Constructions of Vector-Valued Filters and Vector-Valued Wavelets

Abstract

Let a $=(a_1,a_2,\dots ,a_m)$ $\in {ℂ}^m$ be an m-dimensional vector. Then, it can be identified with an $m\times m$ 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.