G-frame representations with bounded operators

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form [Formula: see text], where [Formula: see text] belongs to Hilbert space [Formula: see text] and [Formula: see text] belongs to certain classes of bounded operators. Christensen et al. studied frames for [Formula: see text] with index set [Formula: see text] (or [Formula: see text]), that has representations in the form [Formula: see text] (or [Formula: see text]). As frames of subspaces, fusion frames and generalized translation invariant systems are the special cases of [Formula: see text]-frames, the purpose of this paper is to study and get sufficient conditions for [Formula: see text]-frames [Formula: see text] (or [Formula: see text] having the form [Formula: see text] [Formula: see text] (or [Formula: see text] [Formula: see text]). Also, we get the relation between representations of dual [Formula: see text]-frames with index set [Formula: see text]. Finally, we study stability of [Formula: see text]-frame representations under some perturbations.