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 -entropy where , , is a nondecreasing concave function such that . The -norms fill the gap between the uniform and the variation norms. The original proof of the general properties of -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 -norm of a real function on is the expectation of the mean oscillation of on a subinterval of , chosen in a suitable random process.

**[1]**B. Korenblum,*On a class of Banach spaces of functions associated with the notion of entropy*, manuscript.**[2]**B. Korenblum,*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), no. 2, 215–218. MR**707960**, 10.1090/S0273-0979-1983-15160-1**[3]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776****[4]**Frank F. Bonsall and John Duncan,*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029****[5]**Yitzhak Katznelson,*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1984-0727246-8

Article copyright:
© Copyright 1984
American Mathematical Society