Tight Frame Completions with Prescribed Norms

Let H be a finite dimensional (real or complex) Hilbert space and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {{a_i}} \right\}_{i = 1}^\infty $$\end{document} be a non-increasing sequence of positive numbers. Given a finite sequence of vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F = \left\{ {{f_i}} \right\}_{i = 1}^p$$\end{document} in H we find necessary and sufficient conditions for the existence of r ∈ ℕ ∪ {∞} and a Bessel sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = \left\{ {{g_i}} \right\}_{i = 1}^r$$\end{document} in H such that F ∪ G is a tight frame for H and ‖gi‖2 = ai for every i. Moreover, in this case we compute the minimum r ∈ ℕ ∪ {∞} with this property. We also describe algorithms that perform completions of a given set of vectors to tight frames.