On Factorization of Trigonometric Polynomials

Abstract

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also * -outer, is examined in greater detail. The connection to Agler's model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over R m , m ≥ 1, in terms of rational functions with fixed denominators. Mathematics Subject Classification (2000). 47A68 (Primary), 47A65, 42A05, 30C10, 15A23 (Secondary).

The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

“…Namely, there is a polynomial q(z) = d l=0 q l z l with complex coefficients such that p(z) = |q(z)| 2 , ∀z ∈ T. This yields that any sum of square magnitudes of univariate polynomials can be represented as only one square magnitude of a polynomial. For the case of the multivariate polynomials, Dritschel [6] proved that any (strictly) positive Laurent polynomial on the n-torus T n can be represented as a sum of square magnitudes of polynomials. Calderon and Perpinsky [3] and Rudin [17] proved that it is not always true for the case the polynomial is nonnegative on the torus T n , n ≥ 2.…”

“…A discussion on the Pythagoras number of (complex) Laurent polynomials positive on the n-torus is given in Sect. 6. The Laurent polynomials in that section are considered as sum of square magnitudes of complex-coefficient polynomials.…”

Positive trigonometric polynomials for strong stability of difference equations

“…Note that an alternative proof of Putinar's theorem for trigonometric polynomials, building on ideas of operator theory, can be found in [6], where it is shown that factorable trigonometric polynomials (i.e. those which can be expressed as sum-of-squares) are dense, and in particular, strictly positive polynomials are factorable.…”

Two Variable Orthogonal Polynomials on the BiCircle and Structured Matrices

“…then Delsarte, Genin and Kamp have shown [3] that for a given degree k, This leads to a simple proof of the Matrix Fejér-Reisz factorization Theorem (Helson [13], Dritschel [5], McLean-Woerdeman [16], Geronimo-Lai [10]) which will be useful later.…”

“…As indicated above denotẽ 5) where the (n + 1) × (n + 1)(m + 1) matrixK n,m is given similarly to (3.4) with the roles of n and m interchanged. For the bivariate polynomials φ l n,m (z, w) above we define the reverse polynomials…”