In this paper we introduce and study Besselian $g$-frames. We show that the kernel of associated synthesis operator for a Besselian $g$-frame is finite dimensional. We also introduce $alpha$-dual of a $g$-frame and we get some results when we use the Hilbert-Schmidt norm for the members of a $g$-frame in a finite dimensional Hilbert space.

In this paper, we introduced and characterized the controlled $g$-duals of a frame in separable Hilbert space $\mathcal{H}$ . Afterwards, obtained new $C$-controlled $g$-dual frames from given frames. addition, approximation for was defined some their properties were investigated. Finally, relationship between approximately dual $g$-dual.

The berization of a ne systems via dual Gramian techniques, that was developed in previous papers of the authors, is applied here for the study of a ne frames which have an a ne dual system. Gramian techniques are also used to verify whether a dual pair of a ne frames is a also a pair of bi-orthogonal Riesz bases. A general method for a painless derivation of a dual pair of a ne frames from an ...

Approximately dual frames are studied in the Hilbert space setting. Approximate duals are easier to construct than classical dual frames, and can be tailored to yield almost perfect reconstruction. Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory. An alternative bound is derived for the rich class of Gabor frames,...

Recently, fusion frames and for operators were considered as generalizations of in Hilbert spaces. In this paper, we generalize some the known results frame theory to related a linear bounded operator K which call K-fusion frames. We obtain new by considering with class construct from given ones. also study stability under small perturbations. further give characterizations atomic systems subsp...