Duality of $g$-Bessel sequences and some results about RIP $g$-‎frames

Abstract

‎In this paper‎, ‎first we develop the duality concept for $g$-Bessel sequences‎ ‎and Bessel fusion sequences in Hilbert spaces‎. ‎We obtain some results about dual‎, ‎pseudo-dual ‎and approximate dual of frames and fusion frames‎. ‎We also expand every $g$-Bessel ‎sequence to a frame by summing some elements‎. ‎We define the restricted isometry property for ‎$g$-frames and generalize some results from (B‎. ‎G‎. ‎Bodmann et al‎, ‎Fusion frames and the restricted isometry property‎, ‎Num‎. ‎Func‎. ‎Anal‎. ‎Optim‎. ‎33 (2012) 770-790) to $g$-frame situation‎. ‎Finally we study the stability of $g$-frames under erasure of ‎operators.‎