A. Najati

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.

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.