Control Fusion Frames in Hilbert Spaces and Their Dual

Controlled frames in Hilbert spaces have been introducedby Balazs, Antoine and Grybos to improve the numerical output of inrelation to algorithms for inverting the frame operator. In this paper,we introduce some new concepts and show results on controlled fusionframes for Hilbert spaces. It is shown that controlled fusion framesare a generalization of fusion frames giving a generalized way to obtainnumerical advantage in the sense of preconditioning to check the fusionframe condition. For this end, we introduce the notion of Q-duality forControlled fusion frames. Also, we survey the robustness of controlledfusion frames under some perturbations.