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Convergence rates of cascade algorithms

Author: Rong-Qing Jia
Journal: Proc. Amer. Math. Soc. 131 (2003), 1739-1749
MSC (2000): Primary 39B12, 41A25, 42C40, 65D99
Published electronically: January 15, 2003
MathSciNet review: 1955260
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Abstract: We consider solutions of a refinement equation of the form

\begin{displaymath}\phi = \sum_{\gamma\in\mathbb{Z}^s} a(\gamma) \phi ({M\cdot}-\gamma), \end{displaymath}

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:

\begin{displaymath}\bigl\Vert Q_a^n\psi - \phi \bigr\Vert _p \le C (m^{-1/s})^{\mu n}\quad \forall\, n \in \mathbb{N}, \end{displaymath}

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.

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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Refinement equations, refinable functions, cascade algorithms, subdivision schemes, rates of convergence
Received by editor(s): August 29, 2001
Published electronically: January 15, 2003
Additional Notes: The author was supported in part by NSERC Canada under Grant OGP 121336
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society