Communications in Information and Systems

Volume 21 (2021)

Number 3

On a Fejér–Riesz factorization of generalized trigonometric polynomials

Pages: 371 – 384

DOI: https://dx.doi.org/10.4310/CIS.2021.v21.n3.a3

Authors

Tryphon T. Georgiou (Department of Mechanical and Aerospace Engineering, University of California, Irvine, Calif., U.S.A.)

Anders Lindquist (Department of Automation and the School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China; and Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden)

Abstract

Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejér–Riesz theorem that non-negative trigonometric polynomials can be expressed as the modulus of a polynomial of the same degree evaluated on the unit circle. In the present note we consider a natural generalization of non-negative trigonometric polynomials that are matrix-valued with specified non-trivial poles (i.e., other than at the origin or at infinity). We are interested in the corresponding spectral factors and, specifically, we show that the factorization of trigonometric polynomials can be carried out in complete analogy with the Fejér–Riesz theorem. The affinity of the factorization with the Fejér–Riesz theorem and the contrast to classical spectral factorization lies in the fact that the spectral factors have degree smaller than what standard construction in factorization theory would suggest. We provide two juxtaposed proofs of this fundamental theorem, albeit for the case of strict positivity, one that relies on analytic interpolation theory and another that utilizes classical factorization theory based on the Yacubovich–Popov–Kalman (YPK) positive-real lemma.

Keywords

harmonic analysis in one variable, factorization, trigonometric polynomials, positive-real lemma

2010 Mathematics Subject Classification

42C05, 44A60, 47A68

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This research was partially supported by the National Science Foundation under grants 1807664, 1839441, and the Air Force Office of Scientific Research under grant FA9550-20-1-0029.

Received 20 May 2020

Published 4 June 2021