Study Of Some Questions For Finite Frames

The concept of frame was firstly introduced by Duffin and Schaeffer when they studied nonharmonic Fourier series in1952. Finite frames are overcomplete systems in finite dimensional Hibert space. As similar as a basis, a frame can represent every element in a Hilbert space H. In contrast to a basis, however, the way of representation is not unique because of a frame is a linearly dependent. It also possesses many extensive applications in signal processing, image processing, data compression and so on.This thesis is mainly to discuss some problems for finite frame, and consists of six chapters.Chapter one states background knowledge of frames and operator-valued frames, and sketch the structure and works of the thesis.Chapter two reviews the basic concepts and some results of finite frame.Chapter three lists a characterization of finite and infinite dual frame, respectively, and generalizes the characterization of all finite dual frames.Chapter four, firstly, introuduces the basic concept and character of, scalable frame, then, basing on the result of chapter three, we give bound of scalability factor of scalable frame, and establish relationship between scalability factor and canonical dual frame.Chapter five is one of the main contents. It investigates the sparsity of dual frames by using the concept sparkf(Φ) of matrix Φ. First, a new form of sparsity of optimal sparsity dual frames for spectral tetris frame is established. Secondly, a sufficient and necessary condition of finite frame, which dual frame’s sparsity is minimal, is presented.The final part is Chapter six. First, it gives a characterization of all dual operator-valued frame, and gives related propositions. Secondly, it is proved that the dual frame of operator-valued frame for robust to k erasures is robust to k erasures. Finally, the tight frame{VjTj}jm=1of operator-valued frame{Vj}jm=1, which the frame bound A’ of {VjTj}jm=1is between lower and upper frame bound of the frame{Vj}jm=1, is constructed.