Contractibility of Simple Scaling Sets

Abstract

In this paper, we show that the space of three-interval scaling functions with the induced metric of $L^2(\mathbb R)$ consists of three pathcomponents each of which is contractible and hence, the first fundamental group of these spaces is zero. One method to construct simple scaling sets for $L^2(\mathbb R)$ and $H^2(\mathbb R)$ is described. Further, we obtain a characterization of a method to provide simple scaling sets for higher dimensions with the help of lower dimensional simple scaling sets and discuss scaling sets, wavelet sets and multiwavelet sets for a reducing subspace of $L^2(\mathbb R^n)$. The contractibility of simple scaling sets for different subspaces are also discussed.