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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Szegő's theorem on a bidisc


Author: Takahiko Nakazi
Journal: Trans. Amer. Math. Soc. 328 (1991), 421-432
MSC: Primary 32A35; Secondary 32A37, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1991-1028762-2
MathSciNet review: 1028762
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Abstract: G. Szegö showed that

$\displaystyle \inf \;\int_0^{2\pi } {\vert 1 - f{\vert^2}w\,d\theta /2\pi = \exp \;\int_0^{2\pi } {\log \,w\,d\theta /2\pi } } $

where $ f$ ranges over analytic polynomials with mean value zeros. We study extensions of the Szegö's theorem on the disc to the bidisc. We show that the quantity is a mixed form of an arithmetic mean and a geometric one of $ w$ in some special cases.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1028762-2
Keywords: Szegö's theorem, bidisc, arithmetic mean, geometric mean, prediction error
Article copyright: © Copyright 1991 American Mathematical Society