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Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

Published online by Cambridge University Press:  20 November 2018

Jianbin Yang  and
Song Li
Jianbin Yang
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. Chinae-mail: [email protected]; [email protected]
Song Li
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. Chinae-mail: [email protected]; [email protected]
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Abstract

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We investigate the solutions of refinement equations of the form

$$\phi (x)\,=\,\sum\limits_{\alpha \in {{\mathbb{Z}}^{S}}}{a(\alpha )\,}\phi (Mx\,-\,\alpha ),$$

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 ${{\text{Q}}_{a,M}}$ defined on ${{L}_{p}}({{\mathbb{R}}^{s}})$ by ${{\text{Q}}_{a,M}}{{\phi }_{0}}\,:=\,{{\sum }_{\alpha \in {{\mathbb{Z}}^{s}}}}\,a(\alpha ){{\phi }_{0}}(M\,\cdot \,-\alpha )$. Main results of this paper are related to the convergence rates of ${{(\text{Q}_{a,M}^{n}{{\phi }_{o}})}_{n=1,2,\ldots }}$ in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function ${{\phi }_{0}}$, $\text{Q}_{a,M}^{n}{{\phi }_{0}}$ converges in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with an exponential rate.

Keywords

39B1241A2542C40refinement equationsinfinitely supported maskcascade algorithmsrates of convergence
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 424 - 434
Copyright
Copyright © Canadian Mathematical Society 2012

References

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