Tight frame completions with prescribed norms

Abstract

Let $\hil$ be a finite dimensional (real or complex) Hilbert space and let $\{a_i{\}}_{i=1}^{\infty }$ be a non-increasing sequence of positive numbers. Given a finite sequence of vectors $\f$ in $\hil$ we find necessary and sufficient conditions for the existence of $r\in \NN\cup\{\infty\}$ and a Bessel sequence $\g$ in $\hil$ such that $\cF\cup\cG$ is a tight frame for $\hil$ and $\Vert g_i{\Vert }^2=a_i$ for $1\le i\le r$. Moreover, in this case we compute the minimum $r\in \NN\cup\{\infty\}$ 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 $\cG$.