# A. Rahimi

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### GC-Fusion frames

M. H. Faroughi, A. Rahimi, R. Ahmadi

MFAT **16** (2010), no. 2, 112-119

112-119

In this paper we introduce the generalized continuous version of fusion frame, namely $gc$-fusion frame. Also we get some new results about Bessel mappings and perturbation in this case.

### $g$-frames and stability of $g$-frames in Hilbert spaces

Abbas Najati, M. H. Faroughi, Asghar Rahimi

MFAT **14** (2008), no. 3, 271-286

271-286

Wenchang Sun in his paper [Wenchang Sun, * $G$-frames and $g$-Riesz bases,* J. Math. Anal. Appl. **322** (2006), 437--452] has introduced $g$-frames which are generalized frames and include ordinary frames and many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. In this paper we develop the $g$-frame theory for separable Hilbert spaces and give characterizations of $g$-frames and we show that $g$-frames share many useful properties with frames. We present a version of the Paley-Wiener Theorem for $g$-frames which is in spirit close to results for frames, due to Ole Christensen.

### $pg$-frame in Banach spaces

M. R. Abdollahpour, M. H. Faroughi, A. Rahimi

MFAT **13** (2007), no. 3, 201-210

201-210

For extending the concepts of $p$-frame, frame for Banach spaces and atomic decomposition, we will define the concept of $pg$-frame and $g$-frame for Banach spaces, by which each $f\in X$ ($X$ is a Banach space) can be represented by an unconditionally convergent series $f=\sum g_{i}\Lambda_{i},$ where $\{\Lambda_{i}\}_{i\in J}$ is a $pg$-frame, $\{g_{i}\}\in(\sum\oplus Y_{i}^{*})_{l_q}$ and $\frac{1}{p}+\frac{1}{q}=1$. In fact, a $pg$-frame $\{\Lambda_{i}\}$ is a kind of an overcomplete basis for $X^{*}.$ We also show that every separable Banach space $X$ has a $g$-Banach frame with bounds equal to $1.$

### Continuous frame in Hilbert spaces

A. Rahimi, A. Najati, Y. N. Dehghan

MFAT **12** (2006), no. 2, 170-182

170-182

In this paper we introduce a mean of a continuous frame which is a generalization of discrete frames. Since a discrete frame is a special case of these frames, we expect that some of the results that occur in the frame theory will be generalized to these frames. For such a generalization, after giving some basic results and theorems about these frames, we discuss the following: dual to these frames, perturbation of continuous frames and robustness of these frames to an erasure of some elements.