New results on the rational covariance extension problem with degree constraint

A new theory for the rational covariance extension problem (with degree constraint), or simply the RCEP, has recently emerged with applications in high-resolution spectral estimation, speech synthesis and possibly new applications in time series analysis and system identification. This paper establishes some new theoretical results on the RCEP via an alternative analysis. In one result we show the bijective correspondence between denominator polynomials of non-strictly-positive solutions of the RCEP and the minimizers of a class of (strictly) convex functionals, associated with non-strictly-positive pseudopolynomials, defined on a subset of a finite dimensional space. This result leads to an alternative constructive proof of a theorem of Georgiou on complete parametrization of all solutions of the RCEP and a new geometric proof, with an extension to non-real interpolators, of a homeomorphism derived previously by Blomqvist, Fanizza and Nagamune. We then generalize this homeomorphism to also allow for variation in the covariance data. Our contribution in the generalization is allowing for non-strictly-positive pseudopolynomials. For the special case of strictly positive pseudopolynomials, a stronger property of diffeomorphism (in the context of the Nevanlinna–Pick interpolation with degree constraint) has been shown earlier by Byrnes and Lindquist. The results of this paper have direct analogues to Nevanlinna–Pick interpolation with degree constraint which is of interest to the robust control community.