On the note of C. L. Belna

Author:
Daniel Waterman

Journal:
Proc. Amer. Math. Soc. **80** (1980), 445-447

MSC:
Primary 26A45

MathSciNet review:
581001

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Abstract: The space of functions of ordered harmonic bounded variation (*OHBV*) has been shown by Belna to contain the space of functions of harmonic bounded variation (*HBV*) properly. *OHBV* is a Banach space and *HBV* is a first category subset. The ordered harmonic variation has continuity properties quite different from those of the harmonic variation. The relationship of these classes to the everywhere convergence of Fourier series is discussed.

**[1]**Albert Baernstein II and Daniel Waterman,*Functions whose Fourier series converge uniformly for every change of variable*, Indiana Univ. Math. J.**22**(1972/73), 569–576. MR**0310523****[2]**C. L. Belna,*On ordered harmonic bounded variation*, Proc. Amer. Math. Soc.**80**(1980), no. 3, 441–444. MR**581000**, 10.1090/S0002-9939-1980-0581000-5**[3]**Casper Goffman,*Everywhere convergence of Fourier series*, Indiana Univ. Math. J.**20**(1970/1971), 107–112. MR**0270048****[4]**Casper Goffman and Daniel Waterman,*Functions whose Fourier series converge for every change of variable*, Proc. Amer. Math. Soc.**19**(1968), 80–86. MR**0221193**, 10.1090/S0002-9939-1968-0221193-7**[5]**Casper Goffman and Daniel Waterman,*A characterization of the class of functions whose Fourier series converge for every change of variable*, J. London Math. Soc. (2)**10**(1975), 69–74. MR**0370036****[6]**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****[7]**Daniel Waterman,*On 𝐿-bounded variation*, Studia Math.**57**(1976), no. 1, 33–45. MR**0417355****[8]**Daniel Waterman,*Fourier series of functions of Λ-bounded variation*, Proc. Amer. Math. Soc.**74**(1979), no. 1, 119–123. MR**521884**, 10.1090/S0002-9939-1979-0521884-1

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1980-0581001-7

Article copyright:
© Copyright 1980
American Mathematical Society