具有多项式衰减面具的向量细分格式

具有多项式衰减面具的向量细分方程在刻画小波Riesz基和双正交小波等方面有着重要作用.本文主要研究这类方程解的性质.向量的细分方程具有形式：Ф=∑α∈Zsa（α）（2·-α）,其中Ф=（Ф1,...,Фr）T是定义在Rs上的向量函数,a：=（a（α））α∈Zs是一个具有多项式衰减的r×r矩阵序列称为面具.关于面具a定义一个作用在（Lp（Rs））r上的线性算子Qa,Qaf：=∑α∈Zsa（α）f（2·α）.迭代格式（Qanf）n=1,2,...称为向量细分格式或向量细分算法.本文证明如果具有多项式衰减面具的向量细分格式在（L2（Rs））r中收敛,那么其收敛的极限函数将自动具有多项式衰减.另外,给出了当迭代的初始函数满足一定的条件时的向量细分格式的收敛阶.

Vector subdivision schemes with polynomially decaying masks

We investigate the solutions of vector refinement equations with polynomially decaying masks, which play an important role for characterizations of Riesz wavelet bases and biorthogonal wavelets. A vector refinementequation is of the form： Ф=∑α∈Zsa（α）（2·-α） where Ф=（Ф1,...,Фr）Tis the unknown vector of functions on R^8, a：=（a（α））α∈Zsis a polynomially decaying sequence of r× r matrices called refinement mask. Associated with the mask a is a linear operator Qa defined on （L2（Rs））r by： Qa,Qaf：=∑α∈Zsa（α）f（2·α）. The iterative scheme （Qanf）n=1,2,... is called a vector subdivi- sion scheme or a vector cascade algorithm. In this paper, we shall prove that if the vector subdivision scheme converges in （L2（Rs））r, then its resulting limit function automatically has polynomial decay. Moreover, under some appropriate conditions on the initial function, the convergence rates of the vector subdivision scheme will be given.