Translation and dilation invariant subspaces of \(L^2(\mathbb{R})\) and multiresolution analyses (Q1272845)

If \(\tau=\{V_j\}_{j \in\mathbb{Z}}\) is a multiresolution analysis of \(L^2(\mathbb{R})\) then by definition \(\cup_{j\in \mathbb{Z}}V_j\) is dense in \(L^2(\mathbb{R})\). In this paper the authors consider multiresolution analysis which do not necessarily enjoy this property, and characterize the closure of \(\cup_{j \in\mathbb{Z}}V_j\). One consequence of their development is that in many instances the density hypothesis is redundant. Another consequence is the fact that every multiresolution analysis can be complemented by another one so that the orthogonal direct sum is dense in \(L^2(\mathbb{R})\).