Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

Canadian mathematical bulletin

Jianbin Yang¹  Song Li¹ 

[1] Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China

关键词: refinement equations;    infinitely supported mask;    cascade algorithms;    rates of convergence;   

DOI  :  10.4153/CMB-2011-081-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We investigate the solutions of refinement equations of the form$$phi(x)=sum_{alphainmathbbZ^s}a(alpha):phi(Mx-alpha),$$ where the function $phi$is in $L_p(mathbb R^s)$$(1le pleinfty)$, $a$ is an infinitelysupported sequence on $mathbb Z^s$ called a refinement mask, and$M$ is an $simes s$ integer matrix such that$lim_{noinfty}M^{-n}=0$. Associated with the mask $a$ and $M$ isa linear operator $Q_{a,M}$ defined on $L_p(mathbb R^s)$ by$Q_{a,M} phi_0:=sum_{alphainmathbbZ^s}a(alpha)phi_0(Mcdot-alpha)$. Main results of this paper arerelated to the convergence rates of $(Q_{a,M}^nphi_0)_{n=1,2,dots}$ in $L_p(mathbb R^s)$ with mask $a$ beinginfinitely supported. It is proved that under some appropriateconditions on the initial function $phi_0$, $Q_{a,M}^n phi_0$converges in $L_p(mathbb R^s)$ with an exponential rate.

【 授权许可】

Unknown