Some Studies On The Two-way Parseval Frame Wavelets

Wavelets analysis is a new branch of mathematics which has been developed in recent years, it is the epochal product of the development of Fourier analysis. Daubechies proved that there was no compactly supported orthogonal symmetric Unit-wavelet besides the Haar wavelet. In1994, Goodman put forward the concept of multiwavelet, Multiwavelet is the perfect combination of moothness, compacted support, symmetry, orthogonality which are very important in practical application, So multiwavelet is widely used in practical application. In2006, Yang Shouzhi put forward two dimension refinement equations. Two-direction wavelets are a kind of new wavelets; the general method of constructing two-direction wavelets is a hot research issue at present. Considing the lacking of degree of freedom of orthogonal wavelet, Frame wavelet have been received widespread attention in recent years. In particular, as the promotion of orthogonal wavelet base, Parseval frame wavelet has kept almost all good properties besides orthogonality. It has important significance to research Parseval frame wavelet.This article is about some research of two-direction Parseval frame wavelet. First of all, this article gives the concept of two-direction scaling function and the concept of two-direction low pass filter, discussed some properties of two-direction scaling function and a sufficient and necessary condition to constitute two-direction scaling function. Secondly, it introduces the concept of Parseval frame two-direction wavelet, gives the concept of two-direction high-pass filter and the concept of two-direction filter matrix, discusses some properties of two-direction Parseval frame wavelet associated with two-direction multi-resolution analysis, we obtain a method of constituting Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis. The specific contents of the article is divided into four parts:The first chapter is the introduction. It gives a brief introduction of the generation and development of wavelets analysis, the generation and development and a current research status about two-direction wavelets and frames analysis.The second chapter is about two-direction scaling function associated with two-direction multi-resolution analysis. First of all, it gives two theorems, on the basis of the two-direction multi-resolution analysis, the concept of two-direction scaling functions is introduced. Then, it introduces the concept of two-direction low pass filter, discusses some properties of two-directional scaling functions using two-direction low pass filter. Final, it obtains a sufficient and necessary condition of constituting two-directional scaling function.The third chapter is about some properties of Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis. According to the concept of Parseval frame wavelet it gives the concept of two-direction Parseval frame wavelet and the concept of Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis, two-direction high-pass filter is introduced and according to the discussing of the two-direction scaling function research some properties of Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis.The fourth chapter is about Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis. Firstly, according to the concept of the filter matrix gives the concept of two-directional filter matrix. Using the properties of two-direction filter matrix it obtains a sufficient condition of Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis, then, it introducts a sufficient and necessary condition. Finally, it gives the concept of canonical two-direction low-pass filter matrix, on the basis of which obtains a method of constituting Parseval frame two-direction wavelet associated with two-direction multi-resolution analysis.