In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calderón-Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting.2000 Mathematics Subject Classification. Primary 42C40; Secondary 41A58, 47A58.

Extending work by Hernandez, Labate and Weiss, we present a sufficent condition for a generalized shift-invariant system to be a Bessel sequence or even a frame for L 2 (R d ). In particular, this leads to a sufficient condition for a wave packet system to form a frame. On the other hand, we show that certain natural conditions on the parameters of such a system exclude the frame property.

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.Mathematics Subject Classification (2010). Primary 42C15; Secondary 41A58, 47A58.

Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence ( called the symbol) and synthesis. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled gframes will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.

In this paper, we intend to introduce the concept of c-K-gframes, which are the generalization of K-g-frames. In addition, we prove some new results on c-K-g-frames in Hilbert spaces. Moreover, we define the related operators of c-K-g frames. Then, we give necessary and sufficient conditions on c-K-g-frames to characterize them. Finally, we verify perturbation of c-K-g-frames.2010 Mathematics Subject Classification. Primary 42C15, 42C40.

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Upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the Hilbert space into the positive real numbers. In addition, we study the relationship between redundancy of frames (fusion frames) and dual frames (dual fusion frames). Moreover, we indicate some results about excess of fusion frames. We state the relationship between redundancy of local frames and fusion frames in a particular case. Furthermore, some examples are also given.2000 Mathematics Subject Classification. Primary 42C40; Secondary 41A58, 47A58.