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

Full-text PDF

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.

**[1]**R. R. Coifman,*A real variable characterization of*, Studia Math.**51**(1974), 269-274. MR**0358318 (50:10784)****[2]**R. Dabrowski,*Probability measure representation of norms associated with the notion of entropy*, Proc. Amer. Math. Soc.**90**(1984), 263-268. MR**727246 (86c:46020)****[3]**-,*Bochner integral and continuous functions of bounded entropy*, Preprint, 1986.**[4]**-,*On Fourier coefficients of a continuous periodic function of bounded entropy norm*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), 49-51. MR**919659 (89b:42002)****[5]**P. Duren,*Theory of**spaces*, Academic Press, New York, 1970. MR**0268655 (42:3552)****[6]**P. Koosis,*Introduction to**spaces*, London Math. Soc. Lecture Notes Series, no. 40, Cambridge Univ. Press, 1980. MR**565451 (81c:30062)****[7]**B. Korenblum,*On a class of Banach spaces associated with the notion of entropy*, Trans. Amer. Math. Soc.**290**(1985), 527-553. MR**792810 (87a:46063)****[8]**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), 215-218. MR**707960 (85c:42004)****[9]**C. Sunberg,*Truncations of***B.M.O.***functions*, Indiana Univ. Math. J.**33**(1984).**[10]**K. Yosida,*Functional analysis*, Springer-Verlag, 1974, pp. 130-136. MR**0350358 (50:2851)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
42A45,
42B30,
46E99

Retrieve articles in all journals with MSC: 42A45, 42B30, 46E99

Additional Information

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

Article copyright:
© Copyright 1988
American Mathematical Society