Optimization problems for finite frames

PEDRO MASSEY; RIOS, NOELIA B.; DEMETRIO STOJANOFF

CABA

Congreso; Cuarta escuela sobre Análisis funcional y Geometría; 2018

Instituto Argentino de Matemática "Alberto Calderón"-CONICET

Let a_1 \geq ... a_k > 0 and consider T_d(a) = {(fj)_{j_=1}^k : || f_j||^2 =a_j , 1 ≤ j ≤ k}; notice that T_d(a) is a metric space with its product structure. If S ∈ Md(C)^+ is a positive semidefinite matrix, N is a strictly convex unitarily invariant norm in Md(C) (e.g. p -norms for p ∈ (1, ∞)) consider thefunction P_N defined on T_d(a) such that, for F = (fj)_{j=1}^k ∈ T_d(a),P_N(F) = N(S − S_F) where S_F = \sum_{j=1}^k fj f_j^* ∈ Md(C)+ is the so-called frame operator. The main problem in this context is the study of (local) minimizers of P_N in T_d(a). It turns out that there exist families F^op ∈ T_d(a) that minimize P_N for every strictly convex unitarily invariant norm N. Moreover, local minimizers of P_N are actually global minimizers. If S is an invertible operator and k ≥ d then every (local) minimizer F = {fj}_{j=1}^k of P_N is a set of generators for C^d i.e. a finite frame. In this talk, which is based on joint work with Noelia Rios and Demetrio Stojanoff, we approach these problems by extending some tools from matrix analysis.