Optimal frame completions with prescribed norms for majorization

PEDRO MASSEY; MARIANO RUIZ; DEMETRIO STOJANOFF

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS

BIRKHAUSER BOSTON INC

Lugar: New York; Año: 2014 vol. 20 p. 1111 - 1140

1069-5869

Given a nite sequence of vectors F_0 in a d-dimensional complex Hilbert spaceH we characterize in a complete and explicit way the optimal completions ofF_0 obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequences). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frameoperator of the initial sequence F_0 and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence F_0 we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completionswith prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.