Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T16:00:40.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Translation of Hölder and Lipschitz Functions

Published online by Cambridge University Press:  20 November 2018

H. Mirkil
H. Mirkil*
Affiliation:
Dartmouth College
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

All functions will be complex, periodic, integrable (on [0, 2π]) functions of a real variable x. Moreover, we shall require that every function have mean zero on [0, 2π], so that in particular non-zero constants are excluded.

1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f of bounded variation on [0, 2π]. These f form a Banach space F when each f is normed by its total variation on [0, 2π]. And translations define a natural one-parameter group of isometries on F.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 674 - 685
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Bochner, S., Additive set functions on groups, Ann. Math., 40 (1939), 769–799.Google Scholar
2. de Leeuw, K., Linear spaces with a compact group of operators, 111. J. Math., 2 (1958), 367–377.Google Scholar
3. Hardy, G.H. and Littlewood, J.E., Some properties of fractional integrals I, Math. Zeit., 27 (1927-1928), 565–606.Google Scholar
4. Plessner, A., Eine Kennzeichnung der totalstetig Funktionen, J. reine angew. Math., 160 (1929), 26–32.Google Scholar
5. Silov, G.E., Homogeneous rings of functions, Uspehi Matem. Nauk N.S., 6 (1951), A. M.S. Translation 92.Google Scholar
6. Zygmund, A., Smooth Functions, Duke Math. J., 12 (1945), 47–76.Google Scholar