1 research outputs found
Matrix Extension with Symmetry and Construction of Biorthogonal Multiwavelets
Let (\pP,\wt\pP) be a pair of 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 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 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