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Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

Published:2011-04-25
Printed: Jun 2012
• Jianbin Yang,
Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China
• Song Li,
Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China
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Abstract

We investigate the solutions of refinement equations of the form $$\phi(x)=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\:\phi(Mx-\alpha),$$ where the function $\phi$ is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely supported sequence on $\mathbb Z^s$ called a refinement mask, and $M$ is an $s\times s$ integer matrix such that $\lim_{n\to\infty}M^{-n}=0$. Associated with the mask $a$ and $M$ is a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by $Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\phi_0(M\cdot-\alpha)$. Main results of this paper are related to the convergence rates of $(Q_{a,M}^n \phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$ converges in $L_p(\mathbb R^s)$ with an exponential rate.
 Keywords: refinement equations, infinitely supported mask, cascade algorithms, rates of convergence
 MSC Classifications: 39B12 - Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX] 41A25 - Rate of convergence, degree of approximation 42C40 - Wavelets and other special systems

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