Multiresolution analyses based on fractal functions
The aim of this paper is to construct multiresolution analyses (see the paper for the definition) based on fractal interpolation functions (FIF). The use of finite sets of such functions is motivated by the fact that they are self-affine, i.e. each FIF is a finite union of affine imagines of itself and has non-integral dimension \(d\), which can be used as an additional parameter in characterizing highly complex sets. They obey to a dilation equation and FIF theory is related also to wavelet theory. Section 3 is concerned with multiresolution analyses in \(L^ 2(\mathbb{R})\) and \(C_ 0(\mathbb{R})\). Section 4 is devoted to explicit constructions of the set that generates the multiresolution analyses and with application to decomposition and reconstruction algorithms to concrete examples, namely in \(V_ k\cap L^ 2(\mathbb{R})\) and \(V_ k\cap C_ 0(\mathbb{R})\), where \(\{V_ k\}_{k\geq 0}\) is a nested sequence of linear subspaces considered in Section 3. The examples are illustrated on some figures.
scientific article; zbMATH DE number 93298