The Study Of Gabor Frames For Subspaces Of L～2(R～d)

Let f ∈ L2{R) and a > 0, b > 0. The translation operator Ta and the modulation operator Eb are defined byrespectively .Let g(x) ∈ L2(R) be a fixed function and a > 0, b > 0. A function family of the form {EmbTnag(x)}m,n∈z in called Gabor system for L2(R), where Z is the set of all integers. If Gabor system {EmbTna(x)} m,n∈Z generates a frame for L2(R), then we call it Gabor frame. Since the emerge of wavelet analysis, the research of Gabor frame becomes one of the major active directions gradually. Its content is very rich [1], [2].Whereas, the research of Gabor system in L2(Rd) (d > 1) is less than that in L2(R). There are only a few results which were shown in literature [3].In this thesis , our main purpose is to describe the characterization of a Gabor system which can generate a frame or a frame sequence for L2(Rd) . In details, we give two sufficient conditions with which Gabor system can be a frame or a frame sequence for L2(Rd) , and we compare them . In addition, we conclude that under some special cases ,the sufficient conditions are also the necessary . At the same time , in order to explain the results, some interesting examples are presented. The analogous results in frequency domain will be given in the end of the thesis .