An algorithm for matrix extension and wavelet construction

This paper gives a practical method of extending an n x r matrix P(z), r less than or equal to n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when r is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z is an element of T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.