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Convergence Rates of Cascade Algorithms with Infinitely Supported Masks


Published:20110425
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.
MSC Classifications: 
39B12, 41A25, 42C40 show english descriptions
Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37XX] Rate of convergence, degree of approximation Wavelets and other special systems
39B12  Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37XX] 41A25  Rate of convergence, degree of approximation 42C40  Wavelets and other special systems
