Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace $\cW\subset L^2(\R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(\text{cF})$ induced by finite sequences of vectors $\cF\in \cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $\text{cF}$ and the spectra of $S_{E(\text{cF})}$ is prescribed in each fiber of $\text{Spec}(\text{cW})\subset {\text{T}}^k$. 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 $0<\alpha_1\leq \ldots\leq \alpha_n$wecharacterizethefinitesequences$\cF\in\cW^n$ such that $\mathrm{\parallel }{f}_i{\mathrm{\parallel }}^2={\alpha }_i$, for $1\le i\le n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(\text{cF})$ 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 $P_{\phi }^{\text{cW}}$ induced by $\text{cW}$ and an arbitrary convex function $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$.