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.
Highlights
The classical Fejer-Riesz theorem states [9, Section 1.12] that any nonnegative trigonometric polynomial in z, n−1 p(eiθ) = Λ0 + 2 ak cos(kθ) + bk sin(kθ) k=1 n−1=: ckeikθ k=−(n−1)for Λ0 ∈ R and Λ0 > 0, and ck = ak + ibk ∈ C, k ∈ {1, 2, . . . , k − 1}, i.e., one such that p(eiθ) 0, for all θ ∈ [−π, π], can be written as the square of the modulus of a polynomial g(z) = g0 + g1z + . . . + gn−1zn−1 in z of equal degree, where z is on the unit circle, z = eiθ
For Λ0 ∈ R and Λ0 > 0, and ck = ak + ibk ∈ C, k ∈ {1, 2, . . . , k − 1}, i.e., one such that p(eiθ) 0, for all θ ∈ [−π, π], can be written as the square of the modulus of a polynomial g(z) = g0 + g1z + . . . + gn−1zn−1 in z of equal degree, where z is on the unit circle, z = eiθ
The roots of g(z) in the Fejer-Riesz factorization can be selected outside the unit disc [9, Section 1.12]
Summary
The roots of g(z) in the Fejer-Riesz factorization can be selected outside the unit disc [9, Section 1.12]. Our aim in this work is derive a natural generalization of the Fejer-Riesz factorization that applies to matrix-valued para-conjugate Hermitian rational functions as stated below. V (z) for a unique pair (C, Ω), with C ∈ Rm×n and Ω ∈ Sm + ×m satisfying i) CB = I, ii) Ω is a positive definite matrix, iii) the polynomial det(V(z)), with V(z) := CG(z), has no root in the closed unit disc. At first glance it appears that Theorem 1.1 follows from standard factorization theory for non-negative matrix-valued functions on the unit circle, based on the positive real lemma, see, e.g., [13, Section 6], or [14]. In the rest of the paper we will assert and derive these identities in two different ways
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
