Codes and Expansions (CodEx) Seminar

Pu-Ting Yu (Georgia Tech):
Convergence of Frame Series

If \(\{x_n\}_{n \in \mathbb{N}}\) is a frame for a Hilbert space \(H\), then there exists a canonical dual frame \(\{\widetilde{x}_n\}_{n \in \mathbb{N}}\) such that for every \(x \in H\) we have \(x = \sum \,\langle x,\widetilde{x}_n \rangle \, x_n\), with unconditional convergence of this series. However, if the frame is not a Riesz basis, then there exist alternative duals \(\{y_n\}_{n \in \mathbb{N}}\) and synthesis pseudo-duals \(\{z_n\}_{n \in \mathbb{N}}\) such that \(x = \sum \, \langle x,y_n \rangle \, x_n\) and \(x = \sum \, \langle x,x_n \rangle \, z_n\) for every \(x\). We characterize the frames for which the frame series
\(x = \sum \, \langle x,y_n \rangle\, x_n\)
converges unconditionally for every \(x\) for every alternative dual, and similarly for synthesis pseudo-dual. In particular, we prove that if \(\{x_n\}_{n \in \mathbb{N}}\) does not contain infinitely many zeros then the frame series converges unconditionally for every alternative dual (or synthesis pseudo-dual) if and only if \(\{x_n\}_{n \in \mathbb{N}}\) is a near-Riesz basis. Several other characterizations of near-Riesz basis will also be presented. Finally, using unconditional convergence as a criterion, we study the “classificaition” of alternative duals and show that there does not exist an absolute convergent frame for \(H\).