Two-direction Refinement Equations And Construction Of Wavelets With Good Properties

Wavelet analysis is a newly but quickly developed discipline based on Fourier analysis, which can achieve good localization both in time and frequency. So it is believed that wavelet analysis has double significance, that is, the pro-found theory and broad application. Based on the deep research of two-direction refinement equations, multi-band wavelets and nonseparable wavelets, we inves-tigate L1 solutions of two-direction poly-scale refinement equations, as well as two-direction multivariate refinement equations with finite nonnegative coeffi-cients; then we construct a class of 3-band orthogonal scaling functions with arbitrary high approximation; finally, we present two methods of constructing multivariate nonseparable wavelet whose dilation matrix is non-diagonal. This thesis is organized as follows:In Chapter 1, we outline the development histories of wavelet analysis and current research situations of wavelet theory at home and abroad. Main results of this thesis will be introduced.In Chapter 2, we introduce some related notations and the multiresolution analysis of L2(Rr).In Chapter 3, we study L1-solutions of two-direction poly-scale refinement equation with finite nonnegative coefficients. We prove that the vector space of all L1-solutions of the above equation is at most one-dimensional and consists of compactly supported functions of constant sign. With regard to the L1-solutions of the equation, some simple sufficient conditions for the existence of nontrivial L1-solutions and nonexistence of such solutions are given.In Chapter 4, we study L1-solutions of two-direction multivariate refine-ment equation with finite nonnegative coefficients. The vector space of all L1-solutions of the equation is at most one-dimensional and consists of com-pactly supported functions of constant sign. With regard to the L1-solutions of the equation, a simple sufficient condition for the nonexistence of nontrivial L1-solutions is given and a characterization for the existence of such solutions is presented. At last some simple sufficient conditions (easy for verification) for the existence of nontrivial L1-solutions as well as for the nonexistence of such solutions are shown.In Chapter 5, a class of 3-band orthogonal scaling functions with arbitrary high approximation are constructed. Three examples are given to illustrate the results of this work.In Chapter 6, we present two methods of constructing multivariate nonsepa-rable wavelet whose dilation matrix is non-diagonal. Then we apply the results to construct a class of nonseparable orthonormal wavelet basis for L2(Rr+1). Properties of the new constructed filter banks or new wavelet basis are dis-cussed.