Matrix extension with symmetry and construction of biorthogonal multiwavelets with any integer dilation

In this paper, we investigate the biorthogonal matrix extension problem with symmetry and its application to construction of biorthogonal multiwavelets. Given a pair of biorthogonal matrices , the biorthogonal matrix extension problem is to find a pair of extension matrices of Laurent polynomials with symmetry such that the submatrix of the first r rows of is the given matrix , respectively; Pe and are biorthogonal satisfying ; and Pe and have the same compatible symmetry. We satisfactorily solve the biorthogonal matrix extension problem with symmetry and provide a step-by-step algorithm for constructing the desired pair of extension matrices from the given pair of matrices . Moreover, our results cover the case for paraunitary matrix extension with symmetry (i.e., the given pair satisfies ). 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 biorthogonal matrix extension with symmetry, we obtain a satisfactory algorithm for constructing univariate biorthogonal multiwavelets with symmetry for any dilation factor d from a given pair of biorthogonal d-refinable function vectors with symmetry. Correspondingly, pairs of d-dual filter banks with the perfect reconstruction property and with symmetry can be derived by applying our algorithm to a given pair of d-dual low-pass filters with symmetry. Several examples of symmetric biorthogonal multiwavelets are provided to illustrate our results in this paper.