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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
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
