Solutions Of Multivariate Vector Refinement Equations

In this paper we investigate the multiple vector refinement equations is the formφ(x)=sum fromα∈Z^sα(α)φ(Mx-α), x∈R^s,where the vector of functionsφ=(φ₁,…,φ_r)^T is in (L₁(R^s))^r,α=(α(α))_{α∈Z^s} is infinitesequence of r×r matrices called refinement mask and M is an s×s integer matrixsuch that (?) M^-n=0. We mainly investigate the property of the solutions of multiplevector refinement equations, and we have obtained two results.These results play animportant role in MRA approximation.The first chapter mainly introduce the wavelets' history and main conception,thenthat introduce the main conception of Multi-Resolution Analysis,basic conception ofrefinement equation,multiple wavelets and multivariate wavelets and so on.The main content of the second chapter is the character about the solutions ofmultivariate vector refinement equations,including the basic conception of multivariatevector refinement equations,the character about multivariate refinement equations,thecharacter about multivariate vector refinement equations,where the basic conception ofmultivariate vector refinement equations and the character about multivariate refine-ment equations are the important parts.In the third chapter, there are the application and prospect of wavelet in severalfields,where including the application of wavelet in image compression,the applicationof wavelet transform in image denoising and the technology of image enhancementbased on Multi-Resolution Analysis.In the last part of this paper, I introduce the ridgeletand curvelet which are based on wavelet.