Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the invariance of classes $\Phi BV, \Lambda BV$ under composition


Authors: Pamela B. Pierce and Daniel Waterman
Journal: Proc. Amer. Math. Soc. 132 (2004), 755-760
MSC (2000): Primary 26A45, 42A16
DOI: https://doi.org/10.1090/S0002-9939-03-07129-6
Published electronically: July 31, 2003
MathSciNet review: 2019952
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The necessary and sufficient condition for $g \circ f$ to be in the class $\Phi BV, \Lambda BV$ for every $f$ of that class whose range is in the domain of $g$ is that $g$ be in $\operatorname{Lip}1$.


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

  • [CW] M. Chaika and D. Waterman, On the invariance of certain classes of functions under composition, Proc. Amer. Math. Soc. 43 (1974), 345-348.MR 48:8704
  • [G] C. Goffman, Everywhere convergence of Fourier series, Indiana Univ. Math. J. 20 (1970), 107-112. MR 42:4941
  • [GW] C. Goffman and D. Waterman, Functions whose Fourier series converge for every change of variable, Proc. Amer. Math. Soc. 19 (1968), 80-86. MR 36:4245
  • [GS] A. Garsia and S. Sawyer, On some classes of continuous functions with convergent Fourier series, J. of Math. and Mech. 13 (1964), 586-601. MR 33:7777
  • [J] M. Josephy, Composing functions of bounded variation, Proc. Amer. Math. Soc. 83 (1981), 354-356. MR 83c:26009
  • [MO] J. Musielak and W. Orlicz, On generalized variations I, Studia Math. 18 (1959), 11-41. MR 21:3524
  • [S] R. Salem, Sur un test général pour le convergence uniforme des séries de Fourier, Comptes Rend. Acad. Sci. Paris v. 207 (1938), 662-664.
  • [W1] D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107-117. MR 46:9623
  • [W2] D. Waterman, On $\Lambda$-bounded variation, Studia Math. 57 (1976), 33-45. MR 54:5408
  • [W3] D. Waterman, On the summability of Fourier series of functions of $\Lambda$-bounded variation, Studia Math. 55 (1976), 87-95. MR 53:6212
  • [W4] D. Waterman, Fourier series of functions of $\Lambda$-bounded variation, Proc. Amer. Math. Soc. 74 (1979), 119-123. MR 80j:42010
  • [Y] L. C. Young, Sur une généralisation de la notion de variation de puissance p-ième bornée au sense de M. Wiener, et sur la convergence des séries de Fourier, Comptes Rend. Acad. Sci. Paris 204 (1937), 470-472.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 26A45, 42A16

Retrieve articles in all journals with MSC (2000): 26A45, 42A16


Additional Information

Pamela B. Pierce
Affiliation: Department of Mathematics and Computer Science, The College of Wooster, Wooster, Ohio 44691
Email: ppierce@acs.wooster.edu

Daniel Waterman
Affiliation: Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431-0991
Email: fourier@earthlink.net

DOI: https://doi.org/10.1090/S0002-9939-03-07129-6
Keywords: Bounded variation, Fourier series
Received by editor(s): October 21, 2002
Published electronically: July 31, 2003
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society