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Convergence rates of cascade algorithms
Authors:Rong-Qing Jia
Affiliation:Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Abstract:We consider solutions of a refinement equation of the form

begin{displaymath}phi = sum_{gammainmathbb{Z}^s} a(gamma) phi ({Mcdot}-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 $Q_a psi := sum_{gammainmathbb{Z}^s} a(gamma) psi({Mcdot}-gamma)$. This paper is concerned with the convergence of the cascade algorithm associated with $a$, i.e., the convergence of the sequence $(Q_a^npsi)_{n=1,2,ldots}$ 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}biglVert Q_a^npsi - phi bigrVert _p le C (m^{-1/s})^{mu n}quad forall, n in mathbb{N}, end{displaymath}

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

Keywords:Refinement equations   refinable functions   cascade algorithms   subdivision schemes   rates of convergence
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