Probability measure representation of norms associated with the notion of entropy
Author:
Romuald Dąbrowski
Journal:
Proc. Amer. Math. Soc. 90 (1984), 263268
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 DirichletJordan and the DiniLipschitz 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]
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Korenblum, A generalization of two classical
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9 (1983), no. 2,
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707960 (85c:42004), http://dx.doi.org/10.1090/S027309791983151601
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A.
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Frank
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Yitzhak
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& Sons, Inc., New YorkLondonSydney, 1968. MR 0248482
(40 #1734)
 [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), 215218. 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, SpringerVerlag, 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407272468
PII:
S 00029939(1984)07272468
Article copyright:
© Copyright 1984
American Mathematical Society
