Probability measure representation of norms associated with the notion of entropy
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- by Romuald Dąbrowski PDF
- Proc. Amer. Math. Soc. 90 (1984), 263-268 Request permission
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
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B. Korenblum, On a class of Banach spaces of functions associated with the notion of entropy, manuscript.
- 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, DOI 10.1090/S0273-0979-1983-15160-1
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
- Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 90 (1984), 263-268
- MSC: Primary 46E15; Secondary 42A20
- DOI: https://doi.org/10.1090/S0002-9939-1984-0727246-8
- MathSciNet review: 727246