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\(\widehat{f(k)}\) 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.
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Communicated by J. William Helton
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Bakonyi, M., Rodman, L., Spitkovsky, I.M. et al. Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band. The Journal of Fourier Analysis and Applications 5, 21–44 (1999). https://doi.org/10.1007/BF01274187
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DOI: https://doi.org/10.1007/BF01274187