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Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory. [ABSTRACT FROM AUTHOR]

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 ) are 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 α 1 ≥ … ≥ α n > 0 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 ) 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 P φ W induced by W and an arbitrary convex function φ : R + → R + . [ABSTRACT FROM AUTHOR]

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 compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for F with norm restrictions; as an application, we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible, among oblique duals with norm 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 oblique duality, within the context of consistent sampling. [ABSTRACT FROM AUTHOR]

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame F = { f j } j ∈ I n for C d we compute those dual frames G of F that are optimal perturbations of the canonical dual frame for F under certain restrictions on the norms of the elements of G . On the other hand, we compute those V ⋅ F = { V f j } j ∈ I n – for invertible operators V which are close to the identity – that are optimal perturbations of F . That is, we compute the optimal perturbations of F among frames G = { g j } j ∈ I n that have the same linear relations as F . In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality. [ABSTRACT FROM AUTHOR]

Abstract: We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results. [Copyright &y& Elsevier]