2013/02/15 by Pedro Massey, Massey, Pedro, Mariano Ruiz +3
Engineering · Mathematics · #15A60 #42C15 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Sparse and Compressive Sensing Techniques #Spectral Theory in Mathematical Physics #math.FA #msc:15A60 #msc:42C15
paper · pdf · doi:10.48550/arxiv.1302.3859
25 pages. arXiv admin note: text overlap with arXiv:1206.3588
arxiv created 2013/02/15 · openalex publication_date 2013/02/15 · arxiv updated 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a finite sequence of vectors \mathcal F0 in \Cd we characterize in a complete and explicit way the optimal completions of \mathcal F0 obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence \cF0 and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence \cF0 we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors \mathcal F0 in \Cd we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.