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Let $mathcal F_0={f_i}_{iinmathbb{I}_{n_0}}$ be a finite sequence of vectors in $mathbb C^d$ and let $mathbf{a}=(a_i)_{iinmathbb{I}_k}$ 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 $cal G={g_i}_{iinmathbb{I}_k}$ of vectors in $mathbb C^d$ such that $|g_i|^2=a_i$ for $iinmathbb{I}_k$, and endow the set of completions with the metric $d(cal F,tilde {mathcal F}) =max{ ,|g_i-tilde g_i|: iinmathbb{I}_k}$ 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 Strawns conjecture on the FOD.
Given a finite sequence of vectors $mathcal F_0$ in $C^d$ we characterize in a complete and explicit way the optimal completions of $mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with
Given a finite sequence of vectors $mathcal F_0$ in $C^d$ we describe the spectral and geometrical structure of optimal completions of $mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured w
In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $mathcal F$ we describe the spectral and geometrical structure of optimal completions of $mathcal F$ by a finite family of vectors with prescribed norms,
We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Ha
For a collection of $N$ unit vectors $mathbf{X}={x_i}_{i=1}^N$, define the $p$-frame energy of $mathbf{X}$ as the quantity $sum_{i eq j} |langle x_i,x_j rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimizatio