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.
تحميل البحث