Matrix extension and biorthogonal multiwavelet construction

Suppose that P(z) and are two r × n matrices over the Laurent polynomial ring R[z], where r < n, which satisfy the identity on the unit circle . We develop an algorithm that produces two n × n matrices Q(z) and over R[z], satisfying the identity on such that the submatrices formed by the first r rows of Q(z) and are P(z) and respectively. Our algorithm is used to construct compactly supported biorthogonal multiwavelets from multiresolutions generated by univariate compactly supported biorthogonal scaling functions with an arbitrary dilation parameter m Z, where m >1.