Extensions of matrix valued functions with rational polynomial inverses

Let R_j : |j| m, be a given set of n×n matrices. Necessary and sufficient conditions for the existence and uniqueness of an invertible function F() = F_j^j in the Wiener algebra of n×n matrix valued functions on the unit circle || = 1 such that F_j=R_j for |j| m, and F admits either a right or a left canonical factorization and the matrix Fourier coefficients of F^–1 vanish for |j| > m are presented and discussed. In the special case that the block Toeplitz matrix based on the given R_j is positive definite there is exactly one such extension: the so-called maximum entropy or autoregressive extension of statistical estimation theory. Some special properties of this extension are discussed.