Some Properties Of G-frames In Hilbert Spaces

Frames for Hilbert spaces were formally put forward by R.J.Duffin andA.C.Schaeffer in the fifties of the Twentieth Century in the study of nonharmonicFourier series. Frame is generalization of orthonormal basis. An arbitrary element inthe Hilbert space can be represented by frame, and the frame coefficient may not beunique. Thus, we can select the appropriate coefficient according to actual needs, andobtain some conditions which the orthonormal basis can’t satisfy. With thedevelopment of the frame theory, there appeared many frames, such as boundedquasi-projectors, pseudo-frames and outer frames. Professor Sun gave a unifiedtreatment of these frames, and got more natural generalizations of frames-g-frames.Some properties of the frame theory had been extended to the g-frame theory.The frame theory has been widly applied in fields of signal processing, imageprocessing, signal transmission and so on. It is important to work out the inverse offrame operator in the process of signal reconstruction. But in actual work, it isdifficult,even impossible to work out the inverse of frame operator. Some researchersuse the approach from the finite to the infinite to discuss the problem of theapproximation of the inverse of frame operator.On the basis of the approximation of frame, we discuss the problem of theapproximation of the inverse of g-frame operator by two methods: weak convergenceapproximation and projection method approximation. We apply weak convergenceapproximation theorem and inference to illustrate that near g-Riesz basis is notg-Riesz frame. Then,because the g-frame theory has been further extended tog-continuous frames for Hilbert spaces. We discuss some properties of the alternatedual g-continuous frames.