Analytic interpolation with a degree constraint

The author previously (1983, 1987) showed that there is a correspondence between nonnegative (hermitian) trigonometric polynomials of degree ⩽n and solutions to the standard Nevanlinna-Pick-Caratheodory interpolation problem with n+1 constraints, which are rational and also of degree ⩽n. 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 ⩽n. The conjecture was proven by Byrnes et al. (1995) along with a detailed study of this parametrization, but their result 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