Fractal functions and interpolation. (English) Zbl 0606.41005

Consider a set of data points \(\{x_ i,y_ i)\in I\times R:i=0,...,N\}\), where \(I=[x_ 0,x_ N]\subseteq R\). The author considers continuous functions f:I\(\to R\) which interpolate the given data, i.e. \(f(x_ i)=y_ i\), and such that there exist a compact subset \(K=I\times [a,b]\subseteq R^ 2\) and continuous functions \(w_ n:K\to K\) such that the graph G of f is the unique closed subset of K satisfying \(G=\cup^{N}_{n=1}w_ n(G)\). Such a function f is called a fractal interpolation function. It can occur that the Hausdorff-Besicovitch dimension of G is noninteger, usually for functions f which are Hölder but not differentiable. (For the theory of fractal sets see e.g. K. J. Falconer, The geometry of fractal sets (1985; Zbl 0587.28004.) The fractal interpolation can be used to approximate wilder functions such as temperature in flames, electroencephalograph pen traces, etc. The author also discusses the associated coding theory and measure theory and evaluates explicitly moment integrals for a wide class of fractial interpolation functions.
Reviewer: C.Mustăţa


41A05 Interpolation in approximation theory
94A24 Coding theorems (Shannon theory)


Zbl 0587.28004
Full Text: DOI


[1] M. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London, Ser. A,399:243–275. · Zbl 0588.28002
[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. · Zbl 0571.41010
[3] 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. · Zbl 0613.28008
[4] J. Bellissard (1984): Stability and instability in quantum mechanics. C.N.R.S. (France). Preprint. · Zbl 0584.35024
[5] R. M. Blumenthal, R. G. Getoor (1960):Some theorems on stable processes. Trans. Amer. Math. Soc.,95:263–273. · Zbl 0107.12401
[6] A. S. Besicovitch, H. D. Ursell (1937):Sets of fractional dimension. J. London Math. Soc.,12:18–25. · Zbl 0016.01703
[7] S. Demko, L. Hodges, B. Naylor (1985):Construction of fractal objects with iterated function systems. Computer Graphics,19:271–278.
[8] 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. · Zbl 0674.60093
[9] P. Diaconis, M. Shashahani (1986):Products of random matrices and computer image generation. Contemp. Math.,50:173–182.
[10] K. J. Falconer (1985): The Geometry of Fractal Sets. London: Cambridge University Press. · Zbl 0587.28004
[11] A. Fournier, D. Fussell, L. Carpenter (1982):Computer rendering of stochastic models. Comm. ACM,25.
[12] A. M. Garcia (1962):Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc.,102:409–432. · Zbl 0103.36502
[13] J. Hutchinson (1981):Fractals and self-similarity. Indiana Univ. Math. J.,30:713–747. · Zbl 0598.28011
[14] B. Mandelbrot (1982): The Fractal Geometry of Nature. San Francisco: W. H. Freeman. · Zbl 0504.28001
[15] S. Pelikan (1984):Invariant densities for random maps of the interval. Trans. Amer. Math. Soc.,281:813–825. · Zbl 0532.58013
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