The structure of shift-invariant subspaces of L2(Rn)

Using the range function approach to shift invariant spaces in L-2(R ") we give a simple characterization of frames and Riesz families generated by shifts of it countable set of generators in terms of their behavior on subspaces of l(2)(Z(n)). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function. (C) 2000 Academic Press.