A-inner Product And Its Applications In Gabor Frame

Let f∈L2(R) and a> 0, b> 0. The translation operator Ta and the modulation operator Eb are defined by respectively.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 is called Gabor system for L2(R), where Z is the set of all integers. If Gabor system{EmbTnag(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, and many meaningful results are obtained, but these results are limited to one-dimensional case.In this thesis, our main purpose is to describe bracket product and its applications in Gabor frame. Concretely, we provide a detailed development of the L1 function valued inner product on L2(Rd) known as bracket product. In addition to some of the more basic properties, we show that this inner product has a Bessel inequality and a Riesz representation theorem. We then apply this to Gabor frame to show that there exists " compressed " versions of the frame operator, the frame transform and the preframe oper-ator. Finally, we introduce the notion of an A-frame, and show that there is an equivalence between the frames of translates for this function valued inner product and Gabor frame.