Geodesic
Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables
Jeffrey S. Geronimo1 ; Hugo J. Woerdeman2
1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States
2 Department of Mathematics, Drexel University, Philadelphia, PA 19104, United States
Annals of mathematics, Tome 160 (2004) no. 3, pp. 839-906.

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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.
DOI : 10.4007/annals.2004.160.839

Jeffrey S. Geronimo 1 ; Hugo J. Woerdeman 2

1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States
2 Department of Mathematics, Drexel University, Philadelphia, PA 19104, United States 
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Jeffrey S. Geronimo; Hugo J. Woerdeman. Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables. Annals of mathematics, Tome 160 (2004) no. 3, pp. 839-906. doi : 10.4007/annals.2004.160.839. http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.839/

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