Integrable mean periodic functions on locally compact abelian groups
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- by Inder K. Rana and K. Gowri Navada PDF
- Proc. Amer. Math. Soc. 117 (1993), 405-410 Request permission
Abstract:
Let $G$ be a locally compact abelian group with a Haar measure ${\lambda _G}$. A function $f$ on $G$ is said to be mean-periodic if there exists a nonzero finite regular measure $\mu$ of compact support on $G$ such that $f{\ast }\mu = 0$. It is known that there exist no nontrivial integrable mean periodic functions on ${{\mathbf {R}}^n}$. We show that there exist nontrivial integrable mean periodic functions on $G$ provided $G$ has nontrivial proper compact subgroups. Let $f \in {L_1}(G)$ be mean periodic with respect to a nonzero finite measure $\mu$ of compact support. If $\mu (G) \ne 0$ and ${\lambda _G}(\operatorname {supp} (\mu )) > 0$, then there exists a compact subgroup $K$ of $G$ such that $f{\ast }{\lambda _K} = 0$, i.e., $f$ is mean periodic with respect to ${\lambda _K}$, where ${\lambda _K}$ denotes the normalized Haar measure of $K$. When $G$ is compact, abelian and meterizable, we show that there exists continuous (hence integrable and almost periodic) functions on $G$ that are not mean periodic.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 405-410
- MSC: Primary 43A25; Secondary 43A45
- DOI: https://doi.org/10.1090/S0002-9939-1993-1111221-3
- MathSciNet review: 1111221