Some optimal design problems for nite frames

PEDRO MASSEY; MARIANO RUIZ; DEMETRIO STOJANOFF

Nashville

Congreso; 5th International Conference on Computational Harmonic Analysis; 2014

Vanderbilt University

In this talk we will consider the following two optimal design problems innite frame theory:- Given a redundant frame F={f_i}_{i=1}^n in C^d, we compute the spectral and geometrical structure of minimizers of the frame potential among the dual frames G={g_i}_{i=1}^n for F such that \sum_{i=1}^n || g_i||^2 \geq t and || T_{F#} - T_G||\leq \epsilon, where T_G denotes the synthesis operator of G and F# denotes the canonicaldual of F.- Given a frame F={f_i}_{i=1}^n in C^d, we compute the spectral and geometrical structure of the minimizers of the frame potential among all coherent perturbations VF={V f_i}_{i=1}^n where V is any invertible operator V such that ||V^*V - I||\leq \delta and det(V^*V)\geq s.The motivation of these problems is the search of numerical stable encoding-decoding schemes based on (perturbations) of F. Our approach relies in Lidskii´s type inequalities (both additive and multiplicative) from matrix analysis theory.It turns out that the matrix models behind these two (seemingly unrelated) problems are intimately connected.