Multichannel singular predictor polynomials

The concept of singular predictor polynomials relative to a positive definite block-Toeplitz matrix is considered. These predictors are defined in terms of the classical multichannel predictors in a fairly natural manner. It is shown that the singular predictors satisfy a simple three-term recurrence relation, which gives rise to an efficient Levinson-type algorithm for computing the classical predictors of a given length. An associated Schur-type method can be used to determine the reflection coefficients of the prediction filters. The simplifications resulting from some symmetries in the data are examined, with special emphasis on the centrohermitian structure met in two-variable prediction problems. Finally, the Caratheodory-Fejer interpolation problem for matrix-valued functions is shown to be solvable with the help of a function-theoretic version of the Schur-type algorithm, and a duality relation is exhibited in the special case of lossless functions