Subdivision schemes with polynomially decaying masks

Abstract

In this paper we investigate the L ₂-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 L ₂-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.