On a class of Banach spaces of functions associated with the notion of entropy

Author:
Boris Korenblum

Journal:
Trans. Amer. Math. Soc. **290** (1985), 527-553

MSC:
Primary 46E99; Secondary 30D99, 42A20

MathSciNet review:
792810

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Abstract: A class of function spaces on the circle is introduced which contain all continuous functions of bounded variation but are included in the set of all continuous functions. The corresponding dual spaces consist of certain types of generalized measures. One application of these spaces is a new convergence test for Fourier series which includes both the Dirichlet-Jordan and the Dini-Lipschitz tests.

**[1]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776****[2]**Boris Korenblum,*An extension of the Nevanlinna theory*, Acta Math.**135**(1975), no. 3-4, 187–219. MR**0425124****[3]**Boris Korenblum,*A Beurling-type theorem*, Acta Math.**138**(1976), no. 3-4, 265–293. MR**0447584****[4]**A. M. Garsia and S. Sawyer,*On some classes of continuous functions with convergent Fourier series*, J. Math. Mech.**13**(1964), 589–601. MR**0199634****[5]**Daniel Waterman,*On convergence of Fourier series of functions of generalized bounded variation*, Studia Math.**44**(1972), 107–117. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity. II. MR**0310525****[6]**Robert A. Fefferman,*A theory of entropy in Fourier analysis*, Adv. in Math.**30**(1978), no. 3, 171–201. MR**520232**, 10.1016/0001-8708(78)90036-1

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1985-0792810-2

Article copyright:
© Copyright 1985
American Mathematical Society