The interpolation problem with a degree constraint

Tryphon T. Georgiou

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81 Scopus citations

Abstract

In [6]-[8] it was shown that there is a correspondence between nonnegative (hermitian) trigonometric polynomials of degree ≤n and solutionsto the standard Nevanlinna-Pick-Carathéodory 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 in an insightful work by Byrnes etal. [1], along with a detailed study of this parametrization. However, the result in [1] 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-Fejér interpolation.

Original languageEnglish (US)
Pages (from-to)631-635
Number of pages5
JournalIEEE Transactions on Automatic Control
Volume44
Issue number3
DOIs
StatePublished - 1999

Bibliographical note

Funding Information:
This work was performed in part while A. Shwartz was on sabbatical leave. The hospitality and support of the Mathematical Center, Bell Laboratories, Murray Hill, NJ, and of the Department of Management Science and Information Systems, Graduate School of Business, Rutgers University, New Brunswick, NJ, is gratefully acknowledged.

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