Some New Results Of Frame Perturbation In Hilbert Space

The concept of frame was originally introduced by R. J. Duffin and A. C. Schaeffer when they were working on nonharmonic Fourier series in the fifties of the twentieth century. A frame is a generalized basis for a Hilbert space H, and every element in H can be representated by the frame, but the method of representation is not unique. Frame theory has promptly developed after wavelet analysis's appearance. It has been used in signal processing, image processing, data compressing, sampling theorems, etc. The research contents of frame theory mainly include: frame characters, dual frame characters, perturbation of frames, etc. This thesis is mainly to discuss frame perturbation, and consists of five chapters.Chapter one introduces the origin of frame concept, and sketches the main works and the structure of the thesis.Chapter two lists some basic facts to be used throughout the thesis, and gives some concepts on frame, Riesz basis, near-Riesz basis and Riesz frame. Finally this chapter provides the theorems of judging them.Chapter three is one of main contents. First, we quote two classical results on frame perturbation, and then, we obtain the generalized result of frame perturbation on basis of these classical results. Here we express the main perturbation theorem using operator methods, and it is also proved that some known results are special cases of our result. This chapter also shows that the perturbation theorem is not necessary true if we displace the frame with the frame sequence. But the result is valid if frame is replaced by Riesz basis.Chapter four is another main content, and is devoted to the perturbations of near-Riesz bases and Riesz frames. At last, it is shown that our Riesz frame perturbation theorem covers the known perturbation result.The concepts of Banach frame and atomic decomposition on Banach spaces are introduced in the last chapter. Consequently, perturbation theorems and corollaries of them are also obtained.