Frame completions with prescribed norms: local minimizers and applications

Let be a finite sequence of vectors in and let be a finite sequence of positive numbers, where for . We consider the completions of of the form obtained by appending a sequence of vectors in such that a(2) = a (i) for , and endow the set of completions with the metric where . In this context we show that local minimizers on the set of completions of a convex potential P (phi) , induced by a strictly convex function phi, are also global minimizers. In case that phi(x) = x (2) then P (phi) 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.