Convergence of cascade algorithms associated with nonhomogeneous refinement equations
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- by Rong-Qing Jia, Qingtang Jiang and Zuowei Shen PDF
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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 \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 $\phi _0$.References
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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
- MR Author ID: 292105
- Email: matzuows@leonis.nus.edu.sg
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 415-427
- MSC (2000): Primary 41A58, 42C40; Secondary 41A17, 42C99
- DOI: https://doi.org/10.1090/S0002-9939-00-05567-2
- MathSciNet review: 1707522