Self-affine fractal functions and wavelet series
The authors study functions represented by series of wavelet-type: \[ f(x) = \sum_{g \in G} c_g \psi(g^{-1}(x)), \] where \(G\) is a group generated by affine functions \(L_1, \ldots, L_n\) and \(\psi\) is piecewise affine. They show that the class of self-affine fractal functions previously studied by \textit{M. F. Barnsley} and \textit{S. Demko} [Proc. R. Soc. Lond., Ser. A 399, 243-275 (1985; Zbl 0588.28002)] and \textit{M. F. Barnsley} [Constructive Approximation 2, 303-329 (1986; Zbl 0606.41005)] and classical fractal functions such as Devil's Staircase of Cantor-Lebesgue and Weierstrass-like nowhere differentiable functions can be characterized by such series. As applications, the authors compute the global and local Hölder exponents of affine wavelet-type functions, and they also construct functions with prescribed local Hölder exponents at each point.
scientific article; zbMATH DE number 1417580
scientific article