habib shakoory

Controlled frames in Hilbert spaces have been recently introduced by P. Balazs and etc. for improving the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper we develop a theory based on g-fusion frames on Hilbert spaces, which provides exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we can define analysis, synthesis and frame operators with representation space compatible for (C,C')-Controlled g-fusion frames, which even yield a reconstruction formula. Also, some useful concepts such as Q-dual and perturbation are introduced and investigated.

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.

In this manuscript, we study the relation between K-fusion frame and its local components which leads to the definition of a $C$-controlled $K$-fusion frames, also we extend a theory based on K-fusion frames on Hilbert spaces, which prepares exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we define the analysis, synthesis and frame operator for $C$-controlled $K$-fusion frames, which even yield a reconstruction formula. Also, we define dual of $C$-controlled $K$-fusion frames and study some basic properties and perturbation of them.