2012/06/15 by Pedro Massey, Massey, P., Mariano Ruiz +3 · 1 citation
Mathematics · Engineering · #Mathematical Analysis and Transform Methods #Advanced Numerical Analysis Techniques #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1206.3588
Given a finite sequence of vectors \mathcal F0 in \Cd we describe the spectral and geometrical structure of optimal completions of \mathcal F0 obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to a general convex potential. In particular, our analysis includes the so-called Mean Square Error (MSE) and the Benedetto-Fickus' frame potential. On a first step, we reduce the problem of finding the optimal completions to the computation of the minimum of a convex function in a convex compact polytope in \Rd. As a second step, we show that there exists a finite set (that can be explicitly computed in terms of a finite step algorithm that depends on \cF0 and the sequence of prescribed norms) such that the optimal frame completions with respect to a given convex potential can be described in terms of a distinguished element of this set. As a byproduct we characterize the cases of equality in Lindskii's inequality from matrix theory.