A Study On Some Problem Ofthe Construction Of Wavelets And Multi-wavelets

The wavelet theory has applied widely in many fields, such as the signal analysisand processing, the image compression, detecting the mutation signal, the militaryelectronic countermeasure and so on. The study of the construction and the propertiesof wavelet is the key problem of wavelet analysis. The research of the wavelet theoryin the former is based on the numerical method of harmonic analysis and functionalanalysis in the ‘flat space’. In recent years, the international manifold-value datayields the study of the wavelet on the manifold by the differential geometry method.But the similar study is hardly discussed at home. V.Strela firstly introduced theconcept of two-scale similarity transform (TST). This provides a new way toconstruct multi-wavelets. It is known to all that TST is a benchmark theory of themultiwavelet. And the chosen matrix is a square matrix. For a general matrix withsize m n, it is hardly studied in the relative papers. The orthogonal interpolationmulti-wavelets with dilation factor a and multiplicity r has been studied in the case(a r) while the interpolation multi-wavelets has hardly discussed systematically inthe case (a r). Based on the above present status of the wavelet theory in the homeand abroad, this paper is organized as follows: In chapter2, the orthogonal armletmulti-wavelets with multiplicity r and dilation factor a will be constructed. Inchapter3, the concept of general conjugate two-scale similarity transform (GCTST)is introduced in this paper. GCTST is the most general generalization of TST. Wediscuss how the GCTST change the eigenvalue of the matrix by using the generalinverse matrix theory, which the former authors hardly use. In chapter4, the generalorthogonal interpolation multi-wavelets (a r), especially in the case (a<r), will bediscussed. The obtained interpolation multiscaling function will be orthogonal andbalanced，and the multiwavelets will be orthogonal and interpolated. In chapter5, theexample of wavelet on the manifold will be studied. The local wavelet transform onthe torus will be discussed for the first time, and the reconstruction formula will alsobe offered. Finally the graph of local wavelet on the torus will be shown. In chapter6,the four-direction two-dimension wavelet is presented and discussed systematicallyin cases of tensor product and non-tensor product. The algorithm of the constructionis given.