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
DOI:
https://doi.org/10.1090/S0002-9939-1988-0964862-6
MathSciNet review:
964862
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Abstract | References | Similar Articles | Additional Information
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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1988-0964862-6
Article copyright:
© Copyright 1988
American Mathematical Society