Minimization of convex functionals over frame operators

TL;DR

This paper investigates the minimization of convex functionals over finite sets of vectors in Hilbert spaces, extending known results for the frame potential using majorization techniques and exploring perturbation problems related to frame operators.

Contribution

It introduces new minimization results for convex functionals over frame operators and analyzes perturbations, advancing understanding of frame operator stability.

Findings

Extended known results for the Benedetto-Fickus frame potential.

Developed majorization-based techniques for minimization.

Analyzed perturbation effects on frame operators.

Abstract

We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.