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Translates of L functions and of bounded measures

Published online by Cambridge University Press:  09 April 2009

R. E. Edwards
Affiliation:
Institute of Advanced Studies, Australian National University.
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D. A. Edwards has shown [1] that if X is a locally compact Abelian group and fL, then the translate fa of f varies continuously with α if and only if f is (equal l.a.e. to) a bounded, uniformly continuous function. He remarks that this is a sort of dual to part of a result due to Plessner and Raikov which asserts that an element μ of the space Mb of bounded Radon measures on X belongs to L1 (i.e., is absolutely continuous relative to Haar measure) if and only its translates vary continuously with the group element, the relevant topology on Mb being that defined by the natural norm of Mb as the dual of the space of continuous functions vanishing at infinity. The proof he uses (ascribed to Reiter) applies equally well in both cases, and also to the case in which X is non-Abelian. A brief examination shows that in the latter case it is ultimately immaterial whether left- or right-translates are considered; since the extra complexities of this case are principally terminological, we shall direct no further attention to it.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Edwards, D. A., On translates of L functions, Journal London Math. Soc. 36 (1961), 431432.CrossRefGoogle Scholar
[2]Schwartz, L., Théorie des distributions. Tome II. Act. Sci. et Ind. No. 1122. Paris (1951).Google Scholar
[3]Weil, A., L'intégration dans les groupes topologiques et ses applications. Act. Sci. et Ind. Nos. 869 and 1145, Paris (1941 and 1951).Google Scholar
[4]Edwards, R. E., Derivation of vector-valued functions, Mathematika 5 (1958), 5861.CrossRefGoogle Scholar