On a Fejer-Riesz factorization of generalized trigonometric polynomials

TL;DR

This paper generalizes the Fejer-Riesz theorem to matrix-valued trigonometric polynomials with poles, showing they can be factorized with spectral factors of smaller degree, using two different proof methods.

Contribution

It extends the classical Fejer-Riesz factorization to a broader class of matrix-valued polynomials with poles, providing two independent proofs.

Findings

Spectral factors have smaller degree than standard theory predicts.

Factorization analogous to Fejer-Riesz holds for matrix-valued polynomials with poles.

Two proofs: analytic interpolation and classical positive-real lemma.

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 Fejer-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 Fejer-Riesz theorem. The affinity of the factorization with the Fejer-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.