Fractal functions and interpolation
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. \textit{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.
scientific article; zbMATH DE number 3979666
scientific article