Wavelet Numerical Algorithm Of Convolution-type Singular Integular Operators And Its Application

In this paper, we study the approximation problem and continuity of convolution-type singular integral operators on Besov spaces by using the wavelet numerical algorithm of convolution-type integral operators. This thesis consists of four parts as follows:In Chapter 1, firstly, we introduce the development of the integral operators. Secondly, some numerical algorithms and continuity about singular integral operators are presented. Because the convolution-type Calderón-Zygmund operators such as Hilbert operator and Riesz operator are widely used, we emphasize some wavelet numerical algorithms about convolution-type operators and its applications. Finally, we state the main results of this paper concretely.In Chapter 2, we study the knowledge about wavelet analysis. Firstly, we introduce the development of wavelet analysis.Then we describe the one-dimensional wavelet multi-resolution analysis theory and n-dimensional tensor product space. Finally, based on the one-dimensional wavelet, we discuss and multi-resolution analysis about the n-dimensional tensor product wavelet and get n-dimensional tensor wavelet functions.The third chapter and Chapter 4 are applications of the wavelet algorithm about convolution-type singular integral operator in Chapter 1.1.Using the method based on n-dimensional Daubechies wavelets basis, firstly, we focus on the approximation problem of convolution-type singular integral operators. We obtain the approximation speed 2 ? Rγon Besov spaces , (?)by approximating an convolution-type operator by the band operator which has less coefficients and the wavelet characterization of functions on Besov spaces.Then, we study the continuity of convolution-type operators.By estimating Daubechies wavelet coefficient of distribution kernel, we get the continuity of convolution-type Calderón-Zygmund operators on Besov spaces ,(?)under H(o|¨)rmander condition.