2011/08/22 by Pedro Massey, Massey, Pedro G., Mariano Ruiz +3
Mathematics · #15A60 #42C15 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1108.4412
openalex publication_date 2011/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider two problems in frame theory. On the one hand, given a set of vectors \mathcal F we describe the spectral and geometrical structure of optimal completions of \mathcal F by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus' frame potential. On the other hand, given a fixed frame \mathcal F we describe explicitly the spectral and geometrical structure of optimal frames \mathcal G that are in duality with \mathcal F and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.