Critical Pairs of Sequences of a Mixed Frame Potential

TL;DR

This paper introduces the mixed frame potential, a generalization of the classical frame potential, and characterizes the structure of its critical pairs of sequences, providing conditions for dual frames with prescribed inner products.

Contribution

It extends the classical frame potential to a mixed version and analyzes the critical pairs of sequences, offering new insights into dual frame structures with specified inner products.

Findings

Characterization of critical pairs of sequences for the mixed frame potential

Necessary and sufficient conditions for dual frames with prescribed inner products

Generalization of the classical frame potential to a mixed setting

Abstract

The classical frame potential in a finite dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the start point of a series of new results in frame theory, related to finding tight frames with determined length. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of {\sl mixed frame potential}, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence ${\alpha_m}_{m=1,...,N}$ in $K,$ where $K$ is $\mathbb{R}$ or $\mathbb{C},$ we obtain necessary and sufficient conditions in order to have a dual pair of frames ${f_m}_{m=1,...,N}$, ${g_m}_{m=1,...,N}$ such that $<f_m, g_m>=\alpha_m$ for all $m=1,..., N.$