Positive extensions, Fejér–Riesz factorization and autoregressive filters in two variables

Abstract

In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral Fejér-Riesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal.Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parametrized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem.

Time-relevant stability of 2D systems

“…It is worth mentioning here the important role played by the ω-dependent Bézoutian (20) in the study of stability of 2D filters in the approach of [10,11].…”

Two Variable Orthogonal Polynomials on the BiCircle and Structured Matrices

“…Such structured matrices are called doubly Toeplitz matrices and arise naturally in the bivariate trigonometric moment problem. These types of matrices arose more recently in the work of Geronimo and Woerdeman [8] in their investigation of the bivariate Fejér-Riesz factorization theorem. These authors were able to resolve the question when a strictly positive bivariate trigonometric polynomial of a certain degree can be written as the magnitude squared of a stable polynomial of the same degree.…”

“…The study of orthogonal polynomials on the bicircle with this ordering was begun by Delsarte et.al. [4] and extended in [8]. Given a positive definite linear functional L N,M :…”

“…Because of the structure of C n,m we can associate to L m matrix valued orthogonal polynomials in the following manner [3], [4], [8]. Let {R …”

“…In Section 7 we examine conditions under which the linear functional can be represented as a positive measure supported on the bicircle having the form of one over the magnitude squared of a stable polynomial. This gives a new proof of the Fejér-Riesz result of [8]. Finally in Section 8 examples are given that illustrate various aspects of the theory developed.…”