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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convergence of cascade algorithms associated with nonhomogeneous refinement equations


Authors: Rong-Qing Jia, Qingtang Jiang and Zuowei Shen
Journal: Proc. Amer. Math. Soc. 129 (2001), 415-427
MSC (2000): Primary 41A58, 42C40; Secondary 41A17, 42C99
Published electronically: August 28, 2000
MathSciNet review: 1707522
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Abstract:

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form \begin{equation*}\phi(x) = g(x) + \sum_{\alpha\in\mathbb{Z}^s} a(\alpha) \phi(Mx-\alpha), \qquad x \in \mathbb{R}^s, \end{equation*}where $\phi = (\phi_1,\ldots,\phi_r)^T$ is the unknown, $g = (g_1,\ldots,g_r)^T$ is a given vector of functions on $\mathbb{R}^s$, $M$ is an $s \times s$ dilation matrix, and $a$ is a finitely supported refinement mask such that each $a(\alpha)$ is an $r \times r$ (complex) matrix. Let $\phi_0$ be an initial vector in $(L_2(\mathbb{R}^s))^r$. The corresponding cascade algorithm is given by \begin{equation*}\phi_k := g + \sum_{\alpha\in\mathbb{Z}^s} a(\alpha) \phi_{k-1}({M\kern .12em \cdot} - \alpha), \qquad k=1,2,\ldots. \end{equation*} In this paper we give a complete characterization for the $L_2$-convergence of the cascade algorithm in terms of the refinement mask $a$, the nonhomogeneous term $g$, and the initial vector of functions $\phi_0$.


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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
Email: jia@xihu.math.ualberta.ca

Qingtang Jiang
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260
Address at time of publication: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: qjiang@haar.math.nus.edu.sg

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260
Email: matzuows@leonis.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05567-2
PII: S 0002-9939(00)05567-2
Keywords: Nonhomogeneous refinement equations, cascade algorithms
Received by editor(s): June 29, 1998
Received by editor(s) in revised form: April 13, 1999
Published electronically: August 28, 2000
Additional Notes: The first author was supported in part by NSERC Canada under Grant OGP 121336, and the second and third authors were supported in part by the Wavelets Strategic Research Programme, National University of Singapore.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society