The structure of minimizers of the frame potential on fusion frames

TL;DR

This paper investigates the structure and existence of minimizers of the fusion frame potential, extending classical frame theory to fusion frames, and provides conditions for tight fusion frame existence using Horn-Klyachko inequalities.

Contribution

It characterizes local and global minimizers of the fusion frame potential and establishes necessary and sufficient conditions for tight fusion frames with given parameters.

Findings

Identifies conditions under which tight fusion frames exist.

Relates fusion frame potential minimization to Hadamard product indices.

Provides a comprehensive analysis of fusion frame potential minimizers.

Abstract

In this paper we study the fusion frame potential, that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. The structure of local and global minimizers of this potential is studied, when we restrict the frame potential to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. Thus, we exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko's compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, varying the sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima.