Tight frame completions with prescribed norms

2006/06/13 by Massey, P., Ruiz, M.
#42C15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/0606319

Abstract

Let \hil 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 in \hil we find necessary and sufficient conditions for the existence of r∈ \NN∪\∞\ and a Bessel sequence \g in \hil such that \cF∪\cG is a tight frame for \hil and ‖g_i‖²=a_i for 1≤ i≤ r. Moreover, in this case we compute the minimum r∈ \NN∪\∞\ 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.

Related

• The Unconditional Constants for Hilbert Space Frame Expansions
• Bessel sequences in Hilbert C^∗-modules
• The Schur-Horn theorem for operators and frames with prescribed norms and frame operator
• Frame completions with prescribed norms: local minimizers and\n applications
• Optimal completions of a frame