In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame $cF$ for $hilcongC^d$ we compute those dual frames $cG$ of $cF$ that are optimal perturbations of the canonical dual frame for $cF$ under certain
restrictions on the norms of the elements of $cG$. On the other hand, for a fixed finite frame $cF={f_j}_{jinIn}$ for $hil$ we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame $Vcdot cF={V,f_j}_{jinIn}$ - which is equivalent to $cF$ - is optimal among such perturbations of $cF$. 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 Lidskiis inequality in terms of log-majorization and a characterization of the case of equality.
In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $mathcal F$ we describe the spectral and geometrical structure of optimal completions of $mathcal F$ by a finite family of vectors with prescribed norms,
where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus frame potential. On the other hand, given a fixed frame $mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $mathcal G$ that are in duality with $mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.
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 Klyachkos and Fultons theory on sums of hermitian operators.
