A Research On Perturbations Of Bessel Sequences And Normalized Windowed Fourier Transformation

The theory of frame is an important part in wavelet analysis, and Bessel Se-quence is the basis of frame. In 1952, Duffin and Schaeffer extended the thought of Gabor and put forward the concept of frame in Hilbert space firstly when studying the non-harmorie Fourier analysis. Since then, the study of the frame was payed a wide attention to by many scholars. In recent years, people have made a system-atic study on Bessel sequences, frames and Riesz basis in Hilbert spaces. To make up for lack of Fourier transformation, Gabor proposed and used windowed Fourier transformation in 1946. Windowed Fourier transformation is an important tool to simultaneous time and frequency information. In this paper, generalized perturba-tions and operator perturbations of Bessel sequences in Hilbert space are discussed And both the strong reconstruction formulas and some important properties of normalized windowed Fourier transformation are also obtained. Text is divided into three chapters as follows:In Chapter 1, both brief background and research status on wavelet analysis are introduced, and a brief description of main work and organization of this paper are also listed.In Chapter 2, the basic theory of frames in a Hilbert space is introduced firstly. And then the generalized perturbations are discussed. By adding some new ele-ments, or multiplying some coefficients to a given Bessel sequence so that a new sequence is obtained, the stabilities of Bessel sequences are discussed. Some condi-tions for the obtained sequence to be a Bessel sequence are given. Related examples are listed to illustrate the existence of such perturbations. By analogy, the opera-tor perturbations of Bessel sequences are discussed. The coefficients in generalized perturbations are changed for a series of operators acting on each element.In Chapter 3, by using the theory of functional analysis, a sequence {Fn}n∞=1 of functions defined by the normalized windowed Fourier transformation Twwin f is introduced and proved to be convergent to f in norm. Thus, a strong reconstruction formula by Twwin f is established.