Tight Frame Completions with Prescribed Norms

PEDRO MASSEY; MARIANO RUIZ

Sampling Theory in Signal and Image Processing

SAMPLING PUBLISHING

Año: 2008 vol. 7 p. 1 - 13

1530-6429

Let H be a finite dimensional (real or complex) Hilbert space and let {a_i}_i=1,2,3... be a non-increasing sequence of positive numbers. Given a finite sequence of vectors F = {f_i}i=1,...,p  in H we find necessary and sufficient conditions for the existence of r in N U {\infty} and a Bessel sequence G = {g_i}i=1,...,r  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 {\infty} with this property. We also describe algorithms that perform completions of a given set of vectors to tight frames. Some numerical examples are given.