Contractibility of simple scaling sets

Abstract

In this paper, we show that the space of three-interval scaling functions with the induced metric of L2(ℝ) consists of three path-components each of which is contractible and hence, the first fundamental group of these spaces is zero. One method to construct simple scaling sets for L2(ℝ) and H2(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 L2(ℝn). The contractibility of simple scaling sets for different subspaces are also discussed.