On a natural connection between the entropy spaces and Hardy space
Author:
Romuald Dąbrowski
Journal:
Proc. Amer. Math. Soc. 104 (1988), 812818
MSC:
Primary 42A45; Secondary 42B30, 46E99
MathSciNet review:
964862
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Abstract: In 1983 B. Korenblum [7, 8] introduced a class of Banach function spaces associated with the notion of entropy (we will call these spaces and their norms entropy spaces and entropy norms, respectively). Entropy spaces were used in [8] as a tool for proving a new convergence test for Fourier series which includes classical tests of DirichletJordan and DiniLipschitz. In this paper we construct natural linear operators from the entropy spaces to Hardy space [5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to . As a corollary we obtain a growth condition for Fourier coefficients of a continuous periodic function of bounded entropy norm (as announced in [4]). In particular, we show that if is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent ), and are the Fourier coefficients of , then (resp. ). In §4 we study the relationship between the dual spaces of entropy spaces and space B.M.O. of functions of bounded mean oscillation. In §5 we conjecture that is a direct limit of the entropy spaces in an appropriate sense.
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 R. R. Coifman, A real variable characterization of , Studia Math. 51 (1974), 269274. MR 0358318 (50:10784)
 [2]
 R. Dabrowski, Probability measure representation of norms associated with the notion of entropy, Proc. Amer. Math. Soc. 90 (1984), 263268. MR 727246 (86c:46020)
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 , Bochner integral and continuous functions of bounded entropy, Preprint, 1986.
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 , On Fourier coefficients of a continuous periodic function of bounded entropy norm, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 4951. MR 919659 (89b:42002)
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 P. Duren, Theory of spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
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 P. Koosis, Introduction to spaces, London Math. Soc. Lecture Notes Series, no. 40, Cambridge Univ. Press, 1980. MR 565451 (81c:30062)
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 B. Korenblum, On a class of Banach spaces associated with the notion of entropy, Trans. Amer. Math. Soc. 290 (1985), 527553. MR 792810 (87a:46063)
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 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), 215218. MR 707960 (85c:42004)
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 C. Sunberg, Truncations of B.M.O. functions, Indiana Univ. Math. J. 33 (1984).
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 K. Yosida, Functional analysis, SpringerVerlag, 1974, pp. 130136. MR 0350358 (50:2851)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809648626
PII:
S 00029939(1988)09648626
Article copyright:
© Copyright 1988 American Mathematical Society
