Tight frame completions with prescribed norms

Abstract

Let H be a finite dimensional (real or complex) Hilbert space and let {a_i} ∞_i=1. be a non-increasing sequence of positive numbers. Given a finite sequence of vectors F = {f_i}p_i=1 in H we find necessary and sufficient conditions for the existence of r ∈ N U {∞} and a Bessel sequence G = {g_i}r_i=1 in H such that F U G is a tight frame for H and || g_i ||^2 = a_i for every i. Moreover, in this case we compute the minimum r in NU {∞} with this property. We also describe algorithms that perform completions of a given set of vectors to tight frames.