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Proceedings of the American Mathematical Society
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0727246-8
PII: S 0002-9939(1984)0727246-8
Article copyright: © Copyright 1984 American Mathematical Society