Matrix Extension with Symmetry and Its Applications

In this paper, we are interested in the problems of matrix extension with symmetry, more precisely, the extensions of submatrices of Laurent polynomials satisfying some conditions to square matrices of Laurent polynomials with certain symmetry patterns, which are closely related to the construction of (bi)orthogonal multiwavelets in wavelet analysis and filter banks with the perfect reconstruction property in electronic engineering. We satisfactorily solve the matrix extension problems with respect to both orthogonal and biorthogonal settings. Our results show that the extension matrices do possess certain symmetry patterns and their coefficient supports can be controlled by the given submatrices in certain sense. Moreover, we provide step-by-step algorithms to derive the desired extension matrices. We show that our extension algorithms can be applied not only to the construction of (bi)orthogonal multiwavelets with symmetry, but also to the construction of tight framelets with symmetry and with high order of vanishing moments. Several examples are presented to illustrate the results in this paper.