Convergence of frame series (2203.07047v4)

Abstract: If ${x_n}{n \in \mathbb{N}}$ is a frame for a Hilbert space $H,$ then there exists a canonical dual frame ${\tilde{x_n}}{n \in \mathbb{N}}$ such that for every $x \in H$ we have $x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$ 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 dual) 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.