Direction Refinable Equation And The Cascade Algorithm, A Number Of Issues

Wavelet analysis has intently attracted many researchers'attentions in recent twenty years. It becomes one of important, brandnew and efficient tools in Harmonic Analysis and Signal Processing fields.From the point of mathematical view, in fact, wavelet is in accordance with specific space which is called wavelet basis function on the mathematical expression and approximation. Wavelet basis functions not only have rapid attenuation, full smoothness, energy concentrated in local, but also prossess time-frequency analysis, multiscale zooming, more bases, sparse representations and nonlinear diagonal. Wavelet analysis is widely used in Basic Science, Applied Science, Information Science and Signal Processing. It has not only for a mathematician pays more attention to it, but it is also caused physicists, biologists, engineers and other science fields workers. The theoretical research and application of wavelet analysis becomes more and more deep and rapid.It is known that the two-scale refinement equations play a very important role in wavelet Analysis, Signal Processing and Computer Graphics. A function is refinable function if it satisfies the two-scale refinement equations, the refinable functions play an important role in Wavelet Analysis and Computer Graphics. The cascade algorithm is the main approach to approximate the refinable functions and to study their properties. Two direction refinement equation is more general case than the two-scale refinement equations,and also has a very important role in the construction and application of wavelet.This paper is composed of four parts:The chapter 1 is an introduction which summarizes the emergence and development of wavelet analysis; The chapter 2,we introduce the two-direction equations,two-direction function, Furier techniques and cascade algorithms. The chapter 3,this article will be committed to in the study of the following forms of two-direction refinement equation: Where a are constant real number and a>1,β0<…<βN are also real numbers. {β0,β1,…βN,}∈B(?)R. We prove that the vector space of all L1-solutions of the above equation is at most one dimensional.Next we present sufficient conditions (easy for verification)for the existence of nontrivial L1-solutions of the two-direction refinement. The chapter 4,We investigate the convergence of cascade algorithms associated with an infinite mask by frequency approach.The investigation establish a sufficient and necessity condition, in terms of transition operator and the uniform integrablity, for the convergence of cascade algorithm associated with infinited mask. Moreover, our results apply to the case where the initial function is not compact supported.