On the invariance of classes $\Phi BV, \Lambda BV$ under composition
HTML articles powered by AMS MathViewer
- by Pamela B. Pierce and Daniel Waterman
- Proc. Amer. Math. Soc. 132 (2004), 755-760
- DOI: https://doi.org/10.1090/S0002-9939-03-07129-6
- Published electronically: July 31, 2003
- PDF | Request permission
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
- Milton Chaika and Daniel Waterman, On the invariance of certain classes of functions under composition, Proc. Amer. Math. Soc. 43 (1974), 345–348. MR 330367, DOI 10.1090/S0002-9939-1974-0330367-9
- Casper Goffman, Everywhere convergence of Fourier series, Indiana Univ. Math. J. 20 (1970/71), 107–112. MR 270048, DOI 10.1512/iumj.1970.20.20010
- Casper Goffman and Daniel Waterman, Functions whose Fourier series converge for every change of variable, Proc. Amer. Math. Soc. 19 (1968), 80–86. MR 221193, DOI 10.1090/S0002-9939-1968-0221193-7
- 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
- Michael Josephy, Composing functions of bounded variation, Proc. Amer. Math. Soc. 83 (1981), no. 2, 354–356. MR 624930, DOI 10.1090/S0002-9939-1981-0624930-9
- J. Musielak and W. Orlicz, On generalized variations. I, Studia Math. 18 (1959), 11–41. MR 104771, DOI 10.4064/sm-18-1-11-41
- R. Salem, Sur un test général pour le convergence uniforme des séries de Fourier, Comptes Rend. Acad. Sci. Paris v. 207 (1938), 662–664.
- Daniel Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107–117. MR 310525, DOI 10.4064/sm-44-2-107-117
- Daniel Waterman, On $L$-bounded variation, Studia Math. 57 (1976), no. 1, 33–45. MR 417355, DOI 10.4064/sm-57-1-33-45
- Daniel Waterman, On the summability of Fourier series of functions of $L$-bounded variation, Studia Math. 54 (1975/76), no. 1, 87–95. MR 402391, DOI 10.4064/sm-55-1-87-95
- Daniel Waterman, Fourier series of functions of $\Lambda$-bounded variation, Proc. Amer. Math. Soc. 74 (1979), no. 1, 119–123. MR 521884, DOI 10.1090/S0002-9939-1979-0521884-1
- 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.
Bibliographic Information
- Pamela B. Pierce
- Affiliation: Department of Mathematics and Computer Science, The College of Wooster, Wooster, Ohio 44691
- ORCID: 0000-0002-7495-2990
- Email: ppierce@acs.wooster.edu
- Daniel Waterman
- Affiliation: Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431-0991
- Email: fourier@earthlink.net
- Received by editor(s): October 21, 2002
- Published electronically: July 31, 2003
- Communicated by: Andreas Seeger
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2019952