Matrix Extension with Symmetry and Construction of Biorthogonal Multiwavelets

摘要

Let $(\pP,\wt\pP)$ be a pair of $r \times s$ matrices of Laurent polynomialswith 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 iscompatible. The biorthogonal matrix extension problem with symmetry is to finda pair of $s \times s$ square matrices $(\pP_e,\wt\pP_e)$ of Laurentpolynomials 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 $r$ rowsof $\pP_e,\wt\pP_e$ is the given matrix $\pP,\wt\pP$, respectively), $\pP_e$and $\wt\pP_e$ are biorthogonal satisfying $\pP_e(z)\wt\pP_e^*(z)=I_\mcol$ forall $z\in \CC \bs \{0\}$, and $\pP_e,\wt\pP_e$ have the same compatiblesymmetry. In this paper, we satisfactorily solve this matrix extension problemwith 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 waveletanalysis, electronic engineering, system sciences, and so on. As an applicationof our general results on matrix extension with symmetry, we obtain asatisfactory algorithm for constructing symmetric biorthogonal multiwavelets byderiving high-pass filters with symmetry from any given pair of biorthgonallow-pass filters with symmetry. Several examples of symmetric biorthogonalmultiwavelets are provided to illustrate the results in this paper.