Received: 9 March 2017 / Revised: 14 April 2017 / Accepted: 17 April 2017 / Published: 20 April 2017

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**Abstract**

For a given pair of $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ , which has a certain type of symmetry and satisfies the dual condition ${\overrightarrow{b}\left(z\right)}^{T}\stackrel{}{a}$

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*s*-dimensional real Laurent polynomials
For a given pair of $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ , which has a certain type of symmetry and satisfies the dual condition ${\overrightarrow{b}\left(z\right)}^{T}\overrightarrow{a}\left(z\right)=1$ , an $s\times s$ Laurent polynomial matrix $A\left(z\right)$ (together with its inverse ${A}^{-1}\left(z\right)$ ) is called a symmetric Laurent polynomial matrix extension of the dual pair $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ if $A\left(z\right)$ has similar symmetry, the inverse ${A}^{-1}\left(Z\right)$ also is a Laurent polynomial matrix, the first column of $A\left(z\right)$ is $\overrightarrow{a}\left(z\right)$ and the first row of ${A}^{-1}\left(z\right)$ is ${\left(\overrightarrow{b}\left(z\right)\right)}^{T}$ . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.
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*s*-dimensional real Laurent polynomials
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)