The interpolation problem with a degree constraint

Authors
Citation
Tt. Georgiou, The interpolation problem with a degree constraint, IEEE AUTO C, 44(3), 1999, pp. 631-635
Citations number
12
Categorie Soggetti
AI Robotics and Automatic Control
Journal title
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
ISSN journal
00189286 → ACNP
Volume
44
Issue
3
Year of publication
1999
Pages
631 - 635
Database
ISI
SICI code
0018-9286(199903)44:3<631:TIPWAD>2.0.ZU;2-R
Abstract
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.