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.

FUSION FRAMEs are an extension to FRAMEs that provide a FRAMEwork for applications and providing efficient and robust information processing algorithms. In this article we study the erasure of subspaces of a FUSION FRAME.

The study of the c$k$-FUSIONs FRAMEs shows that the emphasis on the measure spaces introduces a new idea, although some similar properties with the discrete case are raised. Moreover, due to the nature of measure spaces, we have to use new techniques for new results. Especially, the topic of the dual of FRAMEs  which is important for FRAME applications, have been specified  completely for the continuous FRAMEs. After improving and extending the concept of FUSION FRAMEs and continuous FRAMEs, in this paper we introduce continuous $k$-FUSION FRAMEs in Hilbert spaces. Similarly to the c-FUSION FRAMEs, dual of continuous $k$-FUSION FRAMEs may not be defined, we however define the $Q$-dual of continuous $k$-FUSION FRAMEs. Also some new results and the perturbation of continuous $k$-FUSION FRAMEs will be presented.

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.

In this paper we investigate a new notion of bases in Hilbert spaces and similar to FUSION FRAME theory we introduce FUSION bases theory in Hilbert spaces. We also introduce a new definition of FUSION dual sequence associated with a FUSION basis and show that the operators of a FUSION dual sequence are continuous projections. Next we define the FUSION biorthogonal sequence, Bessel FUSION basis, Hilbert FUSION basis and obtain some character-izations of them. we study orthonormal FUSION systems and Riesz FUSION bases for Hilbert spaces. we consider the stability of FUSION bases under small perturbations. We also generalized a result of Paley-Wiener [16] to the situation of FUSION basis.

In this paper, we introduce the notion of woven g-FUSION FRAMEs in Hilbert spaces. Then, we present sufficient conditions for woven g-FUSION FRAMEs in terms of woven FRAMEs in Hilbert spaces. We extend some of the recent results of standard woven FRAMEs and woven FUSION FRAMEs to woven g-FUSION FRAMEs. Also, we study perturbations of woven g-FUSION FRAMEs.

FUSION FRAMEs are a generalized form of FRAMEs in Hilbert spaces. In the present paper we introduce Bessel sub FUSION sequences and sub FUSION FRAMEs and we investigate the relationship between their operations. Also, the definition of the orthogonal complement of sub FUSION FRAMEs and the definition of the completion of Bessel FUSION sequences are provided and several results related with these notions are shown.