Convergence of frame series

TL;DR

This paper investigates the conditions under which frame series in Hilbert spaces converge unconditionally for various duals, characterizing near-Riesz bases as key to universal convergence.

Contribution

It characterizes frames with unconditional convergence of series for all duals, especially near-Riesz bases, and relates the properties of alternative duals to the original frame.

Findings

Unconditional convergence occurs iff the frame is a near-Riesz basis when no infinitely many zeros are present.

All alternative duals and pseudo-duals share the same excess as the original frame.

Frames without infinitely many zeros have universal unconditional convergence for all duals if and only if they are near-Riesz bases.

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.