The interpolation problem with a degree constraint

In [6]-[8] 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 probl em with n + 1 constraints, which are rational and also of degree In. It was conjectured that the correspondence under suitable normalization is biject ive 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 a n insightful work by Byrnes etal. [1], along with a detailed study of this parametrization. However, the result in [1] was shown under a slightly rest rictive assumption that the trigonometric polynomials are positive and acco rdingly, the corresponding solutions have positive real part. The purpose o f the present note Is to extend the result to the case of nonnegative trigo nometric polynomials as well. We present the arguments in the context of th e general Nevanlinna-Pick-Caratheodory-Fejer interpolation.