Convergence rates of cascade algorithms

Abstract:

We consider solutions of a refinement equation of the form \[ \phi = \sum _{\gamma \in \mathbb {Z}^s} a(\gamma ) \phi ({M\cdot }-\gamma ), \] 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: \[ \bigl \| Q_a^n\psi - \phi \bigr \|_p \le C (m^{-1/s})^{\mu n}\quad \forall n \in \mathbb {N}, \] where $m:=|\det M|$ and $C$ is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.