Subspaces with normalized tight frame wavelets in $\mathbb{R}$

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