Journal of Applied Mathematics
Volume 2012 (2012), Article ID 846852, 18 pages
http://dx.doi.org/10.1155/2012/846852
Research Article

Orthogonal Multiwavelet Frames in 𝐿 2 ( 𝑅 𝑑 )

Liu Zhanwei,1 Hu Guoen,2 and Wu Guochang3

1School of Information Engineering, Zhengzhou University, Zhengzhou 450001, China
2Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China
3Department of Applied Mathematics, Henan University of Economics and Law, Zhengzhou 450002, China

Received 27 June 2011; Revised 21 September 2011; Accepted 25 October 2011

Academic Editor: Yuesheng Xu

Copyright © 2012 Liu Zhanwei 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.

Abstract

We characterize the orthogonal frames and orthogonal multiwavelet frames in 𝐿 2 ( 𝑅 𝑑 ) with matrix dilations of the form √ ( 𝐷 𝑓 ) ( 𝑥 ) = | d e t 𝐴 | 𝑓 ( 𝐴 𝑥 ) , where 𝐴 is an arbitrary expanding 𝑑 × 𝑑 matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames. Finally, we give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.