Matrix Extension with Symmetry and Construction of Biorthogonal Multiwavelets

Let $(\pP,\wt\pP)$ be a pair of $r \times s$ matrices of Laurent polynomials with symmetry such that $\pP(z) \wt\pP^*(z)=I_\mrow$ for all $z\in \CC \bs \{0\}$ and both $\pP$ and $\wt\pP$ have the same symmetry pattern that is compatible. The biorthogonal matrix extension problem with symmetry is to find a pair of $s \times s$ square matrices $(\pP_e,\wt\pP_e)$ of Laurent polynomials with symmetry such that $[I_r, \mathbf{0}] \pP_e =\pP$ and $[I_r,\mathbf{0}]\wt\pP_e=\wt\pP$ (that is, the submatrix of the first $r$ rows of $\pP_e,\wt\pP_e$ is the given matrix $\pP,\wt\pP$, respectively), $\pP_e$ and $\wt\pP_e$ are biorthogonal satisfying $\pP_e(z)\wt\pP_e^*(z)=I_\mcol$ for all $z\in \CC \bs \{0\}$, and $\pP_e,\wt\pP_e$ have the same compatible symmetry. In this paper, we satisfactorily solve this matrix extension problem with symmetry by constructing the desired pair of extension matrices $(\pP_e,\wt\pP_e)$ from the given pair of matrices $(\pP,\wt\pP)$. Matrix extension plays an important role in many areas such as wavelet analysis, electronic engineering, system sciences, and so on. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric biorthogonal multiwavelets by deriving high-pass filters with symmetry from any given pair of biorthgonal low-pass filters with symmetry. Several examples of symmetric biorthogonal multiwavelets are provided to illustrate the results in this paper.