## Communications in Mathematical Analysis

### 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.

#### Article information

Source
Commun. Math. Anal. Volume 16, Number 1 (2014), 31-46.

Dates
First available in Project Euclid: 4 November 2013

http://projecteuclid.org/euclid.cma/1383587518

Mathematical Reviews number (MathSciNet)
MR3161734

Zentralblatt MATH identifier
1297.42052

Subjects
Primary: 42C40

#### Citation

Shukla, N. K.; Yadav, G.C.S. Contractibility of Simple Scaling Sets. Commun. Math. Anal. 16 (2014), no. 1, 31--46. http://projecteuclid.org/euclid.cma/1383587518.