Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace $cWsubset L^2(R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(cF)$ induced by finite sequences of vectors $cFin cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $cF$ and the spectra of $S_{E(cF)}$ is prescribed in each fiber of $text{Spec}(cW)subset T^k$. 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 $0<alpha_1leq ldotsleq alpha_n$ we characterize the finite sequences $cFincW^n$ such that $|f_i|^2=alpha_i$, for $1leq ileq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(cF)$ have 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 $P^cW_varphi$ induced by $cW$ and an arbitrary convex function $varphi:R_+rightarrow R_+$.