Frame completions with prescribed norms: local minimizers and applications

Abstract

Let $\mathcal F_0=\{f_i\}_{i\in\mathbb{I}_{n_0}}$ be a finite sequence of vectors in $\mathbb C^d$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $\cal F_0$ of the form $\cal F=(\cal F_0,\cal G)$ obtained by appending a sequence $\cal G=\{g_i\}_{i\in\mathbb{I}_k}$ of vectors in $\mathbb C^d$ such that $\|g_i\|^2=a_i$ for $i\in\mathbb{I}_k$, and endow the set of completions with the metric $d(\cal F,\tilde {\mathcal F}) =\max\{ \,\|g_i-\tilde g_i\|: \ i\in\mathbb{I}_k\}$ where $\tilde {\cal F}=(\cal F_0,\,\tilde {\cal G})$. In this context we show that local minimizers on the set of completions of a convex potential $\text{P}_\varphi$, induced by a strictly convex function $\varphi$, are also global minimizers. In case that $\varphi(x)=x^2$ then $\text{P}_\varphi$ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn's conjecture on the FOD.