Generalization Of Continuous Frames In Hilbert Spaces

The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer in 1952 to study some deep problems in nonharmonic Fourier series. In 1986, frames began to be widely used by Daubechies, Grossmann and Meyer. Discrete frames in a separable Hilbert space have properties similar to an orthonormal basis for expanding arbitrary elements, but the elements of frame can be linearly dependent, so arbitrary element in Hilbert spaces does not have a unique representation, and it is this property that frame theory in signal processing, image processing and data compression is widely used.In 1993, the concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by G. Kaiser and independently by Ali, Antoine and Gazeau. These frames are known as continuous frames that are broader and general situation of discrete frames, so they are called as generalized frames. Firstly, we do some further studies about Riesz-type frames, focus on studying equivalent characterizations, redundancy and perturbation of continuous K-frames in Hilbert spaces, and study some properties of continuous tight K-frames.Firstly, this paper mainly illustrates the research backgrounds and current situations of frames, and introduces the direction of recent study of frame, the relevant knowledge of measurable spaces and frames. In the meantime we introduce main contents and structure of the paper.Secondly, we introduce the knowledge of Riesz-type frames and continuous orthonormal basis, and study the relationship between Riesz-type frames and continuous orthonormal basis.Thirdly, we propose two kinds of equivalent characterizations for continuous K-frames. We also give two sufficient conditions for the remainder of a continuous K-frame after deleting some elements to be a continuous K-frame and a sufficient condition for the remainder to be not a continuous K-frame. We characterize the continuous K-frames by the synthesis operator and a bounder operator associated with two continuous Bessel mappings. Finally, we discuss the perturbation of continuous K-frames in Hilbert spaces.Finally, we propose the concept of continuous tight K-frames, then discuss the equivalent characterizations some properties for continuous tight K-frames. We give the essential condition for continuous tight K-frames at last.