Critical Pairs of Sequences of a Mixed Frame Potential

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 $m=1,..., N.$