TAILIEUCHUNG - Đề tài " Positive extensions, Fej´erRiesz factorization and autoregressive filters in two variables "

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. . | Annals of Mathematics Positive extensions Fej er-Riesz factorization and autoregressive filters in two variables By Jeffrey S. Geronimo and Hugo J. Woerdeman Annals of Mathematics 160 2004 839 906 Positive extensions Fejer-Riesz factorization and autoregressive filters in two variables By JEFFREy S. Geronimo and Hugo J. Woerdeman 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 Fejer-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. Contents 1. Introduction . The main results . The positive extension problem . Two-variable orthogonal polynomials . Fejer-Riesz factorization The research of both authors was partially supported by NSF grants DMS-9970613 JSG and .

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