Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace W⊂L2(Rk)W⊂L2(Rk) we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences E(F)E(F) induced by finite sequences of vectors F∈WnF∈Wn that have a prescribed fine structure i.e., such that the norms of the vectors in FF and the spectra of SE(F)SE(F) are prescribed in each fiber of Spec(W)⊂TkSpec(W)⊂Tk. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given α1≥…≥αn>0α1≥…≥αn>0 we characterize the finite sequences F∈WnF∈Wn such that ‖fi‖2=αi‖fi‖2=αi, for 1≤i≤n1≤i≤n, and such that the fine spectral structure of the shift generated Bessel sequences E(F)E(F) has minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential PφW induced by WW and an arbitrary convex function φ:R+→R+φ:R+→R+.