Multiplicative Lidskii's inequalities and optimal perturbations of frames.

P. MASSEY, M. A. RUIZ Y D. STOJANOFF

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Lugar: Amsterdam; Año: 2015 vol. 436 p. 447 - 464

0024-3795

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame $cF={f_j}_{jinIn}$ for $C^d$ 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,  we compute those $Vcdot cF={V,f_j}_{jinIn}$ - for invertible operators $V$ which are close to the identity - that are optimal perturbations of $cF$. That is, we compute the optimal perturbations of $cF$ among frames $cG={g_j}_{jinIn}$ that have the same linear relations as $cF$. 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 Lidskii´s inequality in terms of log-majorization and a characterization of the case of equality.