Frame completions with prescribed norms: local minimizers and applications

PEDRO GUSTAVO MASSEY; NOELIA BELÉN RIOS; DEMETRIO STOJANOFF

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

Lugar: Berlin; Año: 2018 vol. 44 p. 51 - 86

1019-7168

Let $cF_0={f_i}_{iinmathbb{I}_{n_0}}$ be a finite sequence of vectors in $C^d$ and let $ca=(a_i)_{iinmathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $cF_0$ of the form $cF=(cF_0,cG)$ obtained by appending a sequence $cG={g_i}_{iinmathbb{I}_k}$ of vectors in $C^d$ such that $|g_i|^2=a_i$ for $iinmathbb{I}_k$, and endow the set of completions with the metric $d(cF,ilde cF) =max{ ,|g_i-ilde g_i|: iinmathbb{I}_k}$ where $ilde cF=(cF_0,,ilde cG)$.In this context we show that local minimizers on the set of completions of a convex potential $ext{P}_arphi$, induced by a strictly convex function $arphi$, are also global minimizers. In case that $arphi(x)=x^2$ then $ext{P}_arphi$ 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 completionproblems 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.