Some remarks on functions of -bounded variation

Authors:
S. Perlman and D. Waterman

Journal:
Proc. Amer. Math. Soc. **74** (1979), 113-118

MSC:
Primary 26A45

MathSciNet review:
521883

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Abstract: It is shown that if a function has no external saltus, then its total -variation is independent of its values at points of discontinuity, and a function which is equal to the given function at points of continuity cannot have a lesser total -variation. Necessary and sufficient conditions are determined for one space to contain another and for two spaces to be identical.

**[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]**Casper Goffman,*Everywhere convergence of Fourier series*, Indiana Univ. Math. J.**20**(1970/1971), 107–112. MR**0270048****[3]**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**[4]**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****[5]**-,*The localization principle for double Fourier series*(to appear).**[6]**S. Perlman,*Functions of generalized variation*, Fund. Math.**105**(1979/80), no. 3, 199–211. MR**580582****[7]**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****[8]**-,*On the summability of Fourier series of functions of*-*bounded variation*, Studia Math.**55**(1976), 97-109.**[9]**Daniel Waterman,*On 𝐿-bounded variation*, Studia Math.**57**(1976), no. 1, 33–45. MR**0417355**

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0521883-X

Article copyright:
© Copyright 1979
American Mathematical Society