Matrix extension with symmetry and its application to symmetric orthonormal multiwavelets

Let P bean r × s matrix of Laurent polynomials with symmetry such that P(z)P^* (z) = Ir for all z ∈ ℂ\{0} and the symmetry of P is compatible. The matrix extension problem with symmetry is to find an s × s square matrix P_e of Laurent polynomials with symmetry such that [I_r, 0]P_e = P (that is, the submatrix of the first r rows of P_e is the given matrix P), P_e is paraunitary satisfying P_e(z)P^*(z) = I_s for all z ∈ ℂ\{0}, and the symmetry of P_e is compatible. Moreover, it is highly desirable in many applications that the support of the coefficient sequence of P_e can be controlled by that of P. In this paper, we completely solve the matrix extension problem with symmetry by constructing such a desired matrix P_e from a given matrix P. Furthermore, using a cascade structure, we obtain a complete representation of any r × s paraunitary matrix P having compatible symmetry, which in turn leads to a construction of a desired matrix P_e from a given matrix 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. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.