Multiplicative Lidskii's inequalities and optimal perturbations of frames

Abstract

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame F = {f<SUB>j</SUB>}<SUB>j∈II_n</SUB> for double-struck C<SUP>d</SUP> 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<SUB>j</SUB>}<SUB>j∈II_n</SUB> - 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 = {gf<SUB>j</SUB>}<SUB>j∈II_n</SUB> 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.