Convergence of Frame Series

Abstract

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-duals. In particular, we prove that if $$\{x_n\}_{n \in \mathbb {N}}$$ does not contain infinitely many zeros then the frame series converge unconditionally for every alternative dual (or synthesis pseudo-duals) if and only if $$\{x_n\}_{n \in \mathbb {N}}$$ is a near-Riesz basis. We also prove that all alternative duals and synthesis pseudo-duals have the same excess as their associated frame.