A Study On The Construction Of Multiwavelet、Interval Multiwavelet And Two-direction Vector-valued Wavelets

At present,wavelet analysis is the focus of the first line of scientific research.Multiwavelets have been paid more and more attention because of satisfying this demand.For the tightly supported orthogonal multiwavelets,the multiwavelet is more and more important.The multi-scale functions are used to construct the corresponding multiwavelets from compactly supported orthogonality.So far,there is no trivial construction method,which makes multiwavelet and interval multiwavelet operations quite complicated.Therefore,the construction of multiwavelets and interval multiwavelets is still a hot and difficult problem in the front of wavelet theory.We need to study the new construction condition and method.In this paper,the problem of multiwavelet,interval multiwavelet and bi-directional vector-valued wavelet is considered and studied by Yang Shouzhi ’s thought.This article specifically has the following innovation:1.The current multi-wavelet construction method: the expansion of the imitated unitary matrix,the calculation is quite complex,we found that the existing literature in the method of multi-wavelet has two difficulties: first,how to select the positive definite conditions to meet the coefficient matrix;Second,in the process of calculating matrix inverses H~2=(aI-PiPiT)-1PiPiT,the computational complexity increases with the increase of the order of the matrix.When we recall the two properties of the diagonal matrix,(1)diagonal matrix inversion only need to diagonal elements on the reciprocal,(2)two diagonal matrix can be exchanged when the order of the multiplication,in the second chapter,we give a new Construction conditions and methods,and successfully overcome the above two difficulties.2.In chapter 3,interval multi-wavelet functions and a class of multi-scale functions defined in [0,1] and the scaling function are constructed,and the corresponding construction examples is given.3.In Chapter 4,we focus on the research of two-directional vector-valued multiple wavelet theory.We construct a biorthogonal vector-valued bidirectional multiwavelet(wavelet packet).We also study the bi-directional vector values double positive Multiwavelet(wavelet packet).