Matrix Extension with Symmetry and Its Application to Symmetric Orthonormal Multiwavelets

Let $\mathsf{P}$ be an $r\times s$ matrix of Laurent polynomials with symmetry such that $\mathsf{P}(z)\mathsf{P}^*(z)=I_r$ for all $z\in\mathbb{C}\backslash\{0\}$ and the symmetry of $\mathsf{P}$ is compatible. The matrix extension problem with symmetry is to find an $s\times s$ square matrix $\mathsf{P}_e$ of Laurent polynomials with symmetry such that $[I_r,\mathbf{0}]\mathsf{P}_e =\mathsf{P}$ (that is, the submatrix of the first r rows of $\mathsf{P}_e$ is the given matrix $\mathsf{P}$), $\mathsf{P}_e$ is paraunitary satisfying $\mathsf{P}_e(z)\mathsf{P}_e^*(z)=I_s$ for all $z\in\mathbb{C}\backslash\{0\}$, and the symmetry of $\mathsf{P}_e$ is compatible. Moreover, it is highly desirable in many applications that the support of the coefficient sequence of $\mathsf{P}_e$ can be controlled by that of $\mathsf{P}$. In this paper, we completely solve the matrix extension problem with symmetry by constructing such a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Furthermore, using a cascade structure, we obtain a complete representation of any $r\times s$ paraunitary matrix $\mathsf{P}$ having compatible symmetry, which in turn leads to a construction of a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. 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 orthonormal multiwavelets by deriving high-pass filters with symmetry from any given orthogonal low-pass filters with symmetry. Several examples of symmetric orthonormal multiwavelets are provided to illustrate the results in this paper.

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