Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where $a$ is a finitely supported sequence called the refinement mask. Associated with the mask $a$ is a linear operator $Q_a$ defined on $L_p(\mathbb{R}^s)$by $Q_a \psi := \sum_{\gamma\in\mathbb{Z}^s} a(\gamma) \psi({M\cdot}-\gamma)$. This paper is concerned with the convergence of the cascade algorithm associated with $a$, i.e., the convergence of the sequence $(Q_a^n\psi)_{n=1,2,\ldots}$ in the $L_p$-norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let $\phi$ be the normalized solution of the above refinement equation with the dilation matrix $M$ being isotropic. Suppose $\phi$ lies in the Lipschitz space $\operatorname{Lip} (\mu,L_p(\mathbb{R}^s))$, where $\mu >0$ and $1 \le p \le \infty$. Under appropriate conditions on $\psi$, the following estimate will be established:

where $m:=\vert\det M\vert$ and $C$ is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.