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**Positive extension, Fejér-Riesz factorization and autoregressive filters in two variables.**
*(English)*
Zbl 1084.42019

The positive extension problem for trigonometric polynomials and the spectral Fejér-Riesz factorization are very exciting problems, especially due to the theoretical and practical applications. This is the reason why, along the time, various generalizations were done. In this paper, some of the two-variable problems remaining unresolved are analysed and some solutions are given. In particular, the positive extension problem that appears in the design of causal bivariate autoregressive filters is solved. To do this, after a presentation of the terminology and notational conventions, the main result and the problem are presented in the introduction.

In the second chapter of the paper, the positive extension and stable polynomials are treated. It is shown that the required positive extension exists if and only if a structured partial matrix has a positive definite structured completion satisfying a certain low rank condition. Also a stability test for two-variable polynomials that consists of two one-variable root tests and a single matrix positive definiteness test is obtained. In the third chapter, four applications of the extension results are analysed, concerning two-variable orthogonal polynomials, two-variable stable autoregressive filters, Fejér-Riesz factorization for two-variable trigonometric functions, and inverse formulas for doubly indexed Toeplitz matrices. Also, some numerical results are presented.

In the second chapter of the paper, the positive extension and stable polynomials are treated. It is shown that the required positive extension exists if and only if a structured partial matrix has a positive definite structured completion satisfying a certain low rank condition. Also a stability test for two-variable polynomials that consists of two one-variable root tests and a single matrix positive definiteness test is obtained. In the third chapter, four applications of the extension results are analysed, concerning two-variable orthogonal polynomials, two-variable stable autoregressive filters, Fejér-Riesz factorization for two-variable trigonometric functions, and inverse formulas for doubly indexed Toeplitz matrices. Also, some numerical results are presented.

Reviewer: Ilie Valuşescu (Bucureşti)