We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in L 2 (R k ). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.

In this paper we study some aspects of oblique duality between finite sequences of vectors F and G lying in finite dimensional subspaces W and V, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to F lying in V; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for F under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces V and W has in oblique duality. We apply this analysis to compute those rigid rotations U for W such that the canonical oblique dual of U · F minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations U for W such that the canonical oblique dual pair associated to U ·F minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.

For a given finitely generated shift invariant (FSI) subspace W ⊂ L 2 (R k ) we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences E(F ) induced by finite sequences of vectors F ∈ W n that have a prescribed fine structure i.e., such that the norms of the vectors in F and the spectra of S E(F ) is prescribed in each fiber of Spec(W) ⊂ 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 < α 1 ≤ . . . ≤ α n we characterize the finite sequences F ∈ W n such that f i 2 = α i , for 1 ≤ i ≤ n, and such that the fine spectral structure of the shift generated Bessel sequences E(F ) 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 W ϕ induced by W and an arbitrary convex function ϕ : R + → R + .

Let d = (d j ) j∈Im ∈ N m be a finite sequence (of dimensions) and α = (α i ) i∈In be a sequence of positive numbers (weights), where I k = {1, . . . , k} for k ∈ N. We introduce the (α , d)-designs i.e., families Φ = (F j ) j∈Im such that F j = {f ij } i∈In is a frame for C dj , j ∈ I m , and such that the sequence of non-negative numbers ( f ij 2 ) j∈Im forms a partition of α i , i ∈ I n . We show, by means of a finite-step algorithm, that there exist (α , d)-designs Φ op = (F op j ) that are universally optimal; that is, for every convex function ϕ : [0, ∞) → [0, ∞) then Φ op minimizes the joint convex potential induced by ϕ among (α , d)-designs, namely j∈Im

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