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 cascad...