Positive extensions, Fejér–Riesz factorization and autoregressive filters in two variables
Jeffrey S. Geronimo, Hugo J. Woerdeman · Annals of Mathematics · 2004
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.