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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Probability measure representation of norms associated with the notion of entropy

Author: Romuald Dąbrowski
Journal: Proc. Amer. Math. Soc. 90 (1984), 263-268
MSC: Primary 46E15; Secondary 42A20
MathSciNet review: 727246
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Abstract: One of the applications of Banach spaces introduced by B. Korenblum [1,2] is a new convergence test [2] for Fourier series including both Dirichlet-Jordan and the Dini-Lipschitz tests [3], The norms of the spaces are given in terms of $ \kappa $-entropy where $ \kappa \left( s \right) \geqslant 0$, $ 0{\text{ < }}s \leqslant 1$, is a nondecreasing concave function such that $ \kappa \left( 1 \right) = 1$. The $ \kappa $-norms fill the gap between the uniform and the variation norms. The original proof of the general properties of $ \kappa $-norms uses both combinatorial and approximation arguments which are rather complicated. We give a simple proof introducing a probabilistic representation of the norms so that the $ \kappa $-norm of a real function $ f$ on $ T = R / 2\pi Z$ is the expectation of the mean oscillation of $ f$ on a subinterval of $ T$, chosen in a suitable random process.

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

  • [1] B. Korenblum, On a class of Banach spaces of functions associated with the notion of entropy, manuscript.
  • [2] -, A generalization of two classical convergence tests for Fourier series and some new Banach spaces of functions, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 215-218. MR 707960 (85c:42004)
  • [3] A. Zygmund, Trigonometric series, Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)
  • [4] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin and New York, 1973. MR 0423029 (54:11013)
  • [5] Y. Katznelson, An introduction to harmonic analysis, Wiley, New York 1968. MR 0248482 (40:1734)

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