CDSNS preprint 2000/355 POSITIVE EXTENSIONS AND RIESZ-FEJER FACTORIZATION FOR TWO-VARIABLE TRIGONOMETRIC POLYNOMIALS

In this paper the autoregressive filter problem for bivariate stochastic processes is reduced to a finite positive definite matrix completion problem where the completion is required to satisfy additional low rank conditions. The autoregressive filter problem may also be interpreted as a 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. For the proof a specific two-variable Kronecker theorem is developed, as well as a two-variable Christoffel-Darboux formula. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral Riesz-Fejer factorization of a two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Finally, numerical results are presented for both the autoregressive filter problem as well as the factorization problem.