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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

MSC:

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

Citations:

Zbl 0587.28004
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References:

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