On a natural connection between the entropy spaces and Hardy space

Author:
Romuald Dąbrowski

Journal:
Proc. Amer. Math. Soc. **104** (1988), 812-818

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 Dirichlet-Jordan and Dini-Lipschitz.

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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DOI:
https://doi.org/10.1090/S0002-9939-1988-0964862-6

Article copyright:
© Copyright 1988
American Mathematical Society