Convergence of Frame Series.

If { x n } n ∈ N is a frame for a Hilbert space H, then there exists a canonical dual frame { x ~ n } n ∈ N such that for every x ∈ H we have x = ∑ ⟨ x , x ~ n ⟩ 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 ∈ N and synthesis pseudo-duals { z n } n ∈ N such that x = ∑ ⟨ x , y n ⟩ x n and x = ∑ ⟨ x , x n ⟩ z n for every x. We characterize the frames for which the frame series x = ∑ ⟨ x , y n ⟩ 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 ∈ 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 ∈ 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. [ABSTRACT FROM AUTHOR]