Frame completions with prescribed norms: local minimizers and applications

Abstract

Let F0 = {fi}i∈In0 be a finite sequence of vectors in Cd and let a = (ai)i∈Ik be a finite sequence of positive numbers, where In = {1,...,n} for n ∈ N. We consider the completions of F0 of the form F = (F0, G) obtained by appending a sequence G = {gi}i∈Ik of vectors in Cd such that gi2 = ai for i ∈ Ik, and endow the set of completions with the metric d(F, F˜) = max{ gi − ˜gi : i ∈ Ik} where F˜ = (F0, G˜). In this context we show that local minimizers on the set of completions of a convex potential Pϕ, induced by a strictly convex function ϕ, are also global minimizers. In case that ϕ(x) = x2 then Pϕ 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.