Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Functions of $ \Phi$-bounded variation and Riemann-Stieltjes integration


Author: Michael Schramm
Journal: Trans. Amer. Math. Soc. 287 (1985), 49-63
MSC: Primary 26A45; Secondary 26A42
DOI: https://doi.org/10.1090/S0002-9947-1985-0766206-3
MathSciNet review: 766206
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A notion of generalized bounded variation is introduced which simultaneously generalizes many of those previously examined. It is shown that the class of functions arising from this definition is a Banach space with a suitable norm. Appropriate variation functions are defined and examined, and an analogue of Helly's theorem is estabished. The significance of this class to convergence of Fourier series is briefly discussed. A result concerning Riemann-Stieltjes integrals of functions of this class is proved.


References [Enhancements On Off] (What's this?)

  • [1] R. Leśniewicz and W. Orlicz, On generalized variations (II), Studia Math. 45 (1973), 71-109. MR 0346509 (49:11234)
  • [2] M. Schramm and D. Waterman, On the magnitude of Fourier coefficients, Proc. Amer. Math. Soc. 85 (1982), 407-410. MR 656113 (83h:42008)
  • [3] -, Absolute convergence of Fourier series of functions of $ \Lambda {\text{BV}^{(P)}}$ and $ \phi \Lambda {\text{BV}}$, Acta Math. Hungary 40 (1982), 273-276. MR 686326 (84e:42013)
  • [4] D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107-117. MR 0310525 (46:9623)
  • [5] -, On $ \Lambda $-bounded variation, Studia Math. 57 (1976), 33-45. MR 0417355 (54:5408)
  • [6] L. C. Young, An inequality of Hölder type connected with Stieltjes integration, Acta Math. 67 (1936), 251-282. MR 1555421
  • [7] -, Sur une généralisation de la notion de variation de puissance $ p$-ième bornée au sens de M. Wiener, et sur la convergence des séries de Fourier, Comptes Rendus 204 (1937), 470-472.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A45, 26A42

Retrieve articles in all journals with MSC: 26A45, 26A42


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0766206-3
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society