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
Abstract: One of the applications of Banach spaces introduced by B. Korenblum [1,2] is a new convergence test  for Fourier series including both Dirichlet-Jordan and the Dini-Lipschitz tests , 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.
-  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, https://doi.org/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, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR 0423029
-  Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
- B. Korenblum, On a class of Banach spaces of functions associated with the notion of entropy, manuscript.
- -, 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)
- A. Zygmund, Trigonometric series, Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)
- F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin and New York, 1973. MR 0423029 (54:11013)
- Y. Katznelson, An introduction to harmonic analysis, Wiley, New York 1968. MR 0248482 (40:1734)