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Abstract and Applied Analysis
Volume 2011, Article ID 850850, 13 pages
Research Article

Characterization of Generators for Multiresolution Analyses with Composite Dilations

1School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
2Basic Department, Henan Quality and Engineering Vocational College, Pingdingshan 467000, China
3School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China

Received 20 July 2010; Revised 12 December 2010; Accepted 2 February 2011

Academic Editor: Roman Simon Hilscher

Copyright © 2011 Yuan Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of 𝐿2(𝑛) (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace 𝐿2(𝑆). For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.