The Theory Of Vector-valued Wavelets

Great attention has been given to wavelet analysis, both in theories and applications, since it appeared and comprehensive studies have been made during these years. Vector-valued orthonomal multiwavelets is the answer to the unavoidable disadvantages of mother wavelet, because of its praise-worthy combination of frequency domain and time domain and its properties of orthogonal, short support, and symmetry. Besides,vector-valued orthonomal multiwavelets has a better performance in real application. Although multiwavelet also qualified for frequency domain and time domain, there exist some disadvantages in real use. For exaple, prefilter or balance multiwavelets are required to process signals, which will cause unnecessary troubles. But matrix-valued wavelets and vector-valued multiwavelets can be used directly. In view of its promising application, in this thesis, we do deep and systematic researches on the construction and properties of bivariate vector-valued bi-orthogonal multiwavelets, vec.or-valued wavelets packets and matrix-valued wavelets by applying operator theory, matrix theory and vector subdivision scheme, etc. with several rewarding results.Firstly, the thesis proved the existence of bivariate vector-valued bi-orthogonal wavelets by infinite matrix., showed investigations on the properties and construction approaches of it and an algorithm was provided concerning it. Secondly, based on the notion of two dimension vector-valued, multiresolution analysis and two dimension vector-valued wavelets packets were introduced. We also investigated property of vector-valued wavelets packets and present biorthogonal relationship formule concerning the vector-valued wavelet packets by means of operator theory and integer transformation. What's more, several new Riesz bases of L2 (R2,Cm)were constructed from the orthogonal vector -valued wavelets packets. Besides, the biorthogonality of biorthogonal vector-valued wavelets packets was also discussed. Finally,we researched bivariate matrix-valued biorthogonal wavelets. Sufficient and necessary conditions for the existence of bivariate matrixvalued biorthogonal wavelets were proved. An algorithm of constructing bivariate matrix-valued biorthogonal wavelets was given, then two examples were showed.