.
    Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57, 14–34 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barnsley, M.F., Freiberg, U.: Fractal transformations of harmonic functions. In: Complexity and Nonlinear Dynamics. Proc. SPIE, vol. 6417 (2006) CrossRefGoogle Scholar
  5. 5.
    Berger, M.A.: Random affine iterated function systems: curve generation and wavelets. SIAM Rev. 34, 361–385 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bailey, D.H., Borwein, J.M., Calkin, N.J., Girgensohn, R., Luke, D.R., Moll, V.H.: Experimental Mathematics in Action. AK Peters, Wellesley (2006) Google Scholar
  7. 7.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Massopust, P.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, New York (1995) Google Scholar
  9. 9.
    Massopust, P.: Interpolation and Approximation with Splines and Fractals. Oxford University Press, Oxford (2010) zbMATHGoogle Scholar
  10. 10.
    Narasimhan, R.: Introduction to the Theory of Analytic Spaces. Lecture Notes in Mathematics, vol. 25. Springer, Berlin (1966) zbMATHGoogle Scholar
  11. 11.
    Navascues, M.A.: Fractal polynomial interpolation. Z. Anal. Anwend. 24, 401–414 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Prasad, S.A.: Some aspects of coalescence and superfractal interpolation. Ph.D. Thesis, Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur (March 2011) Google Scholar
  13. 13.
    Scealy, R.: V-variable fractals and interpolation. Ph.D. Thesis, Australian National University (2008) Google Scholar
  14. 14.
    Tosan, E., Guerin, E., Baskurt, A.: Design and reconstruction of fractal surfaces. In: 6th International Conference on Information Visualisation IV, London, UK, July 2002, pp. 311–316. IEEE Comput. Soc., Los Alamitos (2002) Google Scholar
  15. 15.
    Tricot, C.: Curves and Fractal Dimension. Springer, New York (1995) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsAustralian National UniversityCanberraAustralia
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA

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