Tight Frame Completions with Prescribed Norms

Let H be a finite dimensional (real or complex) Hilbert space and let {ai}i=1 be a non-increasing sequence of positive numbers. Given a finite sequence of vectors F = {fi}pi=1 in H we find necessary and sufficient conditions for the existence of r ∈ N∪{∞} and a Bessel sequence G = {gi}i=1 in H such that F ∪ G is a tight frame for H and ‖gi‖ = ai for 1 ≤ i ≤ r. Moreover, in this case we compute the minimum r ∈ N ∪ {∞} with this property. Using recent results on the Schur-Horn theorem, we also obtain a not so optimal but algorithmic computable (in a finite numbers of steps) tight completion sequence G.