Explicit Constructions Of A Class Of Trivariate Nonseprable Wavelets And Bivariate Wavelet Tight Frames

Since 1980's,wavelet analysis has been a popular and concerned field in scientific research.Its application almost involves all branchs in natural science and engineering technology.Nowadays, it has been a powerful tool for exploring and solving many complicated prob-lems in natural science and engineering computation. In this paper,some problems con-cerning constructions of multivariate wavelets and wavelet frame have been given a deepinvestigation and study, and several important results have been obtained. The contents ofthe thesis are organized as follows:For trivariate nonseparable wavelet system, a method of construction of a class com-pactly supported trivariate nonseparable wavelet with a special dilation matrix has beengiven.Furthermore,a way of constructing a class compactly supported trivariate nonseparablewavelet with a kind of special dilation matrix has also been given. Wavelet function obtainedby the method inherits the properties of symmetry and vanishing moment originating fromthe symbol and scaling function. Finally, examples of trivariate nonseparable wavelet havebeen designed.For bivariate wavelet tight frames, bivariate wavelet tight frames with dilation matrix ofanti-diagonal and diagonal matrix have been separately studied. First, sufficient conditionsfor existence of bivariate wavelet tight frames generated by the n functions have been pre-sented, and explicit constructing formula of bivariate wavelet tight frames has been given.Then, the decomposition and reconstruction formulas of bivariate wavelet tight frames havebeen derived. Finally, examples of bivariate wavelet tight frames have been given.