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Integrable mean periodic functions on locally compact abelian groups

Authors: Inder K. Rana and K. Gowri Navada
Journal: Proc. Amer. Math. Soc. 117 (1993), 405-410
MSC: Primary 43A25; Secondary 43A45
MathSciNet review: 1111221
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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.

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Keywords: Locally compact abelian groups, mean periodic functions on groups, almost periodic functions, character group, annihilator, Fourier transform
Article copyright: © Copyright 1993 American Mathematical Society

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