Matrix extension and biorthogonal multiwavelet construction

Abstract Suppose that P(z) and P (z) are two r × n matrices over the Laurent polynomial ring R[z], where r P(z) P (z)∗ = I r on the unit circle T . We develop an algorithm that produces two n × n matrices Q(z) and Q (z) over R[z], satisfying the identity Q(z) Q (z)∗ = I n on T such that the submatrices formed by the first r rows of Q(z) and Q (z) are P(z) and P (z) 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.