Subdivision schemes with polynomially decaying masks

Abstract

In this paper we investigate the L2-solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix, which plays a vital role for characterizations of wavelets and biorthogonal wavelets with infinite support. A vector refinement equation with polynomially decaying masks and a general dilation matrix is the form: $$ \phi(x)=\sum_{\alpha\in\Bbb Z^s}a(\alpha)\medspace\phi(Mx-\alpha),\quad x\in\Bbb R^s, $$ where the vector of functions \(\phi=(\phi_{1},\cdots,\phi_{r})^{T}\) is in \((L_{2}(\Bbb R^s))^{r},\)\(a:=(a(\alpha))_{\alpha\in\Bbb Z^s}\) is a polynomially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that \(\lim_{n\to\infty}M^{-n}=0.\) The corresponding cascade operator on \((L_2(\Bbb R^s))^r\) is given by: $$ Q_{a}f(x):=\sum_{\alpha\in\Bbb Z^s}a(\alpha)f(Mx-\alpha),\quad x\in\Bbb R^s, \quad f=(f_1,...,f_r)^T\in (L_2(\Bbb R^s))^r. $$ The iterative scheme \((Q_a^nf)_{n=1,2,\cdots,}\) is called vector cascade algorithm. In this paper we give a complete characterization of convergence of the sequence \((Q_a^nf)_{n=1,2\cdots}\) in L2-norm. Some properties of the transition operator restricted to a certain linear space are discussed. As an application of convergence, we also obtain a characterization of smoothness of solutions of refinement equation mentioned above for the case r = 1.