Abstract
Let a data set {(x i,y i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x 0,x N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x i)=y i 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.
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References
M. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London, Ser. A,399:243–275.
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.
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.
J. Bellissard (1984): Stability and instability in quantum mechanics. C.N.R.S. (France). Preprint.
R. M. Blumenthal, R. G. Getoor (1960):Some theorems on stable processes. Trans. Amer. Math. Soc.,95:263–273.
A. S. Besicovitch, H. D. Ursell (1937):Sets of fractional dimension. J. London Math. Soc.,12:18–25.
S. Demko, L. Hodges, B. Naylor (1985):Construction of fractal objects with iterated function systems. Computer Graphics,19:271–278.
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.
P. Diaconis, M. Shashahani (1986):Products of random matrices and computer image generation. Contemp. Math.,50:173–182.
K. J. Falconer (1985): The Geometry of Fractal Sets. London: Cambridge University Press.
A. Fournier, D. Fussell, L. Carpenter (1982):Computer rendering of stochastic models. Comm. ACM,25.
A. M. Garcia (1962):Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc.,102:409–432.
J. Hutchinson (1981):Fractals and self-similarity. Indiana Univ. Math. J.,30:713–747.
B. Mandelbrot (1982): The Fractal Geometry of Nature. San Francisco: W. H. Freeman.
S. Pelikan (1984):Invariant densities for random maps of the interval. Trans. Amer. Math. Soc.,281:813–825.
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Communicated by Charles A. Micchelli.
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Barnsley, M.F. Fractal functions and interpolation. Constr. Approx 2, 303–329 (1986). https://doi.org/10.1007/BF01893434
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DOI: https://doi.org/10.1007/BF01893434