Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band
type: Item
instance of: scholarly article
title: Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band
zbMATH Open document ID: 0930.42010
DOI: 10.1007/BF01274187
author: Leiba Rodman
author: Mihály Bakonyi
author: Ilya M. Spitkovsky
author: Hugo J. Woerdeman
published in: The Journal of Fourier Analysis and Applications
publication date: 2000-02-09T00:00:00Z
full work available at URL: https://eudml.org/doc/59595
review text: The paper is concerned with Wiener algebra functions of two variables defined on the bitorus with an absolutely summable Fourier expansion. The problem of finding a positive definite matrix-valued function with certain prescribed coefficients is solved for the case when the coefficients are in a band bordered by two parallel lines that are of equal distance to the origin. A parametrization of such functions is obtained, and there exists one that maximizes the entropy, for which a formula is given. The study contains six sections. The first presents the main results. The irrational case is given in the section 2, using almost periodic function theory, and the rational case is given in section 3, reducing the problem to an operator-valued one-variable case. Some extension of the results are presented in section 4, while in section 5 a version of the extension problem for positive measures is given. In the last section, some of the results are interpreted in the context of spectral estimation of autoregressive moving average processes.
Mathematics Subject Classification ID: 42B99
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 1275885
zbMATH Keywords: positive extensions
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: matrix functions
zbMATH Keywords: band method
zbMATH Keywords: almost periodic functions
zbMATH Keywords: ARMA processes
zbMATH Keywords: Wiener algebra
zbMATH Keywords: bitorus
zbMATH Keywords: entropy
zbMATH Keywords: autoregressive moving average processes
reviewed by: Ilie Valuşescu
MaRDI profile type: Publication
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OpenAlex ID: W2014370019
description: scientific article; zbMATH DE number 1275885