Journal of Applied Mathematics
Volume 2012 (2012), Article ID 130939, 18 pages
http://dx.doi.org/10.1155/2012/130939
Research Article

Constructions of Vector-Valued Filters and Vector-Valued Wavelets

Jianxun He1,2 and Shouyou Huang1,2

1School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

Received 2 March 2012; Revised 16 May 2012; Accepted 30 May 2012

Academic Editor: Jingxin Zhang

Copyright © 2012 Jianxun He and Shouyou Huang. 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

Let a   = ( 𝑎 1 , 𝑎 2 , … , 𝑎 𝑚 ) ∈ ℂ 𝑚 be an m-dimensional vector. Then, it can be identified with an 𝑚 × 𝑚 circulant matrix. By using the theory of matrix-valued wavelet analysis (Walden and Serroukh, 2002), we discuss the vector-valued multiresolution analysis. Also, we derive several different designs of finite length of vector-valued filters. The corresponding scaling functions and wavelet functions are given. Specially, we deal with the construction of filters on symmetric matrix-valued functions space.