IAM   02674
INSTITUTO ARGENTINO DE MATEMATICA ALBERTO CALDERON
Unidad Ejecutora - UE
congresos y reuniones científicas
Título:
Optimal frame completions with prescribed norms
Autor/es:
MASSEY, PEDRO; RUIZ, MARIANO; STOJANOFF, DEMETRIO
Lugar:
Córdoba
Reunión:
Congreso; IV Congreso Latinoamericano de Matemáticos (CLAM); 2012
Institución organizadora:
Unión Matemática de América Latina y el Caribe (UMALCA)
Resumen:
Let \$mathcal H\$ be a \$d\$-dimensional complex Hilbert space, let \$mathcal F_0={f_i}_{i=1}^{n_0}\$ be a finite sequence of vectors in \$mathcal H\$ and let \$mathbf a=(a_i)_{i=n_0+1}^n\$ be a sequence of positive numbers for some \$ngeq max{d,n_0+1}\$. We say that a finite sequence \$mathcal F={f_i}_{i=1}^n\$ of vectors in \$mathcal H\$ is an \$mathbf a\$-completion of \$mathcal F_0\$ if \$|f_i|^2 =a_i\$, \$n_0+1leq ileq n\$. In this context, it is desired to construct \$mathbf a\$-completions \$mathcal F={f_i}_{i=1}^n\$ that minimize a convex functional of the form \$P_g(mathcal F)= ext{tr}(g(S_{mathcal F}))\$, where \$S_{mathcal F}\$ denotes the operator \$sum_{i=1}^n f_iotimes f_i\$ and \$g:[0,infty) ightarrow [0,infty)\$ is a certain strictly convex function (e.g. if \$g(x)=x^2\$ then \$P_g(mathcal F)\$ is known as the Benedetto-Fickus´ frame potential of \$mathcal F\$). It turns out that this problem can be studied in terms of the structure of the set \$\$U_t(S_{mathcal F_0},k)={S_{mathcal F_0}+B: Bin mathcal B(mathcal H)^+, , ext{rank}(B)leq k, , ext{tr}(B)=t} , \$\$ where \$mathcal B(mathcal H)^+\$ denotes the cone of positive operators, \$k=n-n_0\$ and \$t=sum_{i=n_0+1}^na_i\$. Indeed, as a consequence of the structure of \$U_t(S_{mathcal F_0},k)\$ we will show that, under certain hypothesis, there exist frames \$mathcal F\$ that are \$mathbf a\$-completions of \$mathcal F_0\$ and that are optimal with respect to the convex functional \$P_g\$ for every strictly convex function \$g\$.