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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
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.

References [Enhancements On Off] (What's this?)

  • [1] A. G. Miamee, Extension of three theorems for Fourier series on the disc to the torus, Bull. Austral. Math. Soc 33 (1986), 335-350. MR 837478 (88f:60097)
  • [2] T. Nakazi, Extended weak-$ ^{\ast}$ Dirichlet algebras, Pacific J. Math. 81 (1979), 493-513. MR 547616 (82a:46055)
  • [3] -, Two problems in prediction theory, Studia Math. 78 (1984), 7-14. MR 766702 (86i:60122)
  • [4] W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. MR 0255841 (41:501)
  • [5] T. P. Srinivasan and J. K. Wang, Weak-$ ^{\ast}$ Dirichlet algebras, Proc. Internat. Sympos. on Function Algebras (Tulane Univ., 1965), Scott-Foresman, Chicago, Ill., 1966, pp. 216-249. MR 0198282 (33:6441)
  • [6] G. Szegö, Beiträge zur Theorie der Toeplitzschen Formen (Erste Mitteilung), Math. Z. 6 (1920), 167-202. MR 1544404

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Keywords: Szegö's theorem, bidisc, arithmetic mean, geometric mean, prediction error
Article copyright: © Copyright 1991 American Mathematical Society

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