Subspaces with normalized tight frame wavelets in ℝ

Abstract

In this paper we investigate the subspaces of L 2 ( R ) L^2(\mathbb R) which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of R \mathbb R called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of L 2 ( R ) L^2(\mathbb R) that are generated by a Bessel set E E in a special way. We present some results concerning the relation between a Bessel set E E and the corresponding subspace of L 2 ( R ) L^2(\mathbb R) which either has a normalized tight frame wavelet defined by the set function on E E or is generated by E E .