Parseval quasi dual frames

CORACH, GUSTAVO; PEDRO MASSEY; MARIANO RUIZ

CABA

Workshop; Workshop on Infinite Dimensional Analysis Buenos Aires; 2014

Universidad de Buenos Aires

Let $F={f_i}_{i in N}$ be a frame for a complex Hilbert space $mathcal H$, with synthesis operator $T_mathcal F:ell^2(mathbb N)-> H$. Let $ P(H)$ denote the class of Parseval frames $G={g_i}_{i in N}$ for $ H$ i.e. such that $T_G T_G^*=I$. $P(H)$ is a particularly well behaved class of frames for $H$.In this talk, we discuss some results concerning the computation of $$ alpha(F)=inf_{G in P(H)} left{ sup_{x in H,, |x|=1} | sum_{nin mathbb N} langle x,,g_n rangle f_n - x|right}$$ i.e. the infimum of the operator norm of $T_ F,T_ G^*-I$, where $ G$ is a Parseval frame for $H$. $alpha( F)$ is the optimal lower bound for the(normalized) worst-case error in the reconstruction of vectors when analyzed with the Parseval frame and synthesized with $ F$. In case that the infimum is attained on a frame $G in P(H)$ we call it a Parseval quasi-dual of $F$.