Research On Bidirectional Multi-resolution Analysis Based On Locally Compact Abelian Group

The theoretical research of wavelet analysis is a powerful support for the practical application of wavelet analysis,due to the requirements of practical application and the development of mathematics itself,people have constructed different wavelets according to their needs.However,in the practical application process,we find that for the two scale wavelets,except for the Haar wavelet,other single wavelets can not have orthogonality,symmetry,compactness and other good properties at the same time.But the breakthrough of wavelet is to study the orthogonality,symmetry and compactness of wavelets.There is a shortage of single wavelet in this respect,people have introduced the concept of multiwavelets and have invested a lot of research in multiscale functions and mutiwavelets.Compared with scalar wavelets,multiwavelets have superiority.It can extend the multiscale space generated by a scaling function into a space generated by multiple scaling functions.Thus,a greater degree of freedom can be obtained.This not only maintains the good characteristics of time frequency domain,but also has the properties of orthogonality,compactness and symmetry.New results are emerging in the study of wavelets and multiwavelets,which make them more widely applied.Professor Yang Shouzhi first derives the concept of bi-directional refinement functions and bidirectional wavelet.On the basis of one dimension,we study the correlation of the positive and negative masks of the bi-directional refinement functions,and give the conditions for the stable solution of the two scale bi-directional strengthening equation to generate a MRA.The support interval of the two-way finer function determined by the two scale bi-directional strengthening equation is discussed and the definition of orthogonal bi-directional fine function and the corresponding orthogonal bi-directional wavelet are given.Multiresolution analysis based on locally compact Abel group is one of the basic concepts of wavelet theory.At present,some scholar have described the establishment of wavelet analysis on the locally compact Abel group.The first example based on the construction of the orthogonal wavelets on the Cantor dyadic group has been presented,and it has a good multifractal structure.This paper describes the development of wavelet analysis and the research work made by scholars on single wavelet,multiwavelet and bi-directional wavelets,and then introduces the relevant concepts and necessary notation instructions on the Abel group in detail,as well as Fourier transform and Walsh transform on Abel group,a multi-resolution analysis structure based on Abel group and its satisfying properties are proposed.In this paper,the definition of bi-directional fining equation based on the Abel group is given,and the necessary conditions for the bi-directional multiresolution analysis can be generated by the distribution solution of the bi-directional fining equation.Finally,the bi-directional multiresolution analysis is extended to the two dimensional Abel group,which further enriches the wavelet theory on the group.