Optimal reconstruction systems for erasures and for the q-potential

We introduce the $q$-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call $q$-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko's and Fulton's theory on sums of hermitian operators

Aliasing and oblique dual pair designs for consistent sampling

In this paper we study some aspects of oblique duality between finite sequences of vectors \cF and \cG lying in finite dimensional subspaces \cW and \cV, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to \cF lying in \cV; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for \cF under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces \cV and \cW has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for \cW such that the canonical oblique dual of U\cdot \cF minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for \cW such that the canonical oblique dual pair associated to U\cdot \cF minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.Comment: 23 page