Subdivision Schemes With Polynomially Decaying Masks

Refinement equation is the kernel function in wavelet analysis, and MRA playsimportant roles in wavelet. If the solution of the refinement equation has goodproperties and with other conditions, we can construct MRA. Since wavelet can beconstructed via MRA, then we constructed wavelet. Therefore, refinement equationis so important that can't be ignored. The main contributions of this dissertationare based on the refinement equation.In this paper we investigate the multiple vector refinement equations of theform:where the vector of functions (?) is in (?) is apolynomially decaying sequence ofγ×γmatrices called refinement mask and M isan s×s integer matrix such that (?).Associated with the mask a is a Cascade operator on (?) given byThe iterative scheme (?) is called vector cascade algorithm.The theoretical system of the convergence of the subdivision schemes with thefinitely supported mask is very perfect so far, in the process of the proof, it mainlyutilize the joint spectral radius of the operators. But in the characterization of theconvergence of the subdivision schemes with the infinitely supported mask, we canonly use the properties of the compact operator. The main contribution of thisdissertation are summarized as follows:Firstly, some properties of the transition operator restricted to a certain Banachspace are discussed, it's a bounded and compact operator, and the spectral radiusof the transition operator restricted to the space is no less than 1.Secondly, when the mask is a polynomially decaying sequence ofγ×γmatrices,we characterize the L₂-convergence of a vector subdivision scheme with mask a being a polynomially decaying sequence and a general dilation matrix M. For the caseγ= 1, we give the sufficiency condition of the L₂-convergence of the subdivision scheme when the shifts of the initial function are stable.Thirdly, for the caseγ= 1, we characterize the smoothness of solutions of refinement equation, it lies in some Sobolev space.