Positive Matrix Functions on the Bitorus with Prescribed Fourier Coefficients in a Band

M. Bakonyi; L. Rodman; I. M. Spitkovsky; H. J. Woerdeman

doi:10.1007/bf01274187

Positive Matrix Functions on the Bitorus with Prescribed Fourier Coefficients in a Band

M. Bakonyi, L. Rodman, I. M. Spitkovsky, H. J. Woerdeman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let S be a band in Z² bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ¹ 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)
Pages (from-to)	20-44
Number of pages	25
Journal	Journal of Fourier Analysis and Applications
Volume	5
Issue number	1
DOIs	https://doi.org/10.1007/bf01274187
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

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10.1007/bf01274187

Cite this

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title = "Positive Matrix Functions on the Bitorus with Prescribed Fourier Coefficients in a Band",

abstract = "Let S be a band in Z2 bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ1 sequence of matrices {cj}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 ck for k ∈ S. A parameterization is obtained for the set of all positive extensions f of {cj}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.",

keywords = "ARMA processes, Almost periodic functions, Band method, Entropy, Matrix functions on bitorus, Positive extensions, Toeplitz operators, Wiener algebra",

author = "M. Bakonyi and L. Rodman and Spitkovsky, {I. M.} and Woerdeman, {H. J.}",

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publisher = "Birkhauser Boston",

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T1 - Positive Matrix Functions on the Bitorus with Prescribed Fourier Coefficients in a Band

AU - Bakonyi, M.

AU - Rodman, L.

AU - Spitkovsky, I. M.

AU - Woerdeman, H. J.

PY - 1999

Y1 - 1999

N2 - Let S be a band in Z2 bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ1 sequence of matrices {cj}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 ck for k ∈ S. A parameterization is obtained for the set of all positive extensions f of {cj}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.

AB - Let S be a band in Z2 bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ1 sequence of matrices {cj}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 ck for k ∈ S. A parameterization is obtained for the set of all positive extensions f of {cj}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.

KW - ARMA processes

KW - Almost periodic functions

KW - Band method

KW - Entropy

KW - Matrix functions on bitorus

KW - Positive extensions

KW - Toeplitz operators

KW - Wiener algebra

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Positive Matrix Functions on the Bitorus with Prescribed Fourier Coefficients in a Band

Abstract

Keywords

ASJC Scopus subject areas

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