## Abstract

Let S be a band in Z^{2} bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ^{1} sequence of matrices {c_{j}}_{j∈S}, we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients f(k) equal c_{k} for k ∈ S. A parameterization is obtained for the set of all positive extensions f of {c_{j}}_{j∈S}. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes.

Original language | English (US) |
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Pages (from-to) | 20-44 |

Number of pages | 25 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - 1999 |

## Keywords

- ARMA processes
- Almost periodic functions
- Band method
- Entropy
- Matrix functions on bitorus
- Positive extensions
- Toeplitz operators
- Wiener algebra

## ASJC Scopus subject areas

- Analysis
- General Mathematics
- Applied Mathematics