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



Differences of vector-valued
functions on topological groups

Authors: Bolis Basit and A. J. Pryde
Journal: Proc. Amer. Math. Soc. 124 (1996), 1969-1975
MSC (1991): Primary 43A15; Secondary 28B05, 39A05
MathSciNet review: 1317031
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Abstract: Let $G$ be a locally compact group equipped with right Haar measure. The right differences $\triangle _{h} \varphi $ of functions $\varphi $ on $G$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\varphi \in L^{\infty }(G)$ and suppose $\triangle _{h} \varphi \in L^{p} (G)$ for some $1 \leq p < \infty $ and all $h \in G$. We prove that $\Vert \triangle _{h} \varphi \Vert _{p}$ is a right uniformly continuous function of $h$. If $G$ is abelian and the Beurling spectrum $sp(\varphi )$ does not contain the unit of the dual group $\hat {G}$, then we show $\varphi \in L^{p} (G)$. These results have analogues for functions $\varphi : G\to X$, where $X$ is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach $G$-modules.

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

  • 1. B. Basit and M. Emam, Differences of functions in locally convex spaces and applications to almost periodic and almost automorphic functions, Annales Polonici Math. XLI (1983), 193--201. MR 85d:43005
  • 2. B. Basit and A.J. Pryde, Polynomials and functions with finite spectra on locally compact abelian groups, Bull. Austral. Math. Soc. 51 (1994), 33--42. CMP 95:07
  • 3. J.P.R Christensen, Joint continuity of separately continuous functions, Proc. Amer. Math. Soc. 82 (1981), 455--461. MR 82h:54012
  • 4. C. Datry and G. Muraz, Analyse harmonique dans les modules de Banach I: propriétés générales, Bull. Science Mathematique 119 (1995), 299--337.
  • 5. R.E. Edwards, Functional Analysis---Theory and Applications, Holt, Rinehart and Winston Inc., New York, 1965. MR 36:4308
  • 6. F. Galvin, G. Muraz et P. Szeptycki, Fonction aux différence $f(x)-f(a+x)$ continues, C.R.Acad.Sci. Paris, série I 315 (1991), 397--400. MR 94b:39035
  • 7. E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Part I, Springer-Verlag, 1979. MR 81k:43001
  • 8. S. Kwapien, On Banach spaces containing $c_{o}$, Studia Math. 52 (1974), 187--188. MR 50:8627
  • 9. I. Namioka, Separate continuity and joint continuity, Pacific Journal of Math. 51 (1974), 515--531. MR 51:6693
  • 10. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Math. Monographs, Oxford Univ., 1968. MR 46:5933
  • 11. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1966. MR 50:2851

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Additional Information

Bolis Basit
Affiliation: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Email: bbasit(ajpryde)

A. J. Pryde
Affiliation: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Email: bbasit(ajpryde)

Keywords: Differences, weight functions, spectrum, right uniform continuity, $G$-modules, weak continuity, absolutely continuous elements
Received by editor(s): September 21, 1994
Received by editor(s) in revised form: January 4, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society