数学所周五学术报告—On the Extension of Symmetric Laurent Polynomial Matrix and Its Application
主 题: 数学所周五学术报告—On the Extension of Symmetric Laurent Polynomial Matrix and Its Application
报告人: Professor Jianzhong Wang (Sam Houston State University)
时 间: 2017-05-12 15:00-16:00
地 点: 理科1号楼1114
Abstract: For a given pair of $s$-dimensional Laurent polynomial vectors $(\vec{a}(z),\vec{b}(z))$, which has a certain type of symmetry and satisfies the dual condition ${\vec{b}(z)}^T\vec{a}(z)=1$, an $s\times s$ Laurent polynomial matrix $A(z)$ (together with its inverse $A^{-1}(z)$) is called a symmetric Laurent polynomial matrix extension (SLPME) of the dual pair $(\vec{a}(z), \vec{b}(z))$ if (1) each column of $A(z)$ has the similar symmetry as $\vec{a}(z)$, (2) the inverse $A^{-1}(Z)$ also is a Laurent polynomial matrix, (3) the first column of $A(z)$ is $\vec{a}(z)$, and the first row of $A^{-1}(z)$ is $(\vec{b}(z))^T$. The SLPME problems arise in several areas, such as Cuntz algebra, construction of finite impulse response (FIR) multi-band prefect reconstruction filter banks (PRFBs) with linear phases, and symmetric compactly supported multi-wavelets. In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Base on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop the symmetric elementary transformations on the dual pair $(\vec{a}(z), \vec{b}(z))$ for SLPME. We also apply the algorithm in the construction of multi-band SPRFB.
