On The Rate Of Convergence Of Wavelet Series And Related Problems

In this dissertation, we will focus on the convergence rates of Shannon wavelet series of functions, which are members of local ABMV space and its special subspace—local HBMV space.The paper consists of three chapters, the main contents are as follows.In the first chapter, we will carry on a comprehensive summary about the research background and the current development. We’ll introduce the development of wavelets—from Fourier analysis to the theory of wavelet, as well as the basic theory and knowledge of wavelet.In the second chapter, we will concentrate on the convergence rate of Shannon wavelet. First, not only do we introduce some theories about the convergence of wavelet, but we also estimate the convergence degree of Shan-non wavelet series of functions. It’s obvious that functions, which belong to L2(R), is the mean convergence of 2nd order of their Shannon wavelet expan-sion. It is also the truth, that some continuous function will not converge pointwise.1994, Kelly, Kon, Raphael studied the almost-everywhere conver-gence of wavelet expansion. This paper will target the pointwise convergence of Shannon wavelet expansion of functions satisfying some bounded variation features. And, we will discuss the rate of convergence. Concerning the simi-larity between wavelet series and Fourier series, we no longer need all field but a small domain, where the function must satisfy some conditions.1996, and 1997, Sun showed the convergence and the rate of convergence of Shannon wavelet series in terms of BV space and local ABV space. Based on these researches, we will show the convergence rate of Shannon wavelet expansion of functions, which are the members of local ABMV space, larger than previous two spaces.In the third chapter, results of the paper will be strictly proved. One result is about the convergence of functions in local ABMV. Another is for local HBMV. And, that will adjourn the paper.