IEEE Transactions on Automatic Control, Vol.44, No.3, 631-635, 1999
The interpolation problem with a degree constraint
In - it was shown that there is a correspondence between nonnegative (hermitian) trigonometric polynomials of degree less than or equal to n and solutions to the standard Nevanlinna-Pick-Caratheodory interpolation problem with n + 1 constraints, which are rational and also of degree In. It was conjectured that the correspondence under suitable normalization is bijective and thereby, that it results in a complete parametrization of rational solutions of degree less than or equal to n. The conjecture was proven in an insightful work by Byrnes etal. , along with a detailed study of this parametrization. However, the result in  was shown under a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly, the corresponding solutions have positive real part. The purpose of the present note Is to extend the result to the case of nonnegative trigonometric polynomials as well. We present the arguments in the context of the general Nevanlinna-Pick-Caratheodory-Fejer interpolation.