Orthonormal wavelets and shift invariant generalized multiresolution analyses

All wavelets can be associated to a multiresolution-like structure, i.e. an increasing sequence of subspaces of $L^2({\mathbb R})$. We consider the interaction of a wavelet and the shift operator in terms of which of the subspaces in this multiresolution-like structure are invariant under the shift operator. This action defines the notion of the shift invariance property of order $n$. In this paper we show that wavelets of all levels of shift invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral dilation factors.